Composite Characteristic Transport: Propagation and Separation of Riemann–Roch Defects

1. Introduction

Characteristic classes encode geometric information not only through the classes assigned to spaces, sheaves, constructible functions, or other geometric objects, but also through the transformation laws satisfied by those classes under geometric operations. Classical Riemann–Roch theory, singular Riemann–Roch transformations, bivariant theories, motivic characteristic transformations, and Gysin formulas provide several manifestations of this principle [1,2,3,4,5]. A characteristic transformation need not commute with a geometric operation in an uncorrected form. Its functorial behavior may instead involve tangent classes, normal classes, orientations, refined Gysin morphisms, specialization operators, or other correction data prescribed by an established Riemann–Roch-type theorem.

In singular situations, failure of a prescribed transformation law may itself carry geometric information. Verdier specialization, vanishing cycles, Milnor classes, and Hirzebruch–Milnor classes provide important examples in which a discrepancy is supported on singular strata and is identified with independently defined local or global singularity data [6,7,8,9]. The purpose of the present paper is to study what becomes of such information when the underlying geometric construction is composite.

The central object introduced here is not a new characteristic class or a new universal Riemann–Roch transformation. Rather, characteristic transport is a structured record of the behavior of a fixed family of characteristic transformations along a typed sequence of geometric operations. It records, when defined, the reference characteristic operator associated with each stage, the defect of corrected naturality at that stage, and the propagation of that defect through later stages. When geometric support has been independently identified, that support may also be retained. The resulting information is then compared with a prescribed endpoint realization, such as total rational cohomology or its restriction to a chosen Hodge sector.

A principal theme of the paper is therefore the hierarchy

$$\mathrm{geometric}\mathrm{factorization}⟶\mathrm{characteristic}\mathrm{transport}\mathrm{data}⟶\mathrm{prescribed}\mathrm{endpoint}\mathrm{realization}.$$

The first level remembers the intermediate spaces and geometric operations; the second remembers the characteristic operators, stage defects, and their propagation; the third retains only the specified endpoint information. The basic question is where information can be lost when one passes from one level to the next.

1.1. Ambient Conventions and Typed Geometric Operations

Let ${\mathsf{Var}}_{\mathbb{C}}$ denote the category of reduced, separated schemes of finite type over $\mathbb{C}$, with morphisms of $\mathbb{C}$-schemes. Throughout the paper, an object of ${\mathsf{Var}}_{\mathbb{C}}$ is called a complex algebraic variety. Additional hypotheses such as irreducibility, properness, smoothness, projectivity, pure dimensionality, or local complete intersection conditions will be stated whenever they are required.

Unless otherwise stated in this subsection, $X,Y,Z$ denote objects of ${\mathsf{Var}}_{\mathbb{C}}$. The constructions considered below involve operations of different variance. Schematically, if $V\mapsto \mathcal{B}\left(V\right)$ is an assignment of an additive group to each variety for which the indicated operations are defined, then a proper morphism $f:X\to Y$ may induce a covariant homomorphism $f_{*}:\mathcal{B}\left(X\right)\to \mathcal{B}\left(Y\right)$, a smooth morphism $\pi :X\to Y$ may induce a contravariant homomorphism ${\pi }^{*}:\mathcal{B}\left(Y\right)\to \mathcal{B}\left(X\right)$, and a regular or normally nonsingular embedding $i:Z\hookrightarrow X$ may induce a Gysin homomorphism $i^{!}:\mathcal{B}\left(X\right)\to \mathcal{B}\left(Z\right)$. These pushforward, pullback, product, and Gysin structures are standard features of intersection theory and bivariant theories [2,3].

This difference in variance makes it important to distinguish three objects: an underlying geometric morphism, the direction in which a chosen object is transported, and the homomorphism induced on a specified source or target assignment.

A typed transport operation from X to Y will be denoted $u:X\to Y$. The displayed arrow records the direction of transport and does not, by itself, assert that an underlying geometric morphism has direction $X\to Y$. We call X the transport source and Y the transport target of u.

For example, if $f:X\to Y$ is proper, denote the corresponding proper typed transport operation by $u_f^{\mathrm{prop}}:X\to Y$. The symbol $u_f^{\mathrm{prop}}$ denotes the typed operation, not the homomorphism induced on a particular group.

If $\pi :X\to Y$ is smooth, the corresponding pullback-type transport operation is denoted $u_{\pi }^{\mathrm{sm}}:Y\to X$. Its transport direction is opposite to that of the geometric morphism $\pi$, reflecting the contravariance of pullback.

If $i:Z\hookrightarrow X$ is an embedding for which the required Gysin morphism is defined, denote the corresponding Gysin-type transport operation by $u_i^{!}:X\to Z$. Specialization operations are treated in the same way: the typed arrow records the direction of the induced operator, while the underlying geometric datum is recorded separately.

This convention will be used whenever operations of different variance occur in a common composite. In a purely covariant setting, where the direction of transport coincides with the direction of every underlying morphism and no ambiguity can arise, lighter notation such as $F:X\to Y$ may be used.

1.2. Characteristic Transport Data

We now specify the data relative to which characteristic transport is defined.

Definition 1.1  

(Characteristic transport datum). Acharacteristic transport datumis a tuple

$$\mathbb{T}=\left(\mathsf{C},\mathcal{A},\mathcal{H},\Phi ,{Op}_{\mathbb{T}},{\mathcal{T}}_{\Phi }\right)$$

with the following components.

1.

$\mathsf{C}$ is a specified full subcategory of ${\mathsf{Var}}_{\mathbb{C}}$.

2.

For each object $X\in Ob\left(\mathsf{C}\right)$, $\mathcal{A}\left(X\right)$ and $\mathcal{H}\left(X\right)$ are additive groups. The assignments $X\mapsto \mathcal{A}\left(X\right)$ and $X\mapsto \mathcal{H}\left(X\right)$ are called, respectively, thesource assignmentand thecharacteristic target assignment.

3.

For each $X\in Ob\left(\mathsf{C}\right)$, there is an additive homomorphism ${\Phi }_X:\mathcal{A}\left(X\right)\to \mathcal{H}\left(X\right)$. The family $\Phi ={\left\{{\Phi }_X\right\}}_{X\in Ob\left(\mathsf{C}\right)}$ is called thecharacteristic transformation data. No universal naturality property of Φ is assumed.

4.

For each ordered pair $(X,Y)\in Ob\left(\mathsf{C}\right)\times Ob\left(\mathsf{C}\right)$, the set ${Op}_{\mathbb{T}}(X,Y)$ is a specified collection of typed transport operations $u:X\to Y$.

5.

For every $u\in {Op}_{\mathbb{T}}(X,Y)$, there is an additive source-side homomorphism $u_{\mathcal{A}}:\mathcal{A}\left(X\right)\to \mathcal{A}\left(Y\right)$.

6.

For every $u\in {Op}_{\mathbb{T}}(X,Y)$, there is an additivereference characteristic transport operator

$${\mathcal{T}}_{\Phi }\left(u\right):\mathcal{H}\left(X\right)⟶\mathcal{H}\left(Y\right).$$

The operator ${\mathcal{T}}_{\Phi }\left(u\right)$ is the target-side operator selected for comparison with ${\Phi }_Y\circ u_{\mathcal{A}}$. Depending on u, it may be an ordinary pushforward, a corrected pullback, a refined Gysin operator, a specialization operator, or another operator prescribed by an established Riemann–Roch-type comparison theorem.

If $\mathcal{A}\left(X\right)$ or $\mathcal{H}\left(X\right)$ is graded, the notation denotes the relevant total graded group unless a particular component is specified. Any degree shift, dimension shift, or Tate twist associated with an operator is part of its type and will be displayed whenever it enters a comparison.

The word theory will be reserved for assignments carrying the additional axioms of an established homology theory, cohomology theory, bivariant theory, or comparable functorial structure. This distinction is useful because the applications below involve Grothendieck groups, constructible functions, Chow homology, Borel–Moore homology, and bivariant constructions with different variance and functoriality.

Definition 1.2  

(Admissible transport operation). Fix a characteristic transport datum $\mathbb{T}$, let $X,Y\in Ob\left(\mathsf{C}\right)$, and let $u:X\to Y$ be a typed transport operation. The operation u is$\mathbb{T}$-admissibleif $u\in {Op}_{\mathbb{T}}(X,Y)$, the homomorphisms

$$u_{\mathcal{A}}:\mathcal{A}\left(X\right)\to \mathcal{A}\left(Y\right),{\mathcal{T}}_{\Phi }\left(u\right):\mathcal{H}\left(X\right)\to \mathcal{H}\left(Y\right)$$

are defined, and all geometric hypotheses required for those operators and for the comparison formula under consideration are satisfied.

Admissibility is therefore relative to $\mathbb{T}$, not an intrinsic property of a geometric morphism alone. Properness may suffice for one characteristic transformation, whereas smoothness, an lci hypothesis, regularity of an embedding, orientability, or a normally nonsingular condition may be required for another.

The formalism is compatible with, but does not replace, standard bivariant structures. Fulton–MacPherson bivariant theories organize confined pushforwards, independent pullbacks, products, orientations, and Gysin operations in a common framework [2]; refined Gysin maps and their compatibilities are developed systematically in intersection theory [3]. The present datum isolates only the pieces needed to compare characteristic transformations along finite composite constructions.

1.3. One-stage Transport and Characteristic Transport Defects

Fix a characteristic transport datum $\mathbb{T}$, let $X,Y\in Ob\left(\mathsf{C}\right)$, let $u:X\to Y$ be $\mathbb{T}$-admissible, and let $\xi \in \mathcal{A}\left(X\right)$. Then $u_{\mathcal{A}}\left(\xi \right)\in \mathcal{A}\left(Y\right)$, so ${\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)\in \mathcal{H}\left(Y\right)$. Likewise, ${\Phi }_X\left(\xi \right)\in \mathcal{H}\left(X\right)$, so ${\mathcal{T}}_{\Phi }\left(u\right)\left({\Phi }_X\left(\xi \right)\right)\in \mathcal{H}\left(Y\right)$. The two terms in the following definition therefore lie in the same additive group.

Definition 1.3  

(Characteristic transport defect). Thecharacteristic transport defectof ξ along u is

$${\Delta }_{\Phi }(u;\xi ):={\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }\left(u\right)\left({\Phi }_X\left(\xi \right)\right)\in \mathcal{H}\left(Y\right).$$

(1)

The operation u isexact on $\xi$if ${\Delta }_{\Phi }(u;\xi )=0$, and it is$\Phi$-exactif ${\Delta }_{\Phi }(u;\xi )=0$ for every $\xi \in \mathcal{A}\left(X\right)$.

Thus a transport defect is always the difference of two specified elements of the same target group. A nonzero formal difference is not, by itself, assigned geometric significance.

Definition 1.4  

(Geometrically identified defect). Let $u\in {Op}_{\mathbb{T}}(X,Y)$ and $\xi \in \mathcal{A}\left(X\right)$. The defect ${\Delta }_{\Phi }(u;\xi )\in \mathcal{H}\left(Y\right)$ isgeometrically identifiedif an independent theorem identifies it with a separately defined geometric class, or with the image of such a class under a specified homomorphism.

Examples include discrepancies expressed through specialization, vanishing cycles, Milnor-type classes, excess-intersection data, or characteristic classes supported on prescribed singular loci. This distinction between a formal defect and a geometrically identified defect is essential throughout the paper.

The Baum–Fulton–MacPherson Riemann–Roch transformation provides a basic exact example. Let $X,Y\in Ob\left({\mathsf{Var}}_{\mathbb{C}}\right)$ and let $f:X\to Y$ be proper. For a complex algebraic variety V, write $G_0\left(V\right)$ for the Grothendieck group of coherent algebraic sheaves and $A_{*}{\left(V\right)}_{\mathbb{Q}}:=A_{*}\left(V\right){\otimes }_{\mathbb{Z}}\mathbb{Q}$ for rational Chow homology. Baum, Fulton, and MacPherson construct

$${\tau }_V:G_0\left(V\right)⟶A_{*}{\left(V\right)}_{\mathbb{Q}}$$

naturally with respect to proper pushforward [1].

For $f:X\to Y$, let $f_{*}^G:G_0\left(X\right)\to G_0\left(Y\right)$ denote proper pushforward on Grothendieck groups and let $f_{*}^A:A_{*}{\left(X\right)}_{\mathbb{Q}}\to A_{*}{\left(Y\right)}_{\mathbb{Q}}$ denote proper pushforward on Chow homology. Naturality is the typed identity

$${\tau }_Y\circ f_{*}^G=f_{*}^A\circ {\tau }_X.$$

Let $u_f^{\mathrm{prop}}:X\to Y$ be the corresponding typed transport operation. Taking $\mathcal{A}\left(V\right)=G_0\left(V\right)$, $\mathcal{H}\left(V\right)=A_{*}{\left(V\right)}_{\mathbb{Q}}$, ${\Phi }_V={\tau }_V$, ${\left(u_f^{\mathrm{prop}}\right)}_{\mathcal{A}}=f_{*}^G$, and ${\mathcal{T}}_{\tau }\left(u_f^{\mathrm{prop}}\right)=f_{*}^A$, one obtains

$${\Delta }_{\tau }(u_f^{\mathrm{prop}};\xi )=0\mathrm{for}\mathrm{every}\xi \in G_0\left(X\right).$$

If X is nonsingular, let $T_X$ denote its tangent bundle, $td\left(T_X\right)$ its Todd class, and $\left[X\right]\in A_{*}\left(X\right)$ its fundamental cycle. Then

$${\tau }_X\left(\left[{\mathcal{O}}_X\right]\right)=td\left(T_X\right)\cap \left[X\right],$$

where $\left[{\mathcal{O}}_X\right]\in G_0\left(X\right)$ is the class of the structure sheaf [1].

Motivic Chern and Hirzebruch-class transformations provide further examples of characteristic transformations with proper functoriality and normalization formulas on nonsingular varieties [4,5].

Defective stages provide the complementary phenomenon. Schürmann’s generalized Verdier–Riemann–Roch formula for Chern–Schwartz–MacPherson classes identifies defects associated with regular embeddings and suitable local complete intersection situations in terms of specialization and generalized vanishing-cycle data [6]. Milnor and Hirzebruch–Milnor classes similarly measure differences between virtual and functorial characteristic classes in singular settings [7,8,9].

Exact Gysin stages provide another class of examples. Normally nonsingular Gysin formulas for L-type characteristic classes give corrected exact transport under their respective hypotheses [10,11,12].

For a smooth morphism $\pi :X\to Y$ of pure relative complex dimension d, with X and Y pure-dimensional, let $T_{X/Y}$ denote the relative tangent bundle, let y denote the Hirzebruch parameter, and let ${\pi }_{\mathrm{BM}}^{*}$ denote the corresponding smooth pullback in Borel–Moore homology. Previous work of the author establishes the class-level identity

$$I{T}_{y*}\left(X\right)=T_y^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}I{T}_{y*}\left(Y\right)$$

under the hypotheses stated there [13].

The paper therefore treats exact and defective stages in a common formalism, while assigning geometric significance only to independently identified defects.

1.4. Composite Routes and Propagated Defects

Fix a characteristic transport datum $\mathbb{T}$ and a positive integer $r\ge 1$.

Definition 1.5  

(Compatible transport route). Acompatible transport route of length rconsists of objects $X_0,\dots ,X_r\in Ob\left(\mathsf{C}\right)$ and an ordered tuple $\mathcal{R}=(u_1,\dots ,u_r)$ such that each $u_i:X_{i-1}\to X_i$, $1\le i\le r$, is a $\mathbb{T}$-admissible typed transport operation.

We display such a route as

$$\mathcal{R}:X_0\stackrel{u_1}{\to }{X}_1\stackrel{u_2}{\to }\dots \stackrel{u_r}{\to }{X}_r.$$

As above, the arrows record transport direction and need not coincide with the directions of the underlying geometric morphisms.

The route induces the source-side homomorphism

$$u_{\mathcal{A},\mathcal{R}}:={\left(u_r\right)}_{\mathcal{A}}\circ \dots \circ {\left(u_1\right)}_{\mathcal{A}}:\mathcal{A}\left(X_0\right)⟶\mathcal{A}\left(X_r\right)$$

and the stagewise reference characteristic transport operator

$${\mathcal{T}}_{\Phi }\left(\mathcal{R}\right):={\mathcal{T}}_{\Phi }\left(u_r\right)\circ \dots \circ {\mathcal{T}}_{\Phi }\left(u_1\right):\mathcal{H}\left(X_0\right)⟶\mathcal{H}\left(X_r\right).$$

Fix ${\xi }_0\in \mathcal{A}\left(X_0\right)$ and define recursively ${\xi }_i:={\left(u_i\right)}_{\mathcal{A}}\left({\xi }_{i-1}\right)\in \mathcal{A}\left(X_i\right)$ for $1\le i\le r$. For the same range of i, define the stage defect

$${\Delta }_i:={\Delta }_{\Phi }(u_i;{\xi }_{i-1})\in \mathcal{H}\left(X_i\right).$$

Definition 1.6  

(Stagewise route defect). Thestagewise characteristic transport defectof $(\mathcal{R},{\xi }_0)$ is

$${\Delta }_{\Phi }(\mathcal{R};{\xi }_0):={\Phi }_{X_r}\left({\xi }_r\right)-{\mathcal{T}}_{\Phi }\left(\mathcal{R}\right)\left({\Phi }_{X_0}\left({\xi }_0\right)\right)\in \mathcal{H}\left(X_r\right).$$

For $1\le i\le r$, define the propagation operator $P_{i,r}^{\Phi }:\mathcal{H}\left(X_i\right)\to \mathcal{H}\left(X_r\right)$ by

$$P_{i,r}^{\Phi }:=\left\{\begin{array}{cc}{\mathcal{T}}_{\Phi }\left(u_r\right)\circ \dots \circ {\mathcal{T}}_{\Phi }\left(u_{i+1}\right),\hfill & 1\le i<r,\hfill \\ {id}_{\mathcal{H}\left(X_r\right)},\hfill & i=r.\hfill \end{array}\right.$$

Proposition 1.7  

(Finite-route propagation identity). For every compatible route $\mathcal{R}=(u_1,\dots ,u_r)$ and every ${\xi }_0\in \mathcal{A}\left(X_0\right)$,

$${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)={\displaystyle \sum _{i=1}^r}{P}_{i,r}^{\Phi }\left({\Delta }_i\right)\mathrm{in}\mathcal{H}\left(X_r\right).$$

(2)

Proof. 

For each $1\le i\le r$, Definition 1.3 gives

$${\Phi }_{X_i}\left({\xi }_i\right)={\mathcal{T}}_{\Phi }\left(u_i\right)\left({\Phi }_{X_{i-1}}\left({\xi }_{i-1}\right)\right)+{\Delta }_i.$$

Successive substitution yields

$${\Phi }_{X_r}\left({\xi }_r\right)={\mathcal{T}}_{\Phi }\left(\mathcal{R}\right)\left({\Phi }_{X_0}\left({\xi }_0\right)\right)+{\displaystyle \sum _{i=1}^r}{P}_{i,r}^{\Phi }\left({\Delta }_i\right),$$

which is equivalent to (2). □

A nonzero stage defect may therefore survive to the endpoint, be annihilated by a later transport operator, or cancel against other propagated defects. Consequently, ${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)=0$ does not imply ${\Delta }_i=0$ for every i. Endpoint exactness does not determine the intermediate transport history.

1.5. Compressed Composites and Interaction Operators

Fix $X,Y,Z\in Ob\left(\mathsf{C}\right)$, let $u:X\to Y$ and $v:Y\to Z$ be $\mathbb{T}$-admissible typed transport operations, and suppose that the underlying geometry also determines an admissible compressed operation $v\diamond u:X\to Z$. Assume that

$${(v\diamond u)}_{\mathcal{A}}=v_{\mathcal{A}}\circ u_{\mathcal{A}}:\mathcal{A}\left(X\right)⟶\mathcal{A}\left(Z\right).$$

The notation ⋄ distinguishes the compressed geometric operation from ordinary composition of induced homomorphisms.

The operators ${\mathcal{T}}_{\Phi }\left(v\right)\circ {\mathcal{T}}_{\Phi }\left(u\right)$ and ${\mathcal{T}}_{\Phi }(v\diamond u)$ both map $\mathcal{H}\left(X\right)$ to $\mathcal{H}\left(Z\right)$, but need not agree.

Definition 1.8  

(Interaction operator). Thecharacteristic interaction operatorof the pair $(v,u)$ is

$${\mathfrak{I}}_{\Phi }(v,u):={\mathcal{T}}_{\Phi }\left(v\right)\circ {\mathcal{T}}_{\Phi }\left(u\right)-{\mathcal{T}}_{\Phi }(v\diamond u)\in {Hom}_{\mathbf{Ab}}\left(\mathcal{H}\left(X\right),\mathcal{H}\left(Z\right)\right).$$

The pair $(v,u)$ is$\Phi$-coherentif ${\mathfrak{I}}_{\Phi }(v,u)=0$.

For $\xi \in \mathcal{A}\left(X\right)$, the defect of the compressed operation is

$${\Delta }_{\Phi }(v\diamond u;\xi )={\Phi }_Z\left({(v\diamond u)}_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }(v\diamond u)\left({\Phi }_X\left(\xi \right)\right)\in \mathcal{H}\left(Z\right).$$

Proposition 1.9  

(Two-stage compressed-composite identity). For every $\xi \in \mathcal{A}\left(X\right)$,

$$\begin{array}{cc}\hfill {\Delta }_{\Phi }(v\diamond u;\xi )=& {\Delta }_{\Phi }\left(v;u_{\mathcal{A}}\left(\xi \right)\right)+{\mathcal{T}}_{\Phi }\left(v\right)\left({\Delta }_{\Phi }(u;\xi )\right)\hfill \\ \hfill & +{\mathfrak{I}}_{\Phi }(v,u)\left({\Phi }_X\left(\xi \right)\right).\hfill \end{array}$$

(3)

Proof. 

By definition and the identity ${(v\diamond u)}_{\mathcal{A}}=v_{\mathcal{A}}\circ u_{\mathcal{A}}$,

$${\Delta }_{\Phi }(v\diamond u;\xi )={\Phi }_Z\left(v_{\mathcal{A}}{u}_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }(v\diamond u)\left({\Phi }_X\left(\xi \right)\right).$$

Adding and subtracting ${\mathcal{T}}_{\Phi }\left(v\right)\left({\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)\right)$ and ${\mathcal{T}}_{\Phi }\left(v\right){\mathcal{T}}_{\Phi }\left(u\right)\left({\Phi }_X\left(\xi \right)\right)$, and grouping terms, gives Equation (3). □

If $(v,u)$ is $\Phi$-coherent, the interaction term vanishes and

$${\Delta }_{\Phi }(v\diamond u;\xi )={\Delta }_{\Phi }\left(v;u_{\mathcal{A}}\left(\xi \right)\right)+{\mathcal{T}}_{\Phi }\left(v\right)\left({\Delta }_{\Phi }(u;\xi )\right).$$

Thus the interaction operator is not another stage defect. It measures the failure of the stagewise reference operator ${\mathcal{T}}_{\Phi }\left(v\right)\circ {\mathcal{T}}_{\Phi }\left(u\right)$ to agree with the reference operator ${\mathcal{T}}_{\Phi }(v\diamond u)$ assigned directly to the compressed operation. The propagation problem and the compression problem are therefore logically distinct.

1.6. Support Data

Let $X\in Ob\left(\mathsf{C}\right)$, let $S\subseteq X$ be a closed subvariety that is also an object of $\mathsf{C}$, and let $j_S:S\hookrightarrow X$ be the closed immersion. Whenever $\mathcal{H}$ admits the corresponding proper pushforward ${\left(j_S\right)}_{*}:\mathcal{H}\left(S\right)\to \mathcal{H}\left(X\right)$, define

$${\mathcal{H}}_S\left(X\right):=im\left({\left(j_S\right)}_{*}:\mathcal{H}\left(S\right)⟶\mathcal{H}\left(X\right)\right).$$

Definition 1.10  

(Certified support). For $\eta \in \mathcal{H}\left(X\right)$, the closed subvariety $S\subseteq X$ is acertified supportfor η if $\eta \in {\mathcal{H}}_S\left(X\right)$.

No minimality or uniqueness of certified support is asserted. In the applications below, support is supplied by an independent geometric theorem, for example through vanishing cycles, specialization, a Milnor-type class, or a refined Gysin construction.

Now let $\mathcal{R}=(u_1,\dots ,u_r)$ be a compatible route and fix $i\in \{1,\dots ,r\}$. If the stage defect ${\Delta }_i\in \mathcal{H}\left(X_i\right)$ has certified support $S_i\subseteq X_i$, then $P_{i,r}^{\Phi }\left({\Delta }_i\right)\in \mathcal{H}\left(X_r\right)$. The support-propagation problem asks for a geometrically specified closed subvariety $S_{i,r}\subseteq X_r$ such that

$$P_{i,r}^{\Phi }\left({\Delta }_i\right)\in {\mathcal{H}}_{S_{i,r}}\left(X_r\right).$$

For proper pushforward, smooth pullback, and compatible refined Gysin operations, the candidate support is determined respectively by proper image, inverse image, or refined intersection. The precise statements and hypotheses are given in Theorem 5.12.

1.7. Characteristic Transport Profiles

Let $\mathcal{R}=(u_1,\dots ,u_r)$ be a compatible route

$$X_0\stackrel{u_1}{\to }{X}_1\stackrel{u_2}{\to }\dots \stackrel{u_r}{\to }{X}_r$$

and let ${\xi }_0\in \mathcal{A}\left(X_0\right)$. The endpoint defect ${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)\in \mathcal{H}\left(X_r\right)$ does not retain the stage at which each contribution was created. We therefore record the stagewise data before summation.

Definition 1.11  

(Raw characteristic transport profile). Theraw characteristic transport profileof $(\mathcal{R},{\xi }_0)$ is

$${\mathfrak{P}}_{\Phi }^{\mathrm{raw}}(\mathcal{R};{\xi }_0):=\left({\left(X_i\right)}_{i=0}^r,{\left(u_i\right)}_{i=1}^r,{\left({\xi }_i\right)}_{i=0}^r,{\left({\mathcal{T}}_{\Phi }\left(u_i\right)\right)}_{i=1}^r,{\left({\Delta }_i\right)}_{i=1}^r,{\left(P_{i,r}^{\Phi }\right)}_{i=1}^r\right),$$

where ${\xi }_i={\left(u_i\right)}_{\mathcal{A}}\left({\xi }_{i-1}\right)\in \mathcal{A}\left(X_i\right)$, ${\Delta }_i={\Delta }_{\Phi }(u_i;{\xi }_{i-1})\in \mathcal{H}\left(X_i\right)$, and $P_{i,r}^{\Phi }:\mathcal{H}\left(X_i\right)\to \mathcal{H}\left(X_r\right)$.

If $I\subseteq \{1,\dots ,r\}$ is the set of stages for which support has been independently certified and $S_i\subseteq X_i$ is such a support for ${\Delta }_i$ for each $i\in I$, the supported raw profile is obtained by adjoining the family ${\left(S_i\right)}_{i\in I}$.

The raw profile is intentionally sensitive to nontrivial factorization but should not distinguish presentations that differ only trivially.

Definition 1.12  

(Elementary profile equivalence). Two raw profiles are related by anelementary profile equivalenceif one is obtained from the other by either of the following operations:

1.

insertion or deletion of an identity stage $u_X^{\mathrm{id}}:X\to X$ satisfying

$${\left(u_X^{\mathrm{id}}\right)}_{\mathcal{A}}={id}_{\mathcal{A}\left(X\right)},{\mathcal{T}}_{\Phi }\left(u_X^{\mathrm{id}}\right)={id}_{\mathcal{H}\left(X\right)},$$

and ${\Delta }_{\Phi }(u_X^{\mathrm{id}};\xi )=0$ for every $\xi \in \mathcal{A}\left(X\right)$;

2.

replacement of an intermediate object $X_i$ by an isomorphic object $X_i^{\prime }\in Ob\left(\mathsf{C}\right)$, together with transport of the adjacent typed operations, source-side operators, reference characteristic operators, stage classes, defects, and certified supports through the chosen isomorphism $X_i\stackrel{\sim }{\to }{X}_i^{\prime }$.

Because a route is represented by the ordered tuple $(u_1,\dots ,u_r)$, associative rebracketing does not alter the raw route and therefore requires no additional equivalence relation.

Definition 1.13  

(Characteristic transport profile). Thecharacteristic transport profile${\mathfrak{P}}_{\Phi }(\mathcal{R};{\xi }_0)$ is the equivalence class of ${\mathfrak{P}}_{\Phi }^{\mathrm{raw}}(\mathcal{R};{\xi }_0)$ under the equivalence relation generated by elementary profile equivalences.

Thus the profile forgets trivial choices of presentation but remains sensitive to genuinely different nontrivial factorizations.

Let ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ be two routes with common initial object $X_0$ and common initial class ${\xi }_0\in \mathcal{A}\left(X_0\right)$. Whenever their profile types are comparable, write

$$({\mathcal{R}}_1,{\xi }_0){\sim }_{\mathfrak{P}}({\mathcal{R}}_2,{\xi }_0)$$

if their characteristic transport profiles are equal. When ${\xi }_0$ is fixed throughout the discussion, we abbreviate this relation to ${\mathcal{R}}_1{\sim }_{\mathfrak{P}}{\mathcal{R}}_2$.

1.8. Cohomological Realization and Selected-Sector Equivalence

For a complex algebraic variety X, write $H^{•}(X,\mathbb{Q}):={⨁}_{m\ge 0}{H}^m(X,\mathbb{Q})$ for the total singular cohomology of the associated analytic space with rational coefficients.

Whenever a typed operation $u:X\to Y$ admits a specified rational cohomological realization, denote it by ${\rho }_H\left(u\right)$. If degree and Tate twist matter, they form part of its displayed type; thus one may have

$${\rho }_H\left(u\right):H^m(X,\mathbb{Q}\left(a\right))⟶H^n(Y,\mathbb{Q}\left(b\right))$$

for specified integers $m,n,a,b$.

A standard example is supplied by algebraic correspondences. Let X and Y be smooth projective complex varieties, set $d_X:={dim}_{\mathbb{C}}X$, fix $k\in \mathbb{Z}$, and let $\Gamma \in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}$. Let $p_X:X\times Y\to X$ and $p_Y:X\times Y\to Y$ be the projections. For $\alpha \in H^m(X,\mathbb{Q})$, define

$${\Gamma }_{*}\left(\alpha \right):={\left(p_Y\right)}_{*}\left(p_X^{*}\left(\alpha \right)⌣{cl}_{X\times Y}(\Gamma )\right),$$

where

$${cl}_{X\times Y}:C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}⟶H^{2d_X+2k}\left(X\times Y,\mathbb{Q}(d_X+k)\right)$$

is the rational cycle-class map. With these conventions,

$${\Gamma }_{*}:H^m(X,\mathbb{Q})⟶H^{m+2k}(Y,\mathbb{Q}\left(k\right)).$$

This is the standard cohomological action of a correspondence [3].

For a compatible route $\mathcal{R}=(u_1,\dots ,u_r)$, suppose each ${\rho }_H\left(u_i\right)$ is defined and the realization maps are composable. Define

$${\rho }_H\left(\mathcal{R}\right):={\rho }_H\left(u_r\right)\circ \dots \circ {\rho }_H\left(u_1\right).$$

Let ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ have cohomological realizations with common source V and common target W, and let $L\subseteq V$ be a specified $\mathbb{Q}$-linear subspace.

Definition 1.14  

(Selected-sector equivalence). The routes ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ areequivalent on Lif

$${\rho }_H\left({\mathcal{R}}_1\right){{|}_L={\rho }_H\left({\mathcal{R}}_2\right)|}_L:L⟶W.$$

In this case we write ${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$.

In the Hodge-theoretic applications, let $X_0$ be smooth projective and let $p\in {\mathbb{Z}}_{\ge 0}$. The rational Hodge classes of degree $2p$, written with Tate twist p, form

$$H^{2p}{(X_0,\mathbb{Q}\left(p\right))}_{\mathrm{Hdg}}:={Hom}_{\mathrm{MHS}}\left(\mathbb{Q}\left(0\right),H^{2p}(X_0,\mathbb{Q}\left(p\right))\right).$$

Here $\mathrm{MHS}$ denotes the category of rational mixed Hodge structures and $\mathbb{Q}\left(0\right)$ the trivial rational Hodge structure. Since $X_0$ is smooth and projective, these are equivalently the rational cohomology classes of Hodge type $(p,p)$, expressed after Tate twist.

A selected Hodge sector is a specified $\mathbb{Q}$-linear subspace

$$L\subseteq H^{2p}{(X_0,\mathbb{Q}\left(p\right))}_{\mathrm{Hdg}}.$$

The geometric routes are constructed independently of L; only their already-defined cohomological realizations are subsequently restricted to that sector.

The principal comparison is therefore between ${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$ and ${\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2$: the routes may become indistinguishable after restriction to L while retaining distinct characteristic transport histories. This does not assert that characteristic transport is universally finer than total rational cohomology.

1.9. Residual Correspondences and the Hierarchy of Kernels

Let X and Y be smooth projective complex varieties, set $d_X:={dim}_{\mathbb{C}}X$, fix $k\in \mathbb{Z}$, and let ${\Gamma }_1,{\Gamma }_2\in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}$. Define

$$\mathfrak{D}:={\Gamma }_1-{\Gamma }_2\in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}.$$

Let $V\subseteq H^{•}(X,\mathbb{Q})$ be a graded subspace on which the correspondence actions are defined, and let $L\subseteq V$ be a specified $\mathbb{Q}$-linear subspace. For a correspondence $\Gamma$, write ${\rho }_L(\Gamma ):={\Gamma }_{*}{|}_L$. If ${\rho }_L\left({\Gamma }_1\right)={\rho }_L\left({\Gamma }_2\right)$, then ${\rho }_L\left(\mathfrak{D}\right)=0$, so $\mathfrak{D}\in ker\left({\rho }_L\right)$.

Define the total rational cohomological realization

$${\rho }_H:C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}⟶\prod _{m\in \mathbb{Z}}{Hom}_{\mathbb{Q}}\left(H^m(X,\mathbb{Q}),H^{m+2k}(Y,\mathbb{Q}\left(k\right))\right)$$

by

$${\rho }_H(\Gamma )={\left({\Gamma }_{*}{|}_{H^m(X,\mathbb{Q})}\right)}_{m\in \mathbb{Z}}.$$

For smooth projective X and Y, the Künneth decomposition and Poincaré duality identify the rational cohomology class of a correspondence with its total cohomological action. Consequently,

$$ker\left({cl}_{X\times Y}\right)=ker\left({\rho }_H\right)\subseteq ker\left({\rho }_L\right).$$

Thus a residual correspondence may vanish only after restriction to L while remaining visible on total cohomology, or it may already be homologically trivial.

The Jacobian construction of Section 8 realizes the first possibility. Let C and D be smooth projective connected complex curves, let $\pi :C\to D$ be finite, set $n:=deg\left(\pi \right)>1$, and assume $g\left(D\right)\ge 2$. Put $A:=J\left(D\right)$ and $B:=J\left(C\right)$. The morphism $\pi$ induces

$${\pi }^{*}:A⟶B,{Nm}_{\pi }:B⟶A.$$

Define $f:={Nm}_{\pi }$ and $f^{\vee }:={\pi }^{*}$. Then

$$f\circ f^{\vee }={Nm}_{\pi }\circ {\pi }^{*}={\left[n\right]}_A,$$

where ${\left[n\right]}_A:A\to A$ denotes multiplication by n.

Set $g:={dim}_{\mathbb{C}}A=g\left(D\right)$ and define

$$L_A:=H^{2g-2}{\left(A,\mathbb{Q}(g-1)\right)}_{\mathrm{Hdg}}.$$

Let ${\Gamma }_{{\left[n\right]}_A}\in C{H}^g{(A\times A)}_{\mathbb{Q}}$ be the graph of ${\left[n\right]}_A$, and let ${\Delta }_A\in C{H}^g{(A\times A)}_{\mathbb{Q}}$ be the diagonal. Define

$${\mathfrak{D}}_n:={\Gamma }_{{\left[n\right]}_A}-n^2{\Delta }_A\in C{H}^g{(A\times A)}_{\mathbb{Q}}.$$

Section 8 proves

$${\mathfrak{D}}_n\in ker\left({\rho }_{L_A}\right)\setminus ker\left({cl}_{A\times A}\right).$$

Thus selected-sector realization loses information that remains visible to total rational cohomology.

The stronger problem is to construct or detect a nonzero residual $\mathfrak{D}\in ker\left({cl}_{X\times Y}\right)$ by a characteristic, bivariant, Abel–Jacobi-type, or other realization finer than ordinary rational cycle class. No general theorem of that type is claimed here.

1.10. The Comparison Problem

Fix $X_0\in Ob\left(\mathsf{C}\right)$ and ${\xi }_0\in \mathcal{A}\left(X_0\right)$. Let ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ be independently constructed compatible transport routes beginning at $X_0$. Assume their cohomological realizations have common source V and common target W, and let $L\subseteq V$ be a prescribed $\mathbb{Q}$-linear subspace.

Suppose

$${\rho }_H\left({\mathcal{R}}_1\right){{|}_L={\rho }_H\left({\mathcal{R}}_2\right)|}_L,\mathrm{equivalently}{\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2.$$

The comparison problem asks whether independently identified stagewise characteristic data nevertheless imply

$${\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2,$$

with profile comparison taken relative to the common initial class ${\xi }_0$.

