# The Dimension Matching Principle: A Verifiable Criterion for the Algebraicity of Hodge Classes

Badre Mounda, Dongzhe Zheng

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Badre Mounda, Dongzhe Zheng. The Dimension Matching Principle: A Verifiable Criterion for the Algebraicity of Hodge Classes. 2025. hal-05245673v2

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# The Dimension Matching Principle: A Verifiable Criterion for the Algebraicity of Hodge Classes

Badre Mounda

Dongzhe Zheng

## Abstract

In this paper, under the framework of classical Hodge/degeneration theory (VHS/LMHS), we present a verifiable “three-layer screening” mechanism: the first layer (LMHS algebraic window), the second layer (algebraic realization of the central fiber), and the third layer (Gauss–Manin flatness and boundary behavior), and establish a “Dimension Matching Principle” (DMP). Within the framework of this principle, we prove two benchmark results: (A) For smooth quartic K3 surfaces, any rational (1, 1)-class can be constructively realized as an algebraic divisor through a specialization pullback process that controls nodal degenerations (constructive Lefschetz(1, 1)); (B) Under three-dimensional Calabi–Yau nodal degenerations satisfying equivariant and boundary-good conditions, G-invariant (2, 2)-classes are all algebraic classes when dimension matching is satisfied (equivariant strong part). As applications, we verify on two typical families: (i) G-invariant (2, 2) strong part of Dwork quintic under LCSL and toric MPCP paths; (ii) equivariant (2, 2) strong part on Dwork sextic Calabi–Yau fourfolds. We also provide a toric/Lefschetz–Gysin proof of (2, 2)-algebraicity for the Dwork quintic and its mirror, and a boundary-propagation template for nodal degenerations. Combined with DMP, this yields a unified, verifiable toolkit for the Dwork families.

## Contents

<table><tr><td>1 Introduction</td><td>2</td></tr><tr><td>2 Preliminaries and Tools</td><td>3</td></tr><tr><td>2.1 Semistable Degeneration and Limit Mixed Hodge Structure</td><td>3</td></tr><tr><td>2.2 Basic Criteria for Weight 2 and Weight 4</td><td>4</td></tr><tr><td>2.3 Q-Structure, Dimension Matching and Notation Correction</td><td>4</td></tr><tr><td>2.4 Pullback Criterion for Weight 2</td><td>4</td></tr><tr><td>2.5 (2, 2) Algebraicity of Dwork Quintics and Mirror Threefolds</td><td>5</td></tr><tr><td>3 Three-Layer Screening and Dimension Matching Principle</td><td>6</td></tr><tr><td>3.1 LMHS Algebraic Window and Algebraic Realization of Central Fiber</td><td>6</td></tr><tr><td>3.2 Dimension Matching Principle (DMP)</td><td>6</td></tr><tr><td>3.3 Singular Parameters and Algebraicity Propagation</td><td>7</td></tr><tr><td>4 Constructive (1, 1) on K3: Nodal Degeneration Method</td><td>7</td></tr><tr><td>4.1 Nodal Degeneration and Semistable Model of Quartic K3</td><td>8</td></tr><tr><td>4.2 PL Orthogonality, MV Generation and Linear Expression of Limit Classes</td><td>8</td></tr><tr><td>4.3 Quartic K3: Node Upper Bound and Explicit Formula for Specialization</td><td>9</td></tr><tr><td>4.4 Picard–Lefschetz Pairing and Linear System</td><td>11</td></tr><tr><td>4.5 Constructive (1, 1) Theorem</td><td>11</td></tr></table>
<table><tr><td>5 (2, 2) Classes of Threefold Calabi-Yau</td><td>11</td></tr><tr><td>5.1 Compatibility of Specialization and Cycle Class Maps</td><td>12</td></tr><tr><td>5.2 Equivariant Assumptions and Weight 4 Pullback Criterion</td><td>12</td></tr><tr><td>5.3 Setup and Equivariant Algebraic Window</td><td>13</td></tr><tr><td>5.4 Equivariant Strong Part</td><td>13</td></tr><tr><td>5.5 Dwork Quintic LCSL: G-Invariant Description of sp(h<sup>2</sup>)</td><td>14</td></tr><tr><td>5.6 Example: Dwork Quintic Under LCSL Limit and MPCP Decomposition</td><td>14</td></tr><tr><td>5.7 Verification of Equivariant Strong Part for Dwork Quintic</td><td>15</td></tr><tr><td>5.8 Non-equivariant (2, 2) Algebraicity of Mirror Threefolds and Consistency with Equivariant Strong Parts</td><td>15</td></tr></table>

<table><tr><td>6 Equivariant (2, 2) Strong Part on Dwork Sextic Calabi-Yau Fourfolds</td><td>16</td></tr><tr><td>6.1 Primitive Part Invariant Dimension: Jacobian Ring Calculation</td><td>17</td></tr><tr><td>6.2 Equivariant Semistable Degeneration and Algebraic Window</td><td>17</td></tr><tr><td>6.3 Pullback: Algebraic Extension of Window Classes</td><td>18</td></tr><tr><td>6.4 Main Theorem and Matching</td><td>18</td></tr></table>

<table><tr><td>7 Conclusion</td><td>20</td></tr></table>

<table><tr><td>A Technical Details of Flat Extension and Deligne Extension</td><td>21</td></tr><tr><td>A.1 Local Calculation for K3 Single Node Degeneration</td><td>21</td></tr><tr><td>A.2 General Case of Multi-Node Degeneration</td><td>22</td></tr></table>

<table><tr><td>B Verifiable Details of Equivariant Strong Part on Dwork Sextic CY4</td><td>22</td></tr><tr><td>B.1 Invariant Dimension and Flatness of Jacobian Ring</td><td>22</td></tr><tr><td>B.2 Semistable Model, Steenbrink Spectral Sequence and Signs of d<sub>1</sub></td><td>23</td></tr><tr><td>B.3 Pullback Cycles, Primitivity and Intersection Numbers</td><td>25</td></tr></table>

<table><tr><td>C Toric Background and Gysin Exact Sequences</td><td>25</td></tr></table>

# 1 Introduction

Verifiable criteria for the Hodge conjecture in high dimensions and high weights are scarce. In this paper, within the classical VHS/LMHS framework, we propose a three-layer screening mechanism supplemented by a “Dimension Matching Principle” (DMP), which, under controllable degeneration backgrounds, reduces “all $(p, p)$ Hodge classes are algebraic” to three verifiable conditions and one dimension equation. This perspective transforms the traditional static Hodge problem into several verifiable and computable problems, providing a possible new organizing principle for some constructive understanding of the Hodge conjecture.

**Relationship with existing work and acknowledgment** The overall approach and terminology choices of this paper are directly inspired by and influenced by Mounda's two works on “constructive degeneration–limit algebraicity”, which we hereby acknowledge and thank: the first proposed a “constructive degeneration perspective” centered on LMHS and unipotent nilpotent operators, providing general formulations and various examples[Mou25a]; the second gave a constructive proof framework for K3 surfaces independent of Torelli[Mou25b]. Building on this foundation, the main contributions of this paper are: (i) abstracting “constructive degeneration” into verifiable *three-layer screening*, and rigorizing it in LMHS through algebraic windows; (ii) proposing and systematically using the *Dimension Matching Principle* (DMP), connecting “limit algebraic realization + flatness/boundary goodness” with dimension matching on $\mathbb{Q}$-structures in a closed loop.
(I) **LMHS Algebraic Window:** In the weight filtration $W_\bullet$ and limit Hodge filtration $F_\infty^\bullet$ of the limit mixed Hodge structure (LMHS), define

$$W_p(\sigma) := \mathrm{Gr}_{2p}^W H_{\mathrm{lim}}^{2p} \cap F_\infty^p \cap \overline{F_\infty^p} \cap \ker N,$$

which is an “algebraic window” completely determined by nilpotent monodromy $N$ and limit filtrations (for multiple parameters, use $\cap_i \ker N_i$). Through the Clemens–Schmid comparison isomorphism, it corresponds to a specific subspace on the semistable model $X_0^\sim$ of the central fiber.

(II) **Algebraic Realization of Central Fiber:** On the semistable model $X_0^\sim$, $W_p(\sigma)$ corresponds via the Clemens–Schmid isomorphism $\phi$ to a subspace of $H^{2p}(X_0^\sim, \mathbb{Q})(-p)$. If this subspace falls into $cl(CH^p(X_0^\sim))$ (spanned by exceptional curves/exceptional divisors and their intersections), we call it “algebraically realizable”.

(III) **Flatness and Boundary Behavior:** Take $\nabla_{\mathrm{GM}}$-flat sections $\sigma$ to extend candidate classes, making them $(p, p)$ at some point, and examine their Deligne extension and LMHS boundary types; combined with the fixed part theorem and Hodge locus algebraicity (Cattani–Deligne–Kaplan), we obtain algebraicity conclusions for general fibers.

Guided by the three-layer screening and the DMP, we proceed case by case. (i) K3 (weight 2): we give a constructive Lefschetz(1, 1) via controlled nodal degenerations and a specialization-pullback procedure. (ii) $\mathrm{CY}_3$ (weight 4): for the Dwork quintic and its mirror we establish the equivariant strong part within the DMP architecture, and in addition develop a toric/Lefschetz-Gysin line together with a boundary-propagation template that provides an independent and more explicit verification on the same families; the two routes meet on the quintic and sharpen the mirror side. (iii) $\mathrm{CY}_4$ (weight 4): for the Dwork sextic we obtain the equivariant strong part via DMP.

# 2 Preliminaries and Tools

## 2.1 Semistable Degeneration and Limit Mixed Hodge Structure

Let $\pi: \mathcal{X} \to \Delta$ be a one-parameter degeneration, where $\Delta$ is the unit disk, $\mathcal{X}$ is smooth, and $\pi$ is a smooth holomorphic submanifold fibration except possibly at 0. Through standard procedures (base change and blowing up), we may assume $\pi$ is a semistable degeneration: the central fiber $X_0$ is a simple normal crossing (SNC) divisor, and $K_{\mathcal{X}/\Delta} \simeq \mathcal{O}_{\mathcal{X}}$ (in the CY context). Denote the general fiber as $X_t$ ($t \neq 0$).

**Definition 2.1** (LMHS and Unipotent Nilpotent). Let $H^k := R^k \pi_* \mathbb{Q}$ be the local system, with Gauss-Manin connection $\nabla_{\mathrm{GM}}$ and monodromy transformation $T = \exp(N)$ (where $N$ is nilpotent) on $\Delta^*$. The limit mixed Hodge structure (LMHS) is a triple $(H_{\mathrm{lim}}^k, W_\bullet, F_\infty^\bullet)$, where $W_\bullet$ is the $N$-compatible weight filtration and $F_\infty^\bullet$ is the limit Hodge filtration (Deligne–Schmid)[Sch73].

This section will use the following standard tools (proofs omitted, references given):

**Lemma 2.2** (Picard–Lefschetz Formula). If the central fiber has only $A_1$-nodes, then the monodromy $T = \exp(N)$ satisfies $N^2 = 0$, and for any $v \in H^k(X_t, \mathbb{Z})$ we have

$$T(v) = v + \langle v, \gamma \rangle \gamma, \quad N(v) = \langle v, \gamma \rangle \gamma^\vee,$$

where $\gamma$ is the vanishing cycle. For multi-node degenerations, there is a set of commuting $N_i$ and orthogonal $\gamma_i[\mathrm{Dim}17, \mathrm{Mil}68]$.
**Lemma 2.3** (Clemens–Schmid Comparison). Let $\pi$ be a semistable degeneration. Then there exists a comparison isomorphism

$$ \phi : \mathrm{Gr}_{2p}^{W} H_{\mathrm{lim}}^{2p} \xrightarrow{\sim} H^{2p}(X_{0}^{\sim}, \mathbb{Q})(-p), $$

where $X_0^{\sim}$ is the semistable model (obtained by blowing up nodes once or finitely many times). In weight 2 (surfaces/K3) and weight 4 (CY³), these are respectively $\mathrm{Gr}_2^W \simeq H^2(X_0^{\sim})$ and $\mathrm{Gr}_4^W \simeq H^4(X_0^{\sim})$ [Ste76].

**Lemma 2.4** (Hodge Locus Algebraicity (Cattani–Deligne–Kaplan)). Given an integral cohomology class $[\alpha] \in H^{2p}(X_s, \mathbb{Z})$, the set of points $s \in S$ where it is of type $(p, p)$ (Hodge locus) is an algebraic subset of the parameter space $S[GGK12]$.

## 2.2 Basic Criteria for Weight 2 and Weight 4

**Lemma 2.5** (Lefschetz (1, 1) Theorem). For any projective Kähler manifold, all rational (1, 1)-classes are algebraic.

**Lemma 2.6** (Fixed Part Theorem (Deligne)). The monodromy-invariant part of a VHS corresponds to a flat sub-local system; if it is of type $(p, p)$ at one point, then it remains of type $(p, p)$ in a Zariski open set [PS08].

