Equivariant Hodge Atoms at Conifold Degenerations

1. Introduction

Hodge atoms were introduced by Katzarkov–Kontsevich–Pantev–Yu as decomposition-theoretic invariants of smooth projective varieties obtained from the spectral structure of the non-archimedean A-model F-bundle [1]. The construction combines the genus-zero quantum product, the Euler vector field, and the Hodge-theoretic symmetry carried by polarizable rational Hodge structures.

Let ${MT}_{\mathbb{Q}}$ denote the Tannakian Mumford–Tate group of the semisimple tensor category of finite direct sums of polarizable pure rational Hodge structures. The action of ${MT}_{\mathbb{Q}}$ on the Tate Hodge structure $\mathbb{Q}(-1)$ defines the Tate character $f_2:{MT}_{\mathbb{Q}}\to {\mathbf{G}}_{m,\mathbb{Q}}$, and we set $Hod:=ker\left(f_2\right)$. The pro-reductive group Hod governs the $\mathbb{Z}/2$-folded polarizable Hodge structures used in the atom construction [1,2].

For a smooth projective variety X, let $({\mathcal{H}}_X,{\nabla }_X)/B_X$ denote the maximal non-archimedean A-model F-bundle of [1]. The genus-zero Gromov–Witten theory determines the quantum product ★ [3], the F-bundle carries an Euler vector field Eu, and the residual endomorphism $\kappa :=Eu★(-)$ admits the spectral decomposition from which the local Hodge atoms are defined. If a finite group G acts algebraically on X, Cavenaghi–Katzarkov–Kontsevich refine this construction over the $G\times Hod$-fixed locus and retain the induced finite-group representation data of the spectral sectors [4].

The present paper concerns the degeneration side of this picture. Let $\pi :\mathcal{X}\to \Delta$ be a projective one-parameter degeneration of complex threefolds over a sufficiently small analytic disk $\Delta \subset \mathbb{C}$ centered at 0. Set $X_t:={\pi }^{-1}\left(t\right)$ and $X_0:={\pi }^{-1}\left(0\right)$, and assume that $X_t$ is smooth for $t\ne 0$, while $\Sigma :=Sing\left(X_0\right)$ is a finite set of ordinary double points. Write $r:=|\Sigma |$. A threefold ordinary double point is locally analytically equivalent to $z_1^2+z_2^2+z_3^2+z_4^2=0$, and its Milnor fiber has the homotopy type of $S^3$. Its middle vanishing cohomology is therefore one-dimensional [5,6].

The degeneration package used below is built from nearby and vanishing cycles. Let $\mathcal{F}:={\mathbb{Q}}_{\mathcal{X}}\left[3\right]$. With the nearby- and vanishing-cycle normalization fixed in [7,8], the variation morphism determines the corrected perverse object

$$P:=Cone\left({var}_{\mathcal{F}}\right)[-1].$$

The adjective “corrected” refers to replacing the nearby-cycle object alone by this variation-cone extension: the resulting object retains both the intersection-complex sector and the point-supported vanishing contribution.

The corrected perverse object admits a mixed-Hodge-module lift $P^H\in MHM\left(X_0\right)$, with $rat\left(P^H\right)\simeq P$, and, in the finite ordinary-double-point setting, fits into the exact sequence

$$0⟶I{C}_{X_0}^H⟶P^H⟶\underset{p\in \Sigma }{⨁}{i}_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1)⟶0$$

[7,8,9]. Here $I{C}_{X_0}^H$ is the intersection-complex Hodge module and $i_p:\left\{p\right\}\hookrightarrow X_0$ is the closed inclusion. We write $V_p^H:=i_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1)$ and $V_{\mathrm{van}}^H:={⨁}_{p\in \Sigma }{V}_p^H$. Thus

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0.$$

The first term is the rigid sector, while the quotient is the formal node-supported flexible sector. The word “formal” refers to the local direct-sum quotient; it does not assert that the distinguished global extension splits nodewise. Global relations among the nodes may constrain the corresponding gluing data [10].

Suppose now that a finite group G acts algebraically and fiberwise on $\mathcal{X}$, so that $\pi \left(gx\right)=\pi \left(x\right)$ for $g\in G$ and $x\in \mathcal{X}$. Then G preserves $X_0$ and acts on the finite node set $\Sigma$. For $p\in \Sigma$, define $G_p:={Stab}_G\left(p\right)$. Functoriality of vanishing cycles gives an action of $G_p$ on the one-dimensional rational local vanishing space. We denote the resulting rational $G_p$-representation by $V_p$ [11,12].

The problem addressed in this paper is then precise. One must first show that the corrected rigid–flexible carrier itself admits compatible G-linearizations. Once this is done, the symmetry carried by the formal flexible quotient can be reconstructed orbit by orbit from the local pairs $(G_p,V_p)$. The central representation formula is

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p,$$

where $V_{\mathrm{van}}$ is the rational G-representation underlying $V_{\mathrm{van}}^H$. Thus the ordinary node count is refined through

$$\mathrm{node}\mathrm{count}⟶\mathrm{node}G-\mathrm{set}⟶\begin{array}{c}\mathrm{stabilizer}-\mathrm{decorated}\\ \mathrm{vanishing}\mathrm{representation}\end{array}⟶\begin{array}{c}\mathrm{irreducible}\mathrm{symmetry}\\ \mathrm{multiplicities}\end{array}.$$

The integer $r=|\Sigma |$ is recovered as $r={dim}_{\mathbb{Q}}{V}_{\mathrm{van}}$, but it no longer exhausts the flexible-sector information.

A central geometric application shows that the stabilizer decoration is not merely formal. Section 7 constructs an explicit projective $S_4$-equivariant smoothing of a four-nodal cubic threefold of the $S_4$-symmetric type studied in [13]. Its four nodes form a single orbit $S_4/S_3$, but the $S_3$-stabilizer acts on the local vanishing line by the sign character: $V_p\simeq {sgn}_{S_3}$. Consequently,

$$V_{\mathrm{van}}\simeq {Ind}_{S_3}^{S_4}{sgn}_{S_3}\simeq {sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right),$$

and $V_{\mathrm{van}}^{S_4}=0$. This differs from the untwisted permutation representation $\mathbb{Q}[S_4/S_3]\simeq \mathbf{1}\oplus V_{\mathrm{std}}$. Hence even the G-set of nodes does not determine the equivariant vanishing representation: the local stabilizer action is genuine geometric information.

The same four-node $S_4$-geometry appears in the equivariant birational setting of Cavenaghi–Katzarkov–Kontsevich [4]. The explicit one-parameter smoothing developed here therefore provides a concrete degeneration-side realization of a configuration already relevant to smooth equivariant Hodge-atom theory.

The paper does not construct a specialization functor from smooth equivariant Hodge atoms to the limiting profile. Rather, it constructs and computes a finite piece of the degeneration-side equivariant data that such a specialization theory must reproduce. Throughout the remainder of the paper, we use the notation and conventions fixed in Section 1.6.

1.1. Hodge Atoms, Symmetry, and Degeneration

The present construction lies at the interface between smooth equivariant Hodge atoms and the Hodge-theoretic geometry of degeneration. For a smooth projective G-variety X, the smooth equivariant atom theory uses the fixed locus

$$B_X^{G\times Hod}:=\left\{b\in B_X|(g,h)·b=b\mathrm{for}\mathrm{every}(g,h)\in G\times Hod\right\}.$$

Restricting the spectral decomposition of $\kappa =Eu★(-)$ to this locus produces the smooth G-equivariant Hodge atoms [4]. These spectral pieces retain both Hodge-theoretic and finite-group representation data.

An orbit–stabilizer mechanism also occurs in the equivariant singularity theory motivating atoms. Let $L\le G$ and let $(\mathcal{U},f)$ be an L-equivariant isolated singularity germ. The induced G-equivariant germ is formed from $G{\times }_L\mathcal{U}:=(G\times \mathcal{U})/L$, where $\ell ·(g,x):=(g{\ell }^{-1},\ell x)$, and the induced function satisfies ${ind}_L^G\left(f\right)\left([g,x]\right)=f\left(x\right)$. Conversely, when the critical points of a G-equivariant isolated singularity form one G-orbit, the germ is controlled by the stabilizer of one point together with its local equivariant singularity [1].

The present paper realizes the corresponding mechanism for vanishing sectors of a projective degeneration. The intrinsic local datum attached to a node is $(G_p,V_p)$, not merely the orbit $G/G_p$. Transport through the orbit produces ${Ind}_{G_p}^G{V}_p$. The projective $S_4$-cubic of Section 7.3 demonstrates geometrically why the second entry of this pair cannot be discarded: the node orbit is $S_4/S_3$, while the local module is ${sgn}_{S_3}$, so the resulting representation is a twisted induced module rather than the permutation representation.

The mixed-Hodge-module carrier has a different categorical origin from the smooth spectral construction. The objects $I{C}_{X_0}^H$, $P^H$, and $V_{\mathrm{van}}^H$ arise from nearby cycles, vanishing cycles, the variation morphism, and mixed-Hodge-module gluing [7,9,11,12]. The equivariant problem is therefore not solved merely by placing a G-representation on cohomology. The carrier and its structure maps must themselves be lifted to the equivariant mixed-Hodge-module category.

A fiberwise action preserves every fiber, hence preserves $\Sigma =Sing\left(X_0\right)$. For $p\in \Sigma$, the stabilizer $G_p$ preserves the local degeneration germ and therefore acts functorially on $V_p$. Since ${dim}_{\mathbb{Q}}{V}_p=1$, this action has finite image in ${\mathbb{Q}}^{\times }$, necessarily contained in $\{\pm 1\}$. Thus even in the ordinary-double-point case a node may carry a nontrivial local stabilizer character.

The finite collection ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$, taken up to conjugacy under changes of orbit representatives, is the local symmetry datum from which the formal equivariant flexible representation is reconstructed. The distinguished global extension remains a separate piece of data and may still be constrained by global relations [10].

1.2. Geometric Setup

We impose the following geometric hypotheses throughout the paper.

Definition 1 

(Equivariant finite-node conifold degeneration). An equivariant finite-node conifold degeneration consists of a projective flat morphism $\pi :\mathcal{X}\to \Delta$ together with an algebraic fiberwise action of a finite group G on $\mathcal{X}$, subject to the following conditions:

(H1)

for every $t\in {\Delta }^{\times }:=\Delta \setminus \left\{0\right\}$, the fiber $X_t:={\pi }^{-1}\left(t\right)$ is a smooth projective complex threefold;

(H2)

the central fiber $X_0:={\pi }^{-1}\left(0\right)$ is projective and every point of $Sing\left(X_0\right)$ is an ordinary double point;

(H3)

the singular locus $\Sigma :=Sing\left(X_0\right)=\{p_1,\dots ,p_r\}$ is finite, where $r:=|\Sigma |$;

(H4)

the action $G\times \mathcal{X}\to \mathcal{X}$ is algebraic;

(H5)

the action is fiberwise, equivalently $\pi \circ g=\pi$ for every $g\in G$.

No additional hypothesis asserting the existence of $P^H$ is imposed. For degenerations satisfying the preceding finite-node ordinary-double-point hypotheses, the corrected perverse and mixed-Hodge-module constructions recalled above provide P, $P^H$, their realization compatibility, and the rigid–flexible exact sequence [7,8,9]. Section 3 recalls precisely the properties of this construction used in the present paper and then proves their equivariant compatibility.

The local topology and vanishing cohomology of ordinary double points are classical [5,6]. Global smoothing questions for nodal threefolds are treated in [14,15], while nearby and vanishing cycles and their mixed-Hodge-module realizations belong to Saito’s theory [11,12,16].

Because G preserves $\Sigma$, choose representatives $p_1,\dots ,p_s$ of the G-orbits, where $s:=|\Sigma /G|$. Then

$$\Sigma ={\displaystyle \coprod _{a=1}^s}G·p_a,G·p_a\simeq G/G_{p_a},$$

and $r={\sum }_{a=1}^s[G:G_{p_a}]$ by the orbit–stabilizer theorem.

For each representative $p_a$, the local vanishing line $V_{p_a}$ is a one-dimensional rational $G_{p_a}$-module. The standard dimension formula for induction gives

$${dim}_{\mathbb{Q}}{Ind}_{G_{p_a}}^G{V}_{p_a}=[G:G_{p_a}]$$

[17]. Hence

$$r={\displaystyle \sum _{a=1}^s}{dim}_{\mathbb{Q}}{Ind}_{G_{p_a}}^G{V}_{p_a}.$$

This identity is only the dimension shadow of the representation theorem proved later.

For $g\in G$, let $a_g:X_0\to X_0$, $x\mapsto gx$, and use the pullback convention $g^{*}:={\left(a_{g^{-1}}\right)}^{*}$. Section 3 proves that the corrected carrier and its rigid and flexible terms acquire compatible canonical G-linearizations with this convention.

1.3. Statement of the Main Results

The first main result is categorical. It shows that the finite-group symmetry acts on the corrected degeneration carrier itself, rather than only on a cohomological realization.

Theorem 1 proves that $I{C}_{X_0}^H$, $P^H$, and $V_{\mathrm{van}}^H$ carry canonical compatible G-linearizations and that

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0$$

is a short exact sequence in ${MHM}_G\left(X_0\right)$. Forgetting the G-linearizations recovers the corrected non-equivariant rigid–flexible sequence.

The second main result determines the symmetry carried by the formal flexible quotient. For every orbit representative p, Theorem 2 gives

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p.$$

The induced module depends on the full pair $(G_p,V_p)$, not merely on the orbit $G/G_p$. If $V_p$ is trivial, then ${Ind}_{G_p}^G{V}_p\simeq \mathbb{Q}[G/G_p]$, but nontrivial stabilizer characters produce twisted orbitwise representations.

The third main result resolves this representation into irreducible symmetry types. Fix a characteristic-zero splitting field $\mathbf{K}\supset \mathbb{Q}$ for G, and let ${\widehat{G}}_{\mathbf{K}}$ denote the irreducible $\mathbf{K}\left[G\right]$-modules. Put $V_{\mathrm{van},\mathbf{K}}:=V_{\mathrm{van}}{\otimes }_{\mathbb{Q}}\mathbf{K}$. For $\rho \in {\widehat{G}}_{\mathbf{K}}$, define

$$m_{\rho }:={dim}_{\mathbf{K}}{Hom}_{\mathbf{K}\left[G\right]}\left(\rho ,V_{\mathrm{van},\mathbf{K}}\right).$$

Theorem 3, by Frobenius reciprocity, gives

$$m_{\rho }=\sum _{\left[p\right]\in \Sigma /G}{dim}_{\mathbf{K}}{Hom}_{\mathbf{K}\left[G_p\right]}\left({Res}_{G_p}^G\rho ,V_p{\otimes }_{\mathbb{Q}}\mathbf{K}\right).$$

Thus every irreducible multiplicity is determined by the finite local collection ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$.

These data define the equivariant flexible profile ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$, while the canonical G-linearization on $I{C}_{X_0}^H$ defines the equivariant rigid profile ${\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right)$. Their ordered pair is

$${\mathcal{P}}_G^{\lim }\left(\pi \right):=\left({\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right),{\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right).$$

Theorem 5 proves that this limiting rigid–flexible profile is canonically determined by the equivariant degeneration.

The forgetful theorem then shows that this is a refinement of the ordinary conifold profile rather than a replacement for it. If

$${For}_G:{MHM}_G\left(X_0\right)⟶MHM\left(X_0\right)$$

denotes the forgetful functor, Theorem 6 gives

$${For}_G\left({\mathcal{P}}_G^{\lim }\left(\pi \right)\right)={\mathcal{P}}^{\lim }\left(\pi \right)$$

and

$$|\Sigma |={dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=\sum _{\rho \in {\widehat{G}}_{\mathbf{K}}}{m}_{\rho }{dim}_{\mathbf{K}}\rho .$$

The paper also contains a complete projective geometric application. Section 7 constructs an explicit $S_4$-equivariant one-parameter smoothing of a four-nodal cubic threefold. The central fiber belongs to the $S_4$-symmetric four-node family described in [13], while the nearby fibers are smooth projective cubic threefolds with generically free $S_4$-action.

For a node p, the stabilizer is $G_p\simeq S_3$, but the local vanishing line is not the trivial $S_3$-module. Proposition 20 proves $V_p\simeq {sgn}_{S_3}$. Consequently, Theorem 8 gives

$$\begin{array}{cc}\hfill V_{\mathrm{van}}& \simeq {Ind}_{S_3}^{S_4}{sgn}_{S_3}\hfill \\ & \simeq {sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right).\hfill \end{array}$$

In particular, Corollary 10 gives $V_{\mathrm{van}}^{S_4}=0$. The same node $S_4$-set $S_4/S_3$ with trivial stabilizer action would instead give $\mathbb{Q}[S_4/S_3]\simeq \mathbf{1}\oplus V_{\mathrm{std}}$. Thus the projective cubic realizes geometrically a strict distinction between the node G-set and the stabilizer-decorated vanishing representation.

Remark 1 

(Information beyond the node count). The integer $r=|\Sigma |$ determines only the dimension of the formal vanishing representation. The G-set Σ contains additional orbit information, but even this does not determine $V_{\mathrm{van}}$: the local stabilizer representations $V_p$ are essential. The projective $S_4$-cubic of Section 7.3 realizes this distinction with the same four-point orbit $S_4/S_3$ but a nontrivial local sign character.

Remark 2 

(Local carrier versus global gluing). The decomposition

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p$$

describes the formal node-supported quotient. It does not imply an orbitwise or isotypic splitting of

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0.$$

The global extension can be constrained by relations among the nodes [10]. The formal representation and the global gluing class therefore constitute distinct layers of the degeneration data.

1.4. Relation to Equivariant Birational Atom Theory

For a smooth projective G-variety X, the equivariant Hodge-atom theory of Cavenaghi–Katzarkov–Kontsevich is defined from the $G\times Hod$-equivariant spectral construction and is organized by equivariant birational geometry [4]. Its birational relations use equivariant blow-ups in smooth invariant centers and projectivizations of equivariant vector bundles. The corresponding F-bundle decompositions are based on the work of Iritani and Iritani–Koto [18,19], while equivariant weak factorization supplies the birational mechanism [20,21].

The present construction concerns a different operation. It fixes a G-equivariant family $\pi :\mathcal{X}\to \Delta$ and passes from the smooth fibers to a singular central fiber. Nearby cycles, vanishing cycles, monodromy, and the mixed-Hodge-module extension are intrinsic to this specialization problem. Equivariant birational transport and specialization are therefore logically distinct operations.

The four-nodal cubic provides a direct geometric interface between these two settings. Cavenaghi–Katzarkov–Kontsevich consider a singular cubic threefold with four ordinary double points exchanged transitively by $S_4$ in their equivariant birational non-linearizability analysis [4]. The projective family constructed in Theorem 7 realizes a four-nodal $S_4$-cubic of this type as the central fiber of an explicit fiberwise $S_4$-equivariant one-parameter smoothing.

The degeneration-side computation then adds information not contained in the four-point $S_4$-action alone: $(G_p,V_p)=(S_3,{sgn}_{S_3})$, and hence

$$V_{\mathrm{van}}\simeq {Ind}_{S_3}^{S_4}{sgn}_{S_3}.$$

Thus the same geometric class that appears naturally in smooth equivariant birational atom theory also carries a concrete symmetry-resolved limiting vanishing sector on the degeneration side.

Cavenaghi–Katzarkov–Kontsevich also anticipate a parallel theory of specialization for atoms [4], Appendix A. The present paper does not construct such a functor. What it now provides is a nontrivial projective test case together with explicit degeneration-side data that any such specialization theory must reproduce.

For the prospective comparison, let ${\pi }_X:\mathcal{X}\to \Delta$ and ${\pi }_Y:\mathcal{Y}\to \Delta$ be two G-equivariant degenerations. Let ${\mathcal{A}}_G\left(X_t\right)$ and ${\mathcal{A}}_G\left(Y_t\right)$ denote their smooth G-equivariant Hodge-atom data in the sense of [4]. We use ${\mathcal{A}}_G^{\lim }(X_0,{\pi }_X)$ and ${\mathcal{A}}_G^{\lim }(Y_0,{\pi }_Y)$ only as notation for prospective limiting equivariant atomic objects, and write ${Bir}_G$, ${Sp}_G$, and ${Bir}_G^{\lim }$ schematically for smooth equivariant birational transport, prospective equivariant specialization, and prospective limiting birational transport.

Remark 3 

(Future specialization–birational square). With this schematic notation, the eventual compatibility problem has the form

$$\begin{array}{ccc}{\mathcal{A}}_G\left(X_t\right)& \stackrel{{Bir}_G}{\to }& {\mathcal{A}}_G\left(Y_t\right)\\ \downarrow {Sp}_G& & \downarrow {Sp}_G\\ {\mathcal{A}}_G^{\lim }(X_0,{\pi }_X)& \stackrel{{Bir}_G^{\lim }}{\to }& {\mathcal{A}}_G^{\lim }(Y_0,{\pi }_Y).\end{array}$$

None of the three displayed operations is constructed in the present paper, and no commutativity statement is asserted.

The contribution of the present paper is therefore limited but concrete. It constructs the G-linearized degeneration carrier, computes its formal flexible representation, resolves that representation into irreducible symmetry types, and exhibits a projective $S_4$-equivariant degeneration on which the resulting refinement can be calculated completely.

1.5. Scope and Nonclaims

Remark 4 

(Scope). The results of the paper are subject to the following restrictions.

1. 

The acting group G is finite and acts algebraically and fiberwise on $\pi :\mathcal{X}\to \Delta$; in particular, the action on the base is trivial.

2. 

The singularities of $X_0$ are ordinary double points. No theorem is asserted for arbitrary isolated hypersurface singularities or positive-dimensional singular loci.

3. 

The paper constructs a finite-group linearized rigid–flexible carrier in ${MHM}_G\left(X_0\right)$. It does not construct a universal category of degeneration atoms.

4. 

No specialization functor ${Sp}_{\mathrm{Atom}}^G$ from smooth equivariant Hodge atoms to limiting equivariant atomic data is constructed.

5. 

No equivariant birational invariance theorem is proved for ${\mathcal{P}}_G^{\lim }\left(\pi \right)$.

6. 

No Chen–Ruan, inertia-stack, or twisted-sector degeneration atom is constructed. The conditional orbifold extension discussed in [4] is not used.

7. 

The local collection ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$ determines the formal equivariant flexible representation, but not the global extension class represented by $P^H$ [10].

8. 

The present paper establishes the equivariant corrected carrier, its symmetry-resolved formal flexible sector, the G-equivariant limiting mixed Hodge structure and its rational isotypic decomposition, and the explicit projective $S_4$-cubic application. It does not decompose the equivariant extension class into isotypic extension classes nor construct isotypic Picard–Lefschetz interaction matrices or Stokes blocks.

Remark 5 

(Nature of the refinement). The underlying non-equivariant formal flexible object remains

$$V_{\mathrm{van}}^H=\underset{p\in \Sigma }{⨁}{i}_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1).$$

The new datum is its canonical organization as the rational G-representation

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p.$$

Its dimension recovers $|\Sigma |$, while its isomorphism class retains both orbit information and the stabilizer actions on the local vanishing lines. The projective $S_4$-cubic shows that this second layer is geometrically nontrivial. Global extension data, specialization, and birational transport remain separate structures.

1.6. Notation and Conventions

The notation in this subsection is fixed for the remainder of the paper. Unless explicitly stated otherwise, all subsequent sections use these conventions without redefinition.

1.6.1. The Degeneration and Its Singular Fiber

Let $\Delta \subset \mathbb{C}$ be a sufficiently small analytic disk centered at 0, and put ${\Delta }^{\times }:=\Delta \setminus \left\{0\right\}$. We write $\pi :\mathcal{X}\to \Delta$ for an equivariant finite-node conifold degeneration in the sense of Definition 1, with fibers $X_t:={\pi }^{-1}\left(t\right)$ and $X_0:={\pi }^{-1}\left(0\right)$. The singular locus of the central fiber is $\Sigma :=Sing\left(X_0\right)=\{p_1,\dots ,p_r\}$, with $r:=|\Sigma |$, and every $p\in \Sigma$ is an ordinary double point. Unless another dimension is explicitly stated, all degeneration statements concern complex threefolds.

1.6.2. Finite-Group Actions, Node Orbits, and Stabilizers

The symbol G denotes a finite group acting algebraically and fiberwise on $\mathcal{X}$, so that $\pi \circ g=\pi$ for $g\in G$. The action preserves $X_t$, $X_0$, and $\Sigma$.

