POLARIZED HODGE MEZZOPERVERSITIES AND
THE MIDDLE INTERSECTION COMPLEX FOR
FUBINI-STUDY WEDGE METRICS ON
STRATIFIED PROJECTIVE VARIETIES

Jiaming Luo

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Jiaming Luo. POLARIZED HODGE MEZZOPERVERSITIES AND THE MIDDLE INTERSECTION COMPLEX FOR FUBINI-STUDY WEDGE METRICS ON STRATIFIED PROJECTIVE VARIETIES. 2026. hal-05526285

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# POLARIZED HODGE MEZZOPERVERSITIES AND THE MIDDLE INTERSECTION COMPLEX FOR FUBINI-STUDY WEDGE METRICS ON STRATIFIED PROJECTIVE VARIETIES

JIAMING LUO

ABSTRACT. Let $X \subset \mathbb{CP}^N$ be a Thom-Mather stratified projective variety, and endow $X^{\text{reg}}$ with the induced Fubini-Study iterated wedge metric. Closed extensions of the $L^2$ de Rham operator are encoded by mezzoperversities (ideal boundary conditions) along the singular strata. Building on the canonical analytic mezzoperversity $W_{\text{can}}$ defined via iterated Calderón projectors, we supply the Hodge-theoretic input for the *middle criterion*. For each stratum with nontrivial trace bundle we construct an incidence-compatible polarized Hodge structure on the trace bundle; its Weil involution singles out a Lagrangian subbundle $W_{\text{mid}} \subset H$. These data assemble into a canonical self-dual mezzoperversity $W_{\text{mid}}$ with associated Deligne sheaf canonically isomorphic to the middle intersection complex $\text{IC}_{\bar{m}}$. The analytic identification $W_{\text{can}} = W_{\text{mid}}$ is treated in a companion paper.

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Background and aim</td><td>2</td></tr><tr><td>1.2. Canonical Cauchy data and the middle criterion</td><td>3</td></tr><tr><td>1.3. Main results</td><td>3</td></tr><tr><td>1.4. Relation to the “canonical” analytic mezzoperversity</td><td>4</td></tr><tr><td>1.5. Outline of the proof</td><td>5</td></tr><tr><td>1.6. Notation</td><td>5</td></tr><tr><td>1.7. Interfaces used in the argument</td><td>5</td></tr><tr><td>1.8. Logical dependencies and scope</td><td>6</td></tr><tr><td>1.9. Organization</td><td>7</td></tr><tr><td>2. Stratified projective varieties and Fubini-Study wedge metrics</td><td>7</td></tr><tr><td>2.1. Thom-Mather control data and links</td><td>7</td></tr><tr><td>2.2. Iterated wedge metrics</td><td>8</td></tr><tr><td>2.3. Fubini-Study wedge geometry</td><td>8</td></tr><tr><td>3. Mezzoperversities, trace bundles, and Green pairings</td><td>10</td></tr><tr><td>3.1. Local wedge model and tangential operators</td><td>10</td></tr><tr><td>3.2. Trace bundles and the Green pairing</td><td>10</td></tr><tr><td>3.3. Mezzoperversities and incidence compatibility</td><td>11</td></tr><tr><td>3.4. Deligne complexes and the analytic-sheaf dictionary</td><td>12</td></tr><tr><td>3.5. Symplectic incidence and self-adjoint IBC domains</td><td>14</td></tr><tr><td>4. Polarized Hodge modules and intersection complexes</td><td>15</td></tr><tr><td>4.1. Intersection Hodge modules</td><td>15</td></tr><tr><td>4.2. Local systems along strata and link intersection cohomology</td><td>15</td></tr><tr><td>4.3. Polarizations, Weil operators, and duality on strata</td><td>16</td></tr></table>

Date: February 24, 2026.

2020 Mathematics Subject Classification. 14F05, 32U05, 58J52, 55N33.

Key words and phrases. intersection homology, Fubini-Study metric, stratified spaces, mezzoperversity, Hodge module, polarization, Deligne sheaf.
<table>
  <tr>
    <td>4.4.</td>
    <td>Incidence maps and functoriality</td>
    <td>17</td>
  </tr>
  <tr>
    <td>5.</td>
    <td>Hodge structures on trace bundles</td>
    <td>18</td>
  </tr>
  <tr>
    <td>5.1.</td>
    <td>Analytic trace bundles versus link intersection cohomology</td>
    <td>18</td>
  </tr>
  <tr>
    <td>5.2.</td>
    <td>Polarized Hodge structures on the analytic trace bundle</td>
    <td>20</td>
  </tr>
  <tr>
    <td>5.3.</td>
    <td>Polarization versus the Green pairing</td>
    <td>20</td>
  </tr>
  <tr>
    <td>6.</td>
    <td>The Hodge involution and the middle Lagrangian</td>
    <td>20</td>
  </tr>
  <tr>
    <td>6.1.</td>
    <td>The Hodge involution</td>
    <td>21</td>
  </tr>
  <tr>
    <td>6.2.</td>
    <td>Weil operator and Lagrangian eigensubbundles</td>
    <td>21</td>
  </tr>
  <tr>
    <td>6.3.</td>
    <td>Guiding examples</td>
    <td>22</td>
  </tr>
  <tr>
    <td>7.</td>
    <td>Identification with the middle intersection complex</td>
    <td>25</td>
  </tr>
  <tr>
    <td>7.1.</td>
    <td>Local extension data and classification by Lagrangians</td>
    <td>26</td>
  </tr>
  <tr>
    <td>7.2.</td>
    <td>The extension datum determined by the intersection Hodge module</td>
    <td>26</td>
  </tr>
  <tr>
    <td>7.3.</td>
    <td>Outlook: comparison with the analytic canonical mezzoperversity</td>
    <td>28</td>
  </tr>
  <tr>
    <td>Appendix A.</td>
    <td>Symplectic linear algebra for Weil operators</td>
    <td>30</td>
  </tr>
  <tr>
    <td>A.1.</td>
    <td>Anti-symplectic involutions and Lagrangian splittings</td>
    <td>30</td>
  </tr>
  <tr>
    <td>A.2.</td>
    <td>Graphs and local charts</td>
    <td>32</td>
  </tr>
  <tr>
    <td>A.3.</td>
    <td>Fiberwise eigensubbundles</td>
    <td>33</td>
  </tr>
  <tr>
    <td>Appendix B.</td>
    <td>Background on Hodge modules used in the text</td>
    <td>33</td>
  </tr>
  <tr>
    <td>B.1.</td>
    <td>Conventions: realizations, shifts, and functors</td>
    <td>33</td>
  </tr>
  <tr>
    <td>B.2.</td>
    <td>PVHS and Hodge modules on smooth varieties</td>
    <td>34</td>
  </tr>
  <tr>
    <td>B.3.</td>
    <td>Intersection Hodge modules</td>
    <td>34</td>
  </tr>
  <tr>
    <td>B.4.</td>
    <td>Restrictions to strata and induced PVHS</td>
    <td>34</td>
  </tr>
  <tr>
    <td>B.5.</td>
    <td>Projective direct image, intersection cohomology, and polarizations</td>
    <td>35</td>
  </tr>
  <tr>
    <td>Acknowledgements</td>
    <td></td>
    <td>35</td>
  </tr>
  <tr>
    <td>References</td>
    <td></td>
    <td>36</td>
  </tr>
</table>

# 1. INTRODUCTION

## 1.1. Background and aim.
Let $X \subset \mathbb{CP}^N$ be a complex projective variety equipped with a fixed Thom-Mather stratification $\mathcal{S}$ [Pfl01, Ver84, GM88]. Write $X_{\text{reg}}$ for the regular stratum. The induced Fubini-Study metric on $X_{\text{reg}}$ is an *iterated incomplete wedge metric* in the sense of [ALMP18]. One of the guiding problems in analysis on singular spaces is to relate the $L^2$-de Rham theory of such metrics to the topological invariants of $X$, notably middle intersection (co)homology [Che80, GM80, GM83, BBD82, PS91, Sap85].

A central feature of incomplete wedge metrics is that the de Rham operator admits many closed and self-adjoint extensions. Following [ALMP18], these extensions can be organized by *ideal boundary conditions* (IBCs), equivalently by *mezzoperversities* $\mathcal{W}$. Each mezzoperversity assigns to every stratum $Y$ with nontrivial vertical middle-degree harmonic forms a smooth subbundle $W_Y \subset H_Y$ of the corresponding *trace bundle* $H_Y \to Y$; self-dual mezzoperversities are characterized by the additional requirement that each $W_Y$ be Lagrangian for the Green pairing. The resulting IBC de Rham complex is Fredholm and its $L^2$-cohomology is naturally identified with the hypercohomology of a Deligne-type complex $\text{IC}_\mathcal{W}$ on $X$ [ALMP18]; see also the analytic-sheaf identification recorded as Assumption 3.8 below.

For projective varieties equipped with the Fubini-Study metric, one expects a distinguished, geometrically natural choice of IBCs yielding the *middle intersection complex* $\text{IC}_{\bar{m}}$ and hence the classical intersection cohomology $\text{IH}_{\bar{m}}^*(X)$. This is a metric incarnation of the Cheeger-Goresky-MacPherson philosophy [CGM82] that, on suitably structured singular spaces, $L^2$-cohomology should recover middle intersection cohomology.
**1.2. Canonical Cauchy data and the middle criterion.** In the depth-one (projective simple-edge) case, a canonical Cauchy-data bundle and canonical mezzoperversity $W_{\text{can}}$ are constructed via a parameter-dependent Calderón package in [LL26a, Theorems 6.2 and 7.4], yielding a canonical self-adjoint Fredholm realization and the global FS-CGM identification [LL26a, Theorem 7.6]. In arbitrary stratification depth, the iterated Calderón package of [Luo26, Theorems 5.1, 6.3 and 7.6] produces canonical Cauchy-data bundles and a canonical self-adjoint Fredholm realization, whose cohomology identifies with $\text{IH}^*_{W_{\text{can}}}(X)$ by [Luo26, Theorem 7.7].

In the iterated Witt case the choice is forced and agrees with the middle one, hence $\text{IH}^*_{W_{\text{can}}}(X) \cong \text{IH}^*_{\bar{m}}(X)$. In general, a *middle criterion* is required (cf. [Luo26, §9]) to decide whether $W_{\text{can}}$ coincides with the middle perversity data.

The present paper provides the **Hodge/polarization** input for that criterion. Using the algebraic geometry of projective varieties and Saito's theory of polarized Hodge modules [Sai90, Sch14], we construct a canonical family of *middle* Lagrangian boundary conditions $W_{\text{mid}}$ and show that the corresponding Deligne sheaf is the middle intersection complex. The analytic comparison $W_{\text{can}} = W_{\text{mid}}$ is carried out in the companion paper; see §1.4.

**1.3. Main results.** Fix a stratum $Y \subset X$ of codimension $c$ and let $\phi_Y : L_Y \to Y$ be its link fibration. Assume inductively that deeper strata carry already-chosen self-dual boundary conditions (this is automatic for the bottom stratum). Freezing this deeper data, the tangential operator of the de Rham operator along $\phi_Y$ is a family of self-adjoint Fredholm operators $A_Y^{W \to Y} \to Y$; its kernel forms the trace bundle $H_Y \to Y$.

The conceptual contribution of Paper A is to show that the Hodge-theoretic structures carried by the (stratumwise) local systems coming from the intersection Hodge module $\text{IC}_X^H$ [DS90, NA85] can be transferred to these trace bundles in a way compatible with the analytic boundary pairing: the induced polarization agrees (up to a universal sign convention) with the Green pairing, and the associated Weil operator yields a canonical Lagrangian boundary datum $W_{Y,\text{mid}} \subset H_Y$ that is functorial under incidences of strata. This produces a self-dual mezzoperversity $W_{\text{mid}}$ determined purely by Hodge theory, and Theorem 1.2 identifies its Deligne sheaf with the middle intersection complex.

**New ingredients.**

* Trace-bundle realization of Hodge data: we transfer the Hodge-theoretic structures carried by the local systems underlying $\text{IC}_X^H$ to the analytic trace bundles $H_Y \to Y$ associated with the tangential operators of $D_{\text{DR}}$.

* Polarization as an analytic boundary pairing: we identify the induced polarization with the Green pairing (up to a universal sign convention), making the Hodge package effective for ideal boundary conditions.

* Weil operator and canonical middle data: the resulting Weil operator on each $H_Y$ determines a canonical Lagrangian $W_{Y,\text{mid}} = \ker(T_Y - \text{Id})$.

* Incidence functoriality: we prove the compatibility of $\{W_{Y,\text{mid}}\}$ under incidence maps, so that these Lagrangians assemble into a global self-dual mezzoperversity $W_{\text{mid}}$.

**Theorem 1.1 (Polarized Hodge package on trace bundles).** For every stratum $Y$ with nontrivial trace bundle $H_Y$ there exists a canonical polarized variation of (pure) Hodge structure on $H_Y$ with the following properties:

(i) The underlying flat connection is the natural connection induced by the link fibration and the family $A_Y^{W \to Y}$.

(ii) The polarization equals (up to a universal sign convention) the Green pairing on $H_Y$ arising from the boundary term for the de Rham operator.
(iii) The Weil operator canonically determines a smooth involution $T_Y$ on $H_Y$ such that the +1 eigensubbundle

$$W_{Y,\text{mid}} := \ker(T_Y - \text{Id}) \subset H_Y$$

is Lagrangian for the Green pairing.

(iv) (Incidence compatibility) If $Z < Y$ is a deeper stratum and $Y \cap \bar{Z} \neq \emptyset$, then $W_{Y,\text{mid}}$ restricts compatibly to the induced trace bundle data on $Z$ under the natural incidence maps of control data.

The Lagrangian subbundles $W_{Y,\text{mid}}$ assemble into a self-dual mezzoperversity, denoted $W_{\text{mid}}$.

**Theorem 1.2** (Hodge mezzoperversity realizes the middle intersection complex). Let $W_{\text{mid}}$ denote the mezzoperversity obtained by taking $W_{Y,\text{mid}}$ on every stratum $Y$ with nontrivial trace bundle. Then the associated Deligne sheaf (equivalently, the corresponding self-dual extension datum in the sense of [ALMP18]) satisfies

$$\mathrm{IC}_{W_{\text{mid}}} \cong \mathrm{IC}_{\bar{m}}$$

in the derived category of constructible sheaves on $X$. Consequently,

$$\mathrm{IH}_{W_{\text{mid}}}^{\bullet}(X) \cong \mathrm{IH}_{\bar{m}}^{\bullet}(X).$$

**Takeaway.** The upshot is a verifiable Hodge-theoretic construction of the *middle* ideal boundary condition for the Fubini-Study wedge metric, producing a canonical self-dual $L^2$ de Rham complex whose Deligne sheaf is $\mathrm{IC}_{\bar{m}}$.

Figure 1 provides a road map of the construction; the arrows indicate the key constructions/results (not proofs).

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Schematic of the construction and its logical dependencies. Starting from the middle intersection Hodge module $\mathrm{IC}_X^H$, we obtain a polarized Hodge package on the trace bundles (Theorem 1.1; see also Section 5), which determines the middle Lagrangian subbundles $W_{Y,\text{mid}}$ compatible under incidences (Lemma 1.5 and Subsection 3.3). This yields a self-dual mezzoperversity $W_{\text{mid}}$ and an $L^2$ de Rham complex whose Deligne sheaf identifies with the middle intersection complex (Theorem 1.2).</figcaption></figure>

1.4. **Relation to the “canonical” analytic mezzoperversity.** The companion paper proves the analytic identification $W_{\text{can}} = W_{\text{mid}}$ by comparing the Calderón range with the Hodge Lagrangian data and establishing deformation invariance of the Cauchy data. Together with the companion paper, Theorem 1.2 yields the full middle criterion and hence the Cheeger-Goresky-MacPherson identification

$$H_{(2)}^{\bullet}(X_{\text{reg}}; W_{\text{can}}) \cong \mathrm{IH}_{\bar{m}}^{\bullet}(X)$$

for Fubini-Study metrics, where the left-hand side denotes $L^2$ de Rham cohomology computed using the canonical IBC domain determined by $W_{\text{can}}$.
1.5. **Outline of the proof.** The construction of $W_{\text{mid}}$ is intrinsic to the middle intersection Hodge module $\text{IC}_X^H$. At a high level, the argument proceeds as follows.

(1) Using Saito's functoriality for restrictions to strata, we obtain a canonical polarized variation of Hodge structure on the local system underlying $\text{IC}_{\bar{m}}$ along each stratum $Y$ [Sai90].

(2) We identify this local system with the trace bundle $H_Y$ (Section 5). The polarization is shown to coincide with the analytic Green pairing (again up to a universal sign).

(3) The Weil operator attached to the polarized variation produces a canonical Lagrangian sub-bundle $W_{Y,\text{mid}} \subset H_Y$; the key point is that the resulting family is compatible under incidences of strata (Sections 6-7).

(4) Finally, we compare the self-dual extension datum determined by $W_{\text{mid}}$ with the self-dual extension datum coming from $\text{IC}_X^H$. Since the underlying perverse sheaf of $\text{IC}_X^H$ is $\text{IC}_{\bar{m}}$, this yields Theorem 1.2.

1.6. **Notation.** For the reader's convenience, we collect a small list of symbols used throughout the paper.

**Geometry and stratification.**

$X \subset \mathbb{CP}^N$: A complex projective variety.

$\mathcal{S}$: A fixed Thom-Mather stratification of $X$; strata are denoted by $Y, Z, \dots$.

$X^{\text{reg}}$: The regular (top-dimensional smooth) stratum of $X$.

$\phi_Y: \mathcal{L}_Y \to Y$: The link fibration along a stratum $Y$, with fiber $Z_y$ over $y \in Y$; near $Y$ we use wedge coordinates $(r, z, y)$ with $r = \rho_Y$.

$g_{\text{FS}}$: The induced Fubini-Study iterated wedge metric on $X^{\text{reg}}$.

**Analysis and trace bundles.**

$D_{\text{dR}} = d + \delta$: The Hodge-de Rham operator on $(X^{\text{reg}}, g_{\text{FS}})$.

$A_Y^{W>Y}$: The tangential (model) operator on the link fibers along $Y$, determined by deeper boundary data $W^Y = W^{>Y}$.

$H_Y$: The trace bundle along $Y$, defined by $H_Y = \ker(A_Y^{W>Y})$.

$\omega_Y$: The Green pairing on $H_Y$ coming from integration by parts for $D_{\text{dR}}$.

**Mezzoperversities and sheaves.**

$\{W_Y\}_{Y}$: Boundary data $W_Y \subset H_Y$ along strata; a compatible collection defines a mezzoperversity (ideal boundary condition).

$W_{Y,\text{mid}} \subset H_Y$: The distinguished middle Lagrangian subbundle singled out by the normalized Weil involution.

$\mathcal{W}_{\text{can}}, \mathcal{W}_{\text{mid}}$: The canonical analytic mezzoperversity and the Hodge-theoretically defined middle mezzoperversity.