This is a conditional separation problem. The formalism does not manufacture the routes or their selected-sector equivalence; both must be supplied independently. Characteristic transport then asks whether information erased by restriction to L remains visible in the stagewise reference operators, geometrically identified defects, certified supports, or propagated defect classes.

The support-graded separation theorem of Section 6.6 gives sufficient conditions for

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2\mathrm{but}{\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2.$$

It is a separation criterion, not an existence theorem for such pairs.

The singular calculations of Section 9 provide one source of geometric input. In the nodal-cubic example, let $Y:={\mathbf{P}}_{\mathbb{C}}^2$, let $X\subset Y$ be an irreducible plane cubic with a unique ordinary double point $p\in X$, and let $i:X\hookrightarrow Y$ be the regular hypersurface embedding. Let ${\mathbf{1}}_Y$ denote the constant constructible function on Y. In the convention of that section, the associated hypersurface vanishing-cycle function satisfies

$${\mu }_i\left({\mathbf{1}}_Y\right)=-{\mathbf{1}}_{\left\{p\right\}},$$

and the corresponding Chern–Schwartz–MacPherson Gysin defect satisfies

$${\Delta }_{c_{*}}^{!}(i;{\mathbf{1}}_Y)=\left[p\right].$$

If $g:X\to \mathrm{pt}$ is the structure morphism, proper pushforward gives $g_{*}\left[p\right]=\left[\mathrm{pt}\right]$. Thus the calculation produces the chain

$$-{\mathbf{1}}_{\left\{p\right\}}⟼\left[p\right]⟼\left[\mathrm{pt}\right].$$

This gives a concrete nonzero propagated support contribution, although it does not itself construct a second route with the same selected-sector cohomological action.

Characteristic transport therefore functions as a diagnostic layer for composite geometric constructions. It does not prove algebraicity of an arbitrary Hodge class, classify Chow correspondences, or replace Abel–Jacobi theory and other invariants of homologically trivial cycles.

1.11. Principal Results

The main results fall into five groups. Formal identities are separated from geometric propagation and comparison statements that require additional hypotheses.

Composite defect calculus and support propagation. 

Proposition 5.3 develops the two-stage identity represented in the introduction by Equation (3). The identity itself is formal. The geometric content begins with Theorem 5.12, which shows, under its stated hypotheses, that defects with certified support remain supported on the appropriate proper image, smooth inverse image, or refined Gysin intersection after later transport.

Singular propagation and the nodal-cubic calculation. 

Starting from Schürmann’s established one-stage Verdier–Riemann–Roch defect [6], Theorem 9.14 determines its later image under proper, smooth, and compatible Gysin stages while retaining control of propagated support. Theorem 9.24 gives the explicit sequence

$$-{\mathbf{1}}_{\left\{p\right\}}⟼\left[p\right]⟼\left[\mathrm{pt}\right],$$

whose final nonzero class is independently cross-checked using the Euler characteristic of the nodal cubic and the zero-dimensional Fulton–Johnson contribution.

Transport profiles and support-graded separation. 

Section 6 develops the profile ${\mathfrak{P}}_{\Phi }(\mathcal{R};{\xi }_0)$ from the raw stagewise data above. The normalized Hirzebruch profile specializes to Chern-, Todd-, and conditional L-type components whenever the required comparison maps exist. Theorem 6.29 gives a sufficient support-graded criterion for

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2\mathrm{and}{\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2.$$

It is a separation criterion rather than an existence theorem.

Residual correspondences and Jacobian separation. 

For a transfer–return pair on a smooth projective variety X, Section 7 considers a transfer correspondence U, a return correspondence R, a scalar $m\in \mathbb{Q}$ describing the selected-sector action of $R\circ U$, and the diagonal ${\Delta }_X$. The associated residual is

$${\mathfrak{D}}_{U,R}:=R\circ U-m{\Delta }_X.$$

Its selected-sector action vanishes by construction.

For the finite-cover Jacobian construction above, Section 8 proves

$${\mathfrak{D}}_n\in ker\left({\rho }_{L_A}\right)\setminus ker\left({cl}_{A\times A}\right).$$

Thus selected-sector realization is strictly coarser than total rational cohomology in this explicit family. At the same time, exact proper naturality forces the standard reduced defect profiles associated with the Baum–Fulton–MacPherson transformation $\tau$, the Chern–Schwartz–MacPherson transformation $c_{*}$, and the motivic Hirzebruch transformation $T_{y*}$ to collapse in this setting. This calibrates what those profiles do and do not detect.

Mixed smooth–Gysin compatibility. 

Consider the Cartesian square

$$\begin{array}{ccc}{Z}^{\prime }& \stackrel{j}{\to }& X\\ \downarrow {\pi }_Z& & \downarrow \pi \\ Z& \stackrel{i}{\to }& Y,\end{array}$$

where $\pi :X\to Y$ is smooth, $i:Z\hookrightarrow Y$ is the specified Gysin-admissible embedding, $Z^{\prime }:=X{\times }_{Y}Z$, $j:Z^{\prime }\hookrightarrow X$ is the base-change embedding, and ${\pi }_Z:Z^{\prime }\to Z$ is the induced smooth morphism. Section 10 treats the corresponding smooth and Gysin stages as exact corrected transports under the stated hypotheses.

Theorem 10.13 proves

$$j_{I{T}_1}^{\mathrm{RR},!}\circ {\pi }_{I{T}_1}^{\mathrm{RR},*}={\left({\pi }_Z\right)}_{I{T}_1}^{\mathrm{RR},*}\circ i_{I{T}_1}^{\mathrm{RR},!},$$

where the four displayed symbols denote the corrected smooth-pullback and Gysin operators associated with the four maps in the square. The result is an operator-level compatibility of established correction formulas rather than a new universal Gysin theory.

Taken together, these results provide a controlled calculus for transporting independently identified characteristic defects and a factorization-sensitive framework for comparing composite geometric constructions before restriction to a prescribed cohomological realization.

1.12. Scope and Limitations

The formalism is intentionally limited in several respects.

First, ${\Delta }_{\Phi }(u;\xi )$ is defined only after the source assignment $\mathcal{A}$, target assignment $\mathcal{H}$, transformation data $\Phi$, source-side operator $u_{\mathcal{A}}$, and reference operator ${\mathcal{T}}_{\Phi }\left(u\right)$ have all been specified. There is therefore no universal characteristic transport defect attached to an arbitrary geometric morphism.

Second, a nonzero formal defect is not automatically a geometric invariant. Geometric significance is asserted only when an independent theorem identifies it with a separately defined geometric class or with a class having independently controlled support.

Third, selected-sector equivalence ${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$ is external input to the comparison. The formalism does not in general construct pairs of routes with equal selected-sector realization.

Fourth, the Jacobian example proves that, on the residual correspondences considered there,

$$ker\left({\rho }_H\right)⊊ker\left({\rho }_{L_A}\right),$$

but it does not produce a nonzero homologically trivial residual detected by a finer characteristic realization.

Finally, the framework does not classify Chow correspondences, produce algebraic representatives of arbitrary Hodge classes, or replace Abel–Jacobi theory, normal functions, and other invariants designed to study homologically trivial cycles.

1.13. Organization of the Paper

Section 2 separates the inherited Riemann–Roch, bivariant, motivic, singular-defect, Gysin, and classical cycle-theoretic structures from the composite comparison problem and fixes the novelty boundary.

Section 3 establishes the geometric categories, variance conventions, corrected operators, correspondence actions, and selected-sector notation used throughout the remainder of the paper.

Section 4, Section 5 and Section 6 develop the defect calculus, interaction operators, support propagation, comparative profiles, specialization relations, and support-graded separation criteria.

Section 7 introduces transfer–return residuals and distinguishes selected-sector residuals from total-cohomological and homologically trivial residuals.

Section 8 develops explicit Jacobian return data from finite covers of curves, proves selected-sector residual separation by ordinary rational cycle class, and records the collapse of the standard proper defect profiles in that example.

Section 9 propagates Schürmann’s vanishing-cycle defect through later geometric stages and gives the complete nodal-cubic calculation.

Section 10 develops the mixed smooth–Gysin operator comparison and its relation to Gysin coherence.

Section 11 records cancellation, collapse, family dependence, and factorization invariance. Finally, Section 12 summarizes the resulting hierarchy of realizations and isolates the stronger open problem of detecting nonzero homologically trivial residual correspondences by realizations finer than ordinary rational cycle class.

2. Literature and Mathematical Positioning

The constructions used in this paper draw on several established characteristic-class and bivariant formalisms. Singular Riemann–Roch transformations, bivariant products, motivic characteristic transformations, Verdier specialization, Milnor-type defect classes, algebraic correspondences, normally nonsingular Gysin formulas, and smooth intersection-Hirzebruch transfer all precede the present work. The problem studied here begins only after those structures have been fixed: given a composite geometric construction, which characteristic data survive along its transport route, and which distinctions remain visible before the route is replaced by a prescribed cohomological realization or by its restriction to a selected cohomological sector?

Throughout this section, $X,Y,Z$ denote complex algebraic varieties unless stronger hypotheses are stated. We write $A_{*}{\left(X\right)}_{\mathbb{Q}}:=A_{*}\left(X\right){\otimes }_{\mathbb{Z}}\mathbb{Q}$ for rational Chow homology, $G_0\left(X\right)$ for the Grothendieck group of coherent algebraic sheaves, $F\left(X\right)$ for the group of integer-valued algebraically constructible functions, $H_{*}^{\mathrm{BM}}(X,\mathbb{Q})$ for rational Borel–Moore homology, and $K_0(MHM\left(X\right))$ for the Grothendieck group of Saito’s category $MHM\left(X\right)$ of algebraic mixed Hodge modules [14].

2.1. Singular Riemann–Roch and the Classical Corrected Square

Baum, Fulton, and MacPherson construct, for every complex algebraic variety X, a homomorphism

$${\tau }_X:G_0\left(X\right)⟶A_{*}{\left(X\right)}_{\mathbb{Q}}$$

that is natural for proper pushforward [1]. More precisely, if $f:X\to Y$ is proper, let $f_{*}^G:G_0\left(X\right)\to G_0\left(Y\right)$ denote proper pushforward of coherent sheaves and let $f_{*}^A:A_{*}{\left(X\right)}_{\mathbb{Q}}\to A_{*}{\left(Y\right)}_{\mathbb{Q}}$ denote proper pushforward on Chow homology. Then

$${\tau }_Y\circ f_{*}^G=f_{*}^A\circ {\tau }_X.$$

If X is nonsingular, with tangent bundle $T_X$, structure sheaf ${\mathcal{O}}_X$, and fundamental cycle $\left[X\right]$, the normalization is

$${\tau }_X\left(\left[{\mathcal{O}}_X\right]\right)=td\left(T_X\right)\cap \left[X\right].$$

Thus proper pushforward furnishes an exact transport stage for the Baum–Fulton–MacPherson transformation: once the transformation $\tau$ has been fixed, no additional proper-pushforward defect remains.

Fulton–MacPherson bivariant theory provides a broader framework in which confined pushforward, pullback in independent squares, bivariant products, and orientations coexist and satisfy compatibility axioms [2]. If $\mathbb{B}$ and ${\mathbb{B}}^{\prime }$ are bivariant theories, a Grothendieck transformation $\gamma :\mathbb{B}\to {\mathbb{B}}^{\prime }$ is a family of homomorphisms

$${\gamma }_f:\mathbb{B}\left(X\stackrel{f}{\to }Y\right)⟶{\mathbb{B}}^{\prime }\left(X\stackrel{f}{\to }Y\right),$$

indexed by morphisms $f:X\to Y$, that is compatible with the bivariant products, confined pushforwards, and independent pullbacks [2,4].

The present paper does not modify these axioms. Any identity forced solely by bivariant functoriality or by the definition of a Grothendieck transformation is treated as inherited structure rather than as a new composition theorem.

2.2. MacPherson Classes and Motivic Hirzebruch Transformations

MacPherson constructs a natural transformation

$$c_{*}:F\left(X\right)⟶H_{*}^{\mathrm{BM}}(X,\mathbb{Z})$$

that commutes with proper pushforward and, for nonsingular X, satisfies

$$c_{*}\left({\mathbf{1}}_X\right)=c\left(T_X\right)\cap \left[X\right],$$

where ${\mathbf{1}}_X\in F\left(X\right)$ is the constant constructible function and $c\left(T_X\right)$ is the total Chern class [15]. Thus the Chern–Schwartz–MacPherson transformation is another exact proper-pushforward realization.

Brasselet, Schürmann, and Yokura construct the motivic Chern transformation

$$m{C}_y:K_0(Var/X)⟶G_0\left(X\right){\otimes }_{\mathbb{Z}}\mathbb{Z}\left[y\right]$$

and the motivic Hirzebruch transformation

$$T_{y*}:K_0(Var/X)⟶H_{*}^{\mathrm{BM}}(X,\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{Q}\left[y\right],$$

both natural for proper pushforward [16]. Here $K_0(Var/X)$ denotes the relative Grothendieck group of algebraic varieties over X, and y is the Hirzebruch parameter.

The normalized Hirzebruch transformation, usually denoted ${\widehat{T}}_{y*}$, interpolates the standard Chern-, Todd-, and L-type specializations at $y=-1,0,1$, respectively, in the settings in which the corresponding identifications hold [16]. This parameter dependence will later allow several characteristic realizations to be compared within one transport framework.

The mixed-Hodge-module realization underlying motivic Hirzebruch classes factors through $K_0(MHM\left(X\right))$ [14,16]. Passing from $MHM\left(X\right)$ to its Grothendieck group identifies the class of a short exact sequence with the sum of the classes of its subobject and quotient. Consequently, extension data present in the abelian category $MHM\left(X\right)$ are not, in general, retained by the Grothendieck-group class alone. This distinction is relevant later when comparing categorical information with characteristic realizations.

2.3. Bivariant Grothendieck Transformations

Let $f:X\to Y$ and $g:Y\to Z$ be morphisms in a bivariant theory $\mathbb{B}$. If $\alpha \in \mathbb{B}\left(X\stackrel{f}{\to }Y\right)$ and $\beta \in \mathbb{B}\left(Y\stackrel{g}{\to }Z\right)$, their bivariant product lies in $\mathbb{B}\left(X\stackrel{g\circ f}{\to }Z\right)$. For a Grothendieck transformation $\gamma :\mathbb{B}\to {\mathbb{B}}^{\prime }$, compatibility with the product is the identity

$$\gamma (\alpha •\beta )=\gamma \left(\alpha \right)•\gamma \left(\beta \right).$$

This identity is part of the defining structure of a Grothendieck transformation, not a new result of the present paper [2,4]. Motivic bivariant characteristic classes place motivic Chern and Hirzebruch transformations within related bivariant frameworks [5].

Remark 2.1.  

Formal composition is therefore infrastructure. Additional geometric content enters only when one retains information not determined by bivariant multiplication alone, such as the support of a nonzero defect, the behavior of that defect under subsequent stages, comparison among distinct characteristic realizations, or residual information annihilated by a prescribed endpoint realization.

2.4. Schürmann’s Verdier–Riemann–Roch defect

Schürmann’s generalized Verdier–Riemann–Roch theorem supplies the principal inherited model in this paper for a geometrically meaningful nonzero characteristic defect [6]. Let $i:X\hookrightarrow Y$ be a regular embedding satisfying the hypotheses of that theorem and let $\phi \in F\left(Y\right)$. The comparison between restriction of the Chern–Schwartz–MacPherson class to X and the appropriately corrected Gysin transformation of the class on Y contains a discrepancy identified through Verdier specialization to the normal cone and the associated generalized vanishing-cycle term [6].

We use this theorem only as an input. In particular, the existence and geometric identification of the one-stage defect are not novelty claims of the present paper. The new question begins after this defect has been identified: how does it propagate under a later proper pushforward, smooth pullback, or compatible Gysin operation, and what becomes of its support?

The variance convention established in Section 1.1 is important here. If the first stage is Gysin or pullback-type, its transport direction is the direction of the induced operator, not necessarily the direction of the underlying embedding or smooth morphism. Accordingly, later composites are formed at the level of compatible typed transport operations rather than by imposing an artificial covariant composite on contravariant geometry.

Schürmann’s theorem belongs to the broader specialization theory initiated by Verdier. Verdier’s specialization result for Chern classes relates characteristic classes on nearby fibers to characteristic data on a special fiber [17]. In the hypersurface setting, related formulas involving Chern–Schwartz–MacPherson classes, characteristic cycles, and specialization were developed by Aluffi and by Parusiński–Pragacz [18,19]. These results provide the one-stage singular input used in the composite calculations below.

2.5. Milnor and Hirzebruch–Milnor Defect Classes

Let X be a local complete intersection complex algebraic variety for which both a virtual characteristic class $C_{*}^{\mathrm{vir}}\left(X\right)$ and a functorial singular characteristic class $C_{*}\left(X\right)$ are defined in the same homology theory. We use the convention

$${\mathcal{M}}_C\left(X\right):=C_{*}^{\mathrm{vir}}\left(X\right)-C_{*}\left(X\right)$$

for the corresponding Milnor-type defect. Other sign conventions occur in the literature, so the displayed formula fixes the convention used in this paper.

For Chern classes, such differences lead to Milnor classes and to relations among Fulton–Johnson, Chern–Schwartz–MacPherson, and local singularity contributions [18,19,20]. Motivic and Hirzebruch refinements admit descriptions in terms of vanishing cycles and mixed-Hodge-theoretic singularity data [7,8,9].

Under the hypotheses of the established hypersurface and complete-intersection theories, the corresponding Hirzebruch–Milnor defect is supported on the singular locus and admits a vanishing-cycle or mixed-Hodge-theoretic description [8,9].

The present paper does not redefine these singularity defect theories. Instead, it begins with independently identified defects of this type and studies their propagation through additional geometric stages.

2.6. Correspondences, Selected-Sector Kernels, and Classical Cycle-Theoretic Filtrations

Bivariant Chern classes and correspondence-type convolution are established structures [21]. Formal composition of algebraic correspondences is therefore not new here. The comparison relevant to the present paper begins when a correspondence is replaced by its action on only a prescribed cohomological sector.

Let X and Y be smooth projective complex varieties, set $d_X:={dim}_{\mathbb{C}}X$, fix $k\in \mathbb{Z}$, and let

$${\Gamma }_1,{\Gamma }_2\in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}$$

be correspondences of the same codimension. Let $L\subseteq H^{•}(X,\mathbb{Q})$ be a specified graded $\mathbb{Q}$-subspace on which both correspondence actions are defined.

Definition 2.2  

(Selected-sector equivalence for correspondences). The correspondences ${\Gamma }_1$ and ${\Gamma }_2$ arecohomologically equivalent on Lif

$${\left({\Gamma }_1\right)}_{*}{{|}_L={\left({\Gamma }_2\right)}_{*}|}_L.$$

Define the residual correspondence

$$\mathfrak{D}:={\Gamma }_1-{\Gamma }_2.$$

For any correspondence $\Gamma$ of the same codimension, write ${\rho }_L(\Gamma ):={\Gamma }_{*}{|}_L$. Then selected-sector equivalence is equivalent to ${\rho }_L\left(\mathfrak{D}\right)=0$, or $\mathfrak{D}\in ker\left({\rho }_L\right)$.

To compare this selected-sector kernel with the ordinary cohomological kernel, let

$${cl}_{X\times Y}:C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}⟶H^{2d_X+2k}\left(X\times Y,\mathbb{Q}(d_X+k)\right)$$

be the rational cycle-class map, and let ${\rho }_H(\Gamma )$ denote the action of $\Gamma$ on total rational cohomology.

For smooth projective X and Y, the Künneth decomposition and Poincaré duality identify

$$H^{2d_X+2k}\left(X\times Y,\mathbb{Q}(d_X+k)\right)$$

with the corresponding collection of degree-$2k$ operators from the cohomology of X to the cohomology of Y. Consequently,

$$ker\left({cl}_{X\times Y}\right)=ker\left({\rho }_H\right)\subseteq ker\left({\rho }_L\right).$$

The equality of the first two kernels uses the action on total rational cohomology. The final inclusion may be strict because restriction to L forgets the action on the complementary cohomological sectors.

This distinction is central to the explicit Jacobian example. With the notation of Section 8, the residual

$${\mathfrak{D}}_n\in ker\left({\rho }_{L_A}\right)\setminus ker\left({cl}_{A\times A}\right)$$

is invisible on the selected Hodge sector $L_A$ but remains visible to ordinary rational cohomology. It therefore exhibits information loss caused by sector restriction; it does not provide a homologically trivial correspondence detected by a finer invariant.

The genuinely homologically trivial regime belongs to the classical theory of algebraic cycles. For a smooth projective variety X and an integer $p\ge 0$, define

$$C{H}^p{\left(X\right)}_{\mathrm{hom},\mathbb{Q}}:=ker\left({cl}_X:C{H}^p{\left(X\right)}_{\mathbb{Q}}⟶H^{2p}(X,\mathbb{Q}\left(p\right))\right).$$

The Abel–Jacobi construction assigns invariants to homologically trivial cycles in the appropriate intermediate Jacobian, while the Griffiths group measures homologically trivial cycles modulo algebraic equivalence [22]. Conjectural Bloch–Beilinson filtrations seek to organize progressively finer levels of the Chow group beyond ordinary cycle class. Murre’s conjectural Chow–Künneth decomposition and induced filtration provide an influential motivic formulation of this structure [23], while Kimura finite-dimensionality supplies a setting in which nilpotence of homologically trivial correspondences can be studied [24].

The present paper does not claim to refine Abel–Jacobi theory, detect a nonzero Griffiths-group class, or construct a new layer of a Bloch–Beilinson filtration. Its unconditional residual examples occur at the coarser level

$$ker\left({\rho }_L\right)\setminus ker(cl),$$

where restriction to a selected sector has already forgotten information that remains visible to total rational cohomology.

Section 7 formulates the stronger question of whether an independently defined characteristic or bivariant realization can detect a nonzero residual lying in

$$ker(cl).$$

Such a result would enter the genuinely homologically trivial regime. No such finer detection theorem is claimed here.

There is nevertheless a natural interface with bivariant geometry. Characteristic classes in bivariant settings are attached to morphisms, orientations, or geometric presentation data satisfying specified hypotheses; they are not automatically defined as additive invariants of arbitrary Chow correspondences. Correspondences admitting suitable proper–smooth, proper–lci, or comparable geometric presentations therefore provide a natural domain in which finer residual realizations may be sought. The graph and diagonal correspondences used in Section 8 belong to this well-controlled geometric class. What is not automatic, and is not claimed in this paper, is that a characteristic construction defined on such presentations descends to a well-defined additive invariant of the corresponding rational-equivalence class in the Chow group.

2.7. Gysin Restriction and Gysin-coherent Characteristic Classes

Banagl establishes normally nonsingular Gysin formulas for topological L-classes and Hodge-theoretic characteristic classes of singular spaces [10]. Let $g:Y\hookrightarrow X$ be an embedding satisfying the hypotheses of that theory, of complex codimension c, and let $N_{Y/X}$ denote its normal bundle. The corresponding intersection-Hirzebruch specialization at $y=1$ satisfies

$$g^{!}I{T}_{1,*}\left(X\right)=L^{*}\left(N_{Y/X}\right)\cap I{T}_{1,*}\left(Y\right),$$

where $g^{!}$ is the normally nonsingular Gysin homomorphism and $L^{*}\left(N_{Y/X}\right)$ is the cohomological Hirzebruch L-class of the normal bundle [10].

Banagl–Wrazidlo and Banagl–Schürmann–Wrazidlo subsequently organize such formulas into Gysin-coherent characteristic systems and establish corresponding uniqueness and coherence results [11,12]. These results are used in the present paper as exact corrected-transport inputs; their axioms and uniqueness statements are not modified.

2.8. Smooth Transfer for Intersection Hirzebruch Classes

Let $\pi :X\to Y$ be a smooth algebraic morphism of pure relative complex dimension r between pure-dimensional complex algebraic varieties. Neither X nor Y is assumed nonsingular. Let $T_{X/Y}$ denote the relative tangent bundle and

$${\pi }_{\mathrm{BM}}^{*}:H_{*}^{\mathrm{BM}}(Y,\mathbb{Q})⟶H_{*+2r}^{\mathrm{BM}}(X,\mathbb{Q})$$

the smooth Borel–Moore pullback.

Previous work proves the corrected transfer identity

$$I{T}_{y*}\left(X\right)=T_y^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}I{T}_{y*}\left(Y\right)$$

under these hypotheses [13]. Here $I{T}_{y*}\left(V\right)$ denotes the intersection-Hirzebruch class of the pure-dimensional variety V in the setting of that construction, and $T_y^{*}\left(T_{X/Y}\right)$ is the cohomological Hirzebruch class of the relative tangent bundle.

In the transport terminology of Section 1, the corresponding smooth-stage defect is therefore

$${\Delta }_{I{T}_y}^{*}\left(\pi \right):=I{T}_{y*}\left(X\right)-T_y^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}I{T}_{y*}\left(Y\right)=0.$$

Thus smooth intersection-Hirzebruch transport is exact after the prescribed relative-tangent correction.

The same work establishes compatibility of the corrected smooth-transfer formula with composition of smooth morphisms [13]. The later transport calculus therefore uses this compatibility as inherited input rather than as a new smooth composition theorem.

2.9. Novelty Boundary

The preceding literature fixes the mathematical boundary of the present work. Singular Riemann–Roch transformations, bivariant products and Grothendieck transformations, motivic Chern and Hirzebruch classes, Verdier–Riemann–Roch defects, Milnor and Hirzebruch–Milnor classes, correspondence convolution, Gysin coherence, smooth intersection-Hirzebruch transfer, and the classical study of homologically trivial algebraic cycles are established independently of this paper [1,2,4,6,7,8,9,10,11,12,13,16,21,22,23,24].

The new problem is formulated using the transport notation of Section 1. Let $\mathcal{R}=(u_1,\dots ,u_r)$ be a compatible $\mathbb{T}$-admissible route with source $X_0$, and let L be a specified rational cohomological sector on which the endpoint realization ${\rho }_H\left(\mathcal{R}\right)$ is defined.

Definition 2.3  

(Composite characteristic comparison problem). Thecomposite characteristic comparison problemasks which stagewise reference operators, geometrically identified defects, support data, propagated defect classes, and residual data attached to $\mathcal{R}$ remain distinguishable after the route is replaced by the restricted endpoint realization

$${\rho }_H{\left(\mathcal{R}\right)|}_L.$$

The contributions developed from this problem are therefore not the underlying Riemann–Roch, bivariant, motivic, specialization, Gysin, or correspondence formalisms themselves. They concern instead:

1.

propagation of independently identified geometric defects through later compatible transport stages;

2.

retention of stagewise support information under such propagation;

3.

structured comparison of several characteristic realizations along a common composite construction;

4.

sufficient criteria for separating characteristic transport profiles of routes whose prescribed cohomological realizations agree on L;

5.

explicit singular calculations in which a local vanishing-cycle contribution propagates to a nonzero global class; and

6.

a residual framework distinguishing selected-sector invisibility from total cohomological or homologically trivial invisibility.

The residual framework has an unconditional cohomological baseline: ordinary cycle class detects every residual in

$$ker\left({\rho }_L\right)\setminus ker(cl),$$

and Section 8 constructs an explicit family of this type. A genuinely finer residual theorem would require detection of a nonzero class in

$$ker(cl),$$

which lies in the homologically trivial regime. No such characteristic refinement is claimed in the present paper.

Remark 2.4.  

Two logically distinct questions recur throughout the paper.Transport soundnessasks whether a stagewise characteristic comparison is justified by established functorial, Riemann–Roch, Gysin, bivariant, or specialization results.Separationasks whether independently constructed routes remain distinguishable after a prescribed realization has identified them. Sound transport does not imply separation, and separation may occur simply because restriction to a selected sector is strictly coarser than total rational cohomology.

Remark 2.5.  

No construction in this paper is defined from an unknown target Hodge class, an arbitrary Hodge projector, or an unconstructed motivic lift. The geometric operations and their characteristic structures are specified before the selected-sector comparison is made. The new object of study is the behavior of established characteristic data along composite constructions and the information retained before passage to a prescribed endpoint realization. The paper does not claim a new invariant of homologically trivial cycles, a refinement of Abel–Jacobi theory, or a complete invariant of Chow correspondences.

3. Preliminaries and Conventions

This section fixes the geometric categories, variance conventions, characteristic realizations, corrected transport operators, correspondence formalism, and selected cohomological sectors used throughout the paper. We use the characteristic transport terminology of Section 1: every transport operation is interpreted relative to a fixed characteristic transport datum

$$\mathbb{T}=\left(\mathsf{C},\mathcal{A},\mathcal{H},\Phi ,{Op}_{\mathbb{T}},{\mathcal{T}}_{\Phi }\right)$$

as in Definition 1.1. In particular, $\mathcal{A}$ is the source assignment, $\mathcal{H}$ is the characteristic target assignment, and the variance of every operation is part of its specified type.

3.1. Geometric Categories and Admissible Operations

Let ${\mathsf{Var}}_{\mathbb{C}}^{\mathrm{qp}}$ denote the category of reduced, separated, quasi-projective schemes of finite type over $\mathbb{C}$, and let ${\mathsf{SmProj}}_{\mathbb{C}}\subset {\mathsf{Var}}_{\mathbb{C}}^{\mathrm{qp}}$ denote the full subcategory of smooth projective complex varieties. Unless otherwise stated, $X,Y,Z$ denote objects of ${\mathsf{Var}}_{\mathbb{C}}^{\mathrm{qp}}$. Additional properness, smoothness, regularity, local-complete-intersection, pure-dimensionality, or normally nonsingular hypotheses are imposed only where required.

Throughout the remainder of the paper, the geometric category $\mathsf{C}$ occurring in the fixed transport datum $\mathbb{T}$ is taken to be a specified full subcategory of ${\mathsf{Var}}_{\mathbb{C}}^{\mathrm{qp}}$ containing the objects required for the construction under consideration.

The notion of admissibility is the one fixed in Definition 1.2. Thus, for $X,Y\in Ob\left(\mathsf{C}\right)$, a typed transport operation $u:X\to Y$ is $\mathbb{T}$-admissible precisely when $u\in {Op}_{\mathbb{T}}(X,Y)$, the source-side homomorphism $u_{\mathcal{A}}:\mathcal{A}\left(X\right)\to \mathcal{A}\left(Y\right)$ and reference characteristic transport operator ${\mathcal{T}}_{\Phi }\left(u\right):\mathcal{H}\left(X\right)\to \mathcal{H}\left(Y\right)$ are defined, and the geometric hypotheses required for those operators and their comparison are satisfied.

The operations used below fall into four principal geometric classes.

If $f:X\to Y$ is proper, the corresponding transport is covariant, with transport direction $X\to Y$. If $\pi :X\to Y$ is smooth of pure relative complex dimension r, the corresponding pullback transport has direction $Y\to X$; on rational Borel–Moore homology the smooth pullback has the form

$${\pi }_{\mathrm{BM}}^{*}:H_k^{\mathrm{BM}}(Y,\mathbb{Q})⟶H_{k+2r}^{\mathrm{BM}}(X,\mathbb{Q}).$$

Its characteristic correction is determined, in the applications below, by the relative tangent bundle $T_{X/Y}$.

If $i:Z\hookrightarrow X$ is a regular or normally nonsingular embedding for which the required Gysin construction exists, the associated transport has direction $X\to Z$ and is represented on the relevant target by a refined Gysin homomorphism $i^{!}$; its correction is determined by the normal bundle $N_{Z/X}$. Refined Gysin maps and their compatibilities with proper pushforward, flat pullback, and intersection products are standard in intersection theory and bivariant theory [2,3].

Finally, algebraic correspondences act through the standard pullback–intersection–pushforward construction described in Section 3.4. Specialization operations are used only when an appropriate deformation, specialization, or vanishing-cycle theorem supplies the required source-side and characteristic-side maps [6,17].

Remark 3.1.  

Proper pushforward, smooth pullback, refined Gysin pullback, and specialization have different variance and are not formally identified. Mixed constructions are therefore composed as typed transport operations or as their actual induced homomorphisms. An ordinary composite of underlying geometric morphisms is used only when such a composite exists with the required variance.

3.2. Source Assignments and Characteristic Realizations

For $X\in {\mathsf{Var}}_{\mathbb{C}}^{\mathrm{qp}}$, write $G_0\left(X\right):=K_0(Coh\left(X\right))$ for the Grothendieck group of coherent algebraic sheaves. If X is smooth, write $K^0\left(X\right):=K_0(Vect\left(X\right))$ for the Grothendieck group of algebraic vector bundles. We write $K_0(Var/X)$ for the relative Grothendieck group of algebraic varieties over X, $CF\left(X\right)$ for the abelian group of integer-valued algebraically constructible functions, and $K_0(MHM\left(X\right))$ for the Grothendieck group of Saito’s category $MHM\left(X\right)$ of algebraic mixed Hodge modules [14].

The principal characteristic targets used below are rational Chow homology $A_{*}{\left(X\right)}_{\mathbb{Q}}:=A_{*}\left(X\right){\otimes }_{\mathbb{Z}}\mathbb{Q}$ and rational Borel–Moore homology $H_{*}^{\mathrm{BM}}(X,\mathbb{Q})$, with coefficient extensions when the Hirzebruch parameter y is present.

The Baum–Fulton–MacPherson transformation is

$${\tau }_X:G_0\left(X\right)⟶A_{*}{\left(X\right)}_{\mathbb{Q}}.$$

It is natural for proper pushforward and, when X is nonsingular, satisfies

$${\tau }_X\left(\left[{\mathcal{O}}_X\right]\right)=td\left(T_X\right)\cap \left[X\right]$$

[1].

The MacPherson transformation is

$$c_{*}:CF\left(X\right)⟶H_{*}^{\mathrm{BM}}(X,\mathbb{Z}).$$

It is natural for proper pushforward and, when X is nonsingular, satisfies

$$c_{*}\left({\mathbf{1}}_X\right)=c\left(T_X\right)\cap \left[X\right]$$

[15].

The motivic Chern and Hirzebruch transformations are

$$m{C}_y:K_0(Var/X)⟶G_0\left(X\right){\otimes }_{\mathbb{Z}}\mathbb{Z}\left[y\right]$$

and

$$T_{y*}:K_0(Var/X)⟶H_{*}^{\mathrm{BM}}(X,\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{Q}\left[y\right],$$

respectively. Both are natural for proper pushforward [16].

For mixed Hodge modules we use the characteristic transformation

$$MH{T}_{y*}:K_0(MHM\left(X\right))⟶{\mathcal{H}}_y\left(X\right),$$

where

$${\mathcal{H}}_y\left(X\right):=H_{*}^{\mathrm{BM}}(X,\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{Q}[y,y^{-1},{(1+y)}^{-1}].$$

The precise normalization is the one used in the cited mixed-Hodge-module Hirzebruch-class formalism [14,16].

Definition 3.2  

(Intersection Hirzebruch class). Let X be a pure-dimensional complex algebraic variety, and let $I{C}_X^H$ denote the mixed-Hodge-module realization of the intersection complex in the normalization of the cited theory. We write

$$I{T}_{y*}\left(X\right)\in {\mathcal{H}}_y\left(X\right)$$

for the associated intersection Hirzebruch class and $I{T}_{1,*}\left(X\right)$ for its specialization at $y=1$ whenever that specialization is defined [10,13].

Remark 3.3.  

Passage from $MHM\left(X\right)$ to $K_0(MHM\left(X\right))$ is a decategorification. For every short exact sequence

$$0⟶M_1⟶M⟶M_2⟶0$$

in $MHM\left(X\right)$, one has $\left[M\right]=\left[M_1\right]+\left[M_2\right]$ in $K_0(MHM\left(X\right))$. Consequently, characteristic transformations that factor through this Grothendieck group do not in general retain extension data. Equality of such characteristic realizations will therefore never be interpreted as equality of mixed Hodge modules.

3.3. Corrected Riemann–Roch Operators

The abstract reference operator ${\mathcal{T}}_{\Phi }\left(u\right)$ introduced in Definition 1.1 is realized below by the corrected pushforward, pullback, Gysin, or specialization operator appropriate to the geometric stage u. When a classical Riemann–Roch-style notation is useful, we write the corresponding operator as ${\mathcal{O}}_{\Phi }^{\mathrm{RR}}$. This is notation for a specified instance of ${\mathcal{T}}_{\Phi }\left(u\right)$, not a second independent construction.

Definition 3.4  

(Corrected Riemann–Roch operator). Let $u:X\to Y$ be a $\mathbb{T}$-admissible typed transport operation arising from a geometric operation $\mathcal{O}$. A$\Phi$-corrected Riemann–Roch operatoris a target-side homomorphism

$${\mathcal{O}}_{\Phi }^{\mathrm{RR}}:\mathcal{H}\left(X\right)⟶\mathcal{H}\left(Y\right)$$

obtained from the relevant pushforward, pullback, refined Gysin, specialization, or other target-side operator by inserting the correction prescribed by the applicable characteristic-class theorem. In this notation we identify

$${\mathcal{T}}_{\Phi }\left(u\right)={\mathcal{O}}_{\Phi }^{\mathrm{RR}}.$$

The superscript $\mathrm{RR}$ records the presence of the specified Riemann–Roch, tangent, normal, orientation, Gysin, or specialization correction. It does not denote a universal functor. If the characteristic transformation is already natural under a proper morphism $f:X\to Y$, then the corrected operator is simply the corresponding proper pushforward on the characteristic target.

For smooth intersection-Hirzebruch transport, let $\pi :X\to Y$ be smooth of pure relative complex dimension r. For $\eta \in H_{*}^{\mathrm{BM}}(Y,\mathbb{Q})$ in the domain for which the expression is defined, set

$${\pi }_{I{T}_y}^{\mathrm{RR},*}\left(\eta \right):=T_y^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}\left(\eta \right).$$

Here $T_y^{*}\left(T_{X/Y}\right)$ is the cohomological Hirzebruch class of the relative tangent bundle and ${\pi }_{\mathrm{BM}}^{*}$ is smooth Borel–Moore pullback [13].