For $p\in \Sigma$, define $G_p:={Stab}_G\left(p\right)=\{g\in G\mid gp=p\}$. The orbit is $G·p$, while $\left[p\right]$ denotes its class in $\Sigma /G$. Set $s:=|\Sigma /G|$. For orbit representatives $p_1,\dots ,p_s$,

$$\Sigma ={\displaystyle \coprod _{a=1}^s}G·p_a,G·p_a\simeq G/G_{p_a},$$

and $r={\sum }_{a=1}^s[G:G_{p_a}]$.

1.6.3. Mixed Hodge Modules and Equivariant Linearizations

For a complex algebraic variety Z, $MHM\left(Z\right)$ denotes Saito’s abelian category of mixed Hodge modules [12,16].

If G acts algebraically on Z, let $a_g:Z\to Z$ be the automorphism associated with $g\in G$, and adopt the convention $g^{*}:={\left(a_{g^{-1}}\right)}^{*}$. The category ${MHM}_G\left(Z\right)$ consists of objects $M\in MHM\left(Z\right)$ equipped with isomorphisms ${\lambda }_g:g^{*}M\stackrel{\sim }{\to }M$ satisfying

$${\lambda }_1={id}_M,{\lambda }_{gh}={\lambda }_g\circ g^{*}\left({\lambda }_h\right).$$

The Hodge-theoretic formalism is due to Saito [11,12,16]; the analogous finite-group equivariant sheaf formalism is standard [22].

1.6.4. Nearby Cycles, Vanishing Cycles, and Monodromy

We write ${\psi }_{\pi }^H$ and ${\varphi }_{\pi }^H$ for the mixed-Hodge-module nearby- and vanishing-cycle functors with the normalization used in the corrected finite-node construction. Their functoriality is part of Saito’s theory [11,12].

Geometric monodromy on the smooth-fiber cohomology local system over ${\Delta }^{\times }$ is denoted by T. Its unipotent part is $T_u$, and $N:=log{T}_u$ denotes its nilpotent logarithm [23,24].

1.6.5. The Rigid–Flexible Mixed Hodge Objects

Set $\mathcal{F}:={\mathbb{Q}}_{\mathcal{X}}\left[3\right]$. The corrected perverse object is

$$P:=Cone\left({var}_{\mathcal{F}}\right)[-1],$$

with the nearby–vanishing normalization of [7,8].

Its mixed-Hodge-module lift is $P^H\in MHM\left(X_0\right)$, with $rat\left(P^H\right)\simeq P$. Let $I{C}_{X_0}^H$ denote the intersection-complex Hodge module. For $p\in \Sigma$, let $i_p:\left\{p\right\}\hookrightarrow X_0$ be the inclusion and define $V_p^H:=i_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1)$. The total formal vanishing object is $V_{\mathrm{van}}^H:={⨁}_{p\in \Sigma }{V}_p^H$. The corrected mixed-Hodge-module sequence is

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0$$

[7,8,9].

The terms are called the rigid sector, corrected total degeneration carrier, and formal flexible sector, respectively. The adjective “corrected” refers to the variation-cone construction above; the word “formal” indicates that the quotient displays the local node sectors without asserting a splitting of the global extension [10].

1.6.6. Local Vanishing Representations

For $p\in \Sigma$, the stabilizer $G_p$ acts on the local middle vanishing cohomology by functoriality [11,12]. We denote the underlying rational $G_p$-representation by $V_p$. For an ordinary double point, ${dim}_{\mathbb{Q}}{V}_p=1$ [5,6]. Hence the stabilizer action has finite image in ${\mathbb{Q}}^{\times }$, necessarily contained in $\{\pm 1\}$.

After extension to a characteristic-zero splitting field $\mathbf{K}$, the corresponding one-dimensional character may be denoted by ${\chi }_p^{\mathrm{van}}$. The rational G-representation underlying the complete formal vanishing sector is denoted by $V_{\mathrm{van}}$.

1.6.7. Induction, Restriction, and Coefficient Fields

Let $L\le G$ be a subgroup and let W be a finite-dimensional rational L-representation. Define ${Ind}_L^{G}W:=\mathbb{Q}\left[G\right]{\otimes }_{\mathbb{Q}\left[L\right]}W$. For a rational G-representation U, ${Res}_L^{G}U$ denotes its restriction to L. We use the standard conventions for finite-group induction, restriction, semisimplicity, and Frobenius reciprocity [17].

For $p\in \Sigma$, the orbitwise module is ${Ind}_{G_p}^G{V}_p$. If $V_p$ is trivial, then ${Ind}_{G_p}^G{V}_p\simeq \mathbb{Q}[G/G_p]$.

Let ${\widehat{G}}_{\mathbb{Q}}$ denote the set of isomorphism classes of irreducible rational G-representations. Fix a characteristic-zero splitting field $\mathbf{K}\supset \mathbb{Q}$ for G, and let ${\widehat{G}}_{\mathbf{K}}$ denote the set of isomorphism classes of irreducible $\mathbf{K}\left[G\right]$-modules.

Set $V_{\mathrm{van},\mathbf{K}}:=V_{\mathrm{van}}{\otimes }_{\mathbb{Q}}\mathbf{K}$. For $\rho \in {\widehat{G}}_{\mathbf{K}}$, define

$$m_{\rho }:={dim}_{\mathbf{K}}{Hom}_{\mathbf{K}\left[G\right]}\left(\rho ,V_{\mathrm{van},\mathbf{K}}\right).$$

Then

$$V_{\mathrm{van},\mathbf{K}}\simeq \underset{\rho \in {\widehat{G}}_{\mathbf{K}}}{⨁}{\rho }^{\oplus m_{\rho }}$$

by semisimplicity [17].

1.6.8. Equivariant Rigid and Flexible Profiles

The equivariant flexible profile is denoted by ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$. It records the Hodge type of the local vanishing sector together with the nonzero isotypic multiplicities.

The equivariant rigid profile is denoted by ${\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right)$. The limiting equivariant rigid–flexible profile is

$${\mathcal{P}}_G^{\lim }\left(\pi \right):=\left({\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right),{\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right),$$

and its non-equivariant counterpart is denoted by ${\mathcal{P}}^{\lim }\left(\pi \right)$. The full equivariant extension class, the full limiting mixed Hodge structure, and Stokes interaction data are not included in ${\mathcal{P}}_G^{\lim }\left(\pi \right)$.

1.6.9. The Forgetful Operation

The forgetful functor is

$${For}_G:{MHM}_G\left(X_0\right)⟶MHM\left(X_0\right).$$

It removes the G-linearization and retains the underlying mixed Hodge module. The same notation is used for the induced operation on profiles.

1.6.10. Smooth Hodge Atoms and Prospective Limiting Notation

For a smooth projective G-variety X, let ${\mathcal{A}}_G\left(X\right)$ denote its smooth G-equivariant Hodge-atom data in the sense of [4].

The symbol ${MT}_{\mathbb{Q}}$ denotes the Tannakian Mumford–Tate group of polarizable pure rational Hodge structures, while

$$Hod=ker\left(f_2:{MT}_{\mathbb{Q}}\to {\mathbf{G}}_{m,\mathbb{Q}}\right)$$

denotes the pro-reductive symmetry group relevant to the $\mathbb{Z}/2$-folded Hodge structures [1,2,4].

The quantum product is denoted by ★, the Euler vector field by Eu, and $\kappa :=Eu★(-)$. The smooth equivariant atom theory uses the combined $G\times Hod$-action.

The expression ${\mathcal{A}}_G^{\lim }(X_0,\pi )$ denotes only a prospective limiting equivariant atomic object. Likewise, ${Sp}_G$, ${Bir}_G$, and ${Bir}_G^{\lim }$ are schematic notation for prospective specialization, smooth equivariant birational transport, and limiting equivariant birational transport. None is a constructed functor in the present paper.

Remark 6 

(Local representation data and global gluing). The decomposition $V_{\mathrm{van}}^H={⨁}_{p\in \Sigma }{V}_p^H$ records the formal node-supported quotient. Its orbitwise representation theory does not imply an orbitwise or isotypic splitting of

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0.$$

Global cycle relations and extension data remain separate [10].

Remark 7 

(Use of these conventions). Unless explicitly stated otherwise, every subsequent section uses the notation and conventions of Section 1.6. In particular, the symbols π, $\mathcal{X}$, Δ, ${\Delta }^{\times }$, $X_t$, $X_0$, Σ, r, s, G, $G_p$, P, $P^H$, $V_p^H$, $V_p$, $V_{\mathrm{van}}^H$, $V_{\mathrm{van}}$, $\mathbf{K}$, ${\widehat{G}}_{\mathbb{Q}}$, ${\widehat{G}}_{\mathbf{K}}$, $m_{\rho }$, ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$, ${\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right)$, ${\mathcal{P}}_G^{\lim }\left(\pi \right)$, and ${For}_G$ are not redefined locally.

2. Equivariant Finite-Node Conifold Degenerations

We use the notation and conventions of Section 1.6. In particular, $\pi :\mathcal{X}\to \Delta$ is an equivariant finite-node conifold degeneration, G acts algebraically and fiberwise on $\mathcal{X}$, and $\Sigma =Sing\left(X_0\right)$ is the finite G-stable set of ordinary double points of the central fiber.

2.1. Finite Group Actions on One-Parameter Degenerations

Let $a:G\times \mathcal{X}\to \mathcal{X}$ denote the action, and write $a_g\left(x\right):=a(g,x)$. The fiberwise condition is $\pi \circ a_g=\pi$ for $g\in G$.

Proposition 1 

(Fiberwise stability). For every $g\in G$ and $t\in \Delta$, the automorphism $a_g$ restricts to an algebraic automorphism $a_{g,t}:X_t\stackrel{\sim }{\to }{X}_t$. In particular, $X_0$, its smooth locus $X_0^{\mathrm{sm}}$, and its singular locus Σ are G-stable.

Proof. 

If $x\in X_t$, then $\pi \left(a_g\left(x\right)\right)=\pi \left(x\right)=t$, so $a_g\left(x\right)\in X_t$. The inverse of the restricted map is $a_{g^{-1},t}$, hence $a_{g,t}$ is an automorphism.

For $t=0$, the automorphism $a_{g,0}$ preserves $X_0$ and induces isomorphisms of local rings

$${\mathcal{O}}_{X_0,gp}\stackrel{\sim }{\to }{\mathcal{O}}_{X_0,p}.$$

Regularity is invariant under local-ring isomorphism. Thus p is smooth if and only if $gp$ is smooth, so both $X_0^{\mathrm{sm}}$ and its complement $\Sigma$ are G-stable. □

The fiberwise action therefore transports not only the nodes but also their local degeneration data.

Proposition 2 

(Transport of degeneration germs). Let $p\in \Sigma$ and $g\in G$. The automorphism $a_g$ induces an isomorphism of analytic germs over $(\Delta ,0)$,

$$(\mathcal{X},p)\stackrel{\sim }{\to }(\mathcal{X},gp),$$

and hence an isomorphism $(X_0,p)\stackrel{\sim }{\to }(X_0,gp)$. In particular, the ordinary-double-point analytic type is preserved along each G-orbit in Σ.

Proof. 

Analytification of the algebraic automorphism $a_g$ gives an isomorphism of germs

$$(\mathcal{X},p)\stackrel{\sim }{\to }(\mathcal{X},gp).$$

The identity $\pi \circ a_g=\pi$ makes this an isomorphism over $(\Delta ,0)$, and restriction to the central fiber gives the second isomorphism. Since the ordinary-double-point condition is invariant under analytic isomorphism, the final assertion follows. □

Remark 8 

(No freeness hypothesis). No freeness or generic-freeness hypothesis on the G-action is required for the preceding statements. Nontrivial stabilizers of nodes are part of the local structure studied below. Generic freeness enters only in the comparison with the equivariant birational applications of smooth G-equivariant Hodge atoms [4].

2.2. Node Orbits and Stabilizers

The orbit and stabilizer notation is fixed in Section 1.6.

Proposition 3 

(Orbit–stabilizer decomposition of the node set). Choose representatives $p_1,\dots ,p_s$ of the G-orbits in Σ. Then $\Sigma /G=\{\left[p_1\right],\dots ,\left[p_s\right]\}$, and

$$\Sigma ={\displaystyle \coprod _{a=1}^s}G·p_a,G·p_a\simeq G/G_{p_a}$$

as G-sets. Consequently,

$$|\Sigma |={\displaystyle \sum _{a=1}^s}[G:G_{p_a}]=\sum _{\left[p\right]\in \Sigma /G}[G:G_p].$$

Proof. 

For $p\in \Sigma$, the orbit map $G\to G·p$, $g\mapsto gp$, has fibers equal to the left cosets of $G_p$, and therefore induces a G-equivariant bijection $G/G_p\stackrel{\sim }{\to }G·p$. The orbit partition of $\Sigma$ gives the stated disjoint union, and taking cardinalities yields the final identities. □

Remark 9 

(Why stabilizers are part of the local datum). The G-set $G/G_p$ records the permutation of the nodes in an orbit, but not the action of $G_p$ on the local vanishing line. The intrinsic local equivariant datum is therefore the conjugacy class of the pair $(G_p,V_p)$, rather than the orbit $G/G_p$ alone. Conjugacy compatibility is proved in Lemma 1.

Remark 10 

(Dimension shadow). Since every ordinary double point has ${dim}_{\mathbf{Q}}{V}_p=1$, the standard induction formula gives

$${dim}_{\mathbf{Q}}{Ind}_{G_p}^G{V}_p=[G:G_p]$$

[17]. Thus $|\Sigma |={\sum }_{\left[p\right]\in \Sigma /G}[G:G_p]$ is the dimension shadow of the orbitwise representation formula proved in Theorem 2.

2.3. The Local Ordinary Double Point and Its Vanishing Line

Let $p\in \Sigma$. The central-fiber germ $(X_0,p)$ is analytically an ordinary double point, $z_1^2+z_2^2+z_3^2+z_4^2=0$. Its local vanishing cohomology is modeled by the standard smoothing $z_1^2+z_2^2+z_3^2+z_4^2=t$. The Milnor fiber has the homotopy type of $S^3$; hence its reduced rational cohomology is concentrated in degree 3 and is one-dimensional [5,6].

Definition 2 

(Local vanishing Hodge object). For $p\in \Sigma$, let ${\mathsf{V}}_p^H$ denote the rank-one local vanishing Hodge object of the degeneration germ at p. Its point-supported realization is the object $V_p^H=i_{p*}{\mathbf{Q}}_{\left\{p\right\}}^H(-1)$ fixed in Section 1.6, with the threefold normalization of [8]. We write $V_p$ for the underlying one-dimensional rational vanishing representation.

Thus ${\mathsf{V}}_p^H$ is the local vanishing Hodge object, $V_p^H$ its point-supported mixed-Hodge-module realization on $X_0$, and $V_p$ its underlying rational vanishing line.

For $h\in G_p$, Proposition 2 gives an automorphism of the degeneration germ at p. Functoriality of vanishing cycles therefore induces an automorphism of ${\mathsf{V}}_p^H$ and of $V_p$ [11,12].

Proposition 4 

(Stabilizer representation on the vanishing line). For every $p\in \Sigma$, the stabilizer $G_p$ acts naturally on the one-dimensional rational vector space $V_p$. Hence $V_p$ is a one-dimensional $\mathbf{Q}\left[G_p\right]$-module, equivalently a rational character

$${\chi }_p^{\mathrm{van}}:G_p⟶{\mathbf{Q}}^{\times }$$

whose image is contained in $\{\pm 1\}$.

Proof. 

Functoriality of vanishing cycles under automorphisms of the local degeneration germ gives a homomorphism

$$G_p⟶{GL}_{\mathbf{Q}}\left(V_p\right)$$

[11,12]. Since the local ordinary-double-point vanishing space is one-dimensional [5,6], ${GL}_{\mathbf{Q}}\left(V_p\right)\simeq {\mathbf{Q}}^{\times }$. The image of the finite group $G_p$ is a finite subgroup of ${\mathbf{Q}}^{\times }$, and the only roots of unity in $\mathbf{Q}$ are 1 and $-1$. Hence the image lies in $\{\pm 1\}$. □

Remark 11 

(Coefficient convention). The primary local datum is the rational $G_p$-module $V_p$. The notation ${\chi }_p^{\mathrm{van}}$ merely records the corresponding one-dimensional character. Scalar extension to the fixed splitting field $\mathbf{K}$ is used only when the later irreducible decomposition requires it.

The local representations are compatible with transport along node orbits.

Lemma 1 

(Conjugacy compatibility of local vanishing sectors). Let $p\in \Sigma$ and $g\in G$. Transport by g induces an isomorphism

$${\Phi }_g:{\mathsf{V}}_p^H\stackrel{\sim }{\to }{\mathsf{V}}_{gp}^H.$$

Conjugation induces an isomorphism

$$c_g:G_p\stackrel{\sim }{\to }{G}_{gp},c_g\left(h\right)=gh{g}^{-1},$$

and ${\Phi }_g$ intertwines the corresponding stabilizer actions. Equivalently,

$${\Phi }_g(h·v)=\left(gh{g}^{-1}\right)·{\Phi }_g\left(v\right)$$

for $h\in G_p$ and $v\in V_p$.

Proof. 

By Proposition 2, $a_g$ identifies the degeneration germ at p with that at $gp$. Functoriality of the Milnor fiber and vanishing cycles therefore gives ${\Phi }_g:V_p\stackrel{\sim }{\to }{V}_{gp}$, and likewise an isomorphism of the corresponding Hodge objects [5,11,12].

If $h\in G_p$, then $gh{g}^{-1}\in G_{gp}$, so conjugation identifies $G_p$ with $G_{gp}$. The identity $a_g\circ a_h=a_{gh{g}^{-1}}\circ a_g$ and functoriality of vanishing cycles give

$${\Phi }_g\circ h_{*}={\left(gh{g}^{-1}\right)}_{*}\circ {\Phi }_g,$$

which is the required intertwining relation. □

Corollary 1 

(Independence of orbit representative). If $p,q\in \Sigma$ lie in the same G-orbit, then $(G_p,V_p)$ and $(G_q,V_q)$ are conjugate. Consequently,

$${Ind}_{G_p}^G{V}_p\simeq {Ind}_{G_q}^G{V}_q$$

as rational G-representations. Hence the induced representation associated with an orbit $\left[p\right]\in \Sigma /G$ is independent of the chosen representative up to its natural isomorphism.

Proof. 

Choose $g\in G$ with $q=gp$. By Lemma 1, conjugation by g identifies the stabilizer representations. Induction from conjugate subgroups with the transported module gives isomorphic G-representations [17]. □

Remark 12 

(Local versus global information). The local orbit data determine the conjugacy classes ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$, but not the global extension of $I{C}_{X_0}^H$ by $V_{\mathrm{van}}^H$. Global relations among the nodes may constrain that gluing [10]. The assembly of the local data into the formal equivariant flexible carrier is carried out in Section 4.

3. Equivariant Mixed Hodge Modules and Nearby Cycles

We use the notation and conventions of Section 1.6. In particular, $\pi :\mathcal{X}\to \Delta$ is an equivariant finite-node conifold degeneration, G acts algebraically and fiberwise on $\mathcal{X}$, and $X_0={\pi }^{-1}\left(0\right)$.

The purpose of this section is twofold. We first recall explicitly the corrected finite-node degeneration object that is being equivariantized, including the variation-cone construction and its mixed-Hodge-module realization. We then prove that the finite-group action lifts canonically to that entire package. In this form, the equivariant argument uses only the stated functorial properties of nearby cycles, vanishing cycles, variation, intermediate extension, and the corrected rigid–flexible exact sequence.

3.1. Finite-Group Linearizations

Let Z be a complex algebraic variety equipped with an algebraic action of the finite group G. For $g\in G$, let $a_g:Z\to Z$ denote the corresponding automorphism. We use the pullback convention fixed in Section 1.6, $g^{*}:={\left(a_{g^{-1}}\right)}^{*}$, so that, under the canonical coherence identifications, ${\left(gh\right)}^{*}=g^{*}{h}^{*}$.

The category $MHM\left(Z\right)$ is Saito’s abelian category of mixed Hodge modules [12,16]. Pullback by an automorphism is an exact autoequivalence. The finite-group linearization formalism below is the mixed-Hodge-module analogue of the standard equivariant sheaf-theoretic construction [22].

Definition 3 

(G-linearized mixed Hodge module). A G-linearized mixed Hodge module on Z is a pair $\left(M,{\left\{{\lambda }_g\right\}}_{g\in G}\right)$, with $M\in MHM\left(Z\right)$, where ${\lambda }_g:g^{*}M\stackrel{\sim }{\to }M$ is an isomorphism satisfying

$${\lambda }_1={id}_M,{\lambda }_{gh}={\lambda }_g\circ g^{*}\left({\lambda }_h\right)(g,h\in G).$$

A morphism $f:(M,{\lambda }^M)\to (N,{\lambda }^N)$ is a morphism $f:M\to N$ in $MHM\left(Z\right)$ satisfying

$$f\circ {\lambda }_g^M={\lambda }_g^N\circ g^{*}f(g\in G).$$

The resulting category is denoted by ${MHM}_G\left(Z\right)$.

Lemma 2 

(Equivariant kernels and cokernels). Let $f:(M,{\lambda }^M)\to (N,{\lambda }^N)$ be a morphism in ${MHM}_G\left(Z\right)$. Then $ker\left(f\right)$, $im\left(f\right)$, $coim\left(f\right)$, and $coker\left(f\right)$, computed in $MHM\left(Z\right)$, carry canonical G-linearizations for which the canonical maps are G-equivariant.

Proof. 

Let $K:=ker\left(f\right)$, with inclusion $\iota :K\hookrightarrow M$. Since $g^{*}$ is exact, $g^{*}K\simeq ker\left(g^{*}f\right)$. From $f\circ {\lambda }_g^M={\lambda }_g^N\circ g^{*}f$ one obtains $f\circ {\lambda }_g^M\circ g^{*}\iota =0$. Hence the universal property of the kernel gives a unique isomorphism ${\lambda }_g^K:g^{*}K\stackrel{\sim }{\to }K$ such that

$$\iota \circ {\lambda }_g^K={\lambda }_g^M\circ g^{*}\iota .$$

For $g,h\in G$, both ${\lambda }_{gh}^K$ and ${\lambda }_g^K\circ g^{*}\left({\lambda }_h^K\right)$ become ${\lambda }_{gh}^M\circ {\left(gh\right)}^{*}\iota$ after composition with the monomorphism $\iota$. Thus they agree, so K is G-linearized.

For the cokernel, let $C:=coker\left(f\right)$ and $q:N\twoheadrightarrow C$. Exactness of $g^{*}$ gives $g^{*}C\simeq coker\left(g^{*}f\right)$. Since $q\circ {\lambda }_g^N\circ g^{*}f=0$, the universal property of the cokernel gives a unique isomorphism ${\lambda }_g^C:g^{*}C\stackrel{\sim }{\to }C$ satisfying

$${\lambda }_g^C\circ g^{*}q=q\circ {\lambda }_g^N.$$

The cocycle identity follows after composition with the epimorphism $g^{*}q$.

Images and coimages inherit their linearizations by expressing them, respectively, as a kernel of a cokernel and a cokernel of a kernel. □

Proposition 5 

(The equivariant category and the forgetful functor). The category ${MHM}_G\left(Z\right)$ is abelian. The forgetful functor

$${For}_G:{MHM}_G\left(Z\right)⟶MHM\left(Z\right)$$

is faithful and exact. A sequence in ${MHM}_G\left(Z\right)$ is exact if and only if its image under ${For}_G$ is exact in $MHM\left(Z\right)$.

Proof. 

Lemma 2 shows that kernels and cokernels in the underlying category inherit canonical G-linearizations and satisfy the same universal properties in ${MHM}_G\left(Z\right)$.

Since $MHM\left(Z\right)$ is abelian, the canonical map $coim\left(f\right)\to im\left(f\right)$ is an isomorphism on the underlying mixed Hodge modules. It is G-equivariant by construction, and its inverse is therefore G-equivariant as well. Thus ${MHM}_G\left(Z\right)$ is abelian.

The forgetful functor is faithful by definition and preserves kernels and cokernels, hence is exact. Conversely, because kernels and images in the equivariant category are the underlying kernels and images equipped with their induced linearizations, exactness can be checked after applying ${For}_G$. □

3.2. Equivariance of Nearby and Vanishing Cycles

Because the action is fiberwise, every $a_{g^{-1}}:\mathcal{X}\to \mathcal{X}$ is an automorphism over $\Delta$. Functoriality of Saito’s nearby- and vanishing-cycle functors therefore gives, for $M\in MHM\left(\mathcal{X}\right)$, canonical comparison isomorphisms

$$\begin{array}{cc}\hfill {\theta }_{g,M}^{\psi }:g^{*}{\psi }_{\pi }^H\left(M\right)& \stackrel{\sim }{\to }{\psi }_{\pi }^H\left(g^{*}M\right),\hfill \\ \hfill {\theta }_{g,M}^{\varphi }:g^{*}{\varphi }_{\pi }^H\left(M\right)& \stackrel{\sim }{\to }{\varphi }_{\pi }^H\left(g^{*}M\right).\hfill \end{array}$$

These morphisms are natural in M, compatible with monodromy and with the canonical and variation morphisms between nearby and vanishing cycles, and coherent under composition [11,12]. In particular,

$${\theta }_{gh,M}^{\psi }={\theta }_{g,h^{*}M}^{\psi }\circ g^{*}\left({\theta }_{h,M}^{\psi }\right),$$

and similarly for ${\theta }^{\varphi }$.

