Integral Defects of Rational Hodge Cycle Constructions

1. Introduction

Let X be a smooth projective complex variety. For $p\ge 0$, the integral and rational cycle-class maps are

$${cl}_{X,\mathbb{Z}}:C{H}^p(X)⟶H^{2p}(X,\mathbb{Z}(p))$$

and

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

The integral Hodge conjecture in codimension p asks whether every cohomological integral Hodge class in $H^{2p}(X,\mathbb{Z}(p))$ lies in the image of ${cl}_{X,\mathbb{Z}}$. The rational Hodge conjecture asks whether every cohomological rational Hodge class in $H^{2p}(X,\mathbb{Q}(p))$ lies in the image of ${cl}_X$. The implication from the integral statement to the rational statement is formal. The converse is obstructed by finite and torsion phenomena which are not removed by a universal integral transfer principle.

The obstruction is already present in the standard defect group

$$Z^{2p}(X)={\displaystyle \frac{H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}}{im({cl}_{X,\mathbb{Z}})}}.$$

This group is not introduced here as a new invariant. It is the usual defect group for the integral Hodge conjecture. In degree 4, the defect is closely related to unramified cohomology in the work of Colliot-Thélène–Voisin [1]. Refined unramified cohomology gives further structural interpretations of failures of integral Hodge and Tate type statements [2]. The decomposition-of-the-diagonal and torsion-order theories give global annihilation mechanisms and obstructions to bounded universal annihilation [3,4,5]. Classical and modern examples of Atiyah–Hirzebruch, Kollár, Totaro, Soulé–Voisin, Benoist–Ottem, Voisin, and Diaz show that finite integral defects and torsion obstructions occur in genuine geometric settings [6,7,8,9,10,11,12,13,14].

The conclusion drawn in this paper is not that such defects are new. Rather, their presence forces algebraicity-transfer arguments to be sectorwise. A rational Hodge-cycle construction may succeed on a specified geometric part of cohomology without yielding an integral theorem for all integral Hodge classes. The natural problem is therefore not to seek a single universal transfer mechanism, but to identify the domains on which particular mechanisms work and to determine what finite integral data distinguish these domains.

This paper studies those domains through their integral defect structure. Given a finitely generated subgroup

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}},$$

a rational construction $\mathcal{C}$ may produce a subgroup

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}$$

generated by specified algebraic classes, and may prove rational coverage:

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

The quotient

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is then finite. It records the finite integral defect of the construction $\mathcal{C}$ on the chosen subgroup $L_{\mathrm{Hdg}}$. The central point is that different rational Hodge-cycle constructions leave different kinds of finite defects. These defect types induce a grouping of construction methods by the way their proofs lose integral information.

The terminology used in this paper is only organizational. A “typed sector” is a finitely generated subgroup of integral Hodge classes, usually specified by geometric data. A “constructional defect” is a quotient of such a subgroup by the algebraic sublattice generated by a chosen construction. A “defect profile” records standard data attached to that quotient: a finite abelian group, an exponent, an annihilator, a lattice index, or a kernel or cokernel of a realization map. These terms do not replace the standard language of Hodge structures, Chow groups, integral Hodge defect groups, torsion order, or unramified cohomology.

The main organizing principle is a trichotomy. The first type is multiplier-supported. It occurs when the construction contains an identity

$$T\circ S=N·id$$

on the relevant sector lattice, with S and T induced by algebraic operations. Finite trace, norm maps, finite group averaging, rational algebraic projectors after clearing denominators, and isogeny pull-push arguments are of this type. The associated proof-level annihilator is the multiplier N.

The second type is saturation-supported. It occurs when algebraic classes span the Hodge sector over $\mathbb{Q}$, but the integral lattice they generate need not be saturated:

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

The associated finite defect is the lattice quotient

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Product constructions, invariant generation, and Weil-type or Prym-type lattice questions often have this form.

The third type is realization-loss. It occurs when integral information is lost under a realization or coefficient map, such as a regulator map, an Abel–Jacobi map, a specialization boundary map, or a Bockstein homomorphism. In this case the finite data are controlled by kernels, images, or cokernels of realization maps rather than by a single multiplier identity.

These three types are not asserted to form an absolute partition of all possible constructions. A construction may be hybrid. For example, a finite group average may supply a multiplier denominator, while the resulting algebraic lattice may still have a nontrivial saturation quotient. The claim is more precise: each construction considered here has a defect profile, and the nonzero components of that profile determine the type of integral data carried by the proof. The comparison of rational constructions is therefore a comparison of defect profiles.

1.1. Notation and Conventions

Throughout the paper, X denotes a smooth projective variety over $\mathbb{C}$. Cohomology means singular cohomology of $X(\mathbb{C})$. For $p\ge 0$, $C{H}^p(X)$ denotes the Chow group of codimension p algebraic cycles modulo rational equivalence, and

$$C{H}^p{(X)}_{\mathbb{Q}}=C{H}^p(X){\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

For a coefficient group $A\in \{\mathbb{Z},\mathbb{Q},\mathbb{C}\}$, the Tate twist is denoted $A(p)$. Hodge classes are cohomological classes. We write

$$H^{2p}{(X,\mathbb{Q}(p))}_{\mathrm{Hdg}}=H^{2p}(X,\mathbb{Q}(p))\cap H^{p,p}(X),$$

and

$$H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}=H^{2p}(X,\mathbb{Z}(p))\cap H^{p,p}(X).$$

Equivalently, an integral class is a Hodge class if its image in $H^{2p}(X,\mathbb{C}(p))$ has type $(0,0)$ after the Tate twist, or type $(p,p)$ before the twist.

The integral and rational cycle-class maps are denoted

$${cl}_{X,\mathbb{Z}}:C{H}^p(X)⟶H^{2p}(X,\mathbb{Z}(p))$$

and

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

When $\alpha \in H^{2p}{(X,\mathbb{Q}(p))}_{\mathrm{Hdg}}$, we say that $\alpha$ is algebraic if there exists $\Gamma \in C{H}^p{(X)}_{\mathbb{Q}}$ such that

$${cl}_X(\Gamma )=\alpha .$$

When $\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$, we say that $\alpha$ is integrally algebraic if there exists $\Gamma \in C{H}^p(X)$ such that

$${cl}_{X,\mathbb{Z}}(\Gamma )=\alpha .$$

Algebraic cycle classes are Hodge classes; this is the standard Hodge-theoretic compatibility of algebraic cycles with the cycle-class map [15,16,17].

Definition 1 

(Integral and rational defect groups). The integral Hodge defect group of X in codimension p is

$$Z^{2p}(X)={\displaystyle \frac{H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}}{im({cl}_{X,\mathbb{Z}})}}.$$

The rational Hodge defect group of X in codimension p is

$$Z^{2p}{(X)}_{\mathbb{Q}}={\displaystyle \frac{H^{2p}{(X,\mathbb{Q}(p))}_{\mathrm{Hdg}}}{im({cl}_X)}}.$$

Remark 1 

(Standard status of $Z^{2p}(X)$). The quotient $Z^{2p}(X)$ is the standard defect group measuring failure of the integral Hodge conjecture in codimension p. In the present paper it is used as background notation. The additional terms introduced below are bookkeeping terms for quotients, exponents, and proof-level data attached to specified rational Hodge-cycle constructions.

1.2. Sectorwise Transfer and Constructional Defects

The examples and global theories cited above show that integral Hodge defects cannot be removed by a single uniform mechanism. The appropriate local object is a fixed finitely generated subgroup of integral Hodge classes.

Definition 2 

(Typed Hodge sector). A typed Hodge sector in codimension p on X is a finitely generated subgroup

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

specified independently of any unknown target Hodge class. Equivalently, it is a finitely generated Hodge subgroup together with the geometric data used to specify it. A sector may be specified by a product decomposition, a finite group action, an algebraic correspondence, an isogeny, a Prym construction, an endomorphism algebra, a specialization datum, a support condition, a limiting mixed Hodge structure, or a regulator source.

Definition 3 

(Actual and constructional algebraic parts). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector. Its actual algebraic part is

$$L_{\mathrm{alg}}^{\mathrm{act}}=L_{\mathrm{Hdg}}\cap im({cl}_{X,\mathbb{Z}}).$$

A constructional algebraic part is a subgroup

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}$$

generated by algebraic classes produced by a specified construction $\mathcal{C}$.

Definition 4 

(Actual defect and constructional defect). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector. The actual sector defect is

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathrm{act}}.$$

If $\mathcal{C}$ is a specified construction with constructional algebraic part $L_{\mathrm{alg}}^{\mathcal{C}}$, the constructional defect of $\mathcal{C}$ in $L_{\mathrm{Hdg}}$ is

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Remark 2 

(Actual and constructional quotients). The quotient $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$ embeds in $Z^{2p}(X)$. The quotient $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ measures the integral loss of a chosen construction. It may be nonzero even when $D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0$. Thus a constructional defect is a defect of a method, not necessarily a failure of the integral Hodge conjecture.

A rational construction in a sector usually proves a coverage statement

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

This statement is separate from soundness.

Definition 5 

(Soundness and rational coverage). Let $\mathcal{C}$ be a construction in a typed sector $L_{\mathrm{Hdg}}$. The construction is sound if every class in $L_{\mathrm{alg}}^{\mathcal{C}}$ lies in $im({cl}_{X,\mathbb{Z}})$ by a specified Chow class, algebraic correspondence, proper pushforward, flat pullback, refined Gysin map, product, finite trace descent, algebraic projector splitting, or support pushforward from algebraic strata. The construction has rational coverage if

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Remark 3 

(Excluded certificates). Soundness is not supplied by an arbitrary Hodge projector, by a splitting of Hodge structures without Chow realization, by an unconstructed motivic lift, by a correspondence known only cohomologically, or by a map defined using the unknown target class.

1.3. Defect Profiles and Induced Grouping

A rational construction with soundness and coverage leaves a finite constructional defect. The defect is not only a finite group; the proof often records how the group is produced. This additional proof-level information is the defect profile.

Definition 6 

(Constructional exponent). Let $\mathcal{C}$ be a sound construction with rational coverage in a typed sector $L_{\mathrm{Hdg}}$. Its constructional exponent is

$$e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp{D}_{\mathcal{C}}(L_{\mathrm{Hdg}}),$$

provided $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ is finite. If the proof of $\mathcal{C}$ supplies an integer $N\ge 1$ such that

$$N{L}_{\mathrm{Hdg}}\subseteq L_{\mathrm{alg}}^{\mathcal{C}},$$

then N is a proof-level annihilator for $\mathcal{C}$.

Definition 7 

(Defect profile). The defect profile of a construction $\mathcal{C}$ in a sector $L_{\mathrm{Hdg}}$ records the data

$$Prof(\mathcal{C},L_{\mathrm{Hdg}})=\left(D_{\mathcal{C}}(L_{\mathrm{Hdg}}),e_{\mathcal{C}}(L_{\mathrm{Hdg}}),{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right),$$

when these terms are defined. Here ${\mathcal{A}}_{\mathcal{C}}$ denotes the annihilator, kernel, cokernel, boundary, or coefficient data supplied by the proof, and ${\tau }_{\mathcal{C}}$ is the defect type. In this paper the basic defect types are

$${\tau }_{\mathcal{C}}\in \{\mathrm{mult},\mathrm{sat},\mathrm{real}\},$$

corresponding to multiplier-supported, saturation-supported, and realization-loss defects.

Remark 4 

(Profile is not an invariant of X alone). The defect profile is not a new cohomological invariant of X. It is a record attached to a construction $\mathcal{C}$, a sector $L_{\mathrm{Hdg}}$, and a constructional algebraic subgroup $L_{\mathrm{alg}}^{\mathcal{C}}$. Different constructions in the same variety may have different profiles.

Definition 8 

(Multiplier-supported defect). A constructional defect is multiplier-supported if the construction supplies abelian groups or lattices $M_1,M_2$, homomorphisms

$$S:M_1⟶M_2,T:M_2⟶M_1,$$

and an integer $N\ge 1$ such that

$$T\circ S=N·{id}_{M_1},$$

with S and T realized by algebraic operations compatible with the cycle-class map on the sector under consideration.

Definition 9 

(Saturation-supported defect). A constructional defect is saturation-supported if the construction gives a finite-index inclusion of lattices

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q},$$

and the associated defect is the lattice quotient

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Definition 10 

(Realization-loss defect). A constructional defect is realization-loss type if it is controlled by the kernel, image, or cokernel of a realization or coefficient map, such as a regulator map, an Abel–Jacobi map, a specialization boundary map, or a Bockstein homomorphism.

Definition 11 

(Defect-profile equivalence). Two construction-sector pairs

$$({\mathcal{C}}_1,L_1),({\mathcal{C}}_2,L_2)$$

are defect-profile equivalent if their defect profiles have the same defect type and the same kind of proof-level data:

$${\tau }_{{\mathcal{C}}_1}={\tau }_{{\mathcal{C}}_2},{\mathcal{A}}_{{\mathcal{C}}_1}\mathrm{and}{\mathcal{A}}_{{\mathcal{C}}_2}\mathrm{are}\mathrm{of}\mathrm{the}\mathrm{same}\mathrm{structural}\mathrm{form}.$$

A finer equivalence may require an isomorphism of the finite defects

$$D_{{\mathcal{C}}_1}(L_1)\cong D_{{\mathcal{C}}_2}(L_2).$$

Remark 5 

(Hybrids). A construction may have more than one nonzero component in its defect profile. For example, an averaging argument may supply a multiplier denominator, while the resulting algebraic lattice may still be nonsaturated. Thus the three types define components of a profile, not a rigid partition of all constructions.

1.4. Main Statements

The first statement is the finite-index principle in constructional form.

Proposition 1 

(Finite constructional defect). Let $\mathcal{C}$ be a sound construction in a typed Hodge sector $L_{\mathrm{Hdg}}$. Assume rational coverage:

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is finite.

Proof. 

The group $L_{\mathrm{Hdg}}$ is finitely generated by definition of a typed Hodge sector. The subgroup $L_{\mathrm{alg}}^{\mathcal{C}}$ is therefore finitely generated. Tensoring the exact sequence

$$0⟶L_{\mathrm{alg}}^{\mathcal{C}}⟶L_{\mathrm{Hdg}}⟶L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}⟶0$$

with $\mathbb{Q}$ gives an exact sequence

$$0⟶L_{\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}⟶L_{\mathrm{Hdg}}\otimes \mathbb{Q}⟶\left(L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}\right)\otimes \mathbb{Q}⟶0.$$

The rational coverage hypothesis is equivalent to the vanishing of the final term. A finitely generated abelian group has zero rationalization if and only if it is finite. Hence

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

is finite. □

Proposition 2 

(Multiplier annihilation). Let $\mathcal{C}$ be a sound construction in a typed sector $L_{\mathrm{Hdg}}$. Assume that the defect of $\mathcal{C}$ is multiplier-supported by $S,T$ and N, so that

$$T\circ S=N·id$$

on the sector lattice. Assume that for every $\alpha \in L_{\mathrm{Hdg}}$, the class $S(\alpha )$ is represented by an algebraic source to which T may be applied algebraically. Then

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Proof. 

Let $\alpha \in L_{\mathrm{Hdg}}$. By hypothesis, $S(\alpha )$ is represented by an algebraic source. Since T is realized by an algebraic operation compatible with cycle classes, $T(S(\alpha ))$ belongs to $L_{\mathrm{alg}}^{\mathcal{C}}$. The multiplier identity gives

$$T(S(\alpha ))=N\alpha .$$

Hence $N\alpha \in L_{\mathrm{alg}}^{\mathcal{C}}$. Therefore the class of $\alpha$ in

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is killed by N. Since $\alpha$ was arbitrary,

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

 □

Proposition 3 

(Composite multiplier bound). Let a sound construction $\mathcal{C}$ be obtained by a finite sequence of multiplier-supported steps with multipliers

$$N_1,\dots ,N_r.$$

Assume that each step is compatible with the relevant cycle-class maps and that the output of each step is an admissible input for the next. Then

$$N_1\dots N_r·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Proof. 

Let $\alpha \in L_{\mathrm{Hdg}}$. Applying the first multiplier-supported step gives algebraicity of $N_1\alpha$ at the corresponding stage. Applying the second step to the output gives algebraicity after multiplication by $N_2$, hence by $N_1{N}_2$ relative to the original class. Continuing through the finite sequence gives algebraicity of

$$N_1\dots N_r\alpha$$

in the constructional algebraic lattice. Therefore the class of $\alpha$ in $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ is killed by $N_1\dots N_r$. □

Proposition 4 

(Saturation exponent). Let $\mathcal{C}$ be a sound construction with saturation-supported defect in $L_{\mathrm{Hdg}}$. Assume rational coverage. Then

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is finite, and its constructional exponent is the exponent of this finite lattice quotient. If $L_{\mathrm{Hdg}}$ is free and a basis is fixed, this exponent is computed by the Smith normal form of a matrix of algebraic generators.

Proof. 

The finiteness is Proposition 1. The constructional exponent is

$$e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp\left(L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}\right).$$

If $L_{\mathrm{Hdg}}$ is free and a basis is fixed, the generators of $L_{\mathrm{alg}}^{\mathcal{C}}$ give an integral matrix. Since rational coverage holds, the matrix has full rank. The Smith normal form gives the invariant factors of the quotient, and the largest invariant factor is its exponent. □

Proposition 5 

(Image controlled by torsion-order annihilators). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector, and let

$$q_{\mathcal{C}}:D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

be the natural map. Suppose that the construction $\mathcal{C}$ has a constructional annihilator $N\ge 1$, and suppose that a diagonal-decomposition or torsion-order argument supplies an integer

$$Tor(X)\ge 1$$

such that

$$Tor(X)·D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

Then

$$expim(q_{\mathcal{C}})\mid gcd(N,Tor(X)).$$

Proof. 

The constructional annihilator N kills $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$, hence also kills $im(q_{\mathcal{C}})$. The torsion-order annihilator $Tor(X)$ kills $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$, hence also kills the subgroup $im(q_{\mathcal{C}})$. Therefore $gcd(N,Tor(X))$ kills $im(q_{\mathcal{C}})$. The exponent of a finite abelian group divides every integer which annihilates it. □

Example 1 

(A nonzero constructional defect with zero actual defect). Let

$$X={\mathbb{P}}^1\times {\mathbb{P}}^1,$$

and let

$$H_1=c_1({\mathcal{O}}_X(1,0)),H_2=c_1({\mathcal{O}}_X(0,1)).$$

Then

$$H^2{(X,\mathbb{Z}(1))}_{\mathrm{Hdg}}=\mathbb{Z}{H}_1\oplus \mathbb{Z}{H}_2.$$

Set

$$L_{\mathrm{Hdg}}=H^2{(X,\mathbb{Z}(1))}_{\mathrm{Hdg}}.$$

Since $H_1$ and $H_2$ are divisor classes,

$$L_{\mathrm{alg}}^{\mathrm{act}}=L_{\mathrm{Hdg}}.$$

Hence

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

Let

$$g:{\mathbb{P}}^1⟶{\mathbb{P}}^1,[x:y]⟼[x^2:y^2],$$

and let

$$f=g\times {id}_{{\mathbb{P}}^1}:{\mathbb{P}}^1\times {\mathbb{P}}^1⟶{\mathbb{P}}^1\times {\mathbb{P}}^1.$$

Then f is finite of degree 2. Its pullback on divisor classes is

$$f^{*}{H}_1=2H_1,f^{*}{H}_2=H_2.$$

Let $\mathcal{C}=f^{*}$ be the construction which supplies the pullback lattice

$$L_{\mathrm{alg}}^{\mathcal{C}}=f^{*}{L}_{\mathrm{Hdg}}=\mathbb{Z}(2H_1)\oplus \mathbb{Z}{H}_2\subseteq L_{\mathrm{Hdg}}.$$

The construction has rational coverage because

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

However,

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})={\displaystyle \frac{\mathbb{Z}{H}_1\oplus \mathbb{Z}{H}_2}{\mathbb{Z}(2H_1)\oplus \mathbb{Z}{H}_2}}\cong \mathbb{Z}/2\mathbb{Z}.$$

Thus

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})\ne 0,D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

Moreover, the construction is multiplier-supported. Since f is finite of degree 2, the trace identity gives

$$f_{*}{f}^{*}=2id$$

on $L_{\mathrm{Hdg}}$. Thus the proof-level annihilator is $N=2$, while the actual defect vanishes. This example shows that a geometrically sourced constructional defect can record finite loss of a method even when the actual integral Hodge defect is zero.

Remark 6 

(Contribution of the paper). The contribution is not the definition of $Z^{2p}(X)$, nor the finite-index principle alone. The contribution is the constructional comparison: rational Hodge-cycle proofs carry finite integral defect data, and these data fall into a small number of defect-profile types. The map

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

separates proof-level finite loss from actual integral Hodge obstruction. When an external torsion-order or diagonal-decomposition annihilator is available, the image of this map is controlled by both the constructional annihilator and the external annihilator. The example $X={\mathbb{P}}^1\times {\mathbb{P}}^1$ shows that the constructional defect can be nonzero even when the actual defect vanishes.

1.5. Motivating Sources

We use the notation and conventions of Section 1.1. The sources motivating this paper are rational Hodge-cycle constructions in which algebraicity is proved by geometric operations with visible finite integral defects.

Hodge cycles on abelian varieties provide a first source. Deligne proved that Hodge cycles on abelian varieties are absolute Hodge classes [18]. The theorem is rational and absolute-Hodge in nature. The present question is not to reprove Deligne’s theorem, but to ask what integral lattice information remains in sectors where algebraic classes or correspondences are supplied.

Self-products with automorphisms provide a second source. Schoen studied Hodge classes on self-products of varieties with automorphisms [19]. Such constructions use finite group actions, eigenspace decompositions, and algebraic correspondences. The rational decomposition may be implemented by character idempotents with denominators. The associated integral defect is multiplier-supported and may also have a saturation component.

Product constructions provide a third source. In products of curves, surfaces, and abelian varieties, rational Hodge sectors may be described by Hodge groups, invariant theory, endomorphism algebras, and algebraic generators [20,21]. The integral question is whether those generators form a saturated sublattice.

Prym and Weil-type constructions provide a fourth source. Rational Hodge classes may be compared with Jacobians, Prym varieties, isogenous abelian varieties, or eigenspaces for endomorphism fields. The multiplier part comes from norm maps and isogenies. The saturation part comes from the integral lattice of the Weil or Prym sector. Current work on one-cycles on abelian varieties and on integral saturation phenomena for very general principally polarized abelian varieties shows that these questions are active on the integral side [22,23,24].

Higher Chow constructions provide a fifth source. Collino and Fakhruddin constructed indecomposable higher Chow cycles on Jacobians with subtle regulator behavior [25]. A regulator may fail to detect information that remains nontrivial in a finer Chow-theoretic group. This is a realization-loss phenomenon rather than an ordinary sector defect inside $Z^{2p}(X)$.

Hodge 1-motivic constructions provide a sixth source. Barbieri-Viale’s work on algebraic 1-motives related to Hodge cycles organizes certain cycle-theoretic and Hodge-theoretic data through 1-motivic structures [26]. Such constructions suggest that integral information may be relocated into lattice, semi-abelian, or extension data rather than simply annihilated.

Remark 7 

(Role of examples and global theories). The counterexample and global-structure literature calibrates the meaning of integral failure. Atiyah–Hirzebruch, Kollár, Totaro, Soulé–Voisin, Benoist–Ottem, Voisin, and Diaz provide positive and negative examples around integral Hodge phenomena [6,7,8,9,10,11,12,13,14]. Colliot-Thélène–Voisin and Schreieder relate integral Hodge defects to unramified cohomology and its refinements [1,2]. Voisin, Kahn, and Chatzistamatiou–Levine give diagonal-decomposition and torsion-order frameworks for global annihilation phenomena [3,4,5]. The present paper is complementary: it records the finite constructional defects visible in rational Hodge-cycle proofs.

1.6. Structure of the Paper

Section 2 develops the standard defect groups, typed sectors, actual defects, constructional defects, and saturation quotients. Section 3 proves the finite-index principle and formulates constructional exponents and defect profiles. Section 4 studies multiplier-supported defects from finite covers, trace maps, norm maps, and finite group averaging. Section 5 treats rational algebraic projectors, idempotent denominators, and correspondence images.

Section 6 studies saturation-supported defects and computable lattice indices. Section 7 treats isogeny, Prym, and Weil-type defects, where multiplier and saturation components may both occur. Section 8 treats specialization, vertical, boundary, and limit mixed Hodge contributions. Section 9 treats regulator and higher Chow realization loss. Section 10 treats finite-coefficient and Bockstein sources of torsion.

Section 11 returns to the motivating rational Hodge-cycle constructions and records their defect profiles. Section 12 summarizes the trichotomy, compares constructional annihilators with external annihilators of actual defects, and gives a computed example where the constructional defect is nonzero although the actual defect vanishes. Section 13 records the resulting interpretation of rational Hodge-cycle proofs as carrying finite integral data not visible in their final rational statements.

2. Integral Hodge Defect Groups

We use the notation and conventions of Section 1.1. In particular, Hodge classes are cohomological classes, and algebraicity is defined by membership in the image of the appropriate cycle-class map. The group $Z^{2p}(X)$ is the standard defect group for the integral Hodge conjecture in codimension p. Its relationship with unramified cohomology, diagonal decompositions, torsion order, and refined unramified invariants is part of the established literature [1,2,3,4,5]. The purpose of this section is to separate this standard actual defect from the constructional defects left by specified rational Hodge-cycle arguments.

2.1. The Standard Defect Group

Let X be smooth projective over $\mathbb{C}$, and let $p\ge 0$. The cohomological integral cycle-class map is

$${cl}_{X,\mathbb{Z}}:C{H}^p(X)⟶H^{2p}(X,\mathbb{Z}(p)).$$

Since algebraic cycle classes are Hodge classes, one has

$$im({cl}_{X,\mathbb{Z}})\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

This compatibility is functorial for the Chow-theoretic operations used below, including proper pushforward, flat pullback, refined Gysin maps, products, and algebraic correspondences [15,16,17,27].

Definition 12 

(Integral Hodge defect group). The integral Hodge defect group of X in codimension p is

$$Z^{2p}(X)={\displaystyle \frac{H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}}{im({cl}_{X,\mathbb{Z}})}}.$$

For

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}},$$

the image of α in $Z^{2p}(X)$ is denoted by

$${[\alpha ]}_{\mathbb{Z}}.$$

Remark 8 

(Integral algebraicity). For

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}},$$

the following are equivalent:

$$\begin{array}{c}{\alpha }_{\mathbb{Z}}=0,\\ \alpha \in im({cl}_{X,\mathbb{Z}}),\\ \exists \Gamma \in C{H}^p(X)\mathrm{such}\mathrm{that}{cl}_{X,\mathbb{Z}}(\Gamma )=\alpha .\end{array}$$

Thus $Z^{2p}(X)=0$ is the integral Hodge conjecture in codimension p.

Let

$${\rho }_X:H^{2p}(X,\mathbb{Z}(p))⟶H^{2p}(X,\mathbb{Q}(p))$$

be the homomorphism induced by $\mathbb{Z}\hookrightarrow \mathbb{Q}$. It restricts to

$${\rho }_X^{\mathrm{Hdg}}:H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}⟶H^{2p}{(X,\mathbb{Q}(p))}_{\mathrm{Hdg}}.$$

The cycle-class maps are compatible with rationalization:

The left vertical map sends $\Gamma$ to $\Gamma \otimes 1$.

Definition 13 

(Rational Hodge defect group). The rational Hodge defect group of X in codimension p is

$$Z^{2p}{(X)}_{\mathbb{Q}}={\displaystyle \frac{H^{2p}{(X,\mathbb{Q}(p))}_{\mathrm{Hdg}}}{im({cl}_X)}}.$$

For

$${\alpha }_{\mathbb{Q}}\in H^{2p}{(X,\mathbb{Q}(p))}_{\mathrm{Hdg}},$$

the image of ${\alpha }_{\mathbb{Q}}$ in $Z^{2p}{(X)}_{\mathbb{Q}}$ is denoted by

$${[{\alpha }_{\mathbb{Q}}]}_{\mathbb{Q}}.$$

Remark 9 

(Rational algebraicity). For

$${\alpha }_{\mathbb{Q}}\in H^{2p}{(X,\mathbb{Q}(p))}_{\mathrm{Hdg}},$$

the following are equivalent:

$$\begin{array}{c}{{\alpha }_{\mathbb{Q}}}_{\mathbb{Q}}=0,\\ {\alpha }_{\mathbb{Q}}\in im({cl}_X),\\ \exists \Gamma \in C{H}^p{(X)}_{\mathbb{Q}}\mathrm{such}\mathrm{that}{cl}_X(\Gamma )={\alpha }_{\mathbb{Q}}.\end{array}$$

Thus $Z^{2p}{(X)}_{\mathbb{Q}}=0$ is the rational Hodge conjecture in codimension p.

Remark 10 

(Relation with existing theories). The quotient $Z^{2p}(X)$ is not a constructional invariant. It is the actual defect group of the integral Hodge conjecture. In codimension 2, and in related low-codimension settings, unramified cohomology gives refined information about this defect [1,2]. Diagonal decomposition and torsion-order methods give global annihilation results for certain torsion and cycle-theoretic invariants [3,4,5]. The constructional defects studied below are different: they depend on a specified sector and on a specified rational Hodge-cycle construction.

2.2. Rationalization and Torsion

Lemma 1 

(Torsion killed by rationalization). Let A be an abelian group. Then

$$A_{\mathrm{tors}}=ker\left(A⟶A{\otimes }_{\mathbb{Z}}\mathbb{Q}\right).$$

In particular,

$$Z^{2p}{(X)}_{\mathrm{tors}}=ker\left(Z^{2p}(X)⟶Z^{2p}(X){\otimes }_{\mathbb{Z}}\mathbb{Q}\right).$$

Proof. 

Let $a\in A_{\mathrm{tors}}$. Choose $N\ge 1$ such that $Na=0$. Then

$$a\otimes 1=Na\otimes {\displaystyle \frac{1}{N}}=0$$

in $A{\otimes }_{\mathbb{Z}}\mathbb{Q}$. Hence

$$A_{\mathrm{tors}}\subseteq ker\left(A\to A{\otimes }_{\mathbb{Z}}\mathbb{Q}\right).$$

Conversely, suppose that $a\otimes 1=0$. Let

$$\langle a\rangle \subseteq A$$

be the cyclic subgroup generated by a. The induced map

$$\langle a\rangle ⟶\langle a\rangle {\otimes }_{\mathbb{Z}}\mathbb{Q}$$

has kernel equal to the torsion subgroup of $\langle a\rangle$. Since $a\otimes 1=0$, the element a lies in this kernel. Thus a is torsion. This proves the first equality. The second equality is the same statement applied to $A=Z^{2p}(X)$. □

Proposition 6 

(Integral defect and rationalization). Let

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

If there exist $N\ge 1$ and $\Gamma \in C{H}^p(X)$ such that

$${cl}_{X,\mathbb{Z}}(\Gamma )=N\alpha ,$$

then ${[\alpha ]}_{\mathbb{Z}}\in Z^{2p}(X)$ is N-torsion. Hence

$${[\alpha ]}_{\mathbb{Z}}\otimes 1=0\mathrm{in}{Z}^{2p}(X){\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Proof. 

The equality

$${cl}_{X,\mathbb{Z}}(\Gamma )=N\alpha$$

implies

$$N{[\alpha ]}_{\mathbb{Z}}={[N\alpha ]}_{\mathbb{Z}}={[{cl}_{X,\mathbb{Z}}(\Gamma )]}_{\mathbb{Z}}=0$$

in $Z^{2p}(X)$. Thus ${[\alpha ]}_{\mathbb{Z}}$ is N-torsion. The final assertion follows from Lemma 1. □

Remark 11 

(Actual defect versus constructional defect). A nonzero element of $Z^{2p}(X)$ is an actual integral Hodge defect. A torsion element of $Z^{2p}(X)$ is invisible after tensoring with $\mathbb{Q}$. The present paper does not construct new counterexamples to the integral Hodge conjecture. It studies how rational Hodge-cycle constructions can leave finite constructional defects. These constructional defects may map to actual integral Hodge defects, but they may also measure only the finite loss of a chosen method.

2.3. Typed Sectors and Constructional Defects

A rational Hodge-cycle construction usually acts on a specified part of cohomology rather than on all of

$$H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

This specified part is the sector on which a partial algebraicity-transfer mechanism is defined.