## 2.3 $\mathbb{Q}$-Structure, Dimension Matching and Notation Correction

All “dimension matching” in this paper is performed on $\mathbb{Q}$-structures. For a smooth fiber $X_t$ and fixed $p$, define

$$ \rho_p(X_t) := \dim_{\mathbb{Q}}(H^{p,p}(X_t) \cap H^{2p}(X_t, \mathbb{Q})). $$

When $p = 1$, $\rho_1(X_t)$ is the Picard number $\rho(X_t)$. Note that $\rho_p(X_t)$ differs from the Hodge number $h^{p,p}(X_t) = \dim_{\mathbb{C}} H^{p,p}(X_t)$; the former is a $\mathbb{Q}$-dimension, the latter is a complex dimension.

In one-parameter degenerations, only a single nilpotent operator $N$ appears. Therefore, the LMHS algebraic window should be defined as

$$ W_p(\sigma) := \mathrm{Gr}_{2p}^W H_{\mathrm{lim}}^{2p} \cap F_{\infty}^p \cap \overline{F_{\infty}^p} \cap \ker N. $$

For multi-parameter degenerations, if the corresponding Picard–Lefschetz monodromies commute, then there exist commuting $N_i$, and $\ker N$ can be replaced by $\bigcap_i \ker N_i$ (see [Sch73, KU09]).

## 2.4 Pullback Criterion for Weight 2

**Lemma 2.7** (Weight 2 Pullback Criterion). Let $\pi : \mathcal{X} \to \Delta$ be a one-parameter semistable degeneration, $X_t$ be a general fiber, and $X_0^{\sim}$ be the semistable model. If the specialization $\mathrm{sp}([\alpha]) \in H^2(X_0^{\sim}, \mathbb{Q})$ of $[\alpha] \in H^{1,1}(X_t, \mathbb{Q})$ lies in $\mathrm{Pic}(X_0^{\sim}) \otimes \mathbb{Q}$, then $[\alpha]$ is an algebraic class on $X_t$.

*Proof.* By the Clemens–Schmid comparison isomorphism $\mathrm{Gr}_2^W H_{\mathrm{lim}}^{2p} \simeq H^2(X_0^{\sim}, \mathbb{Q})$, $\mathrm{sp}([\alpha])$ is an algebraic class. Take a $\nabla_{\mathrm{GM}}$-flat section extension of $[\alpha]$. At some $t_* \neq 0$, $[\alpha]$ is a (1, 1) class. By the Lefschetz (1, 1) theorem (Lemma 2.5), $[\alpha]$ is an algebraic class on $X_{t_*}$, hence also on general fibers. Technical details of flat extension and Deligne extension are given in Appendix A. $\square$
## 2.5 (2, 2) Algebraicity of Dwork Quintics and Mirror Threefolds

This subsection records two “benchmark results” that will be frequently used in what follows. They are based on standard facts from the weak Lefschetz hyperplane theorem, toric even cohomology, and Gysin exact sequences (see [DK87, Ful93, CLS11, CK99]).

**Proposition 2.8** (Dwork Quintic Threefold). Let $X_ψ = \{x_0^5 + \cdots + x_4^5 - 5ψx_0x_1x_2x_3x_4 = 0\} \subset \mathbb{P}^4$ be the Dwork quintic threefold Calabi-Yau for any smooth parameter $ψ$. Then

$$H^{2,2}(X_ψ, \mathbb{Q}) = \mathbb{Q} \cdot H^2 = \mathrm{cl}(\mathrm{CH}_2(X_ψ))_{\mathbb{Q}}.$$

where $H$ is the hyperplane class. In other words, all rational (2, 2)-classes on $X_ψ$ are algebraic classes.

*Proof.* Applying the weak Lefschetz hyperplane theorem to the inclusion $X_ψ \hookrightarrow \mathbb{P}^4$ (with $\dim X_ψ = 3$), we obtain that the restriction map

$$H^4(\mathbb{P}^4, \mathbb{Q}) \to H^4(X_ψ, \mathbb{Q})$$

is surjective, and the primitive cohomology of $X_ψ$ appears only in degree 3. Since $H^4(\mathbb{P}^4, \mathbb{Q}) = \mathbb{Q} \cdot H^2$, we have $H^4(X_ψ, \mathbb{Q}) = \mathbb{Q} \cdot H^2$, hence $H^{2,2}(X_ψ, \mathbb{Q}) = \mathbb{Q} \cdot H^2$. The class $H^2$ is the algebraic intersection of two hyperplanes, so the proposition follows. □

**Theorem 2.9** (Toric Generation of Mirror Threefolds). Let $T$ be a smooth projective toric fourfold obtained by Greene-Plesser quotient followed by MPCP (or weak-fan) resolution, and let $Y_ψ \subset T$ be a smooth anticanonical hypersurface for any smooth parameter. Then

$$H^{2,2}(Y_ψ, \mathbb{Q}) = \mathrm{cl}(\mathrm{CH}_2(Y_ψ))_{\mathbb{Q}},$$

and the right-hand side is spanned by $\mathbb{Q}$-linear combinations of restriction classes $[S|_{Y_ψ}]$ of torus-invariant surfaces $S = D_i \cap D_j$ in $T$.

*Proof.* We need two standard facts (see [DK87, Ful93, CLS11]).

(A) (Toric even cohomology) For a smooth projective toric variety $T$, we have $H^{2k+1}(T, \mathbb{Q}) = 0$, and the cycle class map $\mathrm{cl}: \mathrm{CH}^*(T)_{\mathbb{Q}} \xrightarrow{\cong} H^{2*}(T, \mathbb{Q})$ is an isomorphism. In particular, $H^4(T, \mathbb{Q}) \cong \mathrm{CH}^2(T)_{\mathbb{Q}}$, and $\mathrm{CH}^2(T)$ is generated by torus-invariant surfaces (intersections of two toric divisors).

(B) (Gysin sequence for smooth hypersurfaces) Let $i: Y_ψ \hookrightarrow T$ denote the inclusion. Then the long exact sequence of the pair gives

$$\dots \to H^4(T, \mathbb{Q}) \xrightarrow{i^*} H^4(Y_ψ, \mathbb{Q}) \to H^5(T, Y_ψ; \mathbb{Q}) \to H^5(T, \mathbb{Q}) \to \dots$$

By the Lefschetz theorem for pairs, $H^5(T, Y_ψ; \mathbb{Q}) \simeq H^3(T, \mathbb{Q}) = 0$, so $i^*: H^4(T, \mathbb{Q}) \to H^4(Y_ψ, \mathbb{Q})$ is surjective.

By (A), $H^4(T, \mathbb{Q})$ is entirely algebraic, and by (B), $H^4(Y_ψ, \mathbb{Q})$ is the image of $H^4(T, \mathbb{Q})$, hence $H^{2,2}(Y_ψ, \mathbb{Q})$ is spanned by restriction classes of torus-invariant surfaces on $Y_ψ$ and is entirely algebraic. □

**Remark 2.10** (Rank and Generation). The mirror quintic threefold satisfies $h^{1,1}(Y_ψ) = h^{2,2}(Y_ψ) = 101$. By the surjectivity in Theorem 2.9 and Poincaré duality, the image of restriction classes of torus-invariant surfaces in $H^{2,2}(Y_ψ, \mathbb{Q})$ has dimension 101, so these algebraic classes indeed generate $H^{2,2}(Y_ψ, \mathbb{Q})$.
# 3 Three-Layer Screening and Dimension Matching Principle

## 3.1 LMHS Algebraic Window and Algebraic Realization of Central Fiber

**Definition 3.1 (Algebraic Window).** Let $\pi$ be a semistable degeneration and $\sigma$ be some approach path (or parameter ray). Define

$$W_p(\sigma) := \mathrm{Gr}_{2p}^W H_{\lim}^{2p} \cap F_{\infty}^p \cap \overline{F_{\infty}^p} \cap \ker N.$$

We call $W_p(\sigma)$ the algebraic window of the LMHS. By Lemma 2.3, via the comparison isomorphism $\phi$, it can be viewed as a subspace of $H^{2p}(X_0^{\sim}, \mathbb{Q})$.

**Definition 3.2 (Algebraic Realization).** If $\phi(W_p(\sigma)) \subset \mathrm{cl}(CH^p(X_0^{\sim})_{\mathbb{Q}}) \subset H^{2p}(X_0^{\sim}, \mathbb{Q})$, then we say $W_p(\sigma)$ is algebraically realizable on the central fiber.

**Lemma 3.3 (Window Algebraic Reachability and Relative Lifting).** Let $\pi : \mathcal{X} \to \Delta$ be a semistable degeneration with SNC central fiber and LMHS window $W_p(\sigma)$ as in Definition 3.1. Then every class $\xi \in W_p(\sigma) \cap \ker N$ that lands in $\phi^{-1}(\mathrm{cl}(CH^p(X_0^{\sim})_{\mathbb{Q}}))$ admits an algebraic $E_1$-representative in Steenbrink's spectral sequence at $q = 2p$, and there exists a relative algebraic cycle $Z \in CH^p(\mathcal{X})_{\mathbb{Q}}$ such that for general $t$,

$$\mathrm{sp}_t(\mathrm{cl}(Z|_{X_t})) = \xi.$$

Equivalently, the window is algebraically reachable from the total space.

*Proof.* For an SNC central fiber $X_0 = \bigcup_{\alpha} X_{\alpha}$, the weight-$2p$ line $E_1^{-r,2p}$ is a direct sum of $H^{2p-2r}(X_0^{[r]})(-r)$ and is generated by algebraic classes on strata $X_0^{[r]}$ (products of hyper-plane classes etc.). If $\xi \in W_p(\sigma) \cap \ker N$ is algebraic on the central fiber, we can choose an $E_1$-representative $\beta$ that is $d_1$-closed (this is precisely the $\ker N$ condition for the window). Interpreting $E_1$ as a Čech-type cycle complex on the SNC cover, $\beta$ glues to an algebraic cycle $Z_0 \in CH^p(X_0)_{\mathbb{Q}}$. By Fulton's specialization construction and Saito's formalism of nearby/vanishing cycles for mixed Hodge modules, the cycle class map commutes with specialization (cf. §5.1); thus there exists a relative $Z \in CH^p(\mathcal{X})_{\mathbb{Q}}$ with $Z|_{X_0} \sim Z_0$ whose cohomology class specializes to $\xi$. This yields the claim. $\square$

**Remark 3.4.** Lemma 3.3 will be invoked in the proof of Theorem 3.5 to replace any implicit use of “surjectivity of specialization on $(p,p)$-layers” by an explicit algebraic lifting from the window.

## 3.2 Dimension Matching Principle (DMP)

**Theorem 3.5 (Dimension Matching Principle).** Let $\pi : \mathcal{X} \to \Delta$ be a one-parameter semistable degeneration. For fixed $p$, assume:

(i) (Algebraic Window) $W_p(\sigma)$ as in Definition 3.1, and $\phi(W_p(\sigma)) \subset \mathrm{cl}(CH^p(X_0^{\sim})_{\mathbb{Q}})$;

(ii) (Flatness/Boundary) For any $[\alpha] \in H^{p,p}(X_t, \mathbb{Q})$ on a general fiber $X_t$, if $\mathrm{sp}([\alpha]) \in W_p(\sigma)$, then there exists a $\nabla_{\mathrm{GM}}$-flat section $\tilde{\alpha}$ extending $[\alpha]$, whose Deligne extension limit at the central fiber is an algebraic class, and after pullback $\tilde{\alpha}(t)$ remains $(p,p)$ and is algebraic in some Zariski open set (for weight 2, this follows from Lefschetz (1,1) and Lemma 2.6; for weight 4, this is given by conditional propositions under equivariant conditions);

(iii) (Dimension Matching) $\mathrm{dim}_{\mathbb{Q}}(W_p(\sigma) \cap H^{2p}(\cdot, \mathbb{Q})) = \rho_p(X_t) = \mathrm{dim}_{\mathbb{Q}}(H^{p,p}(X_t) \cap H^{2p}(X_t, \mathbb{Q}))$.