Before introducing the equivariant structure, we recall precisely the non-equivariant finite-node object to which it will be applied.

Let $\mathcal{F}:={\mathbb{Q}}_{\mathcal{X}}\left[3\right]$ with the normalization used in the corrected conifold construction, and let

$${var}_{\mathcal{F}}:{\varphi }_{\pi }\left(\mathcal{F}\right)⟶{\psi }_{\pi }\left(\mathcal{F}\right)$$

denote the variation morphism. The corrected perverse object is

$$P:=Cone\left({var}_{\mathcal{F}}\right)[-1].$$

The adjective “corrected” refers specifically to this replacement of the nearby-cycle object alone by the variation-cone object, which retains both the invariant intersection-complex contribution and the node-supported vanishing contribution in one extension.

Let

$$rat:MHM\left(X_0\right)⟶Perv(X_0;\mathbb{Q})$$

denote Saito’s exact faithful realization functor.

Proposition 6 

(Recalled corrected finite-node package). Under the standing finite-node ordinary-double-point hypotheses, the corrected degeneration construction has the following properties.

1. 

The shifted cone

$$P=Cone\left({var}_{\mathcal{F}}\right)[-1]$$

is a perverse sheaf on $X_0$.

2. 

There is a canonical mixed-Hodge-module input ${\mathcal{M}}^H\in MHM\left(\mathcal{X}\right)$ with rational realization identified with the input $\mathcal{F}$ in the normalization of the corrected construction. The nearby- and vanishing-cycle variation morphism lifts to

$${var}_{{\mathcal{M}}^H}^H:{\varphi }_{\pi }^H\left({\mathcal{M}}^H\right)⟶{\psi }_{\pi }^H\left({\mathcal{M}}^H\right).$$

Its shifted cone is concentrated in the mixed-Hodge-module heart and defines an object $P^H\in MHM\left(X_0\right)$ satisfying $rat\left(P^H\right)\simeq P$.

3. 

There is a canonical short exact sequence

$$0⟶I{C}_{X_0}^H\stackrel{\iota }{\to }{P}^H\stackrel{q}{\to }{V}_{\mathrm{van}}^H⟶0,$$

where

$$V_{\mathrm{van}}^H=\underset{p\in \Sigma }{⨁}{i}_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1).$$

4. 

The construction is natural with respect to isomorphisms of the degeneration over Δ: an automorphism $a:\mathcal{X}\stackrel{\sim }{\to }\mathcal{X}$ with $\pi \circ a=\pi$ induces compatible isomorphisms on the nearby-cycle object, the vanishing-cycle object, the variation morphism, the corrected carrier $P^H$, the intersection-complex term, and the node-supported quotient.

Proof. 

Items (1)–(3) are the corrected finite-node perverse and mixed-Hodge-module construction [7,8,9]. In particular, the ordinary-double-point vanishing sector is rank one at each node, and the threefold mixed-Hodge-module normalization gives the quotient

$$\underset{p\in \Sigma }{⨁}{i}_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1).$$

For item (4), let a be an automorphism of the degeneration over $\Delta$. Functoriality of nearby and vanishing cycles gives canonical comparison isomorphisms

$$a^{*}{\psi }_{\pi }^H\left({\mathcal{M}}^H\right)\stackrel{\sim }{\to }{\psi }_{\pi }^H\left(a^{*}{\mathcal{M}}^H\right),a^{*}{\varphi }_{\pi }^H\left({\mathcal{M}}^H\right)\stackrel{\sim }{\to }{\varphi }_{\pi }^H\left(a^{*}{\mathcal{M}}^H\right).$$

The variation morphism is a natural transformation of these functors, so the corresponding square commutes. Consequently pullback transports the shifted variation cone to the shifted variation cone of the pulled-back input.

The canonical input ${\mathcal{M}}^H$ is itself transported to the corresponding canonical input under a. Hence the pulled-back corrected carrier is canonically identified with $P^H$. Intermediate extension is functorial under automorphisms of the pair $(X_0,X_0^{\mathrm{sm}})$, while the local ordinary-double-point vanishing objects are transported from p to $a\left(p\right)$. Naturality of the corrected exact sequence therefore identifies its pullback with the same exact sequence. This proves the stated functoriality. □

The proposition isolates all non-equivariant input required below. In particular, no additional hypothesis concerning the existence of $P^H$ is needed beyond the standing finite-node ordinary-double-point assumptions.

Lemma 3 

(Induced linearization on nearby cycles). Let $(M,{\left\{{\lambda }_g\right\}}_{g\in G})\in {MHM}_G\left(\mathcal{X}\right)$. For $g\in G$, define ${\lambda }_g^{\psi }:={\psi }_{\pi }^H\left({\lambda }_g\right)\circ {\theta }_{g,M}^{\psi }$. Then

$${\lambda }_g^{\psi }:g^{*}{\psi }_{\pi }^H\left(M\right)\stackrel{\sim }{\to }{\psi }_{\pi }^H\left(M\right)$$

defines a G-linearization of ${\psi }_{\pi }^H\left(M\right)$.

Proof. 

The maps ${\lambda }_g^{\psi }$ are isomorphisms. For $g,h\in G$, the cocycle identity ${\lambda }_{gh}={\lambda }_g\circ g^{*}\left({\lambda }_h\right)$ and coherence of ${\theta }^{\psi }$ give

$$\begin{array}{cc}\hfill {\lambda }_{gh}^{\psi }& ={\psi }_{\pi }^H\left({\lambda }_g\right)\circ {\psi }_{\pi }^H\left(g^{*}{\lambda }_h\right)\circ {\theta }_{g,h^{*}M}^{\psi }\circ g^{*}\left({\theta }_{h,M}^{\psi }\right).\hfill \end{array}$$

Naturality of ${\theta }^{\psi }$ with respect to ${\lambda }_h:h^{*}M\to M$ gives

$${\psi }_{\pi }^H\left(g^{*}{\lambda }_h\right)\circ {\theta }_{g,h^{*}M}^{\psi }={\theta }_{g,M}^{\psi }\circ g^{*}\left({\psi }_{\pi }^H\left({\lambda }_h\right)\right).$$

Therefore

$${\lambda }_{gh}^{\psi }={\lambda }_g^{\psi }\circ g^{*}\left({\lambda }_h^{\psi }\right).$$

The identity element acts trivially, so the stated maps form a G-linearization. □

Lemma 4 

(Induced linearization on vanishing cycles). Under the hypotheses of Lemma 3, define ${\lambda }_g^{\varphi }:={\varphi }_{\pi }^H\left({\lambda }_g\right)\circ {\theta }_{g,M}^{\varphi }$. Then

$${\lambda }_g^{\varphi }:g^{*}{\varphi }_{\pi }^H\left(M\right)\stackrel{\sim }{\to }{\varphi }_{\pi }^H\left(M\right)$$

defines a G-linearization of ${\varphi }_{\pi }^H\left(M\right)$.

Proof. 

The same argument as in Lemma 3, using the coherence and naturality of ${\theta }^{\varphi }$, gives ${\lambda }_{gh}^{\varphi }={\lambda }_g^{\varphi }\circ g^{*}\left({\lambda }_h^{\varphi }\right)$. Hence ${\varphi }_{\pi }^H\left(M\right)$ is G-linearized. □

Apply the preceding lemmas to the canonical input ${\mathcal{M}}^H$ of Proposition 6. Since the fiberwise automorphisms $a_g$ preserve the degeneration and its canonical input, functoriality supplies canonical isomorphisms ${\lambda }_g^{\mathcal{M}}:g^{*}{\mathcal{M}}^H\stackrel{\sim }{\to }{\mathcal{M}}^H$. Their coherence under composition is inherited from functoriality of pullback, so ${\mathcal{M}}^H$ is canonically G-linearized.

Hence ${\psi }_{\pi }^H\left({\mathcal{M}}^H\right)$ and ${\varphi }_{\pi }^H\left({\mathcal{M}}^H\right)$ carry the linearizations constructed above. Naturality of the variation morphism gives

$${var}_{{\mathcal{M}}^H}^H\circ {\lambda }_g^{\varphi }={\lambda }_g^{\psi }\circ g^{*}\left({var}_{{\mathcal{M}}^H}^H\right).$$

Thus the variation morphism itself is equivariant.

Proposition 7 

(Equivariant nearby- and vanishing-cycle carriers). Under the standing hypotheses, the nearby-cycle and vanishing-cycle objects associated with the corrected finite-node construction inherit canonical G-linearizations. The variation morphism is G-equivariant, and the corrected mixed-Hodge-module carrier defines an object $P^H\in {MHM}_G\left(X_0\right)$.

Proof. 

Lemmas 3 and 4 give canonical G-linearizations on ${\psi }_{\pi }^H\left({\mathcal{M}}^H\right)$ and ${\varphi }_{\pi }^H\left({\mathcal{M}}^H\right)$. By naturality of the variation transformation,

$$\begin{array}{ccc}{g}^{*}{\varphi }_{\pi }^H\left({\mathcal{M}}^H\right)& \stackrel{g^{*}\left({var}_{{\mathcal{M}}^H}^H\right)}{\to }& g^{*}{\psi }_{\pi }^H\left({\mathcal{M}}^H\right)\\ {\lambda }_g^{\varphi }\downarrow \phantom{{\lambda }_g^{\varphi }}& & \phantom{{\lambda }_g^{\psi }}\downarrow {\lambda }_g^{\psi }& & \\ {\varphi }_{\pi }^H\left({\mathcal{M}}^H\right)& \stackrel{{var}_{{\mathcal{M}}^H}^H}{\to }& {\psi }_{\pi }^H\left({\mathcal{M}}^H\right)\end{array}$$

commutes for every $g\in G$.

The corrected carrier $P^H$ is the mixed-Hodge-module object represented by the shifted cone of this variation morphism, as recalled in Proposition 6. The preceding commutative square induces a canonical isomorphism ${\lambda }_g^P:g^{*}{P}^H\stackrel{\sim }{\to }{P}^H$. For $g,h\in G$, the two induced morphisms ${\lambda }_{gh}^P$ and ${\lambda }_g^P\circ g^{*}\left({\lambda }_h^P\right)$ are obtained from the corresponding cocycle identities on the nearby- and vanishing-cycle terms. Functoriality of the cone construction therefore gives

$${\lambda }_{gh}^P={\lambda }_g^P\circ g^{*}\left({\lambda }_h^P\right).$$

Hence $P^H$ is canonically G-linearized. □

Remark 13 

(Canonicity). The G-linearization of Proposition 7 uses only the given fiberwise action, the naturality of nearby and vanishing cycles and of the variation morphism, and the canonical corrected variation-cone construction. No eigenspace decomposition, projector, choice of node generators, or splitting of the rigid–flexible extension is used.

3.3. Equivariance of the Intersection-Complex Sector

Let $U:=X_0^{\mathrm{sm}}$ and $j:U\hookrightarrow X_0$. By Proposition 1, U is G-stable. With the threefold normalization, $I{C}_{X_0}^H=j_{!*}{\mathbf{Q}}_U^H\left[3\right]$. Intermediate extension is functorial under automorphisms of the pair $(X_0,U)$ [12,16].

Proposition 8 

(Canonical linearization of the rigid sector). The intersection-complex Hodge module $I{C}_{X_0}^H$ carries a canonical G-linearization. Thus $I{C}_{X_0}^H\in {MHM}_G\left(X_0\right)$.

Proof. 

The constant Hodge module ${\mathbf{Q}}_U^H\left[3\right]$ carries its canonical G-linearization. For $g\in G$, functoriality of intermediate extension gives

$$g^{*}{j}_{!*}{\mathbf{Q}}_U^H\left[3\right]\simeq j_{!*}{g}^{*}{\mathbf{Q}}_U^H\left[3\right].$$

Composing with the linearization of ${\mathbf{Q}}_U^H\left[3\right]$ gives ${\lambda }_g^{IC}:g^{*}I{C}_{X_0}^H\stackrel{\sim }{\to }I{C}_{X_0}^H$. Functoriality of $j_{!*}$ transfers the cocycle identity from the constant Hodge module to $I{C}_{X_0}^H$. Hence ${\left\{{\lambda }_g^{IC}\right\}}_{g\in G}$ is a G-linearization. □

The formal vanishing quotient is likewise G-stable. Transport by $g\in G$ identifies the local degeneration germ at p with that at $gp$, and hence transports the corresponding point-supported vanishing object.

Proposition 9 

(Canonical linearization of the formal vanishing sector). The mixed Hodge module

$$V_{\mathrm{van}}^H=\underset{p\in \Sigma }{⨁}{V}_p^H$$

carries a canonical G-linearization. Under this linearization, an element $g\in G$ transports the summand supported at p to the summand supported at $gp$.

Proof. 

Lemma 1 gives canonical transport isomorphisms $g^{*}{V}_p^H\stackrel{\sim }{\to }{V}_{gp}^H$ compatible with composition in G. Summing over $p\in \Sigma$ gives

$$g^{*}{V}_{\mathrm{van}}^H\simeq \underset{p\in \Sigma }{⨁}{V}_{gp}^H=V_{\mathrm{van}}^H.$$

The same compatibility gives the cocycle identity of Definition 3. □

3.4. The Equivariant Rigid–Flexible Exact Sequence

By Proposition 6, the corrected finite-node construction gives the canonical short exact sequence

$$0⟶I{C}_{X_0}^H\stackrel{\iota }{\to }{P}^H\stackrel{q}{\to }{V}_{\mathrm{van}}^H⟶0$$

in $MHM\left(X_0\right)$. The three terms now carry the canonical G-linearizations constructed above. It remains to verify that the structure maps are equivariant.

Lemma 5 

(Equivariance of the rigid inclusion). The morphism $\iota :I{C}_{X_0}^H\to P^H$ is a morphism in ${MHM}_G\left(X_0\right)$.

Proof. 

The inclusion $\iota$ is a canonical morphism of the corrected variation-cone construction recalled in Proposition 6. Pullback by $g\in G$ transports the entire corrected construction to itself because g is an automorphism of the degeneration over $\Delta$. Naturality therefore gives

$$\iota \circ {\lambda }_g^{IC}={\lambda }_g^P\circ g^{*}\iota (g\in G).$$

Hence $\iota$ is G-equivariant. □

Lemma 6 

(Equivariance of the flexible quotient). The quotient morphism $q:P^H\to V_{\mathrm{van}}^H$ is a morphism in ${MHM}_G\left(X_0\right)$.

Proof. 

The morphism q is the canonical quotient map in the corrected finite-node exact sequence. Its local component at p is the canonical map to the point-supported vanishing sector $V_p^H$. Pullback by $g\in G$ transports this local component to the corresponding map at $gp$.

By functoriality of nearby and vanishing cycles and Lemma 1, the local quotient maps commute with this transport. Summing over all nodes gives

$$q\circ {\lambda }_g^P={\lambda }_g^{\mathrm{van}}\circ g^{*}q.$$

Thus q is G-equivariant. □

Theorem 1 

(Equivariant rigid–flexible sequence). Let $\pi :\mathcal{X}\to \Delta$ be an equivariant finite-node conifold degeneration satisfying the standing hypotheses. Then the corrected rigid–flexible sequence admits a canonical lift to ${MHM}_G\left(X_0\right)$:

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0.$$

After applying the forgetful functor ${For}_G$, one recovers the corrected non-equivariant finite-node sequence.

Proof. 

Propositions 8, 7, and 9 supply the canonical G-linearizations on the three terms. Lemmas 5 and 6 show that the two structure maps are G-equivariant. Hence the displayed sequence is a sequence in ${MHM}_G\left(X_0\right)$.

After applying ${For}_G$, the sequence is precisely the short exact sequence of Proposition 6. Exactness in ${MHM}_G\left(X_0\right)$ therefore follows from Proposition 5. □

Corollary 2 

(Equivariant extension class). The equivariant rigid–flexible sequence determines an extension class

$$e_{\pi }^G\in {Ext}_{{MHM}_G\left(X_0\right)}^1\left(V_{\mathrm{van}}^H,I{C}_{X_0}^H\right).$$

Its image under the forgetful map on extension groups is the non-equivariant corrected rigid–flexible extension class associated with π.

Proof. 

Theorem 1 is a short exact sequence in the abelian category ${MHM}_G\left(X_0\right)$, and therefore determines the displayed Yoneda extension class. Applying ${For}_G$ gives the corrected non-equivariant short exact sequence of Proposition 6, so the corresponding Yoneda class maps to the non-equivariant corrected extension class. □

Remark 14 

(No isotypic splitting of the extension is asserted). Theorem 1 establishes the equivariant corrected carrier and Corollary 2 records its extension class. The representation-theoretic decomposition of the formal quotient developed later does not, by itself, give a canonical decomposition of $e_{\pi }^G$ or a splitting of the corrected sequence on isotypic sectors. Those questions concern the interaction of the equivariant extension group with the global gluing data and are logically separate from the construction proved here.

4. Orbitwise Structure of the Flexible Sector

We use the notation and conventions of Section 1.6. In particular, $V_{\mathrm{van}}^H$ denotes the node-supported quotient in the equivariant rigid–flexible sequence of Theorem 1, and $V_p^H$ denotes the rank-one point-supported vanishing Hodge object at $p\in \Sigma$.

The purpose of this section is to pass from the canonical G-linearization of the formal flexible quotient to its orbitwise description. The local datum attached to a node is the stabilizer module $(G_p,V_p)$; the main result identifies the contribution of the orbit $G·p$ with induction of this local datum from $G_p$ to G.

4.1. The Node-Supported Quotient

The underlying non-equivariant flexible object is

$$V_{\mathrm{van}}^H\simeq \underset{p\in \Sigma }{⨁}{V}_p^H\simeq \underset{p\in \Sigma }{⨁}{i}_{p*}{\mathsf{V}}_p^H,$$

where ${\mathsf{V}}_p^H$ is the local vanishing Hodge object of Definition 2. The first expression incorporates the point-supported realization into the notation $V_p^H$.

The nodewise decomposition is generally not G-stable term by term: Proposition 9 gives $g^{*}{V}_p^H\simeq V_{gp}^H$. Thus the natural G-stable summands are indexed by the G-orbits in $\Sigma$.

Definition 4 

(Orbitwise vanishing object). Let $\mathcal{O}\subset \Sigma$ be a G-orbit. Define

$$V_{\mathcal{O}}^H:=\underset{q\in \mathcal{O}}{⨁}{V}_q^H.$$

Its underlying rational vanishing representation is denoted by $V_{\mathcal{O}}$.

Since $g\mathcal{O}=\mathcal{O}$ for every $g\in G$, the canonical G-linearization of $V_{\mathrm{van}}^H$ preserves $V_{\mathcal{O}}^H$.

Lemma 7 

(Orbitwise decomposition). There is a canonical decomposition in ${MHM}_G\left(X_0\right)$,

$$V_{\mathrm{van}}^H\simeq \underset{\mathcal{O}\in \Sigma /G}{⨁}{V}_{\mathcal{O}}^H,$$

where $V_{\mathcal{O}}^H$ is supported on the orbit $\mathcal{O}\subset \Sigma$.

Proof. 

The orbit partition $\Sigma ={\coprod }_{\mathcal{O}\in \Sigma /G}\mathcal{O}$ regroups the nodewise decomposition as

$$V_{\mathrm{van}}^H\simeq \underset{\mathcal{O}\in \Sigma /G}{⨁}\left(\underset{q\in \mathcal{O}}{⨁}{V}_q^H\right)=\underset{\mathcal{O}\in \Sigma /G}{⨁}{V}_{\mathcal{O}}^H.$$

For $g\in G$, Proposition 9 gives

$$g^{*}{V}_{\mathcal{O}}^H\simeq \underset{q\in \mathcal{O}}{⨁}{V}_{gq}^H=V_{\mathcal{O}}^H,$$

because $g\mathcal{O}=\mathcal{O}$. These isomorphisms are restrictions of the canonical G-linearization of $V_{\mathrm{van}}^H$, and therefore satisfy the cocycle condition. The decomposition is canonical because the orbit partition is canonical. □

Remark 15 

(Individual nodes and orbitwise sectors). The decomposition $V_{\mathrm{van}}^H={⨁}_{p\in \Sigma }{V}_p^H$ records the support of the underlying non-equivariant object, whereas Lemma 7 records its decomposition into G-stable support sectors. The two decompositions coincide only when every node is fixed by G.

4.2. Induction from the Stabilizer

Fix an orbit $\mathcal{O}=G·p\subset \Sigma$ with stabilizer $G_p$. By Lemma 1, the local vanishing objects along $\mathcal{O}$ are transported from $V_p^H$ through the action of G. The resulting orbitwise object is therefore the mixed-Hodge-module analogue of induction from $G_p$ to G.

For $g\in G$, let $a_g:X_0\to X_0$ be the induced automorphism. Since $a_g$ is an isomorphism, ${\left(a_g\right)}_{*}$ and ${\left(a_{g^{-1}}\right)}^{*}=g^{*}$ are canonically quasi-inverse exact equivalences. We use this identification for point-supported transport below.

Definition 5 

(Induced orbitwise mixed Hodge module). Let $p\in \Sigma$, and choose a set $R\subset G$ of representatives for the left cosets $G/G_p$. Define

$${\mathcal{I}}_p^H\left(R\right):=\underset{r\in R}{⨁}{\left(a_r\right)}_{*}{V}_p^H.$$

The summand ${\left(a_r\right)}_{*}{V}_p^H$ is supported at $rp$.

For $k\in G$ and $r\in R$, write uniquely $kr=r^{\prime }h$, with $r^{\prime }\in R$ and $h\in G_p$. The $G_p$-linearization of $V_p^H$, together with the canonical identification

$$k^{*}{\left(a_r\right)}_{*}{V}_p^H\simeq {\left(a_{kr}\right)}_{*}{V}_p^H={\left(a_{r^{\prime }}\right)}_{*}{\left(a_h\right)}_{*}{V}_p^H,$$

identifies the pulled-back r-summand with the $r^{\prime }$-summand. The resulting G-linearized object is denoted by ${Ind}_{G_p}^G{V}_p^H$.

Lemma 8 

(Well-definedness of the induced orbitwise object). The maps in Definition 5 satisfy the cocycle condition and therefore define an object of ${MHM}_G\left(X_0\right)$. If R and $R^{\prime }$ are two sets of representatives for $G/G_p$, the corresponding G-linearized mixed Hodge modules ${\mathcal{I}}_p^H\left(R\right)$ and ${\mathcal{I}}_p^H\left(R^{\prime }\right)$ are naturally isomorphic.

Proof. 

For $k\in G$ and $r\in R$, write $kr=r^{\prime }h$, with $r^{\prime }\in R$ and $h\in G_p$. Since $a_{kr}=a_{r^{\prime }}\circ a_h$, the $G_p$-linearization of $V_p^H$ gives

$$k^{*}{\left(a_r\right)}_{*}{V}_p^H\stackrel{\sim }{\to }{\left(a_{r^{\prime }}\right)}_{*}{V}_p^H.$$

Summing over $r\in R$ defines

$${\lambda }_k^{\mathrm{ind}}:k^{*}{\mathcal{I}}_p^H\left(R\right)\stackrel{\sim }{\to }{\mathcal{I}}_p^H\left(R\right).$$

For $k_1,k_2\in G$, write $k_{2}r=r_2{h}_2$ and $k_1{r}_2=r_1{h}_1$. Then $k_1{k}_{2}r=r_1{h}_1{h}_2$. The cocycle identity for the $G_p$-linearization of $V_p^H$ therefore gives

$${\lambda }_{k_1{k}_2}^{\mathrm{ind}}={\lambda }_{k_1}^{\mathrm{ind}}\circ k_1^{*}\left({\lambda }_{k_2}^{\mathrm{ind}}\right).$$

If $R^{\prime }$ is another set of representatives, write $r=r^{\prime }{h}_r$, with $r^{\prime }\in R^{\prime }$ and $h_r\in G_p$. The local $G_p$-linearization gives

$${\left(a_r\right)}_{*}{V}_p^H\stackrel{\sim }{\to }{\left(a_{r^{\prime }}\right)}_{*}{V}_p^H.$$

Taking direct sums produces a G-equivariant isomorphism

$${\mathcal{I}}_p^H\left(R\right)\stackrel{\sim }{\to }{\mathcal{I}}_p^H\left(R^{\prime }\right).$$

Thus the induced object is well defined up to natural isomorphism. □

Proposition 10 

(Identification of one orbit with an induced object). Let $\mathcal{O}=G·p$. There is an isomorphism in ${MHM}_G\left(X_0\right)$,

$$V_{\mathcal{O}}^H\simeq {Ind}_{G_p}^G{V}_p^H.$$

Proof. 