Definition 14 

(Typed Hodge sector). A typed Hodge sector of X in codimension p is a finitely generated subgroup

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

specified independently of any unknown target class. Equivalently, it is a finitely generated Hodge subgroup together with the geometric data used to specify it. A sector is geometric if it is specified by data such as a product decomposition, a finite group action, an eigenspace for an algebraic correspondence, a family, a variation of Hodge structure, a Prym construction, an isogeny, a degeneration, a support condition, or a regulator source.

Remark 12 

(Exclusion of tautological sectors). A subgroup generated by an unknown target Hodge class is not a geometric sector in the sense of Definition 14. The sector must be fixed by the construction before any claim of algebraicity is made. No arbitrary Hodge projector, unconstructed motivic lift, correspondence known only cohomologically, or map defined by the target class is used to define $L_{\mathrm{Hdg}}$.

Definition 15 

(Actual algebraic part). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector. Its actual integral algebraic part is

$$L_{\mathrm{alg}}^{\mathrm{act}}=L_{\mathrm{Hdg}}\cap im({cl}_{X,\mathbb{Z}}).$$

The actual sector defect is

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathrm{act}}.$$

Lemma 2 

(Actual sector defect embeds in the standard defect). Let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. The inclusion of $L_{\mathrm{Hdg}}$ into $H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$ induces an injective homomorphism

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})⟶Z^{2p}(X).$$

Proof. 

The inclusion sends $L_{\mathrm{alg}}^{\mathrm{act}}$ into $im({cl}_{X,\mathbb{Z}})$. Therefore it induces a homomorphism

$$\phi :L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathrm{act}}⟶Z^{2p}(X).$$

Let $\alpha \in L_{\mathrm{Hdg}}$ represent a class in $ker(\phi )$. Then the image of $\alpha$ in $Z^{2p}(X)$ is zero, so

$$\alpha \in im({cl}_{X,\mathbb{Z}}).$$

Since also $\alpha \in L_{\mathrm{Hdg}}$, one has

$$\alpha \in L_{\mathrm{Hdg}}\cap im({cl}_{X,\mathbb{Z}})=L_{\mathrm{alg}}^{\mathrm{act}}.$$

Thus $\alpha$ represents zero in $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$. Hence $ker(\phi )=0$. □

Definition 16 

(Constructional algebraic part). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector. A constructional algebraic part for $L_{\mathrm{Hdg}}$ is a subgroup

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}$$

generated by integral cycle classes produced by a specified construction $\mathcal{C}$. The constructional defect of $\mathcal{C}$ is

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Remark 13 

(Actual and constructional defects). The actual quotient

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

measures the failure of integral algebraicity in the sector. The constructional quotient

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

measures the failure of a specified construction to generate the sector integrally. There is a natural surjection

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

because

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}.$$

Thus $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ may be nonzero even when the actual sector defect vanishes.

Definition 17 

(Soundness and coverage in a sector). Let $\mathcal{C}$ be a construction in a typed sector $L_{\mathrm{Hdg}}$. The construction is sound if

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq im({cl}_{X,\mathbb{Z}})$$

by specified Chow classes or by algebraic operations compatible with cycle classes. The construction has rational coverage if

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Remark 14 

(Soundness and coverage are distinct). Soundness is a cycle-class assertion. Coverage is a rational spanning assertion. A rational Hodge-cycle construction must provide both before the finite constructional defect has a definite meaning.

2.4. Saturation and Residual Quotients

Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector, and let

$$L\subseteq L_{\mathrm{Hdg}}$$

be a subgroup. In applications, L will be either $L_{\mathrm{alg}}^{\mathrm{act}}$ or $L_{\mathrm{alg}}^{\mathcal{C}}$.

Definition 18 

(Saturation). The saturation of L in $L_{\mathrm{Hdg}}$ is

$$L^{\mathrm{sat}}=\left\{\alpha \in L_{\mathrm{Hdg}}|\exists N\ge 1\mathrm{such}\mathrm{that}N\alpha \in L\right\}.$$

Lemma 3 

(Basic properties of saturation). The subgroup $L^{\mathrm{sat}}$ satisfies

$$L\subseteq L^{\mathrm{sat}}\subseteq L_{\mathrm{Hdg}}.$$

Moreover,

$$L^{\mathrm{sat}}/L={\left(L_{\mathrm{Hdg}}/L\right)}_{\mathrm{tors}}.$$

Proof. 

The inclusions are immediate from the definition. Let $\alpha \in L^{\mathrm{sat}}$. Then there exists $N\ge 1$ such that $N\alpha \in L$. Hence the class of $\alpha$ in $L_{\mathrm{Hdg}}/L$ is torsion. This gives

$$L^{\mathrm{sat}}/L\subseteq {\left(L_{\mathrm{Hdg}}/L\right)}_{\mathrm{tors}}.$$

Conversely, suppose that the class of $\alpha \in L_{\mathrm{Hdg}}$ is torsion in $L_{\mathrm{Hdg}}/L$. Then there exists $N\ge 1$ such that

$$N\alpha \in L.$$

Thus $\alpha \in L^{\mathrm{sat}}$. Therefore

$${\left(L_{\mathrm{Hdg}}/L\right)}_{\mathrm{tors}}\subseteq L^{\mathrm{sat}}/L.$$

The two inclusions prove the equality. □

Definition 19 

(Saturation quotient and rational residual quotient). Let $L\subseteq L_{\mathrm{Hdg}}$. The saturation quotient is

$$D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)=L^{\mathrm{sat}}/L.$$

The rational residual quotient is

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)=L_{\mathrm{Hdg}}/L^{\mathrm{sat}}.$$

Proposition 7 

(Decomposition by saturation). Let $L\subseteq L_{\mathrm{Hdg}}$. There is a short exact sequence

$$0⟶D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)⟶L_{\mathrm{Hdg}}/L⟶D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)⟶0.$$

Moreover,

$$D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)={\left(L_{\mathrm{Hdg}}/L\right)}_{\mathrm{tors}},$$

and $D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)$ is torsion-free.

Proof. 

The inclusions

$$L\subseteq L^{\mathrm{sat}}\subseteq L_{\mathrm{Hdg}}$$

give the quotient exact sequence

$$0⟶L^{\mathrm{sat}}/L⟶L_{\mathrm{Hdg}}/L⟶L_{\mathrm{Hdg}}/L^{\mathrm{sat}}⟶0.$$

This is the displayed exact sequence. By Lemma 3,

$$L^{\mathrm{sat}}/L={\left(L_{\mathrm{Hdg}}/L\right)}_{\mathrm{tors}}.$$

It remains to prove that $L_{\mathrm{Hdg}}/L^{\mathrm{sat}}$ is torsion-free. Let $\alpha \in L_{\mathrm{Hdg}}$ have torsion image in $L_{\mathrm{Hdg}}/L^{\mathrm{sat}}$. Then there exists $N\ge 1$ such that

$$N\alpha \in L^{\mathrm{sat}}.$$

By the definition of saturation, there exists $M\ge 1$ such that

$$M(N\alpha )\in L.$$

Thus $MN\alpha \in L$, so $\alpha \in L^{\mathrm{sat}}$. Hence the class of $\alpha$ in $L_{\mathrm{Hdg}}/L^{\mathrm{sat}}$ is zero. Therefore $D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)$ is torsion-free. □

Proposition 8 

(Rational coverage and saturation). Let $L\subseteq L_{\mathrm{Hdg}}$ be a subgroup. Then

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$

if and only if

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)=0.$$

In this case

$$L_{\mathrm{Hdg}}/L=D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)$$

is a torsion group. If $L_{\mathrm{Hdg}}$ is finitely generated, then this group is finite.

Proof. 

The equality

$$L\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}$$

holds if and only if for every $\alpha \in L_{\mathrm{Hdg}}$ there exists $N\ge 1$ such that $N\alpha \in L$. This condition is equivalent to

$$L^{\mathrm{sat}}=L_{\mathrm{Hdg}}.$$

By Definition 19, this is equivalent to

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)=0.$$

If $L^{\mathrm{sat}}=L_{\mathrm{Hdg}}$, then

$$L_{\mathrm{Hdg}}/L=L^{\mathrm{sat}}/L=D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L).$$

By Lemma 3, this quotient is torsion. If $L_{\mathrm{Hdg}}$ is finitely generated, then the quotient is finitely generated, and a finitely generated torsion abelian group is finite. □

Corollary 1 

(Integral generation inside a rationally covered sector). Let $L\subseteq L_{\mathrm{Hdg}}$. Assume

$$L\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

Then the following are equivalent:

$$L_{\mathrm{Hdg}}/L=0,D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)=0,L=L_{\mathrm{Hdg}}.$$

Proof. 

Under rational coverage, Proposition 8 gives

$$L_{\mathrm{Hdg}}/L=D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L).$$

Thus the first two conditions are equivalent. The quotient $L_{\mathrm{Hdg}}/L$ vanishes if and only if

$$L=L_{\mathrm{Hdg}}.$$

 □

Remark 15 

(Meaning of the two residual quotients). The quotient $D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)$ measures rational failure of L to cover the sector. The quotient $D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)$ measures the finite integral loss remaining after rational coverage has been achieved. For $L=L_{\mathrm{alg}}^{\mathrm{act}}$, this is an actual sector defect. For $L=L_{\mathrm{alg}}^{\mathcal{C}}$, this is the saturation part of the constructional defect.

2.5. Torsion-Free Sector Convention

Some later constructions begin with an equality after tensoring with $\mathbb{Q}$ and then compare it with an integral equality. Such a comparison requires control of the torsion kernel of rationalization.

Definition 20 

(Torsion-free sector lattice). A typed sector $L_{\mathrm{Hdg}}$ is torsion-free if the rationalization map

$$L_{\mathrm{Hdg}}⟶L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$

is injective.

Lemma 4 

(Integral equality from rational equality in a torsion-free sector). Let $L_{\mathrm{Hdg}}$ be a torsion-free typed sector. Let

$$\alpha ,\beta \in L_{\mathrm{Hdg}}.$$

If

$$\alpha \otimes 1=\beta \otimes 1\mathrm{in}{L}_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q},$$

then

$$\alpha =\beta \mathrm{in}{L}_{\mathrm{Hdg}}.$$

Proof. 

The equality

$$\alpha \otimes 1=\beta \otimes 1$$

is equivalent to

$$(\alpha -\beta )\otimes 1=0.$$

Since the rationalization map is injective on $L_{\mathrm{Hdg}}$, this implies

$$\alpha -\beta =0.$$

Hence $\alpha =\beta$. □

Remark 16 

(Use in multiplier arguments). Whenever a later argument upgrades a rational equality to an integral equality inside a sector, it either assumes the sector lattice is torsion-free, passes to the torsion-free quotient of the sector, or records the remaining torsion term explicitly. This prevents the equality in rational cohomology from being used as an uncontrolled integral equality.

3. The Finite-Index Principle and Defect Profiles

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2. Thus $L_{\mathrm{Hdg}}$ denotes a typed Hodge sector, $L_{\mathrm{alg}}^{\mathrm{act}}$ denotes its actual algebraic part, and $L_{\mathrm{alg}}^{\mathcal{C}}$ denotes the algebraic subgroup generated by a specified construction $\mathcal{C}$. Hodge classes are cohomological classes, and algebraicity means membership in the image of the appropriate cycle-class map.

The purpose of this section is to record the finite-index principle in a form adapted to constructional defects, and then to define the proof-level data which will be used later to compare rational Hodge-cycle constructions.

3.1. Rational Spanning and Finite Quotients

Let A be a finitely generated abelian group. Its torsion subgroup is denoted by $A_{\mathrm{tors}}$, and its rationalization is

$$A_{\mathbb{Q}}=A{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Lemma 5 

(Torsion and rationalization). Let A be a finitely generated abelian group. Then

$$A_{\mathrm{tors}}=ker\left(A⟶A_{\mathbb{Q}}\right).$$

Moreover,

$$A_{\mathbb{Q}}=0⟺A\mathrm{is}\mathrm{finite}.$$

Proof. 

Let $a\in A_{\mathrm{tors}}$. Choose $N\ge 1$ such that $Na=0$. Then

$$a\otimes 1=Na\otimes {\displaystyle \frac{1}{N}}=0$$

in $A_{\mathbb{Q}}$. Hence

$$A_{\mathrm{tors}}\subseteq ker(A\to A_{\mathbb{Q}}).$$

Conversely, assume that $a\otimes 1=0$ in $A_{\mathbb{Q}}$. Let $\langle a\rangle \subseteq A$ be the cyclic subgroup generated by a. The induced map

$$\langle a\rangle ⟶\langle a\rangle {\otimes }_{\mathbb{Z}}\mathbb{Q}$$

has kernel equal to ${\langle a\rangle }_{\mathrm{tors}}$. Since $a\otimes 1=0$, the element a lies in this kernel. Thus a is torsion. Therefore

$$A_{\mathrm{tors}}=ker(A\to A_{\mathbb{Q}}).$$

If A is finite, then $A=A_{\mathrm{tors}}$, so $A_{\mathbb{Q}}=0$. Conversely, if $A_{\mathbb{Q}}=0$, then every element of A lies in $ker(A\to A_{\mathbb{Q}})$. Hence $A=A_{\mathrm{tors}}$. Since A is finitely generated and torsion, A is finite. □

Proposition 9 

(Finite-index principle). Let

$$L\subseteq L_{\mathrm{Hdg}}$$

be finitely generated abelian groups. Then

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$

inside $L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}$ if and only if

$$L_{\mathrm{Hdg}}/L$$

is finite.

Proof. 

Set

$$Q=L_{\mathrm{Hdg}}/L.$$

There is a short exact sequence of abelian groups

$$0⟶L⟶L_{\mathrm{Hdg}}⟶Q⟶0.$$

Since $\mathbb{Q}$ is a flat $\mathbb{Z}$-module, tensoring with $\mathbb{Q}$ gives an exact sequence

$$0⟶L{\otimes }_{\mathbb{Z}}\mathbb{Q}⟶L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}⟶Q{\otimes }_{\mathbb{Z}}\mathbb{Q}⟶0.$$

Thus

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$

if and only if

$$Q{\otimes }_{\mathbb{Z}}\mathbb{Q}=0.$$

The group Q is finitely generated because $L_{\mathrm{Hdg}}$ is finitely generated. By Lemma 5, $Q{\otimes }_{\mathbb{Z}}\mathbb{Q}=0$ if and only if Q is finite. Hence the rational equality is equivalent to the finiteness of $L_{\mathrm{Hdg}}/L$. □

Corollary 2 

(Finite saturation quotient). Let $L\subseteq L_{\mathrm{Hdg}}$ be finitely generated abelian groups. If

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q},$$

then

$$L^{\mathrm{sat}}=L_{\mathrm{Hdg}},$$

and

$$L_{\mathrm{Hdg}}/L=L^{\mathrm{sat}}/L$$

is finite.

Proof. 

By Proposition 9, the quotient

$$Q=L_{\mathrm{Hdg}}/L$$

is finite. Let $\alpha \in L_{\mathrm{Hdg}}$. The class of $\alpha$ in Q has finite order. Hence there exists $N\ge 1$ such that

$$N\alpha \in L.$$

By the definition of saturation in Section 2.4,

$$\alpha \in L^{\mathrm{sat}}.$$

Thus

$$L_{\mathrm{Hdg}}\subseteq L^{\mathrm{sat}}.$$

The opposite inclusion is part of the definition of saturation, so

$$L^{\mathrm{sat}}=L_{\mathrm{Hdg}}.$$

Consequently,

$$L_{\mathrm{Hdg}}/L=L^{\mathrm{sat}}/L,$$

and this group is finite by Proposition 9. □

3.2. Constructional Defects

Let X be smooth projective over $\mathbb{C}$, and let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. Let $\mathcal{C}$ be a specified construction with constructional algebraic part

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}.$$

The constructional defect is

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Definition 21 

(Soundness in a sector). The construction $\mathcal{C}$ is sound in the sector $L_{\mathrm{Hdg}}$ if every class in $L_{\mathrm{alg}}^{\mathcal{C}}$ is represented by specified algebraic data. Equivalently, for every $\alpha \in L_{\mathrm{alg}}^{\mathcal{C}}$, there exists $\Gamma \in C{H}^p(X)$, or an explicitly specified algebraic construction with Chow output $\Gamma \in C{H}^p(X)$, such that

$${cl}_{X,\mathbb{Z}}(\Gamma )=\alpha .$$

Definition 22 

(Rational coverage in a sector). A sound construction $\mathcal{C}$ has rational coverage in $L_{\mathrm{Hdg}}$ if

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Remark 17 

(No tautological coverage). Rational coverage is not obtained by defining $L_{\mathrm{alg}}^{\mathcal{C}}$ from the unknown target class. The subgroup $L_{\mathrm{Hdg}}$ must be fixed geometrically, and the subgroup $L_{\mathrm{alg}}^{\mathcal{C}}$ must be supplied by Chow classes, algebraic correspondences, proper pushforwards, flat pullbacks, refined Gysin morphisms, exterior products, intersection products, finite trace descent, algebraic projector splittings, or support pushforwards from algebraic strata. Arbitrary Hodge projectors, maps defined by the target class, correspondences known only cohomologically, and unconstructed motivic lifts are not used as cycle certificates.

Proposition 10 

(Rational coverage gives finite constructional defect). Let $\mathcal{C}$ be a sound construction in a typed Hodge sector $L_{\mathrm{Hdg}}$. Assume rational coverage:

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is finite. Equivalently,

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}})=0,$$

and

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}).$$

Proof. 

The group $L_{\mathrm{Hdg}}$ is finitely generated by the definition of a typed Hodge sector. The group $L_{\mathrm{alg}}^{\mathcal{C}}$ is a subgroup of $L_{\mathrm{Hdg}}$, hence finitely generated. By Proposition 9, the quotient

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is finite.

The equivalence with

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}})=0$$

is Proposition 8. The equality

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}})$$

then follows from Proposition 7. □

Corollary 3 

(Rational algebraicity as finite-index algebraicity). Let $\mathcal{C}$ be a sound construction with rational coverage in a typed sector $L_{\mathrm{Hdg}}$. Then every class

$$\alpha \in L_{\mathrm{Hdg}}$$

satisfies

$$\exists N_{\alpha }\ge 1,\exists {\Gamma }_{\alpha }\in C{H}^p(X)$$

such that

$${cl}_{X,\mathbb{Z}}({\Gamma }_{\alpha })=N_{\alpha }\alpha .$$

Proof. 

Let $\alpha \in L_{\mathrm{Hdg}}$. By Proposition 10, the quotient

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is finite. Hence the class of $\alpha$ in $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ has finite order. Therefore there exists $N_{\alpha }\ge 1$ such that

$$N_{\alpha }\alpha \in L_{\mathrm{alg}}^{\mathcal{C}}.$$

By soundness of $\mathcal{C}$, there exists ${\Gamma }_{\alpha }\in C{H}^p(X)$, or a specified Chow output ${\Gamma }_{\alpha }$, such that

$${cl}_{X,\mathbb{Z}}({\Gamma }_{\alpha })=N_{\alpha }\alpha .$$

 □

Remark 18 

(What remains after rational coverage). After rational coverage, the remaining question is not whether

$$L_{\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}$$

holds. The remaining question is the structure of the finite group

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}},$$

or equivalently

$$D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}).$$

The later sections identify geometric operations that produce explicit proof-level data for this finite group.

3.3. Constructional Exponents and Proof-Level Annihilators

Definition 23 

(Constructional exponent). Let $\mathcal{C}$ be a sound construction with rational coverage in a typed sector $L_{\mathrm{Hdg}}$. The constructional exponent of $\mathcal{C}$ on $L_{\mathrm{Hdg}}$ is

$$e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp{D}_{\mathcal{C}}(L_{\mathrm{Hdg}}).$$

Definition 24 

(Proof-level annihilator). Let $\mathcal{C}$ be a construction in a typed sector $L_{\mathrm{Hdg}}$. An integer $N\ge 1$ is a proof-level annihilator for $\mathcal{C}$ if

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Equivalently, for every $\alpha \in L_{\mathrm{Hdg}}$,

$$N\alpha \in L_{\mathrm{alg}}^{\mathcal{C}}.$$

Lemma 6 

(Uniform annihilator and cycle classes). Let $\mathcal{C}$ be a sound construction in a typed Hodge sector $L_{\mathrm{Hdg}}$. Let $N\ge 1$. The following are equivalent:

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0$$

and

$$\forall \alpha \in L_{\mathrm{Hdg}},\exists {\Gamma }_{\alpha }\in C{H}^p(X)\mathrm{such}\mathrm{that}{cl}_{X,\mathbb{Z}}({\Gamma }_{\alpha })=N\alpha .$$

Proof. 

Assume first that

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Let $\alpha \in L_{\mathrm{Hdg}}$. The class of $N\alpha$ in

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is zero. Hence

$$N\alpha \in L_{\mathrm{alg}}^{\mathcal{C}}.$$

By soundness of $\mathcal{C}$, there exists ${\Gamma }_{\alpha }\in C{H}^p(X)$, or a specified Chow output ${\Gamma }_{\alpha }$, such that

$${cl}_{X,\mathbb{Z}}({\Gamma }_{\alpha })=N\alpha .$$

Conversely, assume that for every $\alpha \in L_{\mathrm{Hdg}}$ there is a cycle ${\Gamma }_{\alpha }\in C{H}^p(X)$ with

$${cl}_{X,\mathbb{Z}}({\Gamma }_{\alpha })=N\alpha .$$

Then $N\alpha \in L_{\mathrm{alg}}^{\mathcal{C}}$ for every $\alpha \in L_{\mathrm{Hdg}}$, provided these cycles are outputs of the construction $\mathcal{C}$. Hence the class of $N\alpha$ in $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ is zero for every $\alpha$, which is exactly

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

 □

Proposition 11 

(Existence of an abstract annihilator). Let $\mathcal{C}$ be a sound construction with rational coverage in a typed Hodge sector $L_{\mathrm{Hdg}}$. Then there exists $N\ge 1$ such that

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

More precisely,

$$e_{\mathcal{C}}(L_{\mathrm{Hdg}})·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Proof. 

By Proposition 10,

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

is finite. Its exponent

$$e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp{D}_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

is therefore defined and annihilates the group by definition. □

Remark 19 

(Abstract exponent versus proof-level annihilator). The constructional exponent $e_{\mathcal{C}}(L_{\mathrm{Hdg}})$ exists after the finite group $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ is known. A proof-level annihilator is stronger data: it is an integer visible in the construction before the quotient is computed. Degrees, group orders, projector denominators, isogeny degrees, norm exponents, boundary indices, and Bockstein coefficients are examples of proof-level data.

Corollary 4 

(Annihilator implies rational coverage in the generated sector). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector and let $L\subseteq L_{\mathrm{Hdg}}$ be a subgroup. Suppose that there exists $N\ge 1$ such that

$$N{L}_{\mathrm{Hdg}}\subseteq L.$$

Then

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

In particular, if $L=L_{\mathrm{alg}}^{\mathcal{C}}$, then the construction has rational coverage in the sector generated by L and $L_{\mathrm{Hdg}}$.

Proof. 

Let $\alpha \in L_{\mathrm{Hdg}}$. Since

$$N\alpha \in L,$$

one has

$$\alpha \otimes 1=(N\alpha )\otimes {\displaystyle \frac{1}{N}}\in L{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Thus

$$L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}\subseteq L{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

The reverse inclusion follows from $L\subseteq L_{\mathrm{Hdg}}$. Therefore the two rational spans are equal. □

3.4. Defect Profiles

The constructional defect

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

records a finite quotient when rational coverage holds. The proof of the construction may contain more refined data explaining how this quotient is produced.

Definition 25 

(Defect profile). Let $\mathcal{C}$ be a sound construction in a typed sector $L_{\mathrm{Hdg}}$. Assume rational coverage. The defect profile of $\mathcal{C}$ on $L_{\mathrm{Hdg}}$ is the tuple

$$Prof(\mathcal{C},L_{\mathrm{Hdg}})=\left(D_{\mathcal{C}}(L_{\mathrm{Hdg}}),e_{\mathcal{C}}(L_{\mathrm{Hdg}}),{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right),$$

where:

• $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ is the constructional defect;
• $e_{\mathcal{C}}(L_{\mathrm{Hdg}})$ is its exponent;
• ${\mathcal{A}}_{\mathcal{C}}$ is the proof-level annihilator, kernel, cokernel, boundary, saturation, or coefficient data supplied by the proof;
• ${\tau }_{\mathcal{C}}$ is the defect type.

Remark 20 

(Defect profile is constructional). The profile

$$Prof(\mathcal{C},L_{\mathrm{Hdg}})$$

is not an invariant of X alone. It depends on the specified sector, the specified construction, and the specified constructional algebraic subgroup. Different constructions on the same variety may have different defect profiles.

Definition 26 

(Defect-profile equivalence). Two construction-sector pairs

$$({\mathcal{C}}_1,L_1),({\mathcal{C}}_2,L_2)$$

are defect-profile equivalent if their profiles have the same defect type and the same kind of proof-level data:

$${\tau }_{{\mathcal{C}}_1}={\tau }_{{\mathcal{C}}_2},{\mathcal{A}}_{{\mathcal{C}}_1}\mathrm{and}{\mathcal{A}}_{{\mathcal{C}}_2}\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{structural}\mathrm{form}.$$

A finer equivalence may also require an isomorphism

$$D_{{\mathcal{C}}_1}(L_1)\cong D_{{\mathcal{C}}_2}(L_2).$$

Remark 21 

(Induced grouping). Defect-profile equivalence is not asserted to be the unique possible classification of rational Hodge-cycle constructions. It is the grouping induced by the finite integral data carried by the proofs. This is the sense in which later case studies are compared.

3.5. Multiplier, Saturation, and Realization-Loss Types

Definition 27 

(Multiplier-supported defect). Let $\mathcal{C}$ be a construction in a typed sector $L_{\mathrm{Hdg}}$. The constructional defect of $\mathcal{C}$ is multiplier-supported if the proof supplies abelian groups or lattices $M_1,M_2$, homomorphisms

$$S:M_1⟶M_2,T:M_2⟶M_1,$$

and an integer $N\ge 1$ such that

$$T\circ S=N·{id}_{M_1},$$

with S and T realized by algebraic operations compatible with the cycle-class maps on the sector under consideration.

Proposition 12 

(Multiplier-supported annihilation). Let $\mathcal{C}$ be a sound construction in a typed sector $L_{\mathrm{Hdg}}$. Assume that its defect is multiplier-supported by

$$S,T,N.$$

Assume that for every $\alpha \in L_{\mathrm{Hdg}}$, the class $S(\alpha )$ is represented by an algebraic source to which T may be applied algebraically. Then

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Proof. 

Let $\alpha \in L_{\mathrm{Hdg}}$. By hypothesis, $S(\alpha )$ is represented by an algebraic source. Since T is realized by an algebraic operation compatible with cycle classes, the class

$$T(S(\alpha ))$$

belongs to $L_{\mathrm{alg}}^{\mathcal{C}}$. The multiplier identity gives

$$T(S(\alpha ))=N\alpha .$$

Thus

$$N\alpha \in L_{\mathrm{alg}}^{\mathcal{C}}.$$

Therefore the class of $\alpha$ in

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is killed by N. Since $\alpha$ was arbitrary,

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

 □

Proposition 13 

(Composite multiplier bound). Let $\mathcal{C}$ be a sound construction obtained by a finite sequence of multiplier-supported steps with multipliers

$$N_1,\dots ,N_r.$$

Assume that each step is compatible with the relevant cycle-class maps and that the output of each step is an admissible input for the next. Then

$$N_1\dots N_r·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Proof. 

Let $\alpha \in L_{\mathrm{Hdg}}$. Applying the first multiplier-supported step gives algebraicity of $N_1\alpha$ at the corresponding stage. Applying the second step to the output gives algebraicity after multiplication by $N_2$, hence by $N_1{N}_2$ relative to the original class. Continuing through the finite sequence gives algebraicity of

$$N_1\dots N_r\alpha$$

in the constructional algebraic lattice. Hence the class of $\alpha$ in $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ is killed by $N_1\dots N_r$. □

Definition 28 

(Saturation-supported defect). Let $\mathcal{C}$ be a sound construction with rational coverage in $L_{\mathrm{Hdg}}$. The defect of $\mathcal{C}$ is saturation-supported if its proof-level data are given by the finite-index lattice inclusion

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q},$$

and the relevant defect is the finite lattice quotient

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 14 

(Saturation exponent). Let $\mathcal{C}$ be a sound construction with saturation-supported defect in $L_{\mathrm{Hdg}}$. Then

$$e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp\left(L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}\right).$$

If $L_{\mathrm{Hdg}}$ is free and a basis is fixed, then this exponent is computed by the Smith normal form of a matrix whose columns are the coordinates of algebraic generators of $L_{\mathrm{alg}}^{\mathcal{C}}$.

Proof. 

The first equality is the definition of the constructional exponent. If $L_{\mathrm{Hdg}}$ is free and a basis is fixed, the algebraic generators of $L_{\mathrm{alg}}^{\mathcal{C}}$ determine an integral matrix. Rational coverage means that this matrix has full rank. The Smith normal form gives the invariant factors of

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

The largest invariant factor is the exponent. □

Definition 29 

(Realization-loss defect). Let $\mathcal{C}$ be a construction involving a homomorphism

$$\rho :M⟶R$$

between abelian groups attached to algebraic cycles, cohomology, regulators, specialization, Abel–Jacobi maps, or finite-coefficient cohomology. The defect of $\mathcal{C}$ is realization-loss type if its proof-level data are controlled by

$$ker(\rho ),im(\rho ),coker(\rho ),$$

or by a finite-coefficient connecting morphism.

Remark 22 

(Realization-loss is not automatically an integral Hodge defect). A realization-loss defect need not be a subgroup of $Z^{2p}(X)$. Regulator kernels and higher Chow indecomposability quotients, for example, are cycle-theoretic or realization-theoretic defects. They become actual integral Hodge defects only after a specified comparison with the ordinary cycle-class map.

Remark 23 

(Logical direction). The constructions studied later separate four assertions:

$$\begin{array}{c}\mathrm{soundness}\mathrm{of}\mathrm{the}\mathrm{algebraic}\mathrm{sources},\\ \mathrm{rational}\mathrm{coverage}\mathrm{of}\mathrm{the}\mathrm{typed}\mathrm{Hodge}\sec \mathrm{tor},\\ \mathrm{finiteness}\mathrm{of}\mathrm{the}\mathrm{constructional}\mathrm{defect},\\ \mathrm{existence}\mathrm{or}\mathrm{computation}\mathrm{of}\mathrm{proof}-\mathrm{level}\mathrm{data}.\end{array}$$

Soundness is a cycle-class statement. Coverage is a rational spanning statement. Finiteness is a finite-index statement. A proof-level annihilator, lattice index, kernel, cokernel, boundary term, or coefficient source records how the finite integral information is lost.

4. Degree, Norm, and Averaging Defects

We use the notation and conventions of Section 1.1. We also use the notation of Section 2 and Section 3. Thus Hodge classes are cohomological classes, algebraicity means membership in the image of the appropriate cycle-class map, and constructional defects are quotients by algebraic subgroups supplied by specified constructions.

The mechanisms in this section are multiplier-supported in the sense of Definition 27. In each case there are algebraic operations S and T, and an integer $N\ge 1$, such that

$$T\circ S=N·id$$

on the sector under consideration. Finite trace gives $N=deg(f)$. Norm maps in codimension one give the same degree. Finite group averaging gives $N=|G|$. These integers are proof-level annihilators in the sense of Definition 24.

4.1. Finite Covers and Trace

Let

$$f:Y⟶X$$

be a finite surjective morphism of smooth projective complex varieties of degree $d\ge 1$. For $p\ge 0$, flat pullback and proper pushforward give homomorphisms

$$f^{*}:C{H}^p(X)⟶C{H}^p(Y),f_{*}:C{H}^p(Y)⟶C{H}^p(X).$$

The corresponding cohomological maps are

$$f^{*}:H^{2p}(X,\mathbb{Z}(p))⟶H^{2p}(Y,\mathbb{Z}(p)),$$

and

$$f_{*}:H^{2p}(Y,\mathbb{Z}(p))⟶H^{2p}(X,\mathbb{Z}(p)).$$

They are compatible with cycle classes:

$${cl}_{Y,\mathbb{Z}}(f^{*}\Gamma )=f^{*}{cl}_{X,\mathbb{Z}}(\Gamma ),\Gamma \in C{H}^p(X),$$

and

$${cl}_{X,\mathbb{Z}}(f_{*}\Delta )=f_{*}{cl}_{Y,\mathbb{Z}}(\Delta ),\Delta \in C{H}^p(Y).$$

These are the standard functorial compatibilities of Chow groups and cycle-class maps [16,17,27].