Then for a general fiber $X_t$, we have $H^{p,p}(X_t, \mathbb{Q}) = \mathrm{cl}(CH^p(X_t)_{\mathbb{Q}})$, i.e., the Hodge conjecture (or equivariant strong part) of this level holds on $X_t$.
**Proof.** Let $V := H^{p,p}(X_t, \mathbb{Q})$ and $U := W_p(\sigma) \cap H^{2p}(X_t, \mathbb{Q})$. By (iii) we have $\dim_{\mathbb{Q}} U = \dim_{\mathbb{Q}} V = \rho_p(X_t)$. By (i) and Clemens–Schmid, $U$ identifies with the classes on the general fiber whose limits lie in the algebraic subspace of $H^{2p}(X_0^\sim, \mathbb{Q})$. By Lemma 3.3, each $\xi \in U$ is the specialization of some relative algebraic cycle; hence for any $[\alpha] \in V$ with $sp([\alpha]) \in U$, the class $[\alpha]$ can be obtained by pulling back such a relative cycle. Condition (ii) (flatness/boundary) ensures that along a Zariski open set, these pulled-back classes remain of type $(p, p)$ and are algebraic. Finally, since $sp: V \to U$ is a linear map between $\mathbb{Q}$-spaces of equal dimension, its image has the same dimension as $V$; hence for general parameters $sp(V) = U$, and every $[\alpha] \in V$ satisfies the antecedent. The claim follows. $\square$

## 3.3 Singular Parameters and Algebraicity Propagation

In the three-layer screening framework proposed in this section, the third layer emphasizes “flat extension and boundary benignness.” Under controllable singular degenerations (such as conical/nodal degenerations of threefold Calabi–Yau varieties), one can establish a standard “boundary propagation” channel, allowing the algebraicity obtained at the limit to propagate back to nearby smooth fibers.

**Proposition 3.6 (Boundary Algebraicity and Propagation).** Let $\pi: \mathcal{Y} \to \Delta$ be a one-parameter semistable degeneration, with $\mathcal{Y}_t$ as general fibers and $\mathcal{Y}_0$ as the central fiber. Assume that the central fiber, after one or finitely many small/semi-small crepant resolutions, becomes simple normal crossing (SNC) with several exceptional curves/surfaces. Then:

(i) The weight 4 part in the limiting mixed Hodge structure (LMHS) is carried by the cohomology of SNC components of the central fiber, hence is algebraic; exceptional curves/surfaces produced by small crepant resolutions are also algebraic.

(ii) By the Clemens–Schmid sequence, comparison isomorphisms between weight layers identify the algebraic classes from (i) into a subspace of $\text{Gr}_4^W H_{\text{lim}}^4$; then by algebraicity of Hodge orbits [GGK12] and Deligne’s Fixed Part theorem (unipotent flat local subsystems preserve Hodge type on Zariski open sets), we obtain: if some $\beta \in H^{2,2}(\mathcal{Y}_t, \mathbb{Q})$ has limit specialization entering the above algebraic subspace, then on some non-empty Zariski open set in parameter space, $\beta$ remains of (2, 2) type and is an algebraic class.

**Proof Sketch.** (i) follows from Steenbrink’s weight spectral sequence and SNC geometry: the weight graded piece $\text{Gr}_4^W$ of the limit comes from dimension-2 algebraic subvarieties of the central fiber and their intersections (exceptional components are also algebraic). (ii) uses Clemens–Schmid to map algebraic windows to limiting weight layers, then applies algebraicity of Hodge orbits from [GGK12] and Fixed Part (see [PS08]) to ensure flat extension and preservation of Hodge type on Zariski open sets, so that algebraicity can propagate back to general fibers. $\square$

**Remark 3.7.** Proposition 3.6 is completely compatible with the three-layer screening proposed in this chapter: its (i) provides “algebraic realization of algebraic windows in the central fiber,” while its (ii) provides a unified template for “flat extension and boundary benignness.” In Chapter 5’s equivariant strong part and non-equivariant supplement for mirror threefolds, we will repeatedly invoke this propagation mechanism.

# 4 Constructive (1, 1) on K3: Nodal Degeneration Method

In this section, under the context of quartic K3, we give Theorem 4.6, which constructively realizes any rational (1, 1)-class as an algebraic divisor.
## 4.1 Nodal Degeneration and Semistable Model of Quartic K3

Let $X \subset \mathbb{P}^3$ be a smooth quartic K3. Take general points $p_1, \dots, p_k$ and local quartic polynomials $g_i$, constructing the family

$$X_t = \{f_0 + t \sum_{i=1}^{k} \lambda_i g_i = 0\} \subset \mathbb{P}^3 \times \Delta.$$

For general $(\lambda_i)$, the central fiber has $A_1$ nodes at $p_i$ and no other singularities (the upper bound of $k$ is controlled by parameter dimension, classically at most 16 nodes). Blowing up each node once, we get the semistable total space $\mathcal{X} \sim \Delta$ with central fiber $X_0 \sim X_0 \cup \bigcup_i E_i$, where $E_i \simeq \mathbb{P}^1 \times \mathbb{P}^1$ and the intersection of $E_i$ with $X_0$ is a smooth quadric curve. By Mayer-Vietoris, $\mathrm{Pic}(X_0)$ is generated by the hyperplane class $h$ and the various exceptional $(-2)$-curves.

## 4.2 PL Orthogonality, MV Generation and Linear Expression of Limit Classes

**Lemma 4.1 (Orthogonality of Vanishing Cycles and PL Matrix).** Let $X \subset \mathbb{P}^3$ be a smooth quartic K3, and $X_t$ be a one-parameter degeneration with $k$ $A_1$ nodes. For nodes in general position, we can choose vanishing cycles $\{\gamma_i\}_{i=1}^k \subset H_2(X_t, \mathbb{Z})$ such that they are pairwise orthogonal and $\langle \gamma_i, \gamma_j \rangle = -2\delta_{ij}$. Thus the Picard-Lefschetz transformations $T_i = \exp(N_i)$ satisfy $N_i^2 = 0$ and $N_i N_j = 0$ ($i \neq j$).

*Proof Sketch.* The Milnor fiber of each $A_1$ node is $S^2$, and its vanishing cycle has self-intersection $-2$. Vanishing cycles corresponding to different nodes can be taken orthogonal in general position, see[Dim17], [Mil68]. In one-parameter degeneration with locally independent perturbations of nodes, the $N_i$ can commute, see[FS85]. □

**Lemma 4.2 (Mayer-Vietoris and Generation of $\mathrm{Pic}(X_0)$).** Let $X_0 = X_0 \cup \bigcup_{i=1}^k E_i$ be the semistable model, where each $E_i \simeq \mathbb{P}^1 \times \mathbb{P}^1$ intersects $X_0$ in a smooth quadric curve $C_i$. Then there exists a short exact sequence

$$0 \to \mathrm{Pic}(X_0) \to \mathrm{Pic}(X_0) \oplus \left(\bigoplus_i \mathrm{Pic}(E_i)\right) \xrightarrow{\mathrm{res}} \bigoplus_i \mathrm{Pic}(C_i) \to 0,$$

so $\mathrm{Pic}(X_0)$ is generated by $h$ and one $(-2)$-curve class on each $E_i$ (the two rulings contribute only one independent class under gluing). See[PP81], [FS85].

**Proposition 4.3 (Linear Expression of Limit Classes).** Under the conditions of Lemmas 4.1 and 4.2, for any $[\alpha] \in H^{1,1}(X, \mathbb{Q})$, its specialization $\mathrm{sp}([\alpha])$ can be uniquely written as

$$\mathrm{sp}([\alpha]) = m_0 h + \sum_{i=1}^{k} m_i [E_i] \in \mathrm{Pic}(X_0) \otimes \mathbb{Q},$$

where $(m_i)$ is determined by the linear system $\langle [\alpha], \gamma_i \rangle = -2m_i$, and $m_0$ is determined by the degree condition.

*Proof.* By PL and $N_i^2 = 0$, the image of $\mathrm{sp}([\alpha])$ in $\mathrm{Gr}_2^W$ depends only on $\langle [\alpha], \gamma_i \rangle$. By Lemma 4.2, $\mathrm{Pic}(X_0)$ is generated by $h$ and $[E_i]$, and their pairing relationships are determined by gluing and intersection numbers. Thus the linear system has a unique solution, giving the above representation. □

**Remark 4.4 (Node Upper Bound and General Position for Quartic K3).** The node upper bound for smooth quartic surfaces is 16 (see[Dim17], also see related discussions in[BPVdV84]). The dimension of the linear system of quartics is 69, sufficient to arrange $k \le 16$ nodes in general position such that the corresponding vanishing cycles are linearly independent and orthogonal ([Dim17]).
## 4.3 Quartic K3: Node Upper Bound and Explicit Formula for Specialization

**Node Upper Bound** For smooth quartic surfaces $X \subset \mathbb{P}^3$, the number of $A_1$ nodes has upper bound 16 (see [End99]). All discussions of “nodal degeneration” in this paper are within the range $k \le 16$.

**PL Data and Pairing** Taking $k \le 16$ nodes in general position, denote the vanishing cycles $\{\gamma_i\}_{i=1}^k \subset H^2(X, \mathbb{Z})$, which can be chosen to satisfy

$$ \langle \gamma_i, \gamma_j \rangle = -2 \delta_{ij}. \qquad (1) $$

Let $h \in \mathrm{Pic}(X)$ be the hyperplane class ($h^2 = 4$). In subsequent formulas, $\langle \cdot, \cdot \rangle$ is the cup product pairing on $H^2$, which identifies $H_2$ and $H^2$ after Poincaré duality and parity conventions, consistent with (1).

**Explicit Formula for Clemens–Schmid Specialization** Let $\pi : \mathcal{X} \to \Delta$ be a one-parameter semistable degeneration. By resolving nodes point by point, we obtain the semistable model of the central fiber

$$ X_0 \sim = X_0 \cup \bigcup_{i=1}^{k} E_i, \quad E_i \simeq \mathbb{P}^1 \times \mathbb{P}^1. $$

Let $e_i \in \mathrm{Pic}(X_0)$ be the class of an arbitrarily chosen ruling on $E_i$ (the two rulings contribute only one independent direction under gluing, see [PP81, FS85]), and $\pi^*: H^2(X, \mathbb{Q}) \to H^2(X_0, \mathbb{Q})$ be the restriction pullback. Then for any $\alpha \in H^2(X, \mathbb{Q})$, under the Peters–Steenbrink convention (see [PS08, §17.3]), we have the explicit expression for specialization

$$ \mathrm{sp}(\alpha) = \pi^*\alpha - \frac{1}{2} \sum_{i=1}^{k} \langle \alpha, \gamma_i \rangle e_i \in H^2(X_0, \mathbb{Q}). \qquad (2) $$

In particular, if $X$ is a general quartic with $\rho(X) = 1$, then $\alpha = nh$ ($n \in \mathbb{Q}$), and by general position and (1) we can take the pairing $\langle h, \gamma_i \rangle = -2$, so

$$ \mathrm{sp}(nh) = n \left( h - \sum_{i=1}^{k} e_i \right). \qquad (3) $$

This expression is consistent with (2) under the choice of $e_i$ (interchange of two rulings) and overall sign conventions; it also satisfies the numerical consistency of Clemens–Schmid ($\mathrm{sp}\,\alpha)^2 = \alpha^2$.

**Proposition 4.5** (Normalization and the coefficient 1/2 in (2)). With the Picard–Lefschetz convention $\langle \gamma_i, \gamma_j \rangle = -2\delta_{ij}$ and Peters–Steenbrink’s normalization of the Clemens–Schmid isomorphism, the specialization formula

$$ \mathrm{sp}(\alpha) = \pi^*\alpha - \frac{1}{2} \sum_i \langle \alpha, \gamma_i \rangle e_i $$

holds. Moreover, it is uniquely determined by the two numerical compatibilities:

1. $(\mathrm{sp}\,\alpha)^2 = \alpha^2$ in $H^4$;

2. For each exceptional component $E_i \simeq \mathbb{P}^1 \times \mathbb{P}^1$ with glued ruling $e_i$, one has $\langle \mathrm{sp}(\alpha), e_i \rangle = \langle \alpha, \gamma_i \rangle$.
Proof. For clarity we begin with the single-node case (the multi-node case follows by orthogonality and linearity). Let the semistable central fiber be $X_0^\sim = V \cup E$, where $E \simeq \mathbb{P}^1 \times \mathbb{P}^1$ meets $V$ transversely along a smooth quartic curve $C$, and $\pi: V \to X$ is the restriction of the blow-up at the node in the total space. Let $e \in \text{Pic}(X_0^\sim)$ denote the class of one ruling on $E$ (the glued ruling that survives in $\text{Pic}(X_0^\sim)$).