Choose representatives $R\subset G$ for $G/G_p$. The map $R\to \mathcal{O}$, $r\mapsto rp$, is a bijection, and Lemma 1 gives ${\left(a_r\right)}_{*}{V}_p^H\stackrel{\sim }{\to }{V}_{rp}^H$. Hence

$${\mathcal{I}}_p^H\left(R\right)=\underset{r\in R}{⨁}{\left(a_r\right)}_{*}{V}_p^H\stackrel{\sim }{\to }\underset{r\in R}{⨁}{V}_{rp}^H=V_{\mathcal{O}}^H.$$

If $k\in G$ and $kr=r^{\prime }h$, with $h\in G_p$, the induced action sends the r-summand to the $r^{\prime }$-summand through the $G_p$-action of h. Under the displayed identification, this is precisely transport from $rp$ to $k\left(rp\right)=r^{\prime }p$ by Lemma 1. Thus the isomorphism is G-equivariant. Independence of R follows from Lemma 8. □

Theorem 2 

(Orbit–stabilizer formula for the flexible sector). For each orbit representative $p\in \Sigma$, let $V_p^H$ be the rank-one point-supported vanishing Hodge object equipped with its natural $G_p$-linearization. Then

$$V_{\mathrm{van}}^H\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p^H$$

in ${MHM}_G\left(X_0\right)$.

Equivalently, after passing to the underlying rational vanishing representation,

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p$$

as finite-dimensional $\mathbb{Q}\left[G\right]$-modules.

Proof. 

Lemma 7 gives

$$V_{\mathrm{van}}^H\simeq \underset{\mathcal{O}\in \Sigma /G}{⨁}{V}_{\mathcal{O}}^H.$$

For each orbit, Proposition 10 gives $V_{\mathcal{O}}^H\simeq {Ind}_{G_p}^G{V}_p^H$ for any representative $p\in \mathcal{O}$. Combining these isomorphisms proves the first formula.

Changing the representative replaces $(G_p,V_p^H)$ by a conjugate pair. Corollary 1 and Lemma 8 show that the induced object is unchanged up to its natural isomorphism.

Passing to the underlying rational realization converts each orbitwise term into the ordinary induced $\mathbb{Q}\left[G\right]$-module ${Ind}_{G_p}^G{V}_p$, giving the second formula. □

The theorem shows that the formal flexible representation is determined by the finite collection ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$ up to conjugacy. The orbit $G/G_p$ records the permutation of the nodes, while $V_p$ records the action of the stabilizer on the local vanishing direction. These two pieces of information need not be equivalent, as the projective $S_4$-cubic of Section 7.3 demonstrates.

Corollary 3 

(Dimension recovery). The underlying rational dimension of the flexible sector satisfies

$${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=\sum _{\left[p\right]\in \Sigma /G}[G:G_p]{dim}_{\mathbb{Q}}{V}_p.$$

For an ordinary double point, ${dim}_{\mathbb{Q}}{V}_p=1$, and therefore

$${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=\sum _{\left[p\right]\in \Sigma /G}[G:G_p]=|\Sigma |.$$

Proof. 

For a finite subgroup $H\le G$ and a finite-dimensional $\mathbb{Q}\left[H\right]$-module W, ${dim}_{\mathbb{Q}}{Ind}_H^{G}W=[G:H]{dim}_{\mathbb{Q}}W$ [17]. Apply this termwise to Theorem 2. Since ${dim}_{\mathbb{Q}}{V}_p=1$ for an ordinary double point, the final equality follows from Proposition 3. □

Remark 16 

(What the orbit–stabilizer theorem determines). Theorem 2 determines the complete G-representation carried by the formal node-supported quotient $V_{\mathrm{van}}^H$. It does not determine the equivariant extension class

$$e_{\pi }^G\in {Ext}_{{MHM}_G\left(X_0\right)}^1\left(V_{\mathrm{van}}^H,I{C}_{X_0}^H\right)$$

of Corollary 2. The orbitwise decomposition of the quotient therefore does not imply a corresponding decomposition or splitting of the corrected carrier $P^H$.

4.3. The Special Case of Trivial Stabilizer Action

The simplest orbitwise situation occurs when the stabilizer acts trivially on the local vanishing line. This case produces ordinary permutation representations and will serve as a comparison point for the genuinely twisted projective example of Section 7.

Suppose that for an orbit representative p, the stabilizer $G_p$ acts trivially on $V_p$. Then $V_p\simeq \mathbb{Q}$ as a $\mathbb{Q}\left[G_p\right]$-module, and ${Ind}_{G_p}^G{V}_p\simeq \mathbb{Q}[G/G_p]$ is the rational permutation representation on the orbit [17].

Corollary 4 

(Permutation representation case). Assume that $G_p$ acts trivially on $V_p$ for every orbit representative $p\in \Sigma$. Then

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}\mathbb{Q}[G/G_p].$$

Thus the equivariant flexible representation is the direct sum of the permutation representations associated with the node orbits.

Proof. 

Under the hypothesis, every summand in Theorem 2 is ${Ind}_{G_p}^G\mathbb{Q}\simeq \mathbb{Q}[G/G_p]$. □

Corollary 5 

(Free node orbit). Let $\mathcal{O}=G·p$ be a free G-orbit of nodes. Then $G_p=\left\{1\right\}$, and its contribution to $V_{\mathrm{van}}$ is the regular representation $V_{\mathcal{O}}\simeq \mathbb{Q}\left[G\right]$.

Proof. 

For a free orbit, $G_p=\left\{1\right\}$, so $\mathbb{Q}[G/G_p]=\mathbb{Q}\left[G\right]$. Apply Corollary 4. □

Remark 17 

(A transitive four-node $S_4$-orbit). For a transitive four-point $S_4$-orbit with trivial local stabilizer action, one has $G_p\simeq S_3$ and

$$V_{\mathcal{O}}\simeq \mathbb{Q}[S_4/S_3]\simeq \mathbf{1}\oplus V_{\mathrm{std}},$$

where $V_{\mathrm{std}}$ is the three-dimensional standard rational representation [17].

This is the untwisted benchmark. The projective four-nodal $S_4$-cubic constructed in Section 7.3 has the same underlying orbit $S_4/S_3$, but Proposition 20 gives $V_p\simeq {sgn}_{S_3}$. Accordingly, Theorem 8 gives the different representation

$$V_{\mathrm{van}}\simeq {sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right).$$

Thus the four-point $S_4$-set alone does not determine the flexible representation.

Remark 18 

(A pair of nodes exchanged by an involution). If $G=\mathbb{Z}/2$ exchanges two nodes, the orbit is free and contributes the regular representation $\mathbb{Q}[\mathbb{Z}/2]\simeq \mathbf{1}\oplus sgn$. The corresponding symmetric and antisymmetric vanishing directions are worked out explicitly in Section 7.1.

Remark 19 

(Nontrivial stabilizer action). The permutation case does not exhaust the orbitwise theory. If $G_p$ acts nontrivially on $V_p$, then ${Ind}_{G_p}^G{V}_p$ need not be isomorphic to $\mathbb{Q}[G/G_p]$. Thus the intrinsic local datum is the pair $(G_p,V_p)$, not the orbit $G/G_p$ alone.

This distinction is realized globally by the projective $S_4$-equivariant cubic degeneration of Section 7.3, where $(G_p,V_p)=(S_3,{sgn}_{S_3})$. The decomposition of the induced modules into irreducible symmetry types is developed systematically in Section 5.

5. Isotypic Flexible Sectors

We use the notation and conventions of Section 1.6. By Theorem 2, the rational G-representation underlying the formal vanishing sector is

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p.$$

We now decompose this representation into irreducible symmetry types and express their multiplicities directly in terms of the local data $(G_p,V_p)$.

5.1. Rational Irreducible Representations

Since G is finite and $char\left(\mathbb{Q}\right)=0$, Maschke’s theorem implies that $\mathbb{Q}\left[G\right]$ is semisimple [17]. Hence every finite-dimensional rational G-representation is a finite direct sum of irreducibles.

Let ${\widehat{G}}_{\mathbb{Q}}$ denote the isomorphism classes of irreducible finite-dimensional $\mathbb{Q}\left[G\right]$-modules. For $\rho \in {\widehat{G}}_{\mathbb{Q}}$, set $D_{\rho }:={End}_{\mathbb{Q}\left[G\right]}\left(\rho \right)$. By Schur’s lemma, $D_{\rho }$ is a finite-dimensional division algebra over $\mathbb{Q}$ [17].

Definition 6 

(Rational isotypic multiplicity). For $\rho \in {\widehat{G}}_{\mathbb{Q}}$, define the multiplicity of ρ in $V_{\mathrm{van}}$ by

$$m_{\rho }^{\mathbb{Q}}:={dim}_{D_{\rho }}{Hom}_{\mathbb{Q}\left[G\right]}\left(\rho ,V_{\mathrm{van}}\right).$$

The Hom-space is a right $D_{\rho }$-module by precomposition. Semisimplicity gives a noncanonical decomposition

$$V_{\mathrm{van}}\simeq \underset{\rho \in {\widehat{G}}_{\mathbb{Q}}}{⨁}{\rho }^{\oplus m_{\rho }^{\mathbb{Q}}},$$

while the multiplicities themselves are intrinsic.

Proposition 11 

(Uniqueness of rational multiplicities). The integers $m_{\rho }^{\mathbb{Q}}$ are uniquely determined by the isomorphism class of the rational G-module $V_{\mathrm{van}}$. In particular,

$${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=\sum _{\rho \in {\widehat{G}}_{\mathbb{Q}}}{m}_{\rho }^{\mathbb{Q}}{dim}_{\mathbb{Q}}\rho .$$

Proof. 

By semisimplicity, write $V_{\mathrm{van}}\simeq {\rho }^{\oplus n}\oplus W$, where W contains no summand isomorphic to $\rho$. Schur’s lemma gives

$${Hom}_{\mathbb{Q}\left[G\right]}(\rho ,V_{\mathrm{van}})\simeq D_{\rho }^{\oplus n}$$

as right $D_{\rho }$-modules, so $m_{\rho }^{\mathbb{Q}}=n$. Thus the multiplicity is intrinsic. Taking rational dimensions in the semisimple decomposition yields

$${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=\sum _{\rho \in {\widehat{G}}_{\mathbb{Q}}}{m}_{\rho }^{\mathbb{Q}}{dim}_{\mathbb{Q}}\rho .$$

□

For character calculations, pass to the characteristic-zero splitting field $\mathbf{K}$ fixed in Section 1.6. Write ${\widehat{G}}_{\mathbf{K}}$ for the irreducible $\mathbf{K}\left[G\right]$-modules and set $V_{\mathrm{van},\mathbf{K}}:=V_{\mathrm{van}}{\otimes }_{\mathbb{Q}}\mathbf{K}$.

Definition 7 

(Splitting-field multiplicity). For $\sigma \in {\widehat{G}}_{\mathbf{K}}$, define

$$m_{\sigma }^{\mathbf{K}}:={dim}_{\mathbf{K}}{Hom}_{\mathbf{K}\left[G\right]}\left(\sigma ,V_{\mathrm{van},\mathbf{K}}\right).$$

Because $\mathbf{K}$ is a splitting field, ${End}_{\mathbf{K}\left[G\right]}\left(\sigma \right)=\mathbf{K}$, and hence

$$V_{\mathrm{van},\mathbf{K}}\simeq \underset{\sigma \in {\widehat{G}}_{\mathbf{K}}}{⨁}{\sigma }^{\oplus m_{\sigma }^{\mathbf{K}}}$$

[17].

Remark 20 

(Rational and splitting-field decompositions). The sets ${\widehat{G}}_{\mathbb{Q}}$ and ${\widehat{G}}_{\mathbf{K}}$ need not coincide: an irreducible rational representation may become reducible after scalar extension. Thus $m_{\rho }^{\mathbb{Q}}$ and $m_{\sigma }^{\mathbf{K}}$ belong to distinct semisimple decompositions and are not identified term by term without an explicit scalar-extension analysis.

5.2. Multiplicity Formula

The orbit–stabilizer formula reduces the multiplicity calculation to finite representation theory. For each orbit representative p, the local vanishing line $V_p$ is a one-dimensional rational $\mathbb{Q}\left[G_p\right]$-module.

Theorem 3 

(Isotypic multiplicity formula). Let

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p$$

be the flexible representation of Theorem 2. For every $\rho \in {\widehat{G}}_{\mathbb{Q}}$,

$$m_{\rho }^{\mathbb{Q}}=\sum _{\left[p\right]\in \Sigma /G}{dim}_{D_{\rho }}{Hom}_{\mathbb{Q}\left[G_p\right]}\left({Res}_{G_p}^G\rho ,V_p\right),$$

where $D_{\rho }={End}_{\mathbb{Q}\left[G\right]}\left(\rho \right)$.

After extension of scalars to the fixed splitting field $\mathbf{K}$, for every $\sigma \in {\widehat{G}}_{\mathbf{K}}$,

$$m_{\sigma }^{\mathbf{K}}=\sum _{\left[p\right]\in \Sigma /G}{dim}_{\mathbf{K}}{Hom}_{\mathbf{K}\left[G_p\right]}\left({Res}_{G_p}^G\sigma ,V_p{\otimes }_{\mathbb{Q}}\mathbf{K}\right).$$

Proof. 

For $\rho \in {\widehat{G}}_{\mathbb{Q}}$, Definition 6 and Theorem 2 give

$$\begin{array}{cc}\hfill {Hom}_{\mathbb{Q}\left[G\right]}\left(\rho ,V_{\mathrm{van}}\right)& \simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Hom}_{\mathbb{Q}\left[G\right]}\left(\rho ,{Ind}_{G_p}^G{V}_p\right).\hfill \end{array}$$

Frobenius reciprocity identifies each summand with

$${Hom}_{\mathbb{Q}\left[G_p\right]}\left({Res}_{G_p}^G\rho ,V_p\right)$$

[17]. This identification is compatible with the right $D_{\rho }$-action by precomposition. Taking $D_{\rho }$-dimensions proves the rational formula.

After scalar extension,

$$V_{\mathrm{van},\mathbf{K}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G\left(V_p{\otimes }_{\mathbb{Q}}\mathbf{K}\right),$$

since induction commutes with scalar extension. Applying Frobenius reciprocity over $\mathbf{K}$ and taking $\mathbf{K}$-dimensions yields the second formula. □

Corollary 6 

(Trivial stabilizer action). Assume that every $V_p$ is the trivial one-dimensional $\mathbb{Q}\left[G_p\right]$-module. Then, for $\sigma \in {\widehat{G}}_{\mathbf{K}}$,

$$m_{\sigma }^{\mathbf{K}}=\sum _{\left[p\right]\in \Sigma /G}{dim}_{\mathbf{K}}{\left({Res}_{G_p}^G\sigma \right)}^{G_p}.$$

Proof. 

Under the hypothesis, $V_p{\otimes }_{\mathbb{Q}}\mathbf{K}\simeq \mathbf{K}$ with trivial $G_p$-action. Therefore

$${Hom}_{\mathbf{K}\left[G_p\right]}\left({Res}_{G_p}^G\sigma ,\mathbf{K}\right)$$

has dimension equal to the dimension of the $G_p$-invariant subspace of ${Res}_{G_p}^G\sigma$, by semisimplicity over the characteristic-zero field $\mathbf{K}$. Apply Theorem 3. □

Corollary 7 

(Free-orbit contribution). Let $\mathcal{O}=G·p$ be a free orbit of nodes. Its contribution to the splitting-field multiplicity of $\sigma \in {\widehat{G}}_{\mathbf{K}}$ is ${dim}_{\mathbf{K}}\sigma$.

Proof. 

For a free orbit, $G_p=\left\{1\right\}$. The corresponding term in Theorem 3 is therefore

$${dim}_{\mathbf{K}}{Hom}_{\mathbf{K}}(\sigma ,\mathbf{K})={dim}_{\mathbf{K}}\sigma .$$

Equivalently, the orbit contributes the regular representation $\mathbf{K}\left[G\right]$, in which $\sigma$ occurs with multiplicity ${dim}_{\mathbf{K}}\sigma$ [17]. □

Remark 21 

(Local-to-global representation calculation). Theorem 3 depends only on the finite orbit–stabilizer collection $(G_p,V_p)$. The global extension class and any splitting of $P^H$ play no role. The theorem therefore determines the symmetry types of the formal flexible quotient, not an isotypic decomposition of the global rigid–flexible extension.

5.3. The Equivariant Flexible Profile

The representation $V_{\mathrm{van}}$ refines the ordinary flexible count $r=|\Sigma |$. We package this refinement using its intrinsic isotypic multiplicities.

The Hodge realization of every ordinary-double-point vanishing sector has the common Tate-normalized Hodge type ${\tau }_{\mathrm{van}}$, corresponding to the point-supported object $i_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1)$ with the normalization of [8].

Definition 8 

(Equivariant flexible Hodge-atom profile). The rational equivariant flexible profile of π is

$${\mathcal{P}}_{\mathrm{flex},\mathbb{Q}}^G\left(\pi \right):=\left({\tau }_{\mathrm{van}},\left\{(\rho ,m_{\rho }^{\mathbb{Q}})|\rho \in {\widehat{G}}_{\mathbb{Q}},m_{\rho }^{\mathbb{Q}}\ne 0\right\}\right).$$

After extension to the fixed splitting field $\mathbf{K}$, the splitting-field profile is

$${\mathcal{P}}_{\mathrm{flex},\mathbf{K}}^G\left(\pi \right):=\left({\tau }_{\mathrm{van}},\left\{(\sigma ,m_{\sigma }^{\mathbf{K}})|\sigma \in {\widehat{G}}_{\mathbf{K}},m_{\sigma }^{\mathbf{K}}\ne 0\right\}\right).$$

When the coefficient field is clear, we write ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$.

The ordinary flexible count is recovered by dimension:

$$r={dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=\sum _{\rho \in {\widehat{G}}_{\mathbb{Q}}}{m}_{\rho }^{\mathbb{Q}}{dim}_{\mathbb{Q}}\rho ,$$

and after scalar extension,

$$r=\sum _{\sigma \in {\widehat{G}}_{\mathbf{K}}}{m}_{\sigma }^{\mathbf{K}}{dim}_{\mathbf{K}}\sigma .$$

Proposition 12 

(Forgetful dimension of the flexible profile). The dimension map applied to ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$ recovers the ordinary number of node-supported vanishing directions: $dim{V}_{\mathrm{van}}=|\Sigma |$. Thus the equivariant flexible profile refines the non-equivariant flexible rank.

Proof. 

Corollary 3 gives ${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=|\Sigma |$, while Proposition 11 gives

$${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=\sum _{\rho \in {\widehat{G}}_{\mathbb{Q}}}{m}_{\rho }^{\mathbb{Q}}{dim}_{\mathbb{Q}}\rho .$$

Hence the equivariant multiplicities refine the same total dimension by recording its decomposition into symmetry types. □

Theorem 4 

(Determination from local symmetry data). The equivariant flexible profile ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$ is determined by the finite collection of orbit–stabilizer vanishing data ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$, taken up to the conjugacy equivalence of Lemma 1.

Proof. 

Theorem 2 shows that the pairs $(G_p,V_p)$ determine the isomorphism class

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p.$$

Theorem 3 then determines every rational multiplicity $m_{\rho }^{\mathbb{Q}}$ and, after scalar extension, every splitting-field multiplicity $m_{\sigma }^{\mathbf{K}}$. The Hodge type ${\tau }_{\mathrm{van}}$ is fixed by the ordinary-double-point normalization.

Changing an orbit representative replaces $(G_p,V_p)$ by a conjugate pair. By Corollary 1, this does not change the corresponding induced G-representation. Hence the profile depends only on the stated conjugacy classes of local data. □

Corollary 8 

(Equivariant-isomorphism invariance). Let $F:\mathcal{X}\stackrel{\sim }{\to }{\mathcal{X}}^{\prime }$ be a G-equivariant isomorphism of finite-node conifold degenerations over Δ. Then

$${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)={\mathcal{P}}_{\mathrm{flex}}^G\left({\pi }^{\prime }\right).$$

Proof. 

The isomorphism F identifies the node sets as G-sets and, by functoriality of vanishing cycles, identifies the corresponding local stabilizer representations. Thus the collections ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$ agree up to conjugacy. The result follows from Theorem 4. □

Remark 22 

(No birational invariance is asserted). The profile ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$ is invariant under equivariant isomorphism of degenerations. No invariance under G-equivariant birational modification is asserted. In particular, the profile is not identified across equivariant blow-ups, blow-downs, or weak factorizations without an additional compatibility theorem between degeneration and birational transport.

Remark 23 

(Flexible profile versus global extension data). The profile records the semisimple G-representation carried by the formal vanishing quotient. It does not determine the equivariant extension class $e_{\pi }^G$ of Corollary 2. Thus two degenerations may have the same equivariant flexible profile while differing in the way the flexible sector is globally attached to $I{C}_{X_0}^H$. The representation-theoretic profile and the global gluing class are distinct layers of the degeneration data.

6. Equivariant Limiting Hodge-Atom Profiles

We use the notation and conventions of Section 1.6. In particular, $I{C}_{X_0}^H$ is the rigid intersection-complex sector, $V_{\mathrm{van}}^H$ is the formal flexible sector, ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$ is the equivariant flexible profile of Definition 8, and ${For}_G$ denotes the forgetful operation.

The purpose of this section is to package the equivariant structures constructed in Section 3, Section 4 and Section 5 into a canonical degeneration-side profile and to identify precisely what is retained, what is forgotten, and what is recovered after forgetting the finite-group action.

6.1. From Smooth Equivariant Atoms to the Conifold Setting

For a smooth projective complex variety X, Hodge atoms arise from the spectral decomposition of the non-archimedean A-model F-bundle over the Hod-fixed locus [1]. If a finite group G acts algebraically on X, the smooth equivariant construction restricts this spectral theory to the $G\times Hod$-fixed locus and retains the resulting finite-group representation data [4].

The degeneration-side construction has a different categorical origin. No singular non-archimedean F-bundle on $X_0$ is introduced. Instead, the starting point is the corrected mixed-Hodge-module carrier recalled in Proposition 6 and lifted equivariantly in Theorem 1:

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0.$$

Thus the limiting data considered here are extracted from nearby cycles, vanishing cycles, the variation-cone correction, and the resulting mixed-Hodge-module extension. They refine the degeneration-side rigid and flexible sectors rather than defining spectral atoms directly on the singular fiber.

Definition 9 

(Smooth equivariant atomic datum). For a smooth projective G-variety X, let ${\mathcal{A}}_G\left(X\right)$ denote the G-equivariant Hodge-atom datum arising from the $G\times Hod$-equivariant spectral construction of [4].

Definition 10 

(Degeneration-side equivariant profile). For an equivariant finite-node conifold degeneration $\pi :\mathcal{X}\to \Delta$, a degeneration-side equivariant profile is a finite Hodge-theoretic datum extracted from the G-linearized rigid–flexible carrier

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0.$$

In the present paper, the limiting profile retains the equivariant rigid term $I{C}_{X_0}^H$ and the symmetry-resolved formal flexible term $V_{\mathrm{van}}^H$.

The projective $S_4$-equivariant cubic degeneration constructed in Theorem 7 gives a concrete instance of this distinction. Its nearby fibers are smooth projective cubic threefolds with generically free $S_4$-action, while the central fiber has four ordinary double points forming one $S_4$-orbit. On the limiting side, the stabilizer data are $(G_p,V_p)=(S_3,{sgn}_{S_3})$, and hence

$$V_{\mathrm{van}}\simeq {Ind}_{S_3}^{S_4}{sgn}_{S_3}.$$

Thus the degeneration-side profile is not merely a formal receptacle: it can be computed explicitly in a projective family and records information not visible from the node set alone.

Remark 24 

(Smooth atoms and limiting profiles are distinct objects). The datum ${\mathcal{A}}_G\left(X\right)$ is spectral and attached to a smooth projective G-variety, whereas the limiting profile below is extracted from the mixed-Hodge-module degeneration carrier. No functor ${\mathcal{A}}_G\left(X_t\right)\mapsto {\mathcal{P}}_G^{\lim }\left(\pi \right)$ is constructed here. The projective $S_4$-example nevertheless supplies an explicit family in which smooth equivariant geometry and computable degeneration-side representation data occur in the same one-parameter family.