Lemma 7 

(Trace identity). Let $f:Y\to X$ be finite surjective of degree d. Then

$$f_{*}{f}^{*}=d·{id}_{C{H}^p(X)}$$

on $C{H}^p(X)$, and

$$f_{*}{f}^{*}=d·{id}_{H^{2p}(X,\mathbb{Z}(p))}$$

on $H^{2p}(X,\mathbb{Z}(p))$.

Proof. 

It suffices first to check the Chow identity on integral subvarieties. Let $V\subseteq X$ be an integral closed subvariety of codimension p. Since f is finite surjective of degree d, the cycle-theoretic pullback $f^{*}[V]$ is the cycle associated with the finite scheme $Y{\times }_{X}V$. Pushing forward to X gives the degree of the finite map over the generic point of V. This degree is d. Hence

$$f_{*}{f}^{*}[V]=d[V].$$

By additivity, the equality holds on cycles, and therefore on $C{H}^p(X)$.

For cohomology, the trace map for a finite morphism satisfies

$$f_{*}{f}^{*}=d·id$$

on singular cohomology. This equality is compatible with the Chow identity above and with Tate twists. Thus the same identity holds on $H^{2p}(X,\mathbb{Z}(p))$. □

Lemma 8 

(Hodge compatibility of finite trace). Let $f:Y\to X$ be finite surjective. Then

$$f^{*}\left(H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}\right)\subseteq H^{2p}{(Y,\mathbb{Z}(p))}_{\mathrm{Hdg}},$$

and

$$f_{*}\left(H^{2p}{(Y,\mathbb{Z}(p))}_{\mathrm{Hdg}}\right)\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

Proof. 

The morphism f is algebraic and proper. Pullback and pushforward on cohomology are morphisms of Hodge structures with the indicated Tate twists. Therefore they preserve Hodge classes [16,17]. □

Definition 30 

(Trace construction). Let $f:Y\to X$ be finite surjective of degree d. The trace construction ${\mathcal{C}}_{tr}(f)$ in codimension p has typed Hodge sector

$$L_{tr}^p(f)=\left\{\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}|f^{*}\alpha \in im({cl}_{Y,\mathbb{Z}})\right\}.$$

Its constructional algebraic part is

$$L_{tr,\mathrm{alg}}^p(f)=L_{tr}^p(f)\cap im({cl}_{X,\mathbb{Z}}).$$

The associated constructional defect is

$$D_{tr}^p(f)=L_{tr}^p(f)/L_{tr,\mathrm{alg}}^p(f).$$

Proposition 15 

(Degree annihilation). Let $f:Y\to X$ be finite surjective of degree d. Let

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

Assume that $f^{*}\alpha$ is integrally algebraic on Y; equivalently, there exists $\Gamma \in C{H}^p(Y)$ such that

$${cl}_{Y,\mathbb{Z}}(\Gamma )=f^{*}\alpha .$$

Then $d\alpha$ is integrally algebraic on X. More precisely,

$${cl}_{X,\mathbb{Z}}(f_{*}\Gamma )=d\alpha .$$

Consequently,

$$d{[\alpha ]}_{\mathbb{Z}}=0\mathrm{in}{Z}^{2p}(X).$$

Proof. 

By compatibility of proper pushforward with the cycle-class map,

$${cl}_{X,\mathbb{Z}}(f_{*}\Gamma )=f_{*}{cl}_{Y,\mathbb{Z}}(\Gamma ).$$

Using the hypothesis gives

$$f_{*}{cl}_{Y,\mathbb{Z}}(\Gamma )=f_{*}{f}^{*}\alpha .$$

By Lemma 7,

$$f_{*}{f}^{*}\alpha =d\alpha .$$

Hence

$${cl}_{X,\mathbb{Z}}(f_{*}\Gamma )=d\alpha .$$

Thus $d\alpha$ is in the image of ${cl}_{X,\mathbb{Z}}$, and the class ${[\alpha ]}_{\mathbb{Z}}$ is killed by d. □

Corollary 5 

(Trace constructional defect). Let $f:Y\to X$ be finite surjective of degree d. Then

$$d·D_{tr}^p(f)=0.$$

Thus the trace construction is multiplier-supported with proof-level annihilator d.

Proof. 

Let $\alpha \in L_{tr}^p(f)$. By Definition 30, the class $f^{*}\alpha$ is integrally algebraic on Y. Proposition 15 gives

$$d\alpha \in im({cl}_{X,\mathbb{Z}}).$$

Since $d\alpha \in L_{tr}^p(f)$, one has

$$d\alpha \in L_{tr,\mathrm{alg}}^p(f).$$

Thus the image of $\alpha$ in $D_{tr}^p(f)$ is killed by d. □

Remark 24 

(Rational meaning of trace descent). If $f^{*}\alpha$ is integrally algebraic on Y, then Proposition 15 gives

$$d\alpha ={cl}_{X,\mathbb{Z}}(f_{*}\Gamma ).$$

After tensoring with $\mathbb{Q}$,

$$\alpha \otimes 1={cl}_X\left({\displaystyle \frac{1}{d}}{f}_{*}\Gamma \right).$$

Thus trace descent proves rational algebraicity of α, while the integral proof records the multiplier d. Kollár-type examples show that degree phenomena of this form are not merely formal artifacts [7,8].

4.2. Norm Maps

Let $f:Y\to X$ be finite surjective of degree d. For line bundles, there is a norm homomorphism

$${Nm}_f:Pic(Y)⟶Pic(X).$$

It satisfies

$${Nm}_f(f^{*}M)\simeq M^{\otimes d},M\in Pic(X).$$

It is compatible with first Chern classes:

$$c_1({Nm}_f(L))=f_{*}{c}_1(L),L\in Pic(Y).$$

Lemma 9 

(Norm identity in codimension one). Let $f:Y\to X$ be finite surjective of degree d. For every $M\in Pic(X)$,

$$c_1({Nm}_f(f^{*}M))=d{c}_1(M)$$

in $H^2(X,\mathbb{Z}(1))$.

Proof. 

By the defining identity of the norm map,

$${Nm}_f(f^{*}M)\simeq M^{\otimes d}.$$

Taking first Chern classes gives

$$c_1({Nm}_f(f^{*}M))=c_1(M^{\otimes d})=d{c}_1(M).$$

 □

Definition 31 

(Norm construction). Let $f:Y\to X$ be finite surjective of degree d. The norm construction ${\mathcal{C}}_{Nm}(f)$ has typed Hodge sector

$$L_{Nm}(f)=\left\{\alpha \in H^2{(X,\mathbb{Z}(1))}_{\mathrm{Hdg}}|\exists L\in Pic(Y)\mathrm{with}{c}_1(L)=f^{*}\alpha \right\}.$$

Its constructional algebraic part is

$$L_{Nm,\mathrm{alg}}(f)=L_{Nm}(f)\cap im({cl}_{X,\mathbb{Z}}),$$

and its constructional defect is

$$D_{Nm}(f)=L_{Nm}(f)/L_{Nm,\mathrm{alg}}(f).$$

Proposition 16 

(Norm annihilation for divisor classes). Let $f:Y\to X$ be finite surjective of degree d. Let

$$\alpha \in H^2{(X,\mathbb{Z}(1))}_{\mathrm{Hdg}}.$$

Assume that there exists $L\in Pic(Y)$ such that

$$c_1(L)=f^{*}\alpha .$$

Then $d\alpha$ is integrally algebraic on X. More precisely,

$$c_1({Nm}_f(L))=d\alpha .$$

Consequently,

$$d{[\alpha ]}_{\mathbb{Z}}=0\mathrm{in}{Z}^2(X).$$

Proof. 

By compatibility of the norm map with first Chern classes,

$$c_1({Nm}_f(L))=f_{*}{c}_1(L).$$

Using the hypothesis $c_1(L)=f^{*}\alpha$, this becomes

$$c_1({Nm}_f(L))=f_{*}{f}^{*}\alpha .$$

By Lemma 7,

$$f_{*}{f}^{*}\alpha =d\alpha .$$

Hence

$$c_1({Nm}_f(L))=d\alpha .$$

Since ${Nm}_f(L)\in Pic(X)=C{H}^1(X)$, the class $d\alpha$ is integrally algebraic. Therefore $d{[\alpha ]}_{\mathbb{Z}}=0$ in $Z^2(X)$. □

Corollary 6 

(Norm constructional defect). Let $f:Y\to X$ be finite surjective of degree d. Then

$$d·D_{Nm}(f)=0.$$

Thus the norm construction is multiplier-supported with proof-level annihilator d.

Proof. 

Let $\alpha \in L_{Nm}(f)$. By Definition 31, there exists $L\in Pic(Y)$ such that

$$c_1(L)=f^{*}\alpha .$$

By Proposition 16,

$$d\alpha \in im({cl}_{X,\mathbb{Z}}).$$

Thus

$$d\alpha \in L_{Nm,\mathrm{alg}}(f),$$

and the class of $\alpha$ in $D_{Nm}(f)$ is killed by d. □

Remark 25 

(Norm kernels and Prym-type constructions). For covers of curves and for abelian varieties attached to them, Prym-type objects arise from connected components of norm kernels. The integral comparison is controlled by finite kernels and cokernels of norm maps and by associated isogenies. The preceding proposition records the basic cohomological mechanism in codimension one. Later Prym and Weil-type applications use the same multiplier-supported pattern with the relevant isogeny degree, norm-kernel exponent, or polarization index replacing d.

4.3. Finite Group Averaging

Let G be a finite group acting algebraically on a smooth projective complex variety X. For $g\in G$, write

$$g:X⟶X$$

for the corresponding automorphism. For $p\ge 0$, the induced maps

$$g^{*}:C{H}^p(X)⟶C{H}^p(X)$$

and

$$g^{*}:H^{2p}(X,\mathbb{Z}(p))⟶H^{2p}(X,\mathbb{Z}(p))$$

are compatible with cycle classes:

$${cl}_{X,\mathbb{Z}}(g^{*}\Gamma )=g^{*}{cl}_{X,\mathbb{Z}}(\Gamma ),\Gamma \in C{H}^p(X).$$

Let

$$S_G=\sum _{g\in G}{g}^{*}$$

as an endomorphism of $C{H}^p(X)$ and of $H^{2p}(X,\mathbb{Z}(p))$. After tensoring with $\mathbb{Q}$, the operator

$$e_G={\displaystyle \frac{1}{|G|}}{S}_G$$

is the projector onto G-invariants.

Lemma 10 

(Averaging identity). Let

$$H^{2p}{(X,\mathbb{Z}(p))}^G=\{\alpha \in H^{2p}(X,\mathbb{Z}(p))\mid g^{*}\alpha =\alpha \mathrm{for}\mathrm{all}g\in G\}.$$

For every

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}^G,$$

one has

$$S_G(\alpha )=|G|\alpha .$$

Proof. 

Let $\alpha \in H^{2p}{(X,\mathbb{Z}(p))}^G$. Then $g^{*}\alpha =\alpha$ for every $g\in G$. Therefore

$$S_G(\alpha )=\sum _{g\in G}{g}^{*}\alpha =\sum _{g\in G}\alpha =|G|\alpha .$$

 □

Definition 32 

(Averaging construction). Let G act algebraically on X. Define the G-invariant Hodge sector

$$L_G^p(X)=H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}\cap H^{2p}{(X,\mathbb{Z}(p))}^G.$$

Define the averaging constructional algebraic subgroup

$$L_{G,\mathrm{alg}}^{p,\mathrm{avg}}(X)=\left\{\alpha \in L_G^p(X)|\exists \Delta \in C{H}^p(X)\mathrm{such}\mathrm{that}|G|\alpha ={cl}_{X,\mathbb{Z}}\left(\sum _{g\in G}{g}^{*}\Delta \right)\right\}.$$

The averaging constructional defect is

$$D_G^{p,\mathrm{avg}}(X)=L_G^p(X)/L_{G,\mathrm{alg}}^{p,\mathrm{avg}}(X).$$

Proposition 17 

(Averaging annihilation). Let G be a finite group acting algebraically on X. If

$$\alpha \in L_G^p(X)$$

and there exists $\Delta \in C{H}^p(X)$ such that

$$S_G{cl}_{X,\mathbb{Z}}(\Delta )=|G|\alpha$$

in $H^{2p}(X,\mathbb{Z}(p))$, then

$$|G|\alpha ={cl}_{X,\mathbb{Z}}\left(\sum _{g\in G}{g}^{*}\Delta \right).$$

Consequently,

$${|G|[\alpha ]}_{\mathbb{Z}}=0\mathrm{in}{Z}^{2p}(X).$$

Proof. 

By compatibility of cycle classes with algebraic pullback,

$$g^{*}{cl}_{X,\mathbb{Z}}(\Delta )={cl}_{X,\mathbb{Z}}(g^{*}\Delta )$$

for every $g\in G$. Hence

$$\begin{array}{cc}\hfill |G|\alpha & =S_G{cl}_{X,\mathbb{Z}}(\Delta )\hfill \\ & =\sum _{g\in G}{g}^{*}{cl}_{X,\mathbb{Z}}(\Delta )\hfill \\ & =\sum _{g\in G}{cl}_{X,\mathbb{Z}}(g^{*}\Delta )\hfill \\ & ={cl}_{X,\mathbb{Z}}\left(\sum _{g\in G}{g}^{*}\Delta \right).\hfill \end{array}$$

Thus $|G|\alpha$ is integrally algebraic, and the image of $\alpha$ in $Z^{2p}(X)$ is killed by $|G|$. □

Corollary 7 

(Averaging constructional defect). Let G be a finite group acting algebraically on X. Then

$$|G|·D_G^{p,\mathrm{avg}}(X)=0.$$

Thus finite group averaging is multiplier-supported with proof-level annihilator $|G|$.

Proof. 

Let $\alpha \in L_G^p(X)$. If its class is represented in $D_G^{p,\mathrm{avg}}(X)$, then by Definition 32 the subgroup $L_{G,\mathrm{alg}}^{p,\mathrm{avg}}(X)$ contains precisely those classes whose $|G|$-multiples are obtained by averaging algebraic cycles. Hence

$$|G|\alpha \in L_{G,\mathrm{alg}}^{p,\mathrm{avg}}(X)$$

for every class in the constructional image considered by the averaging construction. Therefore the quotient is killed by $|G|$. □

Proposition 18 

(Rational averaging in torsion-free sectors). Let G act algebraically on X. Let

$$L\subseteq L_G^p(X)$$

be a torsion-free typed sector in the sense of Definition 20. Let $\alpha \in L$. Assume that there exists $\Delta \in C{H}^p(X)$ such that

$${\displaystyle \frac{1}{|G|}}{S}_G{cl}_X(\Delta \otimes 1)=\alpha \otimes 1$$

in $H^{2p}(X,\mathbb{Q}(p))$, and assume that

$$S_G{cl}_{X,\mathbb{Z}}(\Delta )\in L.$$

Then

$$|G|\alpha ={cl}_{X,\mathbb{Z}}\left(\sum _{g\in G}{g}^{*}\Delta \right)$$

in L.

Proof. 

The rational equality gives

$$S_G{cl}_X(\Delta \otimes 1)=|G|(\alpha \otimes 1)$$

in $H^{2p}(X,\mathbb{Q}(p))$. The integral classes

$$S_G{cl}_{X,\mathbb{Z}}(\Delta )\mathrm{and}|G|\alpha$$

belong to L by hypothesis and have the same rationalization. Since L is torsion-free, Lemma 4 implies

$$S_G{cl}_{X,\mathbb{Z}}(\Delta )=|G|\alpha$$

in L. The result follows from the computation in Proposition 17. □

Remark 26 

(Denominator of the averaging idempotent). The rational operator

$$e_G={\displaystyle \frac{1}{|G|}}\sum _{g\in G}{g}^{*}$$

is an idempotent on $H^{2p}(X,\mathbb{Q}(p))$. The integral operation available without division is

$$S_G=\sum _{g\in G}{g}^{*}.$$

Thus the averaging construction loses integrality through the denominator $|G|$. When a rational averaging equality is used to obtain an integral statement, the torsion-free sector convention of Section 2.5 is required, or else a residual torsion term must be recorded.

Remark 27 

(Degree, norm, and averaging as multiplier-supported defects). Finite trace, norm, and averaging arguments have the same constructional form. Each supplies algebraic operations S and T with

$$T\circ S=N·id$$

on the sector under consideration. The proof-level annihilator is the degree of a finite morphism, the degree or exponent attached to a norm comparison, or the order of a finite group. Hence these are multiplier-supported defects in the sense of Definition 27.

5. Projector and Correspondence Defects

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2 and the defect-profile terminology of Section 3. All Hodge classes are cohomological classes. All projectors and correspondences in this section are algebraic correspondences. No arbitrary Hodge-theoretic projector is used.

Let X and Y be smooth projective complex varieties of dimensions

$$dimX=n,dimY=m.$$

For an algebraic correspondence

$$\Gamma \in C{H}^r(Y\times X),$$

we write

$${\Gamma }_{*}:C{H}^q(Y)⟶C{H}^{q+r-m}(X)$$

for the induced map on Chow groups. On cohomology, the same correspondence induces

$${\Gamma }_{*}:H^a(Y,\mathbb{Z}(b))⟶H^{a+2(r-m)}(X,\mathbb{Z}(b+r-m)).$$

The Chow-theoretic and cohomological actions are compatible with cycle classes:

$${cl}_{X,\mathbb{Z}}({\Gamma }_{*}\Delta )={\Gamma }_{*}{cl}_{Y,\mathbb{Z}}(\Delta ),\Delta \in C{H}^q(Y).$$

This is the standard cycle-class compatibility for algebraic correspondences [16,17,27].

5.1. Rational Algebraic Projectors

Let X be smooth projective of dimension n. A rational algebraic correspondence of degree 0 is an element

$$\pi \in C{H}^n{(X\times X)}_{\mathbb{Q}}.$$

It acts on Chow groups and cohomology by

$${\pi }_{*}:C{H}^p{(X)}_{\mathbb{Q}}⟶C{H}^p{(X)}_{\mathbb{Q}}$$

and

$${\pi }_{*}:H^{2p}(X,\mathbb{Q}(p))⟶H^{2p}(X,\mathbb{Q}(p)).$$

Definition 33 

(Algebraic rational projector). An algebraic rational projector on X is an element

$$\pi \in C{H}^n{(X\times X)}_{\mathbb{Q}}$$

such that

$$\pi \circ \pi =\pi \mathrm{in}C{H}^n{(X\times X)}_{\mathbb{Q}}.$$

A denominator for π is an integer $N\ge 1$ such that

$$\Pi =N\pi \in C{H}^n(X\times X).$$

Remark 28 

(Algebraicity of the projector). The projector in Definition 33 is an algebraic correspondence with rational coefficients. A projector defined only on the Hodge structure

$$H^{2p}(X,\mathbb{Q}(p))$$

is not used unless it is induced by such an algebraic correspondence.

Let

$$\pi \in C{H}^n{(X\times X)}_{\mathbb{Q}}$$

be an algebraic rational projector with denominator N, and set

$$\Pi =N\pi \in C{H}^n(X\times X).$$

The integral correspondence $\Pi$ acts on integral cohomology:

$${\Pi }_{*}:H^{2p}(X,\mathbb{Z}(p))⟶H^{2p}(X,\mathbb{Z}(p)).$$

After tensoring with $\mathbb{Q}$,

$${\Pi }_{*}=N{\pi }_{*}.$$

Definition 34 

(Projector sector). Let $\pi \in C{H}^n{(X\times X)}_{\mathbb{Q}}$ be an algebraic rational projector, and let $N\ge 1$ be a denominator with

$$\Pi =N\pi \in C{H}^n(X\times X).$$

Define

$$L_{\pi }^p(X)=\left\{\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}|{\pi }_{*}({\alpha }_{\mathbb{Q}})={\alpha }_{\mathbb{Q}}\right\},$$

where

$${\alpha }_{\mathbb{Q}}=\alpha \otimes 1.$$

The actual algebraic part is

$$L_{\pi ,\mathrm{alg}}^{p,\mathrm{act}}(X)=L_{\pi }^p(X)\cap im({cl}_{X,\mathbb{Z}}).$$

Lemma 11 

(Cleared projector identity in torsion-free sectors). Let $\pi \in C{H}^n{(X\times X)}_{\mathbb{Q}}$ be an algebraic rational projector. Let $N\ge 1$ and

$$\Pi =N\pi \in C{H}^n(X\times X).$$

Let

$$L\subseteq L_{\pi }^p(X)$$

be a torsion-free typed sector such that ${\Pi }_{*}L\subseteq L$. If

$$\alpha \in L,$$

then

$${\Pi }_{*}\alpha =N\alpha$$

in L.

Proof. 

Since $\alpha \in L_{\pi }^p(X)$,

$${\pi }_{*}({\alpha }_{\mathbb{Q}})={\alpha }_{\mathbb{Q}}.$$

Because $\Pi =N\pi$, one has

$${\Pi }_{*}({\alpha }_{\mathbb{Q}})=N{\pi }_{*}({\alpha }_{\mathbb{Q}})=N{\alpha }_{\mathbb{Q}}.$$

The classes

$${\Pi }_{*}\alpha \mathrm{and}N\alpha$$

belong to L and have the same rationalization. Since L is torsion-free, Lemma 4 gives

$${\Pi }_{*}\alpha =N\alpha$$

in L. □

Definition 35 

(Projector constructional algebraic subgroup). Let $L\subseteq L_{\pi }^p(X)$ be a torsion-free typed sector with ${\Pi }_{*}L\subseteq L$. Define

$$L_{\pi ,\mathrm{alg}}^{p,\mathcal{C}}(X;L)$$

to be the subgroup of L generated by all classes of the form

$${cl}_{X,\mathbb{Z}}({\Pi }_{*}\Delta ),\Delta \in C{H}^p(X),$$

which lie in L. The corresponding constructional defect is

$$D_{\pi }^{p,\mathcal{C}}(X;L)=L/L_{\pi ,\mathrm{alg}}^{p,\mathcal{C}}(X;L).$$

Proposition 19 

(Projector denominator annihilation). Let $\pi \in C{H}^n{(X\times X)}_{\mathbb{Q}}$ be an algebraic rational projector. Let

$$\Pi =N\pi \in C{H}^n(X\times X)$$

for some $N\ge 1$. Let

$$L\subseteq L_{\pi }^p(X)$$

be a torsion-free typed sector with ${\Pi }_{*}L\subseteq L$. Assume that for every $\alpha \in L$ there exists ${\Delta }_{\alpha }\in C{H}^p(X)$ such that

$${cl}_X({\Delta }_{\alpha }\otimes 1)={\alpha }_{\mathbb{Q}}$$

and such that the rational algebraicity of ${\alpha }_{\mathbb{Q}}$ is obtained by applying ${\pi }_{*}$ to ${\Delta }_{\alpha }\otimes 1$. Then

$$N·D_{\pi }^{p,\mathcal{C}}(X;L)=0.$$

Proof. 

Let $\alpha \in L$. The correspondence $\Pi$ is integral, so

$${\Pi }_{*}{\Delta }_{\alpha }\in C{H}^p(X).$$

Cycle-class compatibility for algebraic correspondences gives

$${cl}_{X,\mathbb{Z}}({\Pi }_{*}{\Delta }_{\alpha })={\Pi }_{*}{cl}_{X,\mathbb{Z}}({\Delta }_{\alpha }).$$

After tensoring with $\mathbb{Q}$, one obtains

$${cl}_X(({\Pi }_{*}{\Delta }_{\alpha })\otimes 1)={\Pi }_{*}{cl}_X({\Delta }_{\alpha }\otimes 1).$$

By the hypothesis

$${cl}_X({\Delta }_{\alpha }\otimes 1)={\alpha }_{\mathbb{Q}},$$

and by $\Pi =N\pi$, the right-hand side is

$$N{\pi }_{*}{\alpha }_{\mathbb{Q}}.$$

Since $\alpha \in L_{\pi }^p(X)$,

$${\pi }_{*}{\alpha }_{\mathbb{Q}}={\alpha }_{\mathbb{Q}}.$$

Thus

$${cl}_X(({\Pi }_{*}{\Delta }_{\alpha })\otimes 1)=N{\alpha }_{\mathbb{Q}}.$$

Both

$${cl}_{X,\mathbb{Z}}({\Pi }_{*}{\Delta }_{\alpha })\mathrm{and}N\alpha$$

belong to L and have the same rationalization. Since L is torsion-free, Lemma 4 gives

$${cl}_{X,\mathbb{Z}}({\Pi }_{*}{\Delta }_{\alpha })=N\alpha .$$

Therefore

$$N\alpha \in L_{\pi ,\mathrm{alg}}^{p,\mathcal{C}}(X;L).$$

The class of $\alpha$ in $D_{\pi }^{p,\mathcal{C}}(X;L)$ is killed by N. Since $\alpha$ was arbitrary,

$$N·D_{\pi }^{p,\mathcal{C}}(X;L)=0.$$

 □

Corollary 8 

(Projector constructional defect). Let $\pi \in C{H}^n{(X\times X)}_{\mathbb{Q}}$ be an algebraic rational projector. Let

$$\Pi =N\pi \in C{H}^n(X\times X)$$

for some $N\ge 1$. Let

$$L\subseteq L_{\pi }^p(X)$$

be a torsion-free typed sector with ${\Pi }_{*}L\subseteq L$. Assume that the hypotheses of Proposition 19 hold. Then

$$N·D_{\pi }^{p,\mathcal{C}}(X;L)=0.$$

Proof. 

This is Proposition 19. □

Remark 29 

(Integral meaning of rational projectors). A rational projector

$$\pi \in C{H}^n{(X\times X)}_{\mathbb{Q}}$$

acts integrally only after choosing a denominator

$$N\ge 1,N\pi \in C{H}^n(X\times X).$$

Thus a rational projector argument is multiplier-supported: the proof-level annihilator is the denominator N. The torsion-free sector convention is needed when the argument upgrades rational projector identities to integral lattice equalities.

5.2. Correspondence Images and Finite Cokernels

Let Y and X be smooth projective complex varieties with

$$dimY=m,dimX=n.$$

Let

$$\Gamma \in C{H}^r(Y\times X)$$

be an algebraic correspondence. Fix integers $a,b,p$ such that

$$a+2(r-m)=2p,b+r-m=p.$$

Then $\Gamma$ induces

$${\Gamma }_{*}:H^a(Y,\mathbb{Z}(b))⟶H^{2p}(X,\mathbb{Z}(p)).$$

Definition 36 

(Algebraic source lattice for a correspondence). Let

$$M_{\mathrm{alg}}\subseteq H^a(Y,\mathbb{Z}(b))$$

be a finitely generated subgroup such that every element of $M_{\mathrm{alg}}$ is represented by an algebraic cycle or by a specified algebraic construction whose cycle-class realization is already known. Define

$$L_{\Gamma ,\mathrm{alg}}={\Gamma }_{*}(M_{\mathrm{alg}})\subseteq H^{2p}(X,\mathbb{Z}(p)).$$

Remark 30 

(Soundness of the source). The subgroup $M_{\mathrm{alg}}$ is part of the input data. It is not defined by the target class in X. Soundness requires that its elements come from Chow classes or from already certified algebraic constructions. The algebraicity of $L_{\Gamma ,\mathrm{alg}}$ follows from correspondence compatibility of the cycle-class map.

Lemma 12 

(Soundness of correspondence images). Assume that

$$M_{\mathrm{alg}}={cl}_{Y,\mathbb{Z}}(B)$$

for some subgroup $B\subseteq C{H}^q(Y)$, where the indices are compatible with

$${\Gamma }_{*}:C{H}^q(Y)⟶C{H}^p(X).$$

Then

$$L_{\Gamma ,\mathrm{alg}}\subseteq im({cl}_{X,\mathbb{Z}}).$$

More precisely, if

$$\mu ={cl}_{Y,\mathbb{Z}}(\Delta ),\Delta \in B,$$

then

$${\Gamma }_{*}\mu ={cl}_{X,\mathbb{Z}}({\Gamma }_{*}\Delta ).$$

Proof. 

Let $\mu ={cl}_{Y,\mathbb{Z}}(\Delta )$ with $\Delta \in B$. By compatibility of algebraic correspondences with the cycle-class map,

$${\Gamma }_{*}\mu ={\Gamma }_{*}{cl}_{Y,\mathbb{Z}}(\Delta )={cl}_{X,\mathbb{Z}}({\Gamma }_{*}\Delta ).$$

Since ${\Gamma }_{*}\Delta \in C{H}^p(X)$, the class ${\Gamma }_{*}\mu$ lies in $im({cl}_{X,\mathbb{Z}})$. Hence

$$L_{\Gamma ,\mathrm{alg}}\subseteq im({cl}_{X,\mathbb{Z}}).$$

 □

Definition 37 

(Correspondence constructional defect). Let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. Let

$$L_{\Gamma ,\mathrm{alg}}\subseteq L_{\mathrm{Hdg}}$$

be a sound correspondence image as in Definition 36. The correspondence constructional defect is

$$D_{\Gamma }(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}.$$

Proposition 20 

(Finite correspondence defect from rational coverage). Let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a finitely generated typed Hodge sector. Let

$$L_{\Gamma ,\mathrm{alg}}\subseteq L_{\mathrm{Hdg}}$$

be a sound correspondence image. Assume rational coverage:

$$L_{\Gamma ,\mathrm{alg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\Gamma }(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}$$

is finite.

Proof. 

The group $L_{\mathrm{Hdg}}$ is finitely generated by hypothesis. The subgroup $L_{\Gamma ,\mathrm{alg}}$ is finitely generated because it is a subgroup of $L_{\mathrm{Hdg}}$. The rational coverage hypothesis is

$$L_{\Gamma ,\mathrm{alg}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

By Proposition 9, the quotient

$$L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}$$

is finite. □

Corollary 9 

(Exponent of a finite correspondence defect). Under the hypotheses of Proposition 20, let

$$N_{\Gamma }=exp{D}_{\Gamma }(L_{\mathrm{Hdg}}).$$

Then

$$N_{\Gamma }·D_{\Gamma }(L_{\mathrm{Hdg}})=0.$$

Equivalently, for every $\alpha \in L_{\mathrm{Hdg}}$, there exists

$${\Delta }_{\alpha }\in C{H}^p(X)$$

constructed from the correspondence image such that

$${cl}_{X,\mathbb{Z}}({\Delta }_{\alpha })=N_{\Gamma }\alpha .$$

Proof. 

Since $D_{\Gamma }(L_{\mathrm{Hdg}})$ is finite by Proposition 20, its exponent $N_{\Gamma }$ is defined and satisfies

$$N_{\Gamma }·D_{\Gamma }(L_{\mathrm{Hdg}})=0.$$

Let $\alpha \in L_{\mathrm{Hdg}}$. The equality above implies

$$N_{\Gamma }\alpha \in L_{\Gamma ,\mathrm{alg}}.$$

By the soundness of the correspondence image, every element of $L_{\Gamma ,\mathrm{alg}}$ is represented by a Chow class on X obtained from the specified algebraic sources by ${\Gamma }_{*}$. Hence there exists ${\Delta }_{\alpha }\in C{H}^p(X)$ with

$${cl}_{X,\mathbb{Z}}({\Delta }_{\alpha })=N_{\Gamma }\alpha .$$

 □

Remark 31 

(Correspondence arguments). Many rational algebraicity arguments have the form

$${\Gamma }_{*}(M_{\mathrm{alg}}\otimes \mathbb{Q})=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

If $M_{\mathrm{alg}}$ is soundly algebraic, then this gives a finite constructional defect in the sector. The integral quotient

$$L_{\mathrm{Hdg}}/{\Gamma }_{*}(M_{\mathrm{alg}})$$

is not detected by the rational equality. It is saturation-supported unless the proof also supplies a multiplier identity for the correspondence image.

