Compatibility of the $L^2$-Hodge structure with Saito's pure Hodge structure on the intersection cohomology of a singular Kähler space (Shentu–Zhao, Conjecture 8.5)

Shentu and Zhao (arXiv:2103.04030) proved that for a compact complex space $X$, a dense Zariski-open subset $X^o\subset X_{\rm reg}$ carrying an $\mathbb{R}$-polarized variation of Hodge structure $\mathbb{V}$ with quasi-unipotent monodromies, and a suitable complete "distinguished" metric $ds^2$ on $X^o$, there is a canonical isomorphism $H^*_{(2)}(X^o,\mathbb{V};ds^2,h)\simeq IH^*(X,\mathbb{V})$ between $L^2$-cohomology and intersection cohomology — a Cheeger–Goresky–MacPherson-type theorem valid for arbitrary complex analytic singularities. When $ds^2$ is Kähler, the $(p,q)$-decomposition of $L^2$-forms endows the left side with a polarized pure Hodge structure, while Saito's theory of Hodge modules endows the right side with one algebraically. The open problem, stated as Conjecture 8.5 of that paper, is whether these two pure Hodge structures coincide under the canonical isomorphism, for every compact Kähler space and every Kähler distinguished metric. The known cases cover isolated singularities with trivial coefficients, the extreme Hodge row, and — by Looijenga (2025) and Ni (2026), resolving the Harris–Zucker conjecture — Shimura varieties with their invariant metric; the case of arbitrary singularities is open.

Background

$L^2$-cohomology is the cohomology of the complex of square-integrable differential forms (with square-integrable exterior derivatives) on a Hermitian manifold; it is the natural analytic cohomology theory for a non-compact or incomplete space such as the regular locus of a singular variety. Intersection cohomology $IH^*$, introduced by Goresky and MacPherson, is a topological cohomology theory for singular spaces built from the middle-perversity intersection complex $IC_X(\mathbb{V})$: the complex of sheaves that extends a local system $\mathbb{V}$ on a dense smooth part while keeping cohomology sheaves of the smallest possible size compatible with the links of the strata; on a smooth space it reduces to ordinary cohomology with $\mathbb{V}$-coefficients. A variation of Hodge structure (VHS) on a complex manifold is a local system of complex vector spaces together with a holomorphic vector bundle, a flat connection $\nabla$, a decreasing Hodge filtration $F^\bullet$, and a flat Hermitian metric $h$ whose graded pieces form a pointwise polarized Hodge structure; local monodromies around the boundary of a compactification are assumed quasi-unipotent (eigenvalues roots of unity), a standard hypothesis forced by the geometry. A Kähler space is a complex space, locally embeddable in $\mathbb{C}^N$, carrying a closed strictly positive $(1,1)$-form.

Cheeger, Goresky and MacPherson (1982) conjectured that for a projective variety $X\subset\mathbb{P}^N$ the $L^2$-cohomology of the regular locus with the Fubini–Study metric is naturally isomorphic to $IH^*(X,\mathbb{C})$; this is known only in special cases — normal surfaces by Hsiang and Pati, isolated singularities by Ohsawa (building on work of Cheeger, Saper and Zucker). In the locally symmetric direction, Zucker's conjecture — $H^*_{(2)}(X,\mathbb{E})\cong IH^*(X^*,\mathbb{E})$ for a Shimura variety $X$ (a quotient $G/\Gamma$ of a Hermitian symmetric domain by an arithmetic lattice), its Baily–Borel compactification $X^*$ (the minimal projective compactification), and a locally homogeneous polarized VHS $\mathbb{E}$ — was proved independently by Looijenga and by Saper–Stern. For a smooth variety with a normal-crossing boundary divisor and a Poincaré-type metric (a complete metric with $|z|^{-2}\log^{-2}|z|$ growth along the divisor), the corresponding statements are due to Zucker, Cattani–Kaplan–Schmid and Kashiwara–Kawai.

Independently, Morihiko Saito's theory of Hodge modules — an algebraic, $\mathcal{D}$-module-theoretic extension of the theory of variations of Hodge structure that carries Deligne's Hodge theory to singular and non-proper spaces — endows $IH^*(X,\mathbb{V})$ of a projective (more generally compact Kähler) variety with a pure Hodge structure, defined algebraically through the filtered de Rham complex of the intersection Hodge module $IC_X(\mathbb{V})$, with $(p,q)$-summands $IH^{p,q}(X,\mathbb{V})$.