The normally nonsingular Gysin case requires a different correction.

Definition 3.5  

(Corrected $I{T}_1$-Gysin operator). Let $i:Z\hookrightarrow X$ be a normally nonsingular regular algebraic embedding satisfying the hypotheses of Banagl’s Gysin theorem for $I{T}_{1,*}$, and let $N_{Z/X}$ denote its normal bundle. Let $L^{*}\left(N_{Z/X}\right)$ denote the cohomological Hirzebruch L-class of the normal bundle. Since its degree-zero term is 1, it is invertible in the rational cohomology ring in the range in which the cap product is used.

Define

$$i_{I{T}_1}^{\mathrm{RR},!}\left(\eta \right):=L^{*}{\left(N_{Z/X}\right)}^{-1}\cap i^{!}\left(\eta \right).$$

Banagl’s Gysin formula

$$i^{!}I{T}_{1,*}\left(X\right)=L^{*}\left(N_{Z/X}\right)\cap I{T}_{1,*}\left(Z\right)$$

therefore gives

$$i_{I{T}_1}^{\mathrm{RR},!}\left(I{T}_{1,*}\left(X\right)\right)=I{T}_{1,*}\left(Z\right)$$

[10].

For a specialization stage, the notation ${Sp}_{\Phi }^{\mathrm{RR}}$ denotes only the corrected specialization operator supplied by the relevant specialization theorem. No universal specialization formula is imposed [6,17].

Definition 3.6  

(Corrected naturality). Let $u:X\to Y$ be $\mathbb{T}$-admissible and $\xi \in \mathcal{A}\left(X\right)$. Corrected naturality for u on ξ is the identity

$${\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)={\mathcal{T}}_{\Phi }\left(u\right)\left({\Phi }_X\left(\xi \right)\right)$$

in $\mathcal{H}\left(Y\right)$.

For contravariant geometric operations, the typed transport direction has already been reversed, so the preceding formula requires no additional variance convention.

Equivalent placements of multiplicative correction classes are identified only when the projection formula and the relevant pullback or Gysin identities prove that they define the same target-side operator [3]. A mere change of normalization cannot therefore be used to manufacture a geometric defect.

Remark 3.7.  

Failure of corrected naturality initially defines only an element of the characteristic target, namely the defect

$${\Delta }_{\Phi }(u;\xi )={\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }\left(u\right)\left({\Phi }_X\left(\xi \right)\right).$$

A geometric interpretation is asserted only when an independent theorem identifies this class with specialization, vanishing-cycle, Milnor-type, excess, support, or analogous geometric data.

3.4. Correspondences and Composition Conventions

Let $X,Y,Z\in {\mathsf{SmProj}}_{\mathbb{C}}$, and set $d_X:={dim}_{\mathbb{C}}X$, $d_Y:={dim}_{\mathbb{C}}Y$, and $d_Z:={dim}_{\mathbb{C}}Z$.

Definition 3.8  

(Degree-k correspondence). For $k\in \mathbb{Z}$, acorrespondence of degree k from X to Yis a class

$$\Gamma \in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}.$$

Let $p_X:X\times Y\to X$ and $p_Y:X\times Y\to Y$ denote the projections. For $\Gamma \in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}$ and $\alpha \in C{H}^p{\left(X\right)}_{\mathbb{Q}}$, define the Chow-theoretic action by

$${\Gamma }_{*}\left(\alpha \right):={\left(p_Y\right)}_{*}\left(p_X^{*}\alpha ·\Gamma \right)\in C{H}^{p+k}{\left(Y\right)}_{\mathbb{Q}}.$$

(4)

Here $p_X^{*}$ is flat pullback, · is the intersection product, and ${\left(p_Y\right)}_{*}$ is proper pushforward [3].

Let

$${cl}_X:C{H}^p{\left(X\right)}_{\mathbb{Q}}⟶H^{2p}(X,\mathbb{Q}\left(p\right))$$

denote the rational cycle-class map. If $\gamma \in H^i(X,\mathbb{Q}\left(j\right))$, define the cohomological action of $\Gamma$ by

$${\Gamma }_{*}\left(\gamma \right):={\left(p_Y\right)}_{*}\left(p_X^{*}\gamma ⌣{cl}_{X\times Y}(\Gamma )\right)\in H^{i+2k}(Y,\mathbb{Q}(j+k)).$$

Here ⌣ denotes cup product and ${\left(p_Y\right)}_{*}$ is the cohomological Gysin pushforward.

Proposition 3.9  

(Cycle-class compatibility). For every $\alpha \in C{H}^p{\left(X\right)}_{\mathbb{Q}}$,

$${cl}_Y\left({\Gamma }_{*}\left(\alpha \right)\right)={\Gamma }_{*}\left({cl}_X\left(\alpha \right)\right).$$

Proof. 

The identity follows from compatibility of the rational cycle-class map with flat pullback, intersection products, and proper pushforward [3]. □

Let

$$p_{12}:X\times Y\times Z\to X\times Y,p_{23}:X\times Y\times Z\to Y\times Z,p_{13}:X\times Y\times Z\to X\times Z$$

denote the indicated projections.

Definition 3.10  

(Composition of correspondences). Let

$$\Gamma \in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}},\Theta \in C{H}^{d_Y+\ell }{(Y\times Z)}_{\mathbb{Q}},$$

where $k,\ell \in \mathbb{Z}$. Their composition, with Γ acting first and Θ second, is

$$\Theta \circ \Gamma :={\left(p_{13}\right)}_{*}\left(p_{12}^{*}\Gamma ·p_{23}^{*}\Theta \right)\in C{H}^{d_X+k+\ell }{(X\times Z)}_{\mathbb{Q}}.$$

Proposition 3.11  

(Compatibility of composition with action). The Chow-theoretic and rational cohomological actions satisfy

$${(\Theta \circ \Gamma )}_{*}={\Theta }_{*}\circ {\Gamma }_{*}.$$

Proof. 

This is the standard projection-formula calculation for algebraic correspondences [3]. □

For $X\in {\mathsf{SmProj}}_{\mathbb{C}}$, let ${\Delta }_X\subseteq X\times X$ denote the diagonal and $\left[{\Delta }_X\right]\in C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$ its cycle class.

Proposition 3.12  

(Diagonal correspondence). The diagonal class $\left[{\Delta }_X\right]$ is the identity correspondence:

$${\left(\left[{\Delta }_X\right]\right)}_{*}=id$$

on both Chow groups and rational cohomology, and composition with $\left[{\Delta }_X\right]$ leaves every compatible correspondence unchanged.

This paper uses algebraic Chow correspondences as its geometric correspondences. A rational cohomology class on $X\times Y$ that is not known to arise from an algebraic cycle is not substituted for an algebraic correspondence.

3.5. Selected Cohomological and Hodge Sectors

Let $X\in {\mathsf{SmProj}}_{\mathbb{C}}$ and let $p\in {\mathbb{Z}}_{\ge 0}$. Write $\mathbb{Q}\left(p\right)$ for the rational Tate Hodge structure of weight $-2p$. Since X is smooth and projective, $H^{2p}(X,\mathbb{Q}\left(p\right))$ is a pure rational Hodge structure of weight 0. Its rational Hodge classes are

$$H^{2p}{(X,\mathbb{Q}\left(p\right))}_{\mathrm{Hdg}}:={Hom}_{{\mathrm{HS}}_{\mathbb{Q}}}\left(\mathbb{Q}\left(0\right),H^{2p}(X,\mathbb{Q}\left(p\right))\right),$$

where ${\mathrm{HS}}_{\mathbb{Q}}$ denotes the category of rational pure Hodge structures. Equivalently, these are the rational degree-$2p$ cohomology classes of Hodge type $(p,p)$, expressed after Tate twist.

The rational cycle-class map

$${cl}_X:C{H}^p{\left(X\right)}_{\mathbb{Q}}⟶H^{2p}(X,\mathbb{Q}\left(p\right))$$

takes algebraic cycles to rational Hodge classes.

Definition 3.13  

(Selected cohomological and Hodge sectors). Aselected cohomological sectorof X is a finite-dimensional $\mathbb{Q}$-linear subspace $L\subseteq H^{•}(X,\mathbb{Q})$. For a fixed $p\in {\mathbb{Z}}_{\ge 0}$, aselected Hodge sector of codimension pis a $\mathbb{Q}$-linear subspace

$$L\subseteq H^{2p}{(X,\mathbb{Q}\left(p\right))}_{\mathrm{Hdg}}.$$

The choice of L includes no algebraic or cohomological projector. All geometric operations and correspondence actions are defined on their ambient theories before restriction to L.

Let $Y\in {\mathsf{SmProj}}_{\mathbb{C}}$, let $d_X:={dim}_{\mathbb{C}}X$, fix $k\in \mathbb{Z}$, and let ${\Gamma }_1,{\Gamma }_2\in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}$ be correspondences of the same degree.

Definition 3.14  

(L-equivalence). The correspondences ${\Gamma }_1$ and ${\Gamma }_2$ areL-equivalent, written

$${\Gamma }_1{\equiv }_L{\Gamma }_2,$$

if

$${\left({\Gamma }_1\right)}_{*}{{|}_L={\left({\Gamma }_2\right)}_{*}|}_L.$$

When ${\Gamma }_1{\equiv }_L{\Gamma }_2$, define their residual correspondence by

$${\mathfrak{D}}_L({\Gamma }_1,{\Gamma }_2):={\Gamma }_1-{\Gamma }_2.$$

To express the same condition functorially, let $W_L\left(Y\right)$ denote the graded rational cohomology space containing the images of all components of L under degree-k correspondences from X to Y. Define

$${\rho }_L:C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}⟶{Hom}_{\mathbb{Q}}\left(L,W_L\left(Y\right)\right)$$

by

$${\rho }_L(\Gamma ):={\Gamma }_{*}{|}_L.$$

Then

$${\Gamma }_1{\equiv }_L{\Gamma }_2⟺{\Gamma }_1-{\Gamma }_2\in ker\left({\rho }_L\right).$$

(5)

The kernel $ker\left({\rho }_L\right)$ is therefore precisely the set of degree-k correspondences whose action vanishes on the selected sector L. Later sections ask whether particular nonzero elements of this kernel remain visible before restriction to L, for example through total cohomology, characteristic transport profiles, or independently identified geometric defects.

For correspondences of fixed degree k, the comparison hierarchy used throughout the paper is

$$\mathrm{equality}\mathrm{in}\mathrm{Chow}⟹\mathrm{equality}\mathrm{of}\mathrm{total}\mathrm{rational}\mathrm{cohomological}\mathrm{action}⟹\mathrm{equality}\mathrm{on}L.$$

No converse is assumed.

Remark 3.15.  

Transport soundness and characteristic separation are logically distinct. Soundness requires a transport identity to follow from established functoriality, Riemann–Roch, Gysin, bivariant, or specialization results. Separation requires an independently established nonzero difference between two characteristic transport profiles after a prescribed cohomological realization has identified their actions. Neither implication is automatic.

4. Characteristic Transport Defects

We use throughout the characteristic transport datum and typed-operation conventions of Section 1 and Section 3. Thus

$$\mathbb{T}=\left(\mathsf{C},\mathcal{A},\mathcal{H},\Phi ,{Op}_{\mathbb{T}},{\mathcal{T}}_{\Phi }\right)$$

denotes a fixed characteristic transport datum, $X,Y\in Ob\left(\mathsf{C}\right)$, and $u:X\to Y$ denotes a $\mathbb{T}$-admissible typed transport operation. Its source-side homomorphism is $u_{\mathcal{A}}:\mathcal{A}\left(X\right)\to \mathcal{A}\left(Y\right)$, while its reference characteristic transport operator is ${\mathcal{T}}_{\Phi }\left(u\right):\mathcal{H}\left(X\right)\to \mathcal{H}\left(Y\right)$.

The characteristic transport defect was introduced in Definition 1.3. For ease of reference in the remainder of the paper, we record the same definition here in the notation used throughout this section.

Definition 4.1  

(Characteristic transport defect). For $u:X\to Y$$\mathbb{T}$-admissible and $\xi \in \mathcal{A}\left(X\right)$, the$\Phi$-transport defectof ξ along u is

$${\Delta }_{\Phi }(u;\xi ):={\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }\left(u\right)\left({\Phi }_X\left(\xi \right)\right)\in \mathcal{H}\left(Y\right).$$

The purpose of the present section is to develop the properties and concrete models of these defects needed in the later composite theory.

4.1. Exact, Defective, and Variance-Specific Transport

Let $u:X\to Y$ be $\mathbb{T}$-admissible and let $\xi \in \mathcal{A}\left(X\right)$.

Definition 4.2  

(Exact and defective transport). The operation u isexact on $\xi$if ${\Delta }_{\Phi }(u;\xi )=0$, anddefective on $\xi$if ${\Delta }_{\Phi }(u;\xi )\ne 0$. It is$\Phi$-exactif ${\Delta }_{\Phi }(u;\xi )=0$ for every $\xi \in \mathcal{A}\left(X\right)$.

Exact transport is calledcorrected exactwhen the reference operator ${\mathcal{T}}_{\Phi }\left(u\right)$ contains a nontrivial Riemann–Roch, tangent, normal-bundle, Gysin, orientation, or specialization correction.

The general definition is variance-safe because the typed arrow $u:X\to Y$ records the direction of the induced transport operator, not necessarily the direction of the underlying geometric morphism. Specialized notation is nevertheless useful for the geometric operations that occur later.

Let $f:X\to Y$ be a proper morphism and let $u_f^{\mathrm{prop}}:X\to Y$ denote the corresponding covariant typed transport operation. When no ambiguity can arise, we abbreviate ${\Delta }_{\Phi }(u_f^{\mathrm{prop}};\xi )$ to ${\Delta }_{\Phi }(f;\xi )$.

Definition 4.3  

(Contravariant and Gysin defect notation). Let $\pi :X\to Y$ be a smooth morphism and let $u_{\pi }^{\mathrm{sm}}:Y\to X$ denote the corresponding pullback-type typed transport operation. For $\eta \in \mathcal{A}\left(Y\right)$, define

$${\Delta }_{\Phi }^{*}(\pi ;\eta ):={\Delta }_{\Phi }\left(u_{\pi }^{\mathrm{sm}};\eta \right)\in \mathcal{H}\left(X\right).$$

Let $i:Z\hookrightarrow X$ be an embedding for which the relevant refined Gysin operation is defined, and let $u_i^{!}:X\to Z$ denote the corresponding Gysin-type typed transport operation. For $\eta \in \mathcal{A}\left(X\right)$, define

$${\Delta }_{\Phi }^{!}(i;\eta ):={\Delta }_{\Phi }\left(u_i^{!};\eta \right)\in \mathcal{H}\left(Z\right).$$

The superscripts ^* and ^! record the underlying geometric variance; they do not define different abstract notions of characteristic transport defect.

Our sign convention throughout is

$$\mathrm{transport}\mathrm{defect}=\mathrm{actual}\mathrm{transformed}\mathrm{characteristic}\mathrm{class}-\mathrm{reference}\mathrm{corrected}\mathrm{characteristic}\mathrm{class}.$$

This convention is maintained in every specialization, smooth-pullback, and refined Gysin calculation below.

Remark 4.4.  

The notation ${\Delta }_{\Phi }$ packages into a common transport language differences that already occur in Verdier–Riemann–Roch, Milnor-class, and Hirzebruch–Milnor theories [6,7,8,9]. The novelty of the present paper lies in their use inside composite transport profiles, not in the existence of one-stage defect formulas.

The codomain of a defect is determined by the characteristic target of its typed operation. Proper Baum–Fulton–MacPherson defects lie in $A_{*}{\left(Y\right)}_{\mathbb{Q}}$, proper Chern–Schwartz–MacPherson defects in $H_{*}^{\mathrm{BM}}(Y,\mathbb{Z})$, motivic Hirzebruch defects in $H_{*}^{\mathrm{BM}}(Y,\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{Q}\left[y\right]$, and mixed-Hodge-module Hirzebruch defects in ${\mathcal{H}}_y\left(Y\right)$ [1,15,16]. Smooth $I{T}_y$-defects associated with $\pi :X\to Y$ lie in the characteristic target attached to X, while normally nonsingular $I{T}_{1,*}$-Gysin defects associated with $i:Z\hookrightarrow X$ lie in the characteristic target attached to Z [10,13].

Remark 4.5.  

The condition ${\Delta }_{\Phi }(u;\xi )\ne 0$ alone does not make a defect a geometric obstruction. Geometric significance is asserted only when an independent theorem identifies the difference with specialization, vanishing-cycle data, a Milnor-type class, excess-intersection data, controlled support, or another independently defined geometric contribution.

4.2. Normalization and Trivial Transport Stages

The transport datum is assumed to contain typed identity operations $u_X^{\mathrm{id}}:X\to X$ for every $X\in Ob\left(\mathsf{C}\right)$.

Definition 4.6  

(Identity normalization). The transport datum $\mathbb{T}$ isidentity-normalizedif

$${\left(u_X^{\mathrm{id}}\right)}_{\mathcal{A}}={id}_{\mathcal{A}\left(X\right)}\mathrm{and}{\mathcal{T}}_{\Phi }\left(u_X^{\mathrm{id}}\right)={id}_{\mathcal{H}\left(X\right)}$$

for every $X\in Ob\left(\mathsf{C}\right)$.

Under identity normalization, ${\Delta }_{\Phi }(u_X^{\mathrm{id}};\xi )=0$ for every $\xi \in \mathcal{A}\left(X\right)$.

Because ${\Phi }_X$, $u_{\mathcal{A}}$, and ${\mathcal{T}}_{\Phi }\left(u\right)$ are additive homomorphisms, the transport defect is additive in the source class:

$${\Delta }_{\Phi }(u;{\xi }_1+{\xi }_2)={\Delta }_{\Phi }(u;{\xi }_1)+{\Delta }_{\Phi }(u;{\xi }_2)$$

for all ${\xi }_1,{\xi }_2\in \mathcal{A}\left(X\right)$. Consequently,

$${\Delta }_{\Phi }(u;n\xi )=n{\Delta }_{\Phi }(u;\xi )$$

for every $n\in \mathbb{Z}$.

The second trivial modification used later is replacement of intermediate objects by isomorphic ones.

Proposition 4.7  

(Naturality under isomorphic replacement). Let

$$\begin{array}{ccc}X& \stackrel{F}{\to }& Y\\ \downarrow a& & \downarrow b\\ X^{\prime }& \stackrel{F^{\prime }}{\to }& Y^{\prime }\end{array}$$

be a commutative square of complex algebraic varieties in which a: X $\stackrel{\sim }{\to }$ X′ and b: Y $\stackrel{\sim }{\to }$ Y′ are isomorphisms.Let u_F : X → Y and uF′ : X′ → Y′ denote the corresponding typed transport operations

Assume that the source assignment, characteristic target assignment, characteristic transformation data, and reference transport operators are functorial under a and b. Denote the induced source-side isomorphisms by

$$a_{\mathcal{A}}:\mathcal{A}\left(X\right)\stackrel{\sim }{\to }\mathcal{A}\left(X^{\prime }\right),b_{\mathcal{A}}:\mathcal{A}\left(Y\right)\stackrel{\sim }{\to }\mathcal{A}\left(Y^{\prime }\right),$$

and the target-side isomorphisms by

$$a_{\mathcal{H}}:\mathcal{H}\left(X\right)\stackrel{\sim }{\to }\mathcal{H}\left(X^{\prime }\right),b_{\mathcal{H}}:\mathcal{H}\left(Y\right)\stackrel{\sim }{\to }\mathcal{H}\left(Y^{\prime }\right).$$

Then, for every $\xi \in \mathcal{A}\left(X\right)$,

$${\Delta }_{\Phi }\left(u_{F^{\prime }};a_{\mathcal{A}}\left(\xi \right)\right)=b_{\mathcal{H}}\left({\Delta }_{\Phi }(u_F;\xi )\right).$$

Proof. 

By functoriality,

$${\left(u_{F^{\prime }}\right)}_{\mathcal{A}}\circ a_{\mathcal{A}}=b_{\mathcal{A}}\circ {\left(u_F\right)}_{\mathcal{A}},$$

$${\Phi }_{X^{\prime }}\circ a_{\mathcal{A}}=a_{\mathcal{H}}\circ {\Phi }_X,{\Phi }_{Y^{\prime }}\circ b_{\mathcal{A}}=b_{\mathcal{H}}\circ {\Phi }_Y,$$

and

$${\mathcal{T}}_{\Phi }\left(u_{F^{\prime }}\right)\circ a_{\mathcal{H}}=b_{\mathcal{H}}\circ {\mathcal{T}}_{\Phi }\left(u_F\right).$$

Therefore

$$\begin{array}{cc}\hfill {\Delta }_{\Phi }\left(u_{F^{\prime }};a_{\mathcal{A}}\left(\xi \right)\right)& ={\Phi }_{Y^{\prime }}\left({\left(u_{F^{\prime }}\right)}_{\mathcal{A}}{a}_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }\left(u_{F^{\prime }}\right)\left({\Phi }_{X^{\prime }}{a}_{\mathcal{A}}\left(\xi \right)\right)\hfill \\ & =b_{\mathcal{H}}\left[{\Phi }_Y\left({\left(u_F\right)}_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }\left(u_F\right)\left({\Phi }_X\left(\xi \right)\right)\right]\hfill \\ & =b_{\mathcal{H}}\left({\Delta }_{\Phi }(u_F;\xi )\right).\hfill \end{array}$$

□

In particular, an isomorphism stage has zero defect whenever its source-side and target-side actions are the corresponding functorial isomorphisms. Identity insertion and isomorphic replacement therefore introduce no intrinsic transport defect. These facts are used when raw routes are later passed to profile-equivalence classes.

4.3. Schürmann’s Defect as the Basic Singular Model

We now record the principal singular one-stage defect used later.

Let $i:X\hookrightarrow Y$ be a regular embedding of complex algebraic varieties satisfying the hypotheses of [6]. Let $N:=N_{X/Y}$ denote the normal bundle, let $\pi :N\to X$ be the bundle projection, and let $k:X\hookrightarrow N$ be the zero section.

Write $CF\left(Y\right)$ for the group of algebraically constructible integer-valued functions on Y, and let ${CF}_{\mathrm{mon}}\left(N\right)$ denote the corresponding group of monodromic constructible functions on N in the specialization formalism of Schürmann. Verdier specialization gives

$${Sp}_{X\setminus Y}:CF\left(Y\right)⟶{CF}_{\mathrm{mon}}\left(N\right)$$

[6,17].

Let $i_{CF}^{*}:CF\left(Y\right)\to CF\left(X\right)$ denote ordinary pullback of constructible functions, and let ${\pi }_{CF}^{*}:CF\left(X\right)\to {CF}_{\mathrm{mon}}\left(N\right)$ denote pullback along the bundle projection.

Definition 4.8  

(Generalized vanishing-cycle transformation). The generalized vanishing-cycle transformation associated with i is

$${\Phi }_i:={Sp}_{X\setminus Y}-{\pi }_{CF}^{*}\circ i_{CF}^{*}:CF\left(Y\right)⟶{CF}_{\mathrm{mon}}\left(N\right).$$

Let

$$c_{*}:CF(-)⟶H_{*}^{\mathrm{BM}}(-,\mathbb{Z})$$

denote the Chern–Schwartz–MacPherson transformation. Let $i_H^{!}$ denote the refined Gysin homomorphism used in [6]. If $c:={codim}_{\mathbb{C}}(X,Y)$, homotopy invariance for the rank-c vector bundle $\pi :N\to X$ gives an isomorphism

$${\pi }_H^{*}:H_{*}^{\mathrm{BM}}\left(X\right)\stackrel{\sim }{\to }{H}_{*+2c}^{\mathrm{BM}}\left(N\right).$$

We write

$$k_H^{!}:={\left({\pi }_H^{*}\right)}^{-1}.$$

Let $c^{*}\left(N\right)$ denote the total cohomological Chern class of the normal bundle. Following the sign convention of this paper, define Schürmann’s comparison term for $\alpha \in CF\left(Y\right)$ by

$${\mathfrak{S}}_i\left(\alpha \right):=c^{*}{\left(N\right)}^{-1}\cap i_H^{!}{c}_{*}\left(\alpha \right)-c_{*}\left(i_{CF}^{*}\alpha \right).$$

Our Gysin-defect convention is

$${\Delta }_{c_{*}}^{!}(i;\alpha )=-{\mathfrak{S}}_i\left(\alpha \right).$$

Lemma 4.9  

(Schürmann’s Verdier–Riemann–Roch defect formula). For every $\alpha \in CF\left(Y\right)$,

$${\mathfrak{S}}_i\left(\alpha \right)=c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right).$$

Equivalently,

$${\Delta }_{c_{*}}^{!}(i;\alpha )=-c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right).$$

Proof. 

Schürmann’s specialization formula gives

$$i_H^{!}{c}_{*}\left(\alpha \right)=k_H^{!}{c}_{*}\left({Sp}_{X\setminus Y}\left(\alpha \right)\right)$$

under the hypotheses of [6]. By definition,

$${Sp}_{X\setminus Y}\left(\alpha \right)={\Phi }_i\left(\alpha \right)+{\pi }_{CF}^{*}{i}_{CF}^{*}\left(\alpha \right).$$

Additivity of $c_{*}$ therefore yields

$$i_H^{!}{c}_{*}\left(\alpha \right)=k_H^{!}{c}_{*}\left({\Phi }_i\left(\alpha \right)\right)+k_H^{!}{c}_{*}\left({\pi }_{CF}^{*}{i}_{CF}^{*}\left(\alpha \right)\right).$$

Smooth Verdier–Riemann–Roch for the bundle projection $\pi :N\to X$ gives

$$k_H^{!}{c}_{*}\left({\pi }_{CF}^{*}{i}_{CF}^{*}\left(\alpha \right)\right)=c^{*}\left(N\right)\cap c_{*}\left(i_{CF}^{*}\left(\alpha \right)\right).$$

Substituting this identity into the preceding equation and capping with $c^{*}{\left(N\right)}^{-1}$ proves the first formula. The second follows from ${\Delta }_{c_{*}}^{!}(i;\alpha )=-{\mathfrak{S}}_i\left(\alpha \right)$.

□

If ${\Phi }_i\left(\alpha \right)=0$, then ${\Delta }_{c_{*}}^{!}(i;\alpha )=0$. When ${\Phi }_i\left(\alpha \right)\ne 0$, Lemma 4.9 identifies the transport defect with independently defined specialization data. It is therefore a geometrically identified defect in the sense of Definition 1.4, rather than an arbitrary formal difference.

For $\alpha ={\mathbf{1}}_Y$ in the geometric situations considered later, the generalized vanishing-cycle contribution is supported over the relevant singular locus. Section 5.3 tracks the resulting support through subsequent transport stages.

4.4. Inherited Exact Transport Models

Several stages used later are exact by established naturality or transfer theorems. We record them together because their role in the composite theory is the same: they provide later operators through which independently identified nonzero defects may propagate.

Proposition 4.10  

(Proper exact transport). Let $f:X\to Y$ be proper and let $u_f^{\mathrm{prop}}:X\to Y$ be the associated typed transport operation.

1.

For every $\xi \in G_0\left(X\right)$,

$${\Delta }_{\tau }\left(u_f^{\mathrm{prop}};\xi \right)=0.$$

2.

For every $\phi \in CF\left(X\right)$,

$${\Delta }_{c_{*}}\left(u_f^{\mathrm{prop}};\phi \right)=0.$$

3.

For every $\eta \in K_0(Var/X)$,

$${\Delta }_{T_{y*}}\left(u_f^{\mathrm{prop}};\eta \right)=0.$$

Proof. 

The three identities are respectively the proper-pushforward naturality of the Baum–Fulton–MacPherson transformation, the Chern–Schwartz–MacPherson transformation, and the motivic Hirzebruch transformation [1,15,16].

□

Proposition 4.11  

(Smooth intersection-Hirzebruch transport). Let $\pi :X\to Y$ be a smooth algebraic morphism of pure relative complex dimension r between pure-dimensional complex algebraic varieties. Let $T_{X/Y}$ denote its relative tangent bundle and ${\pi }_{\mathrm{BM}}^{*}$ the smooth Borel–Moore pullback. Then

$$I{T}_{y*}\left(X\right)=T_y^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}I{T}_{y*}\left(Y\right).$$

Equivalently, with

$${\Delta }_{I{T}_y}^{*}\left(\pi \right):=I{T}_{y*}\left(X\right)-T_y^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}I{T}_{y*}\left(Y\right),$$

one has

$${\Delta }_{I{T}_y}^{*}\left(\pi \right)=0.$$

Proof. 

This is the smooth intersection-Hirzebruch transfer theorem [13].

□

Proposition 4.12  

(Normally nonsingular $I{T}_{1,*}$-Gysin transport). Let $i:Z\hookrightarrow X$ satisfy the hypotheses of Banagl’s $I{T}_{1,*}$-Gysin theorem, and let $N_{Z/X}$ denote its normal bundle. Then

$$i^{!}I{T}_{1,*}\left(X\right)=L^{*}\left(N_{Z/X}\right)\cap I{T}_{1,*}\left(Z\right),$$

or equivalently

$$i_{I{T}_1}^{\mathrm{RR},!}\left(I{T}_{1,*}\left(X\right)\right)=I{T}_{1,*}\left(Z\right).$$

Thus the specialized Gysin defect

$${\Delta }_{I{T}_1}^{!}\left(i\right):=I{T}_{1,*}\left(Z\right)-i_{I{T}_1}^{\mathrm{RR},!}\left(I{T}_{1,*}\left(X\right)\right)$$

vanishes.

Proof. 

The asserted identity is Banagl’s normally nonsingular Gysin formula [10], together with the corrected-operator convention of Definition 3.5.

□

The preceding propositions are inherited exact stages, not new one-stage characteristic identities. Their role below is to transport, preserve, annihilate, or combine defects created at other stages.

4.5. Dependence on Normalization

A change in the presentation of a correction factor must be distinguished from a change in the underlying geometric defect. We formalize the relevant notion of equivalent normalization.

Fix $X,Y\in Ob\left(\mathsf{C}\right)$ and a $\mathbb{T}$-admissible typed transport operation $u:X\to Y$. Suppose $\tilde{\mathcal{H}}\left(X\right)$ and $\tilde{\mathcal{H}}\left(Y\right)$ are additive groups and

$$U_X:\mathcal{H}\left(X\right)\stackrel{\sim }{\to }\tilde{\mathcal{H}}\left(X\right),U_Y:\mathcal{H}\left(Y\right)\stackrel{\sim }{\to }\tilde{\mathcal{H}}\left(Y\right)$$

are additive isomorphisms.

Define transformed characteristic maps by

$${\tilde{\Phi }}_X:=U_X\circ {\Phi }_X,{\tilde{\Phi }}_Y:=U_Y\circ {\Phi }_Y.$$

Definition 4.13  

(Equivalent normalizations). The original and transformed target normalizations areequivalent along uif the transformed reference operator satisfies

$${\tilde{\mathcal{T}}}_{\tilde{\Phi }}\left(u\right)=U_Y\circ {\mathcal{T}}_{\Phi }\left(u\right)\circ U_X^{-1}.$$

Proposition 4.14  

(Covariance of the defect under equivalent normalization). For every $\xi \in \mathcal{A}\left(X\right)$,

$${\Delta }_{\tilde{\Phi }}(u;\xi )=U_Y\left({\Delta }_{\Phi }(u;\xi )\right).$$

In particular,

$${\Delta }_{\tilde{\Phi }}(u;\xi )=0⟺{\Delta }_{\Phi }(u;\xi )=0.$$

Proof. 

By definition,

$$\begin{array}{cc}\hfill {\Delta }_{\tilde{\Phi }}(u;\xi )& ={\tilde{\Phi }}_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)-{\tilde{\mathcal{T}}}_{\tilde{\Phi }}\left(u\right)\left({\tilde{\Phi }}_X\left(\xi \right)\right)\hfill \\ & =U_Y{\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)-U_Y{\mathcal{T}}_{\Phi }\left(u\right)U_X^{-1}{U}_X{\Phi }_X\left(\xi \right)\hfill \\ & =U_Y\left[{\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }\left(u\right)\left({\Phi }_X\left(\xi \right)\right)\right]\hfill \\ & =U_Y\left({\Delta }_{\Phi }(u;\xi )\right).\hfill \end{array}$$

Since $U_Y$ is an isomorphism, defect vanishing and nonvanishing are preserved.

□

A common source of equivalent normalizations is relocation of an invertible multiplicative characteristic class across a proper pushforward.

Lemma 4.15  

(Projection-formula normalization identity). Let $F:X\to Y$ be a proper morphism for which the projection formula holds. Let $C_Y\in H^{\mathrm{even}}(Y,\mathbb{Q})$ and $C_F\in H^{\mathrm{even}}(X,\mathbb{Q})$ be invertible multiplicative classes, and define

$$C_X:=F^{*}{C}_Y⌣C_F.$$

Then, for every $b\in H_{*}^{\mathrm{BM}}(X,\mathbb{Q})$ for which the cap products are defined,

$$C_Y^{-1}\cap F_{*}\left(C_X\cap b\right)=F_{*}\left(C_F\cap b\right).$$

Proof. 

Using $C_X=F^{*}{C}_Y⌣C_F$, the projection formula gives

$$\begin{array}{cc}\hfill F_{*}\left(C_X\cap b\right)& =F_{*}\left(F^{*}{C}_Y\cap (C_F\cap b)\right)\hfill \\ & =C_Y\cap F_{*}\left(C_F\cap b\right).\hfill \end{array}$$

Capping both sides with $C_Y^{-1}$ gives the result.

□

Lemma 4.15 controls equivalent placements of Todd, Hirzebruch, L-type, and analogous invertible multiplicative correction factors whenever the required projection formula applies. Consequently, moving such a correction between equivalent source-side and target-side normalizations cannot create or remove an intrinsic transport defect.

The distinction needed in later sections is therefore between three cases: inherited exact transport, a geometrically identified nonzero defect, and a mere change of equivalent normalization. Only the second contributes new geometric information to a composite characteristic transport profile.

5. Composition and Propagation of Transport Defects

We now study how characteristic transport defects behave under composition. Throughout the section, fix a characteristic transport datum

$$\mathbb{T}=\left(\mathsf{C},\mathcal{A},\mathcal{H},\Phi ,{Op}_{\mathbb{T}},{\mathcal{T}}_{\Phi }\right)$$

as in Definition 1.1. Unless a mixed-variance situation is stated explicitly, let $X,Y,Z\in Ob\left(\mathsf{C}\right)$ and consider compatible admissible typed transport operations $u:X\to Y$ and $v:Y\to Z$.

When u and v arise from covariant geometric operations $F:X\to Y$ and $G:Y\to Z$, respectively, we occasionally use the abbreviations

$$F_{!}:=u_{\mathcal{A}},G_{!}:=v_{\mathcal{A}},$$

and

$$F_{\Phi ,*}^{\mathrm{RR}}:={\mathcal{T}}_{\Phi }\left(u\right),G_{\Phi ,*}^{\mathrm{RR}}:={\mathcal{T}}_{\Phi }\left(v\right).$$

These are not additional structures: they are the source-side and reference target-side operators of the corresponding typed transport operations.

Mixed-variance constructions are always composed in the direction of their induced transport operators. In particular, a Gysin or smooth-pullback stage is never replaced formally by a forward morphism merely to manufacture an ordinary geometric composite.

5.1. The Two-Stage Composition Problem

Let $u:X\to Y$ and $v:Y\to Z$ be compatible $\mathbb{T}$-admissible typed transport operations. Suppose that the underlying geometric construction also determines an admissible compressed operation

$$v\diamond u:X⟶Z$$

whose source-side action satisfies

$${(v\diamond u)}_{\mathcal{A}}=v_{\mathcal{A}}\circ u_{\mathcal{A}}.$$

As in Section 1.5, the symbol ⋄ distinguishes composition of typed geometric operations from ordinary composition of their induced homomorphisms.

Definition 5.1  

(Riemann–Roch composition interaction). TheRiemann–Roch composition interactionof v and u is

$${\mathfrak{I}}_{\Phi }(v,u):={\mathcal{T}}_{\Phi }\left(v\right)\circ {\mathcal{T}}_{\Phi }\left(u\right)-{\mathcal{T}}_{\Phi }(v\diamond u)\in {Hom}_{\mathbf{Ab}}\left(\mathcal{H}\left(X\right),\mathcal{H}\left(Z\right)\right).$$

In the covariant notation associated with geometric operations $F:X\to Y$ and $G:Y\to Z$, this is equivalently

$${\mathfrak{I}}_{\Phi }(G,F)=G_{\Phi ,*}^{\mathrm{RR}}\circ F_{\Phi ,*}^{\mathrm{RR}}-{(G\circ F)}_{\Phi ,*}^{\mathrm{RR}},$$

whenever the compressed operation is represented by the ordinary composite $G\circ F$.

Definition 5.2  

(Riemann–Roch-compatible pair). The compatible pair $(v,u)$, or $(G,F)$ in the preceding covariant notation, is$\Phi$-Riemann–Roch compatibleif

$${\mathfrak{I}}_{\Phi }(v,u)=0.$$

Equivalently,

$${\mathcal{T}}_{\Phi }(v\diamond u)={\mathcal{T}}_{\Phi }\left(v\right)\circ {\mathcal{T}}_{\Phi }\left(u\right).$$

Thus Riemann–Roch compatibility concerns the chosen reference characteristic operators. It is logically independent of the possible vanishing or nonvanishing of the stage defects themselves.