Step 1 (linear decomposition and proportionality). By the Clemens–Schmid comparison and the generation statement in §4.3, we have a vector space decomposition

$$H^2(X_0^\sim, \mathbb{Q}) = \pi^*H^2(X, \mathbb{Q}) \oplus \mathbb{Q} \cdot e.$$

Hence there is a unique linear functional $k: H^2(X, \mathbb{Q}) \to \mathbb{Q}$ with

$$\text{sp}(\alpha) = \pi^*\alpha + k(\alpha)e. \qquad (4)$$

If $\langle \alpha, \gamma \rangle = 0$, then $N\alpha = 0$, so $\alpha$ is single-valued near the boundary and its Deligne extension has limit equal to $\pi^*\alpha$ (see [PS08, §17.3]); thus $k(\alpha) = 0$. It follows that $k$ factors through the vanishing pairing: there exists a constant $c \in \mathbb{Q}$ such that

$$k(\alpha) = c \langle \alpha, \gamma \rangle.$$

Step 2 (compatibility with pairings). Let $\gamma$ be the vanishing cycle and $e$ its corresponding class on the limit side. By the compatibility of Clemens–Schmid with the intersection pairings (see [PS08, §11.3, §17.3]), we have

$$\langle \text{sp}(\alpha), e \rangle = \langle \alpha, \gamma \rangle. \qquad (5)$$

On the other hand, by the projection formula and the fact that $e$ is supported on the exceptional component contracted by $\pi$, we have

$$\langle \pi^*\alpha, e \rangle = \langle \alpha, \pi_*e \rangle = 0. \qquad (6)$$

Moreover, $e$ is a $(-2)$-curve class (in this setting coming from an $A_1$ node and the gluing computation; see [PP81, FS85]), i.e.

$$e^2 = -2. \qquad (7)$$

Combining (4) with (5)–(7) gives

$$\langle \text{sp}(\alpha), e \rangle = \langle \pi^*\alpha, e \rangle + c \langle \alpha, \gamma \rangle e^2 = 0 + c \langle \alpha, \gamma \rangle \cdot (-2) = \langle \alpha, \gamma \rangle,$$

hence $c = -\frac{1}{2}$. Therefore

$$\text{sp}(\alpha) = \pi^*\alpha - \frac{1}{2} \langle \alpha, \gamma \rangle e.$$

If we switch the two rulings on $E$, then $e \mapsto -e$ and $\langle \alpha, \gamma \rangle \mapsto -\langle \alpha, \gamma \rangle$ simultaneously, so the formula is invariant.

For multiple nodes, use the orthogonality of $\{\gamma_i\}$ and linearity to obtain

$$\text{sp}(\alpha) = \pi^*\alpha - \frac{1}{2} \sum_i \langle \alpha, \gamma_i \rangle e_i.$$

□

**General Picard Number Case** If the Picard number is first raised to $\rho(X) = r$ before degeneration, taking a basis $\{h, \ell_1, \dots, \ell_{r-1}\}$ of $\text{Pic}(X) \otimes \mathbb{Q}$, then the coefficients in (2) are uniquely determined by the matrix $((\ell_j, \gamma_i))_{(r-1) \times k}$ and $\langle h, \gamma_i \rangle$, so the linear expression of $\text{sp}: H^2(X, \mathbb{Q}) \to \text{Pic}(X_0^\sim) \otimes \mathbb{Q}$ is still unique.
## 4.4 Picard–Lefschetz Pairing and Linear System

Let $\gamma_i$ be the vanishing cycle corresponding to the $i$-th node, with $\langle \gamma_i, \gamma_j \rangle = -2\delta_{ij}$. For $[\alpha] \in H^{1,1}(X, \mathbb{Q})$, consider the linear system

$$ \langle [\alpha], \gamma_i \rangle = -2m_i, \quad 1 \le i \le k. $$

For a general node configuration, $\{\gamma_i\}$ are linearly independent and mutually orthogonal, so the above system uniquely determines $(m_i) \in \mathbb{Q}^k$. Thus the limit of $[\alpha]$ is

$$ sp([\alpha]) = m_0h + \sum_{i=1}^{k} m_i [E_i] \in \mathrm{Pic}(X_0^{\sim}) \otimes \mathbb{Q}. $$

## 4.5 Constructive (1, 1) Theorem

**Theorem 4.6** (Constructive (1, 1) for K3). Let $X \subset \mathbb{P}^3$ be a smooth quartic K3, and $[\alpha] \in H^{1,1}(X, \mathbb{Q})$ arbitrary. There exists a semistable degeneration with at most 16 nodes in general position such that $sp([\alpha]) \in \mathrm{Pic}(X_0^{\sim}) \otimes \mathbb{Q}$, and after pullback $[\alpha]$ is an algebraic class. More specifically, by §4.3, the number of nodes $k \le 16$, and the specialization $sp(\alpha) = \pi^*\alpha - \frac{1}{2}\sum_i \langle \alpha, \gamma_i \rangle e_i$. When $\rho(X) = 1$, (3) gives $sp(nh) = n(h - \sum_i e_i)$. Through Clemens–Schmid pullback and the Lefschetz (1, 1) theorem, $[\alpha]$ is an algebraic divisor class.

*Proof.* Choose $k$ nodes in general position and $g_i$ such that the PL pairing matrix is diagonal $-2I_k$, so the linear system has a unique solution $(m_i)$. After semistable reduction, Clemens–Schmid gives $\mathrm{Gr}_2^W H_{\lim}^2 \simeq H^2(X_0^{\sim}, \mathbb{Q})$, and by Lemma 4.2 we know $\mathrm{Pic}(X_0^{\sim})$ is generated by $h$ and $[E_i]$, so by Proposition 4.3, $sp([\alpha])$ is an algebraic class. Pulling back to the general fiber $X_t$, $[\alpha]$ is a (1, 1) class, so by Lemma 2.7 it must be algebraic. Specific calculations of $\nabla_{GM}$ flat sections and Deligne extensions for single node degenerations are given in Appendix A.1. $\square$

**Remark 4.7.** Theorem 4.6 does not claim new (1, 1) results (Lefschetz (1, 1) is already known). Its value lies in providing a constructive realization—through the finite-step algorithm of nodal degeneration–specialization–pullback, it explicitly produces an algebraic divisor representing $[\alpha]$. Comparing with Theorem 3.5, here $W_1(\sigma) = \ker N \cap (F_\infty^1 \cap \bar{F}_\infty^1)$, and in this model it dimension-matches with $h^{1,1}(X)$, satisfying the “algebraic realization + pullback” conditions.

# 5 (2, 2) Classes of Threefold Calabi–Yau

This chapter studies (2, 2)-classes of threefold Calabi–Yau varieties at weight 4. First, we establish “strong parts” in the equivariant setting: within the framework of nodal degenerations satisfying mild boundary benignness and equivariant flatness, all $G$-invariant (2, 2)-classes are algebraic (see the preceding sections of this chapter). Second, we provide a non-equivariant supplement for mirror quintic threefolds that is independent of LMHS/DMP: using toric even cohomology and Gysin exact sequences, one can directly prove that $H^{2,2}$ is entirely spanned by algebraic surface restriction classes (see Section 5.8; background tools are summarized in Appendix C). The two approaches corroborate each other and remain consistent with the benchmark conclusions in Section 2.5.

The organization is as follows: the beginning of §5 reviews compatibility of specialization with cycle class homomorphisms, equivariant flatness, and pullback criteria; then we perform window characterization and dimension matching under LCSL/MCP paths; finally, Section 5.8 provides the non-equivariant supplement for mirror threefolds and consistency explanations.
## 5.1 Compatibility of Specialization and Cycle Class Maps

Let $\pi: \mathcal{Y} \to \Delta$ be a one-parameter degeneration, $t \neq 0$ a general point, and 0 the limit. Denote

$$ \mathrm{sp}_{CH}: CH^p(Y_t) \to CH^p(Y_0^{\mathrm{rel}}) \quad \text{and} \quad \mathrm{sp}_H: H^{2p}(Y_t, \mathbb{Q}) \to H^{2p}(Y_0^{\mathrm{rel}}, \mathbb{Q}) $$

as the specializations of Chow groups and cohomology respectively ($Y_0^{\mathrm{rel}}$ represents the relative limit object, such as semistable model or its appropriate decomposition). Then the cycle class map is compatible with specialization, i.e., there exists a commutative diagram

$$ \begin{tikzcd} CH^p(Y_t) \arrow[r, "\mathrm{cl}"] \arrow[d, "\mathrm{sp}_{CH}"] & H^{2p}(Y_t, \mathbb{Q}) \arrow[d, "\mathrm{sp}_H"] \\ CH^p(Y_0^{\mathrm{rel}}) \arrow[r, "\mathrm{cl}"] & H^{2p}(Y_0^{\mathrm{rel}}, \mathbb{Q}) \end{tikzcd} $$

**Corollary 5.1** (Relative lifting for window classes). Under the assumptions of Lemma 3.3, any $\xi \in W_p(\sigma) \cap \ker N$ that is algebraic on the central fiber is the specialization of an algebraic class from a relative cycle $Z \in CH^p(\mathcal{X})_{\mathbb{Q}}$. In particular, the right vertical map $\mathrm{sp}_H$ hits $W_p(\sigma)$ on the nose.

This compatibility can be given by Fulton's specialization construction and Saito's nearby/vanishing cycle functors for mixed Hodge modules (see [Ful98], [Sai90]), combined with the functorial compatibility of LMHS from [PS08]. The compatibility of pushforward/pullback with cohomology/Hodge for small/semismall resolutions can be found in [dCM09].

## 5.2 Equivariant Assumptions and Weight 4 Pullback Criterion

**Assumption A (Equivariant Algebraic Decomposition)** The central fiber $Y_0$ admits a $G$-equivariant projective crepant resolution (small/semismall) $q: \tilde{Y}_0 \to Y_0$, whose exceptional set consists of several smooth surfaces $F_i$. This assumption is guaranteed by MPCP decomposition in toric/mirror symmetric environments (such as Dwork/Fermat type) ([Bat94, CK99]).

**Theorem 5.2** (Existence of $G$-equivariant projective crepant MPCP refinements). In the Dwork/Fermat/Greene-Plesser mirror set-up considered in this paper, there exists a $G$-equivariant projective crepant (small/semismall) toric refinement $q: \tilde{Y}_0 \to Y_0$ as in Assumption A.

*Proof.* Let $\Delta \subset M_{\mathbb{R}}$ be a reflexive polytope, $\Delta^\vee \subset N_{\mathbb{R}}$ its dual reflexive polytope, and $\Sigma = \Sigma(\Delta^\vee)$ the normal fan, giving the (possibly singular Gorenstein) toric ambient variety $X(\Sigma)$. A finite group $G \leq \mathrm{Aut}(N)$ acts preserving $\Delta^\vee$ and $\Sigma$. We construct a $G$-equivariant projective crepant refinement $\Sigma' \succ \Sigma$ from a strictly convex support function.

(1) (Existence of MPCP refinements). As shown by Batyrev, any fine regular star triangulation (FRST) of the lattice points on $\partial\Delta^\vee$ induces a refinement $\Sigma' \succ \Sigma$ whose new rays are generated by lattice points on $\partial\Delta^\vee$ and such that $K_{X(\Sigma')} = f^*K_{X(\Sigma)}$; hence the refinement is crepant. If the triangulation is induced by a strictly convex piecewise linear support function, the morphism $f: X(\Sigma') \to X(\Sigma)$ is projective (see [Bat94, Thm. 4.1.9, Prop. 4.1.2] and [CK99, §4.2]). This yields a projective crepant MPCP refinement.

(2) (G-equivariance). Choose a strictly convex support function $\varphi$ inducing some MPCP refinement $\Sigma'$. Average it over $G$ to obtain $\varphi_G := \frac{1}{|G|} \sum_{g \in G} \varphi \circ g^{-1}$; then $\varphi_G$ is still strictly convex and $G$-invariant, and defines a $G$-invariant regular refinement $\Sigma'_G \succ \Sigma$. Equivalently, one may take the set of new rays $S \subset N$ (given by lattice points on $\partial\Delta^\vee$) and perform star subdivisions in $G$-orbits; the result coincides with the refinement induced by $\varphi_G$. Thus we obtain a $G$-equivariant projective crepant refinement $f: X(\Sigma'_G) \to X(\Sigma)$.

(3) (Small/semismall behavior in our set-up). In the Calabi-Yau hypersurface situation treated here, MPCP refinements are crepant on the ambient and induce crepant resolutions on
the hypersurface; the exceptional locus over the boundary strata has the expected (semi)small behavior by the combinatorics of rays and faces in the reflexive setting (see [CK99, §4.2, §5.1]). In particular, on the central fiber built from toric boundary components, the induced $q: \tilde{Y}_0 \to Y_0$ is $G$-equivariant, projective, and crepant with exceptional set consisting of smooth toric surfaces and their intersections as required in Assumption A.

□

**Remark 5.3.** Thus, throughout §5.2, Assumption A holds for the families we treat; we keep it as an assumption only to emphasize the minimal hypotheses needed by Proposition 5.4.

**Assumption B (Limit Algebraic Realization)** The comparison isomorphism $\phi: \mathrm{Gr}_4^W H_{\mathrm{lim}}^4 \xrightarrow{\sim} H^4(\tilde{Y}_0, \mathbb{Q})$ maps $W_2(\sigma)^G$ into the algebraic part of $H^4(\tilde{Y}_0, \mathbb{Q})^G$ (spanned by $G$-invariant linear combinations of $\{[F_i], [F_i \cdot F_j]\}$).