$\text{IC}_{\mathcal{W}}, \text{IC}_{\bar{m}}$: The Deligne sheaf attached to $\mathcal{W}$ and the middle intersection complex, respectively; $\text{IH}_{\bar{m}}^{\bullet}(X)$ denotes middle intersection cohomology.

$\text{IC}_X^H$: The (graded-polarizable) intersection Hodge module on $X$ whose underlying perverse sheaf is $\text{IC}_{\bar{m}}$.

1.7. **Interfaces used in the argument.** For the reader's convenience (and to make the dependency structure checkable), we isolate the main "interface statements" that connect the Hodge-module input with the analytic trace-bundle formalism of ideal boundary conditions. They are proved in the body of the paper, and they are precisely the bridges used in the proof of Theorem 1.1 and Theorem 1.2.

**Proposition 1.3** (Trace local systems). Let $Y$ be a stratum of $\mathcal{S}$ such that the vertical middle-degree cohomology on the link fibres is nontrivial. Denote by $\text{IC}_X^H$ the (polarized) intersection Hodge module on $X$ and by $\text{IC}_{\bar{m}}$ its underlying perverse sheaf. Then Saito's restriction functor produces a canonical
polarizable variation of Hodge structure along $Y$ whose underlying local system identifies canonically with the trace bundle $H_Y \to Y$ (equivalently, with $\ker(A_Y^{W>Y})$ in the notation of Section 3).

*Proof (pointer).* See Proposition 5.3. □

**Proposition 1.4 (Polarization versus Green pairing).** Under the identification of Proposition 1.3, the polarization of the induced variation of Hodge structure agrees with the analytic Green pairing $\omega_Y$ on $H_Y$, up to a universal sign depending only on the choice of conventions. In particular, $\omega_Y$ is a smooth symplectic form on $H_Y$.

*Proof (pointer).* See Proposition 5.4. □

**Lemma 1.5 (Weil operator and incidence compatibility).** Let $T_Y$ be the normalized Weil operator of the induced polarized variation of Hodge structure along $Y$ (so that $T_Y^2 = \text{Id}$). Then the $+1$-eigensubbundle

$$W_{Y,\text{mid}} := \ker(T_Y - \text{Id}) \subset H_Y$$

is a smooth Lagrangian subbundle for $\omega_Y$. Moreover, for any incidence $Z < Y$ with $Y \cap \bar{Z} \neq \emptyset$, these subbundles are compatible under the natural incidence maps coming from control data. Consequently, the collection $\{W_{Y,\text{mid}}\}$ defines a self-dual mezzoperversity $W_{\text{mid}}$.

*Proof (pointer).* See Lemma 6.1, Lemma 6.2, Proposition 6.3, and Corollary 6.4. □

## 1.8. Logical dependencies and scope.
For the reader's convenience we summarize what is proved here, what is used as input, and what is deferred to companion work. This is meant to make the logical dependence transparent for refereeing.

**Results proved in this paper.** We construct, for every even stratum $Y \in S_{\text{ev}}$, a canonically defined Lagrangian subbundle $W_{Y,\text{mid}} \subset H_Y$ and show that these fit together into a self-dual mezzoperversity $W_{\text{mid}}$ (Sections 5–6). We then identify the resulting self-dual extension datum with that coming from the intersection Hodge module $\text{IC}_X^H$, and conclude that the associated Deligne-type complex satisfies

$$\text{IC}_{W_{\text{mid}}} \cong \text{IC}_{\bar{m}} \quad \text{in } \text{D}_c^b(X; \mathbb{R})$$

(Section 7), conditional only on the standard analytic-sheaf interface recorded as Assumption 3.8.

**External inputs (invoked without reproving).** To keep the dependency structure explicit, we list the few results invoked from the literature (or companion work) without reproving them here.

* Polarized Hodge module input: basic properties of (intersection) Hodge modules, nearby/vanishing cycles, and the induced polarized variations along strata [Sai88, Sai90, Sch14].

* Analytic-sheaf dictionary: the identification of ideal boundary condition de Rham complexes with Deligne-type (intersection) complexes, used only through Assumption 3.8; in Cheeger settings this is established in [ALMP18] and in our projective/Fubini-Study setup it is made explicit in the projective/Fubini-Study setting via the sheaf-theoretic closure of iterated ideal boundary conditions [Luo26, Appendix F and Theorem 7.7]. For quick checking of Assumption 3.8: in our usage, clauses (a) and (c) are covered by [ALMP18, §5], while clause (b) (incidence/transport bookkeeping) is made explicit in [Luo26, Appendix F]; sign conventions are tracked in [Luo26, Appendix D.5].

* Analytic packages on links: the construction and functoriality of canonical Cauchy data bundles / canonical mezzoperversities in the projective Fubini-Study setting (depth one: [LL26a, Theorems 6.2, 7.4, 7.6]; arbitrary depth: [Luo26, Theorems 5.1, 6.3, 7.6, 7.7]; isolated conic case: [LL26b, Theorem 1.1]).

**What is deferred to companion work.** The present paper constructs the *middle* self-dual mezzoperversity $W_{\text{mid}}$ dictated by Hodge-theoretic self-duality. A separate, analytic comparison is required to identify $W_{\text{mid}}$ with the *canonical* mezzoperversity $W_{\text{can}}$ constructed by iterated Calderón packages; this comparison is carried out in the companion work(s) cited above.
## 1.9. Organization.
Section 2 recalls the stratified projective setup and the link fibrations. Section 3 reviews mezzoperversities, trace bundles, and the Green pairing. Section 4 recalls the needed input from Saito’s theory of polarized Hodge modules and explains how intersection Hodge modules produce local PVHS data along strata. Sections 5–7 construct the Hodge package on trace bundles and prove Theorem 1.2. Appendices gather linear-algebraic and Hodge-module background used in the main text.

## 2. STRATIFIED PROJECTIVE VARIETIES AND FUBINI-STUDY WEDGE METRICS

This section fixes the stratified-geometric conventions and isolates the precise amount of wedge geometry needed later for trace bundles, Green pairings, and the “middle” boundary data. The only global input is the standard fact that the ambient Fubini-Study metric restricts to an iterated incomplete wedge metric on $X_{\text{reg}}$ (up to uniform quasi-isometry).

### 2.1. Thom-Mather control data and links.
We fix a Thom-Mather stratification $S$ of the complex projective variety $X \subset \mathbb{CP}^N$, satisfying the frontier condition: if $Y, Z \in S$ and $Z \cap \bar{Y} \neq \emptyset$, then $Z \subset \bar{Y}$. We write $Z < Y$ if $Z \subset \bar{Y} \setminus Y$. For a stratum $Y$ we denote its real codimension by $\text{codim}(Y) = c_Y$ and fix control data in the sense of Thom-Mather: a tubular neighborhood $\mathcal{T}_Y$ together with continuous maps

$$ \pi_Y: \mathcal{T}_Y \to Y, \quad \rho_Y: \mathcal{T}_Y \to [0, \infty), $$

with $Y = \rho_Y^{-1}(0)$, such that $\pi_Y$ restricts to a smooth submersion on each stratum intersecting $\mathcal{T}_Y$ and the control data are compatible on overlaps. We follow the conventions in [ALMP18, §2].

**Remark 2.1 (Existence and choice of control data).** Complex projective varieties admit Whitney stratifications and hence carry Thom-Mather stratifications equipped with compatible control data. We fix one such choice once and for all; all later constructions are invariant under stratified isomorphisms and depend on this choice only through the resulting quasi-isometry class of wedge metrics on $X_{\text{reg}}$. For background on Thom-Mather control data and iterated fibration structures, see [Pfl01, Ch. 1], [Ver84], and [ALMP18, §2].

**Definition 2.2 (Link fibration).** For a stratum $Y$, the link space (or link fibration) is

$$ \phi_Y: \mathcal{L}_Y := \rho_Y^{-1}(1) \cap \mathcal{T}_Y \to Y, \quad \phi_Y := \pi_Y|_{\mathcal{L}_Y}. $$

The fiber over $y \in Y$ is denoted $Z_y$ and called the *link fiber*.

By Thom-Mather theory, $\phi_Y$ is a smooth locally trivial fibration over $Y$ whose fiber $Z_y$ is a compact stratified pseudomanifold of strictly smaller stratification depth than $X$. Moreover, after shrinking $\mathcal{T}_Y$ if necessary, near each $y \in Y$ there are product-type charts identifying $\mathcal{T}_Y$ with $(0, \varepsilon)_r \times (\mathcal{L}_Y)_y$ up to stratified diffeomorphism, where $r := \rho_Y$ is the radial variable.

Geometrically, a neighborhood of $Y$ in $X_{\text{reg}}$ is described by wedge coordinates $(r, z, y)$, where $r = \rho_Y$ is the radial variable, $y \in Y$, and $z \in Z_y$ lies in the link fiber. Figure 2 summarizes the link fibration and the local product-type picture used throughout the wedge analysis.

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Local wedge picture near a stratum $Y$: the link fibration $\phi_Y: \mathcal{L}_Y \to Y$ with fiber $Z_y$, and the product-type identification of $\mathcal{T}_Y \cap X_{\text{reg}}$ by wedge coordinates $(r, z, y)$ with $r = \rho_Y$. See Definition 2.2 and Subsection 2.2.</figcaption></figure>
**Definition 2.3** (Depth and even strata). The depth of the stratification is the maximal length of a strict chain $Y_0 > Y_1 > \cdots > Y_k$ in the poset $(S, <)$.

A stratum $Y$ is called even (for boundary data) if the link fiber $Z_y$ has even real dimension $f_Y := \dim Z_y$ and, recursively, the induced stratification on $Z_y$ satisfies the same parity convention for its singular strata. We write $S_{ev} \subset S$ for the set of even strata.

**Remark 2.4** (Even strata versus genuinely relevant strata). Evenness is a necessary condition for “middle-degree” boundary phenomena, but it is not sufficient: the appearance of boundary data is controlled by the critical-degree kernel of a tangential operator along $\phi_Y$. In later sections we attach to each $Y \in S_{ev}$ a tangential family $A_Y$ (coming from the leading wedge form of $D_{dR}$) and its critical kernel bundle $H_Y$. Only those even strata with $H_Y \neq 0$ contribute to mezzoperversity choices; when $H_Y = 0$ the local $L^2$-de Rham operator is essentially self-adjoint near $Y$ and no boundary condition is needed there (compare [Che80] and [ALMP18]).

2.2. Iterated wedge metrics. Let $Y \in S$ and let $(r, z, y)$ be wedge coordinates on $\mathcal{T}_Y \cap X_{\text{reg}}$, where $r = \rho_Y$, $y \in Y$, and $z \in Z_y$ lies in the link fiber.

A Riemannian metric $g$ on $X_{\text{reg}}$ is called an (incomplete) wedge metric adapted to the control data if, for each stratum $Y$, on $\mathcal{T}_Y \cap X_{\text{reg}}$ it is uniformly quasi-isometric to a model metric of the form

$$ (2.2.1) \qquad g_{w,Y} = dr^2 + r^2 g_{Z_y} + g_Y, $$

where $g_Y$ is a smooth metric on $Y$ and $g_{Z_y}$ is a (possibly iterated) wedge metric on $(Z_y)_{\text{reg}}$ that depends smoothly on $y \in Y$. The metric is called iterated if the same description holds recursively for the family of link fibers. We refer to [ALMP18, §2] for the precise inductive definition and for equivalent formulations in terms of iterated fibration structures.

A basic feature of (2.2.1) is the resulting form of the de Rham operator $D_{dR} := d + \delta$ in these coordinates: schematically,

$$ (2.2.2) \qquad D_{dR} \sim c(dr) \left( \partial_r + \frac{1}{r} A_Y + \text{lower order terms} \right), $$

where $A_Y$ is a first-order self-adjoint tangential operator along the link fibration $\phi_Y$ (depending on the deeper boundary data when the depth is $\ge 2$). The trace bundle $H_Y$ used later is the smooth kernel bundle of this tangential family in the critical degree; it may be trivial (in which case no boundary data occur along $Y$).

**Remark 2.5** (Exact models and bounded perturbations). For the operator-theoretic constructions in later sections we will always work, after possibly shrinking $\mathcal{T}_Y$, with a convenient exact product-type model metric $g_{w,Y}$ of the form (2.2.1) and regard the actual metric (in particular $g_{FS}$) as a uniformly controlled perturbation of this model. This is sufficient because the tangential data (and hence the trace bundles and Green pairings) depend only on the leading wedge form and are stable under such bounded perturbations; see [ALMP18] and the references therein.

**Remark 2.6** (Volume growth and $L^2$-structures). If $\dim Z_y = f_Y$, then the Riemannian density of (2.2.1) is comparable to $r^{f_Y} dr \, dvol_{g_{Z_y}} \, dvol_{g_Y}$. In particular, $L^2$-integrability near $Y$ is governed by the link dimension and the behavior of solutions of (2.2.2) at the critical indicial roots; this is the source of the “middle-degree” phenomena encoded by mezzoperversities.

2.3. Fubini–Study wedge geometry. Let $g_{FS}$ denote the metric induced on $X_{\text{reg}}$ by the ambient Fubini–Study metric on $\mathbb{CP}^N$. The main point for the present paper is that $g_{FS}$ is an iterated incomplete wedge metric adapted to the chosen control data, and that it can be compared uniformly with an exact product-type wedge model.

**Proposition 2.7** (Fubini–Study wedge normal form). For each stratum $Y \in S$ there exist a (possibly smaller) tubular neighborhood $\mathcal{T}_Y$, wedge coordinates $(r, z, y)$ on $\mathcal{T}_Y \cap X_{\text{reg}}$, and an exact wedge model metric $g_{w,Y}$ of the form (2.2.1) such that the following hold on $\mathcal{T}_Y \cap X_{\text{reg}}$:
(i) There exists a constant $K_Y \ge 1$ with

$$K_Y^{-1} g_{w,Y} \le g_{FS} \le K_Y g_{w,Y}.$$

(ii) In the above wedge coordinates one may (and we do) choose the exact model $g_{w,Y}$ so that

$$ (2.3.1) \qquad g_{\mathrm{FS}} = g_{w,Y} + r h_Y, $$

for a smooth symmetric 2-tensor $h_Y$ on $\mathcal{T}_Y \cap X_{\mathrm{reg}}$ with $\|h_Y\|_{g_{w,Y}}$ and $\|\nabla^{g_{w,Y}} h_Y\|_{g_{w,Y}}$ uniformly bounded down to $r=0$. Consequently there exist a bounded bundle endomorphism $B_{Y,0}$ and a first-order wedge differential operator $B_{Y,1}$ (that is, locally a linear combination of the wedge vector fields $\partial_r$, $\partial_{y_j}$ and $\frac{1}{r}\partial_{z_k}$ with coefficients uniformly bounded down to $r=0$) such that

$$ (2.3.2) \qquad D_{\mathrm{dR}}(g_{\mathrm{FS}}) = D_{\mathrm{dR}}(g_{w,Y}) + B_{Y,0} + r B_{Y,1}. $$

In particular, multiplying (2.3.2) by $r$ shows that the rescaled edge operators

$$ rD_{\mathrm{dR}}(g_{\mathrm{FS}}) \quad \text{and} \quad rD_{\mathrm{dR}}(g_{w,Y}) $$

differ by terms that vanish after freezing coefficients at $r=0$; equivalently, they have the same normal operator and indicial family in the wedge calculus. Consequently they determine the same tangential family $A_Y$ in (2.2.2) and the same critical kernel bundle $H_Y = \ker(A_Y)$. See Lemma 2.8 and [ALMP18, §2].

(iii) The family of link metrics $g_{Z_y}$ can be chosen to vary smoothly with $y \in Y$ and, for each $y$, to be itself iterated wedge on the regular part of $Z_y$.

*Proof.* Statements (i) and (iii) are part of the standard wedge package for Thom-Mather stratified spaces with iterated fibration structures; see [ALMP18, §2], [Pfl01].

For the specific case of a complex projective variety equipped with the ambient Fubini-Study metric, [Luo26, Theorem 3.1 and Appendix B] constructs product-type wedge coordinates and an exact model metric $g_{w,Y}$ with smooth dependence on $y \in Y$. Moreover, one may take $g_{w,Y}$ to be obtained by freezing the leading coefficients of $g_{\mathrm{FS}}$ at $r=0$ in these coordinates; this yields the refined expansion (2.3.1) with uniform bounds on $h_Y$ and its first covariant derivative.

To deduce (2.3.2), note that $d$ is metric-independent and only the formal adjoint $\delta$ depends on the Hodge star. The relation (2.3.1) and the bound on $\nabla^{g_{w,Y}} h_Y$ imply that the Hodge stars satisfy $\star_{g_{\mathrm{FS}}} = \star_{g_{w,Y}} + r S_Y$ for a smooth bundle endomorphism $S_Y$ with uniformly bounded coefficients, and the same holds for the connection derivatives needed to write $\delta$. Consequently $\delta(g_{\mathrm{FS}}) - \delta(g_{w,Y})$ decomposes as a bounded zeroth-order term plus $r$ times a first-order wedge term with bounded coefficients (generated by $\partial_r$, $\partial_{y_j}$, and $\frac{1}{r}\partial_{z_k}$ in wedge coordinates), which gives (2.3.2). The co-incidence of normal operators and indicial families is then immediate from (2.3.2); see [ALMP18, §2]. □

**Lemma 2.8** (Rescaled normal data for wedge perturbations). Fix wedge coordinates $(r, z, y)$ near a stratum $Y$. Let $D$ be a first-order wedge differential operator, and let

$$ D' = D + B_0 + r B_1, $$

where $B_0$ is a zeroth-order operator with uniformly bounded coefficients and $B_1 \in \mathrm{Diff}_w^1$ is a first-order wedge differential operator with coefficients uniformly bounded down to $r=0$ (equivalently, a finite sum of bounded-coefficient multiples of $\partial_r$, $\partial_{y_j}$, and $\frac{1}{r}\partial_{z_k}$). Then the rescaled operators $rD$ and $rD'$ have the same normal operator at $r=0$, and consequently $D$ and $D'$ have the same indicial family.

*Proof.* By hypothesis, $rD' = rD + rB_0 + r^2 B_1$, and the last two terms vanish when freezing coefficients at $r=0$ in the definition of the normal operator. The statement about indicial families follows from the Mellin transform description of indicial operators; see [ALMP18, §2] and [Maz91]. □
**Remark 2.9 (Canonicality).** Throughout, “canonical” refers to constructions that are invariant under stratified isomorphisms of $(X, S)$ and under unitary changes of the projective embedding. In particular, the later analytic objects (trace bundles, Green pairings, and canonical domains) depend only on the induced Fubini–Study wedge geometry up to uniform quasi-isometry and bounded perturbation within the class of adapted iterated wedge metrics.

### 3. MEZZOPERVERSITIES, TRACE BUNDLES, AND GREEN PAIRINGS

This section recalls the analytic boundary-data formalism for the iterated wedge de Rham operator and fixes notation for the trace bundles and their Green pairings. We follow [ALMP18] and use the language of ideal boundary conditions (IBCs), equivalently mezzoperversities. Only the features needed later (to compare with the Hodge-module input) are recorded here.