Proposition 5.3  

(Two-stage defect identity). Let $\xi \in \mathcal{A}\left(X\right)$. Then

$$\begin{array}{cc}\hfill {\Delta }_{\Phi }(v\diamond u;\xi )=& {\Delta }_{\Phi }\left(v;u_{\mathcal{A}}\left(\xi \right)\right)+{\mathcal{T}}_{\Phi }\left(v\right)\left({\Delta }_{\Phi }(u;\xi )\right)\hfill \\ \hfill & +{\mathfrak{I}}_{\Phi }(v,u)\left({\Phi }_X\left(\xi \right)\right).\hfill \end{array}$$

(6)

In the covariant notation $u=u_F$, $v=u_G$, this becomes

$${\Delta }_{\Phi }(G\circ F;\xi )={\Delta }_{\Phi }(G;F_{!}\xi )+G_{\Phi ,*}^{\mathrm{RR}}\left({\Delta }_{\Phi }(F;\xi )\right)+{\mathfrak{I}}_{\Phi }(G,F)\left({\Phi }_X\left(\xi \right)\right).$$

Proof. 

By Definition 4.1 and the equality ${(v\diamond u)}_{\mathcal{A}}=v_{\mathcal{A}}\circ u_{\mathcal{A}}$,

$${\Delta }_{\Phi }(v\diamond u;\xi )={\Phi }_Z\left(v_{\mathcal{A}}{u}_{\mathcal{A}}\left(\xi \right)\right)-{\mathcal{T}}_{\Phi }(v\diamond u)\left({\Phi }_X\left(\xi \right)\right).$$

Add and subtract ${\mathcal{T}}_{\Phi }\left(v\right)\left({\Phi }_Y\left(u_{\mathcal{A}}\left(\xi \right)\right)\right)$ and ${\mathcal{T}}_{\Phi }\left(v\right){\mathcal{T}}_{\Phi }\left(u\right)\left({\Phi }_X\left(\xi \right)\right)$. The first resulting difference is ${\Delta }_{\Phi }(v;u_{\mathcal{A}}\left(\xi \right))$; the second is ${\mathcal{T}}_{\Phi }\left(v\right)\left({\Delta }_{\Phi }(u;\xi )\right)$; and the third is ${\mathfrak{I}}_{\Phi }(v,u)\left({\Phi }_X\left(\xi \right)\right)$. This proves (6).

□

Corollary 5.4  

(Compatible composition formula). If $(v,u)$ is Φ-Riemann–Roch compatible, then

$${\Delta }_{\Phi }(v\diamond u;\xi )={\Delta }_{\Phi }\left(v;u_{\mathcal{A}}\left(\xi \right)\right)+{\mathcal{T}}_{\Phi }\left(v\right)\left({\Delta }_{\Phi }(u;\xi )\right).$$

Equivalently, in covariant notation,

$${\Delta }_{\Phi }(G\circ F;\xi )={\Delta }_{\Phi }(G;F_{!}\xi )+G_{\Phi ,*}^{\mathrm{RR}}\left({\Delta }_{\Phi }(F;\xi )\right).$$

Proof. 

Set ${\mathfrak{I}}_{\Phi }(v,u)=0$ in Proposition 5.3.

□

The vanishing of the interaction operator is automatic in several established bivariant settings.

Proposition 5.5  

(Bivariant compatibility criterion). Let $F:X\to Y$ and $G:Y\to Z$ be geometric operations carrying compatible bivariant orientations

$${\theta }_F\in \mathbb{B}\left(X\stackrel{F}{\to }Y\right),{\theta }_G\in \mathbb{B}\left(Y\stackrel{G}{\to }Z\right),$$

and suppose

$${\theta }_{G\circ F}={\theta }_F•{\theta }_G.$$

Let $\gamma :\mathbb{B}\to {\mathbb{B}}^{\prime }$ be a Grothendieck transformation, and assume the corrected characteristic transport operators are induced by the transformed orientations $\gamma \left({\theta }_F\right)$, $\gamma \left({\theta }_G\right)$, and $\gamma \left({\theta }_{G\circ F}\right)$. Then

$${\mathfrak{I}}_{\Phi }(G,F)=0.$$

Proof. 

Compatibility of a Grothendieck transformation with the bivariant product gives

$$\gamma \left({\theta }_{G\circ F}\right)=\gamma ({\theta }_F•{\theta }_G)=\gamma \left({\theta }_F\right)•\gamma \left({\theta }_G\right).$$

Hence the target-side operator induced by the orientation of $G\circ F$ agrees with the composite of the target-side operators induced by ${\theta }_F$ and ${\theta }_G$ [2,4]. Therefore the interaction operator vanishes.

□

Remark 5.6.  

Proposition 5.3 is an additive identity, and Proposition 5.5 is inherited from the axioms of bivariant Grothendieck transformations. The additional geometric content developed below lies in the propagated defect class, its support, and the possibility that these data survive after a coarser realization has identified the endpoint constructions.

5.2. Exact Stages and Composite Exactness

The two-stage identity immediately separates stage exactness from compatibility of the reference operators.

Corollary 5.7  

(Closure of compatible exact transport). Suppose $(v,u)$ is Φ-Riemann–Roch compatible and let $\xi \in \mathcal{A}\left(X\right)$. If

$${\Delta }_{\Phi }(u;\xi )=0$$

and

$${\Delta }_{\Phi }\left(v;u_{\mathcal{A}}\left(\xi \right)\right)=0,$$

then

$${\Delta }_{\Phi }(v\diamond u;\xi )=0.$$

In particular, if u is Φ-exact and v is Φ-exact on the image of $u_{\mathcal{A}}$, then the compressed composite $v\diamond u$ is Φ-exact.

Proof. 

This is Corollary 5.4 with both stage defects equal to zero.

□

Remark 5.8.  

If both stage defects vanish but the reference operators are not Φ-Riemann–Roch compatible, then

$${\Delta }_{\Phi }(v\diamond u;\xi )={\mathfrak{I}}_{\Phi }(v,u)\left({\Phi }_X\left(\xi \right)\right).$$

Thus any residual composite discrepancy is entirely an interaction effect.

Remark 5.9.  

Literal exactness of the compressed composite requires equality of the chosen compressed reference operator with the composite of the stagewise reference operators. If the two expressions differ only by an equivalent normalization, Proposition 4.14 shows that defect vanishing is unchanged.

5.3. Propagation of Supported Defects

We next specialize the certified-support convention of Definition 1.10 to homology-type characteristic targets.

Let $Y\in Ob\left(\mathsf{C}\right)$, let $S\subseteq Y$ be a closed subvariety, and let $j:S\hookrightarrow Y$ denote the closed immersion. Assume that the target assignment admits a proper pushforward $j_{*}:{\mathcal{H}}_{*}\left(S\right)\to {\mathcal{H}}_{*}\left(Y\right)$.

Definition 5.10  

(Supported characteristic class). A class $\eta \in {\mathcal{H}}_{*}\left(Y\right)$ issupported on Sif

$$\eta \in im\left(j_{*}:{\mathcal{H}}_{*}\left(S\right)⟶{\mathcal{H}}_{*}\left(Y\right)\right).$$

Equivalently, η is supported on S if there exists $\tilde{\eta }\in {\mathcal{H}}_{*}\left(S\right)$ such that

$$\eta =j_{*}\tilde{\eta }.$$

This notion records a certified support; no minimality of S is implied.

Assume in addition that the target theory admits cap products by rational cohomology and satisfies the projection formula for proper pushforward.

Lemma 5.11  

(Characteristic corrections do not enlarge support). If $\eta =j_{*}\tilde{\eta }$ is supported on S and $a\in H^{*}(Y,\mathbb{Q})$, then

$$a\cap \eta =j_{*}\left(j^{*}a\cap \tilde{\eta }\right).$$

In particular, multiplication by a characteristic correction class does not enlarge the certified support.

Proof. 

The projection formula gives

$$a\cap j_{*}\tilde{\eta }=j_{*}\left(j^{*}a\cap \tilde{\eta }\right).$$

□

The required projection and base-change identities are standard in intersection-theoretic and bivariant settings [2,3].

Theorem 5.12  

(Propagation of supported defects). Let $Y\in Ob\left(\mathsf{C}\right)$, let $S\subseteq Y$ be closed, and let

$$\delta =j_{*}\tilde{\delta }\in {\mathcal{H}}_{*}\left(Y\right)$$

be supported on S, where $j:S\hookrightarrow Y$ is the closed immersion and $\tilde{\delta }\in {\mathcal{H}}_{*}\left(S\right)$.

Assume the target theory admits the proper pushforward, smooth pullback, and refined Gysin operations used below, together with the corresponding proper–smooth and proper–Gysin base-change identities.

1.

Let $g:Y\to Z$ be proper. Then $g\left(S\right)\subseteq Z$ is closed and $g_{*}\delta$ is supported on $g\left(S\right)$.

2.

Let $\pi :Y^{\prime }\to Y$ be smooth and form the Cartesian square

$$\begin{array}{ccc}{S}^{\prime }& \stackrel{j^{\prime }}{\to }& Y^{\prime }\\ \downarrow {\pi }_S& & \downarrow \pi \\ S& \stackrel{j}{\to }& Y,\end{array}{S}^{\prime }:=Y^{\prime }{\times }_{Y}S.$$

Then ${\pi }_{\mathrm{BM}}^{*}\delta$ is supported on $S^{\prime }$.

3.

Let $i:W\hookrightarrow Y$ be a regular embedding for which refined Gysin pullback is defined, and form the Cartesian square

$$\begin{array}{ccc}{S}_W& \stackrel{j_W}{\to }& W\\ \downarrow i_S& & \downarrow i\\ S& \stackrel{j}{\to }& Y,\end{array}{S}_W:=W{\times }_{Y}S.$$

Assume the refined Gysin base-change identity is valid for this square. Then $i^{!}\delta$ is supported on $S_W$.

Each conclusion remains valid after capping with a multiplicative characteristic correction class defined on the corresponding target.

Proof. 

For the proper case, let $k:g\left(S\right)\hookrightarrow Z$ be the closed immersion and let $g_S:S\to g\left(S\right)$ be the induced proper morphism. Since $g\circ j=k\circ g_S$, functoriality of proper pushforward gives

$$g_{*}\delta =g_{*}{j}_{*}\tilde{\delta }=k_{*}{\left(g_S\right)}_{*}\tilde{\delta }.$$

Hence $g_{*}\delta$ is supported on $g\left(S\right)$.

For the smooth case, proper–smooth base change gives

$${\pi }_{\mathrm{BM}}^{*}{j}_{*}=j_{*}^{\prime }{\left({\pi }_S\right)}_{\mathrm{BM}}^{*}.$$

Therefore

$${\pi }_{\mathrm{BM}}^{*}\delta =j_{*}^{\prime }\left({\left({\pi }_S\right)}_{\mathrm{BM}}^{*}\tilde{\delta }\right),$$

so the pulled-back class is supported on $S^{\prime }$.

For the regular-embedding case, compatibility of refined Gysin pullback with proper pushforward gives

$$i^{!}{j}_{*}={\left(j_W\right)}_{*}{i}_S^{!}.$$

Consequently,

$$i^{!}\delta ={\left(j_W\right)}_{*}\left(i_S^{!}\tilde{\delta }\right),$$

which is supported on $S_W$.

The base-change identities used here are the standard compatibility properties of refined pullback and proper pushforward [2,3]. The final assertion follows from Lemma 5.11.

□

The preceding theorem applies directly to propagated transport defects.

Corollary 5.13  

(Support of a propagated first-stage defect). Let $u:X\to Y$ and $v:Y\to Z$ be a Φ-Riemann–Roch-compatible pair, and let $\xi \in \mathcal{A}\left(X\right)$. Assume

$${\Delta }_{\Phi }\left(v;u_{\mathcal{A}}\left(\xi \right)\right)=0.$$

Suppose the first-stage defect ${\Delta }_{\Phi }(u;\xi )\in {\mathcal{H}}_{*}\left(Y\right)$ is supported on a closed subset $S\subseteq Y$, and suppose the reference operator ${\mathcal{T}}_{\Phi }\left(v\right)$ is a proper pushforward, possibly combined with multiplicative characteristic corrections. Then

$${\Delta }_{\Phi }(v\diamond u;\xi )={\mathcal{T}}_{\Phi }\left(v\right)\left({\Delta }_{\Phi }(u;\xi )\right)$$

is supported on the proper image of S in Z.

In the covariant notation $X\stackrel{F}{\to }Y\stackrel{G}{\to }Z$, this says that if the G-stage is exact on $F_{!}\xi$, then a first-stage defect supported on S propagates to a composite defect supported on $G\left(S\right)$.

Proof. 

The identity follows from Corollary 5.4. The support statement follows from Theorem 5.12 and Lemma 5.11.

□

Remark 5.14.  

The formal two-stage defect identity records how classes add. Theorem 5.12 adds geometric information by controlling the locus on which each propagated defect can be represented.

5.4. Propagation of Specialization and Vanishing-Cycle Terms

Let $i:X\hookrightarrow Y$ be a regular embedding satisfying the hypotheses of Lemma 4.9. Let $N:=N_{X/Y}$, let $\pi :N\to X$ be the normal-bundle projection, and let $k:X\hookrightarrow N$ be the zero section. Recall the generalized vanishing-cycle transformation

$${\Phi }_i={Sp}_{X\setminus Y}-{\pi }_{CF}^{*}{i}_{CF}^{*}$$

from Definition 4.8.

For $\alpha \in CF\left(Y\right)$, write

$${\delta }_i\left(\alpha \right):={\Delta }_{c_{*}}^{!}(i;\alpha )=-c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right).$$

Definition 5.15  

(Support locus of a generalized vanishing-cycle term). For a closed subset $S\subseteq X$, the generalized vanishing-cycle class ${\Phi }_i\left(\alpha \right)\in {CF}_{\mathrm{mon}}\left(N\right)$ issupported over Sif

$$supp{\Phi }_i\left(\alpha \right)\subseteq {\pi }^{-1}\left(S\right).$$

Proposition 5.16  

(Support of the Schürmann defect). If ${\Phi }_i\left(\alpha \right)$ is supported over $S\subseteq X$, then

$${\delta }_i\left(\alpha \right)={\Delta }_{c_{*}}^{!}(i;\alpha )$$

is supported on S.

Proof. 

Let $T:=supp{\Phi }_i\left(\alpha \right)\subseteq {\pi }^{-1}\left(S\right)$. Since ${\Phi }_i\left(\alpha \right)$ vanishes outside T, it is the proper pushforward of a constructible function on T along the closed inclusion $j_T:T\hookrightarrow N$. Proper naturality of the Chern–Schwartz–MacPherson transformation therefore expresses $c_{*}\left({\Phi }_i\left(\alpha \right)\right)$ as a class supported on T.

The zero-section Gysin operation $k_H^{!}$ pulls this support back to

$$X{\times }_{N}T\subseteq S.$$

Hence $k_H^{!}{c}_{*}\left({\Phi }_i\left(\alpha \right)\right)$ is supported on S. Finally, Lemma 5.11 shows that capping with $c^{*}{\left(N\right)}^{-1}$ does not enlarge the support.

□

Remark 5.17.  

If $g:X\to Z$ is proper and ${\delta }_i\left(\alpha \right)$ is supported on $S\subseteq X$, then $g_{*}{\delta }_i\left(\alpha \right)$ is supported on the closed subset $g\left(S\right)\subseteq Z$.

The first stage in this construction is Gysin-type and hence has transport direction $Y\to X$, while the second stage $g:X\to Z$ is covariant. The relevant two-stage transport is therefore the operator

$$g_{*}\circ i_{c_{*}}^{\mathrm{RR},!},$$

not an artificially introduced ordinary morphism $Y\to Z$.

Theorem 5.18  

(Propagation of a Verdier–Riemann–Roch defect through an exact proper stage). Let $i:X\hookrightarrow Y$ satisfy the hypotheses of Lemma 4.9, let $g:X\to Z$ be proper, and let $\alpha \in CF\left(Y\right)$. Then the first-stage Chern–Schwartz–MacPherson Gysin defect propagates through the proper stage as

$$g_{*}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right)=-g_{*}\left(c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right)\right).$$

If ${\Phi }_i\left(\alpha \right)$ is supported over a closed subset $S\subseteq X$, then the propagated defect is supported on $g\left(S\right)\subseteq Z$.

Proof. 

Lemma 4.9 gives

$${\Delta }_{c_{*}}^{!}(i;\alpha )=-c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right).$$

Applying the proper pushforward $g_{*}$ gives the asserted equality. Because proper CSM transport is exact by Proposition 4.10, no additional defect is created at the g-stage.

If ${\Phi }_i\left(\alpha \right)$ is supported over S, then Proposition 5.16 shows that the first-stage defect is supported on S, and Theorem 5.12 shows that its proper pushforward is supported on $g\left(S\right)$.

□

Remark 5.19.  

Theorem 5.18 does not modify Schürmann’s one-stage Verdier–Riemann–Roch theorem. It computes the image and support of the already identified defect after a second exact geometric stage.

Remark 5.20.  

Propagation need not preserve nonvanishing. It may happen that

$${\delta }_i\left(\alpha \right)\ne 0\mathrm{but}{g}_{*}{\delta }_i\left(\alpha \right)=0.$$

Thus nontrivial intermediate singular geometry can become invisible at the endpoint.

5.5. Routes, Collapse, and Finite Propagation

The general notion of a compatible typed transport route was defined in Definition 1.5. In this subsection we specialize to covariant geometric routes for which all underlying morphisms compose in the ordinary sense.

Definition 5.21  

(Covariant geometric route). Afinite covariant geometric route of length $r\ge 1$is a sequence

$$\mathcal{R}:X_0\stackrel{F_1}{\to }{X}_1\stackrel{F_2}{\to }\dots \stackrel{F_r}{\to }{X}_r$$

of admissible covariant geometric operations. Its ordinary endpoint composite is

$$F_{\mathcal{R}}:=F_r\circ \dots \circ F_1.$$

Whenever the typed operations associated with $F_1,\dots ,F_r$ are needed, they are understood to be the corresponding admissible transport stages of $\mathbb{T}$.

Definition 5.22  

(Trivial refactorization). Atrivial refactorizationof a finite route is generated by:

1.

insertion or deletion of an identity stage; and

2.

replacement of an intermediate object by an isomorphic object, together with transport of all adjacent geometric and characteristic data through the chosen isomorphism.

Because a route is represented by an ordered sequence, a change of parenthesization does not alter the route itself. Collapse of a genuinely nontrivial intermediate stage is not a trivial refactorization.

Remark 5.23.  

Under identity normalization, Definition 4.6 gives zero defect for an inserted identity stage. Identity insertion or deletion therefore leaves the endpoint defect unchanged.

Remark 5.24.  

Replacement of an intermediate object by an isomorphic object transports the associated defects through the induced target-side isomorphisms by Proposition 4.7. It therefore introduces no new intrinsic defect.

The substantive operation is collapse of a nontrivial factorization.

Proposition 5.25  

(Controlled collapse of two stages). Let

$$X\stackrel{F}{\to }Y\stackrel{G}{\to }Z$$

be a covariant Φ-Riemann–Roch-compatible pair. For every $\xi \in \mathcal{A}\left(X\right)$,

$${\Delta }_{\Phi }(G\circ F;\xi )={\Delta }_{\Phi }(G;F_{!}\xi )+G_{\Phi ,*}^{\mathrm{RR}}\left({\Delta }_{\Phi }(F;\xi )\right).$$

Thus collapse of the two-stage route to its endpoint composite forgets the ordered pair of stage defects and retains only their propagated sum.

Proof. 

This is Corollary 5.4 in the covariant notation introduced at the beginning of the section.

□

Remark 5.26.  

Let

$$X_0\stackrel{F_1}{\to }{X}_1\stackrel{F_2}{\to }{X}_2\stackrel{F_3}{\to }{X}_3$$

be three covariant stages whose reference operators are strictly compatible under every partial composition. For an initial class ${\xi }_0\in \mathcal{A}\left(X_0\right)$, set

$${\xi }_1:={\left(F_1\right)}_{!}{\xi }_0,{\xi }_2:={\left(F_2\right)}_{!}{\xi }_1,$$

and

$${\Delta }_i:={\Delta }_{\Phi }(F_i;{\xi }_{i-1}),i=1,2,3.$$

Both parenthesizations of the endpoint composite give

$${\Delta }_3+{\left(F_3\right)}_{\Phi ,*}^{\mathrm{RR}}\left({\Delta }_2\right)+{\left(F_3\right)}_{\Phi ,*}^{\mathrm{RR}}{\left(F_2\right)}_{\Phi ,*}^{\mathrm{RR}}\left({\Delta }_1\right).$$

Thus associative regrouping changes neither the endpoint operator nor the endpoint defect.

The same calculation extends to arbitrary finite length.

Corollary 5.27  

(Finite compatible route formula). Let $\mathcal{R}=(F_1,\dots ,F_r)$ be a finite covariant route whose reference characteristic transport operators are strictly compatible under every partial composition. Fix ${\xi }_0\in \mathcal{A}\left(X_0\right)$, and for $1\le i\le r$ define

$${\xi }_i:={\left(F_i\right)}_{!}{\xi }_{i-1},{\Delta }_i:={\Delta }_{\Phi }(F_i;{\xi }_{i-1}).$$

For $1\le i<r$, define the propagation operator

$$P_{i,r}:={\left(F_r\right)}_{\Phi ,*}^{\mathrm{RR}}\circ \dots \circ {\left(F_{i+1}\right)}_{\Phi ,*}^{\mathrm{RR}},$$

and set

$$P_{r,r}:={id}_{\mathcal{H}\left(X_r\right)}.$$

Then

$${\Delta }_{\Phi }(F_{\mathcal{R}};{\xi }_0)={\displaystyle \sum _{i=1}^r}{P}_{i,r}\left({\Delta }_i\right).$$

Proof. 

Apply Proposition 5.25 inductively to the partial composites

$$F_1,F_2\circ F_1,\dots ,F_r\circ \dots \circ F_1.$$

Strict compatibility eliminates every interaction term, leaving precisely the propagated sum displayed above.

□

Corollary 5.27 is the covariant specialization of the finite-route propagation identity (2). It is retained here because the covariant notation is used repeatedly in the later geometric examples.

Remark 5.28.  

Nonzero intermediate defects may yield a zero endpoint defect if their propagated images cancel:

$${\displaystyle \sum _{i=1}^r}{P}_{i,r}\left({\Delta }_i\right)=0$$

does not imply ${\Delta }_i=0$ for every i. Endpoint exactness therefore does not determine the intermediate transport history.

Remark 5.29.  

The defect calculus is invariant under the trivial refactorizations above but intentionally sensitive to collapse of a genuinely nontrivial factorization. This distinction motivates the characteristic transport profiles of Section 6, which retain the stagewise data before endpoint summation.

6. Comparative Characteristic Transport Profiles

Section 5 shows that an endpoint defect can forget how individual discrepancies were created and propagated through a composite construction. A characteristic transport profile retains the stagewise operators, source classes, defects, and propagation maps before those data are collapsed to a single endpoint class. Its purpose is comparative and factorization-sensitive; no completeness claim is made.

Throughout this section, routes are covariant geometric routes unless otherwise stated. Thus, for a route

$$\mathcal{R}:X_0\stackrel{F_1}{\to }{X}_1\stackrel{F_2}{\to }\dots \stackrel{F_r}{\to }{X}_r,$$

the operation $F_i:X_{i-1}\to X_i$ determines an admissible typed transport operation $u_i:X_{i-1}\to X_i$. We use ${\left(F_i\right)}_{!}$ for the source-side operator ${\left(u_i\right)}_{\mathcal{A}}$ and ${\left(F_i\right)}_{\Phi ,*}^{\mathrm{RR}}$ for the reference characteristic transport operator ${\mathcal{T}}_{\Phi }\left(u_i\right)$. These are abbreviations for the typed transport data of Section 1 and Section 3, not additional structures.

Mixed-variance constructions are compared directly at the level of their typed operators, as in Theorem 5.18 and Section 9.

6.1. Definition of a Transport Profile

Fix a characteristic transport datum

$$\mathbb{T}=\left(\mathsf{C},\mathcal{A},\mathcal{H},\Phi ,{Op}_{\mathbb{T}},{\mathcal{T}}_{\Phi }\right)$$

and a finite covariant route $\mathcal{R}=(F_1,\dots ,F_r)$ as above. Let ${\xi }_0\in \mathcal{A}\left(X_0\right)$, and define recursively

$${\xi }_i:={\left(F_i\right)}_{!}{\xi }_{i-1}\in \mathcal{A}\left(X_i\right),1\le i\le r.$$

For the same range of i, set

$$C_i^{\Phi }:={\left(F_i\right)}_{\Phi ,*}^{\mathrm{RR}}={\mathcal{T}}_{\Phi }\left(u_i\right):\mathcal{H}\left(X_{i-1}\right)⟶\mathcal{H}\left(X_i\right)$$

and

$${\delta }_i^{\Phi }:={\Delta }_{\Phi }(F_i;{\xi }_{i-1})\in \mathcal{H}\left(X_i\right).$$

Definition 6.1  

(Profile propagation operator). For integers $1\le i\le j\le r$, define

$$P_{i,i}^{\Phi }:={id}_{\mathcal{H}\left(X_i\right)}$$

and, for $i<j$,

$$P_{i,j}^{\Phi }:=C_j^{\Phi }\circ C_{j-1}^{\Phi }\circ \dots \circ C_{i+1}^{\Phi }:\mathcal{H}\left(X_i\right)⟶\mathcal{H}\left(X_j\right).$$

The terminal propagation of the i-th stage defect is

$${\epsilon }_i^{\Phi }:=P_{i,r}^{\Phi }\left({\delta }_i^{\Phi }\right)\in \mathcal{H}\left(X_r\right).$$

The family $P_{i,j}^{\Phi }$ contains no independent information beyond the stage operators $C_i^{\Phi }$, but it makes intermediate and terminal propagation explicit. In particular,

$$P_{i,k}^{\Phi }=P_{j,k}^{\Phi }\circ P_{i,j}^{\Phi }(1\le i\le j\le k\le r).$$

The stagewise route defect of Section 1 satisfies

$${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)={\displaystyle \sum _{i=1}^r}{\epsilon }_i^{\Phi }$$

without any additional compatibility assumption; this is Proposition 1.7. If, in addition, the ordinary endpoint composite $F_{\mathcal{R}}:=F_r\circ \dots \circ F_1$ is admissible and its reference operator satisfies

$${\left(F_{\mathcal{R}}\right)}_{\Phi ,*}^{\mathrm{RR}}=C_r^{\Phi }\circ \dots \circ C_1^{\Phi },$$

then the compressed endpoint defect agrees with the stagewise route defect:

$${\Delta }_{\Phi }(F_{\mathcal{R}};{\xi }_0)={\Delta }_{\Phi }(\mathcal{R};{\xi }_0)={\displaystyle \sum _{i=1}^r}{\epsilon }_i^{\Phi }.$$

Definition 6.2  

(Single-theory transport profile). The$\Phi$-transport profileof $(\mathcal{R},{\xi }_0)$ is the characteristic transport profile of Definition 1.13, represented in the present covariant notation by the stagewise data

$${\mathfrak{P}}_{\Phi }(\mathcal{R};{\xi }_0)=\left[{\left(X_i\right)}_{i=0}^r,{\left(F_i\right)}_{i=1}^r,{\left({\xi }_i\right)}_{i=0}^r,{\left(C_i^{\Phi }\right)}_{i=1}^r,{\left({\delta }_i^{\Phi }\right)}_{i=1}^r,{\left(P_{i,j}^{\Phi }\right)}_{1\le i\le j\le r}\right],$$

where the brackets indicate passage to the profile-equivalence relation generated by the trivial identifications specified below.

Itsterminal defect profileis the ordered tuple

$${\mathfrak{P}}_{\Phi }^{\mathrm{term}}(\mathcal{R};{\xi }_0):=\left({\epsilon }_1^{\Phi },\dots ,{\epsilon }_r^{\Phi }\right).$$

The additional propagation operators $P_{i,j}^{\Phi }$ are determined by the stage operators and therefore refine the convenient presentation of the profile without changing its underlying equivalence class.

Remark 6.3.  

The endpoint defect records only the sum ${\sum }_i{\epsilon }_i^{\Phi }$. Distinct ordered families of propagated contributions can therefore determine the same endpoint defect. In particular, endpoint exactness does not imply trivial intermediate transport.

6.2. Comparative Characteristic Families

To compare several characteristic realizations along the same geometric route, let Λ be a finite index set and let $\mathcal{S}$ be a common source assignment on the geometric category under consideration.

Definition 6.4  

(Characteristic comparison family). Acharacteristic comparison familyindexed by Λ consists, for each $\lambda \in \Lambda$, of a characteristic transport datum

$${\mathbb{T}}^{\lambda }=\left(\mathsf{C},{\mathcal{A}}^{\lambda },{\mathcal{H}}^{\lambda },{\Phi }^{\lambda },{Op}_{{\mathbb{T}}^{\lambda }},{\mathcal{T}}_{{\Phi }^{\lambda }}\right),$$

together with additive maps

$$q_X^{\lambda }:\mathcal{S}\left(X\right)⟶{\mathcal{A}}^{\lambda }\left(X\right)$$

for every $X\in Ob\left(\mathsf{C}\right)$.

For every route stage $u:X\to Y$ for which the λ-component is defined, the maps $q^{\lambda }$ are required to commute with the source-side transport:

$$q_Y^{\lambda }\circ u_{\mathcal{S}}=u_{{\mathcal{A}}^{\lambda }}\circ q_X^{\lambda }.$$

Here $u_{\mathcal{S}}$ denotes the source-side operation on the common assignment $\mathcal{S}$. A component is included only when the relevant typed operations, source maps, and reference characteristic operators are defined.

Definition 6.5  

(Comparative transport profile). For ${\zeta }_0\in \mathcal{S}\left(X_0\right)$, define

$${\mathfrak{P}}_{\Lambda }(\mathcal{R};{\zeta }_0):={\left({\mathfrak{P}}_{{\Phi }^{\lambda }}\left(\mathcal{R};q_{X_0}^{\lambda }\left({\zeta }_0\right)\right)\right)}_{\lambda \in \Lambda },$$

where only defined components are included.

Thus a comparative profile is not an unrelated collection of invariants: each component is attached to the same geometric route and to the image of the same common source datum.

The relation between two characteristic components is made explicit by the following notion.

Definition 6.6  

(Comparison morphism of characteristic theories). Let $\lambda ,\mu \in \Lambda$. Acomparison morphism from $\lambda$ to $\mu$consists of additive natural maps

$$q_X^{\mu \lambda }:{\mathcal{A}}^{\lambda }\left(X\right)⟶{\mathcal{A}}^{\mu }\left(X\right),s_X^{\mu \lambda }:{\mathcal{H}}^{\lambda }\left(X\right)⟶{\mathcal{H}}^{\mu }\left(X\right)$$

such that

$$s_X^{\mu \lambda }\circ {\Phi }_X^{\lambda }={\Phi }_X^{\mu }\circ q_X^{\mu \lambda }$$

for every object X, and such that for every typed operation $u:X\to Y$ admissible for both components,

$$q_Y^{\mu \lambda }\circ u_{{\mathcal{A}}^{\lambda }}=u_{{\mathcal{A}}^{\mu }}\circ q_X^{\mu \lambda }$$

and

$$s_Y^{\mu \lambda }\circ {\mathcal{T}}_{{\Phi }^{\lambda }}\left(u\right)={\mathcal{T}}_{{\Phi }^{\mu }}\left(u\right)\circ s_X^{\mu \lambda }.$$

Proposition 6.7  

(Comparison of transport defects). Under a comparison morphism from λ to μ, for every $\xi \in {\mathcal{A}}^{\lambda }\left(X\right)$,

$$s_Y^{\mu \lambda }\left({\Delta }_{{\Phi }^{\lambda }}(u;\xi )\right)={\Delta }_{{\Phi }^{\mu }}\left(u;q_X^{\mu \lambda }\left(\xi \right)\right).$$

Proof. 

Using Definition 4.1,

$$\begin{array}{cc}\hfill s_Y^{\mu \lambda }\left({\Delta }_{{\Phi }^{\lambda }}(u;\xi )\right)& =s_Y^{\mu \lambda }{\Phi }_Y^{\lambda }\left(u_{{\mathcal{A}}^{\lambda }}\left(\xi \right)\right)-s_Y^{\mu \lambda }{\mathcal{T}}_{{\Phi }^{\lambda }}\left(u\right)\left({\Phi }_X^{\lambda }\left(\xi \right)\right)\hfill \\ & ={\Phi }_Y^{\mu }{q}_Y^{\mu \lambda }\left(u_{{\mathcal{A}}^{\lambda }}\left(\xi \right)\right)-{\mathcal{T}}_{{\Phi }^{\mu }}\left(u\right)s_X^{\mu \lambda }\left({\Phi }_X^{\lambda }\left(\xi \right)\right)\hfill \\ & ={\Phi }_Y^{\mu }{u}_{{\mathcal{A}}^{\mu }}\left(q_X^{\mu \lambda }\left(\xi \right)\right)-{\mathcal{T}}_{{\Phi }^{\mu }}\left(u\right){\Phi }_X^{\mu }\left(q_X^{\mu \lambda }\left(\xi \right)\right),\hfill \end{array}$$

which is precisely ${\Delta }_{{\Phi }^{\mu }}(u;q_X^{\mu \lambda }\left(\xi \right))$.

□

Remark 6.8.  

Proposition 6.7 is formal once a comparison morphism is specified. Its significance is structural: comparative profiles are linked by actual natural transformations, rather than assembled as an arbitrary list of characteristic invariants.

6.3. Specialization Relations among Profile Components

Let ${\widehat{T}}_{y*}$ denote the normalized motivic Hirzebruch transformation of Brasselet–Schürmann–Yokura [16]. Let

$${ϵ}_X:K_0(Var/X)⟶CF\left(X\right)$$

denote the constructible-function realization.

In the normalized formalism one has

$${ev}_{-1}\circ {\widehat{T}}_{y*}=c_{*}\circ ϵ$$

and

$${ev}_0\circ {\widehat{T}}_{y*}=\tau \circ m{C}_0,$$

with the standard interpretation of the Chern and Todd specializations [16]. A corresponding L-type interpretation at $y=1$ is used only when an independent comparison theorem supplies it in the geometric category under consideration.

Definition 6.9  

(Specialization-compatible transport). Fix $a\in \mathbb{Q}$ for which evaluation at $y=a$ is defined on the source and target coefficient rings. Suppose a specialized characteristic transport datum ${\mathbb{T}}^a$ is given, with characteristic transformation ${\Phi }_a$, source comparison map $q_a$, and target evaluation map ${ev}_a$.

A corrected ${\widehat{T}}_y$-transport operator iscompatible with specialization at aif, for every admissible typed operation $u:X\to Y$,

$${ev}_a\circ {\mathcal{T}}_{{\widehat{T}}_y}\left(u\right)={\mathcal{T}}_{{\Phi }_a}\left(u\right)\circ {ev}_a$$

on the relevant target groups, and the corresponding source-side square with $q_a$ commutes.

Theorem 6.10  

(Specialization of transport defects). If specialization at a is defined on the source and target data and is compatible with the transport operator in the sense of Definition 6.9, then for every admissible $u:X\to Y$ and every source class ξ,

$${ev}_a\left({\Delta }_{{\widehat{T}}_y}(u;\xi )\right)={\Delta }_{{\Phi }_a}\left(u;q_{a,X}\left(\xi \right)\right).$$

Proof. 

The pair $(q_a,{ev}_a)$ is a comparison morphism in the sense of Definition 6.6. Proposition 6.7 therefore applies.

□

Corollary 6.11  

(Chern and Todd specializations). Under the hypotheses of Theorem 6.10,

$${ev}_{-1}\left({\Delta }_{{\widehat{T}}_y}(u;\xi )\right)={\Delta }_{c_{*}}\left(u;{ϵ}_X\left(\xi \right)\right)$$

and

$${ev}_0\left({\Delta }_{{\widehat{T}}_y}(u;\xi )\right)={\Delta }_{\tau }\left(u;m{C}_0\left(\xi \right)\right).$$

Proof. 

Apply Theorem 6.10 to the normalized specializations at $y=-1$ and $y=0$ [16].

□

Corollary 6.12  

(Conditional L-type specialization). Suppose that, on the geometric class under consideration, an independently established comparison identifies

$${ev}_1\circ {\widehat{T}}_{y*}=L_{*}\circ q_1$$

and that the corresponding corrected transport operators commute with evaluation at $y=1$. Then

$${ev}_1\left({\Delta }_{{\widehat{T}}_y}(u;\xi )\right)={\Delta }_{L_{*}}\left(u;q_{1,X}\left(\xi \right)\right).$$

Proof. 

Apply Theorem 6.10 with $a=1$.

□

Remark 6.13.  

The $y=1$ statement is deliberately conditional. No universal identification of the normalized Hirzebruch specialization at $y=1$ with a chosen topological or intersection-homology L-class is assumed for arbitrary singular varieties.

Corollary 6.14  

(Specialization of transport profiles). Let $\mathcal{R}=(F_1,\dots ,F_r)$ be a route for which every stage is specialization-compatible at a. Then, for each $1\le i\le j\le r$,

$${ev}_a\left({\delta }_i^{{\widehat{T}}_y}\right)={\delta }_i^{{\Phi }_a}$$

and

$${ev}_a\circ P_{i,j}^{{\widehat{T}}_y}=P_{i,j}^{{\Phi }_a}\circ {ev}_a.$$

Consequently,

$${ev}_a\left({\epsilon }_i^{{\widehat{T}}_y}\right)={\epsilon }_i^{{\Phi }_a}$$

for every i, and the specialized profile is obtained stagewise from the normalized Hirzebruch profile.