Remark 32 

(Saturation and multiplier readings of correspondence images). The correspondence defect in this subsection is saturation-supported because the only input is rational coverage:

$$L_{\Gamma ,\mathrm{alg}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

If, in addition, the construction supplies an algebraic correspondence ${\Gamma }^{\prime }$ in the opposite direction and an identity

$${\Gamma }^{\prime }\circ \Gamma =Nid$$

on the sector under consideration, then the same correspondence construction has a multiplier-supported component. In that case N is a proof-level annihilator for the corresponding constructional defect. Without such a push–pull identity, the defect is only the finite-index lattice quotient

$$L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}.$$

5.3. Idempotent Denominators

Let A be a finite-dimensional semisimple $\mathbb{Q}$-algebra acting on X by algebraic correspondences. Thus there is a $\mathbb{Q}$-algebra homomorphism

$$A⟶C{H}^n{(X\times X)}_{\mathbb{Q}}.$$

Let

$$e\in A$$

be an idempotent:

$$e^2=e.$$

Write the corresponding algebraic rational correspondence as

$${\pi }_e\in C{H}^n{(X\times X)}_{\mathbb{Q}}.$$

Definition 38 

(Denominator of an algebraic idempotent). A positive integer $N_e\ge 1$ is a denominator of ${\pi }_e$ if

$${\Pi }_e=N_e{\pi }_e\in C{H}^n(X\times X).$$

Remark 33 

(Examples of algebraic idempotents). Algebraic idempotents with denominators include character idempotents for finite group actions, eigenspace idempotents associated with algebraic correspondence actions, and Künneth or motivic projectors when these projectors are represented by algebraic cycles. A cohomological splitting without an algebraic correspondence is not an algebraic idempotent in the sense used here.

Definition 39 

(Idempotent sector). Let ${\pi }_e\in C{H}^n{(X\times X)}_{\mathbb{Q}}$ be an algebraic idempotent. Define

$$L_e^p(X)=\left\{\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}|{({\pi }_e)}_{*}({\alpha }_{\mathbb{Q}})={\alpha }_{\mathbb{Q}}\right\}.$$

Proposition 21 

(Idempotent denominator defect). Let ${\pi }_e\in C{H}^n{(X\times X)}_{\mathbb{Q}}$ be an algebraic idempotent. Let

$${\Pi }_e=N_e{\pi }_e\in C{H}^n(X\times X)$$

for some $N_e\ge 1$. Let

$$L\subseteq L_e^p(X)$$

be a torsion-free typed sector such that ${({\Pi }_e)}_{*}L\subseteq L$. Assume that every $\alpha \in L$ is rationally obtained by applying ${({\pi }_e)}_{*}$ to a Chow class. Then the associated idempotent constructional defect is killed by $N_e$.

Proof. 

This is Proposition 19 applied to the algebraic rational projector

$$\pi ={\pi }_e$$

and the integral correspondence

$$\Pi ={\Pi }_e.$$

For every $\alpha \in L$, the hypothesis gives a Chow class ${\Delta }_{\alpha }\in C{H}^p(X)$ whose rational cycle class maps to ${\alpha }_{\mathbb{Q}}$ after applying ${({\pi }_e)}_{*}$. Applying the integral correspondence ${\Pi }_e$ and using cycle-class compatibility gives

$$N_e\alpha$$

as an integral cycle class in the constructional algebraic subgroup. Hence the class of $\alpha$ in the idempotent constructional defect is killed by $N_e$. Since $\alpha$ was arbitrary, the defect is killed by $N_e$. □

Corollary 10 

(Character-projector denominator). Let a finite group G act algebraically on X. Let

$$\chi :G⟶\mathbb{C}$$

be a rational-valued irreducible character whose associated central idempotent is defined over $\mathbb{Q}$. The idempotent

$$e_{\chi }={\displaystyle \frac{dim\chi }{|G|}}\sum _{g\in G}\chi (g^{-1})g^{*}$$

is a rational algebraic idempotent. If $N_{\chi }\ge 1$ clears the coefficients of $e_{\chi }$, then

$$N_{\chi }{e}_{\chi }$$

is an integral algebraic correspondence. Whenever the hypotheses of Proposition 21 hold, the corresponding constructional defect is annihilated by $N_{\chi }$.

Proof. 

The expression for $e_{\chi }$ is a rational linear combination of the graphs of the algebraic automorphisms $g:X\to X$. Multiplying by an integer $N_{\chi }$ clearing the coefficients gives an integral algebraic correspondence

$$N_{\chi }{e}_{\chi }\in C{H}^n(X\times X).$$

The assertion follows from Proposition 21. □

Remark 34 

(Projectors as finite-loss devices). An algebraic rational projector gives a rational decomposition. Its integral content is obtained only after clearing denominators. Thus a projector argument has two separate components:

$$\begin{array}{c}\mathrm{soundness}\text{:}\mathrm{the}\mathrm{projector}\mathrm{is}\mathrm{represented}\mathrm{by}\mathrm{an}\mathrm{algebraic}\mathrm{correspondence};\\ \mathrm{coverage}\text{:}\mathrm{the}\mathrm{projector}\mathrm{image}\mathrm{spans}\mathrm{the}\mathrm{intended}\mathrm{Hodge}\sec \mathrm{tor}\mathrm{rationally}.\end{array}$$

When both hold, the multiplier-supported part of the defect is controlled by the denominator. Any remaining finite-index quotient is a saturation component of the same constructional profile.

6. Lattice Saturation Defects

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2 and the defect-profile terminology of Section 3. Thus Hodge classes are cohomological classes, algebraicity means membership in the image of the relevant cycle-class map, and constructional defects are quotients by algebraic subgroups supplied by specified constructions.

The constructions in this section are saturation-supported in the sense of Definition 28. The characteristic form is a finite-index inclusion

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q},$$

where $L_{\mathrm{alg}}^{\mathcal{C}}$ is generated by algebraic cycle classes supplied by a specified construction $\mathcal{C}$. The finite quotient

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is the saturation component of the constructional defect. No multiplier identity is assumed in this section.

6.1. Rational Generation Versus Integral Generation

Let X be a smooth projective complex variety, let $p\ge 0$, and let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a torsion-free typed Hodge sector of rank r. Let

$$Z_1,\dots ,Z_s\in C{H}^p(X)$$

be algebraic cycles such that

$${cl}_{X,\mathbb{Z}}(Z_i)\in L_{\mathrm{Hdg}},1\le i\le s.$$

Definition 40 

(Cycle-generated constructional algebraic lattice). The constructional algebraic lattice generated by $Z_1,\dots ,Z_s$ in $L_{\mathrm{Hdg}}$ is

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})={\displaystyle \sum _{i=1}^s}\mathbb{Z}{cl}_{X,\mathbb{Z}}(Z_i)\subseteq L_{\mathrm{Hdg}}.$$

The associated constructional defect is

$$D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•}).$$

Definition 41 

(Rational and integral generation). The cycles $Z_1,\dots ,Z_s$ rationally generate the sector $L_{\mathrm{Hdg}}$ if

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•}){\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

They integrally generate $L_{\mathrm{Hdg}}$ if

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})=L_{\mathrm{Hdg}}.$$

Proposition 22 

(Saturation defect of rational generation). Assume that $Z_1,\dots ,Z_s$ rationally generate $L_{\mathrm{Hdg}}$. Then

$$D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$$

is finite. Moreover,

$$D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})=D_{\mathrm{sat}}\left(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})\right).$$

Proof. 

By Definition 41,

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•}){\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

The group $L_{\mathrm{Hdg}}$ is finitely generated and torsion-free. Therefore Proposition 9 gives that

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$$

is finite.

By Proposition 8, rational coverage is equivalent to

$$D_{\mathrm{rat}}\left(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})\right)=0.$$

Then Proposition 7 gives

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})=D_{\mathrm{sat}}\left(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})\right).$$

 □

Corollary 11 

(Integral generation criterion). Assume that $Z_1,\dots ,Z_s$ rationally generate $L_{\mathrm{Hdg}}$. Then the following are equivalent:

$$\begin{array}{c}{L}_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})=L_{\mathrm{Hdg}},\\ D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})=0,\\ L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})\mathrm{is}\mathrm{saturated}\mathrm{in}{L}_{\mathrm{Hdg}}.\end{array}$$

Proof. 

The equivalence

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})=L_{\mathrm{Hdg}}⟺D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})=0$$

follows from Definition 40.

Assume rational generation. By Proposition 22, the saturation of $L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$ in $L_{\mathrm{Hdg}}$ is $L_{\mathrm{Hdg}}$. Hence $L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$ is saturated in $L_{\mathrm{Hdg}}$ if and only if

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})=L_{\mathrm{Hdg}}.$$

This proves the equivalence of the three conditions. □

Remark 35 

(Soundness and coverage). The cycles $Z_1,\dots ,Z_s$ provide sound algebraic classes because each ${cl}_{X,\mathbb{Z}}(Z_i)$ is the cycle class of an actual Chow class. Rational coverage is the separate assertion that these cycle classes span $L_{\mathrm{Hdg}}\otimes \mathbb{Q}$. The saturation-supported defect measures the finite difference between rational coverage and integral generation.

Proposition 23 

(Annihilator from a finite index). Assume that $Z_1,\dots ,Z_s$ rationally generate $L_{\mathrm{Hdg}}$, and set

$$D=D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}}).$$

Let

$$N=exp(D).$$

Then for every $\alpha \in L_{\mathrm{Hdg}}$ there exist integers $a_1,\dots ,a_s\in \mathbb{Z}$ such that

$$N\alpha ={\displaystyle \sum _{i=1}^s}{a}_i{cl}_{X,\mathbb{Z}}(Z_i).$$

Equivalently,

$$N\alpha ={cl}_{X,\mathbb{Z}}\left({\displaystyle \sum _{i=1}^s}{a}_i{Z}_i\right).$$

Proof. 

Since $N=exp(D)$, one has

$$N·D=0.$$

Let $\alpha \in L_{\mathrm{Hdg}}$. The class of $\alpha$ in

$$D=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$$

is killed by N. Hence

$$N\alpha \in L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•}).$$

By Definition 40, there exist $a_1,\dots ,a_s\in \mathbb{Z}$ such that

$$N\alpha ={\displaystyle \sum _{i=1}^s}{a}_i{cl}_{X,\mathbb{Z}}(Z_i).$$

By additivity of the cycle-class map,

$${\displaystyle \sum _{i=1}^s}{a}_i{cl}_{X,\mathbb{Z}}(Z_i)={cl}_{X,\mathbb{Z}}\left({\displaystyle \sum _{i=1}^s}{a}_i{Z}_i\right).$$

 □

6.2. Product and Invariant-Theoretic Examples

Let X be a smooth projective complex variety. Product decompositions, endomorphism algebras, finite group actions, and algebraic correspondences often specify Hodge sectors. A rational statement usually identifies the sector after tensoring with $\mathbb{Q}$. The integral statement asks whether the algebraic generators form a saturated sublattice.

Definition 42 

(Invariant algebraic sector). Let G be a finite group acting algebraically on X. Let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a torsion-free typed Hodge sector contained in

$$H^{2p}{(X,\mathbb{Z}(p))}^G.$$

Let $Z_1,\dots ,Z_s\in C{H}^p(X)$ satisfy

$${cl}_{X,\mathbb{Z}}(Z_i)\in L_{\mathrm{Hdg}},1\le i\le s.$$

If

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q},$$

then $L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$ is called a rationally generating algebraic invariant lattice in $L_{\mathrm{Hdg}}$.

Proposition 24 

(Invariant saturation defect). In the situation of Definition 42, assume that $L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$ rationally generates $L_{\mathrm{Hdg}}$. Then

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$$

is finite. This finite quotient is the saturation-supported component of the invariant rational-generation construction.

Proof. 

This is Proposition 22 applied to the sector $L_{\mathrm{Hdg}}$ and to the constructional algebraic lattice $L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})$. □

Remark 36 

(Products and Hodge groups). For products of curves, products of surfaces, and self-products with automorphisms, rational Hodge classes may be described using Hodge groups, endomorphism algebras, finite group actions, or algebraic correspondences. Such descriptions can give rational generation of a prescribed sector. The integral question is whether the resulting algebraic lattice is saturated. This is the form in which saturation-supported defects occur in Morrison-type product constructions and in self-product constructions with automorphisms [19,20,21].

Definition 43 

(Correspondence-generated saturation defect). Let $Y_1,\dots ,Y_s$ be smooth projective complex varieties. For each i, let

$${\Gamma }_i\in C{H}^{r_i}(Y_i\times X)$$

be an algebraic correspondence, and let

$$B_i\subseteq C{H}^{q_i}(Y_i)$$

be a subgroup such that

$${\Gamma }_{i,*}{B}_i\subseteq C{H}^p(X).$$

Let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a torsion-free typed sector. Define

$$L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}={\displaystyle \sum _{i=1}^s}{cl}_{X,\mathbb{Z}}\left({\Gamma }_{i,*}{B}_i\right)\subseteq L_{\mathrm{Hdg}}.$$

If

$$L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q},$$

then the correspondence-generated saturation defect is

$$D_{\Gamma }^{\mathrm{sat}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 25 

(Finite correspondence saturation defect). In the situation of Definition 43, the group

$$D_{\Gamma }^{\mathrm{sat}}(L_{\mathrm{Hdg}})$$

is finite.

Proof. 

The subgroup $L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}$ is generated by cycle classes of actual Chow classes on X, namely by the classes of ${\Gamma }_{i,*}{B}_i$. Hence soundness follows from correspondence compatibility of cycle classes [16,17,27]. The rational coverage hypothesis is

$$L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

By Proposition 9,

$$L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}$$

is finite. □

Remark 37 

(No cohomological-only generators). A rational invariant-theoretic description of $H^{2p}{(X,\mathbb{Q}(p))}_{\mathrm{Hdg}}$ does not by itself define an algebraic lattice. The subgroup $L_{\mathrm{alg}}^{\mathcal{C}}$ must be generated by cycle classes of actual Chow classes or by algebraic correspondences applied to already algebraic sources. Arbitrary Hodge projectors and cohomological splittings are not used as algebraic generators.

6.3. Computable Indices

Let L be a free abelian group of rank r, and let

$$v_1,\dots ,v_s\in L$$

be elements whose $\mathbb{Q}$-span is $L{\otimes }_{\mathbb{Z}}\mathbb{Q}$. Let

$$L^{\prime }={\displaystyle \sum _{i=1}^s}\mathbb{Z}{v}_i\subseteq L.$$

Then $L/L^{\prime }$ is finite by Proposition 9.

Definition 44 

(Presentation matrix). Choose a basis

$$e_1,\dots ,e_r$$

of L. Write

$$v_i={\displaystyle \sum _{j=1}^r}{a}_{ji}{e}_j,a_{ji}\in \mathbb{Z}.$$

The matrix

$$A=(a_{ji})\in M_{r\times s}(\mathbb{Z})$$

is the presentation matrix of $L^{\prime }\subseteq L$ with respect to the chosen basis.

Proposition 26 

(Smith normal form and saturation defect). Let $A\in M_{r\times s}(\mathbb{Z})$ be the presentation matrix of $L^{\prime }\subseteq L$. Assume that

$$L^{\prime }\otimes \mathbb{Q}=L\otimes \mathbb{Q}.$$

Then there exist

$$U\in {GL}_r(\mathbb{Z}),V\in {GL}_s(\mathbb{Z}),$$

and integers

$$d_1,\dots ,d_r\ge 1$$

with

$$d_i\mid d_{i+1},1\le i<r,$$

such that $UAV$ has diagonal entries $d_1,\dots ,d_r$ and all remaining entries zero. Moreover,

$$L/L^{\prime }\cong {\displaystyle \underset{i=1}{\overset{r}{⨁}}}\mathbb{Z}/d_i\mathbb{Z}.$$

Proof. 

The Smith normal form theorem gives matrices $U\in {GL}_r(\mathbb{Z})$ and $V\in {GL}_s(\mathbb{Z})$ such that $UAV$ has diagonal form with invariant factors $d_i$ satisfying $d_i\mid d_{i+1}$. Since

$$L^{\prime }\otimes \mathbb{Q}=L\otimes \mathbb{Q},$$

the matrix A has rank r. Thus there are r nonzero invariant factors.

Changing the basis of L by U and changing the generating set of $L^{\prime }$ by V do not change the quotient $L/L^{\prime }$. In the Smith basis, $L^{\prime }$ is generated by

$$d_1{e}_1,\dots ,d_r{e}_r.$$

Therefore

$$L/L^{\prime }\cong {\displaystyle \underset{i=1}{\overset{r}{⨁}}}\mathbb{Z}/d_i\mathbb{Z}.$$

 □

Corollary 12 

(Square full-rank case). Let $L\cong {\mathbb{Z}}^r$, and let $L^{\prime }\subseteq L$ be generated by r elements with presentation matrix

$$A\in M_r(\mathbb{Z}).$$

Assume

$$detA\ne 0.$$

Then $L/L^{\prime }$ is finite and

$$|L/L^{\prime }|=|detA|.$$

Proof. 

By Proposition 26,

$$L/L^{\prime }\cong {\displaystyle \underset{i=1}{\overset{r}{⨁}}}\mathbb{Z}/d_i\mathbb{Z}.$$

Hence

$$|L/L^{\prime }|={\displaystyle \prod _{i=1}^r}{d}_i.$$

The product of the Smith invariant factors equals $|detA|$. Therefore

$$|L/L^{\prime }|=|detA|.$$

 □

Definition 45 

(Computable saturation defect of algebraic generators). Let $L_{\mathrm{Hdg}}$ be a torsion-free typed Hodge sector of rank r, and let $Z_1,\dots ,Z_s\in C{H}^p(X)$ be algebraic cycles such that

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})={\displaystyle \sum _{i=1}^s}\mathbb{Z}{cl}_{X,\mathbb{Z}}(Z_i)\subseteq L_{\mathrm{Hdg}}.$$

Choose a basis $e_1,\dots ,e_r$ of $L_{\mathrm{Hdg}}$. Write

$${cl}_{X,\mathbb{Z}}(Z_i)={\displaystyle \sum _{j=1}^r}{a}_{ji}{e}_j,a_{ji}\in \mathbb{Z}.$$

The matrix

$$A(Z_{•})=(a_{ji})\in M_{r\times s}(\mathbb{Z})$$

is the algebraic generator matrix of $Z_1,\dots ,Z_s$ in $L_{\mathrm{Hdg}}$.

Proposition 27 

(Computation of the saturation defect). Assume that $Z_1,\dots ,Z_s$ rationally generate $L_{\mathrm{Hdg}}$. Let

$$A(Z_{•})$$

be the algebraic generator matrix. If the Smith invariant factors of $A(Z_{•})$ are

$$d_1,\dots ,d_r,d_i\mid d_{i+1},$$

then

$$D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})\cong {\displaystyle \underset{i=1}{\overset{r}{⨁}}}\mathbb{Z}/d_i\mathbb{Z}.$$

In particular,

$$exp{D}_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})=d_r,$$

and

$$\left|D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})\right|={\displaystyle \prod _{i=1}^r}{d}_i.$$

Proof. 

By rational generation,

$$L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

Hence $A(Z_{•})$ has rank r. Apply Proposition 26 to

$$L=L_{\mathrm{Hdg}},L^{\prime }=L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•}).$$

This gives

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•})\cong {\displaystyle \underset{i=1}{\overset{r}{⨁}}}\mathbb{Z}/d_i\mathbb{Z}.$$

By Definition 40,

$$D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}(Z_{•}).$$

The formulas for the exponent and cardinality follow from the invariant factor decomposition. □

Example 2 

(Two generators in a rank-two sector). Let

$$L_{\mathrm{Hdg}}=\mathbb{Z}{e}_1\oplus \mathbb{Z}{e}_2.$$

Let $Z_1,Z_2\in C{H}^p(X)$ have cycle classes

$${cl}_{X,\mathbb{Z}}(Z_1)=2e_1,{cl}_{X,\mathbb{Z}}(Z_2)=e_1+3e_2.$$

Then

$$A(Z_{•})=\left(\begin{array}{cc}2& 1\\ 0& 3\end{array}\right).$$

The determinant is 6. Hence the algebraic classes rationally generate $L_{\mathrm{Hdg}}$, and

$$\left|D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})\right|=6.$$

The Smith normal form has invariant factors 1 and 6. Therefore

$$D_{\mathcal{C}}(Z_{•};L_{\mathrm{Hdg}})\cong \mathbb{Z}/6\mathbb{Z}.$$

Thus rational generation leaves a cyclic saturation-supported defect of order 6.

Remark 38 

(Integral content of rational generation). A rational spanning theorem identifies

$$L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

A saturation computation identifies the finite quotient

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Thus the integral information lost by rational generation is, in favorable cases, an explicit finite abelian group computed by integral linear algebra. The computation is meaningful only after the generators of $L_{\mathrm{alg}}^{\mathcal{C}}$ have been supplied by actual algebraic cycles or by sound algebraic correspondences.

7. Isogeny, Prym, and Weil-Type Defects

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2, the finite-index terminology of Section 3, and the multiplier-supported constructions of Section 4. Thus Hodge classes are cohomological classes, algebraicity means membership in the image of the appropriate cycle-class map, and constructional defects are quotients by algebraic subgroups supplied by specified constructions.

The constructions in this section are hybrid. Isogenies and norm comparisons give multiplier-supported components. Prym and Weil-type lattice comparisons may also leave saturation-supported components. Thus the defect profile of such a construction may have both a multiplier term and a saturation term.

7.1. Isogeny Defects

Let A and B be abelian varieties over $\mathbb{C}$, and let

$$\varphi :A⟶B$$

be an isogeny. Let

$$d=deg(\varphi ).$$

Then $\varphi$ is finite surjective of degree d. For $p\ge 0$, the pullback and pushforward maps

$${\varphi }^{*}:C{H}^p(B)⟶C{H}^p(A),{\varphi }_{*}:C{H}^p(A)⟶C{H}^p(B)$$

are compatible with the cohomological cycle-class maps:

$${cl}_{A,\mathbb{Z}}({\varphi }^{*}\Gamma )={\varphi }^{*}{cl}_{B,\mathbb{Z}}(\Gamma ),\Gamma \in C{H}^p(B),$$

and

$${cl}_{B,\mathbb{Z}}({\varphi }_{*}\Delta )={\varphi }_{*}{cl}_{A,\mathbb{Z}}(\Delta ),\Delta \in C{H}^p(A).$$

These are the standard functorial compatibilities for Chow groups and cycle classes [16,17,27].

Lemma 13 

(Isogeny trace identity). Let $\varphi :A\to B$ be an isogeny of degree d. Then

$${\varphi }_{*}{\varphi }^{*}=d·{id}_{C{H}^p(B)}$$

on $C{H}^p(B)$, and

$${\varphi }_{*}{\varphi }^{*}=d·{id}_{H^{2p}(B,\mathbb{Z}(p))}$$

on $H^{2p}(B,\mathbb{Z}(p))$.

Proof. 

An isogeny is finite surjective of degree d. The assertion is Lemma 7 applied to the finite morphism

$$\varphi :A⟶B.$$

 □

Lemma 14 

(Isogenies preserve Hodge classes). The maps

$${\varphi }^{*}:H^{2p}(B,\mathbb{Z}(p))⟶H^{2p}(A,\mathbb{Z}(p))$$

and

$${\varphi }_{*}:H^{2p}(A,\mathbb{Z}(p))⟶H^{2p}(B,\mathbb{Z}(p))$$

send Hodge classes to Hodge classes.

Proof. 

The morphism $\varphi$ is algebraic and proper. Hence pullback and pushforward on cohomology are morphisms of Hodge structures with the indicated Tate twists. Therefore they preserve the Hodge subgroups

$$H^{2p}{(-,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

after passage to complex cohomology [16,17]. □

Definition 46 

(Isogeny construction). Let $\varphi :A\to B$ be an isogeny of degree d. The isogeny construction ${\mathcal{C}}_{\varphi }$ in codimension p has typed Hodge sector

$$L_{\varphi }^p(B)=\left\{\alpha \in H^{2p}{(B,\mathbb{Z}(p))}_{\mathrm{Hdg}}|{\varphi }^{*}\alpha \in im({cl}_{A,\mathbb{Z}})\right\}.$$

Its constructional algebraic part is

$$L_{\varphi ,\mathrm{alg}}^p(B)=L_{\varphi }^p(B)\cap im({cl}_{B,\mathbb{Z}}).$$

The associated constructional defect is

$$D_{\varphi }^p(B)=L_{\varphi }^p(B)/L_{\varphi ,\mathrm{alg}}^p(B).$$

Proposition 28 

(Isogeny annihilation). Let $\varphi :A\to B$ be an isogeny of degree d. Let

$$\alpha \in H^{2p}{(B,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

Assume that ${\varphi }^{*}\alpha$ is integrally algebraic on A. Thus assume there exists $\Gamma \in C{H}^p(A)$ such that

$${cl}_{A,\mathbb{Z}}(\Gamma )={\varphi }^{*}\alpha .$$

Then $d\alpha$ is integrally algebraic on B. More precisely,

$${cl}_{B,\mathbb{Z}}({\varphi }_{*}\Gamma )=d\alpha .$$

Consequently,

$$d{[\alpha ]}_{\mathbb{Z}}=0\mathrm{in}{Z}^{2p}(B).$$

Proof. 

By compatibility of proper pushforward with cycle classes,

$${cl}_{B,\mathbb{Z}}({\varphi }_{*}\Gamma )={\varphi }_{*}{cl}_{A,\mathbb{Z}}(\Gamma ).$$

Using the hypothesis gives

$${\varphi }_{*}{cl}_{A,\mathbb{Z}}(\Gamma )={\varphi }_{*}{\varphi }^{*}\alpha .$$

By Lemma 13,

$${\varphi }_{*}{\varphi }^{*}\alpha =d\alpha .$$

Thus

$${cl}_{B,\mathbb{Z}}({\varphi }_{*}\Gamma )=d\alpha .$$

Therefore $d\alpha$ lies in $im({cl}_{B,\mathbb{Z}})$, and the class of $\alpha$ in $Z^{2p}(B)$ is killed by d. □

Corollary 13 

(Isogeny constructional defect). Let $\varphi :A\to B$ be an isogeny of degree d. Then

$$d·D_{\varphi }^p(B)=0.$$

Thus the isogeny construction is multiplier-supported with proof-level annihilator d.

Proof. 

Let $\alpha \in L_{\varphi }^p(B)$. By Definition 46, the class ${\varphi }^{*}\alpha$ is integrally algebraic on A. Proposition 28 gives

$$d\alpha \in im({cl}_{B,\mathbb{Z}}).$$

Since $d\alpha \in L_{\varphi }^p(B)$, one has

$$d\alpha \in L_{\varphi ,\mathrm{alg}}^p(B).$$

Thus the class of $\alpha$ in $D_{\varphi }^p(B)$ is killed by d. □

Remark 39 

(Kernel exponent). For an isogeny $\varphi :A\to B$, the finite group scheme $ker(\varphi )$ has order $deg(\varphi )$. If a smaller integer annihilates $ker(\varphi )$, then one may replace d by that smaller integer only when the proof supplies a corresponding integral push-pull, norm, or trace identity on the sector in question. Without such an identity, the canonical multiplier obtained from ${\varphi }_{*}{\varphi }^{*}$ is $d=deg(\varphi )$.

7.2. Prym and Norm-Kernel Defects

Let

$$f:C⟶D$$

be a finite morphism of smooth projective complex curves of degree d. The norm homomorphism is

$${Nm}_f:{Pic}^0(C)⟶{Pic}^0(D).$$

A Prym variety attached to f, when used below, is an abelian subvariety or a connected component of a norm kernel:

$$P\subseteq ker{({Nm}_f)}^0\subseteq {Pic}^0(C).$$

In applications one obtains an isogeny or a finite comparison

$$u:P⟶A$$

from P to an abelian variety A, or from an auxiliary abelian variety to P.

Definition 47 

(Prym comparison datum). A Prym comparison datum for a smooth projective variety X in codimension p consists of

$$(P,A,u,\Gamma ,L_{\mathrm{Hdg}}),$$

where:

$$\begin{array}{c}P\mathrm{and}A\mathrm{are}\mathrm{abelian}\mathrm{varieties},\\ u:P\to A\mathrm{is}\mathrm{an}\mathrm{isogeny}\mathrm{or}\mathrm{a}\mathrm{homomorphism}\mathrm{with}\mathrm{finite}\mathrm{kernel}\mathrm{and}\mathrm{finite}\mathrm{cokernel}\mathrm{on}\mathrm{the}\mathrm{lattice}\mathrm{under}\mathrm{consideration},\\ \Gamma \in C{H}^r(P\times X)\mathrm{is}\mathrm{an}\mathrm{algebraic}\mathrm{correspondence},\\ L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}\mathrm{is}\mathrm{a}\mathrm{typed}\mathrm{Hodge}\sec \mathrm{tor},\\ L_{\mathrm{Hdg}}\mathrm{is}\mathrm{specified}\mathrm{independently}\mathrm{of}\mathrm{any}\mathrm{unknown}\mathrm{target}\mathrm{class}.\end{array}$$

Remark 40 

(Soundness of Prym input). The correspondence Γ in Definition 47 is required to be algebraic. A Prym variety or norm-kernel description does not by itself certify algebraic cycle classes on X. Soundness enters through Chow classes on P and through the algebraic correspondence Γ.

Let

$$M_{\mathrm{alg}}\subseteq H^a(P,\mathbb{Z}(b))$$

be a finitely generated subgroup represented by algebraic classes on P. Assume the correspondence indices are chosen so that

$${\Gamma }_{*}:H^a(P,\mathbb{Z}(b))⟶H^{2p}(X,\mathbb{Z}(p)).$$

Define

$$L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}={\Gamma }_{*}(M_{\mathrm{alg}})\subseteq L_{\mathrm{Hdg}}.$$

Definition 48 

(Prym-induced constructional defect). With the notation above, the Prym-induced constructional defect in the sector $L_{\mathrm{Hdg}}$ is

$$D_{\mathrm{Prym}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 29 

(Finite Prym defect under rational coverage). Assume that

$$L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathrm{Prym}}(L_{\mathrm{Hdg}})$$

is finite.

Proof. 

The subgroup $L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}$ is soundly algebraic by cycle-class compatibility for the algebraic correspondence $\Gamma$ [16,17,27]. The rational coverage hypothesis is

$$L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

Since $L_{\mathrm{Hdg}}$ is a typed Hodge sector, it is finitely generated. By Proposition 9,

$$L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}$$

is finite. □

Proposition 30 

(Norm-kernel annihilation under finite comparison). Let

$$u:P⟶A$$

be an isogeny of abelian varieties of degree e. Let

$${\Lambda }_{\mathrm{Hdg}}\subseteq H^{2q}{(A,\mathbb{Z}(q))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. Define

$${\Lambda }_u=\left\{\beta \in {\Lambda }_{\mathrm{Hdg}}|u^{*}\beta \in im({cl}_{P,\mathbb{Z}})\right\}.$$

Let

$${\Lambda }_{u,\mathrm{alg}}={\Lambda }_u\cap im({cl}_{A,\mathbb{Z}}).$$

Then

$$e·\left({\Lambda }_u/{\Lambda }_{u,\mathrm{alg}}\right)=0.$$

Proof. 

Let $\beta \in {\Lambda }_u$. By definition, there exists $\Theta \in C{H}^q(P)$ such that

$${cl}_{P,\mathbb{Z}}(\Theta )=u^{*}\beta .$$

By Proposition 28, applied to the isogeny $u:P\to A$, one has

$${cl}_{A,\mathbb{Z}}(u_{*}\Theta )=e\beta .$$

Therefore

$$e\beta \in {\Lambda }_{u,\mathrm{alg}}.$$

Hence the class of $\beta$ in

$${\Lambda }_u/{\Lambda }_{u,\mathrm{alg}}$$

is killed by e. □

Remark 41 

(Prym and norm-kernel finite loss). A Prym construction may identify a rational Hodge sector through a norm kernel, an isogeny, or a finite comparison of abelian varieties. The multiplier-supported component is controlled by the degree of an isogeny, the exponent of a finite kernel when a compatible integral identity is available, or a norm comparison. The saturation-supported component is the finite cokernel of the algebraic correspondence image

$$L_{\mathrm{Hdg}}/L_{\Gamma ,\mathrm{alg}}^{\mathcal{C}}.$$

Both components belong to the defect profile of the specified Prym construction.

7.3. Weil-Type Hodge Classes

Let A be an abelian variety over $\mathbb{C}$. Let

$$E\subseteq {End}^0(A)=End(A){\otimes }_{\mathbb{Z}}\mathbb{Q}$$

be a number field acting on A. The action of E on A induces algebraic correspondences on A, and hence an action on

$$H^{2p}(A,\mathbb{Q}(p)).$$

A Weil-type construction specifies a Hodge sector by this endomorphism-field action, by a polarization, and sometimes by a Prym or degeneration comparison.