**3.1. Local wedge model and tangential operators.** Let $Y \in S_{ev}$ be an even stratum with link fibration $\phi_Y: L_Y \to Y$ as in Definition 2.2. On $T_Y \cap X_{reg}$ we may use wedge coordinates $(r, z, y)$, where $r = \rho_Y$, $y \in Y$, and $z \in Z_y$ lies in the link fiber. With respect to a wedge model metric $g_{w,Y}$, the de Rham operator $D_{dR} := d + \delta$ takes the schematic form (2.2.2). The tangential term is a first-order self-adjoint family

$$A_Y \to Y,$$

acting on differential forms on the fibers $Z_y$ and encoding the vertical Hodge–de Rham operator together with the degree shift coming from the wedge metric.

When the stratification depth is $\ge 2$, the definition of $A_Y$ depends on the choices of boundary data on deeper strata $Z < Y$. We package those deeper choices by writing

$$W^{>Y} = W^{>Y}$$

for a fixed collection of self-dual boundary conditions on all strata strictly contained in $\bar{Y} \setminus Y$. Once $W^{>Y}$ is fixed, the tangential operator becomes a well-defined self-adjoint Fredholm family, denoted

$$A_Y^{W^{>Y}}.$$

In the bottom-depth case (no deeper strata) one may take $W^{>Y} = \emptyset$ and recover the usual Cheeger tangential operator.

### 3.2. Trace bundles and the Green pairing.

**Definition 3.1 (Trace bundle).** Fix deeper boundary data $W^{>Y}$ on all strata $Z < Y$. The trace bundle over $Y$ is the smooth vector bundle

$$H_Y := \ker(A_Y^{W^{>Y}}) \to Y.$$

Equivalently, $H_Y$ is the bundle of vertical harmonic forms in the critical (middle) degree on the link fibers $(Z_y)_{reg}$, computed with the induced boundary conditions coming from $W^{>Y}$.

**Remark 3.2 (Flatness and local-system structure).** The family $A_Y^{W^{>Y}}$ is compatible with the fibration $\phi_Y$; in particular, $H_Y$ carries a natural flat connection obtained by parallel transport along $Y$ (using the fibration and the smooth dependence of the kernels). We denote the corresponding local system by the same symbol $H_Y$. Precise constructions (and the required constant-rank / uniform spectral-gap input ensuring that $H_Y$ is a smooth flat bundle) are developed in [ALMP18] and, in the projective/Fubini–Study setting, in [Luo26, Theorem 6.3] (see also [Luo26, Theorem 5.1] for the associated trace/Calderón package in the constant strip).

**Definition 3.3 (Green pairing).** Let $u, v \in \Gamma(H_Y)$ be smooth sections represented by vertical harmonic forms on $Z_y$. The integration-by-parts formula for $D_{dR}$ on a collar $(0, \varepsilon)_r \times L_Y$ produces a bilinear form

$$\omega_Y: H_Y \otimes H_Y \to \mathbb{R},$$
called the *Green pairing*. It is skew-symmetric and nondegenerate on each fiber of $H_Y$; in particular, each fiber $(H_Y)_y$ is a symplectic vector space.

**Remark 3.4** (A model formula and symplecticity). In wedge coordinates one may write $\omega_Y$ explicitly as a boundary-flux pairing. Up to a universal sign convention, it is given by the fiber integral

$$ \omega_Y(u, v) = \int_{Z_y} \langle u, J v \rangle \, d\mathrm{vol}_{Z_y}, $$

where $J$ is the bundle endomorphism on the link-fiber form bundle induced by the inward unit normal for the wedge metric (equivalently, by the principal symbol of $D_{\mathrm{dR}}$ in the normal direction). Since the normal principal symbol is fiberwise invertible, $J$ restricts to an invertible endomorphism of the critical trace space $(H_Y)_y$. Therefore the map $u \mapsto \omega_Y(u, \cdot)$ identifies $(H_Y)_y$ with its dual, so $\omega_Y$ is fiberwise nondegenerate and rk$(H_Y)$ is even.

When working with complex-valued forms we implicitly replace $H_Y$ by its complexification and extend $\omega_Y$ $\mathbb{C}$-bilinearly. We keep the overall sign/normalization implicit and refer to [ALMP18, §3–§4] (see also [BBW93]) for detailed conventions for Green pairings and symplectic boundary data for Dirac-type operators.

A subbundle $W_Y \subset H_Y$ is called *Lagrangian* if it is maximally isotropic for $\omega_Y$.

**3.3. Mezzoperversities and incidence compatibility.** The non-uniqueness of closed and self-adjoint extensions of $D_{\mathrm{dR}}$ is concentrated precisely in the middle-degree kernels $H_Y$. A choice of *ideal boundary condition* amounts to choosing, for every relevant stratum (namely, every even stratum $Y$ for which $H_Y \neq 0$), a Lagrangian subbundle of the corresponding trace bundle that varies smoothly and is compatible under incidence relations.

Incidence morphisms on trace bundles. Let $Z < Y$ be a frontier relation and set $U_{ZY} := Y \cap \mathcal{T}_Z$. The Thom–Mather control data determine a radial retraction $r_Z: U_{ZY} \to Z$ and, over $U_{ZY}$, a canonical smooth map of link fibrations

$$ \iota_{ZY}: \mathcal{L}_Y|_{U_{ZY}} \to \mathcal{L}_Z|_{U_{ZY}} $$

covering $r_Z$; see [ALMP18, §2]. Fix deeper boundary data $W^{>Y}$; it induces deeper data on strata below $Z$ and hence a trace bundle $H_Z = \ker(A_Z^{>Z}) \to Z$. Functionality of the IBC de Rham complexes (equivalently, of the trace/Poisson package) yields a morphism of flat bundles (local systems)

$$ \alpha_{ZY}: H_Y|_{U_{ZY}} \to H_Z|_{U_{ZY}}, $$

called the *incidence morphism*. Moreover, $\alpha_{ZY}$ is compatible with Green pairings in the sense that, after restricting to $U_{ZY}$, it preserves the symplectic structures $\omega_Y$ and $\omega_Z$ up to the standard global sign conventions; see [ALMP18, §3–§4] and [Luo26, Appendix F].

**Definition 3.5** (Mezzoperversity). A *mezzoperversity* on $(X, S)$ is the assignment of a smooth subbundle

$$ W_Y \subset H_Y $$

for every even stratum $Y \in S_{\mathrm{ev}}$ with nontrivial trace bundle, subject to the following *incidence compatibility* condition: for every incidence $Z < Y$ with $U_{ZY} = Y \cap \mathcal{T}_Z \neq \emptyset$, the incidence morphism $\alpha_{ZY}: H_Y|_{U_{ZY}} \to H_Z|_{U_{ZY}}$ satisfies

$$ \alpha_{ZY}(W_Y|_{U_{ZY}}) \subseteq W_Z|_{U_{ZY}}. $$

(If $H_Z = 0$ the condition is vacuous.)

**Convention.** In all applications in this paper, the chosen subbundles $W_Y \subset H_Y$ will be flat (equivalently, they define sub-local systems); the middle choice $W_{Y,\mathrm{mid}}$ has this property by Corollary 7.4.
**Remark 3.6** (Flat versus smooth Lagrangian data). In [ALMP18, §2] each trace bundle $H_Y \to Y$ is viewed as a flat bundle (i.e. a local system), and a self-dual mezzoperversity is often formulated as the choice of a flat Lagrangian subbundle (equivalently, a Lagrangian sub-local system). More generally, a mezzoperversity can be viewed as a choice of a flat subbundle, with self-duality imposing the additional Lagrangian condition. The present definition is phrased in terms of smooth subbundles because the analytic ideal boundary conditions and the incidence condition are naturally expressed at that level. In the sheaf-theoretic classification of self-dual extensions used in Section 7, the relevant extension datum is flat. We will show that the middle choice constructed in Section 6, $W_{Y,\text{mid}} = \ker(T_Y - \text{Id})$, is in fact flat (Corollary 7.4), so that $W_{\text{mid}}$ agrees with the “flat” notion of [ALMP18].

**Remark 3.7** (Self-duality). The condition that a mezzoperversity be self-dual (equivalently, that each $W_Y$ be Lagrangian for the Green pairing) encodes self-duality of the associated extension datum, hence of the resulting Deligne-type complex; see [ALMP18, §4, §8]. In particular, a self-dual mezzoperversity determines a Poincaré duality pairing on the corresponding $L^2$ cohomology groups.

**3.4. Deligne complexes and the analytic-sheaf dictionary.** To a mezzoperversity $\mathcal{W} = \{W_Y\}$ one associates:

* a closed extension of the de Rham differential (hence of $D_{\text{dR}}$) on $X_{\text{reg}}$, usually denoted by an IBC domain $\mathcal{D}_{\mathcal{W}}$, and

* a Deligne-type complex of sheaves $\text{IC}_{\mathcal{W}}$ on $X$.

The construction is functorial under stratified maps and is compatible with Verdier duality; in particular, when $\mathcal{W}$ is self-dual the complex $\text{IC}_{\mathcal{W}}$ is Verdier self-dual.

We isolate as a standing input the analytic-sheaf identification that allows us to pass between the IBC $L^2$ de Rham theory and the corresponding Deligne complex.

**Assumption 3.8** (Analytic-sheaf dictionary). The IBC de Rham complex on $X_{\text{reg}}$ associated with a mezzoperversity $\mathcal{W}$ computes the hypercohomology of the corresponding Deligne-type complex $\text{IC}_{\mathcal{W}}$ on $X$:

$$H_{(2)}^{\bullet}(X_{\text{reg}}; \mathcal{W}) \cong \mathbb{H}^{\bullet}(X; \text{IC}_{\mathcal{W}}).$$

This identification is natural with respect to restriction to open subsets of $X$ and functorial for stratified maps.

Moreover, the Verdier duality pairing on $\text{IC}_{\mathcal{W}}$ corresponds, under the above identification, to the $L^2$ Poincaré pairing on $H_{(2)}^{\bullet}(X_{\text{reg}}; \mathcal{W})$. On the local systems $H_Y$ along strata this correspondence matches the Verdier pairing induced on the boundary/trace data with the Green pairing $\omega_Y$ from Definition 3.3 (up to the standard global sign convention). All of these compatibilities are established in [ALMP18] and, for Fubini-Study wedge metrics, in [Luo26, Appendix F] (see also [Luo26, Appendix D.5] for the corresponding bookkeeping for trace bundles and Green pairings).

**Remark 3.9** (How Assumption 3.8 is used in this paper). For the reader's convenience, we record that Assumption 3.8 is invoked in this paper only through the following consequences:

(a) the identification of $H_{(2)}^{\bullet}(X_{\text{reg}}; \mathcal{W})$ with $\mathbb{H}^{\bullet}(X; \text{IC}_{\mathcal{W}})$, natural for restriction to open subsets of $X$;

(b) compatibility with the restriction/incidence morphisms needed to compare the trace bundle $H_Y = \ker(A_Y^{W \to Y})$ with the local system of middle link intersection cohomology (see Proposition 5.1);

(c) the identification of the Green pairing on $H_{(2)}^{\bullet}$ (hence on trace bundles) with the Verdier pairing on $\text{IC}_{\mathcal{W}}$, up to the universal sign conventions fixed in §3.

No other analytic input is required for the construction of the middle Lagrangian $W_{\text{mid}}$; the remaining arguments are sheaf-theoretic and Hodge-theoretic.
(Quick reference: clauses (a),(c) in [ALMP18, §5]; clause (b) in [Luo26, Appendix F]; signs in [Luo26, Appendix D.5].)

**Proposition 3.10** (Compatibility package implied by Assumption 3.8). Let $\mathcal{W}$ be a (self-dual) mezzoperversity on $X$ and let $\mathrm{IC}_{\mathcal{W}}$ be the associated Deligne-type complex. For each even stratum $Y$, write $Z_y$ for the link fiber at $y \in Y$ and set $d_Y := \frac{1}{2} \dim Z_y$ (independent of $y$). Then the analytic-sheaf correspondence of Assumption 3.8 can be used in this paper through the following functorial compatibilities:

(a) (Cohomology) For every open subset $U \subseteq X$ there is a natural identification

$$H_{(2)}^{\bullet}(U \cap X_{\mathrm{reg}}; \mathcal{W}) \cong \mathbb{H}^{\bullet}(U; \mathrm{IC}_{\mathcal{W}}|_{U}),$$

compatible with restriction in $U$.

(b) (Restriction and incidence) For even strata $Z < Y$ and $U_{ZY} := Y \cap T_Z$, the analytic incidence morphism on trace bundles $\alpha_{ZY}: H_Y|_{U_{ZY}} \to H_Z|_{U_{ZY}}$ corresponds to the morphism of critical stratum local systems induced by restriction of $\mathrm{IC}_{\mathcal{W}}$; equivalently, the diagram

$$\begin{tikzcd}
H_Y|_{U_{ZY}} \arrow[r, "\alpha_{ZY}", "\cong"'] \arrow[d] & H_Z|_{U_{ZY}} \arrow[d] \\
(\mathcal{H}^{-d_Y}(i_Y^*\mathrm{IC}_W)) \otimes_Q \mathbb{R} \arrow[r, "\cong"'] & (\mathcal{H}^{-d_Z}(i_Z^*\mathrm{IC}_W)) \otimes_Q \mathbb{R}
\end{tikzcd}$$

commutes, where the vertical identifications are those coming from Proposition 5.1.

(c) (Pairings) Under the identification of (a) and the trace/critical-local-system identification, the Green pairing on trace data agrees with the Verdier pairing on $\mathrm{IC}_{\mathcal{W}}$ up to the universal sign conventions fixed in §3. In particular, for each even stratum $Y$ the diagram

$$\omega_Y(u, v) = \pm \langle u, v \rangle_{\mathrm{Verdier}} \quad \text{for } u, v \in H_Y,$$

after identifying $H_Y \cong (\mathcal{H}^{-d_Y}(i_Y^*\mathrm{IC}_W)) \otimes_Q \mathbb{R}$ as above. commutes up to the same universal sign.

*Proof.* This proposition is a convenient packaging of the functoriality and duality compatibilities built into the analytic-sheaf correspondence of Assumption 3.8. Precise formulations and sign bookkeeping are carried out in [ALMP18, §5] and [Luo26, Appendix D.5]; see also [Luo26, Appendix F] for the incidence bookkeeping on trace bundles.

For ease of cross-checking, the above three items correspond to the following pieces of the literature:

(a) for (a) (identification of IBC $L^2$ de Rham cohomology with a Deligne-type complex, functorial under restriction), see [ALMP18, §5] (together with the references cited there for the general stratified-wedge setup);

(b) for (b) (compatibility of incidence/restriction morphisms with the critical local systems and trace bundles), see [Luo26, Appendix F] (and the trace-bundle bookkeeping in [Luo26, Appendix D.5]);

(c) for (c) (Verdier pairing versus Green pairing, including the universal sign conventions fixed in §3), see [ALMP18, §5], with the sign convention tracked explicitly in [Luo26, Appendix D.5].

□

**Remark 3.11** (Canonical mezzoperversity). For the Fubini-Study metric, canonical Cauchy data and hence a canonical mezzoperversity $\mathcal{W}_{\mathrm{can}}$ are constructed by Calderón packages on the stratified link fibrations:

* in the isolated conic (depth-0) base case, see [LL26b, Theorem 1.1];

* in the projective simple-edge (depth-1) case, see [LL26a, Theorems 6.2, 7.4, 7.6];

* in arbitrary stratification depth, see [Luo26, Theorems 5.1, 6.3, 7.6, 7.7].
In each case the associated $L^2$ de Rham complex is Fredholm and computes $\mathrm{IH}_{W_{\mathrm{can}}}(X)$. The purpose of this paper is to construct a second, Hodge-theoretically defined, self-dual mezzoperversity $W_{\mathrm{mid}}$ and to show that its Deligne-type complex equals the middle intersection complex $\mathrm{IC}_{\bar{m}}$.

### 3.5. Symplectic incidence and self-adjoint IBC domains.
The previous constructions are set up so that the boundary/trace data behave functorially under incidence relations and encode the adjointness properties of the global $L^2$ de Rham operator. We record two standard facts in a form that can be cited later.

**Lemma 3.12 (Symplecticity of incidence morphisms).** Let $Z < Y$ be a frontier relation and $U_{ZY} = Y \cap \mathcal{T}_Z$. For each connected component of $U_{ZY}$ there exists a sign $\epsilon_{ZY} \in \{\pm 1\}$ such that for all $u, v \in H_Y|_{U_{ZY}}$ one has

$$ \omega_Z(\alpha_{ZY}u, \alpha_{ZY}v) = \epsilon_{ZY} \omega_Y(u, v). $$

In particular, $\alpha_{ZY}$ is fiberwise symplectic up to the standard global sign conventions.

*Proof.* Let $\partial_{dR}$ denote the (wedge) de Rham operator. The Green form is the boundary pairing on maximal domains and depends only on the Cauchy data at each stratum; equivalently, it descends to a nondegenerate pairing on the trace bundle via the Cauchy data map, cf. [ALMP18, Proposition 2.8]. Along a frontier relation $Z < Y$, the incidence morphism $\alpha_{ZY}$ is the restriction/pullback map between the corresponding trace bundles induced by the Thom-Mather control data, and the locality of the boundary pairing implies that the induced pairing is preserved under $\alpha_{ZY}$. The choice of inward-pointing normal (hence the normalization of $J$) produces the locally constant sign $\epsilon_{ZY}$. See [ALMP18, Lemma 5.5] for the boundary-pairing computation and [Luo26, Appendix F] for a direct derivation in the present iterated wedge setting. $\square$

**Proposition 3.13 (Adjoint domains and self-duality).** Let $\mathcal{W} = \{W_Y\}$ be a mezzoperversity and let $\mathcal{D}_\mathcal{W}$ denote the associated IBC domain for the de Rham differential (hence for $D_{dR}$) on $X_{\mathrm{reg}}$. Define the dual mezzoperversity $\mathcal{W}^\sharp = \{W_Y^\sharp\}$ by

$$ W_Y^\sharp := W_Y^{\perp\omega_Y} \quad \text{in } H_Y $$

for each even stratum $Y$ with nontrivial trace bundle. Then the Hilbert adjoint domain satisfies

$$ (\mathcal{D}_\mathcal{W})^* = \mathcal{D}_{\mathcal{W}^\sharp}. $$

Consequently, $D_{dR}$ with domain $\mathcal{D}_\mathcal{W}$ is self-adjoint if and only if $\mathcal{W}$ is self-dual, i.e. $W_Y$ is Lagrangian for every relevant even stratum (equivalently, $W_Y = W_Y^{\perp\omega_Y}$).