Proof. 

The first identity is Theorem 6.10 applied to the i-th stage. The second follows by iterating specialization compatibility through the composite $P_{i,j}^{{\widehat{T}}_y}=C_j^{{\widehat{T}}_y}\circ \dots \circ C_{i+1}^{{\widehat{T}}_y}$. The terminal statement follows by taking $j=r$.

□

Remark 6.15.  

The normalized Hirzebruch profile can organize several specialized characteristic components when the corresponding comparison theorems are available. This provides a useful common parameter space, but no completeness property is claimed.

6.4. Composition and Trivial Refactorization

Let

$${\mathcal{R}}_1:X_0\stackrel{F_1}{\to }\dots \stackrel{F_s}{\to }{X}_s$$

and

$${\mathcal{R}}_2:X_s\stackrel{F_{s+1}}{\to }\dots \stackrel{F_r}{\to }{X}_r$$

be compatible subroutes. Write ${\mathcal{R}}_2★{\mathcal{R}}_1$ for their concatenation.

Fix ${\xi }_0\in \mathcal{A}\left(X_0\right)$, and let ${\xi }_s\in \mathcal{A}\left(X_s\right)$ be the class obtained after transport through ${\mathcal{R}}_1$. Define the stagewise endpoint defects

$$D_1^{\Phi }:={\Delta }_{\Phi }({\mathcal{R}}_1;{\xi }_0)\in \mathcal{H}\left(X_s\right)$$

and

$$D_2^{\Phi }:={\Delta }_{\Phi }({\mathcal{R}}_2;{\xi }_s)\in \mathcal{H}\left(X_r\right).$$

Theorem 6.16  

(Composition of transport profiles). For $1\le i\le s\le j\le r$,

$$P_{i,j}^{\Phi }=P_{s,j}^{\Phi }\circ P_{i,s}^{\Phi }.$$

Moreover,

$${\Delta }_{\Phi }\left({\mathcal{R}}_2★{\mathcal{R}}_1;{\xi }_0\right)=D_2^{\Phi }+P_{s,r}^{\Phi }\left(D_1^{\Phi }\right).$$

If the two subroutes and their concatenation admit compressed endpoint operations whose reference operators are strictly compatible with the stagewise composites, the same formula holds with the corresponding compressed endpoint defects.

Proof. 

The first identity follows immediately from the definitions:

$$P_{s,j}^{\Phi }\circ P_{i,s}^{\Phi }=\left(C_j^{\Phi }\circ \dots \circ C_{s+1}^{\Phi }\right)\circ \left(C_s^{\Phi }\circ \dots \circ C_{i+1}^{\Phi }\right)=P_{i,j}^{\Phi }.$$

For the second identity, write

$$D_1^{\Phi }={\displaystyle \sum _{i=1}^s}{P}_{i,s}^{\Phi }\left({\delta }_i^{\Phi }\right)$$

and

$$D_2^{\Phi }={\displaystyle \sum _{i=s+1}^r}{P}_{i,r}^{\Phi }\left({\delta }_i^{\Phi }\right).$$

Applying $P_{s,r}^{\Phi }$ to the first equality and using the first part of the theorem gives

$$P_{s,r}^{\Phi }\left(D_1^{\Phi }\right)={\displaystyle \sum _{i=1}^s}{P}_{i,r}^{\Phi }\left({\delta }_i^{\Phi }\right).$$

Adding $D_2^{\Phi }$ therefore gives

$${\displaystyle \sum _{i=1}^r}{P}_{i,r}^{\Phi }\left({\delta }_i^{\Phi }\right)={\Delta }_{\Phi }\left({\mathcal{R}}_2★{\mathcal{R}}_1;{\xi }_0\right)$$

by Proposition 1.7.

The final assertion follows when the stagewise route defects are identified with the corresponding compressed endpoint defects.

□

Remark 6.17.  

Theorem 6.16 applies componentwise to a characteristic comparison family whenever the required route operations are defined for each component.

The relevant notion of trivial refactorization has already been fixed in Definition 1.12. We record the specialized terminology used in this section.

Definition 6.18  

(Trivial profile refactorization). Two covariant routes aretrivially profile-equivalentif their raw profile representatives are related by a finite sequence of:

1.

insertions or deletions of identity stages; and

2.

replacements of intermediate objects by isomorphic objects, with all adjacent transport and characteristic data carried through the corresponding isomorphisms.

Changing only the parenthesization of an unchanged ordered sequence does not alter the route and is therefore not treated as a separate refactorization.

Remark 6.19.  

Characteristic transport profiles are invariant under trivial profile-equivalence, up to the canonical identifications supplied by identity normalization and Proposition 4.7.

Remark 6.20.  

Collapse of a genuinely nontrivial factorization is different. Proposition 5.25 replaces separate stage defects by their propagated sum. It can therefore erase information retained by the transport profile and is not included among trivial profile equivalences.

6.5. Selected-sector Equivalence of Geometric Routes

Let X and Y be fixed geometric endpoints, and let $\mathcal{R}(X,Y)$ denote a class of routes whose rational cohomological realizations have a common typed source and target. For notational simplicity, write

$${\rho }_H\left(\mathcal{R}\right):H^{•}(X,\mathbb{Q})⟶H^{•}(Y,\mathbb{Q})$$

for the corresponding total graded rational cohomological realization, with any degree shifts and Tate twists understood as part of the type of ${\rho }_H\left(\mathcal{R}\right)$.

Fix a $\mathbb{Q}$-linear subspace $L\subseteq H^{•}(X,\mathbb{Q})$ on which all route realizations under comparison are defined.

Definition 6.21  

(Selected-sector equivalence of routes). For ${\mathcal{R}}_1,{\mathcal{R}}_2\in \mathcal{R}(X,Y)$, write

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$$

if

$${\rho }_H\left({\mathcal{R}}_1\right){{|}_L={\rho }_H\left({\mathcal{R}}_2\right)|}_L.$$

This agrees with Definition 1.14.

Definition 6.22  

(Total cohomological equivalence of routes). Write

$${\mathcal{R}}_1{\sim }_H{\mathcal{R}}_2$$

if

$${\rho }_H\left({\mathcal{R}}_1\right)={\rho }_H\left({\mathcal{R}}_2\right)$$

on total rational cohomology.

Definition 6.23  

(Endpoint equivalence of routes). Suppose the routes ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ admit specified endpoint geometric realizations. Write

$${\mathcal{R}}_1{\sim }_{\mathrm{end}}{\mathcal{R}}_2$$

if those endpoint geometric realizations agree. For routes represented by algebraic correspondences, this means equality of the endpoint classes in the relevant rational Chow group.

Proposition 6.24  

(Hierarchy of endpoint equivalences). For routes with common typed cohomological source and target,

$${\mathcal{R}}_1{\sim }_{\mathrm{end}}{\mathcal{R}}_2⟹{\mathcal{R}}_1{\sim }_H{\mathcal{R}}_2⟹{\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2,$$

whenever endpoint equality is defined and its rational cohomological realization is functorial.

Proof. 

Equality of the endpoint geometric realization gives equality of its rational cohomological realization. Restricting the resulting equality of total cohomological operators to L gives selected-sector equivalence.

□

Remark 6.25.  

No converse is assumed. In particular, the Jacobian construction of Section 8 exhibits selected-sector equivalence while a residual correspondence remains nonzero on total rational cohomology.

Definition 6.26  

(Profile equivalence). For routes whose characteristic transport profiles are defined in the same comparison family, write

$${\mathcal{R}}_1{\sim }_{\mathfrak{P}}{\mathcal{R}}_2$$

if the corresponding profiles agree after the canonical identifications generated by trivial profile refactorization. For a comparative profile ${\mathfrak{P}}_{\Lambda }$, equality is understood componentwise after applying the specified comparison morphisms wherever such identifications are part of the comparison data.

This agrees with the single-theory profile-equivalence convention of Section 1.7. The central comparison problem is whether

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2\mathrm{while}{\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2.$$

The formalism itself does not construct pairs satisfying the first relation.

6.6. Support-graded Separation after Selected-Sector Realization

Fix a characteristic target assignment $\mathcal{H}$ for which proper pushforward along closed immersions is defined. Let $T\subseteq S\subseteq Y$ be closed subvarieties, with inclusions $i_T:T\hookrightarrow Y$ and $i_S:S\hookrightarrow Y$.

Define

$${\mathcal{H}}_S\left(Y\right):=im\left({\left(i_S\right)}_{*}:\mathcal{H}\left(S\right)⟶\mathcal{H}\left(Y\right)\right)$$

and similarly ${\mathcal{H}}_T\left(Y\right)$. Since $T\subseteq S$, functoriality of proper pushforward gives

$${\mathcal{H}}_T\left(Y\right)\subseteq {\mathcal{H}}_S\left(Y\right).$$

Definition 6.27  

(Support quotient). For closed subvarieties $T\subseteq S\subseteq Y$, define

$${Gr}_{S/T}\mathcal{H}\left(Y\right):={\mathcal{H}}_S\left(Y\right)/{\mathcal{H}}_T\left(Y\right).$$

No assertion is made that these quotients arise from a globally defined filtration indexed by all closed subsets; the notation records only the relative support quotient associated with the fixed inclusion $T\subseteq S$.

Let ${\mathcal{R}}_a\in \mathcal{R}(X,Y)$, $a\in \{1,2\}$, be routes with a common initial source class ${\zeta }_0\in \mathcal{A}\left(X\right)$. Write

$${\epsilon }_{a,i}^{\Phi }\in \mathcal{H}\left(Y\right)$$

for the terminal propagation of the i-th stage defect of ${\mathcal{R}}_a$.

Definition 6.28  

(Terminal characteristic span). Theterminal characteristic spanof $({\mathcal{R}}_a,{\zeta }_0)$ is

$${\mathcal{T}}_{\Phi }({\mathcal{R}}_a;{\zeta }_0):={span}_{\mathbb{Q}}{\left\{{\epsilon }_{a,i}^{\Phi }\right\}}_i\subseteq \mathcal{H}\left(Y\right),$$

whenever $\mathcal{H}\left(Y\right)$ is a $\mathbb{Q}$-vector space. In a merely additive target, the same notation is replaced by the subgroup generated by the terminal contributions.

Theorem 6.29  

(Support-separated characteristic transport). Assume

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2.$$

Let $T⊊S\subseteq Y$ be closed. Suppose that every terminal contribution of ${\mathcal{R}}_2$ lies in ${\mathcal{H}}_T\left(Y\right)$, while every terminal contribution of ${\mathcal{R}}_1$ except one class ${\epsilon }^{\Phi }$ also lies in ${\mathcal{H}}_T\left(Y\right)$. If

$${\epsilon }^{\Phi }\in {\mathcal{H}}_S\left(Y\right)$$

and

$$\left[{\epsilon }^{\Phi }\right]\ne 0\mathrm{in}{Gr}_{S/T}\mathcal{H}\left(Y\right),$$

then

$${\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2.$$

Proof. 

Apply the quotient map

$${\mathcal{H}}_S\left(Y\right)⟶{Gr}_{S/T}\mathcal{H}\left(Y\right)$$

to the terminal profile components. Every contribution of ${\mathcal{R}}_2$ maps to zero, as do all terminal contributions of ${\mathcal{R}}_1$ except ${\epsilon }^{\Phi }$. The latter has nonzero image by hypothesis. Hence the ordered terminal profiles cannot agree, and therefore the full characteristic transport profiles cannot be equivalent.

□

Remark 6.30.  

Theorem 6.29 is a sufficient separation criterion. The two routes and the relation ${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$ are independent geometric inputs; the theorem does not construct them.

There is also an endpoint version of the criterion.

Corollary 6.31  

(Residual characteristic separation). Assume ${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$, and define the endpoint characteristic residual by

$${\mathfrak{R}}_{\Phi }({\mathcal{R}}_1,{\mathcal{R}}_2;{\zeta }_0):={\Delta }_{\Phi }({\mathcal{R}}_1;{\zeta }_0)-{\Delta }_{\Phi }({\mathcal{R}}_2;{\zeta }_0)\in \mathcal{H}\left(Y\right).$$

If, for some closed pair $T\subseteq S\subseteq Y$, its image in

$${Gr}_{S/T}\mathcal{H}\left(Y\right)$$

is nonzero, then

$${\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2.$$

If the routes admit compressed endpoint operations whose reference operators are strictly compatible with their stagewise composites, the same statement may equivalently be written using the compressed endpoint defects.

Proof. 

Equivalent characteristic transport profiles have identical terminal contributions and therefore identical stagewise endpoint defects. Hence a nonzero endpoint residual in any support quotient precludes profile equivalence.

□

The converse phenomenon is also possible: profiles may remain distinct even when their endpoint defects agree.

Proposition 6.32  

(Internal separation with vanishing endpoint residual). Assume

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$$

and

$${\Delta }_{\Phi }({\mathcal{R}}_1;{\zeta }_0)={\Delta }_{\Phi }({\mathcal{R}}_2;{\zeta }_0).$$

Suppose that, for some closed pair $T\subseteq S\subseteq Y$, a terminal propagated contribution of ${\mathcal{R}}_1$ has nonzero image in ${Gr}_{S/T}\mathcal{H}\left(Y\right)$, while every terminal contribution of ${\mathcal{R}}_2$ that is being compared with it has zero image in that quotient. Then

$${\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2.$$

Proof. 

Equality of endpoint defects identifies only the sums

$$\sum _i{\epsilon }_{1,i}^{\Phi }=\sum _j{\epsilon }_{2,j}^{\Phi }.$$

It does not identify the ordered families of terminal propagated contributions. A nonzero support-quotient component on one side that is absent from the other therefore distinguishes the profiles even when the total sums agree.

□

Remark 6.33.  

The separation statements above are one-way criteria. Failure of a chosen characteristic family to distinguish two routes does not imply equality of the routes, equality of their Chow-theoretic endpoints, or equality under a finer realization.

Remark 6.34.  

The conclusion is relative to the selected realization. It is possible to have

$${\rho }_H\left({\mathcal{R}}_1\right){{|}_L={\rho }_H\left({\mathcal{R}}_2\right)|}_L$$

while

$${\mathfrak{P}}_{\Lambda }({\mathcal{R}}_1;{\zeta }_0)\ne {\mathfrak{P}}_{\Lambda }({\mathcal{R}}_2;{\zeta }_0).$$

This does not imply that the comparative transport profile is universally finer than total rational cohomology. The unconditional Jacobian example of Section 7 and Section 8 instead establishes the selected-sector separation

$$ker\left({\rho }_{L_A}\right)\setminus ker\left({cl}_{A\times A}\right)\ne ⌀.$$

Remark 6.35.  

Section 9 supplies independently identified support-controlled defect classes of the type used in Theorem 6.29. The nodal-cubic calculation does not, by itself, supply a second route that is L-equivalent to the first. No projector onto L, map defined from an unknown Hodge class, or unconstructed motivic lift is introduced in order to force separation.

7. Return Correspondences and Residual Geometry

Let $X,Y\in {\mathsf{SmProj}}_{\mathbb{C}}$, with $d_X:={dim}_{\mathbb{C}}X$ and $d_Y:={dim}_{\mathbb{C}}Y$. All Chow groups and correspondences in this section have rational coefficients unless otherwise stated.

The purpose of the section is to distinguish selected-sector invisibility from the stronger condition of invisibility on total rational cohomology. For degree-zero correspondences on a smooth projective variety, total cohomological invisibility is equivalent to homological triviality by Künneth decomposition and Poincaré duality, although we retain both terms to distinguish the corresponding realizations. The Jacobian construction of Section 8 realizes selected-sector invisibility without total cohomological invisibility. Detection of a nonzero residual inside the homologically trivial kernel is a strictly stronger problem and is not claimed here.

7.1. Transfer–return Constructions

Fix $k\in \mathbb{Z}$, and let

$$U\in C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}},R\in C{H}^{d_Y-k}{(Y\times X)}_{\mathbb{Q}}.$$

Thus U is a degree-k correspondence from X to Y, while R is a degree-$(-k)$ correspondence from Y to X, in the convention of Definition 3.8. Their composition is therefore a degree-zero correspondence

$$R\circ U\in C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}.$$

The composition convention is that of Definition 3.10, so U acts first and R second.

Let $L\subseteq H^{•}(X,\mathbb{Q})$ be a specified $\mathbb{Q}$-linear subspace. Whenever Tate twists or degree shifts are relevant, they are understood as part of the typed correspondence action described in Section 3.4.

Definition 7.1  

(Transfer–return pair). The pair $(U,R)$ is atransfer–return pair on Lif there exists a scalar $m\in \mathbb{Q}$ such that

$${(R\circ U)}_{*}{|}_L=m{id}_L.$$

The scalar m is called thereturn scalar on L.

The correspondences U and R are constructed independently of the choice of L. The selected sector enters only after their Chow and cohomological actions have already been defined.

Remark 7.2.  

No projector onto L is included in the definition of a transfer–return pair. The relation ${(R\circ U)}_{*}{|}_L=m{id}_L$ is a property of the restriction of an independently defined correspondence action.

Let ${\Delta }_X\in C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$ denote the diagonal correspondence. Since ${\left({\Delta }_X\right)}_{*}=id$, the scalar correspondence $m{\Delta }_X$ acts by $mid$ on total rational cohomology and therefore on L.

Proposition 7.3.  

For every transfer–return pair $(U,R)$ on L with return scalar m,

$$R\circ U{\equiv }_{L}m{\Delta }_X.$$

Proof. 

By Definition 7.1, ${(R\circ U)}_{*}{|}_L=m{id}_L$, while ${\left(m{\Delta }_X\right)}_{*}{|}_L=m{id}_L$. Hence the two correspondences have the same action on L, which is precisely L-equivalence in the sense of Definition 3.14.

□

Remark 7.4.  

The comparison is therefore between the factored correspondence route

$$X\stackrel{U}{\to }Y\stackrel{R}{\to }X$$

and the one-stage scalar correspondence $m{\Delta }_X$. The arrows here denote correspondence transport, not morphisms of varieties. Equality of their actions on L implies neither equality on total rational cohomology nor equality of their classes in $C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$.

7.2. The Residual Correspondence

Definition 7.5  

(Return residual). For a transfer–return pair $(U,R)$ on L with scalar m, define thereturn residual

$${\mathfrak{D}}_{U,R}:=R\circ U-m{\Delta }_X\in C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}.$$

Let

$${\rho }_L:C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}⟶{Hom}_{\mathbb{Q}}\left(L,H^{•}(X,\mathbb{Q})\right)$$

denote the selected-sector realization ${\rho }_L(\Gamma ):={\Gamma }_{*}{|}_L$. Let ${\rho }_H(\Gamma ):={\Gamma }_{*}$ denote the full graded action of Γ on total rational cohomology.

Lemma 7.6.  

For every transfer–return pair,

$${\mathfrak{D}}_{U,R}\in ker\left({\rho }_L\right).$$

Proof. 

By linearity,

$${\rho }_L\left({\mathfrak{D}}_{U,R}\right)={(R\circ U)}_{*}{{|}_L-m{\left({\Delta }_X\right)}_{*}|}_L=m{id}_L-m{id}_L=0.$$

□

Let

$${cl}_{X\times X}:C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}⟶H^{2d_X}\left(X\times X,\mathbb{Q}\left(d_X\right)\right)$$

denote the rational cohomological cycle-class map. Since $X\times X$ is smooth and projective, Poincaré duality identifies this realization with the corresponding Borel–Moore homology cycle class

$${cl}_{X\times X}^{\mathrm{BM}}:C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}⟶H_{2d_X}^{\mathrm{BM}}\left(X\times X,\mathbb{Q}\right).$$

Accordingly, either realization has the same kernel.

Definition 7.7  

(Residual invisibility levels). A residual correspondence $\mathfrak{D}\in C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$ is:

1.

L-invisibleif ${\rho }_L\left(\mathfrak{D}\right)=0$;

2.

totally cohomologically invisibleif ${\rho }_H\left(\mathfrak{D}\right)=0$;

3.

homologically trivialif

$${cl}_{X\times X}\left(\mathfrak{D}\right)=0,$$

equivalently if ${cl}_{X\times X}^{\mathrm{BM}}\left(\mathfrak{D}\right)=0$.

For smooth projective X, Künneth decomposition and Poincaré duality identify a class in $H^{2d_X}(X\times X,\mathbb{Q}\left(d_X\right))$ with its induced degree-zero endomorphism of total rational cohomology. Therefore

$$ker\left({cl}_{X\times X}\right)=ker\left({\rho }_H\right)\subseteq ker\left({\rho }_L\right).$$

The equality of the first two kernels uses the action onallrational cohomology groups; restriction to L can lose additional information [3].

Remark 7.8.  

Lemma 7.6 establishes only selected-sector invisibility. In particular, it does not imply that ${\mathfrak{D}}_{U,R}$ is homologically trivial or zero in Chow. The Jacobian residual of Section 8 satisfies

$${\mathfrak{D}}_n\in ker\left({\rho }_{L_A}\right)\setminus ker\left({\rho }_H\right).$$

Remark 7.9.  

For degree-zero correspondences on X, the relevant hierarchy is

$$\mathfrak{D}=0\mathrm{in}C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}⟹{cl}_{X\times X}\left(\mathfrak{D}\right)=0⟺{\rho }_H\left(\mathfrak{D}\right)=0⟹{\rho }_L\left(\mathfrak{D}\right)=0.$$

No converse is assumed. The explicit results of this paper establish strictness at the final implication. Detecting a nonzero class in $ker\left({cl}_{X\times X}\right)$ by a realization finer than ordinary rational cohomology is a genuinely stronger problem.

7.3. Residual Realizations and the Cohomological Baseline

The residual correspondence ${\mathfrak{D}}_{U,R}$ lives in a Chow group. Any proposed finer realization must therefore be defined on algebraic correspondences independently of the selected sector L.

Definition 7.10  

(Admissible residual realization). Let

$${Corr}_{\chi }^0\left(X\right)\subseteq C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$$

be a $\mathbb{Q}$-linear subspace of degree-zero correspondences, and let ${\mathcal{C}}_{\chi }(X\times X)$ be an additive target group.

Anadmissible residual realizationis a $\mathbb{Q}$-linear map

$${\chi }_X:{Corr}_{\chi }^0\left(X\right)⟶{\mathcal{C}}_{\chi }(X\times X)$$

satisfying the following requirements:

1.

${\chi }_X$ is defined independently of the choice of L;

2.

${\chi }_X$ arises from a specified geometric, cohomological, characteristic, bivariant, or comparable construction;

3.

whenever support is used, the target carries the required proper pushforwards along the closed immersions under consideration, and ${\chi }_X$ is compatible with those pushforwards.

When the realization is induced by a characteristic theory Φ, we write ${\chi }_{\Phi ,X}$.

Remark 7.11.  

Definition 7.10 is a domain and functoriality requirement, not a detection theorem. Its substantive content begins only after a concrete realization is constructed and proved nonzero on a particular residual. The ordinary rational cycle class is the baseline realization; detecting a nonzero class in its kernel is the stronger problem.

Proposition 7.12  

(The cycle-class residual realization). The rational cycle-class map

$${\chi }_{\mathrm{cl},X}:={cl}_{X\times X}$$

is an admissible residual realization on all of $C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$. Moreover, for every $\Gamma \in C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$,

$${\chi }_{\mathrm{cl},X}(\Gamma )=0⟺{\rho }_H(\Gamma )=0.$$

Proof. 

The cycle-class map is $\mathbb{Q}$-linear, defined independently of L, and compatible with proper pushforward [3]. For a smooth projective variety, Künneth decomposition and Poincaré duality identify the cohomology class of a degree-zero correspondence with its action on total rational cohomology. Hence the two kernels agree.

□

Definition 7.13  

(Visible residual). Let $\mathfrak{D}\in {Corr}_{\chi }^0\left(X\right)$. The residual $\mathfrak{D}$ is$\chi$-visibleif

$${\chi }_X\left(\mathfrak{D}\right)\ne 0.$$

When ${\chi }_X={\chi }_{\Phi ,X}$, we also say that $\mathfrak{D}$ is$\Phi$-characteristically visible.

It follows immediately from Proposition 7.12 that every residual in

$$ker\left({\rho }_L\right)\setminus ker\left({\rho }_H\right)$$

is unconditionally visible to the ordinary rational cycle-class realization. Section 8 provides an explicit infinite family of this type.

Finer characteristic realizations may naturally be sought on correspondences admitting proper–smooth, proper–lci, or comparable geometric presentations [4,5,21]. What is not automatic is descent from a chosen presentation to a well-defined additive invariant of the resulting rational-equivalence class in the Chow group.

To formulate support-sensitive detection, suppose that for every closed subvariety $W\subseteq X\times X$ under consideration the realization target admits a proper pushforward

$${\left(j_W\right)}_{*}^{\chi }:{\mathcal{C}}_{\chi }\left(W\right)⟶{\mathcal{C}}_{\chi }(X\times X),$$

where $j_W:W\hookrightarrow X\times X$ is the inclusion. Define

$${\mathcal{C}}_{\chi ,W}(X\times X):=im\left({\left(j_W\right)}_{*}^{\chi }\right).$$

If $T\subseteq W\subseteq X\times X$ are closed, functoriality gives ${\mathcal{C}}_{\chi ,T}(X\times X)\subseteq {\mathcal{C}}_{\chi ,W}(X\times X)$, and we define the relative support quotient

$${Gr}_{W/T}^{\chi }(X\times X):={\mathcal{C}}_{\chi ,W}(X\times X)/{\mathcal{C}}_{\chi ,T}(X\times X).$$

Theorem 7.14  

(Support-separated residual visibility). Let $(U,R)$ be a transfer–return pair on L with scalar m, and assume

$$R\circ U,{\Delta }_X\in {Corr}_{\chi }^0\left(X\right).$$

Let $T⊊W\subseteq X\times X$ be closed. Suppose

$${\chi }_X(R\circ U)={\eta }_T+{\eta }_W$$

with

$${\eta }_T\in {\mathcal{C}}_{\chi ,T}(X\times X),{\eta }_W\in {\mathcal{C}}_{\chi ,W}(X\times X),$$

and suppose

$$m{\chi }_X\left({\Delta }_X\right)\in {\mathcal{C}}_{\chi ,T}(X\times X).$$

If

$$\left[{\eta }_W\right]\ne 0\mathrm{in}{Gr}_{W/T}^{\chi }(X\times X),$$

then

$${\chi }_X\left({\mathfrak{D}}_{U,R}\right)\ne 0.$$

Proof. 

By linearity of ${\chi }_X$,

$${\chi }_X\left({\mathfrak{D}}_{U,R}\right)={\eta }_T+{\eta }_W-m{\chi }_X\left({\Delta }_X\right).$$

Both ${\eta }_T$ and $m{\chi }_X\left({\Delta }_X\right)$ vanish in the quotient ${Gr}_{W/T}^{\chi }(X\times X)$. Hence the image of ${\chi }_X\left({\mathfrak{D}}_{U,R}\right)$ in this quotient is $\left[{\eta }_W\right]\ne 0$. Therefore ${\chi }_X\left({\mathfrak{D}}_{U,R}\right)\ne 0$.

□

Remark 7.15.  

More generally, any residual whose realization has nonzero image in a support quotient ${Gr}_{W/T}^{\chi }(X\times X)$ is χ-visible.

Remark 7.16.  

Support-separated visibility is a sufficient criterion only. Failure of a chosen realization or support quotient to detect a residual does not imply that the residual vanishes in Chow.

7.4. Residual Detection and Separation

The transfer–return construction naturally determines two correspondence routes with the same selected-sector action.

Definition 7.17  

(Return and scalar routes). For a transfer–return pair $(U,R)$ with scalar m, define

$${\mathcal{R}}_{U,R}:=\left(X\stackrel{U}{\to }Y\stackrel{R}{\to }X\right)$$

and

$${\mathcal{S}}_m:=\left(X\stackrel{m{\Delta }_X}{\to }X\right).$$

The displayed arrows denote correspondence transport. The endpoint correspondences are

$${\Gamma }_{{\mathcal{R}}_{U,R}}=R\circ U,{\Gamma }_{{\mathcal{S}}_m}=m{\Delta }_X.$$

Remark 7.18.  

Every transfer–return pair satisfies

$${\mathcal{R}}_{U,R}{\sim }_L{\mathcal{S}}_m$$

by Proposition 7.3.

The ordinary transport profile records stagewise characteristic data. It is sometimes useful to adjoin an independently defined endpoint realization of the composed correspondence.

Definition 7.19  

(Correspondence-enhanced profile). Let ${\mathfrak{P}}_{\Lambda }(\mathcal{R};{\zeta }_0)$ be a comparative transport profile, and suppose the endpoint correspondence ${\Gamma }_{\mathcal{R}}$ lies in ${Corr}_{\chi }^0\left(X\right)$. Define

$${\mathfrak{P}}_{\Lambda }^{\chi }(\mathcal{R};{\zeta }_0):=\left({\mathfrak{P}}_{\Lambda }(\mathcal{R};{\zeta }_0),{\chi }_X\left({\Gamma }_{\mathcal{R}}\right)\right).$$

The second component is additional endpoint correspondence data; it is not identified with the stagewise transport profile unless an explicit compatibility theorem supplies such an identification.

Theorem 7.20  

(Residual detection separates the return route). If

$${\chi }_X\left({\mathfrak{D}}_{U,R}\right)\ne 0,$$

then

$${\mathcal{R}}_{U,R}{\sim }_L{\mathcal{S}}_m$$

but

$${\mathfrak{P}}_{\Lambda }^{\chi }({\mathcal{R}}_{U,R};{\zeta }_0)\ne {\mathfrak{P}}_{\Lambda }^{\chi }({\mathcal{S}}_m;{\zeta }_0).$$

Proof. 

Selected-sector equivalence follows from Proposition 7.3. The endpoint realization components differ by

$${\chi }_X(R\circ U)-{\chi }_X\left(m{\Delta }_X\right)={\chi }_X\left(R\circ U-m{\Delta }_X\right)={\chi }_X\left({\mathfrak{D}}_{U,R}\right),$$

where $\mathbb{Q}$-linearity gives ${\chi }_X\left(m{\Delta }_X\right)=m{\chi }_X\left({\Delta }_X\right)$. This difference is nonzero by hypothesis, so the enhanced profiles cannot agree.

□

Corollary 7.21  

(Cohomological residual separation). If

$${\rho }_H\left({\mathfrak{D}}_{U,R}\right)\ne 0,$$

then

$${\mathfrak{D}}_{U,R}\in ker\left({\rho }_L\right)\setminus ker\left({\chi }_{\mathrm{cl},X}\right),$$

and the routes ${\mathcal{R}}_{U,R}$ and ${\mathcal{S}}_m$ have distinct cycle-class-enhanced profiles.

Proof. 

Lemma 7.6 gives ${\mathfrak{D}}_{U,R}\in ker\left({\rho }_L\right)$, while Proposition 7.12 gives

$${\rho }_H\left({\mathfrak{D}}_{U,R}\right)\ne 0⟺{\chi }_{\mathrm{cl},X}\left({\mathfrak{D}}_{U,R}\right)\ne 0.$$

Apply Theorem 7.20 with ${\chi }_X={\chi }_{\mathrm{cl},X}$.

□

Remark 7.22.  

The support hypotheses of Theorem 7.14 imply the hypothesis of Theorem 7.20 and therefore separate the correspondence-enhanced return and scalar profiles.

Remark 7.23.  

The unconditional residual separation proved in this paper occurs in

$$ker\left({\rho }_L\right)\setminus ker\left({cl}_{X\times X}\right).$$

A genuinely finer realization would detect a nonzero element of

$$ker\left({cl}_{X\times X}\right)=ker\left({\rho }_H\right).$$

No such characteristic detection theorem is constructed here.

Endpoint residual visibility is not the only possible source of separation. Stagewise profile information may remain nontrivial even when the chosen endpoint realization vanishes.

Theorem 7.24  

(Internal residual separation). Assume

$${\chi }_X\left({\mathfrak{D}}_{U,R}\right)=0.$$

Suppose nevertheless that, in a characteristic component included in the comparative profile, a terminal propagated contribution of ${\mathcal{R}}_{U,R}$ has nonzero image in some support quotient ${Gr}_{S/T}\mathcal{H}\left(X\right)$, while all corresponding contributions of ${\mathcal{S}}_m$ vanish in that quotient. Then

$${\mathcal{R}}_{U,R}{\sim }_L{\mathcal{S}}_m,{\mathcal{R}}_{U,R}\neg {\sim }_{\mathfrak{P}}{\mathcal{S}}_m.$$

Proof. 

The first relation is Proposition 7.3. The second is an instance of Proposition 6.32: equality of the chosen endpoint residual realization does not identify the ordered stagewise transport data.

□

Remark 7.25.  

Endpoint residual visibility and internal route visibility are therefore logically distinct. The former concerns an invariant of the endpoint Chow correspondence; the latter concerns the stagewise characteristic transport history and may survive even when the chosen endpoint residual realization vanishes.

7.5. Return-pair Transport Versus Endpoint Residual Geometry

We finally compare the Chow residual ${\mathfrak{D}}_{U,R}$ with the transport defect of the return route.

Fix a characteristic transport datum

$$\mathbb{T}=\left(\mathsf{C},\mathcal{A},\mathcal{H},\Phi ,{Op}_{\mathbb{T}},{\mathcal{T}}_{\Phi }\right)$$

for which the correspondence operations determined by U, R, and their compressed composite $R\circ U$ are admissible. We use $U_{!}$ and $R_{!}$ for the corresponding source-side operators and $U_{\Phi ,*}^{\mathrm{RR}}$ and $R_{\Phi ,*}^{\mathrm{RR}}$ for their reference characteristic transport operators.

For $\xi \in \mathcal{A}\left(X\right)$, define

$${\delta }_U^{\Phi }\left(\xi \right):={\Delta }_{\Phi }(U;\xi )$$

and

$${\delta }_R^{\Phi }\left(\xi \right):={\Delta }_{\Phi }\left(R;U_{!}\xi \right).$$

Theorem 7.26  

(Return-pair transport-defect formula). For every admissible $\xi \in \mathcal{A}\left(X\right)$,

$${\Delta }_{\Phi }(R\circ U;\xi )={\delta }_R^{\Phi }\left(\xi \right)+R_{\Phi ,*}^{\mathrm{RR}}\left({\delta }_U^{\Phi }\left(\xi \right)\right)+{\mathfrak{I}}_{\Phi }(R,U)\left({\Phi }_X\left(\xi \right)\right).$$

Proof. 

This is Proposition 5.3 applied to the typed correspondence operations associated with U and R.

□

Remark 7.27.  

If the pair $(R,U)$ is Φ-Riemann–Roch compatible, then

$${\Delta }_{\Phi }(R\circ U;\xi )={\delta }_R^{\Phi }\left(\xi \right)+R_{\Phi ,*}^{\mathrm{RR}}\left({\delta }_U^{\Phi }\left(\xi \right)\right).$$

The scalar diagonal route is exact under the expected linearity hypotheses.

Proposition 7.28  

(Scalar-diagonal exactness). Assume $\mathcal{A}\left(X\right)$ and $\mathcal{H}\left(X\right)$ are $\mathbb{Q}$-vector spaces, ${\Phi }_X$ is $\mathbb{Q}$-linear, and the scalar correspondence $m{\Delta }_X$ is admissible with

$${\left(m{\Delta }_X\right)}_{!}=m{id}_{\mathcal{A}\left(X\right)}$$

and

$${\left(m{\Delta }_X\right)}_{\Phi ,*}^{\mathrm{RR}}=m{id}_{\mathcal{H}\left(X\right)}.$$

Then, for every $\xi \in \mathcal{A}\left(X\right)$,

$${\Delta }_{\Phi }(m{\Delta }_X;\xi )=0.$$

Proof. 

By $\mathbb{Q}$-linearity,

$$\begin{array}{cc}\hfill {\Delta }_{\Phi }(m{\Delta }_X;\xi )& ={\Phi }_X\left(m\xi \right)-m{\Phi }_X\left(\xi \right)\hfill \\ & =m{\Phi }_X\left(\xi \right)-m{\Phi }_X\left(\xi \right)\hfill \\ & =0.\hfill \end{array}$$

□

Definition 7.29  

(Return transport residual). For every admissible source class $\xi \in \mathcal{A}\left(X\right)$, define

$${\mathfrak{E}}_{\Phi }(U,R;m;\xi ):={\Delta }_{\Phi }(R\circ U;\xi )-{\Delta }_{\Phi }(m{\Delta }_X;\xi )\in \mathcal{H}\left(X\right).$$

Thus ${\mathfrak{E}}_{\Phi }(U,R;m;\xi )$ compares the endpoint characteristic transport defects of the return correspondence and the scalar diagonal. It is not, by definition, the characteristic realization of the Chow residual.

Remark 7.30.  

Under the hypotheses of Proposition 7.28,

$${\mathfrak{E}}_{\Phi }(U,R;m;\xi )={\delta }_R^{\Phi }\left(\xi \right)+R_{\Phi ,*}^{\mathrm{RR}}\left({\delta }_U^{\Phi }\left(\xi \right)\right)+{\mathfrak{I}}_{\Phi }(R,U)\left({\Phi }_X\left(\xi \right)\right).$$

If the return pair is additionally Φ-Riemann–Roch compatible, the interaction term vanishes.

To relate the transport residual to the Chow residual requires an additional linearity statement at the level of correspondences.