Remark 25 

(No singular F-bundle construction). The present construction defines neither a maximal non-archimedean A-model F-bundle for $X_0$ nor spectral atoms directly on the singular fiber. The limiting profile is a Hodge-theoretic invariant extracted from the corrected mixed-Hodge-module degeneration carrier.

6.2. Rigid and Flexible Equivariant Profiles

The flexible profile was constructed in Section 5. We now record the corresponding equivariant datum carried by the rigid sector.

Let ${\lambda }_g^{IC}:g^{*}I{C}_{X_0}^H\stackrel{\sim }{\to }I{C}_{X_0}^H$, for $g\in G$, be the canonical G-linearization of Proposition 8.

Definition 11 

(Equivariant rigid profile). The equivariant rigid profile of π is the isomorphism class

$${\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right):=\left[I{C}_{X_0}^H,{\left\{{\lambda }_g^{IC}\right\}}_{g\in G}\right]$$

in ${MHM}_G\left(X_0\right)$, together with its induced Hodge-theoretic realization.

Thus ${\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right)$ retains the canonical G-linearization on $I{C}_{X_0}^H$. No decomposition, projector, or splitting is chosen.

The flexible profile ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$ is determined by the orbit–stabilizer formula

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p$$

of Theorem 2, together with the isotypic multiplicities of Theorem 3. After extension to the fixed splitting field $\mathbf{K}$,

$$V_{\mathrm{van},\mathbf{K}}\simeq \underset{\rho \in {\widehat{G}}_{\mathbf{K}}}{⨁}{\rho }^{\oplus m_{\rho }}.$$

The local datum entering this construction is the full stabilizer module $(G_p,V_p)$, rather than only the orbit $G/G_p$. This distinction is realized globally by the projective $S_4$-cubic: ${Ind}_{S_3}^{S_4}\mathbf{1}\simeq \mathbf{1}\oplus V_{\mathrm{std}}$, whereas its actual local vanishing module is ${sgn}_{S_3}$, giving

$${Ind}_{S_3}^{S_4}{sgn}_{S_3}\simeq {sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right)$$

by Theorem 8.

Definition 12 

(Equivariant limiting rigid–flexible profile). The equivariant limiting rigid–flexible profile of π is

$${\mathcal{P}}_G^{\lim }\left(\pi \right):=\left({\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right),{\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right).$$

The extension class

$$e_{\pi }^G\in {Ext}_{{MHM}_G\left(X_0\right)}^1\left(V_{\mathrm{van}}^H,I{C}_{X_0}^H\right)$$

of Corollary 2 is not part of ${\mathcal{P}}_G^{\lim }\left(\pi \right)$. The profile records the equivariant types of the two terms; the global gluing class remains a separate layer.

Proposition 13 

(Canonicity of the rigid profile). The profile ${\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right)$ is canonically determined by the G-equivariant degeneration π.

Proof. 

Proposition 8 gives a canonical G-linearization on $I{C}_{X_0}^H$, functorial under automorphisms of the pair $(X_0,X_0^{\mathrm{sm}})$. Hence no auxiliary choice enters ${\left\{{\lambda }_g^{IC}\right\}}_{g\in G}$, and its isomorphism class in ${MHM}_G\left(X_0\right)$ is determined by $\pi$. □

Proposition 14 

(Canonicity of the flexible profile). The profile ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$ is canonically determined by the G-equivariant degeneration π.

Proof. 

The G-equivariant degeneration determines the G-set $\Sigma$, the stabilizers $G_p$, and the local vanishing modules $V_p$. Changing an orbit representative replaces $(G_p,V_p)$ by a conjugate pair by Lemma 1, without changing the corresponding induced G-representation by Corollary 1.

Theorem 2 therefore determines the isomorphism class of $V_{\mathrm{van}}$, while Theorem 3 determines all rational and splitting-field irreducible multiplicities. The local Hodge type is fixed by the ordinary-double-point normalization in Proposition 6. Hence the flexible profile is canonical. □

Theorem 5 

(Equivariant rigid–flexible atom profile). Let $\pi :\mathcal{X}\to \Delta$ be an equivariant finite-node conifold degeneration satisfying the standing hypotheses. Then the corrected conifold rigid–flexible profile admits the canonical equivariant refinement

$${\mathcal{P}}_G^{\lim }\left(\pi \right)=\left({\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right),{\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right).$$

Its flexible component is determined by

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p,$$

and therefore by the finite collection ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$.

Proof. 

Theorem 1 gives the canonical short exact sequence

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0$$

in ${MHM}_G\left(X_0\right)$.

Propositions 13 and 14 show that the rigid term and the formal flexible quotient determine the two components of ${\mathcal{P}}_G^{\lim }\left(\pi \right)$. The orbit–stabilizer formula is Theorem 2, and dependence on the finite local collection follows from Theorem 4. □

For the projective $S_4$-equivariant cubic degeneration of Theorem 7, the preceding theorem is completely explicit. Since there is one node orbit with stabilizer $S_3$ and $V_p\simeq {sgn}_{S_3}$, the rational flexible profile is

$${\mathcal{P}}_{\mathrm{flex},\mathbb{Q}}^{S_4}\left(\pi \right)=\left({\tau }_{\mathrm{van}},\left\{\left({sgn}_{S_4},1\right),\left(V_{\mathrm{std}}\otimes {sgn}_{S_4},1\right)\right\}\right).$$

Thus the limiting profile of an actual projective degeneration detects the twisted stabilizer contribution and, by Corollary 10, records the fact that $V_{\mathrm{van}}^{S_4}=0$.

Remark 26 

(Information retained by the Paper-I profile). The profile ${\mathcal{P}}_G^{\lim }\left(\pi \right)$ retains the canonical G-equivariant rigid sector and the complete semisimple G-representation carried by the formal flexible sector. In particular, it distinguishes the untwisted and sign-twisted $S_4/S_3$ orbit contributions above.

The profile does not itself contain the full extension class $e_{\pi }^G$, the complete limiting mixed Hodge structure, the Picard–Lefschetz pairing, or Stokes interaction data. These are additional structures on the degeneration rather than components of the profile defined here.

Remark 27 

(No specialization theorem). Theorem 5 is a degeneration-side statement. It does not construct a functor

$${Sp}_{\mathrm{Atom}}^G:{\mathcal{A}}_G\left(X_t\right)⟶{\mathcal{P}}_G^{\lim }\left(\pi \right).$$

The explicit $S_4$-cubic nevertheless provides a concrete test family for such a future comparison: the smooth fibers carry generically free $S_4$-actions, while the singular fiber has the explicitly computed limiting flexible representation ${Ind}_{S_3}^{S_4}{sgn}_{S_3}$. Any specialization theory applicable to this family would therefore have to recover this degeneration-side symmetry data.

6.3. Forgetful Recovery of the Ordinary Conifold Profile

The equivariant construction refines the ordinary rigid–flexible profile by retaining G-linearizations and representation types. Let

$${For}_G:{MHM}_G\left(X_0\right)⟶MHM\left(X_0\right)$$

be the exact forgetful functor of Proposition 5.

Definition 13 

(Ordinary rigid profile). The ordinary rigid profile is ${\mathcal{P}}_{\mathrm{rig}}\left(\pi \right):=\left[I{C}_{X_0}^H\right]$, where the brackets denote the isomorphism class in $MHM\left(X_0\right)$.

Definition 14 

(Ordinary flexible profile). Let ${\tau }_{\mathrm{van}}$ denote the fixed Hodge type of the local ordinary-double-point vanishing sector. The ordinary flexible profile is ${\mathcal{P}}_{\mathrm{flex}}\left(\pi \right):=({\tau }_{\mathrm{van}},|\Sigma |)$.

Definition 15 

(Ordinary limiting rigid–flexible profile). The ordinary limiting rigid–flexible profile is

$${\mathcal{P}}^{\lim }\left(\pi \right):=\left({\mathcal{P}}_{\mathrm{rig}}\left(\pi \right),{\mathcal{P}}_{\mathrm{flex}}\left(\pi \right)\right).$$

These definitions retain the portion of the non-equivariant finite-node rigid–flexible structure relevant to the present profile.

Definition 16 

(Forgetful operation on profiles). The operation ${For}_G$ on ${\mathcal{P}}_G^{\lim }\left(\pi \right)$ is defined componentwise.

On the rigid component,

$${For}_G\left({\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right)\right):={\mathcal{P}}_{\mathrm{rig}}\left(\pi \right).$$

On the flexible component,

$${For}_G\left({\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right):=\left({\tau }_{\mathrm{van}},{dim}_{\mathbb{Q}}{V}_{\mathrm{van}}\right).$$

At the mixed-Hodge-module level, ${For}_G\left(V_{\mathrm{van}}^H\right)={⨁}_{p\in \Sigma }{V}_p^H$. At the representation level, forgetting G removes the orbit and isotypic decomposition and retains only the underlying rational vector space.

Proposition 15 

(Forgetful recovery of the flexible rank). One has ${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=|\Sigma |$. Equivalently, after extension to the splitting field $\mathbf{K}$,

$$\sum _{\rho \in {\widehat{G}}_{\mathbf{K}}}{m}_{\rho }{dim}_{\mathbf{K}}\rho =|\Sigma |.$$

Proof. 

Corollary 3 gives ${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=|\Sigma |$. After scalar extension,

$$V_{\mathrm{van},\mathbf{K}}\simeq \underset{\rho \in {\widehat{G}}_{\mathbf{K}}}{⨁}{\rho }^{\oplus m_{\rho }}.$$

Taking $\mathbf{K}$-dimensions and using preservation of dimension under scalar extension yields

$$|\Sigma |=\sum _{\rho \in {\widehat{G}}_{\mathbf{K}}}{m}_{\rho }{dim}_{\mathbf{K}}\rho .$$

□

Theorem 6 

(Forgetful recovery). For every equivariant finite-node conifold degeneration satisfying the standing hypotheses,

$${For}_G\left({\mathcal{P}}_G^{\lim }\left(\pi \right)\right)={\mathcal{P}}^{\lim }\left(\pi \right).$$

In particular,

$${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=\sum _{\rho \in {\widehat{G}}_{\mathbf{K}}}{m}_{\rho }{dim}_{\mathbf{K}}\rho =|\Sigma |.$$

Proof. 

By Definition 12,

$${\mathcal{P}}_G^{\lim }\left(\pi \right)=\left({\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right),{\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right).$$

Definition 16 gives ${For}_G\left({\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right)\right)={\mathcal{P}}_{\mathrm{rig}}\left(\pi \right)$ and

$${For}_G\left({\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right)=\left({\tau }_{\mathrm{van}},{dim}_{\mathbb{Q}}{V}_{\mathrm{van}}\right).$$

By Proposition 15, ${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=|\Sigma |$. Hence ${For}_G\left({\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right)={\mathcal{P}}_{\mathrm{flex}}\left(\pi \right)$, and therefore

$${For}_G\left({\mathcal{P}}_G^{\lim }\left(\pi \right)\right)={\mathcal{P}}^{\lim }\left(\pi \right).$$

The dimension identity follows from the same proposition. □

Corollary 9 

(Strict refinement of the ordinary flexible count). The ordinary node count $|\Sigma |$ is determined by ${\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)$. The converse need not hold: the integer $|\Sigma |$ does not determine the isomorphism class of the G-representation $V_{\mathrm{van}}$.

Proof. 

The first assertion is Proposition 15. For the second, the orbit–stabilizer formula

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p$$

shows that the representation depends on the orbit decomposition, stabilizers, and local stabilizer modules, none of which is determined by the total cardinality $|\Sigma |$.

The projective $S_4$-cubic gives a stronger geometric manifestation: even after fixing the transitive four-point $S_4$-set $\Sigma \simeq S_4/S_3$, the local stabilizer representation changes the resulting flexible $S_4$-module. The trivial local character gives $\mathbf{1}\oplus V_{\mathrm{std}}$, whereas the geometric sign character gives

$${sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right).$$

□

Remark 28 

(Refinement, not replacement). Theorem 6 shows that forgetting the G-structure recovers the ordinary finite-node rigid–flexible profile. The equivariant construction therefore adds two successive layers of information: the G-set structure of the nodes and the stabilizer representations on their local vanishing lines. Neither changes the underlying non-equivariant degeneration carrier.

Remark 29 

(Extension data remain separate). The forgetful theorem concerns the pair of rigid and flexible profiles. It does not identify the equivariant extension class $e_{\pi }^G$ with this profile or assert that the extension is determined by

$$\left({\mathcal{P}}_{\mathrm{rig}}^G\left(\pi \right),{\mathcal{P}}_{\mathrm{flex}}^G\left(\pi \right)\right).$$

The extension class retains additional global gluing information beyond the semisimple representation-theoretic data recorded by the profile.

7. Examples and Geometric Calculations

We use the notation and conventions of Section 1.6. The examples in this section serve two purposes. The first two isolate elementary representation-theoretic features of the flexible profile: permutation of nodes and dependence on the coefficient field. The third gives a complete projective $S_4$-equivariant conifold degeneration and computes its flexible representation from the geometry of the local stabilizer action. The final examples isolate the local stabilizer mechanism and compare profiles that have the same ordinary flexible rank.

Unless an explicit algebraic degeneration is specified, an example is to be understood as a representation-theoretic model for the local orbit–stabilizer data ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$, rather than as an assertion that every such finite G-set and stabilizer representation occurs as the singular locus of a projective conifold degeneration. The $S_4$-cubic constructed in Section 7.3 is different: it is an explicit projective degeneration satisfying the standing geometric hypotheses and its equivariant flexible profile is computed from the resulting local vanishing representations.

7.1. A Pair of Nodes Exchanged by an Involution

This first example isolates the simplest effect of node permutation. The local stabilizers are trivial, so no additional local character is present; all equivariant information comes from the action of G on the two-node orbit.

Let $G=\mathbb{Z}/2=\langle \sigma \rangle$, and suppose that $\Sigma =\{p_1,p_2\}$ is one free G-orbit with $\sigma p_1=p_2$. Then $G_{p_1}=\left\{1\right\}$, so Corollary 5 gives $V_{\mathrm{van}}\simeq \mathbb{Q}\left[G\right]$.

Let ${\delta }_1,{\delta }_2$ be generators of the two local vanishing lines, chosen so that $\sigma \left({\delta }_1\right)={\delta }_2$ and $\sigma \left({\delta }_2\right)={\delta }_1$. Define ${\delta }_{+}:={\delta }_1+{\delta }_2$ and ${\delta }_{-}:={\delta }_1-{\delta }_2$. Then $\sigma \left({\delta }_{+}\right)={\delta }_{+}$ and $\sigma \left({\delta }_{-}\right)=-{\delta }_{-}$.

Proposition 16 

(Two-node involution profile). Under the preceding hypotheses, $V_{\mathrm{van}}\simeq \mathbf{1}\oplus sgn$, where $\mathbf{1}$ is the trivial rational representation of $\mathbb{Z}/2$ and sgn is the sign representation. The invariant line is generated by ${\delta }_{+}$, and the sign line is generated by ${\delta }_{-}$.

Proof. 

The regular representation $\mathbb{Q}[\mathbb{Z}/2]$ has basis ${\delta }_1,{\delta }_2$, with $\sigma$ interchanging the two basis vectors. Since ${\delta }_1={\displaystyle \frac{1}{2}}({\delta }_{+}+{\delta }_{-})$ and ${\delta }_2={\displaystyle \frac{1}{2}}({\delta }_{+}-{\delta }_{-})$, the vectors ${\delta }_{+},{\delta }_{-}$ form a basis. Their transformation laws give $\mathbb{Q}{\delta }_{+}\simeq \mathbf{1}$ and $\mathbb{Q}{\delta }_{-}\simeq sgn$, and hence the stated decomposition. □

The ordinary flexible count records only ${dim}_{\mathbb{Q}}{V}_{\mathrm{van}}=2$. The equivariant profile resolves this rank into the two symmetry types

$${\mathcal{P}}_{\mathrm{flex}}^{\mathbb{Z}/2}\left(\pi \right)=\left\{(\mathbf{1},1),(sgn,1)\right\},$$

together with the common ordinary-double-point Hodge type ${\tau }_{\mathrm{van}}$. Thus even a free orbit already refines the ordinary node count by separating invariant from anti-invariant vanishing directions.

Remark 30 

(Symmetric and antisymmetric directions). The decomposition distinguishes the invariant combination ${\delta }_1+{\delta }_2$ from the antisymmetric combination ${\delta }_1-{\delta }_2$. This distinction is invisible after applying the forgetful operation ${For}_G$.

7.2. A Cyclic Orbit of Three Nodes

The next example isolates the distinction between rational irreducible representations and their decomposition after extension to a splitting field. Again the node orbit is free, but the rational regular representation does not decompose into three one-dimensional rational characters.

Let $G=\mathbb{Z}/3=\langle \sigma \rangle$, and suppose that $\Sigma =\{p_1,p_2,p_3\}$ is one free orbit with $\sigma p_1=p_2$, $\sigma p_2=p_3$, and $\sigma p_3=p_1$. Then $V_{\mathrm{van}}\simeq \mathbb{Q}[\mathbb{Z}/3]$.

Let ${\delta }_1,{\delta }_2,{\delta }_3$ be the corresponding local vanishing generators. Set ${\delta }_0:={\delta }_1+{\delta }_2+{\delta }_3$ and define

$$W_2:=\left\{a_1{\delta }_1+a_2{\delta }_2+a_3{\delta }_3|a_1+a_2+a_3=0\right\}.$$

Then $W_2$ is a two-dimensional G-stable rational subspace.

Proposition 17 

(Three-node cyclic profile). The rational regular representation of $\mathbb{Z}/3$ decomposes as $V_{\mathrm{van}}\simeq \mathbf{1}\oplus W_2$, where $W_2$ is irreducible over $\mathbb{Q}$.

Proof. 

The line $\mathbb{Q}{\delta }_0$ is fixed by G, while $W_2$ is the kernel of the G-equivariant augmentation map

$$\mathbb{Q}[\mathbb{Z}/3]⟶\mathbb{Q},{\displaystyle \sum _{i=1}^3}{a}_i{\delta }_i⟼{\displaystyle \sum _{i=1}^3}{a}_i.$$

Hence $\mathbb{Q}[\mathbb{Z}/3]=\mathbb{Q}{\delta }_0\oplus W_2$.

If $W_2$ contained a one-dimensional rational subrepresentation, a generator of $\mathbb{Z}/3$ would act on it by a rational cube root of unity. The only rational cube root of unity is 1, but $W_2$ has no nonzero invariant vector. Since ${dim}_{\mathbb{Q}}{W}_2=2$, it follows that $W_2$ is irreducible. □

Let $\mathbf{K}$ contain a primitive cube root of unity ${\zeta }_3$. Then $W_2{\otimes }_{\mathbb{Q}}\mathbf{K}\simeq \chi \oplus {\chi }^{-1}$, where $\chi \left(\sigma \right)={\zeta }_3$.

The underlying node orbit has not changed under scalar extension, but its irreducible representation-theoretic decomposition has. Over $\mathbb{Q}$, the two nontrivial eigendirections form one irreducible two-dimensional sector; over a splitting field they become two distinct one-dimensional sectors.

Remark 31 

(Rational versus splitting-field sectors). Over $\mathbb{Q}$, the two nontrivial complex characters form a single two-dimensional irreducible representation $W_2$. After scalar extension to a splitting field, $W_2$ separates into two one-dimensional eigenspaces. Thus the rational profile and the splitting-field profile must be distinguished as in Remark 20. The relevant representation-theoretic facts are standard [17].

7.3. Four Nodes with Transitive $S_4$-Symmetry

We begin with the untwisted permutation calculation as a benchmark and then construct a projective $S_4$-equivariant cubic degeneration with the same four-point orbit. In the geometric example the stabilizer does not act trivially on the local vanishing line. This produces a different $S_4$-representation and gives a global realization of the distinction between the orbit $G/G_p$ and the full local datum $(G_p,V_p)$.

Let $G=S_4$ act transitively on $\Sigma =\{p_1,p_2,p_3,p_4\}$. For a chosen point $p_1$, one has $G_{p_1}\simeq S_3$ and $G/G_{p_1}\simeq S_4/S_3$.

Assume first, purely as a representation-theoretic comparison model, that $G_{p_1}$ acts trivially on $V_{p_1}$. Then $V_{\mathrm{van}}\simeq \mathbb{Q}[S_4/S_3]$.

Let ${\delta }_1,\dots ,{\delta }_4$ denote the basis vectors associated with the four cosets, set ${\delta }_0:={\delta }_1+{\delta }_2+{\delta }_3+{\delta }_4$, and define

$$V_{\mathrm{std}}:=\left\{{\displaystyle \sum _{i=1}^4}{a}_i{\delta }_i|{\displaystyle \sum _{i=1}^4}{a}_i=0\right\}.$$

Proposition 18 

(Transitive $S_4$ permutation profile). For the transitive four-point $S_4$-action with trivial local stabilizer action, $V_{\mathrm{van}}\simeq \mathbf{1}\oplus V_{\mathrm{std}}$, where $V_{\mathrm{std}}$ is the three-dimensional standard rational representation of $S_4$.

Proof. 

The permutation representation $\mathbb{Q}[S_4/S_3]$ is naturally identified with ${\mathbb{Q}}^4$, with $S_4$ permuting the standard basis. The line $\mathbb{Q}{\delta }_0$ is invariant, while the kernel of the augmentation map

$${\mathbb{Q}}^4⟶\mathbb{Q},{\displaystyle \sum _{i=1}^4}{a}_i{\delta }_i⟼{\displaystyle \sum _{i=1}^4}{a}_i$$

is $V_{\mathrm{std}}$. Hence ${\mathbb{Q}}^4=\mathbb{Q}{\delta }_0\oplus V_{\mathrm{std}}$. The second summand is the standard irreducible representation of $S_4$ [17]. □

We now show that a natural projective four-nodal $S_4$-cubic has the same node orbit but a different local stabilizer representation.

Cheltsov–Tschinkel–Zhang describe the four-nodal $S_4$-symmetric case by the family

$$\begin{array}{cc}\hfill a{x}_5^3& +x_1{x}_2{x}_3+x_1{x}_2{x}_4+x_1{x}_3{x}_4+x_2{x}_3{x}_4\hfill \\ & +x_5^2(x_1+x_2+x_3+x_4)=0,\hfill \end{array}$$

with $S_4$ permuting $x_1,\dots ,x_4$ and fixing $x_5$ [13]. We take the member $a=1$ and verify its singularities directly.

Define

$$\begin{array}{cc}\hfill F_0& :=x_1{x}_2{x}_3+x_1{x}_2{x}_4+x_1{x}_3{x}_4+x_2{x}_3{x}_4\hfill \\ & +x_5^2(x_1+x_2+x_3+x_4)+x_5^3,\hfill \end{array}$$

and let $X_0:=V\left(F_0\right)\subset {\mathbf{P}}^4$. The group $S_4$ acts on ${\mathbf{P}}^4$ by permuting $x_1,\dots ,x_4$ and fixing $x_5$. The polynomial $F_0$ is $S_4$-invariant.

Proposition 19 

(The explicit four-nodal $S_4$-cubic). The singular locus of $X_0$ consists precisely of the four points

$$\begin{array}{cccc}\hfill p_1& =[1:0:0:0:0],\hfill & \hfill p_2& =[0:1:0:0:0],\hfill \\ \hfill p_3& =[0:0:1:0:0],\hfill & \hfill p_4& =[0:0:0:1:0].\hfill \end{array}$$

Each $p_i$ is an ordinary double point, and $S_4$ acts transitively on the set $\Sigma =\{p_1,p_2,p_3,p_4\}$.

Proof. 

Set $z:=x_5$, $s_1:=x_1+x_2+x_3+x_4$, and $s_2:={\sum }_{1\le i<j\le 4}{x}_i{x}_j$. Let

$$e_3:=x_1{x}_2{x}_3+x_1{x}_2{x}_4+x_1{x}_3{x}_4+x_2{x}_3{x}_4,$$

so that $F_0=e_3+z^2{s}_1+z^3$.

For $i=1,\dots ,4$,

$${\displaystyle \frac{\partial e_3}{\partial x_i}}=s_2-x_i(s_1-x_i)=x_i^2-s_1{x}_i+s_2.$$

Therefore

$${\displaystyle \frac{\partial F_0}{\partial x_i}}=x_i^2-s_1{x}_i+s_2+z^2,$$

while ${\displaystyle \frac{\partial F_0}{\partial z}}=2z{s}_1+3z^2$.

Suppose that $[x_1:x_2:x_3:x_4:z]$ is a singular point. Then every $x_i$ is a root of the same quadratic polynomial

$$P\left(u\right):=u^2-s_{1}u+(s_2+z^2).$$

Hence the four numbers $x_1,\dots ,x_4$ assume at most two distinct values.