*Proof.* Fix a stratum $Y$. By [ALMP18, Proposition 2.8], the Cauchy data map identifies the quotient of maximal by minimal domains with the trace bundle, and the Green form induces a nondegenerate skew pairing on this trace bundle; this is precisely our $\omega_Y$. A mezzoperversity $\mathcal{W} = \{W_Y\}$ prescribes a local boundary condition by requiring that the Cauchy data lie in $W_Y$. The general adjoint-domain characterization in terms of the boundary pairing then shows that $v$ lies in $(\mathcal{D}_\mathcal{W})^*$ if and only if its Cauchy data are $\omega_Y$-orthogonal to $W_Y$, i.e. belong to $W_Y^{\perp\omega_Y}$. In depth one, this computation and the self-adjointness of $(D_{dR}, \mathcal{D}_\mathcal{W})$ are carried out explicitly in [ALMP18, Proposition 5.4]. For arbitrary depth, the same conclusion follows by the inductive boundary-pairing analysis [ALMP18, Lemma 5.5 and Theorem 5.6], together with the incidence compatibility encoded in Definition 3.5. Finally, the identification of the corresponding dual mezzoperversity via orthogonal complements and its relation to adjoint domains is recorded in [ALMP18, Proposition 6.1 and Definition 6.2]. For the general Dirac-type boundary-pairing viewpoint, see [BBW93, Ch. 2]. $\square$
## 4. POLARIZED HODGE MODULES AND INTERSECTION COMPLEXES

This section records the Hodge-theoretic input used in the sequel. We work throughout with reduced complex algebraic varieties and use Saito's theory of polarizable (mixed) Hodge modules [Sai88, Sai90]. We only extract two facts: (i) the middle intersection complex carries a canonical Hodge-module lift and (ii) its restrictions to strata produce polarized variations of Hodge structure on the corresponding intersection-cohomology local systems.

**Constructibility along the analytic stratification.** Saito's theory is formulated in the algebraic (hence complex-analytic) category, while our analytic setup uses a fixed Thom-Mather stratification of the underlying real space. We tacitly replace the chosen Thom-Mather stratification by a common refinement with an algebraic Whitney stratification so that all complexes considered below (in particular $IC_{\bar{m}}$ and the underlying perverse sheaves of Hodge modules) are constructible with respect to the resulting stratification. In particular, for each stratum Y the restriction complexes $i_Y^*IC_{\bar{m}}$ have locally constant cohomology sheaves (local systems) along Y.

**4.1. Intersection Hodge modules.** Let X be a complex algebraic variety and let $j: X_{\text{reg}} \hookrightarrow X$ denote the inclusion of the smooth locus. Write $IC_{\bar{m}}$ for the middle (perverse) intersection complex of X in the sense of [GM80, BBD82]. Saito's theory provides a canonical Hodge-module enhancement of $IC_{\bar{m}}$.

**Theorem 4.1 (Saito [Sai90]).** Let X be a reduced complex algebraic variety. There exists a polarizable Hodge module $IC_X^H$ whose underlying perverse sheaf is $IC_{\bar{m}}$ (up to the standard shift convention). Moreover $IC_X^H$ is characterized by extending the constant Hodge module on $X_{\text{reg}}$ and by functoriality with respect to open embeddings.

**Remark 4.2.** We will only use the existence of $IC_X^H$ and the fact that it is *polarizable* and *self-dual* (in the Hodge-module sense). All statements below are insensitive to the cohomological shift conventions, as long as one keeps track of the induced local systems along strata.

The same applies to the Tate twist in the self-duality isomorphism: it only affects the weight bookkeeping and plays no role in the definition of the involutions $T_Y$ and their eigenspace Lagrangians, once the global sign normalization has been fixed (Remark B.7).

**4.2. Local systems along strata and link intersection cohomology.** Fix a Thom-Mather stratification of X and let $Y \subset X$ be a stratum with inclusion $i_Y: Y \hookrightarrow X$. A basic feature of intersection complexes is that their restriction to a stratum is (up to shift) a local system; see [BBD82].

**Proposition 4.3 (Cohomology local systems along a stratum).** Let $Y \subset X$ be a stratum with inclusion $i_Y: Y \hookrightarrow X$. Then for every $k \in \mathbb{Z}$ the cohomology sheaf

$$ \mathcal{H}_Y^k := \mathcal{H}^k(i_Y^*IC_{\bar{m}}) $$

is a finite-rank local system on Y. The formation of these local systems is locally constant along Y and is compatible with refinement of the control data (equivalently, with passing to a common refinement of stratifications as above).

In the even-link situation relevant for trace bundles (i.e. $\dim Z_y$ is even for $y \in Y$), we single out the critical local system

$$ \mathcal{H}_Y := \mathcal{H}_Y^{-d_Y} \quad \text{where} \quad d_Y := \frac{\dim Z_y}{2}. $$

(The integer $d_Y$ is well-defined because $\dim Z_y$ is locally constant on the stratum Y.)

**Proof.** Since $IC_{\bar{m}}$ is constructible with respect to $\mathcal{S}$, the restriction $i_Y^*IC_{\bar{m}}$ has locally constant cohomology sheaves on Y, hence each $\mathcal{H}_Y^k$ is a finite-rank local system. This is standard in the theory of constructible (and perverse) sheaves; see for example [BBD82, GM80, GM83]. Compatibility
with refinement follows from the fact that the intersection complex is invariant under passing to a common refinement of stratifications; see again [BBD82]. □

In the present geometric situation, $\mathcal{H}_Y$ admits a concrete description in terms of the link fibration $\pi_Y: L_Y \to Y$ from Section 2. Let $Z_y := \pi_Y^{-1}(y)$ be the (stratified) link fiber over $y \in Y$.

**Proposition 4.4** (Link description). Let $Y \subset X$ be a stratum and $y \in Y$. Let $\pi_Y: L_Y \to Y$ be the link fibration and write $Z_y = \pi_Y^{-1}(y)$. Then the stalk complex $(i_Y^*\text{IC}_m)_y$ identifies with the (middle) intersection cohomology of the open cone on $Z_y$ (with respect to the induced stratification); see [GM80, GM83, BBD82]. In particular, in the even-link situation relevant for trace bundles (i.e. $\dim Z_y$ is even), the critical stalk of the local system $\mathcal{H}_Y = \mathcal{H}_Y^{-dY}$ from Proposition 4.3 admits a canonical identification

$$ (\mathcal{H}_Y)_y \cong \text{IH}^{dY}(Z_y; \mathbb{Q}), \quad d_Y = \frac{\dim Z_y}{2}, $$

and these identifications vary locally constantly with $y$.

**Remark 4.5.** In Sections 3–6, we will identify the analytic trace bundle $H_Y = \ker(A_Y^{W>Y}) \to Y$ with the real/complex local system underlying the critical stratum local system $\mathcal{H}_Y$ of Proposition 4.3 (after tensoring with $\mathbb{R}$ or $\mathbb{C}$ as appropriate). Proposition 4.4 provides the corresponding link description of the fibers, which will later be obtained analytically from the parameter-dependent Calderón package.

**4.3. Polarizations, Weil operators, and duality on strata.** The Hodge module $\text{IC}_X^H$ is polarizable. Restricting the polarization to a stratum yields a polarized variation of Hodge structure on $\mathcal{H}_Y$.

**Proposition 4.6** (PVHS on $\mathcal{H}_Y$). For every stratum $Y$, the local system $\mathcal{H}_Y$ underlies a polarized variation of Hodge structure (over $\mathbb{Q}$) induced from $\text{IC}_X^H$. In particular, for each $y \in Y$ the fiber $(\mathcal{H}_Y)_y$ carries a pure Hodge structure together with a polarization pairing

$$ Q_Y: (\mathcal{H}_Y)_y \times (\mathcal{H}_Y)_y \to \mathbb{Q}, $$

well-defined up to the conventional Tate twist dictated by Saito’s normalization.

*Proof.* The polarizable Hodge module $\text{IC}_X^H$ restricts to $Y$ via Saito’s functors, and on a smooth stratum these functors land in the subcategory of smooth polarizable Hodge modules. By Saito’s equivalence between smooth polarizable Hodge modules and polarizable variations of Hodge structure, the resulting object is exactly a PVHS on the underlying local system $\mathcal{H}_Y$. The polarization pairing $Q_Y$ is induced from the polarization of $\text{IC}_X^H$ by functoriality of polarizations under restriction. We refer to [Sai90, Sai16, Sch14] for the restriction formalism and the identification of smooth objects with PVHS. □

Let $C_Y$ denote the associated Weil operator on $(\mathcal{H}_Y)_y \otimes \mathbb{C}$. Following the standard Hodge-theoretic convention, we package the polarization into an alternating form that will match the Green pairing introduced in Section 3.2.

**Definition 4.7** (Hodge symplectic form). For $y \in Y$, define a bilinear form

$$ \omega_Y^{\text{Hdg}}(u, v) := Q_Y(u, C_Y v). $$

When $(\mathcal{H}_Y)_y$ is the middle intersection cohomology of an even-dimensional oriented link fiber $Z_y$, the form $\omega_Y^{\text{Hdg}}$ is nondegenerate and skew-symmetric; we therefore view it as a symplectic form on the corresponding trace space.

**Remark 4.8.** The self-duality of $\text{IC}_X^H$ and Verdier duality for perverse sheaves [BBD82] imply that the polarization on $\mathcal{H}_Y$ is compatible with the intersection pairing on link intersection cohomology. This compatibility is the sheaf-theoretic counterpart of the analytic Green pairing compatibility stated in Proposition 5.4 below.
**4.4. Incidence maps and functoriality.** If $Z < Y$ (frontier relation), the control data yield canonical incidence maps between the corresponding link fibrations and hence between the associated local systems on strata. On the sheaf side these arise from the functoriality of restriction maps for $\mathrm{IC}_{\bar{m}}$.

**Proposition 4.9 (Incidence compatibility).** Let $Z < Y$ be strata and consider the corresponding local systems $\mathcal{H}_Y$ on $Y$ and $\mathcal{H}_Z$ on $Z$. There are natural morphisms (incidence maps) between their pullbacks to $Y \cap \mathcal{T}_Z$ which are compatible with the polarization data coming from $\mathrm{IC}_X^H$. Consequently, the family of middle Lagrangians extracted from the Weil operators (Section 6) satisfies the mezzoperversity incidence conditions.

*Proof.* This is a direct consequence of the functoriality of $\mathrm{IC}_X^H$ under restriction and specialization to deeper strata: the induced incidence morphisms $\alpha_{ZY}$ on $U_{ZY} = Y \cap \mathcal{T}_Z$ are morphisms in $\mathrm{D}^b\mathrm{MHM}(U_{ZY})$ and are compatible with the polarizations (cf. Remark B.5 and Remark B.7). Passing to the underlying local systems gives morphisms of polarized variations of Hodge structure. The last assertion, namely that the middle Lagrangians satisfy the mezzoperversity incidence conditions, follows from Lemma 4.10. □

**Lemma 4.10 (Functoriality of the Hodge involution).** Let $Z < Y$ with $Y \cap \bar{Z} \neq \emptyset$ and set

$$U_{ZY} := Y \cap \mathcal{T}_Z.$$

Let

$$\alpha_{ZY}: \mathcal{H}_Y|_{U_{ZY}} \to \mathcal{H}_Z|_{U_{ZY}}$$

be an incidence morphism induced by the restriction/specialization maps for the intersection Hodge module $\mathrm{IC}_X^H$. Then $\alpha_{ZY}$ is a morphism of polarized variations of Hodge structure. In particular it commutes with the normalized Weil operators $T_Y$ and $T_Z$. Equivalently, after identifying the underlying local systems with the trace bundles $H_Y$ and $H_Z$, we have a commutative diagram

$$ \begin{array}{ccc} H_Y|_{U_{ZY}} & \xrightarrow{\alpha_{ZY}} & H_Z|_{U_{ZY}} \\ \downarrow T_Y & & \downarrow T_Z \\ H_Y|_{U_{ZY}} & \xrightarrow{\alpha_{ZY}} & H_Z|_{U_{ZY}}. \end{array} $$

*Proof.* A morphism of mixed Hodge modules (in particular, the restriction and nearby-cycle specialization morphisms used to define $\alpha_{ZY}$) restricts to morphisms on strata which respect the Hodge and weight filtrations and are compatible with the polarizations; see, for instance, [Sch14, Thm. 15.1, §4] and [Sai90]; compare also Remark B.5 and Remark B.7. Passing to the associated local systems yields $\alpha_{ZY}$ as a morphism of PVHS (equivalently, a morphism of smooth polarizable Hodge modules, cf. [Sch14, Thm. 15.1]). Since the Weil operator acts by $i^{p-q}$ on each $(p, q)$-piece and morphisms of Hodge structures preserve the Hodge decomposition, it commutes with $\alpha_{ZY}$; hence $\alpha_{ZY} \circ T_Y = T_Z \circ \alpha_{ZY}$. Consequently, $\alpha_{ZY}$ preserves the $\pm 1$ eigensubbundles of the involutions, and in particular

$$\alpha_{ZY}(W_{Y,\mathrm{mid}}|_{U_{ZY}}) \subseteq W_{Z,\mathrm{mid}}|_{U_{ZY}}.$$

□

**Proposition 4.11 (Control-data description of the incidence morphism).** Write $U_{ZY} = Y \cap \mathcal{T}_Z$ as in Lemma 4.10. On $\mathcal{T}_Z \setminus Z$ the control function $\rho_Z$ determines a radial retraction

$$r_Z: \mathcal{T}_Z \setminus Z \to \mathcal{L}_Z, \quad r_Z(x) := \text{the unique point on the } \rho_Z\text{-ray through } x \text{ with } \rho_Z = 1.$$

Since $\mathcal{L}_Y|_{U_{ZY}} \subset \mathcal{T}_Z \setminus Z$, we obtain a canonical map of fibrations over $U_{ZY}$,

$$\iota_{ZY}: \mathcal{L}_Y|_{U_{ZY}} \to \pi_Z^*\mathcal{L}_Z, \quad \iota_{ZY}(x) := (\phi_Y(x), r_Z(x)),$$
where $\pi_Z^*L_Z \to U_{ZY}$ denotes the pullback of the link fibration $L_Z \to Z$ along $\pi_Z|_{U_{ZY}}: U_{ZY} \to Z$. Equivalently, $\iota_{ZY}$ fits into the commutative diagram

$$ \begin{array}{ccc} L_Y|_{U_{ZY}} & \xrightarrow{\iota_{ZY}} & \pi_Z^*L_Z \\ \downarrow & & \downarrow \\ U_{ZY} & = & U_{ZY}. \end{array} $$

The induced morphism on the middle intersection cohomology local systems of the fibers (with the inherited deeper boundary conditions) agrees with the incidence morphism $\alpha_{ZY}$ after identifying $H_Y$ and $H_Z$ with these local systems via Proposition 5.1 and Assumption 3.8; compare [ALMP18, §2] and [Luo26, Appendix F].

*Proof.* The Thom-Mather control data on $X$ provide the radial retraction $r_Z$ and hence the fibration map $\iota_{ZY}$ over $U_{ZY}$. Since $\iota_{ZY}$ is fiberwise over $U_{ZY}$, it induces a morphism of the middle intersection cohomology local systems of the fibers (with the inherited deeper boundary conditions). Under the identifications of $H_Y$ and $H_Z$ with these local systems from Proposition 5.1 (together with Assumption 3.8), this morphism is precisely the incidence morphism $\alpha_{ZY}$ used in Definition 3.5; see [ALMP18, §2] and [Luo26, Appendix F] for the corresponding constructions in the analytic and sheaf-theoretic models. $\square$

*Proof of Proposition 4.9.* On $U_{ZY}$ the incidence morphism $\alpha_{ZY}$ compares the local systems $\mathcal{H}_Y$ and $\mathcal{H}_Z$. By Lemma 4.10, $\alpha_{ZY}$ commutes with $T_Y$ and $T_Z$. Therefore $\alpha_{ZY}$ carries the +1 eigen-subbundle $\ker(T_Y - \text{Id})$ into $\ker(T_Z - \text{Id})$. Since $W_{Y,\text{mid}}$ and $W_{Z,\text{mid}}$ are defined fiberwise as these eigensubbundles (Proposition 6.3), this is exactly the required incidence compatibility in Definition 3.5. $\square$

**Remark 4.12.** The statements in this section are standard consequences of Saito’s theory, together with the local description of $\text{IC}_{\bar{m}}$ along strata [BBD82, Sai90]. We emphasize them here only as *interfaces* that will be matched to the analytic objects (trace bundles, Green pairings, and Calderón data) constructed later in the paper.

## 5. HODGE STRUCTURES ON TRACE BUNDLES

This section is the main *interface* between the analytic objects from Section 3 (trace bundles and Green pairings) and the Hodge-theoretic local systems extracted from the intersection Hodge module $\text{IC}_X^H$ in Section 4. Throughout, fix an even stratum $Y$ and write $\pi_Y: L_Y \to Y$ for its link fibration; let $Z_y := \pi_Y^{-1}(y)$ be the fiber over $y \in Y$.

**5.1. Analytic trace bundles versus link intersection cohomology.** Recall that, once deeper boundary data have been fixed on strata strictly below $Y$, the de Rham operator admits a tangential family $A_Y^{W>Y} \to Y$ with realized domains depending on those deeper choices, and the associated trace bundle is the kernel bundle $H_Y := \ker(A_Y^{W>Y}) \to Y$ (Section 3.2).

**Inductive hypothesis for Section 5.** We argue by induction on the depth of strata. Fix an even stratum $Y$ and assume that for every stratum $Z < Y$ the middle boundary data $W_{Z,\text{mid}}$ have been constructed and, on the corresponding links, the Deligne-type complex determined by the deeper data agrees with the middle intersection complex:

$$ \text{IC}_{W>z} \cong \text{IC}_{\bar{m}}. $$

In particular, by Assumption 3.8, the IBC de Rham cohomology with deeper domain $W^>Z$ identifies with $\mathbb{H}^\bullet(\cdot; \text{IC}_{\bar{m}})$ on each such link. This hypothesis is used below only to identify trace bundles with local systems of middle link intersection cohomology; the global identification $\text{IC}_{W_{\text{mid}}} \simeq \text{IC}_{\bar{m}}$ is proved later in Section 7 by a separate induction.
**Proposition 5.1** (Trace bundle as middle link intersection cohomology). Assume inductively that all strata $Z < Y$ have already been endowed with the middle Lagrangian boundary data $W_{Z,\text{mid}}$ (hence with a self-dual “deeper” mezzoperversity $W^{>Y} = W^{>Y}$ on the links of $Z_y$). Then:

(i) The family $y \mapsto A_{Y,y}^{W^{>Y}}$ has constant-dimensional kernel and $H_Y = \ker(A_{Y}^{W^{>Y}}) \to Y$ is a smooth flat vector bundle (local system).

(ii) There is a canonical isomorphism of local systems on $Y$

$$H_Y \cong \mathcal{H}_Y \otimes_Q \mathbb{R},$$

where $\mathcal{H}_Y$ is the local system from Proposition 4.3. Equivalently, in the even-link situation relevant for trace bundles,

$$(H_Y)_y \cong \mathrm{IH}_{\bar{m}}^{\dim Z_y} (Z_y; \mathbb{R}) \quad (y \in Y),$$

and these identifications are compatible with parallel transport along $Y$.