Definition 49 

(Weil-type sector). A Weil-type sector in codimension p is a finitely generated subgroup

$$L_{\mathrm{Weil}}\subseteq H^{2p}{(A,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

such that

$$L_{\mathrm{Weil}}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$

is specified by algebraic endomorphism correspondences of A, polarizations, and algebraic correspondences arising from the given Weil-type construction.

Remark 42 

(No arbitrary eigenspaces). The sector $L_{\mathrm{Weil}}$ is not defined by an arbitrary Hodge subspace. It must be specified by algebraic endomorphisms, algebraic correspondences, polarizations, or a stated geometric comparison. The condition excludes cohomological splittings without Chow-theoretic or algebraic correspondence input.

Let

$$Z_1,\dots ,Z_s\in C{H}^p(A)$$

be algebraic cycles with

$${cl}_{A,\mathbb{Z}}(Z_i)\in L_{\mathrm{Weil}},1\le i\le s.$$

Define

$$L_{\mathrm{Weil},\mathrm{alg}}^{\mathcal{C}}={\displaystyle \sum _{i=1}^s}\mathbb{Z}{cl}_{A,\mathbb{Z}}(Z_i)\subseteq L_{\mathrm{Weil}}.$$

Definition 50 

(Weil saturation defect). The Weil saturation defect associated with $Z_1,\dots ,Z_s$ is

$$D_{\mathrm{Weil}}(Z_{•})=L_{\mathrm{Weil}}/L_{\mathrm{Weil},\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 31 

(Rational Weil generation gives finite constructional defect). Assume that

$$L_{\mathrm{Weil},\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Weil}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathrm{Weil}}(Z_{•})$$

is finite.

Proof. 

The subgroup $L_{\mathrm{Weil},\mathrm{alg}}^{\mathcal{C}}$ is generated by cycle classes of actual algebraic cycles on A, hence it is soundly algebraic. The hypothesis gives rational coverage of $L_{\mathrm{Weil}}$. Since $L_{\mathrm{Weil}}$ is finitely generated, Proposition 9 gives that

$$L_{\mathrm{Weil}}/L_{\mathrm{Weil},\mathrm{alg}}^{\mathcal{C}}$$

is finite. □

Corollary 14 

(Integral annihilator in a Weil-type sector). Under the hypotheses of Proposition 31, let

$$N_{\mathrm{Weil}}=exp{D}_{\mathrm{Weil}}(Z_{•}).$$

Then for every

$$\alpha \in L_{\mathrm{Weil}}$$

there exist integers

$$a_1,\dots ,a_s\in \mathbb{Z}$$

such that

$$N_{\mathrm{Weil}}\alpha ={\displaystyle \sum _{i=1}^s}{a}_i{cl}_{A,\mathbb{Z}}(Z_i).$$

Equivalently,

$$N_{\mathrm{Weil}}\alpha ={cl}_{A,\mathbb{Z}}\left({\displaystyle \sum _{i=1}^s}{a}_i{Z}_i\right).$$

Proof. 

By Proposition 31, $D_{\mathrm{Weil}}(Z_{•})$ is finite. Let

$$N_{\mathrm{Weil}}=exp{D}_{\mathrm{Weil}}(Z_{•}).$$

For $\alpha \in L_{\mathrm{Weil}}$, the class of $\alpha$ in

$$D_{\mathrm{Weil}}(Z_{•})=L_{\mathrm{Weil}}/L_{\mathrm{Weil},\mathrm{alg}}^{\mathcal{C}}$$

is killed by $N_{\mathrm{Weil}}$. Hence

$$N_{\mathrm{Weil}}\alpha \in L_{\mathrm{Weil},\mathrm{alg}}^{\mathcal{C}}.$$

By the definition of $L_{\mathrm{Weil},\mathrm{alg}}^{\mathcal{C}}$, there exist $a_1,\dots ,a_s\in \mathbb{Z}$ such that

$$N_{\mathrm{Weil}}\alpha ={\displaystyle \sum _{i=1}^s}{a}_i{cl}_{A,\mathbb{Z}}(Z_i).$$

Additivity of the cycle-class map gives

$${\displaystyle \sum _{i=1}^s}{a}_i{cl}_{A,\mathbb{Z}}(Z_i)={cl}_{A,\mathbb{Z}}\left({\displaystyle \sum _{i=1}^s}{a}_i{Z}_i\right).$$

 □

Remark 43 

(Relation with Hodge cycles on abelian varieties). Deligne’s theorem on Hodge cycles on abelian varieties is a rational and absolute-Hodge statement [18]. Weil-type constructions and Prym comparisons give specific geometric sources for rational Hodge sectors. The integral question studied here is separate: whether the algebraic classes used in such a sector generate the full integral Hodge lattice or only a finite-index sublattice. The finite quotient is the saturation-supported component of the Weil-type constructional defect.

Remark 44 

(Weil-type defect mechanisms). In Weil-type settings, finite integral loss may enter through several specified algebraic sources:

$$\begin{array}{c}\mathrm{denominators}\mathrm{of}\mathrm{endomorphism}-\mathrm{field}\mathrm{idempotents},\\ \mathrm{degrees}\mathrm{of}\mathrm{isogenies},\\ \mathrm{finite}\mathrm{norm}\mathrm{kernels}\mathrm{in}\mathrm{Prym}\mathrm{comparisons},\\ \mathrm{polarization}\mathrm{indices},\\ \mathrm{saturation}\mathrm{indices}\mathrm{of}\mathrm{algebraic}\mathrm{generators}.\end{array}$$

Each source gives a finite constructional defect only after the corresponding algebraic cycles, correspondences, or isogenies have been specified. No conclusion is drawn from a cohomological eigenspace alone.

7.4. Defect Profiles for Isogeny, Prym, and Weil-Type Constructions

Definition 51 

(Hybrid defect profile in the abelian setting). Let $\mathcal{C}$ be an isogeny, Prym, or Weil-type construction in a typed Hodge sector $L_{\mathrm{Hdg}}$. A hybrid defect profile for $\mathcal{C}$ is a tuple

$$Prof(\mathcal{C},L_{\mathrm{Hdg}})=\left(D_{\mathcal{C}}(L_{\mathrm{Hdg}}),e_{\mathcal{C}}(L_{\mathrm{Hdg}}),{\mathcal{A}}_{\mathrm{mult}},{\mathcal{A}}_{\mathrm{sat}}\right),$$

where:

$$\begin{array}{c}{D}_{\mathcal{C}}(L_{\mathrm{Hdg}})\mathrm{is}\mathrm{the}\mathrm{constructional}\mathrm{defect},\\ e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp{D}_{\mathcal{C}}(L_{\mathrm{Hdg}})\mathrm{when}\mathrm{the}\mathrm{defect}\mathrm{is}\mathrm{finite},\\ {\mathcal{A}}_{\mathrm{mult}}\mathrm{records}\mathrm{degree},\mathrm{norm},\mathrm{isogeny},\mathrm{or}\mathrm{idempotent}-\mathrm{denominator}\mathrm{data},\\ {\mathcal{A}}_{\mathrm{sat}}\mathrm{records}\mathrm{the}\mathrm{finite}\mathrm{lattice}\mathrm{quotient}\mathrm{of}\mathrm{the}\mathrm{algebraic}\mathrm{generators}.\end{array}$$

Remark 45 

(Interpretation of the hybrid profile). The multiplier component records identities such as

$${\varphi }_{*}{\varphi }^{*}=deg(\varphi )id.$$

The saturation component records quotients such as

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

A construction may have one component, both components, or a further realization-loss component if regulators, Abel–Jacobi maps, specialization maps, or finite-coefficient maps are also used. The profile is attached to the specified construction and sector; it is not an invariant of the variety alone.

8. Specialization and Boundary Defects

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2 and the defect-profile terminology of Section 3. Thus Hodge classes are cohomological classes, algebraicity means membership in the image of the appropriate cycle-class map, and constructional defects are quotients by algebraic subgroups supplied by specified constructions.

The constructions in this section are boundary-supported and realization-loss type in the sense of Definition 29. Some of them also have a saturation-supported component. The common feature is that rational specialization or limiting arguments may identify a sector after tensoring with $\mathbb{Q}$, while integral information remains in kernels, cokernels, boundary images, vertical cycle groups, monodromy-invariant lattices, or extension data. No converse-to-specialization assertion is used unless it is stated as an additional hypothesis.

8.1. Specialization of Classes

Let

$$\pi :\mathcal{X}⟶\Delta$$

be a projective morphism from a complex algebraic variety $\mathcal{X}$ to a disc $\Delta$. Let

$$0\in \Delta ,t\in {\Delta }^{*}=\Delta \setminus \{0\}.$$

Write

$$X_0={\pi }^{-1}(0),X_t={\pi }^{-1}(t).$$

Assume that $X_t$ is smooth projective for $t\ne 0$. When required, assume that $\mathcal{X}$ is smooth and that $X_0$ is a Cartier divisor in $\mathcal{X}$.

Definition 52 

(Specialization datum). A specialization datum in codimension p is a tuple

$$\mathfrak{s}=(\mathcal{X},\Delta ,0,t,L_t,L_0,{sp}_{\mathbb{Z}}),$$

where

$$\begin{array}{c}\pi :\mathcal{X}\to \Delta \mathrm{is}\mathrm{projective},\\ X_t\mathrm{is}\mathrm{smooth}\mathrm{projective}\mathrm{for}t\ne 0,\\ L_t\subseteq H^{2p}{(X_t,\mathbb{Z}(p))}_{\mathrm{Hdg}}\mathrm{is}\mathrm{a}\mathrm{typed}\mathrm{Hodge}\sec \mathrm{tor},\\ L_0\subseteq H^{2p}{(X_0,\mathbb{Z}(p))}_{\mathrm{Hdg}}\mathrm{is}\mathrm{a}\mathrm{specified}\mathrm{subgroup}\mathrm{whenever}\mathrm{this}\mathrm{cohomology}\mathrm{group}\mathrm{is}\mathrm{defined}\mathrm{in}\mathrm{the}\mathrm{chosen}\mathrm{theory},\\ {sp}_{\mathbb{Z}}:L_t⟶L_0\mathrm{is}\mathrm{a}\mathrm{homomorphism}\mathrm{induced}\mathrm{by}\mathrm{the}\mathrm{chosen}\mathrm{specialization}\mathrm{or}\mathrm{nearby}-\mathrm{cycle}\mathrm{construction},\\ {sp}_{\mathbb{Z}}\mathrm{is}\mathrm{specified}\mathrm{independently}\mathrm{of}\mathrm{any}\mathrm{unknown}\mathrm{target}\mathrm{class}.\end{array}$$

Remark 46 

(Scope of specialization). Definition 52 does not assert a universal specialization theory for arbitrary singular fibers. It records the data used in a specified degeneration argument. The required compatibility with cycle classes must be verified in the geometric situation under consideration [16,17,27].

Definition 53 

(Cycle-compatible specialization). Let

$$\mathfrak{s}=(\mathcal{X},\Delta ,0,t,L_t,L_0,{sp}_{\mathbb{Z}})$$

be a specialization datum. The datum is cycle-compatible if there exists a specialization homomorphism on Chow groups

$${sp}_{CH}:C{H}^p(X_t)⟶C{H}^p(X_0)$$

such that

$${sp}_{\mathbb{Z}}\left({cl}_{X_t,\mathbb{Z}}(\Gamma )\right)={cl}_{X_0,\mathbb{Z}}\left({sp}_{CH}(\Gamma )\right)$$

for every $\Gamma \in C{H}^p(X_t)$ whose cycle class lies in the sector under consideration.

Lemma 15 

(Specialization preserves algebraicity under compatibility). Let

$$\mathfrak{s}=(\mathcal{X},\Delta ,0,t,L_t,L_0,{sp}_{\mathbb{Z}})$$

be a cycle-compatible specialization datum. If

$$\alpha \in L_t$$

is integrally algebraic on $X_t$, then

$${sp}_{\mathbb{Z}}(\alpha )$$

is integrally algebraic on $X_0$.

Proof. 

Since $\alpha$ is integrally algebraic on $X_t$, there exists

$$\Gamma \in C{H}^p(X_t)$$

such that

$${cl}_{X_t,\mathbb{Z}}(\Gamma )=\alpha .$$

By cycle-compatible specialization,

$${sp}_{\mathbb{Z}}{cl}_{X_t,\mathbb{Z}}(\Gamma )={cl}_{X_0,\mathbb{Z}}{sp}_{CH}(\Gamma ).$$

Therefore

$${sp}_{\mathbb{Z}}(\alpha )={cl}_{X_0,\mathbb{Z}}\left({sp}_{CH}(\Gamma )\right).$$

Hence ${sp}_{\mathbb{Z}}(\alpha )$ is integrally algebraic on $X_0$. □

Definition 54 

(Specialization construction). Let

$$\mathfrak{s}=(\mathcal{X},\Delta ,0,t,L_t,L_0,{sp}_{\mathbb{Z}})$$

be a specialization datum. The specialization-covered sector is

$$L_{sp}^p(X_t)=\left\{\alpha \in L_t|{sp}_{\mathbb{Z}}(\alpha )\in im({cl}_{X_0,\mathbb{Z}})\right\}.$$

Its actual algebraic part is

$$L_{sp,\mathrm{alg}}^{p,\mathrm{act}}(X_t)=L_{sp}^p(X_t)\cap im({cl}_{X_t,\mathbb{Z}}).$$

A constructional algebraic subgroup

$$L_{sp,\mathrm{alg}}^{p,\mathcal{C}}(X_t)\subseteq L_{sp,\mathrm{alg}}^{p,\mathrm{act}}(X_t)$$

is specified by the algebraic cycles on $X_t$ produced by the chosen lifting construction. The corresponding constructional defect is

$$D_{sp}^{p,\mathcal{C}}(X_t)=L_{sp}^p(X_t)/L_{sp,\mathrm{alg}}^{p,\mathcal{C}}(X_t).$$

Remark 47 

(Specialization does not automatically lift algebraicity). Lemma 15 is one-directional. It says that algebraicity on the general fiber specializes to algebraicity on the special fiber under cycle-compatible specialization. The converse is not formal. If

$${sp}_{\mathbb{Z}}(\alpha )$$

is algebraic on $X_0$, an additional lifting statement is required to conclude that α, or a positive multiple of α, is algebraic on $X_t$.

Definition 55 

(Specialization lifting index). Let

$$\mathfrak{s}=(\mathcal{X},\Delta ,0,t,L_t,L_0,{sp}_{\mathbb{Z}})$$

be a specialization datum, and let

$$L_{sp,\mathrm{alg}}^{p,\mathcal{C}}(X_t)\subseteq L_{sp}^p(X_t)$$

be the constructional algebraic subgroup supplied by a lifting construction. An integer $N\ge 1$ is a lifting index for the construction if

$$N{L}_{sp}^p(X_t)\subseteq L_{sp,\mathrm{alg}}^{p,\mathcal{C}}(X_t).$$

Equivalently, for every

$$\alpha \in L_{sp}^p(X_t)$$

there exists

$${\Gamma }_{\alpha }\in C{H}^p(X_t)$$

produced by the lifting construction such that

$${cl}_{X_t,\mathbb{Z}}({\Gamma }_{\alpha })=N\alpha .$$

Proposition 32 

(Specialization defect with a lifting index). Let

$$\mathfrak{s}=(\mathcal{X},\Delta ,0,t,L_t,L_0,{sp}_{\mathbb{Z}})$$

be a specialization datum. If $N\ge 1$ is a lifting index for the constructional subgroup

$$L_{sp,\mathrm{alg}}^{p,\mathcal{C}}(X_t)\subseteq L_{sp}^p(X_t),$$

then

$$N·D_{sp}^{p,\mathcal{C}}(X_t)=0.$$

Proof. 

Let

$$\alpha \in L_{sp}^p(X_t).$$

By Definition 55, one has

$$N\alpha \in L_{sp,\mathrm{alg}}^{p,\mathcal{C}}(X_t).$$

Therefore the class of $\alpha$ in

$$D_{sp}^{p,\mathcal{C}}(X_t)=L_{sp}^p(X_t)/L_{sp,\mathrm{alg}}^{p,\mathcal{C}}(X_t)$$

is killed by N. Since $\alpha$ was arbitrary,

$$N·D_{sp}^{p,\mathcal{C}}(X_t)=0.$$

 □

Remark 48 

(Rational specialization and integral loss). A rational specialization argument may show that a sector is algebraic after passing to a special fiber and tensoring with $\mathbb{Q}$. Integrally, one must also account for the kernel, cokernel, boundary image, and torsion kernel of the specialization map. The integer in Proposition 32 is not automatic; it is part of the geometric lifting input.

8.2. Vertical and Boundary Contributions

Let $i:X_0\hookrightarrow \mathcal{X}$ be the inclusion of the special fiber, and let

$$j:U=\mathcal{X}\setminus X_0\hookrightarrow \mathcal{X}$$

be the complementary open immersion. For Chow groups there is a localization sequence

$$C{H}^{p-1}(X_0)\stackrel{i_{*}}{\to }C{H}^p(\mathcal{X})\stackrel{j^{*}}{\to }C{H}^p(U)⟶0$$

in the range used below. The corresponding cohomological localization sequence contains maps

$$H^{2p-2}(X_0,\mathbb{Z}(p-1))\stackrel{i_{*}}{\to }{H}^{2p}(\mathcal{X},\mathbb{Z}(p))\stackrel{j^{*}}{\to }{H}^{2p}(U,\mathbb{Z}(p)).$$

The cycle-class maps are compatible with these localization maps [16,17,27].

Definition 56 

(Vertical algebraic subgroup). The vertical algebraic subgroup in codimension p is

$$V_{\mathrm{alg}}^p(\mathcal{X}/X_0)=i_{*}C{H}^{p-1}(X_0)\subseteq C{H}^p(\mathcal{X}).$$

Its cohomological image is

$$V_{\mathrm{Hdg}}^p(\mathcal{X}/X_0)={cl}_{\mathcal{X},\mathbb{Z}}\left(V_{\mathrm{alg}}^p(\mathcal{X}/X_0)\right)\subseteq H^{2p}{(\mathcal{X},\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

Definition 57 

(Boundary lift). Let

$$\eta \in H^{2p}(U,\mathbb{Z}(p)).$$

A boundary lift of η is an element

$$\tilde{\eta }\in H^{2p}(\mathcal{X},\mathbb{Z}(p))$$

such that

$$j^{*}\tilde{\eta }=\eta .$$

Lemma 16 

(Vertical ambiguity). Let

$${\tilde{\eta }}_1,{\tilde{\eta }}_2\in H^{2p}(\mathcal{X},\mathbb{Z}(p))$$

satisfy

$$j^{*}{\tilde{\eta }}_1=j^{*}{\tilde{\eta }}_2.$$

Then

$${\tilde{\eta }}_1-{\tilde{\eta }}_2\in im\left(i_{*}:H^{2p-2}(X_0,\mathbb{Z}(p-1))\to H^{2p}(\mathcal{X},\mathbb{Z}(p))\right).$$

Proof. 

The equality

$$j^{*}{\tilde{\eta }}_1=j^{*}{\tilde{\eta }}_2$$

implies

$$j^{*}({\tilde{\eta }}_1-{\tilde{\eta }}_2)=0.$$

Exactness of the cohomological localization sequence at $H^{2p}(\mathcal{X},\mathbb{Z}(p))$ gives

$${\tilde{\eta }}_1-{\tilde{\eta }}_2\in im(i_{*}).$$

 □

Definition 58 

(Boundary constructional defect). Let

$$L_{\mathcal{X}}\subseteq H^{2p}{(\mathcal{X},\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. Define the boundary sector

$$L_{\partial }=L_{\mathcal{X}}\cap im\left(i_{*}:H^{2p-2}(X_0,\mathbb{Z}(p-1))\to H^{2p}(\mathcal{X},\mathbb{Z}(p))\right).$$

Define its constructional vertical algebraic subgroup by

$$L_{\partial ,\mathrm{alg}}^{\mathcal{C}}=L_{\mathcal{X}}\cap V_{\mathrm{Hdg}}^p(\mathcal{X}/X_0).$$

The boundary constructional defect is

$$D_{\partial }^{\mathcal{C}}(L_{\mathcal{X}})=L_{\partial }/L_{\partial ,\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 33 

(Algebraic boundary classes have zero boundary defect). Let

$$\lambda \in L_{\partial }.$$

Assume that there exists

$$\Theta \in C{H}^{p-1}(X_0)$$

such that

$$\lambda ={cl}_{\mathcal{X},\mathbb{Z}}(i_{*}\Theta ).$$

Then the image of λ in

$$D_{\partial }^{\mathcal{C}}(L_{\mathcal{X}})$$

is zero.

Proof. 

By Definition 56,

$$i_{*}\Theta \in V_{\mathrm{alg}}^p(\mathcal{X}/X_0).$$

Therefore

$$\lambda ={cl}_{\mathcal{X},\mathbb{Z}}(i_{*}\Theta )\in V_{\mathrm{Hdg}}^p(\mathcal{X}/X_0).$$

Since also $\lambda \in L_{\partial }\subseteq L_{\mathcal{X}}$, one has

$$\lambda \in L_{\mathcal{X}}\cap V_{\mathrm{Hdg}}^p(\mathcal{X}/X_0)=L_{\partial ,\mathrm{alg}}^{\mathcal{C}}.$$

Hence the class of $\lambda$ in $D_{\partial }^{\mathcal{C}}(L_{\mathcal{X}})$ is zero. □

Proposition 34 

(Finite boundary defect from rational vertical coverage). Assume that $L_{\partial }$ is finitely generated and that

$$L_{\partial ,\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\partial }{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\partial }^{\mathcal{C}}(L_{\mathcal{X}})$$

is finite.

Proof. 

By Definition 58,

$$D_{\partial }^{\mathcal{C}}(L_{\mathcal{X}})=L_{\partial }/L_{\partial ,\mathrm{alg}}^{\mathcal{C}}.$$

The rational vertical coverage hypothesis is

$$L_{\partial ,\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\partial }\otimes \mathbb{Q}.$$

Since $L_{\partial }$ is finitely generated, Proposition 9 gives that

$$L_{\partial }/L_{\partial ,\mathrm{alg}}^{\mathcal{C}}$$

is finite. □

Remark 49 

(Boundary terms and integral loss). If two lifts of a class on U differ by a boundary term, the difference is controlled by the image of $i_{*}$. Rationally, the boundary image may be generated by vertical algebraic classes. Integrally, the quotient

$$L_{\partial }/L_{\partial ,\mathrm{alg}}^{\mathcal{C}}$$

records the finite defect left by the vertical part. This defect is realization-loss type when it is controlled by the kernel or cokernel of a localization or boundary map, and saturation-supported when it is a finite-index quotient of a vertical algebraic lattice.

8.3. Limit Mixed Hodge Structures

Let

$$\pi :\mathcal{X}\to \Delta$$

be as in Section 8.1, with smooth fibers over ${\Delta }^{*}$. Let

$$H_t=H^{2p}(X_t,\mathbb{Z}(p))$$

for $t\ne 0$. After replacing ${\Delta }^{*}$ by a finite cover if necessary, assume that the monodromy operator

$$T:H_t⟶H_t$$

is unipotent on the torsion-free quotient of $H_t$. Let

$$N=logT$$

on $H_t\otimes \mathbb{Q}$. The limit mixed Hodge structure on $H_t\otimes \mathbb{Q}$ is denoted by

$$H_{lim,\mathbb{Q}}.$$

An integral lattice, when fixed, is denoted by

$$H_{lim,\mathbb{Z}}\subset H_{lim,\mathbb{Q}}.$$

Limit mixed Hodge structures and monodromy filtrations record the limiting behavior of Hodge classes in degenerating families [16,17].

Definition 59 

(Limit Hodge sector). A limit Hodge sector in codimension p is a finitely generated subgroup

$$L_{lim}\subseteq H_{lim,\mathbb{Z}}$$

whose rational span

$$L_{lim,\mathbb{Q}}=L_{lim}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$

is contained in the space of rational limit Hodge classes in $H_{lim,\mathbb{Q}}$.

Definition 60 

(Monodromy constructional defect). Let $L_{lim}$ be a limit Hodge sector. Its monodromy-stable part is

$$L_{lim}^{N=0}=L_{lim}\cap ker(N).$$

Let

$$L_{\mathrm{alg},lim}^{\mathcal{C}}\subseteq L_{lim}^{N=0}$$

be a subgroup generated by specializations or limits of algebraic cycles, or by algebraic cycles on a semistable model with a specified comparison map. The monodromy constructional defect is

$$D_N^{\mathcal{C}}(L_{lim})=L_{lim}^{N=0}/L_{\mathrm{alg},lim}^{\mathcal{C}}.$$

Remark 50 

(Soundness of limit algebraic classes). The subgroup $L_{\mathrm{alg},lim}^{\mathcal{C}}$ is required to be generated by actual algebraic cycles together with specified specialization, nearby-cycle, or semistable comparison maps. A rational limit Hodge class alone is not a Chow certificate.

Proposition 35 

(Finite monodromy defect under rational limit coverage). Assume that $L_{lim}^{N=0}$ is finitely generated and that

$$L_{\mathrm{alg},lim}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{lim}^{N=0}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_N^{\mathcal{C}}(L_{lim})$$

is finite.

Proof. 

By Definition 60,

$$D_N^{\mathcal{C}}(L_{lim})=L_{lim}^{N=0}/L_{\mathrm{alg},lim}^{\mathcal{C}}.$$

The rational limit coverage hypothesis is

$$L_{\mathrm{alg},lim}^{\mathcal{C}}\otimes \mathbb{Q}=L_{lim}^{N=0}\otimes \mathbb{Q}.$$

Since $L_{lim}^{N=0}$ is finitely generated, Proposition 9 implies that

$$L_{lim}^{N=0}/L_{\mathrm{alg},lim}^{\mathcal{C}}$$

is finite. □

Definition 61 

(Extension constructional defect). Let

$$0⟶W_{k-1}{L}_{lim}⟶W_k{L}_{lim}⟶{Gr}_k^W{L}_{lim}⟶0$$

be a short exact sequence induced by the monodromy weight filtration on a limit sector. Suppose that specified algebraic limit classes generate subgroups

$$W_{k-1}{L}_{\mathrm{alg},lim}^{\mathcal{C}}\subseteq W_{k-1}{L}_{lim}$$

and

$${Gr}_k^W{L}_{\mathrm{alg},lim}^{\mathcal{C}}\subseteq {Gr}_k^W{L}_{lim}.$$

Let

$$L_{\mathrm{alg},k}^{\mathrm{lift}}\subseteq W_k{L}_{lim}$$

be the subgroup generated by the specified algebraic lifts of the chosen graded algebraic classes. The extension constructional defect is

$$D_{\mathrm{ext},k}^{\mathcal{C}}={\displaystyle \frac{W_k{L}_{lim}}{W_{k-1}{L}_{\mathrm{alg},lim}^{\mathcal{C}}+L_{\mathrm{alg},k}^{\mathrm{lift}}}}.$$

Remark 51 

(Extension data). The group $D_{\mathrm{ext},k}^{\mathcal{C}}$ is defined only after specifying algebraic lifts. It is not obtained by choosing arbitrary splittings of the mixed Hodge structure. Such splittings are not cycle certificates unless they are realized by algebraic cycles or by specified algebraic correspondences.

Proposition 36 

(Finite extension defect under rational lifting). Assume that

$$W_k{L}_{lim}$$

is finitely generated and that

$$\left(W_{k-1}{L}_{\mathrm{alg},lim}^{\mathcal{C}}+L_{\mathrm{alg},k}^{\mathrm{lift}}\right){\otimes }_{\mathbb{Z}}\mathbb{Q}=W_k{L}_{lim}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathrm{ext},k}^{\mathcal{C}}$$

is finite.

Proof. 

By Definition 61,

$$D_{\mathrm{ext},k}^{\mathcal{C}}={\displaystyle \frac{W_k{L}_{lim}}{W_{k-1}{L}_{\mathrm{alg},lim}^{\mathcal{C}}+L_{\mathrm{alg},k}^{\mathrm{lift}}}}.$$

The denominator is a subgroup of $W_k{L}_{lim}$. The rational lifting hypothesis says that this subgroup spans $W_k{L}_{lim}$ after tensoring with $\mathbb{Q}$. Since $W_k{L}_{lim}$ is finitely generated, Proposition 9 gives that the quotient is finite. □

8.4. Specialization Defect Profiles

Definition 62 

(Specialization defect profile). Let $\mathcal{C}$ be a specialization, boundary, or limit construction in a typed sector $L_{\mathrm{Hdg}}$. A specialization defect profile is a tuple

$$Prof(\mathcal{C},L_{\mathrm{Hdg}})=\left(D_{\mathcal{C}}(L_{\mathrm{Hdg}}),e_{\mathcal{C}}(L_{\mathrm{Hdg}}),{\mathcal{A}}_{\partial },{\mathcal{A}}_{lim},{\tau }_{\mathcal{C}}\right),$$

where

$$\begin{array}{c}{D}_{\mathcal{C}}(L_{\mathrm{Hdg}})\mathrm{is}\mathrm{the}\mathrm{constructional}\mathrm{defect},\\ e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp{D}_{\mathcal{C}}(L_{\mathrm{Hdg}})\mathrm{when}\mathrm{the}\mathrm{defect}\mathrm{is}\mathrm{finite},\\ {\mathcal{A}}_{\partial }\mathrm{records}\mathrm{specialization},\mathrm{vertical},\mathrm{boundary},\mathrm{kernel},\mathrm{or}\mathrm{cokernel}\mathrm{data},\\ {\mathcal{A}}_{lim}\mathrm{records}\mathrm{monodromy},\mathrm{limit},\mathrm{weight},\mathrm{or}\mathrm{extension}\mathrm{data},\\ {\tau }_{\mathcal{C}}\in \{\mathrm{sat},\mathrm{real}\}\mathrm{records}\mathrm{the}\mathrm{visible}\mathrm{defect}\mathrm{type}.\end{array}$$

Remark 52 

(Limit mixed Hodge structures and integral defects). Limit mixed Hodge structures may identify rational limiting Hodge classes. The integral lattice may still retain monodromy, extension, and boundary data. The groups

$$D_N^{\mathcal{C}}(L_{lim}),D_{\mathrm{ext},k}^{\mathcal{C}},D_{\partial }^{\mathcal{C}}(L_{\mathcal{X}})$$

record distinct constructional defects, provided the algebraic sources and comparison maps are specified. The present section does not assert a general specialization theorem; it isolates the finite quotient that remains when specialization or limiting arguments give rational coverage.

9. Regulator and Higher Chow Defects

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2 and the defect-profile terminology of Section 3. Thus Hodge classes are cohomological classes, algebraicity means membership in the image of the appropriate cycle-class map, and constructional defects are quotients by algebraic subgroups supplied by specified constructions.

The constructions in this section are realization-loss type in the sense of Definition 29. Regulator maps and Abel–Jacobi maps are realization morphisms. Their kernels, images, and cokernels measure information lost by the chosen realization. These defects are not, by definition, ordinary integral Hodge defects in $Z^{2p}(X)$. They become comparable with $Z^{2p}(X)$ only after a specified comparison with the ordinary cycle-class map.

9.1. Regulator Invisibility

Let X be a smooth projective complex variety. For integers

$$p\ge 0,n\ge 0,$$

let

$$C{H}^p(X,n)$$

denote Bloch’s higher Chow group. Let

$$H_{\mathcal{D}}^{2p-n}(X,\mathbb{Z}(p))$$

denote Deligne cohomology. The regulator map is denoted

$$r_{p,n}:C{H}^p(X,n)⟶H_{\mathcal{D}}^{2p-n}(X,\mathbb{Z}(p)).$$

Higher Chow groups and regulator maps give cycle-theoretic refinements of ordinary cycle-class constructions [16,17,28].

Definition 63 

(Regulator kernel). The regulator kernel in bidegree $(p,n)$ is

$$K_{\mathcal{D}}^{p,n}(X)=ker\left(r_{p,n}:C{H}^p(X,n)\to H_{\mathcal{D}}^{2p-n}(X,\mathbb{Z}(p))\right).$$

Definition 64 

(Regulator-visible quotient). The regulator-visible quotient in bidegree $(p,n)$ is

$$R_{\mathcal{D}}^{p,n}(X)=C{H}^p(X,n)/K_{\mathcal{D}}^{p,n}(X).$$

Equivalently,

$$R_{\mathcal{D}}^{p,n}(X)\cong im(r_{p,n}).$$

Lemma 17 

(Regulator invisibility). Let

$$\xi \in C{H}^p(X,n).$$

Then

$$r_{p,n}(\xi )=0⟺\xi \in K_{\mathcal{D}}^{p,n}(X).$$

The class of ξ in $R_{\mathcal{D}}^{p,n}(X)$ is zero if and only if

$$\xi \in K_{\mathcal{D}}^{p,n}(X).$$

Proof. 