**Assumption C (Equivariant Flatness)** For any $\beta \in H^{2,2}(Y_t, \mathbb{Q})^G$, if $\mathrm{sp}(\beta) \in W_2(\sigma)^G$, then there exists a $G$-equivariant $\nabla$-flat section $\tilde{\beta}$, whose Deligne extension limit falls in the algebraic subspace of Assumption B.

**Proposition 5.4 (Weight 4 Equivariant Pullback Criterion).** Under Assumptions A–C and the compatibility of §5.1, $\beta$ is a $G$-invariant algebraic class on the general fiber $Y_t$.

*Proof.* By Assumption B, $\phi(\mathrm{sp}(\beta))$ is an algebraic class on $\tilde{Y}_0$. By the commutative diagram in §5.1, this algebraicity can be transmitted back to the family via specialization/pushforward compatibility, then by the fixed part theorem and Hodge locus algebraicity (Lemmas 2.6, 2.4), the (2,2) type is preserved in a Zariski open set, so $\beta$ is an algebraic class on general fibers. Specific constructions of flat extensions in multi-node cases are given in Appendix A.2. □

## 5.3 Setup and Equivariant Algebraic Window

Let $\pi: \mathcal{Y} \to \Delta$ be a $G$-equivariant $\mathrm{CY}^3$ semistable degeneration, where the central fiber contains only ordinary double points, and its equivariant semistable model is denoted $Y_0^{\sim}$. Consider the weight 4 algebraic window

$$W_2(\sigma) := \mathrm{Gr}_4^W H_{\mathrm{lim}}^4 \cap F_\infty^2 \cap \overline{F_\infty^2} \cap H^4(\cdot)^G,$$

and identify $W_2(\sigma) \subset H^4(Y_0^{\sim}, \mathbb{Q})^G$ through $\phi$.

**Definition 5.5 (Equivariant Algebraic Realization).** If $W_2(\sigma) \subset H^4(Y_0^{\sim}, \mathbb{Q})^G$ lies in the $G$-equivariant algebraic subspace generated by exceptional divisors (new components in resolution or small resolution) and their intersections, then we say $W_2(\sigma)$ is algebraically realizable in the $G$-equivariant sense.

## 5.4 Equivariant Strong Part

**Theorem 5.6 (CY$^3$ Equivariant (2,2) Strong Part).** Let $\pi: \mathcal{Y} \to \Delta$ be a $G$-equivariant $\mathrm{CY}^3$ nodal semistable degeneration satisfying:

(i) (G-Window) $W_2(\sigma) \subset H^4(Y_0^{\sim}, \mathbb{Q})^G$ is algebraically realizable (Definition 5.5);

(ii) (G-Flatness/Pullback) For any $\beta \in H^{2,2}(Y_t, \mathbb{Q})^G$, if $\mathrm{sp}(\beta) \in W_2(\sigma)$, then under Assumptions A–C, by Proposition 5.4, $\beta$ is a $G$-invariant algebraic class in a Zariski open set;

(iii) (G-Dimension Matching) $\mathrm{dim} W_2(\sigma) = \mathrm{dim} H^{2,2}(Y_t, \mathbb{Q})^G$.

Then $H^{2,2}(Y_t, \mathbb{Q})^G = \mathrm{cl}(CH^2(Y_t)\mathbb{Q})^G$, i.e., all $G$-invariant (2,2)-classes are algebraic.
Proof. Same as Theorem 3.5, we just need to apply three-layer screening and dimension matching within the G-invariant subspace. By (i) and (ii), all classes with sp(β) ∈ W$_2(σ)$ are algebraic after pullback; by (iii) matching, all G-invariant (2, 2)-classes satisfy the antecedent, so the conclusion holds. □

## 5.5 Dwork Quintic LCSL: G-Invariant Description of sp(h$^2$)

Taking the LCSL limit $λ = ψ^{-1} → 0$, the central fiber

$$Y_0 = \bigcup_{i=0}^{4} H_i, \quad H_i = \{x_i = 0\} \simeq \mathbb{P}^3$$

is SNC. Denote

$$S_{ij} := H_i \cap H_j \simeq \mathbb{P}^2 \quad (0 \le i < j \le 4), \quad L_{ijk} := H_i \cap H_j \cap H_k \simeq \mathbb{P}^1 \quad (0 \le i < j < k \le 4).$$

Then under Steenbrink's limit spectral sequence and Mayer–Vietoris boundary map $δ$ with the “standard addition-subtraction restriction” convention (see [PS08, §11]), $δ : H^4(Y_0, \mathbb{Q}) → H^6(Y_0^{[1]}, \mathbb{Q})$ acts as

$$\delta\left(\sum_{i<j} c_{ij} [S_{ij}]\right) = \sum_{i<j<k} (c_{ij} + c_{ik} + c_{jk}) [L_{ijk}],$$

while

$$\mathrm{sp}(h^3) = 5 \sum_{i<j<k} [L_{ijk}] \in H^6(Y_0, \mathbb{Q}),$$

where $5 = h^3 = \deg X$. Therefore there exists a unique G-invariant class

$$\mathrm{sp}(h^2) = c \sum_{i<j} [S_{ij}] \quad \text{for some } c \in \mathbb{Q}^\times, \qquad (8)$$

such that $δ \mathrm{sp}(h^2) = \mathrm{sp}(h^3)$. Under the above convention, since each $L_{ijk}$ in $δ$ comes from three pairs $(ij), (ik), (jk)$, we can solve to get $3c = 5$, i.e., $c = 5/3$. If other equivalent conventions are used (e.g., overall scaling of $δ$ or parameter rescaling), then $c$ scales accordingly, but the “G-invariant collinearity” of (8) is invariant, which is sufficient for dimension matching and pullback in DMP.

**Remark 5.7.** There are different conventions for the sign and normalization of $δ$ in the literature. This paper only fixes $c = 5/3$ when needed; all proofs only use the “sp(h$^2$) is collinear with $\sum_{i<j} [S_{ij}]$” conclusion of G-invariance.

## 5.6 Example: Dwork Quintic Under LCSL Limit and MPCP Decomposition

Consider the Dwork quintic three-dimensional CY family

$$X_ψ := \{x_0^5 + \cdots + x_4^5 - 5ψ x_0 x_1 x_2 x_3 x_4 = 0\} \subset \mathbb{P}^4.$$

Let $λ = ψ^{-1}$ and take the large complex structure limit (LCSL) $λ → 0$. The central fiber is

$$Y_0 = \{x_0 x_1 x_2 x_3 x_4 = 0\} = \bigcup_{i=0}^{4} H_i,$$

where $H_i = \{x_i = 0\}$ are coordinate hyperplanes, and $Y_0$ is SNC. This path does not require introducing exceptional surfaces. If we want to characterize exceptional surfaces and their intersections, we can choose a MPCP (maximal projective crepant partial) decomposition in
the toric environment, obtaining a $G$-equivariant projective crepant map $q: \tilde{Y}_0 \to Y_0$ ([Bat94, CK99]). The exceptional components $F_i$ are several toric surfaces, whose intersections are determined by adjacency data of the fan (see intersection theory in [Ful93]).

In the SNC model, the algebraic part of $H^4(Y_0, \mathbb{Q})$ is generated by irreducible components and classes of their intersection subvarieties, for example

$$S_{ij} := H_i \cap H_j \simeq \mathbb{P}^2, \quad 0 \le i < j \le 4.$$

The $G$-invariant combination $\sum_{i<j}[S_{ij}]$ spans a 1-dimensional subspace. On the other hand, the square $h^2 \in H^{2,2}(X_\psi, \mathbb{Q})$ of the hyperplane class $h$ on the general fiber, after specializing to $Y_0$, differs from the above $G$-invariant algebraic combination by only a non-zero rational coefficient (compare intersection numbers with relative hyperplane sections, or calculate the ratio using toric intersection theory, see [CK99, Ful93]).

## 5.7 Verification of Equivariant Strong Part for Dwork Quintic

**Theorem 5.8.** For the Dwork quintic $X_\psi$, take a standard symmetric subgroup $G \subset (\mu_5)^4 \rtimes S_5$ (Greene-Plesser type). Then for any general $\psi$:

(i) $H^{2,2}(X_\psi, \mathbb{Q})^G$ is 1-dimensional and generated by $h^2$;

(ii) Under the LCSL limit, $W_2(\sigma)^G$ is also 1-dimensional; by §5.5, $\text{sp}(h^2)$ is collinear with $\sum_{i<j}[S_{ij}]$. Under the Mayer-Vietoris boundary convention adopted in this paper, $\text{sp}(h^2) = \frac{5}{3} \sum_{i<j}[S_{ij}]$. This ratio does not affect the identification of the $G$-invariant one-dimensional subspace and the matching in DMP;

(iii) The three assumptions of Theorem 3.5 are satisfied (taking $q = \text{Id}$ under the SNC path, or taking $q$ as projective crepant decomposition under toric MPCP): (i) the image of the algebraic window falls in the algebraic part; (ii) pullback is completed by the assumptions of Proposition 5.4 and the compatibility of §5.1; (iii) $\mathbb{Q}$-dimension matching 1 = 1.

Therefore, $G$-invariant (2, 2) classes are all algebraic.

*Proof.* (i) For smooth quintic three-dimensional CY, we have $h^{1,1} = 1$ and $h^{2,2} = h^{1,1} = 1$, so $\rho_2(X_\psi) = 1$, and $H^{2,2}(X_\psi, \mathbb{Q})^G$ is also 1-dimensional, generated by $h^2$.

(ii) Under one-parameter LMHS, $\text{Gr}_4^W$ is related to $\text{ker } N$, and the dimension of the $G$-invariant part of $W_2(\sigma) = \text{Gr}_4^W \cap F_\infty^2 \cap \tilde{F}_\infty^2 \cap \text{ker } N$ is 1 (fixed part theorem and the dimension of $G$-invariant flat subsystems is constant), with generator $\text{sp}(h^2)$. As described in the previous section, $\text{sp}(h^2)$ is collinear with $\sum_{i<j}[S_{ij}]$, and the latter is an algebraic class.

(iii) Thus all three conditions of Theorem 3.5 are satisfied. If we adopt SNC ($q = \text{Id}$), then (i) holds directly; if we adopt toric MPCP ($q$ projective $G$-equivariant), then (i) is guaranteed by Assumptions A-B, and (ii) is given by Proposition 5.4 and the compatibility of §5.1. The conclusion follows. $\square$

**Remark 5.9.** Theorem 5.6 is a “strong part” in the equivariant sense, applicable to highly symmetric families (such as Dwork/Fermat type), where the $G$-invariant subspace has small dimension and clear geometric characteristics, making the three-layer conditions easy to verify. The complete non-equivariant (2, 2) Hodge conjecture still requires deeper tools.

## 5.8 Non-equivariant (2, 2) Algebraicity of Mirror Threefolds and Consistency with Equivariant Strong Parts

This subsection supplements the description of the non-equivariant aspect of mirror quintic threefolds: without relying on LMHS/DMP, one can obtain algebraicity of all (2, 2) classes using toric and Gysin tools. This is consistent with the conclusions of the equivariant strong parts above and provides an independent “baseline” for dimension matching.
**Theorem 5.10.** Let $T$ be a smooth projective toric fourfold obtained by Greene-Plesser quotient followed by MPCP (or weak-fan) resolution, and let $Y_\psi \subset T$ be a smooth anticanonical hypersurface (mirror quintic threefold). Then for any smooth parameter $\psi$,

$$H^{2,2}(Y_\psi, \mathbb{Q}) = \mathrm{cl}(\mathrm{CH}_2(Y_\psi))_{\mathbb{Q}},$$

and the right-hand side is spanned by $\mathbb{Q}$-linear combinations of restriction classes of torus-invariant surfaces $S_{ij} = D_i \cap D_j$ in $T$ to $Y_\psi$.

*Proof.* See Appendix C: the even cohomology of smooth projective toric varieties is entirely generated by algebraic cycles, with $\mathrm{cl}: \mathrm{CH}^*(T)_{\mathbb{Q}} \xrightarrow{\cong} \mathrm{H}^{2\ast}(T, \mathbb{Q})$; on the other hand, the Gysin sequence for the inclusion $i: Y_\psi \hookrightarrow T$ gives a surjection $i^*: \mathrm{H}^4(T, \mathbb{Q}) \to \mathrm{H}^4(Y_\psi, \mathbb{Q})$ in degree 4. Hence $\mathrm{H}^{2,2}(Y_\psi, \mathbb{Q})$ is entirely algebraic and spanned by restriction classes of $S_{ij}$. $\square$

**Remark 5.11 (Consistency with Equivariant Strong Parts).** Under the Mayer-Vietoris convention in the large complex structure limit (LCSL) path, the specialization $\mathrm{sp}(h^2)$ of $h^2$ on $X_\psi$ is collinear with $\sum_{i<j} [S_{ij}]$; using the boundary numbering convention of this manuscript, one can solve for the proportionality constant $c = \frac{5}{3}$ (see Appendix C). This “collinearity” is consistent with the window matching in $G$-invariant directions in the equivariant strong parts of Chapter 5: the equivariant level gives algebraicity of $\mathrm{H}^{2,2}(Y_\psi, \mathbb{Q})^G$, while Theorem 5.10 further shows that the entire non-equivariant $\mathrm{H}^{2,2}(Y_\psi, \mathbb{Q})$ is also spanned by algebraic surface restriction classes.