*Proof.* We separate the argument into two steps.

**Step 1 (constant rank and flatness).** The family $A_{Y}^{W^{>Y}} \to Y$ is obtained by freezing coefficients in a product-type wedge neighborhood and incorporating the already fixed deeper boundary data $W^{>Y}$. Under the inductive hypotheses (in particular, self-duality of the deeper data), $A_{Y,y}^{W^{>Y}}$ is a self-adjoint Fredholm operator with discrete spectrum and a *uniform spectral gap* at 0. Consequently the spectral projection onto $\ker(A_{Y,y}^{W^{>Y}})$ depends smoothly on $y$, so the kernels assemble into a smooth vector bundle $H_Y \to Y$ of locally constant rank. Moreover, the Ehresmann transport of the link fibration $\pi_Y: L_Y \to Y$ identifies the tangential operators along paths in $Y$, and hence induces a flat connection on $H_Y$ (equivalently, $H_Y$ is a local system). For details in the iterated wedge setting we refer to the stability theory in [Luo26, Theorem 6.3 and Theorem 5.1].

**Step 2 (identification with middle link intersection cohomology).** For each $y \in Y$, the maximal-domain asymptotics for $D_{\mathrm{dR}}$ on the cone $(0, \varepsilon)_r \times Z_y$ identify the trace space $\ker(A_{Y,y}^{W^{>Y}})$ with the critical-degree harmonic space for the IBC de Rham complex on $Z_y$ determined by the deeper data $W^{>Y}$ (trace/Poisson package); see [ALMP18, §5] and [Luo26, Appendix D.5]. If $W^{>Y}$ is the *middle* deeper data, then Assumption 3.8 identifies this IBC complex with the middle Deligne extension, hence with the middle intersection complex on the fiber. Therefore, in the even-link situation,

$$(H_Y)_y = \ker(A_{Y,y}^{W^{>Y}}) \cong \mathrm{IH}_{\bar{m}}^{d_Y}(Z_y; \mathbb{R}), \quad d_Y = \frac{1}{2} \dim Z_y.$$

By Proposition 4.4, the group $\mathrm{IH}_{\bar{m}}^{d_Y}(Z_y; \mathbb{R})$ is canonically the realification of the critical stratum local system $(\mathcal{H}_Y)_y$, and the above identification is natural with respect to Ehresmann transport. This yields the claimed isomorphism of local systems on $Y$. $\square$

**Lemma 5.2** (Incidence compatibility of the trace/IH identification). Let $Z < Y$ be even strata and write $U_{ZY} := Y \cap \mathcal{T}_Z$. On $U_{ZY}$ the analytic incidence morphism $\alpha_{ZY}: H_Y|_{U_{ZY}} \to H_Z|_{U_{ZY}}$ induced by the control data (Section 3.2) corresponds, under the identifications of Proposition 5.1, to the natural morphism of critical stratum local systems $\mathcal{H}_Y|_{U_{ZY}} \to \mathcal{H}_Z|_{U_{ZY}}$ coming from restriction of $\mathrm{IC}_{\bar{m}}$. Equivalently, it is the map on middle link intersection cohomology induced by the link fibration and the Thom-Mather control data. In particular, the transported Hodge structures and the forms $\omega_Y^{\mathrm{Hdg}}$ are incidence-compatible.

*Proof.* This is a functoriality statement for the trace/Poisson construction and for the Deligne extension description of $\mathrm{IC}_{\bar{m}}$. On the analytic side, $\alpha_{ZY}$ is defined by restricting asymptotic trace data through the link fibrations and applying the Calderón package; see [ALMP18, §5] and [Luo26, Appendix F]. On the sheaf-theoretic side, the corresponding morphism is induced by the natural restriction maps in the derived category for the pair of strata $Z < Y$. This is precisely the
restriction/incidence compatibility recorded in Proposition 3.10(b) (applied to $\mathcal{W} = \mathcal{W}_{\text{mid}}$ on the relevant links), hence the claim follows. $\square$

## 5.2. Polarized Hodge structures on the analytic trace bundle.

**Proposition 5.3 (PVHS on the trace bundle).** *The trace bundle $H_Y$ carries a canonical polarized variation of pure Hodge structure induced from the intersection Hodge module $\text{IC}_X^H$.*

*Proof.* By Proposition 4.6, the intersection Hodge module $\text{IC}_X^H$ induces a polarizable variation of pure Hodge structure on the critical stratum local system $\mathcal{H}_Y$ (unique up to the conventional Tate twist dictated by Saito's normalization; we suppress twists in the notation). Tensoring with $\mathbb{R}$ yields a PVHS on $\mathcal{H}_Y \otimes_{\mathbb{Q}} \mathbb{R}$. Transporting this PVHS through the canonical identification $H_Y \cong \mathcal{H}_Y \otimes_{\mathbb{Q}} \mathbb{R}$ of Proposition 5.1 produces the desired polarized variation on the analytic trace bundle. Incidence compatibility is ensured by Lemma 5.2. $\square$

## 5.3. Polarization versus the Green pairing.
Recall from Section 3.2 that the analytic trace bundle $H_Y$ carries a canonical symplectic form $\omega_Y$ (the Green pairing) coming from the boundary term in the Green formula for $D_{\text{dR}}$.

**Proposition 5.4 (Polarization equals Green pairing).** *Under the identification of Proposition 5.1, the Hodge symplectic form $\omega_Y^{\text{Hdg}}$ of Definition 4.7 agrees with the analytic Green pairing $\omega_Y$ up to the universal sign determined by the standard Poincaré/Verdier duality conventions.*

*Proof.* Let $k_Y = \frac{1}{2} \dim_{\mathbb{R}} Z_Y$ (independent of $y \in Y$). We again spell out the comparison in two complementary languages.

(a) **Hodge-theoretic side.** A polarization of $\text{IC}_X^H$ yields Verdier self-duality of the underlying intersection complex and, upon restricting to strata, induces the canonical (graded) pairings on the local systems $\mathcal{H}_Y$ (Section 4). Under the link description (Proposition 4.4), this is the topological intersection pairing on the middle link intersection cohomology groups.

(b) **Analytic side.** Green's formula for $D_{\text{dR}}$ yields the boundary pairing $\omega_Y$ on trace data. This construction (and its functoriality in families) is worked out in [ALMP18] and, for Fubini-Study wedge metrics, in [Luo26, Appendix D.5 and Appendix F] (see also [LL26a, Theorem 6.2] in the depth-one case).

(c) **Compatibility and the sign.** Assumption 3.8 asserts that the analytic-sheaf dictionary identifies the IBC de Rham complex with the middle Deligne extension and that it is compatible with Verdier duality. In particular, it transports the Verdier pairing induced by the polarization to the Green boundary pairing on trace data. Thus, under the identification of Proposition 5.1, the symplectic form $\omega_Y^{\text{Hdg}}$ coincides with $\omega_Y$ up to the universal sign fixed by the standard Poincaré/Verdier duality conventions. This identification is exactly the pairing compatibility of Proposition 3.10(c). With the sign bookkeeping as in [ALMP18, §5] and [Luo26, Appendix D.5], this convention amounts to

$$ \omega_Y = (-1)^{k_Y+1} \omega_Y^{\text{Hdg}}. $$

(The sign plays no role in the Lagrangian condition and hence does not affect the resulting self-dual mezzoperversity.) $\square$

## 6. THE HODGE INVOLUTION AND THE MIDDLE LAGRANGIAN

In this section we use the polarized variation of Hodge structure on each trace bundle $H_Y$ (Section 5) together with the Green pairing $\omega_Y$ (Section 3.2) to single out a canonical Lagrangian subbundle $W_{Y,\text{mid}} \subset H_Y$. This is the Hodge-theoretic input for the *middle criterion*.
**6.1. The Hodge involution.** Fix an even stratum $Y \in S_{\text{ev}}$ and write $f_Y := \dim_{\mathbb{R}} Z_y$ for the (real) dimension of the link fiber, independent of $y \in Y$. Set

$$k_Y := \frac{f_Y}{2}.$$

By Section 5, the trace bundle $H_Y$ underlies a polarized variation of Hodge structure; we denote by $C_Y$ its Weil operator on $(H_Y)_y \otimes \mathbb{C}$. Define the *normalized Weil involution* (or *Hodge involution*)

$$(6.1.1) \qquad T_Y := i^{k_Y} C_Y \quad \text{acting on } (H_Y)_y \otimes \mathbb{C}.$$

Since $C_Y$ acts on the Hodge summand $H^{p,q}$ by $i^{p-q}$ and $p+q = k_Y$, the operator $T_Y$ acts by the real scalar $i^{p+q}i^{p-q} = i^{2p} = (-1)^p$; in particular, $T_Y^2 = \text{Id}$ and $T_Y$ preserves the underlying real structure on $(H_Y)_y$. Hence we may (and will) regard $T_Y$ as an $\mathbb{R}$-linear involution on the real trace bundle $H_Y$.

**Lemma 6.1 (Hodge–Riemann sign).** For each $y \in Y$, the involution $T_Y$ is anti-symplectic for the Green pairing:

$$\omega_Y(T_Y u, T_Y v) = -\omega_Y(u, v) \quad (u, v \in (H_Y)_y).$$

In particular, the bilinear form

$$g_Y(u, v) := \omega_Y(u, T_Y v)$$

is symmetric and nondegenerate of split signature.

*Proof.* Let $\omega_Y^{\text{Hdg}}$ be the Hodge-theoretic symplectic form from Definition 4.7. By Proposition 5.4, $\omega_Y$ agrees with $\omega_Y^{\text{Hdg}}$ up to the universal sign dictated by the fixed Poincaré/Verdier duality conventions. More precisely (see [ALMP18, §5] and [Luo26, Appendix D.5]), with $k_Y = \frac{1}{2} \dim_{\mathbb{R}} Z_y$ one has

$$(6.1.2) \qquad \omega_Y = (-1)^{k_Y+1} \omega_Y^{\text{Hdg}}.$$

We first compute the effect of $T_Y$ on $\omega_Y^{\text{Hdg}}$. Write $u = \sum_{p+q=k_Y} u^{p,q}$ and $v = \sum_{p+q=k_Y} v^{p,q}$ with respect to the Hodge decomposition. Since $T_Y$ acts by $(-1)^p$ on $H^{p,q}$, bilinearity gives

$$\omega_Y^{\text{Hdg}}(T_Y u, T_Y v) = \sum_{p,q,p',q'} (-1)^{p+p'} \omega_Y^{\text{Hdg}}(u^{p,q}, v^{p',q'}).$$

The polarization has Hodge type $(-k_Y, -k_Y)$, hence the only nonzero terms occur for $(p', q') = (k_Y - p, k_Y - q)$. For such complementary types one has $p + p' = k_Y$, so $(-1)^{p+p'} = (-1)^{k_Y}$ and therefore

$$(6.1.3) \qquad \omega_Y^{\text{Hdg}}(T_Y u, T_Y v) = (-1)^{k_Y} \omega_Y^{\text{Hdg}}(u, v).$$

Combining (6.1.2) with (6.1.3) yields

$$\omega_Y(T_Y u, T_Y v) = (-1)^{k_Y+1} (-1)^{k_Y} \omega_Y^{\text{Hdg}}(u, v) = -\omega_Y(u, v),$$

which is the anti-symplectic identity.

Finally, symmetry and nondegeneracy of $g_Y(u, v) = \omega_Y(u, T_Y v)$ follow from the general symplectic linear algebra recorded in Appendix A (Lemma A.14), applied fiberwise to the anti-symplectic involution $T_Y$. $\square$

**6.2. Weil operator and Lagrangian eigensubbundles.** We next explain why the +1 eigen-subbundle of the Hodge involution is Lagrangian. The relevant linear algebra is summarized in Appendix A.

**Lemma 6.2 (Symplectic linear algebra).** Let $(H, \omega)$ be a symplectic real vector space and let $T$ be an anti-symplectic involution, i.e. $T^2 = \text{Id}$ and $\omega(Tu, Tv) = -\omega(u, v)$. Set

$$W := \ker(T - \text{Id}), \qquad W^{-} := \ker(T + \text{Id}).$$

Then $H = W \oplus W^{-}$ and both $W$ and $W^{-}$ are Lagrangian subspaces of $(H, \omega)$.
*Proof.* This is Lemma A.1(i)-(iii) in Appendix A.

☐

**Proposition 6.3** (Middle Lagrangian on each stratum). The +1 eigensubbundle

$$W_{Y,\text{mid}} := \ker(T_Y - \text{Id}) \subset H_Y$$

is a smooth Lagrangian subbundle for the Green pairing $\omega_Y$.

*Proof.* Smoothness: by Section 5, $H_Y$ is a smooth vector bundle and the Weil operator $C_Y$ varies smoothly along $Y$; hence so does $T_Y$. Since $T_Y^2 = \text{Id}$, the idempotents

$$P_{\pm} := \frac{1}{2}(\text{Id} \pm T_Y)$$

define smooth bundle endomorphisms, hence $\ker(T_Y \mp \text{Id}) = \text{im}(P_{\pm})$ is a smooth subbundle. Moreover $\text{rk}(P_{\pm}) = \text{tr}(P_{\pm})$ is a continuous $\mathbb{Z}$-valued function on $Y$, hence locally constant. In particular $\ker(T_Y - \text{Id})$ is a smooth subbundle.

Lagrangian property: on each fiber $(H_Y)_y$, Lemma 6.1 shows that $T_Y$ is an anti-symplectic involution for $\omega_Y$. Applying Lemma 6.2 fiberwise yields that the +1 eigenspace $\ker(T_Y - \text{Id})$ is Lagrangian.

☐

**Corollary 6.4** (Hodge mezzoperversity). The assignment $Y \mapsto W_{Y,\text{mid}}$ defines a self-dual mezzoperversity $W_{\text{mid}}$ in the sense of Definition 3.5.

*Proof.* By Proposition 6.3, each $W_{Y,\text{mid}}$ is Lagrangian, hence self-dual with respect to $\omega_Y$. It remains to verify incidence compatibility.

Let $Z < Y$ be a frontier relation. On the sheaf side, Proposition 4.9 shows that the restriction maps between the local systems $\mathcal{H}_Y$ and $\mathcal{H}_Z$ are morphisms of polarized variations of Hodge structure. In particular, the induced incidence morphism on trace bundles (via Proposition 5.1 and Assumption 3.8) commutes with Weil operators and hence with the normalized involutions:

$$\alpha_{ZY} \circ T_Y = T_Z \circ \alpha_{ZY}.$$

Therefore $\alpha_{ZY}$ carries the +1 eigensubbundle of $T_Y$ into that of $T_Z$, i.e.

$$\alpha_{ZY}(W_{Y,\text{mid}}|_{U_{ZY}}) \subseteq W_{Z,\text{mid}}|_{U_{ZY}}.$$

This is precisely the incidence condition in Definition 3.5.

☐

**Remark 6.5** (Flatness of the middle datum). In the analytic and sheaf-theoretic parametrizations of self-dual extensions (e.g. [ALMP18, §2] and [Ban02]), the local extension datum is a flat Lagrangian subbundle of the flat trace bundle $H_Y$, i.e. a Lagrangian sub-local system. Although the Weil operator (hence $T_Y$) is not parallel in general, the eigensubbundle $W_{Y,\text{mid}} = \ker(T_Y - \text{Id})$ is nevertheless flat: it coincides with the local extension datum of the perverse sheaf $\text{IC}_{\bar{m}} \cong \text{rat}(\text{IC}_X^H)$ along $Y$; see Corollary 7.4 below. In particular, $W_{\text{mid}}$ is a mezzoperversity in the sense of [ALMP18].

**6.3. Guiding examples.** The datum $W_{Y,\text{mid}} = \ker(T_Y - \text{Id}) \subset H_Y$ is defined abstractly from the polarized variation of Hodge structure on the trace bundle and the normalized Weil involution (6.1.1). The following two examples record what this choice looks like in familiar projective situations.

**Example 6.6** (Projective cone over a smooth curve). Let $C \subset \mathbb{P}^{N-1}$ be a smooth projective curve of genus $g$ and let $X \subset \mathbb{P}^N$ be the projective cone over $C$ with vertex $v$. Stratify $X$ by $Y := \{v\}$ and $X \setminus \{v\}$. Then the link fibration over $Y$ has fiber $Z_v \simeq C$, hence $f_Y = \dim_{\mathbb{R}} Z_v = 2$ and $k_Y = 1$. The trace space is

$$H_Y \cong IH^1(Z_v; \mathbb{R}) \cong H^1(C; \mathbb{R}),$$
equipped with its polarized Hodge structure of weight 1. On the Hodge summands one has $C_Y|_{H^{1,0}} = i$ and $C_Y|_{H^{0,1}} = -i$, hence the normalized involution $T_Y = i^{k_Y} C_Y = i C_Y$ acts by $-1$ on $H^{1,0}$ and by $+1$ on $H^{0,1}$. Therefore

$$W_{Y,\text{mid}} = \ker(T_Y - \text{Id}) = (H^{0,1}(C) \cap H^1(C; \mathbb{R})),$$

which is Lagrangian for the Green pairing $\omega_Y$. If $g = 0$ then $H_Y = 0$ (the stratum is Witt), so no local boundary datum is required.

**Remark 6.7** (A numerical dimension check in genus $g$). In Example 6.6 one has $\dim_{\mathbb{R}} H_Y = \dim_{\mathbb{R}} H^1(C; \mathbb{R}) = 2g$. Since $T_Y$ is anti-symplectic, Appendix A.1 implies that the $\pm 1$ eigenspaces of $T_Y$ are Lagrangian for $\omega_Y$. In particular,

$$\dim_{\mathbb{R}} W_{Y,\text{mid}} = \dim_{\mathbb{R}} \ker(T_Y - \text{Id}) = g.$$

Thus, in this basic cone case, the middle boundary condition cuts the degree-$k_Y$ Cauchy data by exactly half: for $g = 1$ it singles out a real line, and for $g = 0$ it is trivial (the Witt case).

**Example 6.8** (A trivial simple-edge product model). Let $Y$ be a smooth manifold and let $Z$ be a compact stratified space of even real dimension $2k$ for which $IH^k(Z; \mathbb{R})$ carries a polarized Hodge structure (for instance $Z = C$ as in Example 6.6). Consider the depth-one product model $X := Y \times c(Z)$ with even stratum $Y_{\text{edge}} := Y \times \{*\}$. Then the tangential family is constant along $Y$, and the trace bundle identifies with the constant local system

$$H_{Y_{\text{edge}}} \cong Y \times IH^k(Z; \mathbb{R}).$$

The Weil operator (hence $T_{Y_{\text{edge}}}$) is constant along $Y$, so

$$W_{Y_{\text{edge},\text{mid}}} \cong Y \times \ker(T - \text{Id}) \subset Y \times IH^k(Z; \mathbb{R}).$$

In particular, the middle datum is automatically flat and satisfies the incidence condition.