Shentu and Zhao (2021) proved a Cheeger–Goresky–MacPherson-type theorem at the level of cohomology for arbitrary singularities: for a compact complex space $X$ of pure dimension, a dense Zariski-open $X^o\subset X_{\rm reg}$, an $\mathbb{R}$-polarized VHS $\mathbb{V}$ with quasi-unipotent monodromies (more generally a 0-tame harmonic bundle, i.e. a flat bundle with a pluriharmonic metric of controlled asymptotics), and a complete "distinguished" Hermitian metric $ds^2$ on $X^o$ — their Definition 4.12 asks that, locally on a Whitney stratified resolution $\pi:\tilde X\to X$ with $\pi^{-1}(X\setminus X^o)=\{z_1\cdots z_k=0\}$ for some number $k$ of components, the metric be quasi-isometric to $\sqrt{-1}\partial\bar\partial\varphi_\pi+\omega_0$ for an explicit potential mixing Poincaré-type and iterated-logarithmic growth along the exceptional divisor — the $L^2$ de Rham complex of $(\mathbb{V},h)$-valued forms computes the intersection complex: there is a canonical quasi-isomorphism $\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^2,h}\simeq IC_X(\mathbb{V})$, hence a canonical isomorphism $H^k_{(2)}(X^o,\mathbb{V};ds^2,h)\cong IH^k(X,\mathbb{V})$ for all $k$. When $ds^2$ is Kähler (which can be arranged when $X$ is a Kähler space), complete-Kähler $L^2$-Hodge theory in the style of Zucker and Kashiwara–Kawai gives $H^*_{(2)}$ a polarized pure Hodge structure via the $(p,q)$-decomposition of $L^2$-forms, and Shentu–Zhao prove the Kähler package (Poincaré duality, hard Lefschetz) for it.

Two a priori different pure Hodge structures therefore sit on the same vector space $H^*_{(2)}\cong IH^*$: the analytic one coming from $L^2$-forms, and Saito's algebraic one. Shentu–Zhao's Conjecture 8.5 asserts that they agree. Harris and Zucker earlier formulated exactly this agreement as Conjecture 5.3 of their 2001 Mémoires de la SMF volume, in the Shimura-variety/Baily–Borel setting. The evidence available when the conjecture was made: for isolated singularities with trivial coefficients it follows from Zucker's 1987 work on intersection homology of varieties with isolated singularities; and Shentu–Zhao's Theorem 8.6 — proved via their companion paper giving an $L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module (Saito's $S$-sheaf, the coherent sheaf of "logarithmic" $n$-forms attached to the intersection Hodge module) — establishes the compatibility for the extreme Hodge row $IH^{n+r,q}(X,\mathbb{V})\simeq H^{n+r,q}_{(2)}(X^o,\mathbb{V})$, $q\ge 0$, where $n=\dim X$ and $r$ is the weight of $\mathbb{V}$.

Problem Statement

Let $X$ be a compact Kähler space of pure dimension $n$, let $X^o\subset X_{\rm reg}$ be a dense Zariski-open subset, and let $\mathbb{V}=(\mathcal{V},\nabla,F^\bullet,h)$ be an $\mathbb{R}$-polarized variation of Hodge structure of weight $r$ on $X^o$ with quasi-unipotent local monodromies around $X\setminus X^o$. Let $ds^2$ be a Kähler distinguished metric associated to $(X,X^o)$ in the sense of Shentu–Zhao (arXiv:2103.04030, Definition 4.12): a complete Kähler metric on $X^o$ such that, locally on a Whitney stratified resolution $\pi:\tilde U\to U$ of $(U,X^o\cap U)$ with $\pi^{-1}(U\setminus X^o)=\{z_1\cdots z_k=0\}$ for some $k\le n$, the metric is quasi-isometric to $\sqrt{-1}\partial\bar\partial\varphi_\pi+\omega_0$, where $\omega_0$ is a Hermitian metric on $U$ and $\varphi_\pi$ is an explicit potential with Poincaré-type and iterated-logarithmic growth along the components of the exceptional divisor; such metrics exist whenever $X$ is a compact Kähler space (or a Zariski-open subset of one).

By Shentu–Zhao's Theorem 1.4 there is a canonical isomorphism

$$H^k_{(2)}(X^o,\mathbb{V};ds^2,h)\;\cong\;IH^k(X,\mathbb{V})\quad\text{for every }k,$$

between the $L^2$-cohomology of $X^o$ with coefficients in $\mathbb{V}$ and the intersection cohomology of $X$. Since $ds^2$ is complete Kähler, the $(p,q)$-decomposition of $L^2$-forms endows the left-hand side with a polarized pure Hodge structure, with $(p,q)$-summands $H^{p,q}_{(2)}(X^o,\mathbb{V})$; Saito's theory of polarizable Hodge modules endows the right-hand side with a pure Hodge structure with $(p,q)$-summands $IH^{p,q}(X,\mathbb{V})$.

Prove or disprove (Shentu–Zhao, Conjecture 8.5): under the canonical isomorphism above, the $L^2$-pure Hodge structure on $H^*_{(2)}(X^o,\mathbb{V};ds^2,h)$ coincides with Saito's pure Hodge structure on $IH^*(X,\mathbb{V})$; equivalently, the canonical isomorphism identifies $H^{p,q}_{(2)}(X^o,\mathbb{V})$ with $IH^{p,q}(X,\mathbb{V})$ for every $(p,q)$.

The problem is the general case of arbitrary complex analytic singularities. The following cases are already settled and do not by themselves answer it: isolated singularities with trivial coefficients; the extreme Hodge row $IH^{n+r,q}\simeq H^{n+r,q}_{(2)}$ (Shentu–Zhao, Theorem 8.6); and, with the invariant locally symmetric metric in place of a distinguished metric, Shimura varieties with Baily–Borel compactification (the Harris–Zucker conjecture, proved by Looijenga 2025 and independently by Ni 2026).

References

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