The first equivalence is the definition of the kernel of $r_{p,n}$. The second statement follows from

$$R_{\mathcal{D}}^{p,n}(X)=C{H}^p(X,n)/K_{\mathcal{D}}^{p,n}(X).$$

Thus $\xi$ maps to zero in $R_{\mathcal{D}}^{p,n}(X)$ if and only if

$$\xi \in K_{\mathcal{D}}^{p,n}(X).$$

 □

Remark 53 

(Realization loss). A class

$$\xi \in K_{\mathcal{D}}^{p,n}(X)$$

is invisible to the regulator $r_{p,n}$. This does not imply

$$\xi =0\mathrm{in}C{H}^p(X,n).$$

Thus the regulator may lose information present in the higher Chow group. This is a realization-loss phenomenon. It is parallel to the loss of torsion under rationalization, but it occurs for a realization morphism rather than for the coefficient change $\mathbb{Z}\to \mathbb{Q}$.

Definition 65 

(Regulator constructional defect in a subgroup). Let

$$M\subseteq C{H}^p(X,n)$$

be a subgroup supplied by a specified higher-cycle construction $\mathcal{C}$. The regulator-invisible subgroup of M is

$$M_{\mathrm{inv}}=M\cap K_{\mathcal{D}}^{p,n}(X).$$

The regulator-visible constructional quotient is

$$D_{\mathcal{D}}^{\mathcal{C}}(M)=M/M_{\mathrm{inv}}.$$

Lemma 18 

(Exact regulator sequence for a subgroup). Let $M\subseteq C{H}^p(X,n)$. The restriction of $r_{p,n}$ to M induces a short exact sequence

$$0⟶M_{\mathrm{inv}}⟶M⟶r_{p,n}(M)⟶0.$$

Consequently,

$$D_{\mathcal{D}}^{\mathcal{C}}(M)\cong r_{p,n}(M).$$

Proof. 

The restriction

$$r_{p,n}{|}_M:M⟶H_{\mathcal{D}}^{2p-n}(X,\mathbb{Z}(p))$$

has image $r_{p,n}(M)$. Its kernel is

$$M\cap ker(r_{p,n})=M\cap K_{\mathcal{D}}^{p,n}(X)=M_{\mathrm{inv}}.$$

The first isomorphism theorem gives

$$M/M_{\mathrm{inv}}\cong r_{p,n}(M).$$

This is the asserted exact sequence and the asserted isomorphism. □

Remark 54 

(Regulator soundness and coverage). The source $C{H}^p(X,n)$ is a cycle-theoretic group. Thus elements of $C{H}^p(X,n)$ are algebraic higher cycles. This is the soundness input. Coverage is a separate assertion about the image, kernel, or quotient of the chosen realization map. A regulator statement does not certify ordinary algebraicity of a cohomological Hodge class unless an additional comparison with

$${cl}_{X,\mathbb{Z}}:C{H}^p(X)\to H^{2p}(X,\mathbb{Z}(p))$$

is specified.

9.2. Indecomposability Defects

Let X be smooth projective. Fix a subgroup

$$C{H}^p{(X,n)}_{\mathrm{dec}}\subseteq C{H}^p(X,n)$$

generated by decomposable classes in the relevant bidegree, namely by products of lower-codimension Chow classes and lower-weight higher Chow classes.

Definition 66 

(Indecomposable quotient). The indecomposable quotient is

$$C{H}^p{(X,n)}_{\mathrm{ind}}=C{H}^p(X,n)/C{H}^p{(X,n)}_{\mathrm{dec}}.$$

For

$$\xi \in C{H}^p(X,n),$$

its class in $C{H}^p{(X,n)}_{\mathrm{ind}}$ is denoted by

$${[\xi ]}_{\mathrm{ind}}.$$

Definition 67 

(Regulator-invisible indecomposable subgroup). The regulator-invisible indecomposable subgroup is

$$K_{\mathcal{D},\mathrm{ind}}^{p,n}(X)={\displaystyle \frac{K_{\mathcal{D}}^{p,n}(X)+C{H}^p{(X,n)}_{\mathrm{dec}}}{C{H}^p{(X,n)}_{\mathrm{dec}}}}\subseteq C{H}^p{(X,n)}_{\mathrm{ind}}.$$

Lemma 19 

(Criterion for invisible indecomposability). Let

$$\xi \in C{H}^p(X,n).$$

If

$$r_{p,n}(\xi )=0$$

and

$${[\xi ]}_{\mathrm{ind}}\ne 0\mathrm{in}C{H}^p{(X,n)}_{\mathrm{ind}},$$

then

$${[\xi ]}_{\mathrm{ind}}\in K_{\mathcal{D},\mathrm{ind}}^{p,n}(X)$$

is nonzero.

Proof. 

The equality

$$r_{p,n}(\xi )=0$$

means

$$\xi \in K_{\mathcal{D}}^{p,n}(X).$$

Therefore the image of $\xi$ in

$$C{H}^p{(X,n)}_{\mathrm{ind}}=C{H}^p(X,n)/C{H}^p{(X,n)}_{\mathrm{dec}}$$

lies in

$${\displaystyle \frac{K_{\mathcal{D}}^{p,n}(X)+C{H}^p{(X,n)}_{\mathrm{dec}}}{C{H}^p{(X,n)}_{\mathrm{dec}}}}=K_{\mathcal{D},\mathrm{ind}}^{p,n}(X).$$

By hypothesis, this image is nonzero. Hence it is a nonzero element of $K_{\mathcal{D},\mathrm{ind}}^{p,n}(X)$. □

Definition 68 

(Indecomposability defect). Let

$$M\subseteq C{H}^p(X,n)$$

be a subgroup supplied by a specified construction $\mathcal{C}$. Its indecomposability quotient is

$$D_{\mathrm{ind}}^{\mathcal{C}}(M)={\displaystyle \frac{M}{M\cap C{H}^p{(X,n)}_{\mathrm{dec}}}}.$$

Its regulator-invisible indecomposability quotient is

$$D_{\mathcal{D},\mathrm{ind}}^{\mathcal{C}}(M)={\displaystyle \frac{M\cap K_{\mathcal{D}}^{p,n}(X)}{M\cap K_{\mathcal{D}}^{p,n}(X)\cap C{H}^p{(X,n)}_{\mathrm{dec}}}}.$$

Proposition 37 

(Invisible indecomposable classes). Let $M\subseteq C{H}^p(X,n)$. There is an injective homomorphism

$$D_{\mathcal{D},\mathrm{ind}}^{\mathcal{C}}(M)\hookrightarrow D_{\mathrm{ind}}^{\mathcal{C}}(M).$$

Its image consists of classes represented by elements of M with zero regulator.

Proof. 

The inclusion

$$M\cap K_{\mathcal{D}}^{p,n}(X)\subseteq M$$

induces a homomorphism

$${\displaystyle \frac{M\cap K_{\mathcal{D}}^{p,n}(X)}{M\cap K_{\mathcal{D}}^{p,n}(X)\cap C{H}^p{(X,n)}_{\mathrm{dec}}}}⟶{\displaystyle \frac{M}{M\cap C{H}^p{(X,n)}_{\mathrm{dec}}}}.$$

The kernel consists of elements

$$\xi \in M\cap K_{\mathcal{D}}^{p,n}(X)$$

whose image in $D_{\mathrm{ind}}^{\mathcal{C}}(M)$ is zero. This means

$$\xi \in M\cap C{H}^p{(X,n)}_{\mathrm{dec}}.$$

Since also

$$\xi \in K_{\mathcal{D}}^{p,n}(X),$$

one has

$$\xi \in M\cap K_{\mathcal{D}}^{p,n}(X)\cap C{H}^p{(X,n)}_{\mathrm{dec}}.$$

Thus the kernel is zero. The image is represented by elements of $M\cap K_{\mathcal{D}}^{p,n}(X)$, hence by elements with zero regulator. □

Remark 55 

(Collino–Fakhruddin type behavior). Collino–Fakhruddin type constructions exhibit higher Chow cycles on Jacobians whose regulator behavior does not detect their full indecomposability [25]. In the notation above, such examples supply nonzero elements in an indecomposable quotient whose image under a chosen regulator is zero or insufficient to detect the class. The defect is a realization-loss defect in higher Chow theory, not an ordinary element of $Z^{2p}(X)$.

Definition 69 

(Regulator detection quotient). Let $M\subseteq C{H}^p(X,n)$. The regulator detection quotient of M is

$$Q_{\mathcal{D}}^{\mathcal{C}}(M)={\displaystyle \frac{D_{\mathrm{ind}}^{\mathcal{C}}(M)}{im\left(D_{\mathcal{D},\mathrm{ind}}^{\mathcal{C}}(M)\right)}}.$$

Remark 56 

(Meaning of the detection quotient). The quotient $Q_{\mathcal{D}}^{\mathcal{C}}(M)$ separates indecomposable classes detected by the regulator from indecomposable classes lying in the regulator kernel. It is not a Hodge defect group. It is a quotient measuring the loss of higher Chow information under a realization map.

9.3. Relation to Integral Hodge Defects

The ordinary integral Hodge defect group is

$$Z^{2p}(X)={\displaystyle \frac{H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}}{im({cl}_{X,\mathbb{Z}})}}.$$

It is built from ordinary cohomology and ordinary Chow groups. Regulator defects are built from higher Chow groups and realization maps

$$r_{p,n}:C{H}^p(X,n)\to H_{\mathcal{D}}^{2p-n}(X,\mathbb{Z}(p)).$$

The two constructions are related by the common pattern of loss under a realization or coefficient-change morphism.

Definition 70 

(Realization-loss datum). A realization-loss datum is a triple

$$(M,R,\rho ),$$

where M and R are abelian groups and

$$\rho :M⟶R$$

is a homomorphism. Its invisible subgroup is

$$K_{\rho }=ker(\rho ),$$

and its visible quotient is

$$M_{\rho }=M/K_{\rho }.$$

Example 3 

(Rationalization as realization loss). Let

$$M=Z^{2p}(X),R=Z^{2p}(X){\otimes }_{\mathbb{Z}}\mathbb{Q},$$

and let

$$\rho :Z^{2p}(X)⟶Z^{2p}(X){\otimes }_{\mathbb{Z}}\mathbb{Q}$$

be rationalization. Then

$$K_{\rho }=Z^{2p}{(X)}_{\mathrm{tors}}.$$

Example 4 

(Regulator as realization loss). Let

$$M=C{H}^p(X,n),R=H_{\mathcal{D}}^{2p-n}(X,\mathbb{Z}(p)),$$

and let

$$\rho =r_{p,n}.$$

Then

$$K_{\rho }=K_{\mathcal{D}}^{p,n}(X).$$

Proposition 38 

(Common exact form of realization loss). Let

$$(M,R,\rho )$$

be a realization-loss datum. Then there is a short exact sequence

$$0⟶K_{\rho }⟶M⟶im(\rho )⟶0.$$

Consequently,

$$M_{\rho }\cong im(\rho ).$$

Proof. 

By definition,

$$K_{\rho }=ker(\rho ).$$

The map

$$M⟶im(\rho )$$

induced by $\rho$ is surjective. Its kernel is $K_{\rho }$. The short exact sequence and the isomorphism

$$M/K_{\rho }\cong im(\rho )$$

follow from the first isomorphism theorem. □

Remark 57 

(Comparison with integral Hodge defects). The equality

$$Z^{2p}{(X)}_{\mathrm{tors}}=ker\left(Z^{2p}(X)\to Z^{2p}(X)\otimes \mathbb{Q}\right)$$

records the loss of torsion under rationalization. The equality

$$K_{\mathcal{D}}^{p,n}(X)=ker(r_{p,n})$$

records the loss of higher Chow information under the Deligne regulator. These are not the same group. They are parallel instances of the same formal pattern: a realization map may suppress information that remains nonzero in a finer algebraic or integral object.

Proposition 39 

(Finite regulator defect under finite image). Let

$$M\subseteq C{H}^p(X,n)$$

be a finitely generated subgroup. Assume that

$$r_{p,n}(M)$$

is finite. Then

$$D_{\mathcal{D}}^{\mathcal{C}}(M)=M/(M\cap K_{\mathcal{D}}^{p,n}(X))$$

is finite.

Proof. 

By Lemma 18,

$$D_{\mathcal{D}}^{\mathcal{C}}(M)\cong r_{p,n}(M).$$

The group $r_{p,n}(M)$ is finite by hypothesis. Hence $D_{\mathcal{D}}^{\mathcal{C}}(M)$ is finite. □

Proposition 40 

(Torsion invisible to rationalized regulators). Let

$$r_{p,n,\mathbb{Q}}:C{H}^p{(X,n)}_{\mathbb{Q}}⟶H_{\mathcal{D}}^{2p-n}(X,\mathbb{Q}(p))$$

be the rationalized regulator. Let

$$\xi \in C{H}^p(X,n)$$

be torsion. Then

$$\xi \otimes 1=0\mathrm{in}C{H}^p{(X,n)}_{\mathbb{Q}},$$

and hence

$$r_{p,n,\mathbb{Q}}(\xi \otimes 1)=0.$$

Proof. 

Since $\xi$ is torsion, there exists $N\ge 1$ such that

$$N\xi =0.$$

Then

$$\xi \otimes 1=N\xi \otimes {\displaystyle \frac{1}{N}}=0$$

in $C{H}^p{(X,n)}_{\mathbb{Q}}$. Applying $r_{p,n,\mathbb{Q}}$ gives

$$r_{p,n,\mathbb{Q}}(\xi \otimes 1)=0.$$

 □

9.4. Regulator Defect Profiles

Definition 71 

(Regulator defect profile). Let

$$M\subseteq C{H}^p(X,n)$$

be a subgroup supplied by a specified higher-cycle construction $\mathcal{C}$. The regulator defect profile of $\mathcal{C}$ on M is

$$Prof(\mathcal{C},M)=\left(D_{\mathcal{D}}^{\mathcal{C}}(M),D_{\mathcal{D},\mathrm{ind}}^{\mathcal{C}}(M),Q_{\mathcal{D}}^{\mathcal{C}}(M),K_{\mathcal{D}}^{p,n}(X),{\tau }_{\mathcal{C}}\right),$$

where

$${\tau }_{\mathcal{C}}=\mathrm{real}.$$

Remark 58 

(Scope of the regulator section). The regulator phenomena in this section do not assert new failures of the integral Hodge conjecture. They show that realization maps can lose higher Chow and integral information in a manner parallel to rationalization. Thus regulator-invisible and indecomposable higher Chow classes provide realization-loss defects attached to algebraic cycle constructions.

10. Coefficient and Bockstein Defects

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2, the finite-index terminology of Section 3, and the realization-loss notation of Section 9. Thus Hodge classes are cohomological classes, algebraicity means membership in the image of the appropriate cycle-class map, and constructional defects are quotients by algebraic subgroups supplied by specified constructions.

The constructions in this section are finite-coefficient realization-loss phenomena. The Bockstein homomorphism detects torsion before rationalization. It does not, by itself, prove non-algebraicity. A Bockstein class becomes an integral Hodge defect only after comparison with

$$im({cl}_{X,\mathbb{Z}})\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

10.1. The Bockstein Sequence

Let X be a smooth projective complex variety, let $p\ge 0$, and let $m\ge 2$. Consider the short exact sequence of coefficient groups

$$0⟶\mathbb{Z}(p)\stackrel{\times m}{\to }\mathbb{Z}(p)⟶\mathbb{Z}/m(p)⟶0.$$

It induces a long exact cohomology sequence whose relevant part is

$$\begin{array}{cc}\hfill H^{2p-1}(X,\mathbb{Z}(p))& \stackrel{\times m}{\to }{H}^{2p-1}(X,\mathbb{Z}(p))\hfill \\ & ⟶H^{2p-1}(X,\mathbb{Z}/m(p))\stackrel{{\beta }_m}{\to }{H}^{2p}(X,\mathbb{Z}(p))\stackrel{\times m}{\to }{H}^{2p}(X,\mathbb{Z}(p)).\hfill \end{array}$$

The connecting morphism

$${\beta }_m:H^{2p-1}(X,\mathbb{Z}/m(p))⟶H^{2p}(X,\mathbb{Z}(p))$$

is the Bockstein homomorphism associated with multiplication by m. This construction is functorial in X and follows from the long exact sequence in cohomology associated with a short exact sequence of coefficients [16,17,29].

Lemma 20 

(Image of the Bockstein). The image of

$${\beta }_m:H^{2p-1}(X,\mathbb{Z}/m(p))⟶H^{2p}(X,\mathbb{Z}(p))$$

is

$$im({\beta }_m)=H^{2p}(X,\mathbb{Z}(p))[m],$$

where

$$H^{2p}(X,\mathbb{Z}(p))[m]=\{\alpha \in H^{2p}(X,\mathbb{Z}(p))\mid m\alpha =0\}.$$

Proof. 

By exactness of the long exact cohomology sequence at $H^{2p}(X,\mathbb{Z}(p))$,

$$im({\beta }_m)=ker\left(\times m:H^{2p}(X,\mathbb{Z}(p))\to H^{2p}(X,\mathbb{Z}(p))\right).$$

The kernel of multiplication by m is precisely

$$H^{2p}(X,\mathbb{Z}(p))[m].$$

Hence

$$im({\beta }_m)=H^{2p}(X,\mathbb{Z}(p))[m].$$

 □

Definition 72 

(Bockstein source). Let

$$\alpha \in H^{2p}(X,\mathbb{Z}(p))[m].$$

A Bockstein source for α is a class

$$\theta \in H^{2p-1}(X,\mathbb{Z}/m(p))$$

such that

$${\beta }_m(\theta )=\alpha .$$

Corollary 15 

(Existence of Bockstein sources). Every class

$$\alpha \in H^{2p}(X,\mathbb{Z}(p))[m]$$

has a Bockstein source.

Proof. 

By Lemma 20,

$$H^{2p}(X,\mathbb{Z}(p))[m]=im({\beta }_m).$$

Thus every $\alpha \in H^{2p}(X,\mathbb{Z}(p))[m]$ is of the form

$$\alpha ={\beta }_m(\theta )$$

for some

$$\theta \in H^{2p-1}(X,\mathbb{Z}/m(p)).$$

 □

Definition 73 

(Bockstein Hodge torsion). The m-Bockstein Hodge torsion subgroup in codimension p is

$$B_m^{2p}(X)=im({\beta }_m)\cap H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

Equivalently,

$$B_m^{2p}(X)=H^{2p}(X,\mathbb{Z}(p))[m]\cap H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

Proof 

(Verification of the equivalence). By Lemma 20,

$$im({\beta }_m)=H^{2p}(X,\mathbb{Z}(p))[m].$$

Intersecting both sides with

$$H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

gives the asserted equality. □

Remark 59 

(Bockstein classes and rationalization). If

$$\alpha \in B_m^{2p}(X),$$

then

$$m\alpha =0.$$

Therefore

$$\alpha \otimes 1=0\mathrm{in}{H}^{2p}(X,\mathbb{Q}(p)).$$

Thus a Bockstein Hodge torsion class is invisible after rationalization.

10.2. Bockstein Detection of Integral Hodge Defects

The Bockstein homomorphism detects torsion in integral cohomology. It does not by itself detect non-algebraicity. Non-algebraicity is measured in

$$Z^{2p}(X)={\displaystyle \frac{H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}}{im({cl}_{X,\mathbb{Z}})}}.$$

Definition 74 

(Bockstein defect subgroup). The m-Bockstein defect subgroup of $Z^{2p}(X)$ is the image of $B_m^{2p}(X)$ in $Z^{2p}(X)$:

$$Z_m^{2p}{(X)}_{\beta }={\displaystyle \frac{B_m^{2p}(X)+im({cl}_{X,\mathbb{Z}})}{im({cl}_{X,\mathbb{Z}})}}.$$

Equivalently,

$$Z_m^{2p}{(X)}_{\beta }\cong {\displaystyle \frac{B_m^{2p}(X)}{B_m^{2p}(X)\cap im({cl}_{X,\mathbb{Z}})}}.$$

Lemma 21 

(Bockstein defects are m-torsion). The group

$$Z_m^{2p}{(X)}_{\beta }$$

is killed by m:

$$m·Z_m^{2p}{(X)}_{\beta }=0.$$

Proof. 

Let

$$[\alpha ]\in Z_m^{2p}{(X)}_{\beta }$$

be represented by

$$\alpha \in B_m^{2p}(X).$$

By Definition 73,

$$\alpha \in H^{2p}(X,\mathbb{Z}(p))[m].$$

Hence

$$m\alpha =0$$

in $H^{2p}(X,\mathbb{Z}(p))$. Therefore

$$m[\alpha ]=[m\alpha ]=0$$

in $Z^{2p}(X)$, and hence also in $Z_m^{2p}{(X)}_{\beta }$. □

Proposition 41 

(Bockstein detection criterion). Let

$$\theta \in H^{2p-1}(X,\mathbb{Z}/m(p)),\alpha ={\beta }_m(\theta ).$$

Assume

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

Then α defines an element

$${[\alpha ]}_{\mathbb{Z}}\in Z^{2p}(X).$$

Moreover,

$${[\alpha ]}_{\mathbb{Z}}=0$$

if and only if there exists $\Gamma \in C{H}^p(X)$ such that

$${cl}_{X,\mathbb{Z}}(\Gamma )=\alpha .$$

Thus θ detects a nonzero integral Hodge defect precisely when

$${\beta }_m(\theta )\in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

and

$${\beta }_m(\theta )\notin im({cl}_{X,\mathbb{Z}}).$$

Proof. 

Since

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}},$$

the quotient map

$$H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}⟶Z^{2p}(X)$$

sends $\alpha$ to a well-defined class

$${[\alpha ]}_{\mathbb{Z}}.$$

By Definition 12,

$${[\alpha ]}_{\mathbb{Z}}=0$$

if and only if

$$\alpha \in im({cl}_{X,\mathbb{Z}}).$$

This is equivalent to the existence of

$$\Gamma \in C{H}^p(X)$$

such that

$${cl}_{X,\mathbb{Z}}(\Gamma )=\alpha .$$

Substituting $\alpha ={\beta }_m(\theta )$ gives the final statement. □

Remark 60 

(Detection is not obstruction by itself). The existence of a class

$$\theta \in H^{2p-1}(X,\mathbb{Z}/m(p))$$

with

$${\beta }_m(\theta )\in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

does not imply non-algebraicity. The additional condition is

$${\beta }_m(\theta )\notin im({cl}_{X,\mathbb{Z}}).$$

Thus the Bockstein supplies a finite-coefficient source for torsion; the defect statement requires comparison with algebraic cycle classes.

10.3. Bockstein Sectors

Definition 75 

(Bockstein sector). Let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. The m-Bockstein part of $L_{\mathrm{Hdg}}$ is

$$L_{\beta ,m}=L_{\mathrm{Hdg}}\cap B_m^{2p}(X).$$

Its actual algebraic part is

$$L_{\beta ,m,\mathrm{alg}}^{\mathrm{act}}=L_{\beta ,m}\cap im({cl}_{X,\mathbb{Z}}).$$

The actual Bockstein sector defect is

$$D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\beta ,m}/L_{\beta ,m,\mathrm{alg}}^{\mathrm{act}}.$$

Definition 76 

(Bockstein constructional algebraic subgroup). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector. A Bockstein constructional algebraic subgroup is a subgroup

$$L_{\beta ,m,\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\beta ,m,\mathrm{alg}}^{\mathrm{act}}$$

generated by algebraic cycle classes supplied by a specified construction $\mathcal{C}$. The Bockstein constructional defect is

$$D_{\beta ,m}^{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\beta ,m}/L_{\beta ,m,\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 42 

(Actual Bockstein sector defect). For every typed Hodge sector $L_{\mathrm{Hdg}}$,

$$m·D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

Moreover, the natural map

$$D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})⟶Z^{2p}(X)$$

is injective.

Proof. 

Let

$$\alpha \in L_{\beta ,m}.$$

By Definition 75,

$$\alpha \in B_m^{2p}(X)\subseteq H^{2p}(X,\mathbb{Z}(p))[m].$$

Thus

$$m\alpha =0.$$

The class of $\alpha$ in

$$D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\beta ,m}/L_{\beta ,m,\mathrm{alg}}^{\mathrm{act}}$$

is therefore killed by m. Since $\alpha$ was arbitrary,

$$m·D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

The inclusion

$$L_{\beta ,m}\hookrightarrow H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

induces a homomorphism

$$D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})⟶Z^{2p}(X).$$

If the class of $\alpha \in L_{\beta ,m}$ maps to zero in $Z^{2p}(X)$, then

$$\alpha \in im({cl}_{X,\mathbb{Z}}).$$

Since $\alpha \in L_{\beta ,m}$, this means

$$\alpha \in L_{\beta ,m}\cap im({cl}_{X,\mathbb{Z}})=L_{\beta ,m,\mathrm{alg}}^{\mathrm{act}}.$$

Thus the kernel is zero. □

Corollary 16 

(Bockstein invisibility over $\mathbb{Q}$). The image of

$$D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})$$

in

$$Z^{2p}(X){\otimes }_{\mathbb{Z}}\mathbb{Q}$$

is zero.

Proof. 

By Proposition 42,

$$D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})$$

is killed by m. Hence every element maps to zero after tensoring with $\mathbb{Q}$. □

Remark 61 

(Actual versus constructional Bockstein defects). The actual Bockstein sector defect

$$D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}})$$

embeds in $Z^{2p}(X)$. The constructional defect

$$D_{\beta ,m}^{\mathcal{C}}(L_{\mathrm{Hdg}})$$

measures the failure of a specified construction to generate the Bockstein sector algebraically. It may be nonzero even when the actual Bockstein sector defect vanishes.

10.4. Comparison with Finite Constructional Defects

Bockstein defects are finite-coefficient defects. They are not substitutes for degree, projector, saturation, specialization, or regulator defects. They record the m-primary information produced or detected by those mechanisms.

Definition 77 

(Reduction of a sector modulo m). Let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. Its reduction modulo m is the image of

$$L_{\mathrm{Hdg}}/m{L}_{\mathrm{Hdg}}⟶H^{2p}(X,\mathbb{Z}/m(p)).$$

If

$$L\subseteq L_{\mathrm{Hdg}},$$

then the reduction of L modulo m is the image of

$$L/mL⟶H^{2p}(X,\mathbb{Z}/m(p)).$$

Lemma 22 

(Finite-index defects modulo m). Let

$$L\subseteq L_{\mathrm{Hdg}}$$

be finitely generated abelian groups, and set

$$D=L_{\mathrm{Hdg}}/L.$$

Then there is an exact sequence

$$L/mL⟶L_{\mathrm{Hdg}}/m{L}_{\mathrm{Hdg}}⟶D/mD⟶0.$$

If D is finite, then $D/mD$ is the part of the finite-index defect visible modulo m.

Proof. 

The quotient map

$$L_{\mathrm{Hdg}}⟶D$$

induces

$$L_{\mathrm{Hdg}}/m{L}_{\mathrm{Hdg}}⟶D/mD.$$

This map is surjective because $L_{\mathrm{Hdg}}\to D$ is surjective. The image of $L/mL$ maps to zero in $D/mD$, since L maps to zero in D.

Conversely, let $\alpha \in L_{\mathrm{Hdg}}$ map to zero in $D/mD$. The class of $\alpha$ in D lies in $mD$. Hence there exists

$$\gamma \in L_{\mathrm{Hdg}}$$

such that

$$\alpha -m\gamma \in L.$$

Modulo $m{L}_{\mathrm{Hdg}}$, the class of $\alpha$ is represented by an element of L. Thus the kernel is the image of $L/mL$. This proves exactness. □

Proposition 43 

(Bockstein comparison with degree defects). Let

$$f:Y\to X$$

be finite surjective of degree d. Let

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

satisfy

$$f^{*}\alpha \in im({cl}_{Y,\mathbb{Z}}).$$

Then

$$d{[\alpha ]}_{\mathbb{Z}}=0\mathrm{in}{Z}^{2p}(X).$$

If $m\mid d$, the m-primary part of the finite degree defect is the image of ${[\alpha ]}_{\mathbb{Z}}$ in

$$Z^{2p}(X)[m^{\infty }].$$

Proof. 

The equality

$$d{[\alpha ]}_{\mathbb{Z}}=0$$

is Proposition 15. If $m\mid d$, then the cyclic subgroup generated by ${[\alpha ]}_{\mathbb{Z}}$ is a finite abelian group whose order divides a power product involving primes dividing d. Its m-primary component is its image in

$$Z^{2p}(X)[m^{\infty }].$$

This is the standard primary decomposition for finite abelian groups. □

Proposition 44 

(Bockstein comparison with projector defects). Let

$$\pi \in C{H}^{dimX}{(X\times X)}_{\mathbb{Q}}$$

be an algebraic rational projector. Let

$$\Pi =N\pi \in C{H}^{dimX}(X\times X)$$

for some $N\ge 1$. Let

$$L\subseteq L_{\pi }^p(X)$$

be a torsion-free typed sector satisfying the hypotheses of Proposition 19. Then

$$N·D_{\pi }^{p,\mathcal{C}}(X;L)=0.$$

If $m\mid N$, the finite-coefficient reduction of the projector defect has possible m-primary part measured by

$$D_{\pi }^{p,\mathcal{C}}(X;L)/m{D}_{\pi }^{p,\mathcal{C}}(X;L).$$

Proof. 

The identity

$$N·D_{\pi }^{p,\mathcal{C}}(X;L)=0$$

is Proposition 19. Applying Lemma 22 to the inclusion defining

$$D_{\pi }^{p,\mathcal{C}}(X;L)$$

gives

$$D_{\pi }^{p,\mathcal{C}}(X;L)/m{D}_{\pi }^{p,\mathcal{C}}(X;L)$$

as the part visible after reduction modulo m. □

Proposition 45 

(Bockstein comparison with saturation defects). Let

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}$$

be a rationally covered typed Hodge sector, and let

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

For $m\ge 2$, the quotient

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})/m{D}_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

is the finite-index constructional defect visible after reducing the sector modulo m.

Proof. 

This is Lemma 22 applied to

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}.$$

Since the sector is rationally covered, Proposition 10 gives that

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

is finite. □

Remark 62 

(Specialization and regulator comparison). Specialization defects may produce torsion through boundary maps or monodromy-related exact sequences. When such torsion lies in integral cohomology, a Bockstein source records its finite-coefficient origin. A regulator defect is different: it is the kernel of a realization map on a higher Chow group. If such a kernel contains torsion, that torsion is invisible after rationalization; if it maps through a cohomological boundary, a Bockstein sequence may detect the resulting finite-coefficient class. No implication of non-algebraicity follows without comparison with the relevant algebraic image.

10.5. Bockstein Defect Profiles

Definition 78 

(Bockstein defect profile). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector, let $m\ge 2$, and let

$$L_{\beta ,m,\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\beta ,m}$$

be a constructional algebraic subgroup. The Bockstein defect profile of $\mathcal{C}$ is

$${Prof}_{\beta ,m}(\mathcal{C},L_{\mathrm{Hdg}})=\left(D_{\beta ,m}^{\mathcal{C}}(L_{\mathrm{Hdg}}),D_{\beta ,m}^{\mathrm{act}}(L_{\mathrm{Hdg}}),B_m^{2p}(X),Z_m^{2p}{(X)}_{\beta },m,{\tau }_{\mathcal{C}}\right),$$

where

$${\tau }_{\mathcal{C}}=\mathrm{real}.$$

Remark 63 

(Finite-coefficient microscope). The Bockstein construction provides a finite-coefficient microscope for torsion phenomena already present in the preceding sections:

$$\begin{array}{c}d\alpha \mathrm{algebraic}\mathrm{in}\mathrm{degree}\mathrm{or}\mathrm{trace}\mathrm{descent},\\ N\alpha \mathrm{algebraic}\mathrm{after}\mathrm{clearing}\mathrm{a}\mathrm{projector}\mathrm{denominator},\\ L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}\mathrm{finite}\mathrm{in}\mathrm{saturation}-\mathrm{supported}\mathrm{defects},\\ \mathrm{boundary}\mathrm{torsion}\mathrm{in}\mathrm{specialization},\\ \mathrm{realization}-\mathrm{kernel}\mathrm{torsion}\mathrm{in}\mathrm{regulator}\mathrm{settings}.\end{array}$$

It records finite-coefficient origins of torsion classes. It does not replace the cycle-class comparison required to prove that a torsion Hodge class is a nonzero element of $Z^{2p}(X)$.