Definition 7.31  

(Correspondence-linear transport realization). Let

$$\mathcal{C}\subseteq C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$$

be a $\mathbb{Q}$-linear subspace such that every $\Gamma \in \mathcal{C}$ determines an admissible typed correspondence operation $u_{\Gamma }:X\to X$. Fix $\xi \in \mathcal{A}\left(X\right)$.

A map

$${\Psi }_{\Phi ,\xi }:\mathcal{C}⟶\mathcal{H}\left(X\right)$$

iscorrespondence-linear at $\xi$if it is $\mathbb{Q}$-linear and satisfies

$${\Psi }_{\Phi ,\xi }(\Gamma )={\Delta }_{\Phi }(u_{\Gamma };\xi )$$

for every $\Gamma \in \mathcal{C}$.

When no ambiguity can arise, we abbreviate ${\Delta }_{\Phi }(u_{\Gamma };\xi )$ to ${\Delta }_{\Phi }(\Gamma ;\xi )$.

Correspondence-linearity is an additional hypothesis. It is not implied merely by additivity of the characteristic transformation in the source class.

Proposition 7.32  

(Comparison with the Chow residual). Suppose a correspondence-linear realization ${\Psi }_{\Phi ,\xi }$ is defined on a $\mathbb{Q}$-linear subspace $\mathcal{C}$ containing $R\circ U$ and ${\Delta }_X$. Then

$${\mathfrak{E}}_{\Phi }(U,R;m;\xi )={\Psi }_{\Phi ,\xi }\left({\mathfrak{D}}_{U,R}\right).$$

Proof. 

Since $\mathcal{C}$ is $\mathbb{Q}$-linear, ${\mathfrak{D}}_{U,R}=R\circ U-m{\Delta }_X$ also belongs to $\mathcal{C}$. Therefore

$$\begin{array}{cc}\hfill {\Psi }_{\Phi ,\xi }\left({\mathfrak{D}}_{U,R}\right)& ={\Psi }_{\Phi ,\xi }(R\circ U)-m{\Psi }_{\Phi ,\xi }\left({\Delta }_X\right)\hfill \\ & ={\Delta }_{\Phi }(R\circ U;\xi )-{\Delta }_{\Phi }(m{\Delta }_X;\xi )\hfill \\ & ={\mathfrak{E}}_{\Phi }(U,R;m;\xi ),\hfill \end{array}$$

where the second equality uses correspondence-linearity, including

$${\Psi }_{\Phi ,\xi }\left(m{\Delta }_X\right)=m{\Psi }_{\Phi ,\xi }\left({\Delta }_X\right)={\Delta }_{\Phi }(m{\Delta }_X;\xi ).$$

□

Remark 7.33.  

The Chow residual

$${\mathfrak{D}}_{U,R}\in C{H}^{d_X}{(X\times X)}_{\mathbb{Q}}$$

and the transport residual

$${\mathfrak{E}}_{\Phi }(U,R;m;\xi )\in \mathcal{H}\left(X\right)$$

belong to different mathematical levels. They are related only when an explicit compatibility statement such as Proposition 7.32 is available.

Remark 7.34.  

The conditions

$${(R\circ U)}_{*}{|}_L=m{id}_L,$$

$${\chi }_X\left({\mathfrak{D}}_{U,R}\right)\ne 0,$$

and

$${\mathfrak{E}}_{\Phi }(U,R;m;\xi )\ne 0$$

concern three distinct levels: selected cohomological action, endpoint residual geometry, and characteristic transport history. The first does not imply either of the latter two, and no implication between the final two is assumed without an explicit compatibility theorem.

Remark 7.35.  

Residual visibility is not an algebraicity theorem for an arbitrary Hodge class. The correspondences U, R, ${\Delta }_X$, and ${\mathfrak{D}}_{U,R}$ are algebraic Chow correspondences before any selected-sector comparison is made. The question is which realizations retain distinctions among those already algebraic correspondences.

8. Abelian Varieties and Jacobian Factorizations

The Jacobian setting provides a rigid test of the characteristic transport formalism. A factored construction

$$A⟶J\left(C\right)⟶A$$

may have the same endpoint morphism as a direct multiplication map while retaining a nontrivial intermediate Jacobian. At the same time, tangent bundles of complex abelian varieties are trivial, so the standard tangent-dependent characteristic corrections occurring in the examples below collapse. The setting therefore separates two logically different questions: whether a nontrivial factorization is visible to a chosen characteristic transport profile, and whether restriction to a selected cohomological sector loses information that remains visible on total rational cohomology.

The conclusions are correspondingly calibrated. Standard proper-pushforward defect profiles do not detect the intermediate Jacobian. By contrast, the return correspondence and a scalar multiple of the diagonal can agree on a selected Hodge sector while remaining distinct on total rational cohomology. The latter gives an unconditional instance of the residual-separation framework of Section 7, detected by the ordinary rational cycle-class realization. It does not produce a homologically trivial residual detected by an invariant finer than rational cohomology.

8.1. Geometric Setup

Let $(A,\Theta )$ be a principally polarized complex abelian variety of dimension g, and let C be a smooth connected projective complex curve of genus h. Let $\iota :C\to A$ be a morphism whose image generates A, meaning that the smallest abelian subvariety containing the translated curve $\iota \left(C\right)-\iota \left(c_0\right)$ is A, for a chosen base point $c_0\in C$.

Let $a_C:C\to J\left(C\right)$ be the Abel–Jacobi map normalized by $a_C\left(c_0\right)=0$. By the universal property of the Jacobian, after translating ι so that $\iota \left(c_0\right)=0$, there is a unique homomorphism of abelian varieties

$$f:J\left(C\right)⟶A$$

such that

$$f\circ a_C=\iota$$

[25].

Let

$${\lambda }_A:A\stackrel{\sim }{\to }\widehat{A},{\lambda }_J:J\left(C\right)\stackrel{\sim }{\to }\widehat{J\left(C\right)}$$

denote the homomorphisms induced by the principal polarizations of A and $J\left(C\right)$, respectively. The dual homomorphism of $f:J\left(C\right)\to A$ is $\widehat{f}:\widehat{A}\to \widehat{J\left(C\right)}$.

Definition 8.1  

(Polarized dual Jacobian map). Thepolarized dualof f is the homomorphism

$$f^{\vee }:={\lambda }_J^{-1}\circ \widehat{f}\circ {\lambda }_A:A⟶J\left(C\right).$$

Definition 8.2  

(Jacobian return datum). AJacobian return datum of coefficient qconsists of

$$(A,\Theta ,C,\iota ,f,f^{\vee },q),q\in {\mathbb{Z}}_{>0},$$

satisfying

$$f\circ f^{\vee }={\left[q\right]}_A,$$

where ${\left[q\right]}_A:A\to A$ denotes multiplication by q on the abelian variety A.

Remark 8.3.  

The scalar-return relation $f\circ f^{\vee }={\left[q\right]}_A$ is geometric input. It is not obtained from a Hodge projector, from a chosen target Hodge class, or from any map defined after selecting the cohomological sector used below. Proposition 8.6 gives an explicit family of such return data with $q>1$.

The graph of $f^{\vee }:A\to J\left(C\right)$ is a correspondence of degree $h-g$, while the graph of $f:J\left(C\right)\to A$ has degree $g-h$. Their composition is therefore a degree-zero correspondence on A.

Proposition 8.4  

(Graph composition). For every Jacobian return datum,

$${\Gamma }_f\circ {\Gamma }_{f^{\vee }}={\Gamma }_{{\left[q\right]}_A}\in C{H}^g{(A\times A)}_{\mathbb{Q}}.$$

Proof. 

The composition of graph correspondences is the graph of the composite morphism. Since $f\circ f^{\vee }={\left[q\right]}_A$, the result follows from the correspondence-composition convention of Definition 3.10 [3].

□

The scalar-return identity also constrains the geometry of the two homomorphisms.

Proposition 8.5.  

Let $(A,\Theta ,C,\iota ,f,f^{\vee },q)$ be a Jacobian return datum. Then:

1.

$f:J\left(C\right)\to A$ is surjective;

2.

$ker\left(f^{\vee }\right)\subseteq A\left[q\right]$, and hence $ker\left(f^{\vee }\right)$ is finite;

3.

$B:=f^{\vee }\left(A\right)\subseteq J\left(C\right)$ is an abelian subvariety of dimension g;

4.

$f^{\vee }$ factors as

$$A\stackrel{e}{\to }BjJ\left(C\right),$$

where $e:A\to B$ is an isogeny and $j:B\hookrightarrow J\left(C\right)$ is the inclusion.

Proof. 

Because ${\left[q\right]}_A=f\circ f^{\vee }$ is surjective, f is surjective. If $a\in ker\left(f^{\vee }\right)$, then

$${\left[q\right]}_A\left(a\right)=f\left(f^{\vee }\left(a\right)\right)=0,$$

so $ker\left(f^{\vee }\right)\subseteq A\left[q\right]$. Since the q-torsion subgroup $A\left[q\right]$ is finite, $ker\left(f^{\vee }\right)$ is finite.

The image of a homomorphism of abelian varieties is an abelian subvariety [25]. Hence $B=f^{\vee }\left(A\right)$ is an abelian subvariety of $J\left(C\right)$. Because $f^{\vee }:A\to B$ has finite kernel, $dimB=dimA=g$, and the induced surjective homomorphism $e:A\to B$ is an isogeny.

□

8.2. Explicit Return Data from Finite Covers

The scalar-return relation arises naturally from norm and pullback homomorphisms between Jacobians.

Let $\pi :C\to D$ be a finite nonconstant morphism of smooth connected projective complex curves, and set $n:=deg\left(\pi \right)>1$. Choose $c_0\in C$, set $d_0:=\pi \left(c_0\right)$, and let $a_C:C\to J\left(C\right)$ and $a_D:D\to J\left(D\right)$ be the Abel–Jacobi maps normalized by $a_C\left(c_0\right)=0$ and $a_D\left(d_0\right)=0$.

Set $A:=J\left(D\right)$, equipped with its canonical principal polarization ${\Theta }_D$, and define

$$\iota :=a_D\circ \pi :C⟶A.$$

Since π is finite and nonconstant, it is surjective; consequently $\iota \left(C\right)=a_D\left(D\right)$, whose image generates $J\left(D\right)$.

The morphism π induces the pullback homomorphism

$${\pi }^{*}:J\left(D\right)={Pic}^0\left(D\right)⟶J\left(C\right)={Pic}^0\left(C\right)$$

and the norm homomorphism

$${Nm}_{\pi }:J\left(C\right)⟶J\left(D\right).$$

With the chosen base points,

$${Nm}_{\pi }\circ a_C=a_D\circ \pi ,$$

so the homomorphism induced by ι through the universal property of $J\left(C\right)$ is

$$f={Nm}_{\pi }.$$

Proposition 8.6  

(Explicit Jacobian return data from finite covers). Let $\pi :C\to D$ be as above and set $f:={Nm}_{\pi }:J\left(C\right)\to J\left(D\right)$. Under the canonical principal polarizations, norm and pullback are dual homomorphisms, so

$$f^{\vee }={\pi }^{*}:J\left(D\right)⟶J\left(C\right)$$

[25]. Moreover,

$${Nm}_{\pi }\circ {\pi }^{*}={\left[n\right]}_{J\left(D\right)}.$$

Consequently,

$$\left(J\left(D\right),{\Theta }_D,C,\iota ,{Nm}_{\pi },{\pi }^{*},n\right)$$

is a Jacobian return datum of coefficient $q=n$.

Proof. 

The equality $f^{\vee }={\pi }^{*}$ is the standard duality between norm and pullback under the canonical principal polarizations [25].

Let $\mathcal{L}\in {Pic}^0\left(D\right)$. Since π is finite of degree n, the norm satisfies

$${Nm}_{\pi }\left({\pi }^{*}\mathcal{L}\right)\simeq {\mathcal{L}}^{\otimes n}.$$

Under the group law on $J\left(D\right)={Pic}^0\left(D\right)$, tensor power by n is multiplication by n. Hence

$${Nm}_{\pi }\circ {\pi }^{*}={\left[n\right]}_{J\left(D\right)}.$$

Taking $A=J\left(D\right)$ gives the asserted return datum with $q=n$.

□

Corollary 8.7.  

Let $\pi :C\to D$ be a finite morphism of smooth connected projective curves of degree $n>1$, with $g\left(D\right)\ge 2$. Then $A:=J\left(D\right)$, $f:={Nm}_{\pi }$, and $f^{\vee }:={\pi }^{*}$ define a Jacobian return datum with coefficient $q=n>1$.

For $g:=g\left(D\right)=dimJ\left(D\right)$, the selected sector used below is the codimension-$(g-1)$ Hodge sector

$$L_A=H^{2g-2}{\left(A,\mathbb{Q}(g-1)\right)}_{\mathrm{Hdg}},$$

the degree in which cohomology classes of algebraic curves on A lie.

Proof. 

Apply Proposition 8.6.

□

Remark 8.8.  

The identity ${Nm}_{\pi }\circ {\pi }^{*}={\left[n\right]}_{J\left(D\right)}$ also holds when $g\left(D\right)=1$. In that case $g=1$ and

$$L_A=H^0{(A,\mathbb{Q}\left(0\right))}_{\mathrm{Hdg}}=\mathbb{Q},$$

so the selected sector reduces to degree-zero cohomology. The principal selected-sector examples below are therefore taken with $g\left(D\right)\ge 2$.

8.3. Scalar Return on the Selected Hodge Sector

Let $(A,\Theta ,C,\iota ,f,f^{\vee },q)$ be a Jacobian return datum and assume $g:=dimA\ge 2$. Define

$$L_A:=H^{2g-2}{\left(A,\mathbb{Q}(g-1)\right)}_{\mathrm{Hdg}}.$$

This is the rational Hodge sector of codimension $g-1$; algebraic curves on A have cycle classes in this degree.

The action of multiplication by q on the cohomology of an abelian variety is standard [25].

Lemma 8.9.  

For every integer $k\ge 0$,

$${\left[q\right]}_A^{*}=q^{k}id\mathrm{on}{H}^k(A,\mathbb{Q}).$$

Proof. 

Multiplication by q acts as $qid$ on $H^1(A,\mathbb{Q})$, and the cohomology algebra of the complex abelian variety A satisfies

$$H^{•}(A,\mathbb{Q})\simeq {\bigwedge }^{•}{H}^1(A,\mathbb{Q})$$

[25]. The result follows by taking exterior powers.

□

Proposition 8.10.  

For $0\le k\le 2g$,

$${\left[q\right]}_{A,*}=q^{2g-k}id\mathrm{on}{H}^k(A,\mathbb{Q}).$$

The same scalar acts on the corresponding Tate-twisted group.

Proof. 

The isogeny ${\left[q\right]}_A:A\to A$ has degree $q^{2g}$. The push–pull identity for a finite morphism gives

$${\left[q\right]}_{A,*}\circ {\left[q\right]}_A^{*}=q^{2g}id.$$

By Lemma 8.9, ${\left[q\right]}_A^{*}=q^{k}id$ on $H^k(A,\mathbb{Q})$. Hence

$${\left[q\right]}_{A,*}=q^{2g-k}id.$$

Tensoring with a Tate structure does not change this scalar.

□

Corollary 8.11  

(Quadratic scalar return). For every Jacobian return datum,

$${(f\circ f^{\vee })}_{*}{|}_{L_A}=q^2{id}_{L_A}.$$

Proof. 

Since $f\circ f^{\vee }={\left[q\right]}_A$, apply Proposition 8.10 in degree $k=2g-2$.

□

Definition 8.12  

(Jacobian selected-sector residual). Define

$${\mathfrak{D}}_q:={\Gamma }_{{\left[q\right]}_A}-q^2{\Delta }_A\in C{H}^g{(A\times A)}_{\mathbb{Q}},$$

where ${\Gamma }_{{\left[q\right]}_A}$ is the graph of ${\left[q\right]}_A$ and ${\Delta }_A$ is the diagonal correspondence.

Proposition 8.13.  

The residual ${\mathfrak{D}}_q$ is invisible on the selected Hodge sector:

$${\left({\mathfrak{D}}_q\right)}_{*}{|}_{L_A}=0.$$

Equivalently,

$${\mathfrak{D}}_q\in ker\left({\rho }_{L_A}\right).$$

Proof. 

By Corollary 8.11, ${\Gamma }_{{\left[q\right]}_A}$ acts as $q^2{id}_{L_A}$, while $q^2{\Delta }_A$ acts by the same scalar. Their difference therefore acts trivially on $L_A$.

□

Selected-sector invisibility is strictly weaker than invisibility on total cohomology.

Proposition 8.14.  

If $q>1$, then

$${\left({\mathfrak{D}}_q\right)}_{*}\ne 0\mathrm{on}{H}^{•}(A,\mathbb{Q}).$$

Consequently,

$${cl}_{A\times A}\left({\mathfrak{D}}_q\right)\ne 0,$$

and hence

$${\mathfrak{D}}_q\ne 0\mathrm{in}C{H}^g{(A\times A)}_{\mathbb{Q}}.$$

Proof. 

Consider top-degree cohomology $H^{2g}(A,\mathbb{Q}\left(g\right))$. By Proposition 8.10,

$${\left[q\right]}_{A,*}=id$$

on this group, whereas $q^2{\Delta }_A$ acts as $q^{2}id$. Hence

$${\left({\mathfrak{D}}_q\right)}_{*}=(1-q^2)id$$

on $H^{2g}(A,\mathbb{Q}\left(g\right))$, which is nonzero for $q>1$.

By Proposition 7.12, a degree-zero correspondence on a smooth projective variety has zero rational cycle class if and only if its action on total rational cohomology is zero. Therefore ${cl}_{A\times A}\left({\mathfrak{D}}_q\right)\ne 0$, and the Chow class ${\mathfrak{D}}_q$ is nonzero.

□

Remark 8.15.  

For $q>1$,

$${\mathfrak{D}}_q\in ker\left({\rho }_{L_A}\right)\setminus ker\left({\chi }_{\mathrm{cl},A}\right),$$

where

$${\chi }_{\mathrm{cl},A}={cl}_{A\times A}$$

is the ordinary rational cycle-class residual realization of Proposition 7.12. Equivalently, one may use the corresponding Borel–Moore cycle class via Poincaré duality.

Thus the selected-sector realization is strictly coarser than the ordinary cycle-class realization in this example. The residual ${\mathfrak{D}}_q$ is not homologically trivial: total rational cohomology already detects it. The example therefore establishes information loss under restriction to $L_A$, not the existence of an invariant finer than total cohomology.

8.4. Riemann–Roch Simplifications for Abelian Varieties

The characteristic correction terms simplify sharply on abelian varieties.

Lemma 8.16.  

Let B be a complex abelian variety of dimension d. Then

$$T_B\simeq {\mathcal{O}}_B^{\oplus d}.$$

Consequently,

$$c\left(T_B\right)=td\left(T_B\right)=L^{*}\left(T_B\right)={\widehat{T}}_y^{*}\left(T_B\right)=1.$$

Proof. 

Translation by points of B identifies all tangent spaces with $T_{B,0}$, giving $T_B\simeq B\times T_{B,0}\simeq {\mathcal{O}}_B^{\oplus d}$. The displayed characteristic classes are multiplicative and normalized to 1 on a trivial bundle [16,25].

□

Recall the factorization

$$A\stackrel{e}{\to }BjJ\left(C\right)$$

of $f^{\vee }$ from Proposition 8.5.

Proposition 8.17.  

With the preceding notation:

1.

the isogeny $e:A\to B$ is finite étale and has zero relative tangent bundle;

2.

the normal bundle of B in $J\left(C\right)$ is trivial of rank $h-g$:

$$N_{B/J\left(C\right)}\simeq {\mathcal{O}}_B^{\oplus (h-g)};$$

3.

the surjective homomorphism $f:J\left(C\right)\to A$ is smooth of pure relative dimension $h-g$, and

$$T_{J\left(C\right)/A}\simeq {\mathcal{O}}_{J\left(C\right)}^{\oplus (h-g)}.$$

Proof. 

Over $\mathbb{C}$, an isogeny of abelian varieties is finite étale, proving the first assertion [25].

Both $T_B$ and $T_{J\left(C\right)}{|}_B$ are translation-trivial bundles. The differential of the inclusion $j:B\hookrightarrow J\left(C\right)$ identifies at every point, by translation, with the fixed inclusion $T_{B,0}\hookrightarrow T_{J\left(C\right),0}$. Hence the quotient bundle

$$N_{B/J\left(C\right)}\simeq T_{J\left(C\right)}{|}_B/T_B$$

is the trivial bundle associated with the fixed quotient vector space $T_{J\left(C\right),0}/T_{B,0}$, of dimension $h-g$.

Similarly, the differential of the surjective homomorphism $f:J\left(C\right)\to A$ is translation invariant. Its kernel is the constant vector subspace $ker\left(d{f}_0\right)\subseteq T_{J\left(C\right),0}$, of dimension $h-g$. Since homomorphisms of complex abelian varieties are smooth onto their images, f is smooth and its relative tangent bundle is the corresponding trivial bundle.

□

Corollary 8.18.  

All multiplicative Todd, normalized Hirzebruch, and L-class corrections associated with $T_{J\left(C\right)/A}$ are equal to 1. Likewise, the Chern, Todd, normalized Hirzebruch, and L-class corrections associated with $N_{B/J\left(C\right)}$ are equal to 1.

Proof. 

Apply the corresponding multiplicative characteristic classes to the trivial bundles in Proposition 8.17.

□

Remark 8.19.  

Trivial tangent and normal corrections do not imply that every conceivable route invariant is trivial. They remove only the information carried by those multiplicative correction factors.

8.5. Characteristic Profile of the Factored Route

Consider the factored morphism route

$${\mathcal{R}}_J:A\stackrel{f^{\vee }}{\to }J\left(C\right)\stackrel{f}{\to }A$$

and the direct route

$${\mathcal{R}}_q:A\stackrel{{\left[q\right]}_A}{\to }A.$$

Their endpoint morphisms agree because $f\circ f^{\vee }={\left[q\right]}_A$.

The full characteristic transport profile contains the stage operators and propagation data. For the present calibration it is useful to retain only the defect components.

Definition 8.20  

(Reduced defect profile). For a characteristic transport profile ${\mathfrak{P}}_{\Phi }(\mathcal{R};\xi )$, define itsreduced defect profileby

$${\mathfrak{D}}_{\Phi }(\mathcal{R};\xi ):=\left({\left({\delta }_i^{\Phi }\right)}_i,{\left({\epsilon }_i^{\Phi }\right)}_i\right),$$

where ${\delta }_i^{\Phi }$ are the stage defects and ${\epsilon }_i^{\Phi }$ their terminal propagations.

Thus ${\mathfrak{D}}_{\Phi }$ deliberately forgets the stage operators themselves; it is used only to calibrate whether the standard defect components detect the intermediate Jacobian.

Proposition 8.21  

(Proper-defect calibration for the Jacobian factorization). Let

$$\Phi \in \{\tau ,c_{*},T_{y*}\}.$$

For every admissible source class ξ in the corresponding source assignment, the proper-pushforward defect of each stage $f^{\vee }$, f, and ${\left[q\right]}_A$ is zero. Consequently,

$${\mathfrak{D}}_{\Phi }({\mathcal{R}}_J;\xi )={\mathfrak{D}}_{\Phi }({\mathcal{R}}_q;\xi )=0.$$

Proof. 

The morphisms $f^{\vee }:A\to J\left(C\right)$, $f:J\left(C\right)\to A$, and ${\left[q\right]}_A:A\to A$ are proper because their sources and targets are projective. Apply Proposition 4.10, which records proper exact transport for the Baum–Fulton–MacPherson transformation τ, the Chern–Schwartz–MacPherson transformation $c_{*}$, and the motivic Hirzebruch transformation $T_{y*}$. □

Corollary 8.22.  

For $\Phi \in \{\tau ,c_{*},T_{y*}\}$ and every admissible source class,

$${\Delta }_{\Phi }(f\circ f^{\vee };\xi )={\Delta }_{\Phi }({\left[q\right]}_A;\xi )=0.$$

Proof. 

The endpoint morphisms agree and proper transport is exact for each of the three characteristic transformations.

□

Remark 8.23.  

Proposition 8.21 is a calibration result. Exact proper functoriality removes the intermediate Jacobian from the reduced proper-defect profile. This does not imply that every characteristic or geometric realization of the factorization is trivial.

The same geometry also produces exact stages of different variance.

Proposition 8.24  

(Smooth $I{T}_y$-transport along the norm map). The surjective homomorphism $f:J\left(C\right)\to A$ is smooth of pure relative complex dimension $h-g$. Hence the smooth intersection-Hirzebruch transfer formula gives

$$I{T}_{y*}\left(J\left(C\right)\right)=T_y^{*}\left(T_{J\left(C\right)/A}\right)\cap f_{\mathrm{BM}}^{*}I{T}_{y*}\left(A\right).$$

Since $T_{J\left(C\right)/A}\simeq {\mathcal{O}}_{J\left(C\right)}^{\oplus (h-g)}$,

$$T_y^{*}\left(T_{J\left(C\right)/A}\right)=1,$$

and therefore

$$I{T}_{y*}\left(J\left(C\right)\right)=f_{\mathrm{BM}}^{*}I{T}_{y*}\left(A\right).$$

Equivalently,

$${\Delta }_{I{T}_y}^{*}\left(f\right)=0.$$

Proof. 

Proposition 8.17 gives smoothness and triviality of the relative tangent bundle. The smooth intersection-Hirzebruch transfer theorem therefore applies [13].

□

Proposition 8.25  

(Normally nonsingular $I{T}_{1,*}$-Gysin transport). Let $j:B\hookrightarrow J\left(C\right)$ be the abelian-subvariety inclusion from Proposition 8.5. Under the hypotheses of Banagl’s normally nonsingular Gysin theorem,

$$j^{!}I{T}_{1,*}\left(J\left(C\right)\right)=I{T}_{1,*}\left(B\right).$$

Proof. 

Banagl’s Gysin formula gives

$$j^{!}I{T}_{1,*}\left(J\left(C\right)\right)=L^{*}\left(N_{B/J\left(C\right)}\right)\cap I{T}_{1,*}\left(B\right)$$

[10]. By Proposition 8.17, $N_{B/J\left(C\right)}$ is trivial, so $L^{*}\left(N_{B/J\left(C\right)}\right)=1$.

□

Remark 8.26.  

The smooth formula associated with $f:J\left(C\right)\to A$ is contravariant at the level of transport, whereas the Gysin formula associated with $j:B\hookrightarrow J\left(C\right)$ has its own refined pullback variance. They are not combined into an artificial covariant route.

8.6. Direct, Factored, and Scalar Transport

Two mathematically different comparisons occur in the Jacobian construction.

First, the factored route

$${\mathcal{R}}_J:A\stackrel{f^{\vee }}{\to }J\left(C\right)\stackrel{f}{\to }A$$

and the direct route

$${\mathcal{R}}_q:A\stackrel{{\left[q\right]}_A}{\to }A$$

have the same endpoint morphism.

Second, the degree-zero correspondences ${\Gamma }_{{\left[q\right]}_A}$ and $q^2{\Delta }_A$ have the same action only after restriction to the selected Hodge sector $L_A$. The first comparison is therefore an endpoint-functoriality comparison, whereas the second is a genuine selected-sector residual comparison.

Proposition 8.27  

(Endpoint rigidity of the direct and factored routes). Every invariant depending only on the endpoint morphism assigns the same value to ${\mathcal{R}}_J$ and ${\mathcal{R}}_q$.

Proof. 

Both routes have endpoint morphism

$$f\circ f^{\vee }={\left[q\right]}_A.$$

□

Corollary 8.28  

(Collapse of standard defect transport). For $\Phi \in \{\tau ,c_{*},T_{y*}\}$,

$${\mathfrak{D}}_{\Phi }({\mathcal{R}}_J;\xi )={\mathfrak{D}}_{\Phi }({\mathcal{R}}_q;\xi )=0$$

for every admissible source class ξ.

Proof. 

Apply Proposition 8.21. □

Remark 8.29.  

Any invariant that distinguishes the factored route ${\mathcal{R}}_J$ from the direct route ${\mathcal{R}}_q$ must retain information not annihilated by ordinary proper functoriality. The mere existence of the intermediate Jacobian does not force a nonzero proper transport defect.

The selected-sector comparison is different and does produce a nonzero residual.

Theorem 8.30  

(Selected-sector scalar separation). Assume $q>1$. Then

$${\Gamma }_f\circ {\Gamma }_{f^{\vee }}{\equiv }_{L_A}{q}^2{\Delta }_A.$$

However, the two correspondences have distinct actions on total rational cohomology. Consequently,

$${\Gamma }_f\circ {\Gamma }_{f^{\vee }}\ne q^2{\Delta }_A$$

in $C{H}^g{(A\times A)}_{\mathbb{Q}}$.

Proof. 

Proposition 8.4 gives

$${\Gamma }_f\circ {\Gamma }_{f^{\vee }}={\Gamma }_{{\left[q\right]}_A}.$$

Proposition 8.13 gives

$${\Gamma }_{{\left[q\right]}_A}{\equiv }_{L_A}{q}^2{\Delta }_A.$$

On the other hand, Proposition 8.14 shows that

$${\mathfrak{D}}_q={\Gamma }_{{\left[q\right]}_A}-q^2{\Delta }_A$$

acts nontrivially on total rational cohomology. Hence its rational cohomological cycle class is nonzero, so ${\mathfrak{D}}_q\ne 0$ in $C{H}^g{(A\times A)}_{\mathbb{Q}}$. □

Remark 8.31.  

The equality

$$f\circ f^{\vee }={\left[q\right]}_A$$

is equality of endpoint morphisms. By contrast,

$${\left[q\right]}_{A,*}{|}_{L_A}=q^2{id}_{L_A}$$

is an equality only after restricting the cohomological action of ${\left[q\right]}_A$ to $L_A$. The latter comparison leaves the nonzero residual

$${\mathfrak{D}}_q={\Gamma }_{{\left[q\right]}_A}-q^2{\Delta }_A.$$

Thus equality on the selected Hodge sector does not imply equality of total rational cohomological actions and therefore does not imply equality of the underlying Chow correspondences.

The residual formalism of Section 7 applies unconditionally at the ordinary cohomological baseline.

Theorem 8.32  

(Cycle-class detection of the Jacobian residual). Assume $q>1$. Then

$${\mathfrak{D}}_q\in ker\left({\rho }_{L_A}\right)\setminus ker\left({\chi }_{\mathrm{cl},A}\right),$$

where

$${\chi }_{\mathrm{cl},A}={cl}_{A\times A}.$$

Consequently, selected-sector realization identifies ${\Gamma }_{{\left[q\right]}_A}$ and $q^2{\Delta }_A$, whereas the ordinary rational cycle-class realization separates them.

Proof. 

Proposition 8.13 gives ${\mathfrak{D}}_q\in ker\left({\rho }_{L_A}\right)$, while Proposition 8.14 gives ${cl}_{A\times A}\left({\mathfrak{D}}_q\right)\ne 0$. □

Combining the preceding theorem with the finite-cover construction gives a concrete family of selected-sector residuals.

Corollary 8.33  

(Finite-cover residual separation). Let $\pi :C\to D$ be a finite morphism of smooth connected projective curves of degree $n>1$, with $g\left(D\right)\ge 2$. Set

$$A:=J\left(D\right),f:={Nm}_{\pi },f^{\vee }:={\pi }^{*}.$$

Then

$$f\circ f^{\vee }={\left[n\right]}_A.$$

Writing

$${\mathfrak{D}}_n:={\Gamma }_{{\left[n\right]}_A}-n^2{\Delta }_A,$$

one has

$${\mathfrak{D}}_n\in ker\left({\rho }_{L_A}\right)\setminus ker\left({\chi }_{\mathrm{cl},A}\right).$$

Proof. 

Proposition 8.6 gives a Jacobian return datum with $q=n$, and Theorem 8.32 applies. □

Finer characteristic detection is logically separate from this cohomological baseline.

Proposition 8.34.  

Let ${\chi }_{\Phi ,A}$ be an admissible characteristic residual realization in the sense of Definition 7.10, defined on ${\mathfrak{D}}_q$. If

$${\chi }_{\Phi ,A}\left({\mathfrak{D}}_q\right)\ne 0,$$

then ${\Gamma }_{{\left[q\right]}_A}$ and $q^2{\Delta }_A$ remain separated by the Φ-characteristic residual realization even though

$${\Gamma }_{{\left[q\right]}_A}{\equiv }_{L_A}{q}^2{\Delta }_A.$$

Proof. 

Their difference is ${\mathfrak{D}}_q$. By Proposition 8.13, this residual acts trivially on $L_A$, whereas it has nonzero ${\chi }_{\Phi ,A}$-realization by hypothesis. □

Remark 8.35.  

The Jacobian calculation has two distinct outcomes.

First, for $q>1$,

$${\mathfrak{D}}_q\in ker\left({\rho }_{L_A}\right)\setminus ker\left({\chi }_{\mathrm{cl},A}\right),$$

so the residual framework of Section 7 is instantiated unconditionally by the ordinary rational cycle-class realization.

Second,

$${\mathfrak{D}}_q\notin ker\left({\chi }_{\mathrm{cl},A}\right).$$

The example therefore does not enter the homologically trivial kernel. Whether a characteristic, bivariant, Abel–Jacobi-type, or other finer realization detects a nonzero residual in $ker\left({\chi }_{\mathrm{cl},A}\right)$ is a stronger problem not resolved here.

Remark 8.36.  

The Jacobian construction serves simultaneously as a calibration and a separation example. Exact proper characteristic transport identifies what the standard reduced defect profiles fail to see: the intermediate Jacobian. The residual

$${\mathfrak{D}}_q={\Gamma }_{{\left[q\right]}_A}-q^2{\Delta }_A$$

then identifies what restriction to the selected Hodge sector $L_A$ forgets: information that remains visible to total rational cohomology and ordinary cycle class.

The example therefore does not show that characteristic transport is finer than total rational cohomology. Rather, it establishes the cohomological baseline from which any genuinely finer residual theory must proceed.

9. Singular Composite Constructions

This section begins with an independently identified singular characteristic defect and studies its behavior under subsequent geometric operations. The one-stage input is Schürmann’s Verdier–Riemann–Roch formula, which expresses the failure of corrected Chern–Schwartz–MacPherson Gysin naturality through Verdier specialization and generalized vanishing cycles [6]. The new composite problem begins after that defect has been identified: we determine its image and control its support under later proper, smooth, and compatible Gysin stages.

The final subsection gives a complete calculation for a nodal plane cubic. There a local vanishing-cycle contribution produces a nonzero point-supported CSM defect, and that defect remains nonzero after proper propagation to a point.

Throughout the section, let $i:X\hookrightarrow Y$ be a regular embedding, let $N:=N_{X/Y}$ be its normal bundle, let $\pi :N\to X$ be the bundle projection, let $k:X\hookrightarrow N$ be the zero section, and let $\alpha \in CF\left(Y\right)$.

9.1. Regular Embeddings and the Schürmann Defect

Recall from Definition 4.8 that the generalized vanishing-cycle transformation associated with i is

$${\Phi }_i={Sp}_{X\setminus Y}-{\pi }_{CF}^{*}\circ i_{CF}^{*}.$$

Here ${Sp}_{X\setminus Y}$ is Verdier specialization to the normal bundle, $i_{CF}^{*}:CF\left(Y\right)\to CF\left(X\right)$ is restriction of constructible functions, and ${\pi }_{CF}^{*}:CF\left(X\right)\to {CF}_{\mathrm{mon}}\left(N\right)$ is pullback along the bundle projection.

Let $i_H^{!}$ denote the refined Gysin morphism on the homology theory used in Schürmann’s formula, and define the corrected CSM Gysin operator by

$$i_{c_{*}}^{\mathrm{RR},!}\left(\eta \right):=c^{*}{\left(N\right)}^{-1}\cap i_H^{!}\left(\eta \right).$$

With the sign convention of Definition 4.3, the corresponding transport defect is

$${\Delta }_{c_{*}}^{!}(i;\alpha )=c_{*}\left(i_{CF}^{*}\alpha \right)-c^{*}{\left(N\right)}^{-1}\cap i_H^{!}{c}_{*}\left(\alpha \right).$$

The specialization framework originates in Verdier’s specialization theorem for Chern classes and is closely related to hypersurface formulas developed by Aluffi and Parusiński–Pragacz [17,18,19]. We use Schürmann’s Verdier–Riemann–Roch identity as the precise one-stage input.

Definition 9.1  

(Schürmann signed defect). Schürmann’s signed Verdier–Riemann–Roch difference is

$${\mathfrak{S}}_i\left(\alpha \right):=c^{*}{\left(N\right)}^{-1}\cap i_H^{!}{c}_{*}\left(\alpha \right)-c_{*}\left(i_{CF}^{*}\alpha \right).$$

Thus

$${\Delta }_{c_{*}}^{!}(i;\alpha )=-{\mathfrak{S}}_i\left(\alpha \right).$$

Proposition 9.2  

(Schürmann). For every $\alpha \in CF\left(Y\right)$,

$${\mathfrak{S}}_i\left(\alpha \right)=c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right).$$

Equivalently,

$${\Delta }_{c_{*}}^{!}(i;\alpha )=-c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right).$$

Proof. 

This is Lemma 4.9, which records Schürmann’s generalized Verdier–Riemann–Roch defect formula [6]. □

When i is a hypersurface embedding, the generalized specialization term can be expressed through the corresponding hypersurface vanishing-cycle function.

Definition 9.3  

(Hypersurface vanishing-cycle function). Assume ${codim}_{Y}X=1$. Let ${\mu }_i\left(\alpha \right)\in CF\left(X\right)$ denote the hypersurface vanishing-cycle function in Schürmann’s convention, characterized by

$${\Phi }_i\left(\alpha \right)=\left({\pi }_{CF}^{*}-k_{CF,*}\right){\mu }_i\left(\alpha \right),$$

where $k_{CF,*}:CF\left(X\right)\to {CF}_{\mathrm{mon}}\left(N\right)$ is proper pushforward along the zero section.