**Remark 6.9** (Half-dimensionality in the product case). If $\dim_{\mathbb{R}} IH^k(Z; \mathbb{R}) = 2m$, then each fiber of $H_{Y_{\text{edge}}}$ is a $2m$-dimensional symplectic vector space (for the Green pairing), and $W_{Y_{\text{edge},\text{mid}}}$ has fiber rank $m$. Equivalently, the normalized involution $T_{Y_{\text{edge}}}$ imposes exactly $m$ independent linear boundary conditions in middle degree.

**Remark 6.10** (A fully explicit numerical family: abelian fibers). In Example 6.8 take as link fiber a $k$-dimensional complex abelian variety $A$ (so $\dim_{\mathbb{R}} A = 2k$), hence $IH^k(A; \mathbb{R}) = H^k(A; \mathbb{R})$. Since $A$ is a real torus $(S^1)^{2k}$, one has the explicit binomial count

$$\dim_{\mathbb{R}} H^k(A; \mathbb{R}) = \binom{2k}{k}.$$

Therefore the middle datum has fiber rank

$$\text{rank}(W_{Y_{\text{edge},\text{mid}}}) = \frac{1}{2} \binom{2k}{k}.$$

For instance, $k = 1$ gives $\dim_{\mathbb{R}} H^1 = 2$ and $\text{rank}(W) = 1$ (elliptic curve), while $k = 2$ gives $\dim_{\mathbb{R}} H^2 = 6$ and $\text{rank}(W) = 3$ (abelian surface).

**Example 6.11** (A global hypercohomology check in a projective product cone). Let $B = \mathbb{P}^d$ be a smooth projective space and let $A \subset \mathbb{P}^{N-1}$ be a $k$-dimensional complex abelian variety (as in Remark 6.10), with projective cone $C(A) \subset \mathbb{P}^N$ and vertex $v$. Consider the projective variety

$$X := B \times C(A) \subset \mathbb{P}^d \times \mathbb{P}^N,$$

stratified by the simple-edge stratum $Y_{\text{edge}} := B \times \{v\}$ and its complement. This is exactly the product model of Example 6.8 (with $Y = B$ and $Z = A$), so the middle trace bundle along $Y_{\text{edge}}$ is
the constant local system $B \times H^k(A; \mathbb{R})$ and the canonical middle datum is the constant Lagrangian subbundle of rank $\frac{1}{2}\binom{2k}{k}$ from Remark 6.10.

Because the stratification and control data on $X$ are products, the associated Deligne-type complex on $X$ factorizes as an external tensor product, and one has a Künneth-type decomposition of hypercohomology:

$$ \mathbb{H}^m(X; \mathrm{IC}_{W_{\mathrm{edge},\mathrm{mid}}}) \cong \bigoplus_{p+q=m} \mathbb{H}^p(B; \mathbb{R}) \otimes \mathbb{H}^q(C(A); \mathrm{IC}_{W_{\mathrm{mid}}}), $$

where $\mathrm{IC}_{W_{\mathrm{mid}}}$ denotes the depth-one Deligne complex on the cone factor determined by the fiberwise Lagrangian $W = \ker(T - \mathrm{Id}) \subset \mathbb{H}^k(A; \mathbb{R})$.

For the cone $C(A)$, the only dependence on the middle datum occurs in degree $k$: one has

$$ \mathbb{H}^q(C(A); \mathrm{IC}_{W_{\mathrm{mid}}}) \cong \begin{cases} \mathbb{H}^q(A; \mathbb{R}), & 0 \le q < k, \\ W, & q = k, \\ 0, & q > k, \end{cases} $$

so the Poincaré series of $\mathbb{H}^\bullet(X; \mathrm{IC}_{W_{\mathrm{edge},\mathrm{mid}}})$ is explicitly

$$ P_X(t) = P_B(t) \cdot \left( \sum_{q=0}^{k-1} \binom{2k}{q} t^q + \frac{1}{2} \binom{2k}{k} t^k \right), \qquad P_B(t) = 1 + t^2 + \cdots + t^{2d}. $$

In particular, this model provides a fully checkable global numerical test: the canonical middle datum replaces the middle binomial coefficient $\binom{2k}{k}$ by its half, while all degrees $< k$ agree with the ordinary cohomology of the link fiber.

**Remark 6.12** (Low-dimensional expansions of the global numerical test). In Example 6.11 the hypercohomology Poincaré series takes the form

$$ P_X(t) = P_{\mathbb{P}d}(t) \cdot F_k(t), \qquad P_{\mathbb{P}d}(t) = 1 + t^2 + \cdots + t^{2d}, $$

where

$$ F_k(t) := \sum_{q=0}^{k-1} \binom{2k}{q} t^q + \frac{1}{2} \binom{2k}{k} t^k. $$

For small values of $k$, the coefficient vector of $F_k(t)$ (hence the global ranks in degrees $< k$ and in degree $k$) is completely explicit:

<table><thead><tr><th>k</th><th>F<sub>k</sub>(t)</th></tr></thead><tbody><tr><td>1</td><td>1 + t,</td></tr><tr><td>2</td><td>1 + 4t + 3t<sup>2</sup>,</td></tr><tr><td>3</td><td>1 + 6t + 15t<sup>2</sup> + 10t<sup>3</sup>.</td></tr></tbody></table>

In particular, for $d=1$ (so $B = \mathbb{P}^1$ and $P_{\mathbb{P}1}(t) = 1+t^2$) one obtains the fully expanded global series

$$ P_X(t) = (1 + t^2) F_k(t) = \begin{cases} 1 + t + t^2 + t^3, & k = 1, \\ 1 + 4t + 4t^2 + 4t^3 + 3t^4, & k = 2, \\ 1 + 6t + 16t^2 + 16t^3 + 15t^4 + 10t^5, & k = 3. \end{cases} $$

These expansions provide a “numerical checksum”: the only deviation from ordinary cohomology occurs in the middle term, where $\binom{2k}{k}$ is replaced by $\frac{1}{2}\binom{2k}{k}$. Similarly, for $d=2$ (so $B = \mathbb{P}^2$ and $P_{\mathbb{P}2}(t) = 1 + t^2 + t^4$) one gets the fully expanded global series

$$ P_X(t) = (1 + t^2 + t^4) F_k(t) = \begin{cases} 1 + t + t^2 + t^3 + t^4 + t^5, & k = 1, \\ 1 + 4t + 4t^2 + 4t^3 + 4t^4 + 4t^5 + 3t^6, & k = 2, \\ 1 + 6t + 16t^2 + 16t^3 + 16t^4 + 16t^5 + 15t^6 + 10t^7, & k = 3. \end{cases} $$
Similarly, for $d = 3$ (so $B = \mathbb{P}^3$ and $P_{\mathbb{P}^3}(t) = 1 + t^2 + t^4 + t^6$) one gets the fully expanded global series

$$P_X(t) = (1+t^2+t^4+t^6)F_k(t) = \begin{cases} 1+t+t^2+t^3+t^4+t^5+t^6+t^7, & k=1, \\ 1+4t+4t^2+4t^3+4t^4+4t^5+4t^6+4t^7+3t^8, & k=2, \\ 1+6t+16t^2+16t^3+16t^4+16t^5+16t^6+16t^7+15t^8+10t^9, & k=3. \end{cases}$$

More generally, writing $F_k(t) = \sum_{q=0}^{k} a_q t^q$ with $a_q = \binom{2k}{q}$ for $q < k$ and $a_k = \frac{1}{2} \binom{2k}{k}$, one has the explicit coefficient formula

$$P_X(t) = \sum_{j \ge 0} \left( \sum_{r=0}^{d} a_{j-2r} \right) t^j, \quad a_m := 0 \text{ for } m \notin \{0, \dots, k\}.$$

**Example 6.13** (An iterated cone model with a genuine frontier relation). Let $C \subset \mathbb{P}^{N-1}$ be a smooth projective curve and let $S \subset \mathbb{P}^N$ be the projective cone over $C$ with vertex $w$ as in Example 6.6. Let $X \subset \mathbb{P}^{N+1}$ be the projective cone over $S$ with vertex $v$. Then the singular locus of $X$ contains the generator line $L = \overline{vw} \cong \mathbb{P}^1$. Stratify $X$ by the depth-two chain

$$Z := \{v\} < Y := L \setminus \{v\} < X_{\text{reg}}.$$

Choose Thom-Mather control data adapted to this stratification. For $y \in Y$, the link fibration $\pi_Y: L_Y \to Y$ has (projective) fiber canonically identified with $C$, so $f_Y = \dim_{\mathbb{R}} C = 2$ and $k_Y = 1$. Thus the middle trace bundle along $Y$ is the (locally constant) system

$$H_Y \cong Y \times IH^1(C; \mathbb{R}) \cong Y \times H^1(C; \mathbb{R}),$$

and the normalized involution $T_Y$ agrees fiberwise with the one computed in Example 6.6. In particular,

$$W_{Y,\text{mid}} = \ker(T_Y - \text{Id}) = Y \times (H^{0,1}(C) \cap H^1(C; \mathbb{R})).$$

At the deeper stratum $Z = \{v\}$ the link fiber is $S$, so $f_Z = \dim_{\mathbb{R}} S = 4$ and $k_Z = 2$; the trace space is

$$H_Z \cong IH^2(S; \mathbb{R}),$$

equipped with its induced polarized Hodge structure and normalized involution $T_Z$. For the frontier relation $Z < Y$, set $U_{ZY} := Y \cap T_Z$ as in the general discussion of incidence morphisms on trace bundles. The control data give a canonical map of link fibrations $\iota_{ZY}: L_Y|_{U_{ZY}} \to L_Z|_{U_{ZY}}$ and hence an induced incidence morphism

$$\alpha_{ZY}: H_Y|_{U_{ZY}} \to H_Z|_{U_{ZY}}.$$

On the Hodge-theoretic side, $\alpha_{ZY}$ is induced by restriction for $\text{IC}_X^H$ (Proposition 4.9), hence is a morphism of polarized variations of Hodge structure. Therefore it commutes with Weil operators and with the normalized involutions:

$$\alpha_{ZY} \circ T_Y = T_Z \circ \alpha_{ZY}.$$

Consequently $\alpha_{ZY}$ carries the +1-eigensubbundle into the +1-eigenspace and one obtains the incidence inclusion

$$\alpha_{ZY}(W_{Y,\text{mid}}|_{U_{ZY}}) \subseteq W_{Z,\text{mid}}|_{U_{ZY}},$$

which is exactly the compatibility required in Definition 3.5.

## 7. IDENTIFICATION WITH THE MIDDLE INTERSECTION COMPLEX

In this section we identify the Deligne sheaf attached to the Hodge mezzoperversity $W_{\text{mid}}$ with the middle intersection complex. The argument is local along each even stratum and uses the general classification of *self-dual* Deligne/IBC extensions by Lagrangian data in the trace bundles.
**7.1. Local extension data and classification by Lagrangians.** Let $Y \in S_{ev}$ and let $\mathcal{T}_Y$ be a sufficiently small distinguished tubular neighborhood with complement $U := \mathcal{T}_Y \setminus Y$. Assume that a self-dual mezzoperversity (equivalently, a self-dual collection of local ideal boundary conditions) has already been fixed on $U$; we denote the corresponding self-dual Deligne sheaf on $U$ by $\mathrm{IC}_{W>Y}$. Recall from Section 3.2 that $Y$ carries a flat symplectic bundle $(H_Y, \omega_Y)$ of middle-degree trace data.

Here and below, a flat subbundle of $H_Y$ means a subbundle preserved by the canonical flat connection, equivalently a sub-local system.

**Proposition 7.1 (Lagrangian classification of self-dual perverse extensions).** Fix the data on $U$ as above. Then Verdier self-dual perverse sheaves $\mathcal{P}$ on $\mathcal{T}_Y$ whose restriction to $U$ is identified with $\mathrm{IC}_{W>Y}$ are classified (up to unique isomorphism over $U$) by choices of a flat Lagrangian subbundle

$$W_Y \subset (H_Y, \omega_Y)$$

satisfying the usual incidence compatibility along strata in $\bar{Y} \setminus Y$. Under this correspondence, the extension $\mathcal{P}$ is canonically identified with the Deligne-type sheaf $\mathrm{IC}_{W>Y \cup W_Y}$ obtained by adjoining $W_Y$ as the new local extension datum along $Y$.

*Proof.* We briefly recall the local model underlying the classification. After shrinking $\mathcal{T}_Y$ we may assume, using the conic control data, that $\mathcal{T}_Y$ is stratified-locally a product $Y \times c(Z)$ and $U = \mathcal{T}_Y \setminus Y \simeq Y \times (0, 1) \times Z$, where $Z$ is the link of $Y$.

Any perverse sheaf $\mathcal{P}$ on $\mathcal{T}_Y$ whose restriction to $U$ is identified with $\mathrm{IC}_{W>Y}$ is determined, up to unique isomorphism over $U$, by the additional local extension datum along $Y$; equivalently, by a choice of a sub-local system $W_Y \subset H_Y$ inside the middle trace local system (Section 3.2). In particular, the adjective “flat” is intrinsic: it means precisely that the datum varies locally constantly along $Y$.

Verdier self-duality imposes that this local datum be Lagrangian for the induced symplectic form, and conversely a flat Lagrangian sub-local system yields a self-dual Deligne-type extension. In Banagl’s formalism of self-dual sheaves on non-Witt spaces, this is exactly the additional choice needed to specify a self-dual intersection-homology-type extension at a non-Witt stratum; see [Ban02, §3–§6] and the comparison with analytic boundary conditions in [ABL$^+$15, §2–§4]. On the analytic side, ALMP package the same datum as a choice of a flat Lagrangian subbundle of the middle-degree harmonic trace bundle, with self-duality equivalent to the Lagrangian condition for the Green symplectic pairing; see [ALMP18, §3, §6].

Finally, in our projective/Fubini-Study setting, Assumption 3.8 identifies the analytic and sheaf-theoretic parametrizations on the tubular neighborhood $\mathcal{T}_Y$, hence transports the standard classification to the present statement. $\square$

**Remark 7.2.** The self-duality isomorphism $\mathcal{P} \simeq \mathbb{D}(\mathcal{P})$ is unique only up to a global sign. Changing this sign exchanges the two complementary Lagrangian choices determined by the associated anti-symplectic involution on the trace bundle; throughout we fix the normalization coming from the polarization of $\mathrm{IC}_X^H$ (Section 4).

**7.2. The extension datum determined by the intersection Hodge module.** We now show that the polarized Hodge module $\mathrm{IC}_X^H$ singles out the middle local extension datum $W_{Y,\mathrm{mid}}$ from Section 6.

**Proposition 7.3 (Extension datum of $\mathrm{IC}_X^H$).** Let $Y \in S_{ev}$. Under the identification of Proposition 5.1, the local self-dual extension datum of the perverse sheaf $\mathrm{rat}(\mathrm{IC}_X^H) \simeq \mathrm{IC}_{\bar{m}}$ along $Y$ corresponds to the middle Lagrangian subbundle

$$W_{Y,\mathrm{mid}} = \ker(T_Y - \mathrm{Id}) \subset H_Y,$$

where $T_Y$ is the normalized Weil involution from (6.1.1).
**Proof.** By Theorem B.1, $\mathrm{rat}(\mathrm{IC}_X^H)$ is the middle intersection complex $\mathrm{IC}_{\bar{m}}$ and is Verdier self-dual; the self-duality isomorphism is induced by the polarization of $\mathrm{IC}_X^H$. To make the extraction of the local extension datum completely explicit, let $i_Y: Y \hookrightarrow \mathcal{T}_Y$ and $j_Y: U = \mathcal{T}_Y \setminus Y \hookrightarrow \mathcal{T}_Y$. Under the identification of Proposition 5.1 (together with the definition of $H_Y$ in Section 3.2), the trace local system may be viewed as

$$ \mathrm{rat}(\mathrm{IC}_X^H)|_{\mathcal{T}_Y} \xrightarrow{i_Y^*} i_Y^*\mathrm{rat}(\mathrm{IC}_X^H) \xrightarrow{\mathcal{H}^{-d_Y}} \mathcal{H}^{-d_Y}(i_Y^*\mathrm{rat}(\mathrm{IC}_X^H)) \xrightarrow{\mathrm{Prop. 5.1}} \mathcal{H}_Y. $$

Equivalently,

$$ H_Y \cong \mathcal{H}^{-d_Y}(i_Y^*(\mathrm{IC}_{\bar{m}})) = \mathcal{H}^{-d_Y}(i_Y^*\mathrm{rat}(\mathrm{IC}_X^H)). $$

By Proposition 7.1, the local extension datum of a Verdier self-dual perverse extension $\mathcal{P}$ over $\mathcal{T}_Y$ is a flat Lagrangian sub-local system $W_Y \subset H_Y$; Here “flat” means that $W_Y$ is preserved by the natural flat connection on $H_Y$, equivalently that it defines a sub-local system; for the middle choice $W_{Y,\mathrm{mid}}$ this flatness is recorded explicitly in Corollary 7.4 below. for the particular choice $\mathcal{P} = \mathrm{IC}_{\bar{m}}$ we now identify this $W_Y$ with the +1-eigensubbundle of $\mathcal{T}_Y$.

Restricting to $Y$ and passing to the corresponding trace local system therefore equips $H_Y$ with (i) a polarized variation of Hodge structure (Proposition 5.3) and (ii) a canonical symplectic pairing coming from the Verdier duality pairing. By Proposition 5.4, this Verdier pairing agrees with the analytic Green pairing $\omega_Y$ (up to the universal convention sign, which does not affect the Lagrangian condition).

In the local classification of Proposition 7.1, a Verdier self-dual extension is encoded by a Lagrangian subbundle $W_Y \subset (H_Y, \omega_Y)$. For the particular self-dual extension $\mathrm{IC}_{\bar{m}}$ endowed with the polarization-induced self-duality, the Hodge involution $T_Y$ is anti-symplectic for $\omega_Y$ (Lemma 6.1). The induced decomposition of $H_Y$ into $\pm 1$ eigensubbundles of $T_Y$ is therefore a Lagrangian splitting (Lemma 6.2). The polarization normalization fixes which of these two eigensubbundles is the gluing datum: changing the polarization by a sign exchanges the two complementary choices, cf. Remark 7.2. With the convention in (6.1.1), the resulting datum is precisely the +1 eigensubbundle $\ker(T_Y - \mathrm{Id}) = W_{Y,\mathrm{mid}}$. $\square$

**Corollary 7.4 (Flatness of the middle Lagrangian).** For every even stratum $Y \in S_{\mathrm{ev}}$, the middle eigensubbundle $W_{Y,\mathrm{mid}} = \ker(T_Y - \mathrm{Id}) \subset H_Y$ is a flat Lagrangian subbundle of the flat symplectic bundle $(H_Y, \omega_Y)$; equivalently, it is a Lagrangian sub-local system of $H_Y$.