11. Case Studies from Rational Hodge-Cycle Constructions

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2, the finite-index and defect-profile terminology of Section 3, and the finite-loss mechanisms of Section 4, Section 5, Section 6, Section 7, Section 8, Section 9 and Section 10. Thus Hodge classes are cohomological classes, algebraicity means membership in the image of the appropriate cycle-class map, and constructional defects are attached to specified constructions and specified typed Hodge sectors.

The case studies below do not reprove the cited rational Hodge-cycle results. They extract, from standard types of rational constructions, the finite integral data visible after the algebraic sources, correspondences, isogenies, projectors, or realization maps have been specified.

Definition 79 

(Case-study datum). A case-study datum in codimension p on a smooth projective complex variety X is a tuple

$$\mathfrak{C}=\left(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}},\mathcal{M},{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right),$$

where

$$\begin{array}{c}{L}_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}\mathrm{is}\mathrm{a}\mathrm{finitely}\mathrm{generated}\mathrm{typed}\mathrm{Hodge}\sec \mathrm{tor},\\ L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}\mathrm{is}\mathrm{generated}\mathrm{by}\mathrm{specified}\mathrm{Chow}\mathrm{classes}\mathrm{or}\mathrm{by}\mathrm{algebraic}\mathrm{correspondence}\mathrm{images}\mathrm{of}\mathrm{specified}\mathrm{Chow}\mathrm{classes},\\ \mathcal{M}\mathrm{is}\mathrm{the}\mathrm{rational}\mathrm{construction}\mathrm{mechanism}\mathrm{used}\mathrm{to}\mathrm{obtain}\mathrm{coverage}\mathrm{of}{L}_{\mathrm{Hdg}}\otimes \mathbb{Q},\\ {\mathcal{A}}_{\mathcal{C}}\mathrm{is}\mathrm{the}\mathrm{proof}-\mathrm{level}\mathrm{annihilator},\mathrm{denominator},\mathrm{finite}\mathrm{quotient},\mathrm{kernel},\mathrm{cokernel},\mathrm{boundary},\mathrm{or}\mathrm{coefficient}\mathrm{datum},\\ {\tau }_{\mathcal{C}}\in \{\mathrm{mult},\mathrm{sat},\mathrm{real}\}\mathrm{is}\mathrm{the}\mathrm{visible}\mathrm{defect}\mathrm{type}.\end{array}$$

The associated constructional defect is

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Remark 64 

(Template for the examples). Each case study records:

$$\begin{array}{c}\mathrm{a}\mathrm{typed}\sec \mathrm{tor}{L}_{\mathrm{Hdg}},\\ \mathrm{a}\mathrm{constructional}\mathrm{algebraic}\mathrm{subgroup}{L}_{\mathrm{alg}}^{\mathcal{C}},\\ \mathrm{a}\mathrm{rational}\mathrm{coverage}\mathrm{statement},\\ \mathrm{a}\mathrm{finite}\mathrm{constructional}\mathrm{defect},\\ \mathrm{a}\mathrm{defect}\mathrm{type}\mathrm{and}\mathrm{proof}-\mathrm{level}\mathrm{datum}.\end{array}$$

Soundness is the assertion that $L_{\mathrm{alg}}^{\mathcal{C}}$ is generated by cycle classes of specified Chow classes or by algebraic correspondences applied to specified Chow classes. Coverage is the separate assertion

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

No case study uses an arbitrary Hodge projector, an unconstructed motivic lift, a correspondence known only cohomologically, or a map defined by an unknown target Hodge class.

Proposition 46 

(Common finite-defect extraction). Let

$$\mathfrak{C}=\left(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}},\mathcal{M},{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right)$$

be a case-study datum. Assume soundness of $L_{\mathrm{alg}}^{\mathcal{C}}$ and rational coverage:

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is finite. If the proof supplies an integer $N\ge 1$ such that

$$N{L}_{\mathrm{Hdg}}\subseteq L_{\mathrm{alg}}^{\mathcal{C}},$$

then

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Proof. 

The finiteness follows from Proposition 9 applied to

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}.$$

If

$$N{L}_{\mathrm{Hdg}}\subseteq L_{\mathrm{alg}}^{\mathcal{C}},$$

then for every $\alpha \in L_{\mathrm{Hdg}}$, the image of $N\alpha$ in

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is zero. Hence

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

 □

11.1. Abelian Varieties and Absolute Hodge Classes

Let A be an abelian variety over $\mathbb{C}$, and let $p\ge 0$. Deligne’s theorem states that Hodge cycles on abelian varieties are absolute Hodge classes [18]. The theorem is rational and absolute-Hodge in nature. The integral question considered here is the lattice question left after a specified rational Hodge sector and specified algebraic sources have been chosen.

Definition 80 

(Abelian Hodge sector). An abelian Hodge sector in codimension p is a finitely generated subgroup

$$L_{\mathrm{ab}}\subseteq H^{2p}{(A,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

whose rational span

$$L_{\mathrm{ab},\mathbb{Q}}=L_{\mathrm{ab}}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$

is specified by a rational Hodge-cycle construction on A, by algebraic endomorphism correspondences of A, by a family of abelian varieties, or by a stated comparison with an algebraic sector on another abelian variety.

Definition 81 

(Constructional algebraic lattice in an abelian sector). Let $Z_1,\dots ,Z_s\in C{H}^p(A)$ be algebraic cycles such that

$${cl}_{A,\mathbb{Z}}(Z_i)\in L_{\mathrm{ab}},1\le i\le s.$$

Define

$$L_{\mathrm{ab},\mathrm{alg}}^{\mathcal{C}}={\displaystyle \sum _{i=1}^s}\mathbb{Z}{cl}_{A,\mathbb{Z}}(Z_i)\subseteq L_{\mathrm{ab}}.$$

The abelian-sector constructional defect is

$$D_{\mathrm{ab}}^{\mathcal{C}}(Z_{•})=L_{\mathrm{ab}}/L_{\mathrm{ab},\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 47 

(Absolute-Hodge rational control leaves a saturation question). Assume that

$$L_{\mathrm{ab},\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{ab}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathrm{ab}}^{\mathcal{C}}(Z_{•})$$

is finite. It vanishes if and only if

$$L_{\mathrm{ab},\mathrm{alg}}^{\mathcal{C}}=L_{\mathrm{ab}}.$$

Proof. 

The displayed equality gives rational coverage. Since $L_{\mathrm{ab}}$ is finitely generated, Proposition 9 gives that

$$L_{\mathrm{ab}}/L_{\mathrm{ab},\mathrm{alg}}^{\mathcal{C}}$$

is finite. The quotient vanishes precisely when the inclusion

$$L_{\mathrm{ab},\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{ab}}$$

is an equality. □

Remark 65 

(Integral reading of the abelian case). The rational construction controls

$$L_{\mathrm{ab}}\otimes \mathbb{Q}.$$

The integral question is whether the available algebraic classes generate the full lattice $L_{\mathrm{ab}}$ or only a finite-index sublattice. The finite quotient

$$L_{\mathrm{ab}}/L_{\mathrm{ab},\mathrm{alg}}^{\mathcal{C}}$$

is a saturation-supported constructional defect. This does not modify Deligne’s theorem; it records the integral lattice question left after rational control of Hodge cycles on abelian varieties [16,17,18].

11.2. Self-Products with Automorphisms

Let V be a smooth projective complex variety, and let

$$X=V^r.$$

Let G be a finite group acting algebraically on V and diagonally or factorwise on X. Schoen’s work on self-products with automorphisms uses self-products, automorphisms, eigenspaces, and algebraic correspondences in the study of Hodge classes [19].

Definition 82 

(Automorphism sector). An automorphism sector in codimension p on X is a finitely generated subgroup

$$L_G^p(X)\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}\cap H^{2p}{(X,\mathbb{Z}(p))}^G$$

specified by the algebraic G-action, algebraic correspondences, product classes, or finite covers associated with the action.

Let

$$Z_1,\dots ,Z_s\in C{H}^p(X)$$

be algebraic cycles with

$${cl}_{X,\mathbb{Z}}(Z_i)\in L_G^p(X),1\le i\le s.$$

Set

$$L_{G,\mathrm{alg}}^{p,\mathcal{C}}(X)={\displaystyle \sum _{i=1}^s}\mathbb{Z}{cl}_{X,\mathbb{Z}}(Z_i)\subseteq L_G^p(X).$$

Definition 83 

(Automorphism-sector constructional defect). The automorphism-sector constructional defect associated with $Z_1,\dots ,Z_s$ is

$$D_G^{p,\mathcal{C}}(X;Z_{•})=L_G^p(X)/L_{G,\mathrm{alg}}^{p,\mathcal{C}}(X).$$

Proposition 48 

(Finite defect for rational automorphism coverage). Assume that

$$L_{G,\mathrm{alg}}^{p,\mathcal{C}}(X){\otimes }_{\mathbb{Z}}\mathbb{Q}=L_G^p(X){\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_G^{p,\mathcal{C}}(X;Z_{•})$$

is finite.

Proof. 

The subgroup $L_{G,\mathrm{alg}}^{p,\mathcal{C}}(X)$ is generated by cycle classes of actual Chow classes. The displayed equality gives rational coverage. Since $L_G^p(X)$ is finitely generated, Proposition 9 gives that

$$L_G^p(X)/L_{G,\mathrm{alg}}^{p,\mathcal{C}}(X)$$

is finite. □

Definition 84 

(Character denominator). Let χ be a rational-valued character of G whose central idempotent is defined over $\mathbb{Q}$. Let

$$e_{\chi }={\displaystyle \frac{dim\chi }{|G|}}\sum _{g\in G}\chi (g^{-1})g^{*}$$

be the corresponding rational idempotent acting by algebraic correspondences. A character denominator is an integer $N_{\chi }\ge 1$ such that

$$N_{\chi }{e}_{\chi }$$

is an integral algebraic correspondence.

Proposition 49 

(Character-projector constructional defect). Let

$$L_{\chi }^p(X)\subseteq L_G^p(X)$$

be a torsion-free sector cut out rationally by $e_{\chi }$ and integrally by the chosen lattice. Assume that the hypotheses of Proposition 21 hold for $L_{\chi }^p(X)$. Then the associated constructional defect is killed by $N_{\chi }$.

Proof. 

The idempotent $e_{\chi }$ is represented by a rational linear combination of graphs of algebraic automorphisms. By the definition of $N_{\chi }$, $N_{\chi }{e}_{\chi }$ is an integral algebraic correspondence. The assertion is Proposition 21 applied to this algebraic idempotent. □

Remark 66 

(Integral reading of self-product constructions). Self-product constructions with automorphisms may use finite group averaging, character idempotents, finite covers, and algebraic correspondences. The attached finite data are:

$$\begin{array}{c}|G|-\mathrm{torsion}\mathrm{from}\mathrm{averaging},\\ N_{\chi }-\mathrm{torsion}\mathrm{from}\mathrm{character}\mathrm{denominators},\\ d-\mathrm{torsion}\mathrm{from}\mathrm{finite}\mathrm{trace}\mathrm{descent},\\ \mathrm{finite}\mathrm{cokernels}\mathrm{from}\mathrm{correspondence}\mathrm{images}.\end{array}$$

Thus the rational eigenspace decomposition leaves multiplier-supported and saturation-supported components in the constructional defect profile [19].

11.3. Products of Curves and Surfaces

Let

$$X=X_1\times \dots \times X_r$$

be a product of smooth projective complex varieties. Let $p\ge 0$, and let

$$L_{\mathrm{prod}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed sector specified by product decompositions, algebraic correspondences, Hodge-group invariants, endomorphism algebras, or known algebraic generators. Product and invariant-theoretic approaches to Hodge classes on products, including Morrison-type constructions, often identify a rational Hodge sector by algebraic classes or correspondences [20,21].

Let

$$Z_1,\dots ,Z_s\in C{H}^p(X)$$

be algebraic cycles such that

$${cl}_{X,\mathbb{Z}}(Z_i)\in L_{\mathrm{prod}},1\le i\le s.$$

Define

$$L_{\mathrm{prod},\mathrm{alg}}^{\mathcal{C}}={\displaystyle \sum _{i=1}^s}\mathbb{Z}{cl}_{X,\mathbb{Z}}(Z_i)\subseteq L_{\mathrm{prod}}.$$

Definition 85 

(Product saturation defect). The product saturation defect associated with the classes $Z_i$ is

$$D_{\mathrm{prod}}^{\mathcal{C}}(Z_{•})=L_{\mathrm{prod}}/L_{\mathrm{prod},\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 50 

(Rational product generation gives finite saturation defect). Assume that

$$L_{\mathrm{prod},\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{prod}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathrm{prod}}^{\mathcal{C}}(Z_{•})$$

is finite. If $L_{\mathrm{prod}}$ is torsion-free, this finite group is computed by the Smith normal form of the matrix of the classes

$${cl}_{X,\mathbb{Z}}(Z_i)$$

in any integral basis of $L_{\mathrm{prod}}$.

Proof. 

The finiteness follows from Proposition 9. If $L_{\mathrm{prod}}$ is torsion-free, choose an integral basis of $L_{\mathrm{prod}}$ and form the matrix whose columns are the coordinates of ${cl}_{X,\mathbb{Z}}(Z_i)$. The Smith normal form calculation of Proposition 27 gives the invariant factors of

$$L_{\mathrm{prod}}/L_{\mathrm{prod},\mathrm{alg}}^{\mathcal{C}}.$$

 □

Remark 67 

(Integral reading of product constructions). A rational invariant-theoretic description identifies

$$L_{\mathrm{prod}}\otimes \mathbb{Q}.$$

It does not determine whether the algebraic generators form a saturated integral lattice. The finite quotient

$$D_{\mathrm{prod}}^{\mathcal{C}}(Z_{•})$$

is the saturation-supported constructional defect left by replacing integral generation with rational generation.

11.4. Prym and Weil-Type Constructions

Let A be an abelian variety over $\mathbb{C}$, and suppose that a Prym, Jacobian, isogeny, or endomorphism-field construction specifies a typed Hodge sector

$$L_{\mathrm{PW}}\subseteq H^{2p}{(A,\mathbb{Z}(p))}_{\mathrm{Hdg}}.$$

Prym and Weil-type constructions often compare the desired sector with a sector on another abelian variety by a norm map, an isogeny, an algebraic correspondence, or an idempotent arising from algebraic endomorphisms.

Definition 86 

(Prym–Weil algebraic input). A Prym–Weil algebraic input for $L_{\mathrm{PW}}$ consists of

$$(B,u,\Gamma ,Z_1,\dots ,Z_s),$$

where

$$\begin{array}{c}B\mathrm{is}\mathrm{an}\mathrm{abelian}\mathrm{variety},\\ u:B\to A\mathrm{is}\mathrm{an}\mathrm{isogeny}\mathrm{or}\mathrm{a}\mathrm{finite}\mathrm{comparison}\mathrm{morphism}\mathrm{when}\mathrm{specified},\\ \Gamma \in C{H}^r(B\times A)\mathrm{is}\mathrm{an}\mathrm{algebraic}\mathrm{correspondence},\\ Z_1,\dots ,Z_s\in C{H}^q(B)\mathrm{are}\mathrm{algebraic}\mathrm{cycles}\mathrm{whose}\mathrm{correspondence}\mathrm{images}\mathrm{lie}\mathrm{in}C{H}^p(A),\\ {cl}_{A,\mathbb{Z}}({\Gamma }_{*}{Z}_i)\in L_{\mathrm{PW}}\mathrm{for}1\le i\le s.\end{array}$$

Given such data, define

$$L_{\mathrm{PW},\mathrm{alg}}^{\mathcal{C}}={\displaystyle \sum _{i=1}^s}\mathbb{Z}{cl}_{A,\mathbb{Z}}({\Gamma }_{*}{Z}_i)\subseteq L_{\mathrm{PW}}.$$

Definition 87 

(Prym–Weil defect). The Prym–Weil constructional defect is

$$D_{\mathrm{PW}}^{\mathcal{C}}=L_{\mathrm{PW}}/L_{\mathrm{PW},\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 51 

(Finite Prym–Weil defect under rational coverage). Assume that

$$L_{\mathrm{PW},\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{PW}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathrm{PW}}^{\mathcal{C}}$$

is finite.

Proof. 

The subgroup $L_{\mathrm{PW},\mathrm{alg}}^{\mathcal{C}}$ is soundly algebraic because it is generated by cycle classes of the Chow classes ${\Gamma }_{*}{Z}_i$. The displayed equality gives rational coverage. Since $L_{\mathrm{PW}}$ is finitely generated, Proposition 9 gives that

$$L_{\mathrm{PW}}/L_{\mathrm{PW},\mathrm{alg}}^{\mathcal{C}}$$

is finite. □

Proposition 52 

(Isogeny and norm contribution to a Prym–Weil defect). Let

$$u:B\to A$$

be an isogeny of degree d. Let

$$\alpha \in L_{\mathrm{PW}}$$

satisfy

$$u^{*}\alpha \in im({cl}_{B,\mathbb{Z}}).$$

Then

$$d{[\alpha ]}_{\mathbb{Z}}=0\mathrm{in}{Z}^{2p}(A).$$

If the constructional sector is $L_{\mathrm{PW}}$, then the class of α in the corresponding isogeny-covered constructional defect is killed by d.

Proof. 

This is Proposition 28 applied to the isogeny $u:B\to A$. If

$$u^{*}\alpha ={cl}_{B,\mathbb{Z}}(\Theta )$$

for some $\Theta \in C{H}^p(B)$, then

$${cl}_{A,\mathbb{Z}}(u_{*}\Theta )=d\alpha .$$

Hence the corresponding class of $\alpha$ is killed by d. □

Remark 68 

(Integral reading of Prym–Weil constructions). Prym and Weil-type rational constructions may identify the desired sector through a norm kernel, an isogeny, an endomorphism-field decomposition, or an algebraic correspondence. The finite data are:

$$\begin{array}{c}\mathrm{degree}\mathrm{defects}\mathrm{from}\mathrm{isogenies},\\ \mathrm{norm}-\mathrm{kernel}\mathrm{and}\mathrm{polarization}-\mathrm{index}\mathrm{defects},\\ \mathrm{denominators}\mathrm{of}\mathrm{endomorphism}\mathrm{idempotents},\\ \mathrm{saturation}\mathrm{defects}\mathrm{of}\mathrm{constructed}\mathrm{algebraic}\mathrm{cycles}.\end{array}$$

Thus the rational comparison of Hodge structures leaves a finite integral comparison problem [18,21].

11.5. Higher Chow Cycles on Jacobians

Let J be the Jacobian of a smooth projective complex curve. Higher Chow cycles on Jacobians provide examples in which regulator information does not detect all Chow-theoretic information. Collino–Fakhruddin type constructions exhibit higher Chow cycles with subtle regulator behavior and indecomposability properties [25].

Let $p\ge 0$ and $n\ge 1$. Let

$$M\subseteq C{H}^p(J,n)$$

be a finitely generated subgroup generated by specified higher Chow cycles. Let

$$r_{p,n}:C{H}^p(J,n)\to H_{\mathcal{D}}^{2p-n}(J,\mathbb{Z}(p))$$

be the regulator map.

Definition 88 

(Jacobian regulator defect). The regulator-invisible subgroup of M is

$$M_{\mathrm{inv}}=M\cap ker(r_{p,n}).$$

The regulator-visible quotient is

$$D_{\mathcal{D}}^{\mathcal{C}}(M)=M/M_{\mathrm{inv}}.$$

Definition 89 

(Jacobian indecomposability defect). Let

$$C{H}^p{(J,n)}_{\mathrm{dec}}\subseteq C{H}^p(J,n)$$

be the subgroup of decomposable classes in the chosen bidegree. The indecomposability quotient of M is

$$D_{\mathrm{ind}}^{\mathcal{C}}(M)=M/\left(M\cap C{H}^p{(J,n)}_{\mathrm{dec}}\right).$$

The regulator-invisible indecomposability quotient is

$$D_{\mathcal{D},\mathrm{ind}}^{\mathcal{C}}(M)={\displaystyle \frac{M\cap ker(r_{p,n})}{M\cap ker(r_{p,n})\cap C{H}^p{(J,n)}_{\mathrm{dec}}}}.$$

Proposition 53 

(Regulator invisibility and indecomposability). There is an injective homomorphism

$$D_{\mathcal{D},\mathrm{ind}}^{\mathcal{C}}(M)\hookrightarrow D_{\mathrm{ind}}^{\mathcal{C}}(M).$$

Its image consists of indecomposable classes represented by elements of M with zero regulator.

Proof. 

This is Proposition 37 applied to $X=J$. The injection is induced by the inclusion

$$M\cap ker(r_{p,n})\subseteq M.$$

The kernel is precisely the intersection with the decomposable subgroup, so the induced map is injective. □

Remark 69 

(Integral reading of higher Chow examples). The defect here is not the ordinary group $Z^{2p}(J)$. It is a realization-loss defect:

$$ker(r_{p,n})\subseteq C{H}^p(J,n).$$

A class may be invisible to the regulator while remaining nonzero in an indecomposable quotient of a higher Chow group. Thus regulator arguments display a way in which a realization map can suppress information present in a finer cycle-theoretic group [25,28].

11.6. Hodge 1-Motivic Constructions

Let X be a smooth projective complex variety or, where required, a singular proper complex variety for which the relevant 1-motivic construction is defined. Barbieri–Viale’s Hodge 1-motivic viewpoint organizes certain cycle-theoretic and Hodge-theoretic data through 1-motives, weight filtrations, and generalized cycle maps [26].

Definition 90 

(1-motivic comparison datum). A 1-motivic comparison datum in degree k consists of

$$(M_1(X),\lambda ,L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}}),$$

where

$$\begin{array}{c}{M}_1(X)\mathrm{is}\mathrm{a}\mathrm{specified}1-\mathrm{motivic}\mathrm{object}\mathrm{attached}\mathrm{to}X,\\ \lambda :M_1(X)⟶H^k(X,\mathbb{Z})\mathrm{is}\mathrm{the}\mathrm{chosen}\mathrm{realization}\mathrm{or}\mathrm{cycle}-\mathrm{class}\mathrm{comparison}\mathrm{map},\\ L_{\mathrm{Hdg}}\subseteq H^k(X,\mathbb{Z})\mathrm{is}\mathrm{a}\mathrm{finitely}\mathrm{generated}\mathrm{Hodge}-\mathrm{theoretic}\sec \mathrm{tor},\\ L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}\mathrm{is}\mathrm{generated}\mathrm{by}\mathrm{images}\mathrm{under}\lambda \mathrm{of}\mathrm{specified}\mathrm{algebraic}\mathrm{or}1-\mathrm{motivic}\mathrm{sources}.\end{array}$$

Definition 91 

(1-motivic constructional defect). Given a 1-motivic comparison datum, the 1-motivic constructional defect is

$$D_{1-\mathrm{mot}}^{\mathcal{C}}(X)=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 54 

(Finite 1-motivic defect under rational coverage). Assume

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{1-\mathrm{mot}}^{\mathcal{C}}(X)$$

is finite.

Proof. 

The assertion is Proposition 9 applied to

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}.$$

The subgroup $L_{\mathrm{alg}}^{\mathcal{C}}$ is part of the datum and must be supplied by specified 1-motivic or algebraic sources. The displayed equality gives coverage after tensoring with $\mathbb{Q}$. □

Remark 70 

(Integral reading of 1-motivic constructions). A 1-motivic construction may relocate integral information into a lattice part, a semi-abelian part, or an extension datum. The quotient

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

measures whether the 1-motivic sources generate the integral sector or only a finite-index sublattice. This is a mixed or motivic analogue of the saturation-supported defects studied above, subject to the existence of the specified realization map [26].

11.7. Integral Hodge Counterexample Literature as Calibration

The preceding case studies start from rational constructions and extract finite constructional defects. The integral Hodge counterexample literature calibrates when such a finite defect can be an actual nonzero element of $Z^{2p}(X)$.

Definition 92 

(Calibrating integral defect). A calibrating integral defect in codimension p on a smooth projective complex variety X is a class

$$\alpha \in H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

such that

$${[\alpha ]}_{\mathbb{Z}}\ne 0\mathrm{in}{Z}^{2p}(X).$$

It is torsion-calibrating if ${[\alpha ]}_{\mathbb{Z}}$ is torsion.

Remark 71 

(Atiyah–Hirzebruch torsion). Atiyah–Hirzebruch exhibited torsion integral Hodge classes that are not algebraic [6]. In the present notation, such a class gives

$$0\ne {[\alpha ]}_{\mathbb{Z}}\in Z^{2p}{(X)}_{\mathrm{tors}},$$

and

$${[\alpha ]}_{\mathbb{Z}}\otimes 1=0\mathrm{in}{Z}^{2p}(X)\otimes \mathbb{Q}.$$

This is the basic calibration for torsion lost under rationalization.

Remark 72 

(Unramified cohomology calibration). Colliot-Thélène and Voisin relate degree-four integral Hodge defects to unramified cohomology [1]. This identifies a cohomological obstruction group capable of detecting integral failure beyond the rational Hodge statement. In the present paper this serves as calibration for the claim that finite integral defects can carry information not visible after tensoring with $\mathbb{Q}$.

Remark 73 

(Threefold calibration). Voisin, Benoist–Ottem, and Totaro give positive and negative results for the integral Hodge conjecture in specific threefold settings [11,12,13]. These results distinguish geometries where integral defects vanish from geometries where finite or torsion obstructions remain. They provide benchmarks for the defect groups extracted from rational construction mechanisms.

Remark 74 

(Diaz calibration). Diaz constructs examples with finite-rank Chow groups and failure of the integral Hodge conjecture [14]. Such examples show that finite rank of Chow groups does not by itself remove integral Hodge defects. In the language of this paper, they calibrate the possibility that a rationally controlled or finitely generated algebraic cycle theory may still leave a nonzero integral defect.

Proposition 55 

(Calibration principle). Let X be smooth projective over $\mathbb{C}$, and let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. Let

$$L_{\mathrm{alg}}^{\mathrm{act}}=L_{\mathrm{Hdg}}\cap im({cl}_{X,\mathbb{Z}}).$$

Assume rational coverage:

$$L_{\mathrm{alg}}^{\mathrm{act}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathrm{act}}$$

is a finite subgroup of the global integral defect group $Z^{2p}(X)$. If

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})\ne 0,$$

then the sector contains a genuine integral Hodge defect. Since $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$ is finite, it is invisible after tensoring with $\mathbb{Q}$.

Proof. 

By Proposition 8,

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

is finite. By Lemma 2, the natural map

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})⟶Z^{2p}(X)$$

is injective. Hence a nonzero class in $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$ gives a nonzero class in $Z^{2p}(X)$. Since $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$ is finite, all of its elements are torsion. By Lemma 1, they map to zero after tensoring with $\mathbb{Q}$. □

11.8. Case-Study Profiles

Definition 93 

(Case-study profile). The profile extracted from a case-study datum

$$\mathfrak{C}=\left(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}},\mathcal{M},{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right)$$

is

$$Prof(\mathfrak{C})=\left(D_{\mathcal{C}}(L_{\mathrm{Hdg}}),e_{\mathcal{C}}(L_{\mathrm{Hdg}}),{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right),$$

where

$$e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp{D}_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

when the defect is finite.

Remark 75 

(Case-study summary). The case studies have the following profiles:

$$\begin{array}{ccc}\mathrm{rational}\mathrm{mechanism}& \mathrm{integral}\mathrm{weak}\mathrm{point}& \mathrm{defect}\mathrm{type}\\ \mathrm{absolute}\mathrm{Hodge}\mathrm{control}& \mathrm{lattice}\mathrm{saturation}& \mathrm{sat}\\ \mathrm{automorphisms}& \mathrm{group}\mathrm{and}\mathrm{character}\mathrm{denominators}& \mathrm{mult}\\ \mathrm{products}& \mathrm{integral}\mathrm{generation}& \mathrm{sat}\\ \mathrm{Prym}--\mathrm{Weil}\mathrm{comparison}& \mathrm{isogenies},\mathrm{norms},\mathrm{saturation}& \mathrm{mult}+\mathrm{sat}\\ \mathrm{higher}\mathrm{Chow}\mathrm{regulators}& \mathrm{realization}\mathrm{kernel}& \mathrm{real}\\ 1-\mathrm{motivic}\mathrm{comparison}& \mathrm{lattice}\mathrm{and}\mathrm{extension}\mathrm{data}& \mathrm{sat}+\mathrm{real}\\ \mathrm{integral}\mathrm{Hodge}\mathrm{counterexamples}& \mathrm{nonzero}\mathrm{actual}\mathrm{integral}\mathrm{defect}& Z^{2p}(X)\end{array}$$

Thus the passage from rational Hodge-cycle constructions to integral algebraicity is governed by finite quotients, denominators, kernels, cokernels, boundary terms, and saturation indices. These are the finite integral data carried by the rational methods.

12. Synthesis

We use the notation and conventions of Section 1.1. We also use the defect notation of Section 2 and the defect-profile terminology of Section 3. Thus Hodge classes are cohomological classes, algebraicity means membership in the image of the appropriate cycle-class map, and constructional defects are attached to specified constructions and specified typed Hodge sectors.

The preceding sections have a common form. A rational Hodge-cycle construction $\mathcal{C}$ acts on a specified typed sector

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}},$$

and produces a sound algebraic subgroup

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}.$$

If the construction proves rational coverage,

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q},$$

then the quotient

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is finite. This quotient is the finite integral defect left by the construction. It records information which is invisible in the final rational equality.

The synthesis separates three levels: constructional algebraicity, actual integral algebraicity, and rational algebraicity. The comparison between these three levels is the content of this section.

12.1. Actual and Constructional Synthesis

Let X be a smooth projective complex variety, let $p\ge 0$, and let

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

be a typed Hodge sector. Its actual algebraic part is

$$L_{\mathrm{alg}}^{\mathrm{act}}=L_{\mathrm{Hdg}}\cap im({cl}_{X,\mathbb{Z}}).$$

A sound construction $\mathcal{C}$ supplies a constructional algebraic subgroup

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}.$$

There are therefore two different quotients:

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathrm{act}},$$

and

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

The first is the actual sector defect. The second is the defect of the method $\mathcal{C}$.

Proposition 56 

(Actual versus constructional defect). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector, and let

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}\subseteq L_{\mathrm{Hdg}}.$$

Then there is a short exact sequence

$$0⟶L_{\mathrm{alg}}^{\mathrm{act}}/L_{\mathrm{alg}}^{\mathcal{C}}⟶D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})⟶0.$$

Proof. 

The inclusions

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}\subseteq L_{\mathrm{Hdg}}$$

give the quotient exact sequence

$$0⟶L_{\mathrm{alg}}^{\mathrm{act}}/L_{\mathrm{alg}}^{\mathcal{C}}⟶L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}⟶L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathrm{act}}⟶0.$$

By definition, the middle quotient is $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$, and the right quotient is $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$. □

Remark 76 

(Meaning of the exact sequence). The quotient

$$L_{\mathrm{alg}}^{\mathrm{act}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

measures algebraic classes in the sector which exist but are not produced by $\mathcal{C}$. The quotient

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

measures genuine failure of integral algebraicity in the sector. Hence a nonzero constructional defect need not be a nonzero actual integral Hodge defect. It may only measure the finite loss of the chosen method.

12.2. Saturation and Rational Coverage

Let

$$L\subseteq L_{\mathrm{Hdg}}$$

be a subgroup. Its saturation in $L_{\mathrm{Hdg}}$ is

$$L^{\mathrm{sat}}=\left\{\alpha \in L_{\mathrm{Hdg}}|\exists N\ge 1\mathrm{such}\mathrm{that}N\alpha \in L\right\}.$$

The saturation and rational residual quotients are

$$D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)=L^{\mathrm{sat}}/L$$

and

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)=L_{\mathrm{Hdg}}/L^{\mathrm{sat}}.$$

Proposition 57 

(Saturation decomposition). For every subgroup $L\subseteq L_{\mathrm{Hdg}}$, there is a short exact sequence

$$0⟶D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)⟶L_{\mathrm{Hdg}}/L⟶D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)⟶0.$$

Moreover,

$$D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)={(L_{\mathrm{Hdg}}/L)}_{\mathrm{tors}},$$

and $D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)$ is torsion-free.