Corollary 9.4  

(Hypersurface defect formula). For a regular hypersurface embedding,

$${\mathfrak{S}}_i\left(\alpha \right)=c^{*}{\left(N\right)}^{-1}\cap c_{*}\left({\mu }_i\left(\alpha \right)\right),$$

and therefore

$${\Delta }_{c_{*}}^{!}(i;\alpha )=-c^{*}{\left(N\right)}^{-1}\cap c_{*}\left({\mu }_i\left(\alpha \right)\right).$$

Proof. 

This is the hypersurface specialization of Proposition 9.2 [6]. □

The vanishing-cycle support controls the support of the corresponding CSM defect.

Definition 9.5  

(Support of the singular defect). Let $S\subseteq X$ be closed, with inclusion $j_S:S\hookrightarrow X$. The regular-embedding defect ${\Delta }_{c_{*}}^{!}(i;\alpha )$ issupported on Sif

$${\Delta }_{c_{*}}^{!}(i;\alpha )\in im\left({\left(j_S\right)}_{*}:H_{*}^{\mathrm{BM}}(S,\mathbb{Z})⟶H_{*}^{\mathrm{BM}}(X,\mathbb{Z})\right).$$

This is the Borel–Moore-homology specialization of the certified-support notion in Definition 1.10.

Proposition 9.6.  

Let $S\subseteq X$ be closed. If

$$supp{\Phi }_i\left(\alpha \right)\subseteq {\pi }^{-1}\left(S\right),$$

then

$${\Delta }_{c_{*}}^{!}(i;\alpha )$$

is supported on S.

Proof. 

The constructible function ${\Phi }_i\left(\alpha \right)$ is supported on ${\pi }^{-1}\left(S\right)$, so proper naturality of $c_{*}$ expresses $c_{*}\left({\Phi }_i\left(\alpha \right)\right)$ as a homology class supported on ${\pi }^{-1}\left(S\right)$. The zero-section Gysin map $k_H^{!}$ restricts this support to

$$X{\times }_N{\pi }^{-1}\left(S\right)=S.$$

Finally, capping with $c^{*}{\left(N\right)}^{-1}$ does not enlarge support by Lemma 5.11. The claim follows from Proposition 9.2. □

Remark 9.7.  

Proposition 9.2 and Corollary 9.4 are inherited one-stage results. The composite problem begins only after ${\Delta }_{c_{*}}^{!}(i;\alpha )$ has been independently identified. The remaining question is how this class and its certified support behave under subsequent admissible transport.

9.2. Propagation Through a Second Geometric Stage

Because the first stage is Gysin-type, its transport direction is $Y\to X$, opposite to the direction of the embedding $i:X\hookrightarrow Y$. Consequently, the constructions below are typed operator routes rather than ordinary composites of forward geometric morphisms. For example, if $g:X\to Z$ is proper, the characteristic transport is

$$g_{*}\circ i_{c_{*}}^{\mathrm{RR},!},$$

not an artificial morphism composite $g\circ i$.

9.2.1. Proper Propagation

Let $g:X\to Z$ be proper.

Definition 9.8  

(Properly propagated singular defect).

Define

$${\Delta }_{c_{*}}^{g\leftarrow i}\left(\alpha \right):=g_{*}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right)\in H_{*}^{\mathrm{BM}}(Z,\mathbb{Z}).$$

Proposition 9.9  

(Proper propagation).

If

$$supp{\Phi }_i\left(\alpha \right)\subseteq {\pi }^{-1}\left(S\right)$$

for a closed subset $S\subseteq X$, then

$${\Delta }_{c_{*}}^{g\leftarrow i}\left(\alpha \right)=-g_{*}\left(c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right)\right),$$

and the propagated class is supported on the closed subset $g\left(S\right)\subseteq Z$.

Proof. 

The displayed identity follows by applying $g_{*}$ to Proposition 9.2. By Proposition 9.6, the first-stage defect is supported on S, and proper support propagation in Theorem 5.12 gives support on $g\left(S\right)$. Since g is proper, $g\left(S\right)$ is closed. □

For a hypersurface embedding, Corollary 9.4 gives the specialized formula

$${\Delta }_{c_{*}}^{g\leftarrow i}\left(\alpha \right)=-g_{*}\left(c^{*}{\left(N\right)}^{-1}\cap c_{*}\left({\mu }_i\left(\alpha \right)\right)\right).$$

9.2.2. Smooth Propagation

Let $p:X^{\prime }\to X$ be smooth of pure relative complex dimension r, and let $T_p:=T_{X^{\prime }/X}$ be its relative tangent bundle. Denote smooth Borel–Moore pullback by

$$p_{\mathrm{BM}}^{*}:H_k^{\mathrm{BM}}(X,\mathbb{Z})⟶H_{k+2r}^{\mathrm{BM}}(X^{\prime },\mathbb{Z}).$$

Smooth Verdier–Riemann–Roch for CSM classes uses the corrected target operator

$$p_{c_{*}}^{\mathrm{RR},*}\left(\eta \right):=c^{*}\left(T_p\right)\cap p_{\mathrm{BM}}^{*}\left(\eta \right)$$

[6].

Definition 9.10  

(Smoothly propagated singular defect).

Define

$${\Delta }_{c_{*}}^{p\leftarrow i}\left(\alpha \right):=p_{c_{*}}^{\mathrm{RR},*}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right).$$

Proposition 9.11  

(Smooth propagation).

One has

$${\Delta }_{c_{*}}^{p\leftarrow i}\left(\alpha \right)=-c^{*}\left(T_p\right)\cap p_{\mathrm{BM}}^{*}\left(c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right)\right).$$

If the initial defect is supported on a closed subset $S\subseteq X$, then the propagated defect is supported on

$$p^{-1}\left(S\right)\subseteq X^{\prime }.$$

Proof. 

Apply the corrected smooth operator $p_{c_{*}}^{\mathrm{RR},*}$ to the identity of Proposition 9.2. The support statement follows from the smooth case of Theorem 5.12; the additional factor $c^{*}\left(T_p\right)$ does not enlarge support by Lemma 5.11.

□

9.2.3. Propagation Through a Second Gysin Stage

Let $j:W\hookrightarrow X$ be a regular embedding with normal bundle $M:=N_{W/X}$, and assume the refined Gysin morphism required below is defined. Set

$$j_{c_{*}}^{\mathrm{RR},!}\left(\eta \right):=c^{*}{\left(M\right)}^{-1}\cap j_H^{!}\left(\eta \right).$$

Proposition 9.12  

(Gysin propagation under exact second-stage transport).

Assume the second Gysin stage is corrected exact on $i_{CF}^{*}\alpha$, namely

$${\Delta }_{c_{*}}^{!}\left(j;i_{CF}^{*}\alpha \right)=0.$$

Then

$$\begin{array}{cc}& c_{*}\left(j_{CF}^{*}{i}_{CF}^{*}\alpha \right)-j_{c_{*}}^{\mathrm{RR},!}{i}_{c_{*}}^{\mathrm{RR},!}{c}_{*}\left(\alpha \right)\hfill \\ & =j_{c_{*}}^{\mathrm{RR},!}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right).\hfill \end{array}$$

Proof. 

Corrected exactness of the j-stage gives

$$c_{*}\left(j_{CF}^{*}{i}_{CF}^{*}\alpha \right)=j_{c_{*}}^{\mathrm{RR},!}{c}_{*}\left(i_{CF}^{*}\alpha \right).$$

By Definition 4.1,

$$c_{*}\left(i_{CF}^{*}\alpha \right)=i_{c_{*}}^{\mathrm{RR},!}{c}_{*}\left(\alpha \right)+{\Delta }_{c_{*}}^{!}(i;\alpha ).$$

Substituting and using additivity of $j_{c_{*}}^{\mathrm{RR},!}$ gives the result.

□

Corollary 9.13.  

Assume the refined Gysin base-change hypotheses of Theorem 5.12. If ${\Delta }_{c_{*}}^{!}(i;\alpha )$ is supported on $S\subseteq X$, then

$$j_{c_{*}}^{\mathrm{RR},!}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right)$$

is supported on

$$W{\times }_{X}S.$$

Proof. 

The refined Gysin pullback is supported on $W{\times }_{X}S$ by Theorem 5.12. Capping with $c^{*}{\left(M\right)}^{-1}$ does not enlarge that support by Lemma 5.11.

□

9.3. A Composite Vanishing-Cycle Transport Formula

The preceding cases have a common interpretation. The first-stage defect does not acquire a new geometric meaning after composition: its origin remains Schürmann’s specialization class. The composite statement records the image of this already-identified class under a later transport operator and controls the support on which that image can survive.

Theorem 9.14  

(Composite vanishing-cycle transport). Assume

$$supp{\Phi }_i\left(\alpha \right)\subseteq {\pi }^{-1}\left(S\right)$$

for a closed subset $S\subseteq X$. Then:

1.

if $g:X\to Z$ is proper, then

$$g_{*}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right)=-g_{*}\left(c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right)\right),$$

and this class is supported on $g\left(S\right)$;

2.

if $p:X^{\prime }\to X$ is smooth, then

$$p_{c_{*}}^{\mathrm{RR},*}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right)=-p_{c_{*}}^{\mathrm{RR},*}\left(c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right)\right),$$

and this class is supported on $p^{-1}\left(S\right)$;

3.

if $j:W\hookrightarrow X$ is a regular embedding for which the second CSM Gysin stage is corrected exact and the required refined base-change identities hold, then

$$j_{c_{*}}^{\mathrm{RR},!}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right)=-j_{c_{*}}^{\mathrm{RR},!}\left(c^{*}{\left(N\right)}^{-1}\cap k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right)\right),$$

and this class is supported on $W{\times }_{X}S$.

Proof. 

The common first-stage formula is Proposition 9.2. The proper and smooth claims follow from Propositions 9.9 and 9.11, respectively. The Gysin claim follows from Proposition 9.12 and Corollary 9.13.

□

Corollary 9.15  

(Hypersurface composite formula). Assume ${codim}_{Y}X=1$. Then each formula in Theorem 9.14 is obtained by replacing

$$k_H^{!}\left(c_{*}\left({\Phi }_i\left(\alpha \right)\right)\right)$$

with

$$c_{*}\left({\mu }_i\left(\alpha \right)\right).$$

Equivalently, if $\mathcal{T}$ denotes any one of the three applicable corrected propagation operators

$$g_{*},p_{c_{*}}^{\mathrm{RR},*},j_{c_{*}}^{\mathrm{RR},!},$$

then

$$\mathcal{T}\left({\Delta }_{c_{*}}^{!}(i;\alpha )\right)=-\mathcal{T}\left(c^{*}{\left(N\right)}^{-1}\cap c_{*}\left({\mu }_i\left(\alpha \right)\right)\right).$$

Proof. 

Apply Corollary 9.4 before applying the second-stage transport operator.

□

Remark 9.16.  

Schürmann’s theorem identifies the one-stage singular defect. Theorem 9.14 does not alter that identification; it determines how the already identified class propagates and where its image can be supported after a second admissible stage.

This support information interfaces directly with the separation criteria of Section 6. If a propagated singular contribution has nonzero image in a support quotient ${Gr}_{S^{\prime }/T}\mathcal{H}$, while every contribution from a comparison route maps into the lower-support subgroup, then Theorem 6.29 applies. The existence of the comparison route and its selected-sector equivalence are additional hypotheses: the composite vanishing-cycle theorem supplies the geometric defect data but does not manufacture a second cohomologically equivalent route.

9.4. Worked Singular Example

We now carry out the preceding mechanism for an isolated singularity.

Let $Y:={\mathbf{P}}_{\mathbb{C}}^2$, and let $X\subset Y$ be an irreducible plane cubic with a unique ordinary double point p. The inclusion $i:X\hookrightarrow Y$ is a regular hypersurface embedding with

$$N_{X/Y}\simeq {\mathcal{O}}_X\left(3\right).$$

Take $\alpha :={\mathbf{1}}_Y\in CF\left(Y\right)$.

Lemma 9.17.  

For the nodal cubic,

$${\mu }_i\left({\mathbf{1}}_Y\right)=-{\mathbf{1}}_{\left\{p\right\}}.$$

Proof. 

The node is an $A_1$ plane-curve hypersurface singularity with Milnor number 1. Its Milnor fiber $F_p$ has the homotopy type of a circle, so $\chi \left(F_p\right)=0$ [26]. In Schürmann’s hypersurface convention,

$${\mu }_i\left({\mathbf{1}}_Y\right)\left(p\right)=\chi \left(F_p\right)-1=-1,$$

while the vanishing-cycle function is zero on the smooth locus of X [6]. Hence ${\mu }_i\left({\mathbf{1}}_Y\right)=-{\mathbf{1}}_{\left\{p\right\}}$.

□

Proposition 9.18  

(Initial nodal defect). The nodal cubic has CSM Gysin defect

$${\Delta }_{c_{*}}^{!}(i;{\mathbf{1}}_Y)=\left[p\right]\in H_0^{\mathrm{BM}}(X,\mathbb{Z}).$$

Equivalently,

$${\mathfrak{S}}_i\left({\mathbf{1}}_Y\right)=-\left[p\right].$$

Proof. 

By Corollary 9.4,

$${\Delta }_{c_{*}}^{!}(i;{\mathbf{1}}_Y)=-c^{*}{\left(N_{X/Y}\right)}^{-1}\cap c_{*}\left({\mu }_i\left({\mathbf{1}}_Y\right)\right).$$

Lemma 9.17 gives

$$c_{*}\left({\mu }_i\left({\mathbf{1}}_Y\right)\right)=-c_{*}\left({\mathbf{1}}_{\left\{p\right\}}\right)=-\left[p\right].$$

Since $\left[p\right]$ is zero-dimensional, every positive-degree component of $c^{*}{\left(N_{X/Y}\right)}^{-1}$ acts trivially on it. Therefore

$$c^{*}{\left(N_{X/Y}\right)}^{-1}\cap \left[p\right]=\left[p\right],$$

and hence

$${\Delta }_{c_{*}}^{!}(i;{\mathbf{1}}_Y)=\left[p\right].$$

The identity for ${\mathfrak{S}}_i$ follows from Definition 9.1.

□

The same defect can be expressed as the difference between the functorial CSM class and the virtual Fulton–Johnson class.

Definition 9.19  

(Fulton–Johnson class of the nodal cubic). Define

$$c^{\mathrm{FJ}}\left(X\right):=c^{*}\left(T_{{\mathbf{P}}^2}{|}_X\right)c^{*}{\left(N_{X/{\mathbf{P}}^2}\right)}^{-1}\cap \left[X\right].$$

Equivalently,

$$c^{\mathrm{FJ}}\left(X\right)=c^{*}\left(T_X^{\mathrm{vir}}\right)\cap \left[X\right],T_X^{\mathrm{vir}}:=T_{{\mathbf{P}}^2}{|}_X-N_{X/{\mathbf{P}}^2}$$

in $K^0\left(X\right)$.

Proposition 9.20.  

For the nodal cubic,

$$c_{*}\left({\mathbf{1}}_X\right)-c^{\mathrm{FJ}}\left(X\right)=\left[p\right].$$

Proof. 

Because $Y={\mathbf{P}}^2$ is smooth,

$$c_{*}\left({\mathbf{1}}_Y\right)=c^{*}\left(T_{{\mathbf{P}}^2}\right)\cap \left[{\mathbf{P}}^2\right]$$

by MacPherson normalization [15]. Refined Gysin pullback gives

$$i_H^{!}{c}_{*}\left({\mathbf{1}}_Y\right)=c^{*}\left(T_{{\mathbf{P}}^2}{|}_X\right)\cap \left[X\right].$$

Hence the corrected Gysin term is

$$c^{*}{\left(N_{X/{\mathbf{P}}^2}\right)}^{-1}\cap i_H^{!}{c}_{*}\left({\mathbf{1}}_Y\right)=c^{\mathrm{FJ}}\left(X\right).$$

By the definition of the CSM Gysin defect,

$${\Delta }_{c_{*}}^{!}(i;{\mathbf{1}}_Y)=c_{*}\left({\mathbf{1}}_X\right)-c^{\mathrm{FJ}}\left(X\right).$$

Proposition 9.18 now yields the result [3,6].

□

Let $g:X\to Spec\mathbb{C}$ be the structure morphism.

Proposition 9.21  

(Propagation of the nodal defect). The point-supported defect remains nonzero after proper propagation:

$$g_{*}\left({\Delta }_{c_{*}}^{!}(i;{\mathbf{1}}_Y)\right)=\left[\mathrm{pt}\right].$$

Proof. 

By Proposition 9.18, ${\Delta }_{c_{*}}^{!}(i;{\mathbf{1}}_Y)=\left[p\right]$. Proper pushforward of the fundamental class of the closed point gives

$$g_{*}\left[p\right]=\left[\mathrm{pt}\right].$$

□

The same result admits an independent check using the global characteristic numbers of the singular cubic.

Lemma 9.22.  

The nodal cubic satisfies

$$\chi \left(X\right)=1.$$

Proof. 

The normalization $\nu :{\mathbf{P}}^1\to X$ identifies precisely two distinct points of ${\mathbf{P}}^1$ to form the node. Therefore

$$\chi \left(X\right)=\chi \left({\mathbf{P}}^1\right)-1=2-1=1.$$

□

Lemma 9.23.  

The zero-dimensional component of $c^{\mathrm{FJ}}\left(X\right)$ has degree zero.

Proof. 

For the cubic hypersurface $X\subset {\mathbf{P}}^2$,

$$T_X^{\mathrm{vir}}=T_{{\mathbf{P}}^2}{|}_X-{\mathcal{O}}_X\left(3\right).$$

If H denotes the hyperplane class, then

$$c_1\left(T_{{\mathbf{P}}^2}{|}_X\right){=3H|}_X=c_1\left({\mathcal{O}}_X\left(3\right)\right).$$

Hence

$$c_1\left(T_X^{\mathrm{vir}}\right)=0.$$

Since X is one-dimensional,

$$c^{\mathrm{FJ}}\left(X\right)=\left[X\right]+c_1\left(T_X^{\mathrm{vir}}\right)\cap \left[X\right],$$

so its zero-dimensional component is zero [3]. □

Theorem 9.24  

(Complete nodal-cubic transport calculation). For the source-side operator route

$$CF\left({\mathbf{P}}^2\right)\stackrel{i_{CF}^{*}}{\to }CF\left(X\right)\stackrel{g_{*}}{\to }CF(Spec\mathbb{C}),$$

one has

$${\mu }_i\left({\mathbf{1}}_{{\mathbf{P}}^2}\right)=-{\mathbf{1}}_{\left\{p\right\}},$$

$${\Delta }_{c_{*}}^{!}\left(i;{\mathbf{1}}_{{\mathbf{P}}^2}\right)=\left[p\right],$$

and

$$g_{*}{\Delta }_{c_{*}}^{!}\left(i;{\mathbf{1}}_{{\mathbf{P}}^2}\right)=\left[\mathrm{pt}\right].$$

Moreover,

$$g_{*}{c}_{*}\left({\mathbf{1}}_X\right)=\left[\mathrm{pt}\right]$$

and

$$g_{*}{c}^{\mathrm{FJ}}\left(X\right)=0,$$

so

$$g_{*}\left(c_{*}\left({\mathbf{1}}_X\right)-c^{\mathrm{FJ}}\left(X\right)\right)=\left[\mathrm{pt}\right].$$

Proof. 

The vanishing-cycle identity is Lemma 9.17. The first-stage defect and its proper propagation are Propositions 9.18 and 9.21.

For the independent global calculation, MacPherson naturality for the proper map g gives

$$g_{*}{c}_{*}\left({\mathbf{1}}_X\right)=c_{*}\left(g_{*}{\mathbf{1}}_X\right).$$

Since

$$g_{*}{\mathbf{1}}_X=\chi \left(X\right){\mathbf{1}}_{\mathrm{pt}},$$

MacPherson normalization and Lemma 9.22 yield

$$g_{*}{c}_{*}\left({\mathbf{1}}_X\right)=\chi \left(X\right)\left[\mathrm{pt}\right]=\left[\mathrm{pt}\right]$$

[15].

By Lemma 9.23, the zero-dimensional component of $c^{\mathrm{FJ}}\left(X\right)$ has degree zero. The one-dimensional component pushes forward to zero in Borel–Moore homology of a point, so

$$g_{*}{c}^{\mathrm{FJ}}\left(X\right)=0.$$

Subtracting gives

$$g_{*}\left(c_{*}\left({\mathbf{1}}_X\right)-c^{\mathrm{FJ}}\left(X\right)\right)=\left[\mathrm{pt}\right],$$

in agreement with the transported local calculation. □

Remark 9.25.  

The nodal cubic realizes the singular transport mechanism through the explicit chain

$${\mu }_i\left({\mathbf{1}}_Y\right)=-{\mathbf{1}}_{\left\{p\right\}}⟹{\Delta }_{c_{*}}^{!}(i;{\mathbf{1}}_Y)=\left[p\right]⟹g_{*}\left[p\right]=\left[\mathrm{pt}\right].$$

Thus a local singular contribution identified by vanishing cycles survives as a nonzero global characteristic discrepancy after a second geometric stage.

Remark 9.26.  

The class $\left[\mathrm{pt}\right]$ is not determined by the formal two-stage composition identity alone. Its value depends on the local $A_1$ singularity, the Milnor-fiber calculation, and Schürmann’s geometric identification of the first-stage defect. The example therefore provides a concrete realization of the support-propagation mechanism underlying the transport-profile formalism.

It does not, by itself, furnish two routes having the same selected cohomological realization and different characteristic profiles. Such a separation statement requires an independently constructed comparison route satisfying the relevant selected-sector equivalence.

10. Smooth and Gysin Transport

Smooth pullback and normally nonsingular Gysin pullback provide two complementary contravariant directions for corrected characteristic transport. Smooth transport is corrected by relative tangent data, whereas Gysin transport is corrected by normal-bundle data. The principal result of this section identifies the corresponding corrected operators across a Cartesian smooth–Gysin square.

The smooth input is the class-level intersection-Hirzebruch transfer theorem: if $\pi :X\to Y$ is a smooth algebraic morphism of pure relative complex dimension between pure-dimensional complex algebraic varieties, then the intersection-Hirzebruch class transforms by the relative tangent correction [13]. Neither X nor Y is required to be nonsingular. On the Gysin side, exactness is invoked only under the hypotheses of Banagl’s algebraic $I{T}_{1,*}$-Gysin theorem [10].

The two one-stage formulas are inherited. The new comparison developed here is their compatibility in a Cartesian square once refined base change and the corresponding tangent- and normal-bundle identities are imposed.

10.1. Smooth Transport

Let $\pi :X\to Y$ be smooth of pure relative complex dimension r, and write $T_{\pi }:=T_{X/Y}$ for its relative tangent bundle. Let

$${\pi }_{\mathrm{BM}}^{*}:H_k^{\mathrm{BM}}(Y,\mathbb{Q})⟶H_{k+2r}^{\mathrm{BM}}(X,\mathbb{Q})$$

denote smooth Borel–Moore pullback.

Definition 10.1  

(Corrected smooth $I{T}_y$-operator). Define

$${\pi }_{I{T}_y}^{\mathrm{RR},*}\left(\eta \right):=T_y^{*}\left(T_{\pi }\right)\cap {\pi }_{\mathrm{BM}}^{*}\left(\eta \right),$$

for every η in the relevant intersection-Hirzebruch target group on Y.

This is the concrete realization, for smooth $I{T}_y$-transport, of the reference operator ${\mathcal{T}}_{\Phi }\left(u_{\pi }^{\mathrm{sm}}\right)$ introduced in Section 1.

Proposition 10.2  

(Smooth intersection-Hirzebruch transfer). Let $\pi :X\to Y$ be a smooth algebraic morphism of pure relative complex dimension between pure-dimensional complex algebraic varieties. Then

$$I{T}_{y*}\left(X\right)=T_y^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}I{T}_{y*}\left(Y\right)={\pi }_{I{T}_y}^{\mathrm{RR},*}\left(I{T}_{y*}\left(Y\right)\right).$$

Equivalently,

$${\Delta }_{I{T}_y}^{*}\left(\pi \right)=0.$$

Proof. 

This is the class-level smooth transfer theorem for intersection-Hirzebruch classes [13].

□

Remark 10.3.  

Proposition 10.2 is inherited input. Only the class-level transformation law is used here; no object-level statement identifying a smooth pullback of intersection Hodge modules is required.

Now let

$$X\stackrel{\pi }{\to }Y\stackrel{\sigma }{\to }Z$$

be composable smooth algebraic morphisms. Their relative tangent bundles fit into the canonical exact sequence

$$0⟶T_{X/Y}⟶T_{X/Z}⟶{\pi }^{*}{T}_{Y/Z}⟶0.$$

Remark 10.4.  

Multiplicativity and pullback naturality of the cohomological Hirzebruch class give

$$T_y^{*}\left(T_{X/Z}\right)=T_y^{*}\left(T_{X/Y}\right)⌣{\pi }^{*}{T}_y^{*}\left(T_{Y/Z}\right).$$

This is the bundle-theoretic identity underlying composition of the corrected smooth operators [13,16].

Proposition 10.5  

(Composition of smooth corrected transport). For composable smooth morphisms as above,

$${(\sigma \circ \pi )}_{I{T}_y}^{\mathrm{RR},*}={\pi }_{I{T}_y}^{\mathrm{RR},*}\circ {\sigma }_{I{T}_y}^{\mathrm{RR},*}.$$

Proof. 

Let η lie in the relevant target group on Z. Functoriality of smooth Borel–Moore pullback and the projection formula for cap products give

$$\begin{array}{cc}\hfill {\pi }_{I{T}_y}^{\mathrm{RR},*}{\sigma }_{I{T}_y}^{\mathrm{RR},*}\left(\eta \right)& =T_y^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}\left(T_y^{*}\left(T_{Y/Z}\right)\cap {\sigma }_{\mathrm{BM}}^{*}\eta \right)\hfill \\ & =T_y^{*}\left(T_{X/Y}\right)\cap {\pi }^{*}{T}_y^{*}\left(T_{Y/Z}\right)\cap {(\sigma \circ \pi )}_{\mathrm{BM}}^{*}\eta \hfill \\ & =T_y^{*}\left(T_{X/Z}\right)\cap {(\sigma \circ \pi )}_{\mathrm{BM}}^{*}\eta ,\hfill \end{array}$$

where the final equality is Remark 10.4. The right-hand side is ${(\sigma \circ \pi )}_{I{T}_y}^{\mathrm{RR},*}\left(\eta \right)$.

□

Remark 10.6.  

Consequently, exact smooth $I{T}_y$-transport is closed under composition: the individual stage defects vanish, and the corrected reference operators compose strictly.

10.2. Gysin Transport

Let $i:Z\hookrightarrow Y$ be an algebraic normally nonsingular embedding of complex codimension c, let $N_i:=N_{Z/Y}$ be its normal bundle, and let

$$i^{!}:H_k^{\mathrm{BM}}(Y,\mathbb{Q})⟶H_{k-2c}^{\mathrm{BM}}(Z,\mathbb{Q})$$

denote the corresponding Borel–Moore Gysin homomorphism.

Definition 10.7  

(Corrected $I{T}_1$-Gysin operator). Define

$$i_{I{T}_1}^{\mathrm{RR},!}\left(\eta \right):=L^{*}{\left(N_i\right)}^{-1}\cap i^{!}\left(\eta \right).$$

The class $L^{*}\left(N_i\right)$ has degree-zero term 1, so it is invertible in rational cohomology in the finite-dimensional range relevant to the cap product.

Proposition 10.8  

(Banagl). Under the hypotheses of Banagl’s $I{T}_{1,*}$-Gysin theorem,

$$i^{!}I{T}_{1,*}\left(Y\right)=L^{*}\left(N_i\right)\cap I{T}_{1,*}\left(Z\right).$$

Equivalently,

$$i_{I{T}_1}^{\mathrm{RR},!}\left(I{T}_{1,*}\left(Y\right)\right)=I{T}_{1,*}\left(Z\right),$$

and hence

$${\Delta }_{I{T}_1}^{!}\left(i\right)=0.$$

Proof. 

This is Banagl’s normally nonsingular Gysin formula [10].

□

For comparison with the more general notion of Gysin-coherent characteristic systems, let $c{\ell }_{*}\left(Y\right)\in H_{*}^{\mathrm{BM}}(Y,\mathbb{Q})$, and let $c{\ell }^{*}\left(N_i\right)\in H^{*}(Z,\mathbb{Q})$ be an invertible multiplicative characteristic class.

Definition 10.9  

(Gysin identity). The pair $(c{\ell }^{*},c{\ell }_{*})$ satisfies theGysin identity along iif

$$i^{!}c{\ell }_{*}\left(Y\right)=c{\ell }^{*}\left(N_i\right)\cap c{\ell }_{*}\left(Z\right).$$

Remark 10.10.  

Define

$$i_{c\ell }^{\mathrm{RR},!}\left(\eta \right):=c{\ell }^{*}{\left(N_i\right)}^{-1}\cap i^{!}\left(\eta \right).$$

Then the Gysin identity is equivalent to

$$i_{c\ell }^{\mathrm{RR},!}\left(c{\ell }_{*}\left(Y\right)\right)=c{\ell }_{*}\left(Z\right),$$

and therefore to vanishing of the corresponding corrected Gysin transport defect.

Remark 10.11.  

This is only a transport-language reformulation of an established Gysin transformation law. The Gysin-coherence axioms, their functorial structure, and the corresponding uniqueness results are those of [11,12].

10.3. Mixed smooth–Gysin Composition

Let $\pi :X\to Y$ be smooth and $i:Z\hookrightarrow Y$ normally nonsingular. Form the Cartesian square

$$\begin{array}{ccc}W& \stackrel{j}{\to }& X\\ \downarrow {\pi }_Z& & \downarrow \pi \\ Z& \stackrel{i}{\to }& Y,\end{array}W:=X{\times }_{Y}Z.$$

Then πZ : W → Z is smooth. Under the standard stability hypotheses for normally nonsingular embeddings under smooth base change, j : W → X is normally nonsingular.

The two typed transport paths from Y to W are

$$Y\stackrel{u_{\pi }^{\mathrm{sm}}}{\to }X\stackrel{u_j^{!}}{\to }W$$

and

$$Y\stackrel{u_i^{!}}{\to }Z\stackrel{u_{{\pi }_Z}^{\mathrm{sm}}}{\to }W.$$

Thus the comparison is between two operator routes with common source and target, not between ordinary composites of geometric morphisms.

Lemma 10.12.  

There are canonical vector-bundle identifications

$$N_{W/X}\simeq {\pi }_Z^{*}{N}_{Z/Y}$$

and

$$T_{W/Z}\simeq j^{*}{T}_{X/Y}.$$

Consequently,

$$L^{*}\left(N_{W/X}\right)={\pi }_Z^{*}{L}^{*}\left(N_{Z/Y}\right)$$

and

$$L^{*}\left(T_{W/Z}\right)=j^{*}{L}^{*}\left(T_{X/Y}\right).$$

Proof. 

The normal-bundle identity is the standard base-change identification for a regular or normally nonsingular embedding pulled back along a smooth morphism. The relative tangent identity follows from the Cartesian description of the smooth base change. Naturality of the cohomological L-class then gives the characteristic-class identities [3].

□

At $y=1$, the normalized cohomological Hirzebruch class of a complex vector bundle agrees with its Hirzebruch L-class. Accordingly, define

$${\pi }_{I{T}_1}^{\mathrm{RR},*}\left(\eta \right):=L^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}\left(\eta \right),$$

and similarly

$${\left({\pi }_Z\right)}_{I{T}_1}^{\mathrm{RR},*}\left(\zeta \right):=L^{*}\left(T_{W/Z}\right)\cap {\left({\pi }_Z\right)}_{\mathrm{BM}}^{*}\left(\zeta \right).$$

Theorem 10.13  

(Mixed smooth–Gysin compatibility). Assume the refined smooth–Gysin base-change identity

$$j^{!}\circ {\pi }_{\mathrm{BM}}^{*}={\left({\pi }_Z\right)}_{\mathrm{BM}}^{*}\circ i^{!}$$

for the Cartesian square above. Then the corrected $I{T}_1$-operators satisfy

$$j_{I{T}_1}^{\mathrm{RR},!}\circ {\pi }_{I{T}_1}^{\mathrm{RR},*}={\left({\pi }_Z\right)}_{I{T}_1}^{\mathrm{RR},*}\circ i_{I{T}_1}^{\mathrm{RR},!}.$$

Proof. 

Let $\eta \in H_{*}^{\mathrm{BM}}(Y,\mathbb{Q})$. By definition,

$$\begin{array}{cc}\hfill j_{I{T}_1}^{\mathrm{RR},!}{\pi }_{I{T}_1}^{\mathrm{RR},*}\left(\eta \right)& =L^{*}{\left(N_{W/X}\right)}^{-1}\cap j^{!}\left(L^{*}\left(T_{X/Y}\right)\cap {\pi }_{\mathrm{BM}}^{*}\eta \right)\hfill \\ & =L^{*}{\left(N_{W/X}\right)}^{-1}\cap j^{*}{L}^{*}\left(T_{X/Y}\right)\cap j^{!}{\pi }_{\mathrm{BM}}^{*}\eta ,\hfill \end{array}$$

where compatibility of refined Gysin pullback with cap product is used in the second equality.

By the assumed base-change identity and Lemma 10.12,

$$\begin{array}{cc}\hfill j_{I{T}_1}^{\mathrm{RR},!}{\pi }_{I{T}_1}^{\mathrm{RR},*}\left(\eta \right)& ={\pi }_Z^{*}{L}^{*}{\left(N_{Z/Y}\right)}^{-1}\cap L^{*}\left(T_{W/Z}\right)\cap {\left({\pi }_Z\right)}_{\mathrm{BM}}^{*}{i}^{!}\eta \hfill \\ & =L^{*}\left(T_{W/Z}\right)\cap {\left({\pi }_Z\right)}_{\mathrm{BM}}^{*}\left(L^{*}{\left(N_{Z/Y}\right)}^{-1}\cap i^{!}\eta \right)\hfill \\ & ={\left({\pi }_Z\right)}_{I{T}_1}^{\mathrm{RR},*}{i}_{I{T}_1}^{\mathrm{RR},!}\left(\eta \right).\hfill \end{array}$$

The middle equality uses compatibility of smooth pullback with cap product.

□

Corollary 10.14  

(Path independence on $I{T}_{1,*}$). Assume, in addition, that $X,Y,Z,W$ are pure-dimensional, π and ${\pi }_Z$ have pure relative dimension, and i and j satisfy the hypotheses of Banagl’s $I{T}_{1,*}$-Gysin theorem. Then

$$I{T}_{1,*}\left(W\right)=j_{I{T}_1}^{\mathrm{RR},!}{\pi }_{I{T}_1}^{\mathrm{RR},*}\left(I{T}_{1,*}\left(Y\right)\right)$$

and

$$I{T}_{1,*}\left(W\right)={\left({\pi }_Z\right)}_{I{T}_1}^{\mathrm{RR},*}{i}_{I{T}_1}^{\mathrm{RR},!}\left(I{T}_{1,*}\left(Y\right)\right).$$

Hence

$$j_{I{T}_1}^{\mathrm{RR},!}{\pi }_{I{T}_1}^{\mathrm{RR},*}\left(I{T}_{1,*}\left(Y\right)\right)={\left({\pi }_Z\right)}_{I{T}_1}^{\mathrm{RR},*}{i}_{I{T}_1}^{\mathrm{RR},!}\left(I{T}_{1,*}\left(Y\right)\right).$$

Proof. 

Smooth transport along π and ${\pi }_Z$ is exact by Proposition 10.2, specialized to $y=1$. Gysin transport along i and j is exact by Proposition 10.8. Theorem 10.13 identifies the two corrected composite operators.

□

The next statement separates the formal propagation of stage defects from the geometric base-change comparison above.

Proposition 10.15  

(Concentration of mixed transport defect). Let Φ be a characteristic transformation for which the smooth stage $\pi :X\to Y$ and the Gysin stage $j:W\hookrightarrow X$ are admissible. Let $\eta \in \mathcal{A}\left(Y\right)$, and define

$${\delta }_{\pi }:={\Delta }_{\Phi }^{*}(\pi ;\eta )\in \mathcal{H}\left(X\right)$$

and

$${\delta }_j:={\Delta }_{\Phi }^{!}\left(j;{\pi }_{\mathcal{A}}^{*}\left(\eta \right)\right)\in \mathcal{H}\left(W\right),$$

where ${\pi }_{\mathcal{A}}^{*}:\mathcal{A}\left(Y\right)\to \mathcal{A}\left(X\right)$ is the source-side smooth pullback.

For the typed transport route

$$Y\stackrel{u_{\pi }^{\mathrm{sm}}}{\to }X\stackrel{u_j^{!}}{\to }W,$$

the stagewise route defect is

$${\Delta }_{\Phi }({\mathcal{R}}_{\pi ,j};\eta )={\delta }_j+j_{\Phi }^{\mathrm{RR},!}\left({\delta }_{\pi }\right).$$

If a compressed operator for the two-stage route is defined and its reference operator agrees with $j_{\Phi }^{\mathrm{RR},!}\circ {\pi }_{\Phi }^{\mathrm{RR},*}$, then the same expression is the compressed two-stage defect. In particular, if either stage is exact, the total defect is the propagated defect of the other stage.

Proof. 