**Proof.** By Proposition 7.1, the local extension datum attached to any Verdier self-dual perverse extension over $\mathcal{T}_Y$ is given by a flat Lagrangian subbundle of $H_Y$. Applying this to the polarized self-dual perverse sheaf $\mathrm{IC}_{\bar{m}} \cong \mathrm{rat}(\mathrm{IC}_X^H)$ and using Proposition 7.3, we identify its extension datum with $W_{Y,\mathrm{mid}}$. Hence $W_{Y,\mathrm{mid}}$ is flat. $\square$

**Proof of Theorem 1.2. Induction convention.** The induction in this section is purely sheaf-theoretic (on the depth of the stratification for Deligne extensions) and is logically independent of the local-on-links induction used in Section 5. In particular, the proof below uses only the sheaf-theoretic classification of self-dual extensions from Proposition 7.1 together with the Hodge-theoretic identification of the relevant extension data; it does not require any further analytic input from Assumption 3.8. Both $\mathrm{IC}_{W_{\mathrm{mid}}}$ and $\mathrm{IC}_{\bar{m}}$ are Verdier self-dual perverse sheaves on $X$ extending the same local system on $X_{\mathrm{reg}}$. Proceed by induction on the depth of the stratification. Assume that the two sheaves have been identified on the complement of the union of strata of depth $\ge d$, and let $Y$ be a stratum of minimal depth among those remaining. Over a sufficiently small $\mathcal{T}_Y$, the induction hypothesis identifies the restrictions of the two sheaves to $U = \mathcal{T}_Y \setminus Y$. Proposition 7.1 then reduces the comparison on $\mathcal{T}_Y$ to comparing their local Lagrangian extension data along $Y$. By definition of $W_{\mathrm{mid}}$ this datum is $W_{Y,\mathrm{mid}}$ (which is flat by Corollary 7.4), and by Proposition 7.3
the same holds for $IC_{\bar{m}}$. Hence the two sheaves coincide on $\mathcal{T}_Y$; iterating over strata proves the global identification

$$IC_{W_{\text{mid}}} \cong IC_{\bar{m}}.$$

Taking hypercohomology gives $\text{IH}_{W_{\text{mid}}}^{\bullet}(X) \cong \text{IH}_{\bar{m}}^{\bullet}(X)$. $\square$

<figure><img src="image_4.png" /></figure>

**Proof of Theorem 1.1.** The polarized variation of Hodge structure on $H_Y$ (together with the flat connection in (i)) is constructed in Proposition 5.3 using the identification with the critical stratum local system (Proposition 5.1). The polarization comparison (ii) is Proposition 5.4. The normalized Weil operator defines the involution $T_Y$ in Lemma 6.1, and the Lagrangian property in (iii) follows from Lemma 6.2 and Proposition 6.3. Finally, incidence compatibility (iv) is proved in Proposition 4.9 and summarized in Corollary 6.4. $\square$

<figure><img src="image_5.png" /></figure>

**7.3. Outlook: comparison with the analytic canonical mezzoperversity.** The remaining step in the program is to compare the analytic canonical mezzoperversity coming from iterated Calderón packages with the *Hodge* middle mezzoperversity $W_{\text{mid}}$ constructed in this paper.

**Proposition 7.5 (Comparison theorem; proved in the companion paper).** Let $X \subset \mathbb{CP}^N$ be a complex projective variety endowed with a Thom-Mather stratification and equipped on $X_{\text{reg}}$ with the induced Fubini-Study wedge metric. Let $W_{\text{can}} = \{W_{Y,\text{can}}\}$ be the canonical mezzoperversity defined by the Calderón-range Cauchy data bundles produced by the iterated parameter-dependent Calderón package, and let $W_{\text{mid}} = \{W_{Y,\text{mid}}\}$ be the middle mezzoperversity defined here from the Hodge involutions $T_Y$. Then $W_{\text{can}} = W_{\text{mid}}$. Equivalently, for every even stratum $Y$ one has an equality of Lagrangian subbundles

$$W_{Y,\text{can}} = W_{Y,\text{mid}} \subset H_Y = \ker(A_Y^{W>Y}).$$

In particular, the canonical self-adjoint Fredholm extension of the de Rham operator determined by $W_{\text{can}}$ has $L^2$ cohomology canonically isomorphic to the middle intersection cohomology $\text{IH}_{\bar{m}}^{\bullet}(X)$, and carries the polarized Hodge structure described in Theorem 1.1.

**Proof (precise pointer).** The comparison $W_{\text{can}} = W_{\text{mid}}$ is established in the companion paper [Luo26, §§7–9], building on the depth-0 and depth-1 base cases [LL26b, Theorem 1.1] and [LL26a, Theorems 7.4 and 7.6]. For the reader’s convenience we isolate the analytic inputs (and where they are proved) that are used there to compare the Lagrangian Cauchy-data bundles:

(1) *Iterated parameter-dependent Calderón package on a neighborhood of an even stratum.* For a fixed even stratum $Y$ and fixed deeper boundary data $W^{>Y}$, the package produces tangential families $A_{Y,t}^{W>Y}$ and Calderón projectors $C_{Y,t}$ for a smooth deformation of iterated wedge metrics near $Y$; see [Luo26, Theorems 5.1 and 6.3].

* **Hypotheses (keywords; see the cited results for precise statements).** Exact (iterated) wedge structure near $Y$ with fixed Thom-Mather control data; a fixed deeper boundary datum $W^{>Y}$ so that the induced tangential operator $A_{Y,t}^{W>Y}$ on link fibers is self-adjoint with discrete spectrum; and full ellipticity of the associated wedge Dirac family in the iterated calculus.

(2) *Deformation invariance and transport of Calderón ranges.* The Calderón ranges $W_{Y,\text{can}}(t) = \text{ran}(C_{Y,t})$ form a smooth Lagrangian subbundle of the trace bundle $H_Y$ and are canonically identified for different $t$ by parallel transport; this is recorded in Lemma 7.6 and proved in the companion paper using the norm-resolvent stability mechanism in the conic model [LL26b, Theorem 1.1] together with the parametric package [Luo26, Theorem 6.3].

* **Hypotheses (keywords).** A common (graph) domain for $A_{Y,t}^{W>Y}$ and norm-resolvent continuity in $t$; stability of the small spectrum near 0 (equivalently, locally constant dim $\ker(A_{Y,t}^{W>Y})$); smooth dependence of the Calderón projectors $C_{Y,t}$ in the parameter; and compatibility with the flat transport on $H_Y$ coming from the link fibration.
(3) *Identification of the limiting model Cauchy data.* After transporting to $t = 0$ (the exact wedge model), the limit Cauchy data coincides with the Hodge middle Lagrangian $W_{Y,\text{mid}} = \ker(T_Y - \text{Id})$; see Lemma 7.7. This step uses the trace realization $H_Y \cong \mathcal{H}_Y \otimes \mathbb{R}$ (Proposition 5.1) and the identification of the polarization with the Green pairing (Proposition 5.4), together with the explicit model computation in [Luo26, Appendix D.5 and Appendix F] (and, in depth 1, [LL26a, Theorem 7.6]).

• **Hypotheses (keywords).** At $t=0$ an *exact* wedge model so that normal/indicial data are explicit; identification of $\ker(A_{Y,0}^{W^{>Y}})$ with the critical link intersection co-homology local system (ALMP); and the compatibility of the polarization with the Green pairing and of the Weil operator with incidence (Propositions 5.1 and 5.4 and Lemma 4.10).

(4) *Assembly by depth induction.* Assuming the comparison holds on all deeper strata, the incidence morphisms $\alpha_{ZY}$ and the iterated Calderón package promote the pointwise comparison on $Y$ to an equality of mezzoperversities on a neighborhood of $Y$, and iterating over strata yields the global equality $W_{\text{can}} = W_{\text{mid}}$; see [Luo26, Appendix F and Theorem 7.7].

• **Hypotheses (keywords).** Inductive control of deeper strata (fixed $W^{>Y}$) and symplectic incidence morphisms $\alpha_{ZY}$ on trace bundles; preservation of the Calderón ranges under incidence; and the uniqueness of the local extension datum in terms of flat Lagrangian subbundles (Section 7).

Combining (1)–(4) gives the equality of Lagrangian subbundles $W_{Y,\text{can}} = W_{Y,\text{mid}}$ for every even stratum $Y$, hence $W_{\text{can}} = W_{\text{mid}}$ and the stated consequences for $L^2$ de Rham cohomology. $\square$

The companion paper proves Proposition 7.5 by isolating a small list of analytic inputs and then assembling them by induction on the stratification depth using the incidence formalism (Proposition 4.11 and Lemma 4.10).

*Roadmap.* Fix an even stratum $Y$ and deeper boundary data $W^{>Y}$. The proof proceeds by:

(1) *Deformation of tangential families and Calderón ranges.* One interpolates between an exact model wedge metric $g_{w,Y}$ and the Fubini-Study wedge metric $g_{FS}$ by a smooth one-parameter family $g_t$ and shows that the associated tangential operators $A_{Y,t}^{W^{>Y}}$ and Calderón projectors $C_{Y,t}$ vary in a controlled (in fact, norm-resolvent continuous) manner. This produces a smooth family of Lagrangian Cauchy-data bundles $W_{Y,\text{can}}(t) = \text{ran}(C_{Y,t})$ and a canonical parallel-transport identification in $t$ (Lemma 7.6).

(2) *Model computation at $t=0$.* After transporting $W_{Y,\text{can}}(t)$ to the model fiber $t=0$, one computes the limit Cauchy data for the exact wedge model and identifies it with the Hodge-theoretic middle Lagrangian $W_{Y,\text{mid}} = \ker(T_Y - \text{Id})$. This step uses the trace realization $H_Y \cong \mathcal{H}_Y \otimes \mathbb{R}$ (Proposition 5.1) and the identification of the polarization with the Green pairing (Proposition 5.4) (Lemma 7.7).

(3) *Assembly by depth induction.* Assuming the comparison holds on all deeper strata, the incidence maps $\alpha_{ZY}$ promote the pointwise comparison on each $Y$ to an equality of mezzoperversities on a neighborhood of $Y$; iterating over strata yields the global equality $W_{\text{can}} = W_{\text{mid}}$.

We record these inputs for convenient reference.

**Lemma 7.6** (Deformation invariance of Calderón ranges; proved in the companion paper). *Fix an even stratum $Y$ and deeper boundary data $W^{>Y}$. Let $g_t$ be a smooth one-parameter family of iterated incomplete wedge metrics on a neighborhood of $Y$ which interpolates between an exact model wedge metric $g_{w,Y}$ and the Fubini-Study wedge metric $g_{FS}$. Let $A_{Y,t}^{W^{>Y}}$ be the associated tangential operators acting on the trace bundle $H_Y = \ker(A_{Y,t}^{W^{>Y}})$. Then the family $A_{Y,t}^{W^{>Y}}$ is norm-resolvent continuous in $t$, and the corresponding Calderón projectors $C_{Y,t}$ vary smoothly in $t$. In particular,*
the ranges ran($C_{Y,t}$) assemble into a smooth Lagrangian subbundle of $H_Y$, canonically identified for different $t$ by the induced parallel transport.

**Lemma 7.7** (Limiting Cauchy data and the Hodge Lagrangian; proved in the companion paper). With the notation of Lemma 7.6, transport the Lagrangian bundle $W_{Y,\text{can}}(t) = \text{ran}(C_{Y,t})$ to $t = 0$ using the canonical parallel transport in $H_Y$. For the exact model wedge metric $g_{w,Y}$ one has

$$ \lim_{t \to 0} W_{Y,\text{can}}(t) = W_{Y,\text{mid}} = \ker(T_Y - \text{Id}) \subset H_Y. $$

Equivalently, in the Lagrangian chart of Corollary A.5 determined by the splitting $H_Y = W_{Y,\text{mid}} \oplus W_{Y,\text{mid}}^-$, the associated graph maps $S_{Y,t}: W_{Y,\text{mid}} \to W_{Y,\text{mid}}^-$ satisfy $S_{Y,t} \to 0$ as $t \to 0$.

**Lemma 7.8** (Base cases in depth 0 and 1). If X has isolated conical singularities (depth 0), then $W_{\text{can}} = W_{\text{mid}}$ by [LL26b, Theorem 1.1]. If X is a projective simple-edge space (depth 1), then $W_{\text{can}} = W_{\text{mid}}$ by [LL26a, Theorems 7.4 and 7.6].

Together with the parameter-dependent Calderón package at arbitrary depth [Luo26, Theorems 5.1 and 6.3], these inputs imply the global equality $W_{\text{can}} = W_{\text{mid}}$ and hence the identification of the canonical $L^2$ de Rham cohomology with $\text{IH}_{\bar{m}}(X)$ in Proposition 7.5.

# APPENDIX A. SYMPLECTIC LINEAR ALGEBRA FOR WEIL OPERATORS

This appendix records the symplectic linear algebra underlying the discussion of normalized Weil involutions and Cauchy data in Section 6.2. Throughout, $(H, \omega)$ denotes a finite-dimensional real symplectic vector space. For background on these standard symplectic linear-algebra facts, see [MS17].

## A.1. Anti-symplectic involutions and Lagrangian splittings.

**Lemma A.1.** Let $T: H \to H$ be an anti-symplectic involution, i.e. $T^2 = \text{Id}$ and

$$ \omega(Tu, Tv) = -\omega(u, v) \quad \text{for all } u, v \in H. $$

Set

$$ H^+ := \ker(T - \text{Id}), \quad H^- := \ker(T + \text{Id}). $$

Then:

(1) $H = H^+ \oplus H^-$.

(2) Both $H^+$ and $H^-$ are isotropic, i.e. $\omega|_{H^\pm} = 0$.

(3) Both $H^+$ and $H^-$ are Lagrangian, and the pairing

$$ H^+ \times H^- \to \mathbb{R}, \quad (x, y) \mapsto \omega(x, y) $$

is perfect. Equivalently, the map

$$ \Phi: H^- \to (H^+)^\vee, \quad y \mapsto (x \mapsto \omega(x, y)) $$

is an isomorphism.

(4) The bilinear form

$$ g(u, v) := \omega(u, Tv) $$

is symmetric and nondegenerate. With respect to the decomposition in (1) one has

$$ g(x_+ + x_-, y_+ + y_-) = \omega(x_-, y_+) - \omega(x_+, y_-), \quad x_\pm \in H^\pm, y_\pm \in H^\pm, $$

so in particular $g|_{H^+} = g|_{H^-} = 0$ and $\text{sign}(g) = (n, n)$ where $\dim H = 2n$.
*Proof.* Since $T^2 = \text{Id}$, the minimal polynomial of $T$ splits with simple roots $\pm 1$, hence $H = \ker(T - \text{Id}) \oplus \ker(T + \text{Id}) = H^+ \oplus H^-$, proving (1).

If $x, y \in H^+$ then $Tx = x$ and $Ty = y$, so

$$ \omega(x, y) = \omega(Tx, Ty) = -\omega(x, y), $$

hence $\omega(x, y) = 0$. The same argument with $Tx = -x$ and $Ty = -y$ gives $\omega|_{H^-} = 0$, proving (2).

To prove (3), consider the map $\Phi: H^- \to (H^+)^\vee$ defined above. If $y \in H^-$ satisfies $\Phi(y) = 0$, then $\omega(x, y) = 0$ for all $x \in H^+$. Since $H^-$ is isotropic, also $\omega(H^-, y) = 0$, hence $\omega(H, y) = 0$ and nondegeneracy of $\omega$ forces $y = 0$. Thus $\Phi$ is injective. Exchanging the roles of $H^+$ and $H^-$ and using the same argument shows that $x \mapsto (y \mapsto \omega(x, y))$ gives an injection $H^+ \hookrightarrow (H^-)^\vee$, hence $\dim H^+ \le \dim H^-$ and $\dim H^- \le \dim H^+$, so $\dim H^+ = \dim H^- = n$. Therefore $H^\pm$ are maximal isotropic, i.e. Lagrangian, and the cross pairing is perfect; equivalently, $\Phi$ is an isomorphism.

Finally, for symmetry of $g$ we compute using skew-symmetry of $\omega$ and the anti-symplectic identity:

$$ g(v, u) = \omega(v, Tu) = -\omega(Tu, v) = \omega(u, Tv) = g(u, v). $$

For nondegeneracy, if $g(u, \cdot) = 0$ then $\omega(u, T\cdot) = 0$. Since $T$ is an isomorphism, this implies $\omega(u, \cdot) = 0$, hence $u = 0$. The displayed decomposition formula follows from $T(y_+ + y_-) = y_+ - y_-$ and (2). Choose dual bases $\{e_i\}_{i=1}^n \subset H^+$ and $\{f_i\}_{i=1}^n \subset H^-$ such that $\omega(e_i, f_j) = \delta_{ij}$. Then, with respect to the ordered basis $(e_1, \dots, e_n, f_1, \dots, f_n)$, the matrix of $g(u, v) = \omega(u, Tv)$ is $\begin{pmatrix} 0 & -I \\ -I & 0 \end{pmatrix}$. Setting $u_i = e_i + f_i$ and $v_i = e_i - f_i$, one computes $g(u_i, u_i) = -2$, $g(v_i, v_i) = 2$, and $g(u_i, v_j) = 0$; thus $g$ has split signature $(n, n)$. $\square$

**Lemma A.2 (Cotangent model).** In the situation of Lemma A.1, the isomorphism $\Phi: H^- \stackrel{\sim}{\to} (H^+)^\vee$ identifies $(H, \omega, T)$ with the standard triple

$$ H^+ \oplus (H^+)^\vee, \quad \omega_{\text{can}}((x, \xi), (y, \eta)) := \eta(x) - \xi(y), \quad T_{\text{can}}(x, \xi) := (x, -\xi). $$

In particular, there exists a symplectic basis $(e_1, \dots, e_n, f_1, \dots, f_n)$ such that

$$ \omega(e_i, f_j) = \delta_{ij}, \quad \omega(e_i, e_j) = \omega(f_i, f_j) = 0, $$

and

$$ Te_i = e_i, \quad Tf_i = -f_i \quad (1 \le i \le n). $$

*Proof.* By Lemma A.13 the map $\Phi: H^- \to (H^+)^\vee$ is an isomorphism. Consider

$$ \Psi: H^+ \oplus H^- \to H^+ \oplus (H^+)^\vee, \quad (x, y) \mapsto (x, \Phi(y)). $$

Using $\omega|_{H^\pm} = 0$ from Lemma A.12, we have for $(x, y), (x', y') \in H^+ \oplus H^-$:

$$ \omega((x, y), (x', y')) = \omega(x, y') - \omega(x', y) = \Phi(y')(x) - \Phi(y)(x') = \omega_{\text{can}}(\Psi(x, y), \Psi(x', y')). $$

Moreover, $\Psi \circ T = T_{\text{can}} \circ \Psi$ since $T$ acts by $+1$ on $H^+$ and by $-1$ on $H^-$. This proves the identification. Choosing any basis $(e_i)$ of $H^+$ and taking the $\Phi^{-1}$-dual basis $(f_i)$ of $H^-$ gives the stated symplectic basis. $\square$

**Proposition A.3 (Classification).** Anti-symplectic involutions on $(H, \omega)$ are in bijection with ordered pairs $(L, L')$ of transverse Lagrangian subspaces of $H$. More precisely:

(1) If $T$ is an anti-symplectic involution, then

$$ (L, L') := (\ker(T - \text{Id}), \ker(T + \text{Id})) $$

is a pair of transverse Lagrangians.