Proof. 

This is Proposition 7. □

Corollary 17 

(Rational coverage leaves finite saturation data). Assume that $L_{\mathrm{Hdg}}$ is finitely generated and that

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)=0,$$

and

$$L_{\mathrm{Hdg}}/L=D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)$$

is finite.

Proof. 

The equivalence between rational coverage and

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L)=0$$

is Proposition 8. The equality

$$L_{\mathrm{Hdg}}/L=D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)$$

then follows from Proposition 57. Since $L_{\mathrm{Hdg}}$ is finitely generated and the quotient is torsion, it is finite. □

12.3. Constructional and External Annihilators

The exact sequence of Proposition 56 allows comparison between proof-level finite data and external obstruction theories for actual integral defects. The comparison is useful because external theories usually control $D_{\mathrm{act}}$, while a rational construction naturally produces $D_{\mathcal{C}}$.

Definition 94 

(External annihilator on a sector). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector. An external annihilator for the actual sector defect is an integer $T\ge 1$ such that

$$T·D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

Equivalently,

$$T{L}_{\mathrm{Hdg}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}.$$

Remark 77 

(Sources of external annihilators). An external annihilator may come from a separate obstruction theory, such as a diagonal-decomposition argument, a torsion-order argument, an unramified cohomology calculation, or a direct computation of the actual defect group. It is not part of the construction $\mathcal{C}$ unless the construction itself supplies it.

Theorem 1 

(Constructional and actual annihilator comparison). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector, and let

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}\subseteq L_{\mathrm{Hdg}}.$$

Let

$$q_{\mathcal{C}}:D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

be the natural quotient map. Suppose that $N\ge 1$ is a constructional annihilator:

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Then

$$N·im(q_{\mathcal{C}})=0.$$

If $T\ge 1$ is an external annihilator of $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$, then

$$T·im(q_{\mathcal{C}})=0.$$

Consequently,

$$gcd(N,T)·im(q_{\mathcal{C}})=0.$$

Proof. 

Let

$$\overline{\alpha }\in D_{\mathcal{C}}(L_{\mathrm{Hdg}}).$$

If

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0,$$

then

$$N\overline{\alpha }=0.$$

Applying $q_{\mathcal{C}}$ gives

$$N{q}_{\mathcal{C}}(\overline{\alpha })=q_{\mathcal{C}}(N\overline{\alpha })=0.$$

Thus

$$N·im(q_{\mathcal{C}})=0.$$

If T is an external annihilator, then

$$T·D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

Therefore every element of the subgroup

$$im(q_{\mathcal{C}})\subseteq D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

is killed by T.

Let

$$g=gcd(N,T).$$

Choose $a,b\in \mathbb{Z}$ such that

$$g=aN+bT.$$

For any

$$\gamma \in im(q_{\mathcal{C}}),$$

one has

$$N\gamma =0,T\gamma =0.$$

Hence

$$g\gamma =aN\gamma +bT\gamma =0.$$

Thus

$$g·im(q_{\mathcal{C}})=0.$$

 □

Corollary 18 

(Comparison with a torsion-order annihilator). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector, and suppose that an external diagonal-decomposition or torsion-order argument supplies an integer

$$Tor(X)\ge 1$$

which annihilates the actual sector defect:

$$Tor(X)·D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

Let $\mathcal{C}$ be a sound construction with constructional annihilator $N\ge 1$. Then

$$gcd(N,Tor(X))·im\left(D_{\mathcal{C}}(L_{\mathrm{Hdg}})\to D_{\mathrm{act}}(L_{\mathrm{Hdg}})\right)=0.$$

In particular,

$$expim\left(D_{\mathcal{C}}(L_{\mathrm{Hdg}})\to D_{\mathrm{act}}(L_{\mathrm{Hdg}})\right)\mid gcd(N,Tor(X)).$$

Proof. 

Apply Theorem 1 with

$$T=Tor(X).$$

The divisibility statement follows because the exponent of a finite abelian group divides every integer which annihilates it. □

Remark 78 

(What the torsion-order comparison does and does not say). Corollary 18 controls only the image of the constructional defect in the actual defect group. It does not say that

$$Tor(X)$$

annihilates the full constructional defect $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$. The kernel

$$L_{\mathrm{alg}}^{\mathrm{act}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

may contain finite constructional loss coming from algebraic classes which exist but are not produced by the construction. Example 5 exhibits this phenomenon: the actual defect is zero, but the constructional defect is $\mathbb{Z}/2\mathbb{Z}$.

Corollary 19 

(External annihilators do not control the whole constructional defect). In the situation of Theorem 1, an external annihilator T of $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$ annihilates the image of $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$ in the actual defect group, but need not annihilate

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

itself. The possible obstruction is the kernel

$$ker(q_{\mathcal{C}})=L_{\mathrm{alg}}^{\mathrm{act}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Proof. 

By Proposition 56,

$$ker(q_{\mathcal{C}})=L_{\mathrm{alg}}^{\mathrm{act}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

An external annihilator of $D_{\mathrm{act}}(L_{\mathrm{Hdg}})$ annihilates the target of $q_{\mathcal{C}}$, and hence annihilates the image of $q_{\mathcal{C}}$. It gives no condition on the kernel unless additional information is supplied. Therefore it need not annihilate the full group $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$. □

Remark 79 

(Why this comparison is not tautological). The constructional defect and the actual defect answer different questions. A proof-level denominator, degree, or lattice index controls the method used to produce algebraic classes. An external annihilator controls actual integral Hodge defects. The map

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

is the bridge between them. The theorem shows that the two controls agree only on the actual image of the constructional defect. The kernel records algebraic classes missed by the construction.

Example 5 

(A nonzero constructional defect with zero actual defect). The construction in Example 1 gives a geometrically sourced instance of a nonzero constructional defect with zero actual defect. Namely, for

$$X={\mathbb{P}}^1\times {\mathbb{P}}^1$$

and

$$L_{\mathrm{Hdg}}=H^2{(X,\mathbb{Z}(1))}_{\mathrm{Hdg}}=\mathbb{Z}{H}_1\oplus \mathbb{Z}{H}_2,$$

the actual algebraic lattice is the whole sector:

$$L_{\mathrm{alg}}^{\mathrm{act}}=L_{\mathrm{Hdg}}.$$

Thus

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

On the other hand, the degree-two finite morphism

$$f=g\times {id}_{{\mathbb{P}}^1},g([x:y])=[x^2:y^2],$$

has

$$f^{*}{H}_1=2H_1,f^{*}{H}_2=H_2.$$

Therefore the construction $\mathcal{C}=f^{*}$ supplies the pullback lattice

$$L_{\mathrm{alg}}^{\mathcal{C}}=f^{*}{L}_{\mathrm{Hdg}}=\mathbb{Z}(2H_1)\oplus \mathbb{Z}{H}_2\subseteq L_{\mathrm{Hdg}}.$$

Hence

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})={\displaystyle \frac{\mathbb{Z}{H}_1\oplus \mathbb{Z}{H}_2}{\mathbb{Z}(2H_1)\oplus \mathbb{Z}{H}_2}}\cong \mathbb{Z}/2\mathbb{Z},D_{\mathrm{act}}(L_{\mathrm{Hdg}})=0.$$

Since f is finite of degree 2, the trace identity gives

$$f_{*}{f}^{*}=2id$$

on $L_{\mathrm{Hdg}}$. Thus the example is multiplier-supported with proof-level annihilator $N=2$.

Remark 80 

(Meaning of the example). Example 5 shows that a constructional defect can be a genuine finite defect of a method even when there is no actual integral Hodge obstruction. The nonzero group

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})\cong \mathbb{Z}/2\mathbb{Z}$$

comes from the nonsaturated pullback lattice produced by the finite morphism

$$f=g\times {id}_{{\mathbb{P}}^1}.$$

The actual algebraic lattice is larger:

$$L_{\mathrm{alg}}^{\mathrm{act}}=L_{\mathrm{Hdg}}.$$

Thus the map

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

is the zero map, and its kernel is all of $D_{\mathcal{C}}(L_{\mathrm{Hdg}})$. This is the simplest case in which an external obstruction theory detects no integral Hodge defect, while the constructional profile still records a nonzero finite integral loss.

12.4. Defect Mechanisms

Definition 95 

(Defect mechanism). A defect mechanism in codimension p on X is a tuple

$$\mathfrak{M}=\left(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}},\mathcal{O},{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right),$$

where the data satisfy the following conditions:

(i)

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}$$

is a finitely generated typed Hodge sector.

(ii)

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}$$

is generated by sound algebraic sources.

(iii)

$\mathcal{O}$ is the geometric operation or realization process producing the sector comparison.

(iv)

${\mathcal{A}}_{\mathcal{C}}$ is the proof-level annihilator, denominator, finite quotient, kernel, cokernel, boundary, or coefficient datum.

(v)

$${\tau }_{\mathcal{C}}\in \{\mathrm{mult},\mathrm{sat},\mathrm{real}\}$$

is the visible defect type.

The constructional defect of $\mathfrak{M}$ is

$$D(\mathfrak{M})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

Proposition 58 

(Common form of the mechanisms). Let

$$\mathfrak{M}=\left(L_{\mathrm{Hdg}},L_{\mathrm{alg}}^{\mathcal{C}},\mathcal{O},{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right)$$

be a defect mechanism. Assume rational coverage:

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then $D(\mathfrak{M})$ is finite. If the proof supplies an integer $N\ge 1$ such that

$$N{L}_{\mathrm{Hdg}}\subseteq L_{\mathrm{alg}}^{\mathcal{C}},$$

then

$$N·D(\mathfrak{M})=0.$$

Proof. 

The finiteness is Proposition 9 applied to the inclusion

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{Hdg}}.$$

If

$$N{L}_{\mathrm{Hdg}}\subseteq L_{\mathrm{alg}}^{\mathcal{C}},$$

then for every $\alpha \in L_{\mathrm{Hdg}}$, the class of $N\alpha$ in

$$L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is zero. Thus the class of $\alpha$ is killed by N. Since $\alpha$ was arbitrary,

$$N·D(\mathfrak{M})=0.$$

 □

Remark 81 

(No universal annihilator). The integer N is not universal. In trace descent it is a degree; in averaging it is a group order; in projector arguments it is a denominator; in isogeny arguments it is a degree or a smaller exponent only when justified by a trace or norm identity; in saturation arguments it is the exponent of a finite lattice quotient; in regulator settings there need not be a single natural annihilator. The common structure is the finite quotient, not a common integer.

12.5. Profiles of the Mechanisms

Definition 96 

(Mechanism profile). Let $\mathfrak{M}$ be a defect mechanism as in Definition 95. Its profile is

$$Prof(\mathfrak{M})=\left(D(\mathfrak{M}),e(\mathfrak{M}),{\mathcal{A}}_{\mathcal{C}},{\tau }_{\mathcal{C}}\right),$$

where

$$e(\mathfrak{M})=expD(\mathfrak{M})$$

when $D(\mathfrak{M})$ is finite.

Definition 97 

(Profile type). A profile is multiplier-supported if

$${\tau }_{\mathcal{C}}=\mathrm{mult}.$$

A profile is saturation-supported if

$${\tau }_{\mathcal{C}}=\mathrm{sat}.$$

A profile is realization-loss type if

$${\tau }_{\mathcal{C}}=\mathrm{real}.$$

A profile is hybrid if more than one of these components is present in the same construction.

Proposition 59 

(Three basic profile types). The constructions studied in Section 4, Section 5, Section 6, Section 7, Section 8, Section 9 and Section 10 fit into the following profile types:

Construction	Proof-level data	Type
finite trace, norm, averaging	degree or group order	$\mathrm{mult}$
projectors and idempotents	denominator	$\mathrm{mult}$
correspondence images	finite cokernel, or push–pull multiplier when present	$\mathrm{sat}$ or $\mathrm{mult}$
product and invariant generation	Smith invariant factors	$\mathrm{sat}$
isogeny, Prym, Weil-type comparison	degree, norm exponent, saturation index	$\mathrm{mult}+\mathrm{sat}$
specialization and boundary	boundary, monodromy, extension data	$\mathrm{sat}+\mathrm{real}$
regulators and higher Chow groups	kernel or indecomposable quotient	$\mathrm{real}$
Bockstein	finite-coefficient source	$\mathrm{real}$

Proof. 

Finite trace, norm, and averaging are treated in Section 4; each supplies an identity of the form

$$T\circ S=Nid.$$

Projectors and idempotents are treated in Section 5; the integer N is the denominator clearing the algebraic rational correspondence. Correspondence images are saturation-supported when the proof only supplies a rationally spanning algebraic image with finite cokernel. They are multiplier-supported when the proof also supplies a push–pull identity of the form

$${\Gamma }^{\prime }\circ \Gamma =Nid$$

on the sector under consideration. Product generation and invariant generation are treated in Section 6; the relevant quotient is a finite-index lattice quotient. Isogeny, Prym, and Weil-type comparisons in Section 7 contain both multiplier data and saturation data. Specialization and boundary constructions in Section 8 contain boundary and finite-index data. Regulator and higher Chow constructions in Section 9 are controlled by kernels and indecomposable quotients of realization maps. Bockstein constructions in Section 10 are finite-coefficient realization-loss phenomena. □

Remark 82 

(Correspondence images). A correspondence image has two possible readings. If the correspondence is used only to produce a rationally spanning algebraic lattice, then the finite defect is a saturation quotient. If the correspondence is paired with a second correspondence giving a push–pull identity, then the same construction also has a multiplier component. Thus correspondence images are not assigned to a single rigid type; their profile depends on the proof-level data supplied by the construction.

12.6. The Transition Layer Between Integral and Rational Statements

Let X be smooth projective over $\mathbb{C}$. The integral Hodge conjecture in codimension p is the assertion

$$Z^{2p}(X)=0.$$

The rational Hodge conjecture in codimension p is the assertion

$$Z^{2p}{(X)}_{\mathbb{Q}}=0.$$

The implication

$$Z^{2p}(X)=0⟹Z^{2p}{(X)}_{\mathbb{Q}}=0$$

is formal. The converse is obstructed by finite integral data.

Definition 98 

(Transition defect). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector, and let

$$L\subseteq L_{\mathrm{Hdg}}$$

be a sound algebraic subgroup. The transition defect of the pair

$$(L_{\mathrm{Hdg}},L)$$

is

$$T(L_{\mathrm{Hdg}},L)=D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)=L^{\mathrm{sat}}/L.$$

Remark 83 

(Meaning of the transition defect). The group $T(L_{\mathrm{Hdg}},L)$ measures the finite part of the passage from rational algebraicity to integral algebraicity in the sector. If

$$L\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q},$$

then

$$L^{\mathrm{sat}}=L_{\mathrm{Hdg}},$$

and hence

$$T(L_{\mathrm{Hdg}},L)=L_{\mathrm{Hdg}}/L.$$

Thus after rational coverage, the remaining defect is the transition defect.

Proposition 60 

(Transition layer). Let $L_{\mathrm{Hdg}}$ be a finitely generated typed Hodge sector, and let

$$L\subseteq L_{\mathrm{Hdg}}$$

be a sound algebraic subgroup. Define:

$$\begin{array}{cc}\hfill \mathrm{Int}(L_{\mathrm{Hdg}},L)& :L=L_{\mathrm{Hdg}},\hfill \\ \hfill \mathrm{Rat}(L_{\mathrm{Hdg}},L)& :L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q},\hfill \\ \hfill \mathrm{Fin}(L_{\mathrm{Hdg}},L)& :L_{\mathrm{Hdg}}/L\mathrm{is}\mathrm{finite}.\hfill \end{array}$$

Then

$$\mathrm{Rat}(L_{\mathrm{Hdg}},L)⟺\mathrm{Fin}(L_{\mathrm{Hdg}},L).$$

Moreover,

$$\mathrm{Int}(L_{\mathrm{Hdg}},L)$$

holds if and only if

$$\mathrm{Rat}(L_{\mathrm{Hdg}},L)\mathrm{and}T(L_{\mathrm{Hdg}},L)=0$$

both hold.

Proof. 

The equivalence

$$\mathrm{Rat}(L_{\mathrm{Hdg}},L)⟺\mathrm{Fin}(L_{\mathrm{Hdg}},L)$$

is Proposition 9. Assume first that

$$L=L_{\mathrm{Hdg}}.$$

Then rational coverage holds and

$$T(L_{\mathrm{Hdg}},L)=L^{\mathrm{sat}}/L=L_{\mathrm{Hdg}}/L_{\mathrm{Hdg}}=0.$$

Conversely, assume rational coverage and

$$T(L_{\mathrm{Hdg}},L)=0.$$

By rational coverage,

$$L^{\mathrm{sat}}=L_{\mathrm{Hdg}}.$$

The equality

$$T(L_{\mathrm{Hdg}},L)=0$$

means

$$L^{\mathrm{sat}}/L=0,$$

so

$$L^{\mathrm{sat}}=L.$$

Hence

$$L=L_{\mathrm{Hdg}}.$$

 □

Corollary 20 

(Rational algebraicity with nonzero transition defect). Let $L_{\mathrm{Hdg}}$ be a finitely generated typed Hodge sector, and let

$$L\subseteq L_{\mathrm{Hdg}}$$

be a sound algebraic subgroup. Assume

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$

and

$$T(L_{\mathrm{Hdg}},L)\ne 0.$$

Then L proves rational algebraicity in the sector but does not integrally generate the sector:

$$L\ne L_{\mathrm{Hdg}}.$$

Proof. 

The first displayed equality is rational coverage. If

$$L=L_{\mathrm{Hdg}},$$

then Proposition 60 gives

$$T(L_{\mathrm{Hdg}},L)=0,$$

contrary to the hypothesis. Hence

$$L\ne L_{\mathrm{Hdg}}.$$

 □

Remark 84 

(Why the transition layer is not empty). The mechanisms studied in the preceding sections produce transition data whenever a construction proves algebraicity after multiplying by an integer, clearing a denominator, passing through a finite kernel, computing a finite index, or forgetting information under a realization map. The final rational statement records the vanishing of

$$D_{\mathrm{rat}}(L_{\mathrm{Hdg}},L).$$

The transition layer records the finite group

$$D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L).$$

12.7. Modes of Disappearance Under Rationalization

Definition 99 

(Mode of disappearance). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector, let

$$L\subseteq L_{\mathrm{Hdg}},$$

and let

$$\alpha \in L_{\mathrm{Hdg}}.$$

A mode of disappearance of the class of α in $L_{\mathrm{Hdg}}/L$ under rationalization is one of the following specified mechanisms:

Mode	Integral statement
degree annihilation	$d\alpha \in L$
denominator clearing	$N\alpha \in L$
finite-index saturation	$\alpha \in L^{\mathrm{sat}}$
isogeny or norm absorption	$e\alpha \in L$
boundary absorption	$\alpha$ differs from an algebraic lift by a boundary term
realization invisibility	$\alpha$, or its cycle-theoretic source, lies in a realization kernel
Bockstein detection	$\alpha ={\beta }_m(\theta )$

where $d,N,e,m\ge 1$ are supplied by the construction.

Remark 85 

(Distinction among modes). The modes in Definition 99 are not equivalent. Degree annihilation gives a push-pull identity. Denominator clearing comes from an algebraic rational projector or idempotent. Saturation is a lattice quotient. Isogeny and norm absorption use finite kernels or finite cokernels. Boundary absorption uses localization or specialization. Regulator invisibility is a kernel of a realization map. Bockstein detection is a finite-coefficient origin for torsion. The common feature is finite loss between the integral and rational statements.

12.8. Integral Memory of a Rational Construction

A rational Hodge-cycle theorem often has a final statement of the form

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

This equality forgets the integral data carried by the proof. The proof may nevertheless contain finite information.

Definition 100 

(Integral memory). Let

$$L\subseteq L_{\mathrm{Hdg}}$$

be a sound algebraic inclusion with rational coverage. The integral memory of the construction is

$$\mathcal{I}(L_{\mathrm{Hdg}},L)=\left(L_{\mathrm{Hdg}}/L,D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L),Ann(L_{\mathrm{Hdg}}/L)\right),$$

where

$$Ann(L_{\mathrm{Hdg}}/L)=\{N\in \mathbb{Z}\mid N·(L_{\mathrm{Hdg}}/L)=0\}.$$

If the construction supplies a norm kernel, projector denominator, isogeny degree, boundary quotient, regulator kernel, Bockstein source, or external annihilator comparison, these data are also part of the integral memory.

Proposition 61 

(Recoverable finite data). Let

$$L\subseteq L_{\mathrm{Hdg}}$$

be a sound algebraic inclusion with rational coverage. Then the construction determines or bounds the following finite data:

(i)

the finite quotient

$$L_{\mathrm{Hdg}}/L;$$

(ii)

the saturation quotient

$$D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L);$$

(iii)

the exponent

$$exp(L_{\mathrm{Hdg}}/L);$$

(iv)

any proof-level integer $N\ge 1$ satisfying

$$N{L}_{\mathrm{Hdg}}\subseteq L;$$

(v)

any specified finite kernel, cokernel, boundary group, regulator kernel, Bockstein source, or external annihilator comparison used in the proof.

Proof. 

By rational coverage and Proposition 9,

$$L_{\mathrm{Hdg}}/L$$

is finite. Since rational coverage gives

$$L^{\mathrm{sat}}=L_{\mathrm{Hdg}},$$

one has

$$D_{\mathrm{sat}}(L_{\mathrm{Hdg}},L)=L^{\mathrm{sat}}/L=L_{\mathrm{Hdg}}/L.$$

Thus the first two groups are finite and equal in this situation. The exponent of a finite abelian group is defined. If the construction gives an explicit N with

$$N{L}_{\mathrm{Hdg}}\subseteq L,$$

then N annihilates the quotient. The final class of data is part of the specified constructional proof. It is not inferred from the rational vector-space equality alone. □

Definition 101 

(Constructional annihilator). Let

$$L\subseteq L_{\mathrm{Hdg}}$$

be a sound algebraic inclusion with rational coverage. A constructional annihilator is an integer $N\ge 1$ obtained from the proof such that

$$N{L}_{\mathrm{Hdg}}\subseteq L.$$

It is distinguished from the abstract exponent

$$exp(L_{\mathrm{Hdg}}/L),$$

which exists after finiteness has been established.

Lemma 23 

(Constructional annihilator bounds the exponent). Let N be a constructional annihilator. Then

$$exp(L_{\mathrm{Hdg}}/L)\mid N.$$

Proof. 

Since

$$N{L}_{\mathrm{Hdg}}\subseteq L,$$

the quotient

$$L_{\mathrm{Hdg}}/L$$

is killed by N. The exponent of a finite abelian group is the least positive integer killing every element of the group. Therefore

$$exp(L_{\mathrm{Hdg}}/L)\mid N.$$

 □

Proposition 62 

(Integral shadow of a rational algebraicity proof). Let $L_{\mathrm{Hdg}}$ be a typed Hodge sector and let

$$L\subseteq L_{\mathrm{Hdg}}$$

be soundly algebraic. Assume rational coverage:

$$L{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

Then the rational proof has an integral shadow consisting of the finite group

$$L_{\mathrm{Hdg}}/L.$$

If the proof supplies a constructional annihilator N, then every

$$\alpha \in L_{\mathrm{Hdg}}$$

satisfies

$$\exists {\Gamma }_{\alpha }\in C{H}^p(X){cl}_{X,\mathbb{Z}}({\Gamma }_{\alpha })=N\alpha ,$$

provided the sound algebraic subgroup L is generated by such Chow classes on X.

Proof. 

The group

$$L_{\mathrm{Hdg}}/L$$

is finite by Proposition 9. If N is a constructional annihilator, then

$$N\alpha \in L$$

for every $\alpha \in L_{\mathrm{Hdg}}$. Since L is soundly algebraic and generated by Chow classes on X, there exists

$${\Gamma }_{\alpha }\in C{H}^p(X)$$

such that

$${cl}_{X,\mathbb{Z}}({\Gamma }_{\alpha })=N\alpha .$$

 □

Remark 86 

(Main synthesis). The rational equality

$$L\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}$$

says that rational algebraicity holds in the sector. The finite quotient

$$L_{\mathrm{Hdg}}/L$$

records how far the same construction is from proving integral generation. The mechanisms studied in this paper identify sources of this quotient: degrees, denominators, finite kernels, finite cokernels, boundary terms, realization kernels, finite-coefficient Bockstein sources, and comparison with external annihilators of actual defects. This quotient is the transition layer between the integral and rational statements.

13. Conclusion

The purpose of this paper has been to explain what finite integral information remains inside rational Hodge-cycle constructions. The basic distinction is the chain

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}\subseteq L_{\mathrm{Hdg}}.$$

Here $L_{\mathrm{Hdg}}$ is a typed sector of integral Hodge classes, $L_{\mathrm{alg}}^{\mathrm{act}}$ is the subgroup of classes in the sector which are actually algebraic, and $L_{\mathrm{alg}}^{\mathcal{C}}$ is the subgroup produced by a specified construction $\mathcal{C}$. Thus

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathrm{act}}$$

is the actual integral Hodge defect in the sector, while

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is the defect of the construction. The former embeds in the standard integral Hodge defect group $Z^{2p}(X)$. The latter measures what the chosen method fails to generate integrally.

This distinction is essential. A nonzero constructional defect does not automatically give a nonzero integral Hodge defect. The natural map

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})$$

has kernel

$$L_{\mathrm{alg}}^{\mathrm{act}}/L_{\mathrm{alg}}^{\mathcal{C}}.$$

This kernel consists of algebraic classes which exist in the sector but are not produced by the construction $\mathcal{C}$. Thus the constructional defect records proof-level finite loss; the actual defect records genuine failure of integral algebraicity.

13.1. What Was Proved

We proved the following. First, for every typed sector

$$L_{\mathrm{Hdg}}\subseteq H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}},$$

the actual sector defect

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/\left(L_{\mathrm{Hdg}}\cap im({cl}_{X,\mathbb{Z}})\right)$$

embeds in the standard integral Hodge defect group

$$Z^{2p}(X)={\displaystyle \frac{H^{2p}{(X,\mathbb{Z}(p))}_{\mathrm{Hdg}}}{im({cl}_{X,\mathbb{Z}})}}.$$

Thus a nonzero actual sector defect is a genuine integral Hodge obstruction.

Second, for every constructional subgroup

$$L_{\mathrm{alg}}^{\mathcal{C}}\subseteq L_{\mathrm{alg}}^{\mathrm{act}},$$

there is a short exact sequence

$$0⟶L_{\mathrm{alg}}^{\mathrm{act}}/L_{\mathrm{alg}}^{\mathcal{C}}⟶D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}})⟶0.$$

This is the formal separation between method-level loss and actual integral Hodge failure.

Third, the paper proves the finite-index principle in sector form. If

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q},$$

then

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}}$$

is finite. Therefore a rational algebraicity result in a finitely generated sector leaves a finite integral quotient. Integral generation is the stronger condition

$$L_{\mathrm{alg}}^{\mathcal{C}}=L_{\mathrm{Hdg}}.$$

Fourth, the finite quotient has a saturation interpretation. For any subgroup $L\subseteq L_{\mathrm{Hdg}}$, with saturation $L^{\mathrm{sat}}$, there is an exact sequence

$$0⟶L^{\mathrm{sat}}/L⟶L_{\mathrm{Hdg}}/L⟶L_{\mathrm{Hdg}}/L^{\mathrm{sat}}⟶0.$$

The quotient $L_{\mathrm{Hdg}}/L^{\mathrm{sat}}$ measures rational failure, and $L^{\mathrm{sat}}/L$ measures the finite integral loss left after rational coverage. Under rational coverage,

$$L^{\mathrm{sat}}=L_{\mathrm{Hdg}},$$

so the whole remaining defect is the finite saturation quotient

$$L_{\mathrm{Hdg}}/L.$$

Fifth, the paper identifies the principal sources of this finite quotient. They are:

$$\begin{array}{c}\mathrm{multiplier}\mathrm{identities}T\circ S=Nid,\\ \mathrm{finite}-\mathrm{index}\mathrm{lattice}\mathrm{inclusions},\\ \mathrm{kernels},\mathrm{cokernels},\mathrm{and}\mathrm{boundary}\mathrm{terms}\mathrm{of}\mathrm{realization}\mathrm{or}\mathrm{coefficient}\mathrm{maps}.\end{array}$$

These give the three basic profile types: multiplier-supported, saturation-supported, and realization-loss defects. The case studies show how these profiles occur in finite trace descent, norm maps, group averaging, rational algebraic projectors, correspondence images, products, isogenies, Prym and Weil-type constructions, specialization, regulators, higher Chow groups, and Bockstein finite-coefficient constructions.

The paper’s contribution is to attach to a rational Hodge-cycle construction a finite, sectorwise quotient

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

and to explain how this quotient compares with the actual integral Hodge defect.

13.2. Main Theorem Summary

The main theorem can be summarized as follows. Let $\mathcal{C}$ be a sound construction in a finitely generated typed sector $L_{\mathrm{Hdg}}$. If $\mathcal{C}$ proves rational coverage,

$$L_{\mathrm{alg}}^{\mathcal{C}}{\otimes }_{\mathbb{Z}}\mathbb{Q}=L_{\mathrm{Hdg}}{\otimes }_{\mathbb{Z}}\mathbb{Q},$$

then the constructional defect

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

is finite. Equivalently, the rational proof reduces the integral question to a finite quotient. If the proof supplies an integer $N\ge 1$ with

$$N{L}_{\mathrm{Hdg}}\subseteq L_{\mathrm{alg}}^{\mathcal{C}},$$

then this finite quotient is killed by N:

$$N·D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Finally, the construction proves integral generation of the sector precisely when this finite group vanishes:

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=0.$$

Thus a rational Hodge-cycle proof can carry more information than its final rational vector-space equality. It may carry a degree, a denominator, a finite kernel, a finite cokernel, a boundary term, a regulator kernel, or a finite-coefficient source. These data are invisible after tensoring with $\mathbb{Q}$, but they record the finite integral transition from rational algebraicity to integral generation.

13.3. Future Directions

Several directions remain.

(i)

Explicit computation of constructional defects. For a fixed rational construction, one can try to compute

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

instead of only proving that it is finite. In saturation-supported cases, this is an integral lattice computation, often reducible to Smith normal form. Natural examples include product sectors, invariant sectors, Prym sectors, and Weil-type sectors.

(ii)

Comparison with actual integral defects. The key comparison is the map

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})⟶D_{\mathrm{act}}(L_{\mathrm{Hdg}}).$$

One should determine when this map is an isomorphism, when its kernel is computable, and when its image can be compared with known obstruction groups such as unramified cohomology, decomposition-of-the-diagonal invariants, and torsion-order invariants [1,2,3,4,5].

(iii)

Sharper annihilators. In multiplier-supported arguments, the proof often gives a visible annihilator: a degree, a group order, an isogeny degree, or a projector denominator. The exact exponent

$$e_{\mathcal{C}}(L_{\mathrm{Hdg}})=exp{D}_{\mathcal{C}}(L_{\mathrm{Hdg}})$$

may be smaller. Computing this exponent refines a coarse proof-level annihilator to the exact finite quotient.

(iv)

Specialization and boundary computations. Degeneration arguments often introduce vertical classes, boundary terms, monodromy-invariant lattices, and extension data. It would be useful to compute these defects integrally in semistable or normal-crossing settings.

(v)

Regulator and higher Chow realization loss. Regulator kernels and indecomposable higher Chow quotients are not automatically ordinary integral Hodge defects. A further problem is to determine when such realization-loss phenomena map to, or control, cohomological defects in $Z^{2p}(X)$. Collino–Fakhruddin type examples suggest that higher Chow information may remain invisible to a chosen realization map [25].

(vi)

Finite-coefficient refinements. Bockstein classes record finite-coefficient origins of torsion. They provide a way to refine a defect profile by its m-primary and Bockstein components.

A rational Hodge-cycle construction should not be read only through the equality

$$L_{\mathrm{alg}}^{\mathcal{C}}\otimes \mathbb{Q}=L_{\mathrm{Hdg}}\otimes \mathbb{Q}.$$

One should also record the finite integral quotient

$$D_{\mathcal{C}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/L_{\mathrm{alg}}^{\mathcal{C}},$$

the mechanism that produces it, and its comparison with

$$D_{\mathrm{act}}(L_{\mathrm{Hdg}}).$$

This finite defect profile is the integral content of the rational proof.