Apply the two-stage route identity of Corollary 5.4, or equivalently the finite-route propagation identity Proposition 1.7, to the typed route $(u_{\pi }^{\mathrm{sm}},u_j^{!})$.

□

Remark 10.16.  

The individual smooth-transfer and Gysin formulas are inherited from [10,13]. The genuinely mixed statement is Theorem 10.13: the tangent and normal corrections commute across the Cartesian square because the underlying smooth-pullback and refined Gysin operators satisfy base change and because the corresponding tangent and normal bundles are related by pullback.

10.4. Compatibility with Gysin Coherence

The preceding calculation has an abstract analogue for a Gysin-coherent characteristic system. We record only the operator structure needed for characteristic transport.

Let $\mathcal{G}$ be a class of admissible normally nonsingular inclusions equipped with a Gysin-coherent characteristic system $(c{\ell }^{*},c{\ell }_{*})$ in the sense of [11,12].

Definition 10.17  

(Corrected operator for a Gysin-coherent system). For $i:Z\hookrightarrow Y$ in $\mathcal{G}$, with normal bundle $N_i$, define

$$i_{c\ell }^{\mathrm{RR},!}\left(\eta \right):=c{\ell }^{*}{\left(N_i\right)}^{-1}\cap i^{!}\left(\eta \right).$$

Remark 10.18.  

For a fixed admissible inclusion i, the Gysin identity

$$i^{!}c{\ell }_{*}\left(Y\right)=c{\ell }^{*}\left(N_i\right)\cap c{\ell }_{*}\left(Z\right)$$

is equivalent to

$$i_{c\ell }^{\mathrm{RR},!}\left(c{\ell }_{*}\left(Y\right)\right)=c{\ell }_{*}\left(Z\right).$$

Thus a Gysin-coherent stage is corrected exact in the transport terminology.

Remark 10.19.  

The preceding statement introduces no new Gysin-coherence axiom. The coherence conditions and uniqueness results are precisely those established in [11,12].

Remark 10.20.  

Consequently, admissible coherent Gysin stages contribute zero corrected stage defect. Identity insertion, isomorphic replacement, and compatible composition preserve this vanishing by the general transport results of Section 4 and Section 5.

Now suppose that an admissible smooth morphism $\pi :X\to Y$ carries an invertible multiplicative correction $C_{\pi }$ and corrected operator

$${\pi }_{c\ell }^{\mathrm{RR},*}\left(\eta \right):=C_{\pi }\cap {\pi }^{*}\left(\eta \right).$$

Here ${\pi }^{*}$ denotes the smooth pullback in the underlying target theory. Assume that these correction classes are functorial under the smooth base changes under consideration.

Theorem 10.21  

(Compatibility of mixed transport with Gysin coherence). Consider a Cartesian square

$$\begin{array}{ccc}W& \stackrel{j}{\to }& X\\ \downarrow {\pi }_Z& & \downarrow \pi \\ Z& \stackrel{i}{\to }& Y.\end{array}$$

Assume that π and ${\pi }_Z$ are admissible smooth morphisms, $i,j\in \mathcal{G}$, and the underlying target operators satisfy

$$j^{!}\circ {\pi }^{*}={\pi }_Z^{*}\circ i^{!}.$$

Assume further that

$$c{\ell }^{*}\left(N_j\right)={\pi }_Z^{*}c{\ell }^{*}\left(N_i\right)$$

and

$$C_{{\pi }_Z}=j^{*}{C}_{\pi }.$$

Then

$$j_{c\ell }^{\mathrm{RR},!}\circ {\pi }_{c\ell }^{\mathrm{RR},*}={\left({\pi }_Z\right)}_{c\ell }^{\mathrm{RR},*}\circ i_{c\ell }^{\mathrm{RR},!}.$$

If the smooth and Gysin stages are corrected exact on the characteristic classes under consideration, both mixed routes have zero total corrected defect.

Proof. 

Let η lie in the common source target group on Y. Then

$$\begin{array}{cc}\hfill j_{c\ell }^{\mathrm{RR},!}{\pi }_{c\ell }^{\mathrm{RR},*}\left(\eta \right)& =c{\ell }^{*}{\left(N_j\right)}^{-1}\cap j^{!}\left(C_{\pi }\cap {\pi }^{*}\eta \right)\hfill \\ & =c{\ell }^{*}{\left(N_j\right)}^{-1}\cap j^{*}{C}_{\pi }\cap j^{!}{\pi }^{*}\eta \hfill \\ & ={\pi }_Z^{*}c{\ell }^{*}{\left(N_i\right)}^{-1}\cap C_{{\pi }_Z}\cap {\pi }_Z^{*}{i}^{!}\eta \hfill \\ & =C_{{\pi }_Z}\cap {\pi }_Z^{*}\left(c{\ell }^{*}{\left(N_i\right)}^{-1}\cap i^{!}\eta \right)\hfill \\ & ={\left({\pi }_Z\right)}_{c\ell }^{\mathrm{RR},*}{i}_{c\ell }^{\mathrm{RR},!}\left(\eta \right).\hfill \end{array}$$

The second and fourth equalities use compatibility of Gysin and smooth pullback with cap products. The third uses the assumed base-change identity and the two correction-class identities.

If every stage is corrected exact, each stage defect vanishes. The route defect formula of Proposition 10.15 therefore gives zero total defect along either path.

□

Remark 10.22.  

Gysin coherence supplies exact Gysin stages and compatible corrected operators inside a characteristic transport profile. A nonzero profile contribution must therefore arise from a stage outside the relevant exactness hypotheses, from failure of the required mixed compatibility, or from additional geometric information not determined by the coherent Gysin transformation law itself.

Remark 10.23.  

No new uniqueness theorem for Gysin-coherent characteristic classes is claimed in this section. The contribution is the transport-level operator comparison under the stated smooth base-change hypotheses.

11. Comparison Phenomena and Limitations

Characteristic transport profiles retain stagewise information that may be lost after passage to an endpoint defect or to a coarser realization. This additional information is nevertheless neither automatic nor complete. The preceding sections already exhibit two opposite behaviors. In Section 9, an independently identified singular defect survives propagation and remains nonzero at the endpoint. In Section 8, exact proper functoriality forces the standard reduced defect profiles to vanish despite the presence of a nontrivial intermediate Jacobian.

The purpose of this section is to isolate the corresponding limitations: propagated defects may cancel or be annihilated, exact theories may collapse nontrivial factorizations, separation depends on the chosen characteristic family, and the full profile is naturally attached to a factorization class rather than merely to an endpoint operation. We use the composition and profile formalism of Section 5 and Section 6 without repeating its construction.

11.1. Cancellation and Support

Let $\mathcal{R}=(F_1,\dots ,F_r)$ be a finite compatible covariant route, fix ${\xi }_0\in \mathcal{A}\left(X_0\right)$, and let ${\delta }_i^{\Phi }\in \mathcal{H}\left(X_i\right)$ and ${\epsilon }_i^{\Phi }\in \mathcal{H}\left(X_r\right)$ denote its stage defects and terminal propagated contributions. The stagewise route defect is

$${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)={\displaystyle \sum _{i=1}^r}{\epsilon }_i^{\Phi }$$

by Proposition 1.7.

If the route also admits an admissible compressed endpoint operation $F_{\mathcal{R}}:=F_r\circ \dots \circ F_1$ whose reference operator is strictly compatible with the stagewise reference operators, then

$${\Delta }_{\Phi }(F_{\mathcal{R}};{\xi }_0)={\Delta }_{\Phi }(\mathcal{R};{\xi }_0).$$

Thus cancellation should first be regarded as a property of the stagewise route defect; it becomes a statement about the compressed endpoint defect when this compatibility hypothesis is imposed.

Definition 11.1  

(Nontrivial intermediate transport). The route $(\mathcal{R},{\xi }_0)$ hasnontrivial intermediate transportif ${\delta }_i^{\Phi }\ne 0$ for at least one stage i. It hasnontrivial propagated transportif ${\epsilon }_i^{\Phi }\ne 0$ for at least one i.

A nonzero stage defect need not yield a nonzero terminal contribution: a later reference operator may annihilate it. Even when several terminal contributions are nonzero, their sum may vanish.

Definition 11.2  

(Characteristic transport cancellation). The route $(\mathcal{R},{\xi }_0)$ exhibitscharacteristic transport cancellationif

$${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)=0$$

while ${\epsilon }_i^{\Phi }\ne 0$ for at least one i.

When the compressed endpoint operation is strictly compatible with the stagewise reference operators, this is equivalently the condition

$${\Delta }_{\Phi }(F_{\mathcal{R}};{\xi }_0)=0$$

with at least one nonzero terminal contribution.

The two-stage case separates cancellation from annihilation particularly clearly.

Proposition 11.3  

(Cancellation and annihilation). Let

$$X\stackrel{F}{\to }Y\stackrel{G}{\to }Z$$

be a Φ-Riemann–Roch-compatible covariant pair, and let $\xi \in \mathcal{A}\left(X\right)$. Then

$${\Delta }_{\Phi }(G\circ F;\xi )=0$$

if and only if

$${\Delta }_{\Phi }(G;F_{!}\xi )=-G_{\Phi ,*}^{\mathrm{RR}}\left({\Delta }_{\Phi }(F;\xi )\right).$$

If the G-stage is exact on $F_{!}\xi$, then

$${\Delta }_{\Phi }(G\circ F;\xi )=0⟺{\Delta }_{\Phi }(F;\xi )\in ker\left(G_{\Phi ,*}^{\mathrm{RR}}\right).$$

Proof. 

By Corollary 5.4,

$${\Delta }_{\Phi }(G\circ F;\xi )={\Delta }_{\Phi }(G;F_{!}\xi )+G_{\Phi ,*}^{\mathrm{RR}}\left({\Delta }_{\Phi }(F;\xi )\right).$$

The first equivalence follows by rearrangement. If the G-stage is exact, the first summand vanishes, giving the second equivalence.

□

Thus two distinct mechanisms can erase intermediate information. Inannihilation, a later transport operator sends a nonzero earlier defect to zero. Incancellation, two or more nonzero propagated contributions sum to zero.

Support data can obstruct cancellation.

Proposition 11.4  

(Support obstruction to cancellation). Let $T\subseteq S\subseteq X_r$ be closed, and assume every terminal contribution ${\epsilon }_i^{\Phi }$ belongs to ${\mathcal{H}}_S\left(X_r\right)$. Suppose that for some index j,

$$\left[{\epsilon }_j^{\Phi }\right]\ne 0\mathrm{in}{Gr}_{S/T}\mathcal{H}\left(X_r\right),$$

while

$${\epsilon }_i^{\Phi }\in {\mathcal{H}}_T\left(X_r\right)\mathrm{for}\mathrm{every}i\ne j.$$

Then

$${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)\ne 0.$$

If a strictly compatible compressed endpoint operation exists, then also

$${\Delta }_{\Phi }(F_{\mathcal{R}};{\xi }_0)\ne 0.$$

Proof. 

The route defect is

$${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)=\sum _i{\epsilon }_i^{\Phi }.$$

Passing this equality to ${Gr}_{S/T}\mathcal{H}\left(X_r\right)$ kills every summand except $\left[{\epsilon }_j^{\Phi }\right]$. Hence

$$\left[{\Delta }_{\Phi }(\mathcal{R};{\xi }_0)\right]=\left[{\epsilon }_j^{\Phi }\right]\ne 0,$$

so the route defect itself is nonzero. Under strict compatibility, the compressed endpoint defect equals the route defect.

□

Remark 11.5.  

In a fixed support quotient, cancellation therefore requires at least two nonzero graded contributions capable of cancelling one another. A single nonzero graded contribution cannot disappear through summation with terms lying entirely in the lower-support subgroup.

Remark 11.6.  

No general existence theorem for geometric cancellation or annihilation is claimed. The nodal-cubic calculation of Section 9.4 exhibits neither phenomenon: its point-supported defect survives proper propagation as the nonzero class $\left[\mathrm{pt}\right]$.

11.2. Profile Collapse

The opposite extreme occurs when every stage defect vanishes.

Definition 11.7  

(Defect-trivial route). A route $(\mathcal{R},{\xi }_0)$ is$\Phi$-defect-trivialif

$${\delta }_i^{\Phi }=0\mathrm{for}\mathrm{every}\mathrm{stage}i.$$

It isglobally $\Phi$-defect-trivialif this holds for every admissible initial source class.

Proposition 11.8  

(Collapse under exact transport). If $(\mathcal{R},{\xi }_0)$ is Φ-defect-trivial, then every terminal propagated contribution vanishes and hence

$${\mathfrak{D}}_{\Phi }(\mathcal{R};{\xi }_0)=0.$$

Its stagewise route defect also vanishes:

$${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)=0.$$

If, in addition, an admissible compressed endpoint operation exists and its reference operator is strictly compatible with the stagewise reference operators, then

$${\Delta }_{\Phi }(F_{\mathcal{R}};{\xi }_0)=0.$$

In particular, a route consisting entirely of proper morphisms has zero reduced defect profile for

$$\Phi \in \{\tau ,c_{*},T_{y*}\}.$$

Consequently, the direct and factored Jacobian routes of Section 8 satisfy

$${\mathfrak{D}}_{\Phi }({\mathcal{R}}_J;\xi )={\mathfrak{D}}_{\Phi }({\mathcal{R}}_q;\xi )=0$$

for these three characteristic transformations.

Proof. 

If every stage defect vanishes, then ${\epsilon }_i^{\Phi }=P_{i,r}^{\Phi }\left({\delta }_i^{\Phi }\right)=0$ for every i. Hence the reduced profile and the stagewise route defect vanish.

For proper stages, exactness follows from Proposition 4.10, which records proper exact transport for τ, $c_{*}$, and $T_{y*}$. The Jacobian assertion is Proposition 8.21.

□

We use the following terminology for data that have already forgotten all factorization information.

Definition 11.9  

(Endpoint-determined invariant). A route datum $I(\mathcal{R};\xi )$ isendpoint-determined, or equivalently anendpoint invariant, if whenever two admissible routes have the same endpoint operation and the same initial source datum, they have the same value of I.

Remark 11.10.  

Endpoint-equivalent routes necessarily have identical endpoint-determined data. In particular, an invariant depending only on a compressed endpoint morphism or endpoint Chow correspondence cannot distinguish two factorizations of that same endpoint.

Remark 11.11.  

Profile collapse is a limitation of the chosen characteristic realization, not a contradiction in the transport formalism. Exact proper naturality cannot by itself retain the presence of a nontrivial intermediate Jacobian: the corresponding defect components vanish because the relevant transformation laws are exact.

Remark 11.12.  

Suppose a normalized Hirzebruch profile has vanishing stage defects. Whenever a specialization is compatible in the sense of Theorem 6.10, every specialized defect component also vanishes by Corollary 6.14.

11.3. Dependence on the Characteristic Family

A comparative profile depends on the family of characteristic realizations chosen for comparison. Enlarging the family can preserve or increase separating power, but cannot destroy a distinction already visible in a smaller family.

Definition 11.13  

($\Lambda$-profile equivalence). Let Λ be a characteristic comparison family. For routes for which the corresponding comparative profiles are defined, write

$${\mathcal{R}}_1{\sim }_{{\mathfrak{P}}_{\Lambda }}{\mathcal{R}}_2$$

if

$${\mathfrak{P}}_{\Lambda }({\mathcal{R}}_1;{\zeta }_0)={\mathfrak{P}}_{\Lambda }({\mathcal{R}}_2;{\zeta }_0)$$

after the canonical identifications allowed by profile equivalence.

Definition 11.14  

($\Lambda$-separation). The routes ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ are$\Lambda$-separatedif

$${\mathcal{R}}_1\neg {\sim }_{{\mathfrak{P}}_{\Lambda }}{\mathcal{R}}_2.$$

Proposition 11.15  

(Monotonicity under enlargement). Let $\Lambda \subseteq {\Lambda }^{\prime }$, and assume the Λ-profile is obtained from the ${\Lambda }^{\prime }$-profile by forgetting the components indexed by ${\Lambda }^{\prime }\setminus \Lambda$. If

$${\mathcal{R}}_1{\sim }_{{\mathfrak{P}}_{{\Lambda }^{\prime }}}{\mathcal{R}}_2,$$

then

$${\mathcal{R}}_1{\sim }_{{\mathfrak{P}}_{\Lambda }}{\mathcal{R}}_2.$$

Equivalently,

$${\mathcal{R}}_1\neg {\sim }_{{\mathfrak{P}}_{\Lambda }}{\mathcal{R}}_2⟹{\mathcal{R}}_1\neg {\sim }_{{\mathfrak{P}}_{{\Lambda }^{\prime }}}{\mathcal{R}}_2.$$

Proof. 

Equality of the larger comparative profiles implies equality after restriction to the subfamily Λ. Taking the contrapositive gives the second statement.

□

Definition 11.16  

(Strictly more separating family). An inclusion $\Lambda \subseteq {\Lambda }^{\prime }$ isstrictly more separating on a class of routesif there exist routes in that class such that

$${\mathcal{R}}_1{\sim }_{{\mathfrak{P}}_{\Lambda }}{\mathcal{R}}_2$$

but

$${\mathcal{R}}_1\neg {\sim }_{{\mathfrak{P}}_{{\Lambda }^{\prime }}}{\mathcal{R}}_2.$$

Remark 11.17.  

If ${\Lambda }^{\prime }=\Lambda \cup \left\{\mu \right\}$ and two routes have identical Λ-components but distinct μ-components, then ${\Lambda }^{\prime }$ is strictly more separating on every route class containing that pair.

Remark 11.18.  

Suppose the Chern, Todd, or conditional L-type components arise from a normalized ${\widehat{T}}_y$-profile by specialization as in Section 6.3. Equality of the ${\widehat{T}}_y$-profiles then forces equality of each specialization for which the corresponding comparison theorem is defined. Those specialized components cannot distinguish a pair that the full normalized ${\widehat{T}}_y$-profile has already identified.

Remark 11.19.  

No finite characteristic family considered in this paper is asserted to be complete. Equality of every chosen characteristic component may still fail to imply equality of the underlying routes, endpoint correspondences, or finer categorical realizations.

Remark 11.20.  

A characteristic transformation that factors through $K_0(MHM\left(X\right))$ depends only on the Grothendieck class of the mixed Hodge module. In particular, if two objects have the same class in $K_0(MHM\left(X\right))$, no collection of realizations depending only on that same Grothendieck class can recover extension data that were already lost under decategorification [14,16].

This does not say that all transformations out of $K_0(MHM\left(X\right))$ have identical separating power; it says only that none can reconstruct categorical data that the passage to the Grothendieck group has already identified.

11.4. Factorization Dependence and Geometric Invariance

The transport profile is intended to ignore trivial changes in presentation while retaining information about genuinely different nontrivial factorizations.

The notion of endpoint invariant is the one fixed in Definition 11.9. We now distinguish it from factorization-class and presentation-dependent data.

Definition 11.21  

(Factorization-class invariant). A route datum is afactorization-class invariantif it is unchanged under:

1.

insertion or deletion of an identity stage; and

2.

replacement of an intermediate object by an isomorphic object, with the adjacent geometric and characteristic data transported through the chosen isomorphism.

Since a route is represented by an ordered sequence of stages, a change of parenthesization does not change the route itself and is not treated as an additional equivalence operation.

Definition 11.22  

(Route-presentation datum). A datum that can change under identity insertion or deletion or under isomorphic replacement of an intermediate object is called aroute-presentation datum.

Remark 11.23.  

Whenever an admissible compressed endpoint operation $F_{\mathcal{R}}$ is defined and the chosen reference operator for that compressed operation depends only on $F_{\mathcal{R}}$, the compressed endpoint defect

$${\Delta }_{\Phi }(F_{\mathcal{R}};{\xi }_0)$$

is endpoint-determined.

By contrast, the stagewise route defect ${\Delta }_{\Phi }(\mathcal{R};{\xi }_0)$ is defined directly from the route and agrees with the compressed endpoint defect only when the relevant interaction terms vanish.

The full characteristic transport profile is intentionally not an endpoint invariant.

Proposition 11.24.  

Let

$$\mathcal{R}:X\stackrel{F}{\to }Y\stackrel{G}{\to }Z$$

and let

$${\mathcal{R}}^{\prime }:X\stackrel{G\circ F}{\to }Z$$

be the collapsed route, with the compressed operation admissible. Then

$$\mathcal{R}{\sim }_{\mathrm{end}}{\mathcal{R}}^{\prime }.$$

If the pair is Φ-Riemann–Roch compatible, the collapsed endpoint defect is

$${\Delta }_{\Phi }(G\circ F;\xi )={\Delta }_{\Phi }(G;F_{!}\xi )+G_{\Phi ,*}^{\mathrm{RR}}\left({\Delta }_{\Phi }(F;\xi )\right).$$

Thus the collapsed route retains only the propagated sum and need not retain the two separate stage defects. Consequently, the full stagewise transport profiles of $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$ need not agree.

Proof. 

The endpoint operations agree by construction. The defect formula is Proposition 5.25. The full profile of $\mathcal{R}$ retains the separate stage data, whereas the one-stage route ${\mathcal{R}}^{\prime }$ does not.

□

Theorem 11.25  

(Factorization invariance of the transport profile). Routes related by trivial profile refactorizations have canonically identified characteristic transport profiles.

Proof. 

Identity insertion contributes the identity source and target operators and zero stage defect by Definition 4.6. Isomorphic replacement transports defects through the canonical target-side isomorphisms by Proposition 4.7. These are precisely the elementary profile identifications of Definition 1.12.

□

Definition 11.26  

(Factorization class). Thefactorization classof a route $\mathcal{R}$, written

$${\left[\mathcal{R}\right]}_{\mathrm{fac}},$$

is its equivalence class under the trivial refactorizations of Definition 11.21.

Remark 11.27.  

By Theorem 11.25, the comparative characteristic transport profile descends from a written route $\mathcal{R}$ to its factorization class ${\left[\mathcal{R}\right]}_{\mathrm{fac}}$.

When all indicated realizations exist, the resulting information hierarchy is

$$\mathcal{R}⟼{\left[\mathcal{R}\right]}_{\mathrm{fac}}⟼F_{\mathcal{R}}⟼{\rho }_H\left(\mathcal{R}\right)⟼{\rho }_H{\left(\mathcal{R}\right)|}_L.$$

Here $F_{\mathcal{R}}$ denotes the endpoint geometric operation or correspondence represented by the route, and ${\rho }_H\left(\mathcal{R}\right)$ denotes its specified total rational cohomological realization.

Proposition 11.28  

(Information-loss hierarchy). The successive passages in the preceding hierarchy may forget, respectively:

1.

trivial presentation data within a fixed factorization;

2.

nontrivial intermediate factorization data while retaining only the endpoint operation;

3.

geometric or Chow-theoretic information lying in the kernel of total rational cohomological realization; and

4.

cohomological information carried outside the selected sector L.

None of the converse reconstructions is assumed in general.

Proof. 

The first arrow is the quotient by trivial refactorization. The second forgets the intermediate stages and retains only the endpoint operation. The third applies the specified total cohomological realization, whose kernel may be nontrivial. The fourth restricts the resulting operator to L.

□

Remark 11.29.  

The characteristic transport profile is therefore naturally attached to an admissible factorization class, rather than merely to an endpoint operation. Its factorization sensitivity is deliberate.

Remark 11.30.  

No invariance is imposed under replacement of a route by a genuinely different nontrivial factorization having the same endpoint. Detecting differences between such factorizations is one of the purposes of the transport-profile construction.

12. Consequences and Further Directions

The preceding results provide more than a formal calculus of composition, but substantially less than a universal theory of residual algebraic cycles. Independently identified characteristic defects can be propagated through composite geometric constructions with controlled support; characteristic transport profiles retain stagewise information before endpoint collapse; and restriction to a prescribed cohomological sector can identify correspondences that remain distinct on total rational cohomology.

Two calculations delimit the present scope. For the nodal cubic, the vanishing-cycle contribution produces the explicit local-to-global chain

$$-{\mathbf{1}}_{\left\{p\right\}}⟼\left[p\right]⟼\left[\mathrm{pt}\right].$$

For the Jacobian construction, a Jacobian return datum of coefficient $q>1$ produces

$${\mathfrak{D}}_q={\Gamma }_{{\left[q\right]}_A}-q^2{\Delta }_A\in ker\left({\rho }_{L_A}\right)\setminus ker\left({cl}_{A\times A}\right).$$

Thus selected-sector realization can lose information already visible to ordinary rational cohomology. The stronger problem of detecting a nonzero residual in $ker(cl)$ by a finer characteristic, bivariant, or other geometric realization remains open.

12.1. Residual Geometry and Characteristic Equivalence

Let ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ be admissible routes with common typed source and target, let L be a selected cohomological sector of their common source, and let Λ be a characteristic comparison family for which both comparative profiles are defined.

Definition 12.1  

(Characteristic equivalence relative to L and $\Lambda$). Write

$${\mathcal{R}}_1{\sim }_{\mathrm{char},L,\Lambda }{\mathcal{R}}_2$$

if both

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$$

and

$${\mathcal{R}}_1{\sim }_{{\mathfrak{P}}_{\Lambda }}{\mathcal{R}}_2.$$

Thus characteristic equivalence relative to L and Λ is simply the intersection of selected-sector equivalence and Λ-profile equivalence.

Remark 12.2.  

On any class of routes for which both constituent relations are equivalence relations, their intersection is again an equivalence relation. No additional geometric theorem is contained in this observation.

The resulting comparison hierarchy is immediate but useful for locating where information may be lost.

Proposition 12.3  

(Comparison hierarchy). Let ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ have common source and target. If

$${\left[{\mathcal{R}}_1\right]}_{\mathrm{fac}}={\left[{\mathcal{R}}_2\right]}_{\mathrm{fac}},$$

then

$${\mathcal{R}}_1{\sim }_{\mathrm{char},L,\Lambda }{\mathcal{R}}_2.$$

Moreover,

$${\mathcal{R}}_1{\sim }_{\mathrm{char},L,\Lambda }{\mathcal{R}}_2⟹{\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2.$$

Hence

$${\left[{\mathcal{R}}_1\right]}_{\mathrm{fac}}={\left[{\mathcal{R}}_2\right]}_{\mathrm{fac}}⟹{\mathcal{R}}_1{\sim }_{\mathrm{char},L,\Lambda }{\mathcal{R}}_2⟹{\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2.$$

Under the hypotheses of Theorem 6.29, the second implication can be strict.

Proof. 

Equality of factorization classes gives canonically identified transport profiles by Theorem 11.25. It also preserves the endpoint geometric realization and therefore its restriction to L. Hence the routes are characteristically equivalent relative to L and Λ.

The second implication is part of Definition 12.1. Under the hypotheses of Theorem 6.29, one can have ${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$ while ${\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2$, proving possible strictness.

□

Remark 12.4.  

Characteristic equivalence is relative to the chosen sector L and comparison family Λ. It is not asserted to imply equality of geometric routes, equality of endpoint Chow correspondences, equality of motives, or equality in any finer categorical realization.

We now isolate the residuals that are invisible to L but visible to a specified additional realization. Let X and Y be smooth projective, set $d_X:={dim}_{\mathbb{C}}X$, fix $k\in \mathbb{Z}$, and let $L\subseteq H^{•}(X,\mathbb{Q})$. The selected-sector realization is

$${\rho }_L:C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}⟶{Hom}_{\mathbb{Q}}\left(L,H^{•}(Y,\mathbb{Q})\right),{\rho }_L(\Gamma ):={\Gamma }_{*}{|}_L.$$

In this subsection we use the evident degree-k, X-to-Y extension of Definition 7.10: an admissible residual realization is a $\mathbb{Q}$-linear map defined on a specified $\mathbb{Q}$-linear class of degree-k algebraic correspondences in $C{H}^{d_X+k}{(X\times Y)}_{\mathbb{Q}}$, constructed independently of the selected sector L, with the same geometric and functoriality requirements imposed there.

Definition 12.5  

(Characteristically detectable residual locus). Let

$$\chi :{Corr}_{\chi }(X,Y)⟶{\mathcal{C}}_{\chi }(X,Y)$$

be an admissible additive residual realization defined on a $\mathbb{Q}$-linear class ${Corr}_{\chi }(X,Y)$ of algebraic correspondences. Define

$${\mathcal{D}}_{\chi ,L}(X,Y):=\left\{\Gamma \in ker\left({\rho }_L\right)\cap {Corr}_{\chi }(X,Y):\chi (\Gamma )\ne 0\right\}.$$

The notation ${\mathcal{D}}_{\chi ,L}(X,Y)$ denotes a detectablelocus, not in general a vector subspace: the nonvanishing condition $\chi (\Gamma )\ne 0$ need not be preserved under addition.

Remark 12.6.  

If ${\mathcal{D}}_{\chi ,L}(X,Y)\ne ⌀$, then the joint realization

$$\Gamma ⟼\left({\rho }_L(\Gamma ),\chi (\Gamma )\right)$$

contains information not determined by ${\rho }_L$ alone. Indeed, an element of ${\mathcal{D}}_{\chi ,L}(X,Y)$ has zero selected-sector realization and nonzero χ-realization.

The ordinary rational cycle class gives the unconditional instance proved in Section 8. For every Jacobian return datum with $q>1$,

$${\mathfrak{D}}_q\in {\mathcal{D}}_{\mathrm{cl},L_A}(A,A),$$

where ${\chi }_{\mathrm{cl},A}={cl}_{A\times A}$. Consequently,

$${\mathcal{D}}_{\mathrm{cl},L_A}(A,A)\ne ⌀.$$

Remark 12.7.  

More generally, let $(U,R)$ be a transfer–return pair on L. If

$${\chi }_X\left({\mathfrak{D}}_{U,R}\right)\ne 0,$$

then

$${\mathfrak{D}}_{U,R}\in {\mathcal{D}}_{\chi ,L}(X,X),$$

because Lemma 7.6 gives ${\rho }_L\left({\mathfrak{D}}_{U,R}\right)=0$.

Remark 12.8.  

No classification of $ker\left({\rho }_L\right)$ is claimed. For smooth-projective correspondences, the Künneth–Poincaré-duality comparison established earlier gives

$$ker\left({cl}_{X\times Y}\right)=ker\left({\rho }_H\right)\subseteq ker\left({\rho }_L\right).$$

The explicit Jacobian residuals lie in

$$ker\left({\rho }_L\right)\setminus ker\left({\rho }_H\right).$$

Thus they establish selected-sector information loss but do not enter the homologically trivial kernel.

Remark 12.9.  

Endpoint residual geometry and route-level transport residuals remain distinct. An endpoint residual such as ${\mathfrak{D}}_{U,R}$ is a Chow correspondence obtained after comparing endpoint actions, whereas a transport profile retains the stagewise defects and their propagation histories before endpoint collapse. No identification between these levels is made without an explicit compatibility result such as Proposition 7.32.

12.2. Hodge-theoretic Positioning

Let X be smooth projective and let

$$\alpha \in H^{2p}{(X,\mathbb{Q}\left(p\right))}_{\mathrm{Hdg}}$$

be a rational Hodge class. Its algebraicity remains the classical condition

$$\alpha \in im\left({cl}_X:C{H}^p{\left(X\right)}_{\mathbb{Q}}⟶H^{2p}(X,\mathbb{Q}\left(p\right))\right).$$

Characteristic transport neither changes this criterion nor constructs a Chow lift of an arbitrary Hodge class.

Definition 12.10  

(Hodge-to-Chow geometric route). AHodge-to-Chow geometric routeis a finite admissible algebraic-geometric construction satisfying the following conditions: its geometric stages and endpoint are defined independently of any unknown target Hodge class; its endpoint carries a specified cycle-theoretic or correspondence-theoretic realization; and its already-defined rational cohomological realization can subsequently be restricted to a prescribed Hodge sector.

Thus selection of a Hodge sector is a comparison operation performedafterconstruction of the geometric route. It does not supply the route itself.

Remark 12.11.  

If two Hodge-to-Chow geometric routes satisfy

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$$

but

$${\mathcal{R}}_1\neg {\sim }_{\mathfrak{P}}{\mathcal{R}}_2,$$

then their common action on L does not determine their characteristic transport histories.

Remark 12.12.  

A nonzero characteristic transport defect does not prove that a Hodge class is algebraic. Conversely, vanishing of a transport defect does not construct an algebraic cycle whose cycle class is a prescribed Hodge class.

Remark 12.13.  

The Hodge-theoretic role of characteristic transport in the present paper is diagnostic. It can retain singularity, support, tangent, normal, specialization, and factorization information that may disappear after a geometric construction is replaced by its action on a selected Hodge sector. Any use toward an algebraicity problem would require an additional argument connecting those retained data to the existence of an algebraic cycle.

12.3. Open Comparison Problems

The preceding results suggest several natural problems. None is required for the unconditional transport and residual results proved above.

Question 12.14.  

Does there exist a finite characteristic comparison family that separates every pair of routes in a specified geometrically natural class, modulo admissible factorization equivalence?

Any such question necessarily depends on the route class and on the admissible characteristic theories; no universal finite family is proposed here.

Question 12.15.  

For which classes of geometric routes does a normalized ${\widehat{T}}_y$-transport profile determine, by specialization, all corresponding Chern-, Todd-, and L-type profile components for which the relevant comparison theorems exist?

The conditional qualification at $y=1$ remains essential for singular spaces.

Question 12.16.  

Can characteristic transport profiles be organized by a bivariant, motivic, or cobordism-theoretic orientation from which the stage corrections, interaction operators, and propagation maps arise functorially?

Fulton–MacPherson bivariant theory, Grothendieck transformations, and motivic bivariant characteristic classes provide natural starting points [2,4,5].

Question 12.17.  

Which characteristic transport defects admit intrinsic geometric descriptions in terms of vanishing cycles, specialization, normal cones, excess intersections, singular strata, or comparable local geometric data?

Schürmann’s Verdier–Riemann–Roch formula provides one established model of such a geometric identification [6]; the present paper does not assert that every formal transport defect has such an interpretation.

Question 12.18.  

For which route classes, selected sectors L, and characteristic families Λ does

$${\mathcal{R}}_1{\sim }_L{\mathcal{R}}_2$$

force

$${\mathcal{R}}_1{\sim }_{\mathrm{char},L,\Lambda }{\mathcal{R}}_2?$$

Equivalently, one asks when restriction to L already determines all profile information represented by the chosen family Λ.

Question 12.19.  

Does there exist a natural admissible residual realization

$$\chi :{Corr}_{\chi }(X,Y)⟶{\mathcal{C}}_{\chi }(X,Y)$$

on a geometrically meaningful class of algebraic correspondences, together with a nonzero correspondence

$$D\in ker\left({cl}_{X\times Y}\right),$$

such that

$$\chi \left(D\right)\ne 0?$$

Equivalently, can such a realization detect a nonzero homologically trivial algebraic correspondence?

This is the genuinely finer residual problem. The ordinary cycle-class realization detects the explicit residuals constructed here only because they lie in

$$ker\left({\rho }_L\right)\setminus ker\left(cl\right).$$

It cannot, by definition, detect a class in its own kernel.

Remark 12.20.  

Suppose, in addition, that χ is defined on the same correspondence space as the ordinary cycle-class map and satisfies

$$ker\left(\chi \right)\subseteq ker(cl).$$

If Question 12.19 has a positive answer for a class D, then $D\in ker(cl)$ but $D\notin ker\left(\chi \right)$. Hence

$$ker\left(\chi \right)⊊ker(cl).$$

Thus such a realization would be strictly finer than ordinary rational cycle class at the level of kernels.

A positive answer to Question 12.19 would enter the classical homologically trivial regime discussed in Section 2.6, where Abel–Jacobi-type invariants, Griffiths groups, and conjectural Bloch–Beilinson filtrations provide the standard surrounding framework [22,23].

Remark 12.21.  

None of Questions 12.14–12.19 is required for the finite-route propagation identity, support-propagation theorem, singular composite calculation, mixed smooth–Gysin compatibility, or Jacobian selected-sector separation proved in the paper.

12.4. Final Perspective

The paper distinguishes three levels of information:

$$\mathrm{geometric}\mathrm{factorization}⟶\mathrm{characteristic}\mathrm{transport}⟶\mathrm{prescribed}\mathrm{cohomological}\mathrm{realization}.$$

A geometric factorization records the intermediate spaces and operations. Characteristic transport records how established characteristic structures behave along those stages, including corrected operators, geometrically identified defects, certified support, and propagation. A prescribed cohomological realization retains only the information visible to the chosen endpoint realization, possibly after further restriction to a selected sector.

The nodal-cubic and Jacobian calculations mark two different boundaries of this framework. For the nodal cubic,

$$-{\mathbf{1}}_{\left\{p\right\}}⟼\left[p\right]⟼\left[\mathrm{pt}\right]$$

shows that a local vanishing-cycle contribution can remain visible after global propagation. For the Jacobian family, exact proper transport makes the standard reduced defect profile insensitive to the intermediate Jacobian, while the selected Hodge sector loses the nonzero residual

$${\mathfrak{D}}_q={\Gamma }_{{\left[q\right]}_A}-q^2{\Delta }_A\in ker\left({\rho }_{L_A}\right)\setminus ker\left({cl}_{A\times A}\right).$$

Characteristic transport should therefore be viewed as an intermediate diagnostic structure, not as a universal invariant of algebraic cycles. It does not replace Chow-theoretic algebraicity, Abel–Jacobi theory, or the study of homologically trivial cycles. Its role is to organize and propagate independently justified characteristic data along composite constructions and to make explicit which information is retained before passage to a coarser endpoint realization.

The unresolved frontier begins precisely where the unconditional Jacobian example stops: constructing a natural realization capable of detecting a nonzero algebraic correspondence lying in

$$ker\left(cl\right).$$

Such a result would move the residual theory from selected-sector information loss into the genuinely homologically trivial regime.