# 6 Equivariant (2, 2) Strong Part on Dwork Sextic Calabi-Yau Fourfolds

In this section, we rigorously implement the “window-pullback-matching (DMP)” paradigm in a four-dimensional example, giving a high-dimensional and non-trivial equivariant algebraicity result. Let

$$X_{\psi} = \{[x_0 : \cdots : x_5] \in \mathbb{P}^5 \mid f_{\psi}(x) = 0\}, \quad f_{\psi}(x) = \sum_{i=0}^{5} x_i^6 - 6\psi x_0 x_1 x_2 x_3 x_4 x_5,$$

where the parameter $\psi$ is taken in an open set $S \subset \mathcal{A}^1$ excluding the discriminant locus, ensuring $X_\psi$ is smooth. Let

$$G := \{(\zeta_0, \dots, \zeta_5) \in \mu_6^6 \mid \prod_{i=0}^{5} \zeta_i = 1\} \rtimes S_6$$

act on the entire family $X \to S$ by coordinate phases and permutations. Denote $H \in \mathrm{H}^2(X_\psi, \mathbb{Q})$ as the restriction of the hyperplane class.

To state the “strong part”, we fix the large complex structure limit (LCSL) direction $\psi \to \infty$. For the family $X \to S$, we take a $G$-equivariant finite base change and standard semistable reduction to get $\pi: \mathcal{X} \to \Delta$, with central fiber $X_0$ being the normal crossing union of six coordinate hyperplanes. Denote

$$Y^{[0]} = \bigsqcup_{i} \mathbb{P}^4, \quad Y^{[1]} = \bigsqcup_{i<j} \mathbb{P}^3, \quad Y^{[2]} = \bigsqcup_{i<j<k} \mathbb{P}^2$$

as the disjoint unions of intersections of various orders. Steenbrink’s weight spectral sequence $E_1^{-p,q} = \mathrm{H}^{q-2p}(Y^{[p]})(-p) \Rightarrow \mathrm{H}^q(X_t)$ has its “algebraic window” at $q=4$ generated by

$$E_1^{0,4} = \mathrm{H}^4(Y^{[0]}), \quad E_1^{-1,4} = \mathrm{H}^2(Y^{[1]})(1), \quad E_1^{-2,4} = \mathrm{H}^0(Y^{[2]})(2)$$

classes, all coming from algebraic cycles.
**Definition 6.1** (Equivariant Strong Part). Under the above semistable degeneration, define the equivariant strong part on general fibers as

$$ \mathrm{Strong}_{G}^{2,2}(X_t) := \mathrm{sp}_t^{-1}(W_{\mathrm{alg}}^{G}) \subset \mathrm{H}^{2,2}(X_t, \mathbb{Q})^{G}, $$

where $W_{\mathrm{alg}}^G \subset H_{\mathrm{lim}}^4(X_t)^G$ is the $G$-invariant part of the image of $E_1^{0,4} \oplus E_1^{-1,4} \oplus E_1^{-2,4}$ in the limit mixed Hodge structure, and $\mathrm{sp}_t$ is the limit specialization map. Hereafter, “strong part” refers to this definition relative to the oriented LCSL.

## 6.1 Primitive Part Invariant Dimension: Jacobian Ring Calculation

Let $S = \mathbb{Q}[x_0, \dots, x_5]$, $J(f_\psi)$ be the Jacobian ideal of $f_\psi$, and $R_\psi := S/J(f_\psi)$. Griffiths theory gives

$$ H_{\mathrm{prim}}^{2,2}(X_\psi, \mathbb{Q}) \cong (R_\psi)_{12}. $$

The $G$ action is induced by coordinate phases and permutations, varying algebraically with $\psi$.

**Proposition 6.2.** For any $\psi \in S$, we have $\dim_{\mathbb{Q}} H_{\mathrm{prim}}^{2,2}(X_\psi, \mathbb{Q})^G = 1$. More specifically, at $\psi = 0$ (Fermat fiber),

$$ (R_0)_{12}^G = \mathbb{Q} \cdot [x_0^2 x_1^2 x_2^2 x_3^2 x_4^2 x_5^2]. $$

*Proof.* At $\psi = 0$, we have $J(f_0) = (x_0^5, \dots, x_5^5)$, so $(R_0)_{12}$ is generated by monomials $x^b = x_0^{b_0} \cdots x_5^{b_5}$ satisfying $0 \le b_i \le 4$ and $\sum b_i = 12$. The phase group $K := \{(\zeta_i) \in \mu_6^6 \mid \prod \zeta_i = 1\}$ acts on $x^b$ with character $\chi_{b+1}(\zeta) = \prod \zeta_i^{b_i+1}$. $\chi_{b+1}$ is trivial on $K$ if and only if $b_i + 1$ are all equal in $\mathbb{Z}/6\mathbb{Z}$; given $0 \le b_i \le 4$ and sum 12, the unique solution is $b = (2, 2, 2, 2, 2, 2)$. This monomial is clearly $S_6$-invariant, so $\dim(R_0)_{12}^G = 1$.

Since the $G$ action is induced by fixed linear automorphisms, the dimension of the $G$-invariant subspace on $H^4(X_\psi)$ is locally constant in $\psi$; combining the above calculation with semicontinuity, the conclusion holds for general $\psi$, and by connectedness for all $\psi \in S$. $\square$

**Theorem 6.3** (Zariski-open constancy of $G$-invariant primitive rank). Let $S$ be the parameter line with the discriminant removed. Then there exists a Zariski open dense subset $U \subset S$ such that for all $\psi \in U$,

$$ \dim_{\mathbb{Q}} H_{\mathrm{prim}}^{2,2}(X_\psi, \mathbb{Q})^G = 1. $$

*Proof.* By Proposition B.2 (Appendix B), $V^G := R_{\mathrm{prim}}^4 \pi_* \mathbb{Q}$ fixed by $G$ is a sub-local system of constant rank on each connected component of $S$. Semicontinuity implies that the Hodge (2, 2)-part of $V^G$ has constant dimension on a Zariski open subset $U \subset S$. The explicit enumeration at the Fermat point (Lemma B.1) gives rank 1, hence the same holds on $U$. $\square$

**Remark 6.4.** By Lefschetz decomposition, $H^4(X_\psi, \mathbb{Q})^G = \mathbb{Q} \cdot H^2 \oplus H_{\mathrm{prim}}^4(X_\psi, \mathbb{Q})^G$ is an orthogonal direct sum, so

$$ \dim_{\mathbb{Q}} H^{2,2}(X_\psi, \mathbb{Q})^G \ge 1 (\text{ambient } H^2) + 1 (\text{primitive invariant}) = 2. $$

## 6.2 Equivariant Semistable Degeneration and Algebraic Window

**Lemma 6.5** (Semistable Model and Window). There exists a $G$-equivariant finite base change and standard semistable reduction such that the central fiber

$$ X_0 = \bigcup_{i=0}^{5} H_i, \quad H_i = \{x_i = 0\} \cong \mathbb{P}^4 $$
is a normal crossing union, with double intersections $H_i \cap H_j \cong \mathbb{P}^3$ and triple intersections $H_i \cap H_j \cap H_k \cong \mathbb{P}^2$. In Steenbrink's spectral sequence at $q = 4$,

$$E_1^{0,4} = H^4(\bigsqcup \mathbb{P}^4), \quad E_1^{-1,4} = H^2(\bigsqcup \mathbb{P}^3)(1), \quad E_1^{-2,4} = H^0(\bigsqcup \mathbb{P}^2)(2)$$

are entirely generated by algebraic classes, and the $G$ action on each level is via permutation of components and natural pullbacks. Denote $W_{\text{alg}} \subset H_{\text{lim}}^4$ as the image of these three levels in $E_{\infty}$. Then $W_{\text{alg}}$ satisfies on its $G$-invariant part

$$\dim_{\mathbb{Q}} W_{\text{alg}}^G \geq 2,$$

where one dimension is spanned by the ambient class (limit of $H^2$ on each $\mathbb{P}^4$), and another dimension is spanned by the matching class from “constant weight assignment” on double intersections.

**Proof Outline.** $X_0$ is the union of coordinate hyperplanes, with intersections being linear spaces, so the cohomology groups of each $E_1$-level are generated by algebraic classes; $G$ acts on $X_0$ by permuting these linear components. On $G$-invariant subspaces, $E_1^{0,4,G}$, $E_1^{-1,4,G}$, $E_1^{-2,4,G}$ are each one-dimensional (taking constant values on each orbit).

The generator of $E_1^{-1,4,G}$ can be understood as Cartier data of “hyperplane class on each $\mathbb{P}^3$ double intersection taking the same coefficient”. By Mayer–Vietoris/Čech interpretation, there exists a non-zero matching combination whose $d_1$-boundary is zero, thus giving a non-zero limit class in $E_{\infty}$; this provides the second direction of $W_{\text{alg}}^G$. The ambient direction clearly exists. Their linear independence can be determined by projection to components or through limit pairing (see next section). Specific construction of the semistable model and differential calculation of the spectral sequence are given in Appendix B.2. $\square$

## 6.3 Pullback: Algebraic Extension of Window Classes

**Lemma 6.6 (Pullback).** Let $\xi \in W_{\text{alg}}^G$. Then there exists a $G$-invariant algebraic 2-cycle $Z$ on the total space $\mathcal{X}$ (allowing consideration after finite base change) such that for general $t$,

$$\mathrm{cl}(Z|_{X_t}) \in H^{2,2}(X_t, \mathbb{Q})^G, \quad \mathrm{sp}_t(\mathrm{cl}(Z|_{X_t})) = \xi.$$

**Proof Sketch.** In the SNC case, the cycle class map is compatible with Steenbrink's spectral sequence, and the $E_1$ levels come from algebraic classes. Take an $E_1$-representative of $\xi$, first glue it into an algebraic 2-cycle on the central fiber using matching data, then use the compatibility of specialization/lifting (relative Chow group moving lemma and Saito-type specialization) to get a relative algebraic cycle $Z$ on the total space. Apply $G$-averaging to $Z$ to get the $G$-invariant conclusion. Explicit construction of matching cycles and technical details of pullback are given in Appendix B.3. $\square$

## 6.4 Main Theorem and Matching

**Theorem 6.7 (Equivariant Strong Part on Dwork Sextic CY4).** There exists a Zariski open set $U \subset S$ in the parameter space such that for all $\psi \in U$:

(i) $H^{2,2}(X_{\psi}, \mathbb{Q})^G = \mathbb{Q} \cdot H^2 \oplus \mathbb{Q} \cdot \Theta_{\psi}$, where $\Theta_{\psi} \in H^{2,2}(X_{\psi}, \mathbb{Q})^G$ is some $G$-invariant primitive class;

(ii) $\Theta_{\psi}$ can be constructed by algebraic cycles, and its limit $\mathrm{sp}_{\psi}(\Theta_{\psi}) \in W_{\text{alg}}^G$ is the “matching class” direction in Lemma 6.5;

(iii) Therefore $\mathrm{Strong}_{G}^{2,2}(X_{\psi}) = H^{2,2}(X_{\psi}, \mathbb{Q})^G$.
Proof. By Proposition 6.2 and Lefschetz decomposition, dim $H^{2,2}(X_\psi, \mathbb{Q})^G \ge 2$, and $H_{\text{prim}}^{2,2}(X_\psi, \mathbb{Q})^G$ is one-dimensional. Taking the Fermat point $\psi = 0$, $H_{\text{prim}}^{2,2}(X_0, \mathbb{Q})^G$ is generated by the residue class corresponding to $[x_0^2 \cdots x_5^2]$; along $S$ it forms a locally trivial $G$-invariant sub-local system, giving the primitive invariant direction for general $\psi$.

On the other hand, Lemma 6.5 gives two linearly independent limit directions in $W_{\text{alg}}^G$: the limit of ambient class $\tilde{H}^2$ and the matching class $\tilde{\Pi}$. By Lemma 6.6, there exist $G$-invariant algebraic cycles $Z_H, Z_\Pi$ such that

$$ \text{sp}_\psi(\text{cl}(Z_H|_{X_\psi})) = \tilde{H}^2, \quad \text{sp}_\psi(\text{cl}(Z_\Pi|_{X_\psi})) = \tilde{\Pi}. $$

The former equals the ambient class $H^2$ on general fibers. The latter gives some $G$-invariant (2,2) class $\Theta_\psi$ on general fibers. If there exist $\alpha, \beta \in \mathbb{Q}$ such that $\alpha H^2 + \beta \Theta_\psi = 0$, then taking the limit gives $\alpha \tilde{H}^2 + \beta \tilde{\Pi} = 0$, contradicting linear independence, so $\{H^2, \Theta_\psi\}$ are linearly independent.