We first consider $z=0$. If all four coordinates are equal to a number c, then $s_1=4c$ and $s_2=6c^2$, and the equation $c^2-s_{1}c+s_2=0$ becomes $3c^2=0$. Thus $c=0$, which does not define a point of projective space.

Assume therefore that two distinct values $\alpha$ and $\beta$ occur with multiplicities m and $4-m$, respectively. Since they are the two roots of $P\left(u\right)$, Vieta’s relation gives $\alpha +\beta =s_1$. On the other hand, $s_1=m\alpha +(4-m)\beta$. Hence

$$(m-1)\alpha +(3-m)\beta =0.$$

If $m=1$, then $\beta =0$, so exactly one of the four coordinates is nonzero. Up to projective rescaling this gives one of $p_1,\dots ,p_4$. The case $m=3$ is identical.

If $m=2$, then $\alpha +\beta =0$, so $\beta =-\alpha$. In this case $s_2={\alpha }^2+{\beta }^2+4\alpha \beta =-2{\alpha }^2$, whereas Vieta gives $s_2=\alpha \beta =-{\alpha }^2$. Thus $\alpha =0$, again impossible in projective space. Therefore the only singular points with $z=0$ are $p_1,\dots ,p_4$.

Now suppose $z\ne 0$. Rescale projectively so that $z=1$. The equation ${\displaystyle \frac{\partial F_0}{\partial z}}=0$ gives $2s_1+3=0$, hence $s_1=-{\displaystyle \frac{3}{2}}$. Again the $x_i$ assume at most two values.

Suppose first that two distinct roots $\alpha ,\beta$ occur with multiplicities m and $4-m$. The same comparison of the two expressions for $s_1$ gives

$$(m-1)\alpha +(3-m)\beta =0.$$

For $m=1$, one obtains $\beta =0$. Then $s_2=0$, whereas the constant term of P is $s_2+1=1$, so Vieta would require $\alpha \beta =1$, contradicting $\beta =0$. The case $m=3$ is identical. For $m=2$, one obtains $\alpha +\beta =0$, which contradicts $s_1=-3/2$.

It remains to consider the case in which all four $x_i$ are equal to c. Then $4c=-{\displaystyle \frac{3}{2}}$, so $c=-{\displaystyle \frac{3}{8}}$. But

$${\displaystyle \frac{\partial F_0}{\partial x_i}}=3c^2+1={\displaystyle \frac{91}{64}}\ne 0.$$

Thus there are no singular points with $z\ne 0$.

We have therefore proved

$$Sing\left(X_0\right)=\{p_1,p_2,p_3,p_4\}.$$

It remains to verify that these singularities are ordinary double points. By $S_4$-symmetry it suffices to work at $p_1$. On the affine chart $x_1=1$, set $u=x_2$, $v=x_3$, $w=x_4$, and $z=x_5$. The local defining function is

$$\begin{array}{cc}\hfill f_0(u,v,w,z)& =uv+uw+vw+uvw\hfill \\ & +z^2(1+u+v+w)+z^3.\hfill \end{array}$$

Its quadratic part is $q(u,v,w,z)=uv+uw+vw+z^2$. The Hessian matrix of q is

$$Hess\left(q\right)=\left(\begin{array}{cccc}0& 1& 1& 0\\ 1& 0& 1& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 2\end{array}\right).$$

The upper-left $3\times 3$ block has eigenvalues $2,-1,-1$, and hence $detHess\left(q\right)=4\ne 0$. Thus q is nondegenerate. By the holomorphic Morse lemma, the germ $(X_0,p_1)$ is an ordinary double point. The same conclusion holds at the remaining three nodes by the transitive $S_4$-action. □

We next embed $X_0$ into an explicit one-parameter $S_4$-equivariant smoothing. Let $F_{\mathrm{sm}}:=x_1^3+x_2^3+x_3^3+x_4^3+x_5^3$. The Fermat cubic $V\left(F_{\mathrm{sm}}\right)\subset {\mathbf{P}}^4$ is smooth and $S_4$-invariant.

Consider the family

$$\mathcal{X}:=\left\{(\left[x\right],t)\in {\mathbf{P}}^4\times \Delta |F_0\left(x\right)+t{F}_{\mathrm{sm}}\left(x\right)=0\right\},$$

with projection $\pi :\mathcal{X}\to \Delta$.

Theorem 7 

(An explicit projective $S_4$-equivariant conifold degeneration). After shrinking the disk Δ around 0, the morphism $\pi :\mathcal{X}\to \Delta$ is projective and flat, its general fiber is a smooth projective cubic threefold, and its central fiber is the four-nodal cubic of Proposition 19. The permutation action of $S_4$ on $x_1,\dots ,x_4$, with $x_5$ and t fixed, defines an algebraic fiberwise $S_4$-action on $\mathcal{X}$. Moreover, after shrinking Δ, the total space $\mathcal{X}$ is smooth.

Proof. 

Both $F_0$ and $F_{\mathrm{sm}}$ are invariant under permutation of $x_1,\dots ,x_4$. Hence $F_0\left(gx\right)+t{F}_{\mathrm{sm}}\left(gx\right)=F_0\left(x\right)+t{F}_{\mathrm{sm}}\left(x\right)$ for every $g\in S_4$. Thus $S_4$ preserves $\mathcal{X}$, and since t is fixed the action is fiberwise.

The family is projective because $\mathcal{X}$ is a closed hypersurface of ${\mathbf{P}}^4\times \Delta$, which is projective over $\Delta$.

We next verify flatness. It is enough to work locally on a standard affine chart of ${\mathbf{P}}^4$. Before imposing the hypersurface equation, the coordinate ring has the form $A=\mathbb{C}[t,y_1,y_2,y_3,y_4]$. The family is locally defined by one equation $f_0+t{f}_{\mathrm{sm}}$, so its coordinate ring is $B:=A/(f_0+t{f}_{\mathrm{sm}})$. We show that B is torsion-free over $\mathbb{C}\left[t\right]$.

Fix $c\in \mathbb{C}$. Suppose that $(t-c)b=0$ in B. Choose a representative $b\in A$. Then for some $r\in A$,

$$(t-c)b=(f_0+t{f}_{\mathrm{sm}})r.$$

Reducing modulo $t-c$ gives $0=(f_0+c{f}_{\mathrm{sm}})\overline{r}$ in the polynomial ring $\mathbb{C}[y_1,y_2,y_3,y_4]$, which is an integral domain. The polynomial $f_0+c{f}_{\mathrm{sm}}$ is nonzero because $F_0$ and $F_{\mathrm{sm}}$ are linearly independent cubic forms. Hence $\overline{r}=0$, so $r=(t-c)r_1$ for some $r_1\in A$. Substitution gives

$$(t-c)b=(t-c)(f_0+t{f}_{\mathrm{sm}})r_1.$$

Since A is a domain, $b=(f_0+t{f}_{\mathrm{sm}})r_1$. Thus $b=0$ in B. Hence multiplication by every $t-c$ is injective on B. Since every nonzero polynomial in $\mathbb{C}\left[t\right]$ factors into linear factors, B is torsion-free over $\mathbb{C}\left[t\right]$. Because $\mathbb{C}\left[t\right]$ is a principal ideal domain, B is flat over $\mathbb{C}\left[t\right]$. Restriction to the analytic disk preserves flatness.

It remains to show that all sufficiently small noncentral fibers are smooth. Consider the projective pencil

$$\left\{[s{F}_0+t{F}_{\mathrm{sm}}]|[s:t]\in {\mathbf{P}}^1\right\}$$

in the projective space of cubic forms on ${\mathbf{P}}^4$. The locus of singular cubic forms is a proper Zariski-closed discriminant subset. The pencil is not contained in that discriminant because $F_{\mathrm{sm}}=0$ is the smooth Fermat cubic. Therefore the intersection of the pencil with the discriminant is a finite subset of ${\mathbf{P}}^1$. One singular parameter is the central value corresponding to $F_0$. Shrinking $\Delta$ around 0, we may exclude every other singular member. Hence $X_t:={\pi }^{-1}\left(t\right)$ is smooth for every $t\in {\Delta }^{\times }$.

Finally, consider the total space. At a point lying over $t\ne 0$, the fiber $X_t$ is smooth, so the differential in the projective-coordinate directions is already nonzero. The same is true at every smooth point of the central fiber.

It therefore remains only to inspect the four points $(p_i,0)$. The total-space equation is $F_0\left(x\right)+t{F}_{\mathrm{sm}}\left(x\right)=0$, and

$${\displaystyle \frac{\partial }{\partial t}}\left(F_0+t{F}_{\mathrm{sm}}\right)=F_{\mathrm{sm}}.$$

At every coordinate node $p_i$, $F_{\mathrm{sm}}\left(p_i\right)=1$. Hence

$${\displaystyle \frac{\partial }{\partial t}}\left(F_0+t{F}_{\mathrm{sm}}\right)(p_i,0)=1\ne 0.$$

Thus the total space is smooth at the four points lying over the nodes. After the preceding shrinking of $\Delta$, the total space $\mathcal{X}$ is smooth.

The geometric hypotheses of Definition 1 are therefore satisfied. The corrected finite-node mixed-Hodge-module carrier used throughout the paper is the one associated with such a finite-node conifold degeneration in [8]. □

The preceding proof shows more than the existence of a smoothing: the deformation direction is nonzero at each node. We record the precise local form because it is needed to compute the stabilizer representation.

At $p_1$, use the coordinates $u=x_2$, $v=x_3$, $w=x_4$, and $z=x_5$ on the chart $x_1=1$. The family has local equation $f_0(u,v,w,z)+th(u,v,w,z)=0$, where

$$\begin{array}{cc}\hfill f_0& =uv+uw+vw+uvw\hfill \\ & +z^2(1+u+v+w)+z^3,\hfill \end{array}$$

and $h=1+u^3+v^3+w^3+z^3$. Since $h\left(0\right)=1$, the function h is a holomorphic unit near the origin. The total space is locally the graph $t=-f_0/h$. Thus the degeneration map is represented locally by the holomorphic function $\tau (u,v,w,z):=-f_0(u,v,w,z)/h(u,v,w,z)$. Its quadratic part is $-q$, where $q=uv+uw+vw+z^2$.

The stabilizer of $p_1$ is $G_{p_1}=\{\sigma \in S_4\mid \sigma \left(1\right)=1\}\simeq S_3$. It acts by permutation of $u,v,w$ and fixes z. Both $f_0$ and h, and hence $\tau$, are $S_3$-invariant.

For the local character computation, we use an equivariant form of the holomorphic Morse lemma.

Lemma 9 

(Finite-group equivariant holomorphic Morse lemma). Let Γ be a finite group acting linearly on a complex vector space E, and let $\phi :(E,0)\to (\mathbb{C},0)$ be a Γ-invariant holomorphic germ with a nondegenerate critical point at 0. Let $q_{\phi }$ denote the quadratic part of φ. Then there is a Γ-equivariant biholomorphic germ $\Phi :(E,0)\stackrel{\sim }{\to }(E,0)$ such that $\phi \circ \Phi =q_{\phi }$.

Proof. 

Write $\phi =q_{\phi }+r$, with $r\in {\mathfrak{m}}^3$, where $\mathfrak{m}$ is the maximal ideal at 0. For $s\in [0,1]$, define ${\phi }_s:=q_{\phi }+sr$. Every ${\phi }_s$ has the same nondegenerate Hessian at the origin.

We construct a time-dependent holomorphic vector field $X_s$ satisfying $d{\phi }_s\left(X_s\right)=-r$. Since $d{\phi }_s\left(0\right)=0$, there is a holomorphic matrix $B_s\left(z\right)$ such that $\nabla {\phi }_s\left(z\right)=B_s\left(z\right)z$. The matrix $B_s\left(0\right)$ is the Hessian of $q_{\phi }$, and is therefore invertible. After restricting to a sufficiently small neighborhood, $B_s\left(z\right)$ is invertible uniformly for $s\in [0,1]$.

Because $r\in {\mathfrak{m}}^3$, we may write $r\left(z\right)={\sum }_j{z}_j{c}_j\left(z\right)$, where each $c_j\in {\mathfrak{m}}^2$. Let $c\left(z\right)$ be the corresponding column vector and define

$$X_s\left(z\right):=-{\left(B_s{\left(z\right)}^T\right)}^{-1}c\left(z\right).$$

Then $d{\phi }_s\left(X_s\right)=-r$.

The vector field $X_s$ need not be $\Gamma$-equivariant. Average it:

$${\overline{X}}_s\left(z\right):={\displaystyle \frac{1}{|\Gamma |}}\sum _{\gamma \in \Gamma }{\gamma }^{-1}{X}_s\left(\gamma z\right).$$

Because both ${\phi }_s$ and r are $\Gamma$-invariant, $d{\phi }_s\left({\overline{X}}_s\right)=-r$. Moreover, ${\overline{X}}_s\left(\gamma z\right)=\gamma {\overline{X}}_s\left(z\right)$, so ${\overline{X}}_s$ is $\Gamma$-equivariant.

Let ${\Phi }_s$ be the local flow determined by

$${\displaystyle \frac{d}{ds}}{\Phi }_s={\overline{X}}_s\circ {\Phi }_s,{\Phi }_0=id.$$

Since ${\overline{X}}_s$ vanishes to order at least two at the origin, after shrinking the neighborhood the flow exists for $0\le s\le 1$ and gives biholomorphic germs. The flow is $\Gamma$-equivariant.

Now

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{ds}}\left({\phi }_s\left({\Phi }_s\left(z\right)\right)\right)& ={\displaystyle \frac{\partial {\phi }_s}{\partial s}}\left({\Phi }_s\left(z\right)\right)+d{\phi }_s\left({\overline{X}}_s\right)\left({\Phi }_s\left(z\right)\right)\hfill \\ & =r\left({\Phi }_s\left(z\right)\right)-r\left({\Phi }_s\left(z\right)\right)\hfill \\ & =0.\hfill \end{array}$$

Therefore ${\phi }_1\left({\Phi }_1\left(z\right)\right)={\phi }_0\left(z\right)=q_{\phi }\left(z\right)$. Since ${\phi }_1=\phi$, the map $\Phi :={\Phi }_1$ has the required properties. □

We now apply the lemma to the local degeneration at $p_1$.

Proposition 20 

(The $S_3$-stabilizer acts by the sign character). For the projective degeneration of Theorem 7, the stabilizer $G_{p_1}\simeq S_3$ acts on the one-dimensional rational middle vanishing space at $p_1$ by the sign representation: $V_{p_1}\simeq {sgn}_{S_3}$. By conjugacy, the same statement holds at every node.

Proof. 

The local degeneration map is the $S_3$-invariant holomorphic germ $\tau =-f_0/h$, whose quadratic part is $-q=-(uv+uw+vw+z^2)$. Multiplication of the defining function by $-1$ does not affect the stabilizer representation on the vanishing line, so we work with $q=uv+uw+vw+z^2$.

By Lemma 9, the germ is $S_3$-equivariantly analytically equivalent to its nondegenerate quadratic part. It therefore suffices to compute the action on the vanishing homology of q.

Consider the real permutation representation of $S_3$ on the coordinates $(u,v,w)$. Decompose

$${\mathbb{R}}^3=\ell \oplus W,$$

where $\ell =\mathbb{R}(1,1,1)$ is the trivial representation and $W=\{(u,v,w)\in {\mathbb{R}}^3\mid u+v+w=0\}$ is the two-dimensional real standard representation.

Let $s:=(u+v+w)/\sqrt{3}$ be the normalized coordinate on ℓ, and choose an orthonormal basis of W, with coordinates $r_1,r_2$. Since

$$uv+uw+vw={\displaystyle \frac{{(u+v+w)}^2-(u^2+v^2+w^2)}{2}},$$

one obtains $uv+uw+vw=s^2-{\displaystyle \frac{1}{2}}(r_1^2+r_2^2)$. Hence $q=s^2-{\displaystyle \frac{1}{2}}{r}_1^2-{\displaystyle \frac{1}{2}}{r}_2^2+z^2$.

Make the complex-linear change of variables

$$y_1=s,y_2={\displaystyle \frac{i}{\sqrt{2}}}{r}_1,y_3={\displaystyle \frac{i}{\sqrt{2}}}{r}_2,y_4=z.$$

Then $q=y_1^2+y_2^2+y_3^2+y_4^2$. Because the same scalar is applied to both coordinates of W, this change commutes with the $S_3$-action. In these coordinates $S_3$ acts through real orthogonal matrices: it acts trivially on $y_1$ and $y_4$, and through the real two-dimensional standard representation on $(y_2,y_3)$.

For $ϵ>0$, consider the standard quadratic Milnor fiber

$$Q_{ϵ}:=\left\{y\in {\mathbb{C}}^4|y_1^2+y_2^2+y_3^2+y_4^2=ϵ\right\}.$$

Write $y=x+i\xi$, with $x,\xi \in {\mathbb{R}}^4$. The defining equation becomes

$${\left|x\right|}^2-{\left|\xi \right|}^2=ϵ,x·\xi =0.$$

Consequently $x\ne 0$. Writing $u_0:=x/\left|x\right|\in S^3$, one has $\xi \in T_{u_0}{S}^3$ and $\left|x\right|=\sqrt{ϵ+{\left|\xi \right|}^2}$. Thus

$$T{S}^3⟶Q_{ϵ},(u_0,\xi )⟼\sqrt{ϵ+{\left|\xi \right|}^2}{u}_0+i\xi$$

is an $S_3$-equivariant diffeomorphism. The zero section corresponds to the real vanishing sphere

$$S_{ϵ}^3=\left\{y\in {\mathbb{R}}^4|y_1^2+y_2^2+y_3^2+y_4^2=ϵ\right\}.$$

Hence $H_3(Q_{ϵ},\mathbb{Q})\simeq H_3(S^3,\mathbb{Q})\simeq \mathbb{Q}$, in agreement with the standard ordinary-double-point Milnor-fiber calculation [5,6].

For a real orthogonal transformation $A:{\mathbb{R}}^4\to {\mathbb{R}}^4$, the induced self-map of $S^3$ has degree $deg\left(A{|}_{S^3}\right)=det\left(A\right)$. Therefore the action on the vanishing generator is multiplication by $det\left(A\right)$.

In the present case the tangent representation of $S_3$ is the direct sum of the three-dimensional permutation representation on $(u,v,w)$ and the trivial representation on z. Its determinant is therefore the sign character:

$$det\left({\sigma |}_{T_{p_1}}\right)=sgn\left(\sigma \right).$$

A transposition acts by $-1$ on the vanishing generator, while a three-cycle acts by $+1$. Hence $V_{p_1}\simeq {sgn}_{S_3}$.

The local representations at the other nodes are obtained by conjugation, so Lemma 1 gives the same stabilizer character at each point of the $S_4$-orbit. □

The actual geometric $S_4$-cubic therefore differs from the untwisted permutation benchmark of Proposition 18.

Theorem 8 

(Flexible representation of the projective $S_4$-cubic smoothing). For the projective $S_4$-equivariant conifold degeneration of Theorem 7,

$$V_{\mathrm{van}}\simeq {Ind}_{S_3}^{S_4}{sgn}_{S_3}.$$

Moreover,

$$V_{\mathrm{van}}\simeq {sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right)$$

as rational $S_4$-representations.

Proof. 

The four nodes form one $S_4$-orbit, and the stabilizer of $p_1$ is $S_3$. By Proposition 20, $V_{p_1}\simeq {sgn}_{S_3}$. The orbit–stabilizer formula of Theorem 2 therefore gives

$$V_{\mathrm{van}}\simeq {Ind}_{S_3}^{S_4}{sgn}_{S_3}.$$

Let ${sgn}_{S_4}$ denote the sign representation of $S_4$. Its restriction to the stabilizer $S_3$ is ${Res}_{S_3}^{S_4}{sgn}_{S_4}={sgn}_{S_3}$. The tensor identity for induction gives

$$\begin{array}{cc}\hfill {Ind}_{S_3}^{S_4}{sgn}_{S_3}& \simeq {Ind}_{S_3}^{S_4}\left({Res}_{S_3}^{S_4}{sgn}_{S_4}\right)\hfill \\ & \simeq {sgn}_{S_4}\otimes {Ind}_{S_3}^{S_4}\mathbf{1}\hfill \\ & \simeq {sgn}_{S_4}\otimes \mathbb{Q}[S_4/S_3].\hfill \end{array}$$

By Proposition 18, $\mathbb{Q}[S_4/S_3]\simeq \mathbf{1}\oplus V_{\mathrm{std}}$. Therefore

$$\begin{array}{cc}\hfill V_{\mathrm{van}}& \simeq {sgn}_{S_4}\otimes \left(\mathbf{1}\oplus V_{\mathrm{std}}\right)\hfill \\ & \simeq {sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right).\hfill \end{array}$$

The two summands are irreducible rational $S_4$-representations of dimensions 1 and 3, respectively [17]. □

Corollary 10 

(The explicit $S_4$-cubic has no invariant vanishing direction). For the preceding projective degeneration, $V_{\mathrm{van}}^{S_4}=0$. In particular, the flexible representation is not isomorphic to the permutation representation $\mathbb{Q}[S_4/S_3]\simeq \mathbf{1}\oplus V_{\mathrm{std}}$.

Proof. 

By Theorem 8,

$$V_{\mathrm{van}}\simeq {sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right).$$

Neither summand is trivial, so no $S_4$-invariant vector occurs.

Equivalently, Frobenius reciprocity gives

$$\begin{array}{cc}\hfill {Hom}_{S_4}\left(\mathbf{1},{Ind}_{S_3}^{S_4}{sgn}_{S_3}\right)& \simeq {Hom}_{S_3}\left(\mathbf{1},{sgn}_{S_3}\right)\hfill \\ & =0.\hfill \end{array}$$

By contrast, $\mathbb{Q}[S_4/S_3]={Ind}_{S_3}^{S_4}\mathbf{1}$ contains the trivial representation once. □

The rational equivariant flexible profile of this projective example is therefore

$${\mathcal{P}}_{\mathrm{flex},\mathbb{Q}}^{S_4}\left(\pi \right)=\left({\tau }_{\mathrm{van}},\left\{\left({sgn}_{S_4},1\right),\left(V_{\mathrm{std}}\otimes {sgn}_{S_4},1\right)\right\}\right).$$

Its forgetful dimension is $1+3=4=|\Sigma |$.

Thus the same four-point $S_4$-orbit $S_4/S_3$ supports two different representation-theoretic possibilities:

$$\begin{array}{cc}\hfill \mathrm{trivial}\mathrm{stabilizer}\mathrm{character}\text{:}& \mathbf{1}\oplus V_{\mathrm{std}},\hfill \\ \hfill \mathrm{geometric}\mathrm{cubic}\mathrm{stabilizer}\mathrm{character}\text{:}& {sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right).\hfill \end{array}$$

The distinction is entirely due to the local $S_3$-module $V_p$. It is therefore invisible both to the ordinary node count and to the underlying $S_4$-set of nodes.

The smooth fibers also lie naturally in the class of generically free finite-group actions used in smooth equivariant birational geometry.

Proposition 21 

(Generic freeness on the smooth fibers). For every sufficiently small $t\ne 0$, the $S_4$-action on the smooth cubic threefold $X_t$ is generically free.

Proof. 

Fix a nonidentity element $g\in S_4$. The projective fixed locus ${Fix}_{{\mathbf{P}}^4}\left(g\right)$ is the union of the projectivizations of the eigenspaces of the corresponding linear permutation operator. Since $g\ne 1$, this is a proper closed subset of ${\mathbf{P}}^4$, consisting of finitely many proper projective linear subspaces.

The smooth cubic $X_t\subset {\mathbf{P}}^4$ is an irreducible hypersurface and is not contained in any proper projective linear subspace. Hence $X_t^g=X_t\cap {Fix}_{{\mathbf{P}}^4}\left(g\right)$ is a proper closed subset of $X_t$.

Since $S_4$ is finite, ${\bigcup }_{1\ne g\in S_4}{X}_t^g$ is a proper closed subset of $X_t$. Its complement consists precisely of points with trivial stabilizer. Therefore the $S_4$-action is generically free. □

Remark 32 

(Geometric scope of the $S_4$ calculation). Unlike the untwisted permutation calculation at the beginning of this subsection, the sign-twisted profile is realized by an explicit projective one-parameter degeneration. The central fiber is a four-nodal cubic threefold with transitive $S_4$-action, and the nearby fibers are smooth projective cubic threefolds with generically free $S_4$-action.