(2) Conversely, given transverse Lagrangians $H = L \oplus L'$, the linear map

$$ T_{L,L'}: H \to H, \quad T_{L,L'}(u + v) := u - v \quad (u \in L, v \in L') $$

is an anti-symplectic involution with eigenspaces $L$ and $L'$.

These constructions are inverse to one another.
*Proof.* The first part is Lemma A.1 (1–3). For the second part, $T_{L,L'}^2 = \text{Id}$ is immediate. For $u, u' \in L$ and $v, v' \in L'$ we have $\omega(u, u') = \omega(v, v') = 0$ since $L, L'$ are Lagrangian, hence

$$ \omega(T_{L,L'}(u+v), T_{L,L'}(u'+v')) = \omega(u-v, u'-v') = -\omega(u, v') - \omega(v, u') = -\omega(u+v, u'+v'). $$

Thus $T_{L,L'}$ is anti-symplectic. The eigenspace statement is clear, and the inverse property follows from uniqueness of the $\pm 1$ splitting. $\square$

**A.2. Graphs and local charts.** Fix transverse Lagrangians $H = L \oplus L'$. The symplectic form induces an isomorphism

$$ \iota: L' \xrightarrow{\sim} L^{\vee}, \quad \iota(l')(x) := \omega(x, l') \quad (x \in L). $$

**Lemma A.4 (Graphs of symmetric maps).** Let $S: L \to L'$ be a linear map and let

$$ \Gamma(S) := \{x + Sx : x \in L\} \subset H $$

be its graph. Then $\Gamma(S)$ is Lagrangian if and only if the bilinear form

$$ b_S(x, y) := \omega(x, Sy) \quad (x, y \in L) $$

is symmetric, equivalently $\iota \circ S: L \to L^{\vee}$ is a symmetric linear map.

Moreover, a subspace $\Lambda \subset H$ is Lagrangian and transverse to $L$ if and only if $\Lambda = \Gamma(S)$ for a unique $S: L \to L'$ with $b_S$ symmetric.

*Proof.* Since $L$ is isotropic, for $x, y \in L$ we compute

$$ \omega(x + Sx, y + Sy) = \omega(x, Sy) + \omega(Sx, y), $$

because $\omega(x, y) = 0$ and $\omega(Sx, Sy) = 0$ (as $L'$ is isotropic). Using skew-symmetry $\omega(Sx, y) = -\omega(y, Sx)$, the right-hand side equals $b_S(x, y) - b_S(y, x)$. Hence $\Gamma(S)$ is isotropic if and only if $b_S$ is symmetric. Since $\dim \Gamma(S) = \dim L = n = \frac{1}{2} \dim H$, isotropicity is equivalent to being Lagrangian.

If $\Lambda$ is transverse to $L$, the projection $H = L \oplus L' \to L$ restricts to an isomorphism $\Lambda \xrightarrow{\sim} L$, so $\Lambda$ is the graph of a unique linear map $S: L \to L'$. The first part shows that $\Lambda$ is Lagrangian precisely when $b_S$ is symmetric. $\square$

**Corollary A.5 (Bundle charts for Lagrangian subbundles).** Let $(E, \omega)$ be a smooth symplectic vector bundle over a manifold $B$ and let $L, L' \subset E$ be smooth Lagrangian subbundles with $E = L \oplus L'$. Write $\iota: L' \to L^{\vee}$ for the bundle isomorphism induced by $\omega$. Then the assignment

$$ S \mapsto \Gamma(S) = \{x + Sx : x \in L\} \subset E $$

restricts to a bijection between

(1) smooth bundle maps $S: L \to L'$ such that $\iota \circ S: L \to L^{\vee}$ is fiberwise symmetric, and

(2) smooth Lagrangian subbundles $\Lambda \subset E$ transverse to $L$.

In particular, Lagrangian transversality to a fixed $L$ is an open condition and the corresponding set of Lagrangian subbundles is modeled on symmetric bundle maps $L \to L^{\vee}$.

*Proof.* All statements are fiberwise, hence follow from Lemma A.4 once we justify smoothness. If $\Lambda \subset E$ is a smooth subbundle transverse to $L$, the projection $E = L \oplus L' \to L$ restricts to a smooth bundle isomorphism $\Lambda \xrightarrow{\sim} L$, whose inverse defines a smooth bundle map $S: L \to L'$ with $\Lambda = \Gamma(S)$. Conversely, if $S$ is smooth then $\Gamma(S)$ is a smooth subbundle. Finally, Lagrangianity is a fiberwise condition and is equivalent to symmetry of $\iota \circ S$ by Lemma A.4. $\square$
### A.3. Fiberwise eigensubbundles.

**Corollary A.6.** Let $(E, \omega)$ be a smooth symplectic vector bundle over a manifold $B$ and let $T: E \to E$ be a smooth anti-symplectic involution, i.e. $T^2 = \text{Id}$ and $\omega(Tu, Tv) = -\omega(u, v)$ fiberwise. Then the eigenbundles

$$E^{\pm} := \ker(T \mp \text{Id})$$

are smooth Lagrangian subbundles and $E = E^+ \oplus E^-$.

*Proof.* The spectral projectors $P_{\pm} = \frac{1}{2}(\text{Id} \pm T)$ are smooth idempotent bundle maps. Their ranks are locally constant because $\text{rank}(P_{\pm}) = \text{tr}(P_{\pm})$ and $\text{tr}(P_{\pm})$ is a continuous integer-valued function on $B$. Hence $E^{\pm} = \text{im}(P_{\pm}) = \ker(T \mp \text{Id})$ are smooth subbundles and $E = E^+ \oplus E^-$. Fiberwise Lagrangianity follows from Lemma A.1 (2-3). □

**Remark A.7 (Interface with Weil operators).** In the main text, $E = H_Y \to Y$ is the flat bundle of harmonic Cauchy data on the link fibration over a stratum $Y$, $\omega = \omega_Y$ is the Green form, and $T = T_Y$ is the normalized Weil involution constructed in Sections 5-7. Corollary A.6 then yields smooth Lagrangian eigensubbundles $\ker(T_Y - \text{Id})$ and $\ker(T_Y + \text{Id})$. When comparing an arbitrary Lagrangian boundary condition with the canonical splitting coming from $T_Y$, Corollary A.5 provides a convenient local model: after choosing a complementary Lagrangian, nearby Lagrangian subbundles are graphs of symmetric bundle maps.

For the reader's convenience, we record the main places where the results of this appendix are invoked:

* Lemma A.1 is used in Lemma 6.2 and in the proof of Lemma 6.1 (to deduce symmetry and nondegeneracy of $g_Y(u, v) = \omega_Y(u, T_Y v)$).

* Corollary A.6 provides a bundle-level criterion for smooth eigenbundles; it underlies the smoothness part of Proposition 6.3.

* Corollary A.5 gives the standard Lagrangian chart by graphs of symmetric maps; it is the chart used in Lemma 7.7 (via the graph maps $S_{Y,t}$).

## APPENDIX B. BACKGROUND ON HODGE MODULES USED IN THE TEXT

We summarize the pieces of Saito's theory of (mixed) Hodge modules that are invoked in the main text, with emphasis on (i) the existence and basic properties of the intersection Hodge module $\text{IC}_X^H$ and (ii) the induced polarized variations of Hodge structure (PVHS) carried by the local systems obtained by restricting $\text{IC}_X^H$ to strata. Our standing references are [Sai88, Sai90] (foundations), [BBD82] (intersection complexes and perverse sheaves), and [Sch14, §2–§4] (a convenient overview).

### B.1. Conventions: realizations, shifts, and functors.
For a reduced complex algebraic variety $X$ we denote by $\text{MHM}(X)$ the abelian category of graded-polarizable mixed Hodge modules on $X$ and by $\text{D}^b\text{MHM}(X)$ its bounded derived category. The main structural point is that objects of $\text{MHM}(X)$ simultaneously refine (a) constructible complexes/perverse sheaves with $\mathbb{Q}$-coefficients and (b) filtered regular holonomic $\mathcal{D}_X$-modules, and that all operations in the six-functor formalism are available in the Hodge-module setting. More precisely, there are faithful “realization” functors

$$\text{rat: } \text{MHM}(X) \to \text{Perv}(X; \mathbb{Q}), \quad \text{DR: } \text{MHM}(X) \to \text{D}_c^b(X; \mathbb{C}),$$

compatible with the standard functors on the target categories; see [Sai90, Sch14, Kas84]. In particular, for an algebraic morphism $f: X \to Y$ there are functors

$$f_*, f!, f^*, f', \otimes, \text{Hom}, \mathbb{D}$$

Here $\otimes$ and $\text{Hom}$ denote the derived tensor product and internal Hom in $\text{D}^b\text{MHM}(-)$, and $\mathbb{D}$ denotes Verdier duality; under rat these correspond to the usual tensor product, internal Hom, and Verdier duality on constructible complexes. See [Sch14, §4]. When $X$ is smooth of complex
dimension $d$, we write $\mathbb{Q}_X^H$ for the constant Hodge module normalized so that $\mathrm{rat}(\mathbb{Q}_X^H) = \mathbb{Q}_X[d]$ is
perverse.

B.2. **PVHS and Hodge modules on smooth varieties.** Let $Y$ be smooth and connected. A polarizable variation of Hodge structure (PVHS) on $Y$ (with $\mathbb{Q}$-structure) determines, and is determined by, a polarizable pure Hodge module on $Y$ whose underlying perverse sheaf is a shifted local system. Concretely, if $\mathcal{V}$ is a PVHS of weight $w$ on $Y$, then there is a polarizable pure Hodge module $\mathcal{V}_Y^H$ such that $\mathrm{rat}(\mathcal{V}_Y^H) = \mathcal{V}[d]$ and whose weight is $w+d$ ([Sai90, Sch14]). Conversely, any polarizable pure Hodge module on $Y$ with smooth underlying perverse sheaf arises in this way. This is the mechanism used repeatedly in Sections 4 and 5 to pass from restrictions of $\mathrm{IC}_X^H$ to PVHS on strata.

Equivalently, smooth graded-polarizable mixed Hodge modules on $Y$ correspond to graded-polarizable admissible variations of mixed Hodge structure on $Y$; in the pure case this is precisely a PVHS. See [Sch14, §7] and [Sai16].

B.3. **Intersection Hodge modules.** Let $X$ be a reduced complex algebraic variety and $j: X_{\mathrm{reg}} \hookrightarrow X$ the inclusion of the smooth locus. Write $\mathrm{IC}_{\bar{m}}$ for the middle intersection complex of $X$ (with $\mathbb{Q}$-coefficients) as in [BBD82]. Saito constructs a canonical Hodge-module lift of $\mathrm{IC}_{\bar{m}}$.

**Theorem B.1** (Saito [Sai90]). There exists a polarizable pure Hodge module $\mathrm{IC}_X^H \in \mathrm{MHM}(X)$ such that $\mathrm{rat}(\mathrm{IC}_X^H) \cong \mathrm{IC}_{\bar{m}}$ (up to the standard shift convention). Furthermore, $\mathrm{IC}_X^H$ has strict support equal to $X$; equivalently, it is the intermediate extension $j_{\bullet*}\mathbb{Q}_{X_{\mathrm{reg}}}^H$ (for any fixed normalization on $X_{\mathrm{reg}}$). Moreover $\mathrm{IC}_X^H$ is uniquely characterized by:

(a) $\mathrm{IC}_X^H|_{X_{\mathrm{reg}}} \cong \mathbb{Q}_{X_{\mathrm{reg}}}^H$ as pure Hodge modules on $X_{\mathrm{reg}}$, and

(b) functoriality with respect to open embeddings (equivalently, the intermediate-extension property).

The Hodge module $\mathrm{IC}_X^H$ is Verdier self-dual:

$$ \mathbb{D}(\mathrm{IC}_X^H) \cong \mathrm{IC}_X^H(w_X) $$

for a suitable Tate twist $w_X$ depending only on the dimension convention.

**Remark B.2.** Different references place the intersection complex in slightly different degrees. Our arguments only depend on two robust consequences of Theorem B.1: (1) the underlying perverse sheaf of $\mathrm{IC}_X^H$ is $\mathrm{IC}_{\bar{m}}$ and (2) $\mathrm{IC}_X^H$ is polarizable and self-dual in the Hodge-module sense. In particular, the precise cohomological shift convention plays no role as long as one keeps track of the induced local systems along strata.

B.4. **Restrictions to strata and induced PVHS.** Fix a Thom-Mather stratification of $X$ and let $Y \subset X$ be a stratum with inclusion $i_Y: Y \hookrightarrow X$. The perverse sheaf $\mathrm{IC}_{\bar{m}}$ restricts to $Y$ as a shifted local system; this is recorded in Proposition 4.3. The corresponding statement for $\mathrm{IC}_X^H$ upgrades these local systems to PVHS.

**Proposition B.3** (Stratum PVHS from $\mathrm{IC}_X^H$). Let $Y$ be a stratum of $X$. Then the restriction $i_Y^*\mathrm{IC}_X^H$ is an object of $\mathrm{D}^b\mathrm{MHM}(Y)$ whose perverse cohomology objects lie in $\mathrm{MHM}(Y)$, and their realizations are the corresponding perverse cohomology sheaves of $i_Y^*\mathrm{IC}_{\bar{m}}$. Since $Y$ is smooth and $i_Y^*\mathrm{IC}_{\bar{m}}$ is locally constant up to finite shifts (Proposition 4.3), these Hodge modules are in fact smooth graded-polarizable mixed Hodge modules on $Y$. Equivalently, they correspond to graded-polarizable admissible variations of (mixed) Hodge structure on $Y$; see [Sch14, Thm. 15.1] and [Sai16].

In particular, the critical local system $\mathcal{H}_Y$ of Proposition 4.3 carries a canonical polarizable variation of pure Hodge structure, unique up to the shift convention of Remark B.2. Its polarization is induced functorially from the polarization of $\mathrm{IC}_X^H$.
**Remark B.4** (Matching the local systems $\mathcal{H}_Y^k$). In Proposition 4.3 we define the local systems $\mathcal{H}_Y^k = \mathcal{H}^k(i_Y^*\mathrm{IC}_{\bar{m}})$ on the smooth stratum $Y$. Since $i_Y^*\mathrm{IC}_{\bar{m}}$ is locally constant up to finite shifts, these local systems are precisely the underlying $\mathbb{Q}$-local systems of the smooth Hodge modules provided by Proposition B.3 (after the usual perverse shift convention on $Y$ recorded in Remark B.2). In particular, the critical local system $\mathcal{H}_Y = \mathcal{H}_Y^{-dY}$ underlies a polarizable PVHS, and its Weil operator is the involution $T_Y$ used in Section 6 (see also Remark B.7).

**Remark B.5.** All incidence morphisms between the stratum local systems (such as $\alpha_{ZY}$ in Lemma 4.10) are induced by functoriality of the restriction and nearby/vanishing-cycle functors in $\mathrm{D}^\mathrm{b}\mathrm{MHM}(-)$. Concretely, on $U_{ZY} = Y \cap \mathcal{T}_Z$ one may choose a local defining function $g$ for $Z$ inside $\mathcal{T}_Z$; then $\alpha_{ZY}$ is the underlying $\mathbb{Q}$-local system of the morphism in $\mathrm{D}^\mathrm{b}\mathrm{MHM}(U_{ZY})$ obtained from the canonical specialization maps

$$i_Y^*\mathrm{IC}_X^\mathrm{H} \to \psi_g(i_Z^*\mathrm{IC}_X^\mathrm{H})$$

(and the associated perverse cohomology), see [Sch14, §4]. After passing to the associated $\mathbb{Q}$-local systems, these morphisms are morphisms of PVHS and hence commute with the corresponding Weil operators. This is the Hodge-theoretic input behind the functoriality statement used in Lemma 4.10.

**B.5. Projective direct image, intersection cohomology, and polarizations.** Assume now that $X$ is projective. Saito's theory equips the intersection cohomology groups of $X$ with canonical Hodge structures compatible with the topological intersection complex.

**Theorem B.6** (Saito; decomposition and hard Lefschetz). Let $f: X \to \mathrm{pt}$ be the structure morphism of a projective variety. Then $Rf_*\mathrm{IC}_X^\mathrm{H} \in \mathrm{D}^\mathrm{b}\mathrm{MHM}(\mathrm{pt})$ decomposes as a direct sum of its perverse cohomology objects, and each cohomology group

$$\mathrm{IH}^k(X; \mathbb{Q}) := \mathrm{H}^k(\mathrm{rat}(Rf_*\mathrm{IC}_X^\mathrm{H}))$$

carries a functorial pure Hodge structure of weight $k$. Under rat, this decomposition is compatible with the corresponding decomposition of $Rf_*\mathrm{IC}_{\bar{m}}$ in $\mathrm{D}_c^\mathrm{b}(\mathrm{pt}; \mathbb{Q})$ (hence recovers the BBD decomposition theorem [BBD82]), and the resulting Hodge structures on $\mathrm{IH}^k(X; \mathbb{Q})$ are canonical. Moreover, for a choice of ample line bundle on $X$, the corresponding Lefschetz operator satisfies hard Lefschetz and the associated primitive parts admit polarizations in the sense of Hodge theory. See [Sai90, dCM05].

**Remark B.7.** The polarizations in Theorem B.6 determine the Weil operators on the Hodge structures $\mathrm{IH}^k(X; \mathbb{Q})$ and, more generally, on the PVHS appearing in Proposition B.3. These operators are the source of the involutions $T_Y$ used in Section 6; Appendix A records the purely symplectic linear algebra needed to pass from $T_Y$ and the Green pairing to canonical Lagrangian subbundles.

Note that the self-duality/polarization is unique only up to a global sign; changing this sign replaces the associated Weil operator by its negative and exchanges the two complementary Lagrangian choices coming from the anti-symplectic involution. Throughout the paper we fix the normalization induced by the polarization of $\mathrm{IC}_X^\mathrm{H}$; see Remark 7.2.

## ACKNOWLEDGEMENTS

I would like to thank the authors of the foundational works that shaped the analytic and sheaf-theoretic approaches to stratified spaces and intersection cohomology, in particular J. Cheeger, M. Goresky, R. MacPherson, and their successors. I am grateful to P. Pardon and M. Stern for their influential analysis on complex varieties with singularities, and to P. Albin, E. Leichtnam, R. Mazzeo, and P. Piazza for developing the mezzoperversity framework and the analytic theory of ideal boundary conditions on stratified spaces. I also thank L. Saper and M. Banagl for important contributions to $\mathrm{L}^2$-cohomology and self-dual sheaf theory, which provided valuable perspective for the present work.
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