Finally, using the “one-dimensional primitive invariant” from Proposition 6.2 to show that the primitive component of $\Theta_\psi$ is non-zero and corresponds to the Jacobian ring class at the Fermat point; by Lefschetz orthogonality ($H^4 = \mathbb{Q}H^2 \oplus H_{\text{prim}}^4$ orthogonal direct sum),

$$ \int_{X_\psi} \Theta_\psi \wedge H^2 = 0, \quad \int_{X_\psi} H^4 = \text{deg } X_\psi = 6, $$

we see $\{H^2, \Theta_\psi\}$ are also independent under intersection pairing. Therefore

$$ \dim H^{2,2}(X_\psi, \mathbb{Q})^G \le 1 \text{ (ambient)} + 1 \text{ (primitive)} = 2, $$

matching the lower bound to get (i). Combined with pullback construction and definition, (ii)(iii) immediately hold. To ensure local constancy and exclude possible jumps, take $U$ after removing a Zariski closed set.

Verifiable calculations for each step are detailed in Appendix B. $\square$

Detailed combinatorial enumeration is given in Appendix B.1.

**Remark 6.8 (Constructive and Verifiable Key Points).**

(a) Jacobian ring invariant calculation is purely combinatorial: enumerate $b \in \{0, 1, 2, 3, 4\}^6$ with $\sum b_i = 12$, filter for those where $b_i + 1$ are constant in $\mathbb{Z}/6\mathbb{Z}$, only getting $b = (2, \dots, 2)$.

(b) Window directions can be described by Čech–de Rham: the $G$-invariant generator of $E_1^{-1,4}$ is given by hyperplane classes on each $\mathbb{P}^3$ taking “constant weight assignment”, with boundary $d_1$ being zero in the invariant direction, thus defining the limit class $\tilde{\Pi}$.

(c) Pullback cycle $Z_\Pi$ can be obtained by gluing relative Cartier data in the total space; $G$-averaging ensures equivariance. Its class on general fibers is $\Theta_\psi$.

**Corollary 6.9 (Equality of Equivariant Strong Part).** On the open set $U$ in Theorem 6.7, $\text{Strong}_G^{2,2}(X_\psi) = H^{2,2}(X_\psi, \mathbb{Q})^G$, and

$$ \text{Strong}_G^{2,2}(X_\psi) = \mathbb{Q} \cdot H^2 \oplus \mathbb{Q} \cdot \Theta_\psi $$

is generated by cohomology classes of two explicit algebraic cycles.
# 7 Conclusion

Within the LMHS/VHS framework we introduced a verifiable three-layer screening and the Dimension Matching Principle (DMP). The results were established case by case: (i) for K3 (weight 2), we gave a constructive Lefschetz(1, 1) via controlled nodal degenerations and specialization-pullback; (ii) for $CY_3$ (weight 4), on the Dwork quintic and its mirror we proved the equivariant strong part within the DMP architecture, and additionally provided a toric/Lefschetz-Gysin argument together with a boundary-propagation template that independently cross-checks and makes the generators explicit on the same families; (iii) for $CY_4$ (weight 4), on the Dwork sextic we obtained the equivariant strong part via DMP in this work.

Overall, the approach realizes

Dimension Matching+Verifiable Three-Layer Conditions $\implies$ Algebraicity of corresponding $(p, p)$-classes,

throughout relying on auditable tools (Clemens-Schmid, Picard-Lefschetz, Lefschetz(1, 1), fixed part theorem, Hodge locus algebraicity). On $CY_3$ these ingredients assemble into a coherent toolkit that complements DMP by explicit toric/Gysin constructions, while on $CY_4$ the conclusions are derived from DMP alone.

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# A Technical Details of Flat Extension and Deligne Extension

## A.1 Local Calculation for K3 Single Node Degeneration

Let

$$ \pi : \mathcal{X} \subset \mathbb{P}^3 \times \Delta \longrightarrow \Delta, \quad \mathcal{X}_t = \{f_t(x) = 0\}, \quad f_t = f_0 - t \cdot g, $$

where $t$ is the disk coordinate and $p \in \mathcal{X}_0$ is the unique $A_1$ node. Take local coordinates $(u, v, w)$ such that

$$ f_t = u^2 + v^2 + w^2 - t. $$

Let $\gamma \in H^2(\mathcal{X}_t, \mathbb{Z})$ be the vanishing cycle, and complete to a basis $\{\gamma, \eta_1, \dots, \eta_{21}\}$ satisfying

$$ \langle \gamma, \gamma \rangle = -2, \quad \langle \gamma, \eta_i \rangle = 0. $$

Picard–Lefschetz gives the monodromy

$$ T(\alpha) = \alpha + \langle \alpha, \gamma \rangle \gamma, \quad N = \log T = T - \text{id}, \quad N^2 = 0. $$

Thus the Gauss–Manin connection in this basis is written as

$$ \nabla_{\mathrm{GM}} = d + \frac{N}{2\pi i} \frac{dt}{t}. $$

Solving the flatness equation $\nabla_{\mathrm{GM}} s = 0$ gives

$$ s(t) = \alpha_0 + \frac{\log t}{2\pi i} N \alpha_0, \quad \alpha_0 \in H^{1,1}(\mathcal{X}_{t_0}). $$
Deligne's canonical extension takes

$$ \tilde{s}(t) := \exp\left(-\frac{\log t}{2\pi i} N\right) s(t) = \alpha_0, $$

so the limit value $\tilde{s}(0)$ lies in $W_2 \cap F^1 \cap \overline{F^1}$, and $N\alpha_0 = \langle \alpha_0, \gamma \rangle$ $\gamma$ remains of type (1, 1).

## A.2 General Case of Multi-Node Degeneration

Consider

$$ \mathcal{X}_{\lambda}: \quad y^2 = Q_{+}(x)Q_{-}(x) + \lambda^2 R(x) \subset \mathbb{P}(1, 1, 1, 1, 4) \times \Delta_{\lambda}, $$

taking base change $t = \lambda^m$ (for some $m \in \mathbb{Z}_{>0}$) to ensure quasi-unipotent monodromy. Let the semistable model of the central fiber be $Y_0 = V \cup W$, with intersection being a smooth surface $Z = V \cap W$, and a series of vanishing cycles $\{\gamma_i\}$ of $A_1$ nodes appearing along $Z$. Then

$$ T = \exp(N), \quad N = \sum_{i} N_{i}, \quad N_{i}^{2} = 0, \quad N_{i}N_{j} = 0 \ (i \neq j), $$

The Gauss-Manin connection is

$$ \nabla_{GM} = d + \frac{N}{2\pi i} \frac{dt}{t} = d + \frac{\sum_{i} N_{i}}{2\pi i} \frac{dt}{t}. $$

For given $\alpha_0 \in H^{2,2}(X_{t_0}, \mathbb{Q})^G$, the flat section is

$$ s(t) = \alpha_0 + \frac{\log t}{2\pi i} \sum_{i} \langle \alpha_0, \gamma_i \rangle \gamma_i, $$

and the Deligne extension is

$$ \tilde{s}(t) = \exp\left(-\frac{\log t}{2\pi i} N\right) s(t) = \alpha_0 $$

The limit at $t = 0$ lies in the $G$-invariant algebraic subspace of $W_2 \cap F^2 \cap \overline{F^2}$.

# B Verifiable Details of Equivariant Strong Part on Dwork Sex-tic CY4

## B.1 Invariant Dimension and Flatness of Jacobian Ring

**Lemma B.1** (Enumeration at Fermat Point and Uniqueness of $G$-Invariants). Let $\psi = 0$, $f_0 = \sum_{i=0}^{5} x_i^6$, $R_0 = S/J(f_0)$. Then

$$ H_{\text{prim}}^{2,2}(X_0, \mathbb{Q}) \cong (R_0)_{12}, $$

A basis is given by monomials $x^b = x_0^{b_0} \cdots x_5^{b_5}$ satisfying $0 \le b_i \le 4$ and $\sum b_i = 12$. Under the action of the phase group

$$ K := \{ (\zeta_0, \dots, \zeta_5) \in \mu_6^6 \mid \prod \zeta_i = 1 \} $$

$x^b$ has character $\chi_{b+1}(\zeta) = \prod \zeta_i^{b_i+1}$. $\chi_{b+1}$ is trivial on $K$ if and only if $b_i + 1$ are all equal in $\mathbb{Z}/6\mathbb{Z}$. Thus the unique $K$-invariant (hence $G$-invariant) element is

$$ b = (2, 2, 2, 2, 2, 2), \quad x^b = x_0^2 x_1^2 x_2^2 x_3^2 x_4^2 x_5^2. $$
*Proof.* The key is that the character group of $K$ is $(\mathbb{Z}/6\mathbb{Z})^6/\langle(1, \dots, 1)\rangle$. Let $a_i := b_i + 1 \in \{1, 2, 3, 4, 5\}$. $\chi_a$ restricted to $K$ is trivial if and only if $a$ is zero in the quotient group, i.e., $a_i \equiv a_j (\text{mod } 6)$. Combined with $\sum b_i = 12$ and $0 \le b_i \le 4$, we only get $b_i \equiv 2$, so $b = (2, \dots, 2)$ is unique. $\square$

For machine verification, we give a list of all “types” of $b$ arranged in non-decreasing order ($b \in \{0, 1, 2, 3, 4\}^6$, $\sum b_i = 12$; each row represents an $S_6$-orbit class):

<table>
  <tbody>
    <tr>
      <td>(1)</td>
      <td>[4,4,4,0,0,0]</td>
    </tr>
    <tr>
      <td>(2)</td>
      <td>[4,4,2,2,0,0]</td>
    </tr>
    <tr>
      <td>(3)</td>
      <td>[4,4,2,1,1,0]</td>
    </tr>
    <tr>
      <td>(4)</td>
      <td>[4,4,1,1,1,1]</td>
    </tr>
    <tr>
      <td>(5)</td>
      <td>[4,3,3,2,0,0]</td>
    </tr>
    <tr>
      <td>(6)</td>
      <td>[4,3,3,1,1,0]</td>
    </tr>
    <tr>
      <td>(7)</td>
      <td>[4,3,2,2,1,0]</td>
    </tr>
    <tr>
      <td>(8)</td>
      <td>[4,3,2,1,1,1]</td>
    </tr>
    <tr>
      <td>(9)</td>
      <td>[4,2,2,2,1,1]</td>
    </tr>
    <tr>
      <td>(10)</td>
      <td>[4,2,2,2,2,0]</td>
    </tr>
    <tr>
      <td>(11)</td>
      <td>[3,3,3,3,0,0]</td>
    </tr>
    <tr>
      <td>(12)</td>
      <td>[3,3,3,2,1,0]</td>
    </tr>
    <tr>
      <td>(13)</td>
      <td>[3,3,3,1,1,1]</td>
    </tr>
    <tr>
      <td>(14)</td>
      <td>[3,3,2,2,2,0]</td>
    </tr>
    <tr>
      <td>(15)</td>
      <td>[3,3,2,2,1,1]</td>
    </tr>
    <tr>
      <td>(16)</td>
      <td>[3,2,2,2,2,1]</td>
    </tr>
    <tr>
      <td>(17)</td>
      <td>[2,2,2,2,2,2]</td>
    </tr>
    <tr>
      <td>(18)</td>
      <td>[4,3,1,1,1,2]</td>
    </tr>
  </tbody>
</table>

Only row (17) satisfies $b_i \equiv b_j (\text{mod } 6)$, thus giving the unique $G$-invariant.

**Proposition B.2 (Flatness and Rank Constancy of Invariants).** Let $\pi : \mathcal{X} \to S$ be a smooth family, $V := R_{\text{prim}}^4 \pi_* \mathbb{Q}$ be the $\mathbb{Q}$-local system of primitive cohomology. The $G$ action induces a fixed linear representation $\rho : G \to \text{Aut}(V_s)$ (for each $s \in S$). Then $V^G := \{v \in V \mid \rho(g)v = v, \forall g \in G\}$ is a sub-local system of $V$ with constant rank over $S$. Therefore

$$ \dim_{\mathbb{Q}} H_{\text{prim}}^{2,2}(X_{\psi}, \mathbb{Q})^{G} $$

is constant in $\psi$. Combined with Lemma B.1, this dimension is 1 for general $\psi$.