Cavenaghi–Katzarkov–Kontsevich prove that a singular cubic threefold with four ordinary double points exchanged by $S_4$ is not birationally equivalent to ${\mathbf{P}}^3$ equipped with an $S_4$-linearizable action [4]. The central fiber constructed above satisfies the geometric hypotheses of that result. Consequently, the same four-node $S_4$-configuration appears both in the smooth-equivariant-atom birational setting and in the degeneration-side calculation developed here.

The present calculation adds information not visible from the transitive four-point action alone: the local node stabilizer is $S_3$, its action on the local vanishing line is the sign character, and hence the degeneration carries the twisted induced representation

$${Ind}_{S_3}^{S_4}{sgn}_{S_3}.$$

Thus this example gives a projective geometric realization of the principle that the intrinsic local equivariant datum is the pair $(G_p,V_p)$, rather than the orbit $G/G_p$ alone.

7.4. A Fixed Node with Nontrivial Stabilizer Action

The preceding $S_4$-cubic gives a global projective realization of a nontrivial stabilizer character. The following local model isolates the same mechanism in its simplest possible form: a node is fixed, but its one-dimensional vanishing space transforms nontrivially.

Consider the local ordinary-double-point smoothing

$$\mathcal{U}:=\left\{(z_1,z_2,z_3,z_4,t)\in {\mathbf{C}}^4\times \Delta |z_1^2+z_2^2+z_3^2+z_4^2=t\right\},$$

with projection ${\pi }_{\mathrm{loc}}:\mathcal{U}\to \Delta$. Let $H=\mathbb{Z}/2=\langle \sigma \rangle$ act by

$$\sigma (z_1,z_2,z_3,z_4,t)=(-z_1,z_2,z_3,z_4,t).$$

The unique singular point of the central fiber is $p=0$, and $H_p=H$.

For $t>0$ real and sufficiently small, the real vanishing sphere is

$$S_t^3:=\left\{(x_1,x_2,x_3,x_4)\in {\mathbf{R}}^4|x_1^2+x_2^2+x_3^2+x_4^2=t\right\},$$

and the involution restricts to reflection in the first coordinate.

Proposition 22 

(Nontrivial stabilizer action on a local vanishing line). For the local smoothing ${\pi }_{\mathrm{loc}}$ above, the induced H-action on the one-dimensional middle vanishing homology, and hence on the corresponding rational vanishing line, is the sign representation.

Proof. 

The Milnor fiber of the ordinary double point deformation retracts onto $S_t^3$ [5,6]. The involution

$$(x_1,x_2,x_3,x_4)⟼(-x_1,x_2,x_3,x_4)$$

is induced by a linear transformation of ${\mathbf{R}}^4$ of determinant $-1$. Its restriction to the oriented sphere has degree $-1$, and therefore acts by multiplication by $-1$ on $H_3(S_t^3,\mathbb{Q})\simeq \mathbb{Q}$. Hence the local vanishing representation is sgn. □

The local datum is therefore $(H_p,V_p)=(\mathbb{Z}/2,sgn)$, whereas the same fixed node with trivial local action would give $(\mathbb{Z}/2,\mathbf{1})$. Both have the same orbit cardinality, namely one. Their difference lies entirely in the stabilizer action on the local vanishing line.

More generally, suppose $H\le G$ occurs as the stabilizer of a node and $V_p$ is a one-dimensional nontrivial rational H-module. The orbitwise contribution is ${Ind}_H^G{V}_p$, whereas the untwisted contribution is $\mathbb{Q}[G/H]$.

Proposition 23 

(Twisted versus untwisted orbit contribution). Let $H\le G$ and let χ be a nontrivial one-dimensional rational representation of H. If

$${Ind}_H^G\chi \not\simeq {Ind}_H^G\mathbf{1},$$

then two node orbits with the same underlying G-set $G/H$, but with local stabilizer representations χ and $\mathbf{1}$, have distinct equivariant flexible representations.

Proof. 

By Theorem 2, the two local data contribute ${Ind}_H^G\chi$ and ${Ind}_H^G\mathbf{1}=\mathbb{Q}[G/H]$, respectively. By hypothesis these G-modules are nonisomorphic, although both have dimension $[G:H]$. Hence the orbit cardinality and underlying G-set do not determine the equivariant flexible contribution. □

The local reflection example provides the elementary model for the determinant mechanism used in Proposition 20: an orientation-reversing linear symmetry of the real vanishing sphere acts by $-1$ on its fundamental class.

Remark 33 

(Local model versus global realization). Proposition 22 is a local analytic ordinary-double-point calculation and is retained because it isolates the stabilizer-character mechanism without the additional geometry of a projective family. The $S_4$-cubic of Section 7.3 shows that nontrivial stabilizer actions on vanishing lines are not merely local possibilities: they occur in an explicit projective equivariant conifold degeneration.

7.5. Separation of Equal-Node-Count Profiles

The preceding calculations show two different sources of refinement over the ordinary node count. Orbit structure can change the representation even when local stabilizer actions are trivial, and the projective $S_4$-cubic shows that the stabilizer representation can further change the answer even after the node G-set has been fixed.

We finish with the simplest model separation at fixed node count.

Let $G=\mathbb{Z}/2=\langle \sigma \rangle$. Consider two G-sets of cardinality two.

In the first model, ${\Sigma }_A=\{p_1,p_2\}$ is one free orbit. By Proposition 16, $V_{\mathrm{van}}^A\simeq \mathbf{1}\oplus sgn$.

In the second model, ${\Sigma }_B=\{q_1,q_2\}$ consists of two fixed nodes, with trivial stabilizer action at both nodes. Each orbit therefore contributes $\mathbf{1}$, so $V_{\mathrm{van}}^B\simeq {\mathbf{1}}^{\oplus 2}$.

Proposition 24 

(Equal-node-count separation at the representation level). The two model configurations satisfy $|{\Sigma }_A|=|{\Sigma }_B|=2$, but $V_{\mathrm{van}}^A\not\simeq V_{\mathrm{van}}^B$ as rational $\mathbb{Z}/2$-representations.

Proof. 

Both representations have dimension two, but the first contains sgn with multiplicity one whereas the second contains none. Equivalently,

$$dim{\left(V_{\mathrm{van}}^A\right)}^G=1,dim{\left(V_{\mathrm{van}}^B\right)}^G=2.$$

Since invariant-space dimension is preserved by G-module isomorphism, the two representations are not isomorphic. □

Thus the forgetful profiles agree at the level of the flexible rank,

$${For}_G\left({\mathcal{P}}_{\mathrm{flex}}^G\left(A\right)\right)={For}_G\left({\mathcal{P}}_{\mathrm{flex}}^G\left(B\right)\right),$$

while ${\mathcal{P}}_{\mathrm{flex}}^G\left(A\right)\ne {\mathcal{P}}_{\mathrm{flex}}^G\left(B\right)$.

Remark 34 

(Geometric realization of equal-count separation). Proposition 24 remains a representation-theoretic comparison between two admissible local orbit–stabilizer configurations; no pair of projective $\mathbb{Z}/2$-equivariant degenerations realizing these two particular configurations is asserted here.

The geometric status of the stabilizer refinement is nevertheless stronger than in the model-only discussion: the projective $S_4$-degeneration of Section 7.3 realizes globally a nontrivial local stabilizer character and shows that the representation carried by the vanishing sector need not be the permutation representation determined by the node G-set.

Remark 35 

(What the examples establish). The examples separate three levels of information.

The ordinary profile retains only the total dimension $|\Sigma |$. The orbit decomposition additionally retains the G-set structure of Σ. The full flexible profile retains, beyond both of these, the stabilizer representations $V_p$ through

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p.$$

The two-node involution example shows how permutation of nodes produces symmetric and antisymmetric directions. The cyclic example shows why rational and splitting-field decompositions must be distinguished. The untwisted four-point $S_4$-model gives $\mathbf{1}\oplus V_{\mathrm{std}}$, whereas the explicit projective four-nodal $S_4$-cubic has the same underlying $S_4/S_3$ node orbit but local stabilizer character ${sgn}_{S_3}$, and therefore carries

$${sgn}_{S_4}\oplus \left(V_{\mathrm{std}}\otimes {sgn}_{S_4}\right).$$

This is a global geometric realization of information that is invisible in the node G-set itself. The fixed-node reflection gives the elementary local model for the same determinant mechanism. Finally, Proposition 24 shows explicitly that the ordinary flexible count can fail even to determine the orbit-level representation.

Thus the sequence of examples realizes the strict hierarchy

$$\mathit{node}\mathit{count}⟶\mathit{node}G\text{-}\mathit{set}⟶\mathit{stabilizer}\text{-}\mathit{decorated}\mathit{vanishing}\mathit{representation},$$

with a loss of information at each forgetful step.

8. Basic Compatibility with Monodromy and Global Relations

We use the notation and conventions of Section 1.6. The purpose of this section is to record the first structures beyond the formal flexible representation that are forced by equivariance. We show not only that the finite-group action commutes with monodromy, but also that it acts by automorphisms of the limiting mixed Hodge structure and that the rational G-isotypic components are sub-mixed-Hodge structures preserved by the nilpotent monodromy operator.

The equivariant extension class and the global relation layer remain separate. No isotypic decomposition of the equivariant extension class, Picard–Lefschetz pairing, or Stokes data is asserted here.

8.1. Commutation with Monodromy

Fix $t_0\in {\Delta }^{\times }$, and let ${\mathbb{H}}^k:=R^k{\pi }_{*}\mathbb{Q}{|}_{{\Delta }^{\times }}$ be the rational cohomology local system of the smooth fibers in degree k. Since $\pi$ is smooth and projective over ${\Delta }^{\times }$, ${\mathbb{H}}^k$ underlies a polarizable rational variation of Hodge structure of weight k [23].

The fiberwise G-action induces, for every $g\in G$, an automorphism $g_{\mathbb{H}}:{\mathbb{H}}^k\stackrel{\sim }{\to }{\mathbb{H}}^k$ of this variation. In particular, $g_{\mathbb{H}}$ is flat and its restriction to every fiber is a morphism of pure Hodge structures. On the fiber over $t_0$, we write simply $g:H^k(X_{t_0},\mathbb{Q})\to H^k(X_{t_0},\mathbb{Q})$.

Let $\gamma \in {\pi }_1({\Delta }^{\times },t_0)$ be the positively oriented generator, and let $T:H^k(X_{t_0},\mathbb{Q})\stackrel{\sim }{\to }{H}^k(X_{t_0},\mathbb{Q})$ denote geometric monodromy along $\gamma$.

Proposition 25 

(Basic monodromy equivariance). For every $g\in G$, $gT=Tg$ on $H^k(X_{t_0},\mathbb{Q})$.

After a finite base change for which the monodromy is unipotent, let $T_u$ denote the unipotent monodromy and set $N:=log{T}_u$. Then $gN=Ng$ for every $g\in G$.

Proof. 

Because the action is fiberwise, every $g\in G$ induces an automorphism of ${\mathcal{X}|}_{{\Delta }^{\times }}$ over the identity of ${\Delta }^{\times }$, and hence an automorphism of the local system ${\mathbb{H}}^k$. Naturality of parallel transport gives $g\circ P_{\gamma }=P_{\gamma }\circ g$. Since $P_{\gamma }=T$, one obtains $gT=Tg$.

The monodromy theorem gives quasi-unipotence of T. After finite base change one may work with its unipotent part $T_u$ [23,24]. Since g commutes with T, it commutes with the semisimple and unipotent factors of the Jordan decomposition and therefore with $T_u$.

Because $T_u-id$ is nilpotent,

$$N=log{T}_u=\sum _{j\ge 1}{\displaystyle \frac{{(-1)}^{j+1}}{j}}{(T_u-id)}^j$$

is a finite polynomial in $T_u-id$. Hence g commutes with N. □

Corollary 11 

(Stability of monodromy kernels and images). For every integer $j\ge 1$, the subspaces $ker\left(N^j\right)$ and $im\left(N^j\right)$ are G-stable.

Proof. 

By Proposition 25, $N^{j}g=g{N}^j$. Hence $N^j\left(gv\right)=0$ whenever $N^j\left(v\right)=0$, proving stability of $ker\left(N^j\right)$. Likewise, if $v=N^j\left(w\right)$, then $gv=N^j\left(gw\right)$, so $im\left(N^j\right)$ is G-stable. □

We now pass from monodromy equivariance to the limiting mixed Hodge structure. After the finite base change above, fix the usual identification of the fibers of the pulled-back local system on the universal cover of ${\Delta }^{\times }$ with a rational vector space $H_{\lim }^k$. Let $W_{•}\left(N\right)$ denote the monodromy weight filtration centered at k, and let $F_{\infty }^{•}$ denote the limiting Hodge filtration. The classical limiting Hodge theory of Schmid and Steenbrink gives a mixed Hodge structure

$$\left(H_{\lim }^k,W_{•}\left(N\right),F_{\infty }^{•}\right),$$

and $N:H_{\lim }^k\to H_{\lim }^k$ is a morphism of this limiting mixed Hodge structure of type $(-1,-1)$ [23,24].

Theorem 9 

(Equivariant limiting mixed Hodge structure). For every $g\in G$, the induced automorphism of $H_{\lim }^k$ preserves both the monodromy weight filtration and the limiting Hodge filtration: $g{W}_{\ell }\left(N\right)=W_{\ell }\left(N\right)$ for every ℓ, and $g{F}_{\infty }^p=F_{\infty }^p$ for every p. Consequently,

$$g:\left(H_{\lim }^k,W_{•}\left(N\right),F_{\infty }^{•}\right)\stackrel{\sim }{\to }\left(H_{\lim }^k,W_{•}\left(N\right),F_{\infty }^{•}\right)$$

is an automorphism of the limiting mixed Hodge structure. Moreover, N is G-equivariant and is a morphism of the limiting mixed Hodge structure of type $(-1,-1)$.

Proof. 

The G-action on ${\mathcal{X}|}_{{\Delta }^{\times }}$ induces an action by automorphisms of the polarizable variation of Hodge structure ${\mathbb{H}}^k$. Thus every $g\in G$ is flat and preserves the Hodge filtration on every smooth fiber.

By Proposition 25, $gN=Ng$. The monodromy weight filtration $W_{•}\left(N\right)$ is uniquely characterized by the nilpotent operator N, the center k, and the isomorphisms induced by powers of N on the corresponding graded pieces. Applying g to this filtration produces another filtration with the same defining properties because g commutes with N. Uniqueness therefore gives $g{W}_{\ell }\left(N\right)=W_{\ell }\left(N\right)$ for every ℓ.

It remains to treat the limiting Hodge filtration. Pass to the universal cover of ${\Delta }^{\times }$, choose a coordinate z, and use flat trivialization to regard the fiberwise Hodge filtration as a holomorphic family

$$F_z^{•}\subset H_{\lim }^k{\otimes }_{\mathbb{Q}}\mathbb{C}.$$

Since g acts by a morphism of the variation, $g{F}_z^p=F_z^p$ for every z. The limiting Hodge filtration is obtained from the nilpotent-orbit expression

$$F_{\infty }^p=\underset{Imz\to +\infty }{lim}exp(-zN)F_z^p$$

in the standard normalization [23]. Since $gN=Ng$,

$$\begin{array}{cc}\hfill g\left(exp(-zN)F_z^p\right)& =exp(-zN)g{F}_z^p\hfill \\ & =exp(-zN)F_z^p.\hfill \end{array}$$

Passing to the limit gives $g{F}_{\infty }^p=F_{\infty }^p$.

Thus g preserves the rational structure, the weight filtration, and the Hodge filtration, and therefore acts by an automorphism of the limiting mixed Hodge structure.

Finally, the statement that N is a morphism of type $(-1,-1)$ is part of the classical limiting mixed Hodge structure theorem [23,24], while its G-equivariance is Proposition 25. □

The preceding theorem permits a rational isotypic refinement of the limiting mixed Hodge structure itself. This is stronger than merely decomposing the formal vanishing representation.

For $\rho \in {\widehat{G}}_{\mathbb{Q}}$, let $e_{\rho }\in \mathbb{Q}\left[G\right]$ denote the primitive central idempotent corresponding to the rational irreducible representation $\rho$, and define $H_{\lim ,\rho }^k:=e_{\rho }{H}_{\lim }^k$.

Corollary 12 

(Rational isotypic limiting mixed Hodge structures). There is a canonical decomposition

$$H_{\lim }^k=\underset{\rho \in {\widehat{G}}_{\mathbb{Q}}}{⨁}{H}_{\lim ,\rho }^k$$

into rational G-isotypic components. Each $H_{\lim ,\rho }^k$ is a sub-mixed-Hodge structure of $H_{\lim }^k$, with filtrations

$$W_{\ell }{H}_{\lim ,\rho }^k=H_{\lim ,\rho }^k\cap W_{\ell }\left(N\right),$$

and

$$F^p\left(H_{\lim ,\rho }^k{\otimes }_{\mathbb{Q}}\mathbb{C}\right)=\left(H_{\lim ,\rho }^k{\otimes }_{\mathbb{Q}}\mathbb{C}\right)\cap F_{\infty }^p.$$

Moreover, $N\left(H_{\lim ,\rho }^k\right)\subseteq H_{\lim ,\rho }^k$ for every ρ. Hence the nilpotent monodromy operator decomposes as

$$N=\underset{\rho \in {\widehat{G}}_{\mathbb{Q}}}{⨁}{N}_{\rho },$$

where $N_{\rho }:=N{|}_{H_{\lim ,\rho }^k}$ is a morphism of type $(-1,-1)$ on the corresponding rational isotypic limiting mixed Hodge structure.

Proof. 

By Theorem 9, every element $g\in G$ acts by an automorphism of the limiting mixed Hodge structure. Consequently every element of the rational group algebra $\mathbb{Q}\left[G\right]$ acts by an endomorphism of the limiting mixed Hodge structure.

In particular, the primitive central idempotent $e_{\rho }$ is an idempotent endomorphism preserving both $W_{•}\left(N\right)$ and $F_{\infty }^{•}$. Its image $H_{\lim ,\rho }^k=im\left(e_{\rho }\right)$ and the complementary image $im(1-e_{\rho })$ therefore inherit the induced mixed Hodge filtrations. Thus $H_{\lim ,\rho }^k$ is a sub-mixed-Hodge structure.

Semisimplicity of $\mathbb{Q}\left[G\right]$ [17] gives

$$1=\sum _{\rho \in {\widehat{G}}_{\mathbb{Q}}}{e}_{\rho }$$

with mutually orthogonal central idempotents. Hence

$$H_{\lim }^k=\underset{\rho \in {\widehat{G}}_{\mathbb{Q}}}{⨁}{e}_{\rho }{H}_{\lim }^k.$$

Finally, Proposition 25 implies that N commutes with every element of $\mathbb{Q}\left[G\right]$, and therefore with each $e_{\rho }$. Thus $N{e}_{\rho }=e_{\rho }N$, which gives $N\left(H_{\lim ,\rho }^k\right)\subseteq H_{\lim ,\rho }^k$. The restriction $N_{\rho }$ remains of type $(-1,-1)$ because N has that type on the full limiting mixed Hodge structure. □

Remark 36 

(Scope of the monodromy statement). Theorem 9 and Corollary 12 give a canonical G-equivariant limiting mixed Hodge structure and its rational isotypic decomposition. This is a statement about the smooth-fiber variation and its limiting Hodge data.

It does not identify these isotypic pieces with an isotypic decomposition of the corrected extension class $e_{\pi }^G$, nor does it determine the Picard–Lefschetz interaction pairing or Stokes blocks. Those require additional compatibility with the global rigid–flexible gluing.

8.2. Interaction with Global Node Relations

The orbit–stabilizer formula of Theorem 2 gives the formal node-supported representation

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p.$$

This determines the G-representation carried by the direct sum of local vanishing sectors, but not the global manner in which those sectors are attached to the rigid term.

The non-equivariant finite-node theory contains a separate global relation layer: relations among the local node data can constrain the gluing of the flexible sector to $I{C}_{X_0}^H$ [10]. We therefore distinguish the formal local carrier from any quotient imposed by global relations.

Definition 17 

(Formal local flexible carrier). The formal local flexible carrier of the equivariant degeneration is

$$V_{\mathrm{loc}}^G\left(\pi \right):=V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p.$$

Note that, no global relation quotient is included in $V_{\mathrm{loc}}^G\left(\pi \right)$. Let $R\left(\pi \right)\subseteq V_{\mathrm{van}}$ denote a global relation space arising from a relation construction of the type considered in [10], whenever such a relation space is defined in the chosen realization.

Definition 18 

(G-functorial relation construction). A global relation construction is called G-functorial if $gR\left(\pi \right)=R\left(\pi \right)$ for every $g\in G$.

Proposition 26 

(Equivariance of a functorial relation space). Assume that the global relation construction is G-functorial in the sense of Definition 18. Then $R\left(\pi \right)$ is a G-subrepresentation of $V_{\mathrm{van}}$, and

$$V_{\mathrm{glob}}^G\left(\pi \right):=V_{\mathrm{van}}/R\left(\pi \right)$$

inherits a canonical G-representation.

Proof. 

The hypothesis $gR\left(\pi \right)=R\left(\pi \right)$ means exactly that $R\left(\pi \right)$ is stable under the G-action, hence is a G-subrepresentation.

Let $q:V_{\mathrm{van}}\to V_{\mathrm{van}}/R\left(\pi \right)$ be the quotient map. Define $g·q\left(v\right):=q\left(gv\right)$. If $q\left(v\right)=q\left(v^{\prime }\right)$, then $v-v^{\prime }\in R\left(\pi \right)$. Since $R\left(\pi \right)$ is G-stable, $g(v-v^{\prime })\in R\left(\pi \right)$, and hence $q\left(gv\right)=q\left(g{v}^{\prime }\right)$. Thus the action is well defined and gives the asserted quotient representation. □

Remark 37 

(Conditional nature of the global quotient). Proposition 26 is conditional on G-functoriality of the chosen relation construction. The present paper does not define a universal relation space $R\left(\pi \right)$ and does not prove a representation-valued defect formula.

Remark 38 

(Formal carrier does not imply free global gluing). The decomposition

$$V_{\mathrm{van}}\simeq \underset{\left[p\right]\in \Sigma /G}{⨁}{Ind}_{G_p}^G{V}_p$$

concerns the formal node-supported quotient. It does not imply that $P^H$ is obtained by independently extending $I{C}_{X_0}^H$ by the orbitwise summands.

The extension class

$$e_{\pi }^G\in {Ext}_{{MHM}_G\left(X_0\right)}^1\left(V_{\mathrm{van}}^H,I{C}_{X_0}^H\right)$$

may encode coupling among different orbitwise sectors. Such global gluing information is invisible in the semisimple representation $V_{\mathrm{van}}$ alone.

Likewise, the rational isotypic decomposition of the limiting mixed Hodge structure from Corollary 12 does not by itself force a corresponding decomposition of this extension class.

Remark 39 

(No defect comparison). No equality is asserted between multiplicities in $R\left(\pi \right)$ and any classical nodal defect. Likewise, no representation-valued invariant ${Def}_G(\Sigma )$ is defined here.

8.3. What Remains for the Equivariant Gluing Theory

The results above now isolate four distinct equivariant structures:

$$V_{\mathrm{van}},e_{\pi }^G,\left(H_{\lim }^k,W_{•}\left(N\right),F_{\infty }^{•}\right),R\left(\pi \right).$$

The formal flexible representation and the limiting mixed Hodge structure have been decomposed equivariantly in the present paper. The remaining problem is to understand how these decompositions interact with the global extension and relation data.

Remark 40 

(Isotypic extension problem). The existence of

$$e_{\pi }^G\in {Ext}_{{MHM}_G\left(X_0\right)}^1\left(V_{\mathrm{van}}^H,I{C}_{X_0}^H\right)$$

does not itself produce extension classes indexed by irreducible G-representations.

The rational isotypic decompositions of $V_{\mathrm{van}}$ and of $H_{\lim }^k$ now provide canonical symmetry sectors on the two representation-theoretic sides of the degeneration. A further analysis must determine whether these sectors induce useful decompositions of the equivariant extension group and how global gluing can couple them.

Remark 41 

(Representation-theoretic vanishing problem). For an irreducible rational symmetry type ρ, one may ask whether a corresponding component $e_{\pi ,\rho }^G$, once defined, is forced to vanish when no compatible ρ-type data occur in the rigid sector or in the relevant limiting Hodge-theoretic piece. No such vanishing criterion is proved here.

Remark 42 

(Equivariant limiting mixed Hodge structure). Theorem 9 and Corollary 12 show that the limiting mixed Hodge structure itself admits a canonical rational G-isotypic decomposition and that N preserves every isotypic summand.

What remains is not the existence of this decomposition, but its interaction with the corrected rigid–flexible carrier. In particular, the present paper does not identify the isotypic pieces of $H_{\lim }^k$ with components of the extension class $e_{\pi }^G$ or with a corresponding decomposition of the global relation space.

Remark 43 

(Equivariant Picard–Lefschetz and Stokes data). The G-action preserves the vanishing-cycle space on which the Picard–Lefschetz transformations act. The equivariant LMHS established above provides natural symmetry sectors in which one may study the Picard–Lefschetz interaction pairing, its possible isotypic blocks, and the associated Stokes-type data.

No such interaction-block decomposition is constructed here. It requires additional information beyond the existence of the G-isotypic limiting mixed Hodge structures.

9. Consequences and Comparison with Smooth Equivariant Atoms

We use the notation and conventions of Section 1.6. The preceding results show that equivariance refines the ordinary finite-node profile in two distinct ways: it records the G-set structure of the nodes and, beyond that, the stabilizer action on each local vanishing line. We first express this refinement in the rational representation ring and then compare the degeneration-side construction with smooth $G\times Hod$- equivariant Hodge atoms.

9.1. What Equivariance Adds

The ordinary flexible profile retains only the rank $r=|\Sigma |={dim}_{\mathbb{Q}}{V}_{\mathrm{van}}$, whereas Theorem 2 determines the full rational G-representation $V_{\mathrm{van}}$ from the local data ${\left\{(G_p,V_p)\right\}}_{\left[p\right]\in \Sigma /G}$. Thus the successive levels of information are

$$\mathrm{node}\mathrm{count}<\mathrm{node}G-\mathrm{set}<\mathrm{stabilizer}-\mathrm{decorated}\mathrm{vanishing}\mathrm{representation}.$$

Let $R_{\mathbb{Q}}\left(G\right)$ denote the Grothendieck group of finite-dimensional rational G-representations.

Definition 19 

(Equivariant flexible representation class). The equivariant flexible representation class of π is ${\mathfrak{v}}_G\left(\pi \right):=\left[V_{\mathrm{van}}\right]\in R_{\mathbb{Q}}\left(G\right)$.

By Theorem 2, this class is determined by the orbit–stabilizer vanishing data. Equivalently, if

$$V_{\mathrm{van}}\simeq \underset{\rho \in {\widehat{G}}_{\mathbb{Q}}}{⨁}{\rho }^{\oplus m_{\rho }^{\mathbb{Q}}},$$

then

$${\mathfrak{v}}_G\left(\pi \right)=\sum _{\rho \in {\widehat{G}}_{\mathbb{Q}}}{m}_{\rho }^{\mathbb{Q}}\left[\rho \right].$$

Hence ${\mathfrak{v}}_G\left(\pi \right)$ records precisely the rational irreducible multiplicities of the formal flexible sector [17].

Proposition 27 

(Rank as the forgetful shadow). Under the dimension homomorphism ${dim}_{\mathbb{Q}}:R_{\mathbb{Q}}\left(G\right)\to \mathbb{Z}$, one has ${dim}_{\mathbb{Q}}{\mathfrak{v}}_G\left(\pi \right)=|\Sigma |$. Equivalently, after extension to the fixed splitting field $\mathbf{K}$,

$$\sum _{\rho \in {\widehat{G}}_{\mathbf{K}}}{m}_{\rho }{dim}_{\mathbf{K}}\rho =|\Sigma |.$$

Proof. 

Since ${\mathfrak{v}}_G\left(\pi \right)=\left[V_{\mathrm{van}}\right]$, the first identity is Corollary 3. The second follows by taking dimensions after scalar extension to $\mathbf{K}$. □

The dimension map therefore forgets genuine geometric information. The projective $S_4$-cubic of Section 7.3 gives the strongest example in the paper: its node set is the transitive $S_4$-set $S_4/S_3$, but its local $S_3$-module is the sign representation. Consequently its flexible representation is the twisted module of Theorem 8, rather than the untwisted permutation representation of Proposition 18. Thus even the node G-set does not determine ${\mathfrak{v}}_G\left(\pi \right)$.

9.2. Relation with $G\times \mathrm{Hod}$-Equivariant Atoms

For a smooth projective G-variety X, the equivariant Hodge-atom construction uses the spectral decomposition of the non-archimedean A-model F-bundle with its combined $G\times Hod$-symmetry [1,4]. The degeneration-side construction of the present paper has a different origin: it is built from the corrected variation-cone mixed-Hodge-module carrier of Theorem 1.

The finite-node degeneration canonically carries three related but distinct equivariant structures. First, the rigid and formal flexible terms $I{C}_{X_0}^H$ and $V_{\mathrm{van}}^H$ have canonical G-linearizations. Second, the formal flexible term has representation class ${\mathfrak{v}}_G\left(\pi \right)$, determined by the local stabilizer data. Third, Theorem 9 and Corollary 12 give a G-equivariant limiting mixed Hodge structure whose rational isotypic components are sub-mixed-Hodge structures preserved by N.

Thus both the smooth and degeneration-side theories retain finite-group symmetry together with Hodge-theoretic information, but they do so through different geometric constructions. The projective $S_4$-family of Section 7.3 places these two settings in one one-parameter geometry: the nearby fibers are smooth projective $S_4$-varieties, while the central fiber has the explicitly computed twisted flexible representation of Theorem 8.

The comparison should not be interpreted as a categorical identification. The present paper constructs no specialization functor from smooth $G\times Hod$-equivariant atoms to ${\mathcal{P}}_G^{\lim }\left(\pi \right)$, and proves no equivariant birational invariance theorem for the limiting profile. The projective $S_4$-cubic instead provides a concrete test case for any future specialization theory: such a theory must recover the stabilizer-decorated limiting representation of Theorem 8, not merely the four-node $S_4$-set or its ordinary node count.

10. Outlook and Further Directions

We use the notation and conventions of Section 1.6.

The preceding sections isolate three canonical equivariant layers of a finite-node conifold degeneration: the G-linearized corrected rigid–flexible carrier, the stabilizer-decorated representation carried by its formal flexible sector, and the G-equivariant limiting mixed Hodge structure. The projective $S_4$-cubic of Section 7.3 shows that the representation-theoretic refinement is realized geometrically, while Theorem 9 and Corollary 12 show that the limiting mixed Hodge structure itself admits compatible rational symmetry sectors. The remaining problems concern the interaction among these structures and their relation to specialization and equivariant birational geometry.

10.1. Further Equivariant Structure

The first unresolved layer is the equivariant extension class

$$e_{\pi }^G\in {Ext}_{{MHM}_G\left(X_0\right)}^1\left(V_{\mathrm{van}}^H,I{C}_{X_0}^H\right)$$

of Corollary 2. The isotypic decomposition of the formal flexible representation does not by itself produce a corresponding decomposition of $e_{\pi }^G$. Global relations may couple different node orbits or symmetry sectors, so the extension contains information not encoded by the semisimple representation class ${\mathfrak{v}}_G\left(\pi \right)$ [10].

The same issue arises in comparing the corrected carrier with the limiting mixed Hodge structure. Section 8 proves that the latter decomposes canonically into rational G-isotypic sub-mixed-Hodge structures preserved by N. What remains is not the existence of these symmetry sectors, but their relationship with the vanishing representation and with $e_{\pi }^G$. In particular, the present results do not identify an isotypic component of the limiting mixed Hodge structure with a corresponding component of the global rigid–flexible extension.

A further refinement would incorporate interaction data among vanishing cycles. The Picard–Lefschetz transformations act on the same vanishing space carrying the G-action, suggesting a decomposition of the interaction pairing into symmetry sectors. The associated Stokes-type structures considered on the conifold side [8] should likewise be compared with these blocks. Such a refinement would record not only which irreducible symmetry types occur, but also how they interact.

Finally, the global relation space considered in Section 8.2 may itself carry representation-theoretic information. When $R\left(\pi \right)\subseteq V_{\mathrm{van}}$ is G-functorial, Proposition 26 gives induced G-representations on both $R\left(\pi \right)$ and $V_{\mathrm{van}}/R\left(\pi \right)$. Whether their classes in $R_{\mathbb{Q}}\left(G\right)$ admit a useful interpretation as equivariant refinements of classical nodal relation or defect data remains open.

Remark 44 

(Separation of the remaining structures). The formal flexible representation, the equivariant extension class, the G-equivariant limiting mixed Hodge structure, and the global relation data are distinct layers of the degeneration. Section 8 establishes the equivariant structure needed to study them, but no theorem simultaneously identifies or decomposes all four layers.

10.2. Specialization and Birational Transport

The broader problem suggested by the finite-node theory is specialization of equivariant Hodge atoms. For a smooth projective G-variety X, the smooth theory produces equivariant atomic data ${\mathcal{A}}_G\left(X\right)$ from the spectral decomposition of the non-archimedean A-model F-bundle with its $G\times Hod$-symmetry [4]. The degeneration-side objects constructed here have a different origin: they arise from nearby and vanishing cycles, the corrected mixed-Hodge-module carrier, and limiting Hodge theory.

A future specialization theory would therefore have to compare smooth spectral data with an enhanced limiting object containing at least the equivariant rigid–flexible profile and the G-equivariant limiting mixed Hodge structure. The projective $S_4$-cubic provides a concrete test case: any such theory must recover the twisted stabilizer-decorated flexible representation of Theorem 8, rather than merely the four-node $S_4$-set or its ordinary node count.

This test case is especially relevant because the same four-nodal $S_4$-geometry occurs in the equivariant birational setting considered by Cavenaghi–Katzarkov–Kontsevich [4]. Thus the family of Section 7.3 supplies an explicit geometry on which a future relation between equivariant birational transport and specialization can be tested.

To formulate the prospective compatibility problem, let ${\pi }_X:\mathcal{X}\to \Delta$ and ${\pi }_Y:\mathcal{Y}\to \Delta$ be G-equivariant degenerations with smooth fibers $X_t$ and $Y_t$ for $t\ne 0$. Retain the schematic notation ${Bir}_G$, ${Sp}_G$, and ${Bir}_G^{\lim }$ of Section 1.6.10. The organizing diagram is

$$\begin{array}{ccc}{\mathcal{A}}_G\left(X_t\right)& \stackrel{{Bir}_G}{\to }& {\mathcal{A}}_G\left(Y_t\right)\\ \downarrow {Sp}_G& & \downarrow {Sp}_G\\ {\mathcal{A}}_G^{\lim }(X_0,{\pi }_X)& \stackrel{{Bir}_G^{\lim }}{\to }& {\mathcal{A}}_G^{\lim }(Y_0,{\pi }_Y).\end{array}$$

The vertical arrows would compare smooth spectral data with limiting degeneration data, while the lower horizontal arrow would describe birational transport after specialization. The eventual compatibility problem is whether these operations can be constructed together with a canonical comparison

$${Sp}_G\circ {Bir}_G\simeq {Bir}_G^{\lim }\circ {Sp}_G.$$

Remark 45 

(Status of the compatibility problem). The prospective specialization and limiting birational operations in the preceding diagram are not constructed here, and no commutativity theorem is asserted. The present paper instead supplies explicit degeneration-side constraints—including the projective $S_4$-example—that any such theory must reproduce.

Specialization and birational transport are therefore logically distinct. Smooth equivariant birational transport is governed by equivariant blow-up and projective-bundle relations [4,18,19], whereas specialization must compare spectral F-bundle data with nearby, vanishing, and limiting-Hodge data.

10.3. Beyond Ordinary Double Points

The orbit–stabilizer mechanism does not intrinsically require the local vanishing representation to have rank one. That condition is specific to ordinary double points.

For a more general isolated hypersurface singularity p, let $G_p={Stab}_G\left(p\right)$ and let $V_p$ denote its rational local vanishing representation, assuming that the group action preserves the local degeneration germ. In general, ${dim}_{\mathbb{Q}}{V}_p$ may exceed one, so $V_p$ may carry a genuinely higher-dimensional $G_p$-representation.

If these local vanishing objects admit a compatible global equivariant assembly, their orbitwise contribution should again be governed by induction from $G_p$ to G, now retaining both the local Milnor rank and the stabilizer action on the vanishing cohomology. The finite-node ODP case already shows why the stabilizer module matters even at rank one: the projective $S_4$-cubic has a nontrivial local stabilizer character.

Two additional ingredients would be required for a general theorem. The local problem is to determine the mixed-Hodge-theoretic vanishing module and its $G_p$-action for the singularity type under consideration. The global problem is to construct an analogue of the corrected carrier $P^H$ and prove a functorial equivariant local-to-global assembly. Only after both steps are available does the orbitwise representation become part of a genuine limiting Hodge-theoretic theory.

Remark 46 

(Scope beyond the ODP case). No generalization theorem for arbitrary isolated hypersurface singularities is proved here. The orbitwise form suggested by Theorem 2 identifies the expected representation-theoretic structure, but a broader theory requires the appropriate local mixed Hodge modules, a corrected global degeneration carrier, and their equivariant compatibility.

The ordinary-double-point case is therefore a useful test model rather than merely the lowest-rank example: its local vanishing sectors are simple enough to compute explicitly, yet the projective $S_4$-example already shows nontrivial stabilizer behavior. Extending this structure to higher Milnor rank and relating it to smooth equivariant Hodge atoms under specialization are natural next problems.

Appendix A.1. Rational Representations and Semisimplicity

Let G be a finite group and let $\mathbb{Q}\left[G\right]$ denote its rational group algebra.

Proposition A1 

(Maschke semisimplicity). Every finite-dimensional rational G-representation is semisimple. Equivalently, $\mathbb{Q}\left[G\right]$ is a semisimple algebra.

Proof. 

This is Maschke’s theorem, since $char\left(\mathbb{Q}\right)=0$ and hence $\left|G\right|$ is invertible in $\mathbb{Q}$ [17]. □

Let ${\widehat{G}}_{\mathbb{Q}}$ denote the set of isomorphism classes of irreducible rational G-representations.

Definition A1 

(Endomorphism division algebra). For $\rho \in {\widehat{G}}_{\mathbb{Q}}$, set $D_{\rho }:={End}_{\mathbb{Q}\left[G\right]}\left(\rho \right)$.

By Schur’s lemma, $D_{\rho }$ is a finite-dimensional division algebra over $\mathbb{Q}$; it need not equal $\mathbb{Q}$.

Definition A2 

(Rational multiplicity). For a finite-dimensional rational G-representation V and $\rho \in {\widehat{G}}_{\mathbb{Q}}$, define

$$m_{\rho }^{\mathbb{Q}}\left(V\right):={dim}_{D_{\rho }}{Hom}_{\mathbb{Q}\left[G\right]}(\rho ,V),$$

where the Hom-space is a right $D_{\rho }$-module by precomposition.

Proposition A2 

(Multiplicity and dimension). For every finite-dimensional rational G-representation V,

$$V\simeq \underset{\rho \in {\widehat{G}}_{\mathbb{Q}}}{⨁}{\rho }^{\oplus m_{\rho }^{\mathbb{Q}}\left(V\right)},{dim}_{\mathbb{Q}}V=\sum _{\rho \in {\widehat{G}}_{\mathbb{Q}}}{m}_{\rho }^{\mathbb{Q}}\left(V\right){dim}_{\mathbb{Q}}\rho .$$

Proof. 

Semisimplicity gives a decomposition into irreducible rational G-modules. Schur’s lemma identifies the number of copies of $\rho$ with ${dim}_{D_{\rho }}{Hom}_{\mathbb{Q}\left[G\right]}(\rho ,V)$. The dimension formula follows by taking rational dimensions [17]. □

Appendix A.2. Splitting Fields and Complex Irreducibles

Let $\mathbf{K}\supset \mathbb{Q}$ be the characteristic-zero splitting field fixed in Section 1.6. For a rational G-representation V, set $V_{\mathbf{K}}:=V{\otimes }_{\mathbb{Q}}\mathbf{K}$. Let ${\widehat{G}}_{\mathbf{K}}$ denote the irreducible $\mathbf{K}\left[G\right]$-modules. Since $\mathbf{K}$ is a splitting field, ${End}_{\mathbf{K}\left[G\right]}\left(\sigma \right)=\mathbf{K}$ for $\sigma \in {\widehat{G}}_{\mathbf{K}}$, and therefore

$$V_{\mathbf{K}}\simeq \underset{\sigma \in {\widehat{G}}_{\mathbf{K}}}{⨁}{\sigma }^{\oplus m_{\sigma }^{\mathbf{K}}},m_{\sigma }^{\mathbf{K}}={dim}_{\mathbf{K}}{Hom}_{\mathbf{K}\left[G\right]}(\sigma ,V_{\mathbf{K}}).$$

Remark A1 

(Rational and splitting-field irreducibles). An irreducible rational representation need not remain irreducible after extension of scalars. Thus ${\widehat{G}}_{\mathbb{Q}}$ and ${\widehat{G}}_{\mathbf{K}}$ are distinct indexing sets, and the rational multiplicities $m_{\rho }^{\mathbb{Q}}$ should not be identified termwise with the splitting-field multiplicities $m_{\sigma }^{\mathbf{K}}$. The example $G=\mathbb{Z}/3$ in Section 7.2 illustrates this distinction: the two nontrivial one-dimensional characters over a splitting field combine into one two-dimensional irreducible rational representation.

Appendix A.3. Induction, Restriction, and Frobenius Reciprocity

Let $H\le G$. For a rational H-representation W, set ${Ind}_H^{G}W:=\mathbb{Q}\left[G\right]{\otimes }_{\mathbb{Q}\left[H\right]}W$, and for a rational G-representation V, let ${Res}_H^{G}V$ denote its restriction to H.

Proposition A3 

(Dimension of induction). For every finite-dimensional rational H-representation W, ${dim}_{\mathbb{Q}}{Ind}_H^{G}W=[G:H]{dim}_{\mathbb{Q}}W$.

Proof. 

As a right $\mathbb{Q}\left[H\right]$-module, $\mathbb{Q}\left[G\right]$ is free of rank $[G:H]$, with basis given by any set of left-coset representatives. Tensoring with W gives the formula. □

Proposition A4 

(Frobenius reciprocity). Let V be a rational G-representation and W a rational H-representation. There is a natural isomorphism

$${Hom}_{\mathbb{Q}\left[G\right]}\left(V,{Ind}_H^{G}W\right)\simeq {Hom}_{\mathbb{Q}\left[H\right]}\left({Res}_H^{G}V,W\right).$$

The analogous statement holds over the splitting field $\mathbf{K}$.

Proof. 

For finite groups, induction and coinduction are naturally isomorphic, so induction is also right adjoint to restriction. This gives the displayed form of Frobenius reciprocity; scalar extension gives the corresponding statement over $\mathbf{K}$ [17]. □

This is the representation-theoretic input used in Theorem 3.

Appendix A.4. Permutation Representations

For $H\le G$, let $\mathbb{Q}[G/H]$ denote the rational permutation representation on the left coset space $G/H$.

Proposition A5 

(Permutation representation as induction). Let ${\mathbf{1}}_H$ denote the trivial one-dimensional rational H-representation. Then $\mathbb{Q}[G/H]\simeq {Ind}_H^G{\mathbf{1}}_H$.

Proof. 

Under ${Ind}_H^G{\mathbf{1}}_H=\mathbb{Q}\left[G\right]{\otimes }_{\mathbb{Q}\left[H\right]}\mathbb{Q}$, the assignment $g\otimes 1\mapsto e_{gH}$ is a well-defined G-equivariant isomorphism. □

For $H=\left\{1\right\}$, this is the regular representation $\mathbb{Q}\left[G\right]$.

Remark A2 

(Use in the flexible sector). If $G_p$ acts trivially on the local vanishing line $V_p$, then ${Ind}_{G_p}^G{V}_p\simeq \mathbb{Q}[G/G_p]$. For a nontrivial local stabilizer character the induced representation need not be a permutation representation. The projective $S_4$-cubic of Section 7.3 gives the geometric example

$${Ind}_{S_3}^{S_4}{sgn}_{S_3}\not\simeq \mathbb{Q}[S_4/S_3].$$

Appendix B.1. Local Analytic Model

Let $p\in \Sigma$. Analytically, the central-fiber germ is $z_1^2+z_2^2+z_3^2+z_4^2=0$, with standard smoothing $z_1^2+z_2^2+z_3^2+z_4^2=t$. For $0<\left|t\right|\ll \epsilon$, let

$$F_{p,t}:=\left\{z_1^2+z_2^2+z_3^2+z_4^2=t\right\}\cap B_{\epsilon }.$$

The Milnor fiber has the homotopy type of $S^3$, so

$${\tilde{H}}^j(F_{p,t},\mathbb{Q})=0(j\ne 3),{\tilde{H}}^3(F_{p,t},\mathbb{Q})\simeq {\tilde{H}}_3(F_{p,t},\mathbb{Q})\simeq \mathbb{Q}$$

[5,6].

Appendix B.2. Vanishing Generator and Sign Convention

For $t>0$ real and sufficiently small, the real vanishing sphere is

$$S_{p,t}^3:=\left\{x\in {\mathbb{R}}^4|x_1^2+x_2^2+x_3^2+x_4^2=t\right\}.$$

Choose an orientation and let ${\delta }_p\in H_3(F_{p,t},\mathbb{Q})$ be its fundamental class. The associated vanishing line is $V_p:=\mathbb{Q}{\delta }_p$. Although ${\delta }_p$ is determined only up to sign, $V_p$ and its stabilizer representation are intrinsic.

Remark A3 

(Basis independence of the stabilizer character). If $h\in G_p$ acts by $h{\delta }_p={\epsilon }_h{\delta }_p$, then replacing ${\delta }_p$ by $-{\delta }_p$ leaves ${\epsilon }_h$ unchanged. Consequently the one-dimensional $G_p$-module $V_p$ is independent of the choice of oriented generator.

Appendix B.3. Perverse and Mixed-Hodge-Module Normalization

Since ${dim}_{\mathbb{C}}{X}_0=3$, for $U=X_0^{\mathrm{sm}}$ and $j:U\hookrightarrow X_0$ we use $I{C}_{X_0}^H=j_{!*}{\mathbb{Q}}_U^H\left[3\right]$. For $p\in \Sigma$, with $i_p:\left\{p\right\}\hookrightarrow X_0$, the point-supported local vanishing object is $V_p^H:=i_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1)$, and $V_{\mathrm{van}}^H={⨁}_{p\in \Sigma }{V}_p^H$. Thus the corrected rigid–flexible sequence has the normalization

$$0⟶I{C}_{X_0}^H⟶P^H⟶V_{\mathrm{van}}^H⟶0$$

[8].

Remark A4 

(Tate normalization). The twist $(-1)$ in $V_p^H$ is the fixed threefold convention of [8]. No alternate Tate normalization is used in this paper.

Remark A5 

(Local Hodge object and underlying representation). We distinguish the local vanishing Hodge object ${\mathsf{V}}_p^H$, its point-supported realization $V_p^H=i_{p*}{\mathbb{Q}}_{\left\{p\right\}}^H(-1)$, and the underlying rational vanishing representation $V_p$.

Appendix B.4. Stabilizer Action on the Vanishing Line

Every $h\in G_p$ induces an automorphism of the local degeneration germ and hence an automorphism of $V_p$. Since $V_p$ is one dimensional, write $h_{*}={\epsilon }_p\left(h\right){id}_{V_p}$, where ${\epsilon }_p\left(h\right)\in {\mathbb{Q}}^{\times }$.

Proposition A6 

(Rational stabilizer character). The map ${\epsilon }_p:G_p\to \{\pm 1\}$ is a group homomorphism and determines the rational $G_p$-module $V_p$.

Proof. 

Functoriality gives ${\epsilon }_p\left(h_1{h}_2\right)={\epsilon }_p\left(h_1\right){\epsilon }_p\left(h_2\right)$. Since $G_p$ is finite, the image consists of finite-order elements of ${\mathbb{Q}}^{\times }$, hence is contained in $\{\pm 1\}$. □

After scalar extension to $\mathbf{K}$, the same representation is denoted by ${\chi }_p^{\mathrm{van}}$.

Remark A6 

(No preferred generator). The character ${\epsilon }_p$ is intrinsic even though the oriented generator ${\delta }_p$ is not. Accordingly, the main text formulates stabilizer statements in terms of $V_p$ or ${\chi }_p^{\mathrm{van}}$, not a preferred generator.

Appendix B.5. Compatibility Under Transport of Nodes

If $g\in G$ and $p\in \Sigma$, transport by g identifies the local degeneration germs at p and $gp$, while conjugation identifies $G_p$ with $G_{gp}$. The induced isomorphism ${\Phi }_g:V_p\stackrel{\sim }{\to }{V}_{gp}$ satisfies

$${\Phi }_g\left(hv\right)=\left(gh{g}^{-1}\right){\Phi }_g\left(v\right),h\in G_p,v\in V_p,$$

by Lemma 1. Hence an orbit carries the well-defined conjugacy class of the pair $(G_p,V_p)$, which is the local input to ${Ind}_{G_p}^G{V}_p$.