*Proof.* For each $g \in G$, $\text{Fix}(g) := \text{ker}(\text{id} - \rho(g)) \subset V$ is a sub-local system (kernel formation commutes with pullback). Thus $V^G = \bigcap_{g \in G} \text{Fix}(g)$ is the intersection of finitely many sub-local systems, still a sub-local system with locally constant rank. Connectedness gives global constancy. Matching with the calculation at $\psi = 0$ proves the claim. $\square$

## B.2 Semistable Model, Steenbrink Spectral Sequence and Signs of $d_1$

Consider $t = \psi^{-1}$, total equation

$$ \mathcal{X} = \left\{ ([x], t) \in \mathbb{P}^5 \times \Delta \mid t \cdot \sum_{i=0}^{5} x_i^6 - 6x_0x_1x_2x_3x_4x_5 = 0 \right\}. $$

When $t = 0$, the central fiber $X_0 = \bigcup_{i=0}^5 H_i$ is the union of coordinate hyperplanes (reduced), with intersections being linear subspaces

$$ H_{i_0 \cdots i_p} := H_{i_0} \cap \cdots \cap H_{i_p} \cong \mathbb{P}^5 - p. $$
This is an SNC divisor. To obtain a smooth total space, we can perform a series of G-equivariant blowups along each $H_{i_0 \cdots i_p} \times \{0\}$ from highest to lowest intersection depth; this process is the standard “toroidal-semistable reduction” (toric/toroidal) procedure.

The relevant levels of Steenbrink's spectral sequence at $q = 4$ are

$$E_1^{0,4} = H^4(\bigsqcup \mathbb{P}^4), \quad E_1^{-1,4} = H^2(\bigsqcup \mathbb{P}^3)(1), \quad E_1^{-2,4} = H^0(\bigsqcup \mathbb{P}^2)(2),$$

The differential $d_1: E_1^{-p,4} \to E_1^{-p+1,4}$ is the “alternating sum”.

**Lemma B.3** (Signs of $d_1$ and Vanishing on G-Invariant Subspaces). Let $i < j < k$. Take the constant basis element $1_{ijk} \in H^0(H_{ijk})$ on $E_1^{-2,4}$, and basis elements $h_{pq} \in H^2(H_{pq})$ on $E_1^{-1,4}$ representing the hyperplane class of $\mathbb{P}^3$. Then

$$d_1(1_{ijk}) = (+1) h_{ij} + (-1) h_{ik} + (+1) h_{jk}.$$

Similarly, $d_1: H^2(H_{ij}) \to H^4(H_i) \oplus H^4(H_j)$ satisfies (for $i < j$)

$$d_1(h_{ij}) = (+h_i^2) \oplus (-h_j^2).$$

On the G-invariant diagonal subspaces

$$A_2 := E_1^{-2,4,G} \cong \mathbb{Q}, \quad A_1 := E_1^{-1,4,G} \cong \mathbb{Q}, \quad A_0 := E_1^{0,4,G} \cong \mathbb{Q}$$

both $d_1: A_2 \to A_1$ and $d_1: A_1 \to A_0$ are zero maps.

**Proof.** The signs come from the alternating sum convention of simplicial boundaries. For $A_2 \to A_1$: adding constant functions on all triple intersections and projecting to the G-invariant subspace amounts to aggregating coefficients for each edge (pq) in all triangles containing it. Due to complete symmetry and alternating signs, the total coefficient for each edge is zero, so $d_1|_{A_2} = 0$.

For $A_1 \to A_0$: adding constant $h_{ij}$ on all edges, let $v = \sum_{i<j} h_{ij}$, then

$$d_1(v) = \sum_{i<j} (h_i^2 \oplus (-h_j^2)).$$

On the diagonal projection (summing to the direction of $\sum_i h_i^2$), the coefficient is

$$\sum_{i=0}^{5} (\#\{j > i\} - \#\{k < i\}) = \sum_{i=0}^{5} (5 - 2i) = 30 - 2 \cdot 15 = 0.$$

So $d_1|_{A_1} = 0$.

□

**Proposition B.4** (Dimension Matching of $W_{alg}^G$). Take $q = 4$. Then $E_2^{-2,4,G} \cong \mathbb{Q}$, $E_2^{-1,4,G} \cong \mathbb{Q}$, $E_2^{0,4,G} \cong \mathbb{Q}$, and $d_r$ vanishes on G-invariant subspaces for $r \ge 2$ (dimension and weight obstruction). Therefore $W_{alg}^G$ contributes two dimensions in the limit: one from $E_\infty^{0,4,G}$ (ambient direction), and one from $E_\infty^{-1,4,G}$ (matching direction).

**Proof.** By Lemma B.3, $d_1$ is zero in invariant directions, so $E_2^{\bullet,4,G} = E_1^{\bullet,4,G}$. Higher differentials $d_r: E_r^{-p,4} \to E_r^{-p+r,4-r+1}$ on G-invariant subspaces have targets beyond visible levels if they exceed, hence are zero; under current dimension and weight constraints, no non-trivial arrows can reach, so $E_\infty = E_2$ on invariant subspaces. The conclusion follows. □

Although $E_\infty^{-2,4,G} \cong \mathbb{Q}$, this subspace satisfies $N \ne 0$ in the LMHS, so it does not belong to $W_{alg}^G := W_4 \cap F^2 \cap \bar{F}^2 \cap \ker N$; thus $W_{alg}^G$ is actually spanned only by $E_\infty^{0,4,G}$ and $E_\infty^{-1,4,G}$, with dimension exactly 2.
## B.3 Pullback Cycles, Primitivity and Intersection Numbers

The “matching direction” of $E_{\infty}^{-1,4,G}$ obtained in the previous section can be used to construct relative algebraic 2-cycles in the total space through Čech gluing of Cartier data and pull back to general fibers.

**Definition B.5 (Matching Relative Cartier 1-Cocycle).** In the semistable model $\pi : \mathcal{X} \to \Delta$, denote $D_{ij} \subset H_{ij}$ as the class of relative hyperplane sections (generator of Pic on $H_{ij} \cong \mathbb{P}^3$), let

$$ \mathcal{Z} := \sum_{i<j} \iota_{ij,*}(D_{ij}) \in Z^2(\mathcal{X}_0). $$

By the $d_1|_{A_1} = 0$ from Lemma B.3, $\mathcal{Z}$ is $d_1$-closed in the Steenbrink complex (matching). By compatibility of specialization and cycle class maps, there exists a relative algebraic 2-cycle $Z_{\Pi} \in Z^2(\mathcal{X})$ in the total space such that $Z_{\Pi}|_{X_0} \sim \mathcal{Z}$, and defines the class on general fibers

$$ \Theta_{\psi} := \mathrm{cl}(Z_{\Pi}|_{X_{\psi}}) \in H^{2,2}(X_{\psi}, \mathbb{Q})^{G}. $$

**Lemma B.6 (Primitivity and Orthogonality).** For general $\psi$, the primitive component of $\Theta_{\psi}$ is non-zero, and

$$ \int_{X_{\psi}} \Theta_{\psi} \wedge H^2 = 0. $$

*Proof.* The limit $\mathrm{sp}_{\psi}(\Theta_{\psi})$ is precisely the generator of $E_{\infty}^{-1,4,G}$ from Proposition B.4, from which we can see its primitive component is non-zero (otherwise the limit would only fall in the ambient direction). Orthogonality comes from Lefschetz primitive decomposition and continuity of limit pairing: the matching direction and ambient direction in $W_{\mathrm{alg}}^G$ are orthogonal under limit pairing, and pullback preserves this orthogonality. $\square$

**Proposition B.7 (Intersection Number Verification).** Let $H$ be the hyperplane class. Then for all general $\psi$,

$$ \int_{X_{\psi}} H^4 = 6, \quad \int_{X_{\psi}} \Theta_{\psi} \wedge H^2 = 0, \quad \int_{X_{\psi}} \Theta_{\psi} \wedge \Theta_{\psi} > 0. $$

*Proof.* First formula: $[X_{\psi}] = 6H$ in $A^1(\mathbb{P}^5)$, so $\int_{X_{\psi}} H^4 = \int_{\mathbb{P}^5} H^4 \cdot [X_{\psi}] = 6 \int_{\mathbb{P}^5} H^5 = 6$. Second formula: see Lemma B.6. Third formula: $\Theta_{\psi}$ lies in the primitive (2,2) subspace, Hodge-Riemann bilinear relation gives positive definiteness (the sign here is $(-1)^2 = +1$), so self-intersection is positive. $\square$

**Remark B.8 (Note on “Explicit Algebraic Equations”).** The intuitive candidate $Z'_{\Pi}(\psi) = \sum_{i<j} (X_{\psi} \cap \{x_i = x_j = 0\})$ is $G$-invariant with clear configuration, but cohomologically equals some multiple of the ambient class $H^2$, and cannot give the primitive direction. Therefore we adopt the “matching Cartier 1-cocycle” gluing method in Definition B.5 to produce $\Theta_{\psi}$. This method essentially uses Čech data on SNC covers to construct relative cycles in $CH^2(\mathcal{X})$, whose limit is a representative of the class in $E_{\infty}^{-1,4,G}$, thereby guaranteeing its primitivity and its independence from $H^2$.

## C Toric Background and Gysin Exact Sequences

This appendix summarizes two standard tools needed for mirror threefolds and fixes the Mayer-Vietoris normalization convention under LCSL in this manuscript.
**Lemma C.1** (Toric Even Cohomology and Cycle Class Isomorphism). Let $T$ be a smooth projective toric variety. Then for any $k$, $H^{2k+1}(T, \mathbb{Q}) = 0$, and the cycle class homomorphism

$$ \mathrm{cl}: \mathrm{CH}^*(T)_{\mathbb{Q}} \xrightarrow{\cong} \mathrm{H}^{2*}(T, \mathbb{Q}) $$

is an isomorphism. In particular, $H^4(T, \mathbb{Q}) \cong \mathrm{CH}^2(T)_{\mathbb{Q}}$, and $\mathrm{CH}^2(T)$ is generated by torus-invariant surfaces $D_i \cap D_j$.

*Proof.* See [DK87, Ful93, CLS11]. □

**Lemma C.2** (Gysin Surjection). Let $i: Y \hookrightarrow T$ be a smooth divisor (e.g., anticanonical hyper-surface), where $T$ is a smooth projective toric variety. Then the restriction map

$$ i^*: \mathrm{H}^4(T, \mathbb{Q}) \to \mathrm{H}^4(Y, \mathbb{Q}) $$

is surjective.

*Proof.* Consider the long exact sequence for the pair $(T, Y)$:

$$ \dots \to \mathrm{H}^4(T, \mathbb{Q}) \xrightarrow{i^*} \mathrm{H}^4(Y, \mathbb{Q}) \to \mathrm{H}^5(T, Y; \mathbb{Q}) \to \mathrm{H}^5(T, \mathbb{Q}) \to \dots $$

By the Lefschetz theorem for pairs, $\mathrm{H}^5(T, Y; \mathbb{Q}) \simeq \mathrm{H}^3(T, \mathbb{Q})$; for toric $T$, $\mathrm{H}^3(T, \mathbb{Q}) = 0$ (Lemma C.1), so $i^*$ is surjective. □

**Remark C.3** (Mayer–Vietoris Normalization under LCSL). In the LCSL of the Dwork quintic mirror family $Y_\psi$, the central fiber is the union of five coordinate hyperplanes. Let $S_{ij} = H_i \cap H_j \simeq \mathbb{P}^2$ and $L_{ijk} = H_i \cap H_j \cap H_k \simeq \mathbb{P}^1$. Using the Steenbrink–Mayer–Vietoris “alternating restriction” convention (see [PS08]), we have

$$ \delta \left( \sum_{i<j} c_{ij}[S_{ij}] \right) = \sum_{i<j<k} (c_{ij} + c_{ik} + c_{jk}) [L_{ijk}], \quad \mathrm{sp}(h^3) = 5 \sum_{i<j<k} [L_{ijk}]. $$

From $\delta \mathrm{sp}(h^2) = \mathrm{sp}(h^3)$, we solve $3c = 5$, i.e., $\mathrm{sp}(h^2) = \frac{5}{3} \sum_{i<j} [S_{ij}]$. Different literature may have different symbols/normalizations leading to different proportionality constants, but the fact that “$\mathrm{sp}(h^2)$ and $\sum_{i<j} [S_{ij}]$ are collinear” in $G$-invariant directions is invariant, and this fact suffices for dimension matching and pullback algebraicity.

**Proposition C.4** (MV-normalization yields the coefficient 5/3). Under the alternating restriction convention used in [PS08], the unique $G$-invariant solution to $\delta \mathrm{sp}(h^2) = \mathrm{sp}(h^3)$ is

$$ \mathrm{sp}(h^2) = \frac{5}{3} \sum_{i<j} [S_{ij}]. $$

*Proof.* Each triple line $L_{ijk}$ appears in the boundary of exactly three faces $(ij)$, $(ik)$, $(jk)$ with coefficients $+1$, hence $\delta(\sum c_{ij}[S_{ij}]) = \sum_{i<j<k} (c_{ij}+c_{ik}+c_{jk})[L_{ijk}]$. Since $\mathrm{sp}(h^3) = 5 \sum_{i<j<k} [L_{ijk}]$ and $c_{ij} \equiv c$ by $G$-invariance, we get $3c = 5$. □