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

For a compact Kähler space $X$ with singularities, Shentu and Zhao constructed a class of complete Kähler "distinguished" metrics on the regular locus $X_{\rm reg}$ for which $L^2$-cohomology canonically computes intersection cohomology, $H^k_{(2)}(X_{\rm reg},ds^2)\simeq IH^k(X)$, with coefficients in any polarized variation of Hodge structure with quasi-unipotent local monodromies. The $(p,q)$-decomposition of $L^2$-forms then endows these groups with a pure Hodge structure, and Saito's theory of Hodge modules endows $IH^*(X)$ with another pure Hodge structure. The problem is Shentu–Zhao's Conjecture 8.5: whether these two pure Hodge structures coincide under the canonical $L^2$-intersection-cohomology isomorphism. Only the top Hodge piece and the case of isolated singularities with trivial coefficients are known; the general case remains open.

Background

Intersection cohomology $IH^*(X)$, introduced by Goresky and MacPherson, restores Poincaré duality for singular complex varieties: it is the hypercohomology of the intersection complex $IC_X$, a distinguished self-dual complex of sheaves (a perverse sheaf, i.e. a constructible complex satisfying support and co-support conditions) that behaves like the constant sheaf on a smooth space. In the early 1980s Cheeger, Goresky and MacPherson conjectured that on a projective variety $X\subset\mathbb{P}^N$ the $L^2$-cohomology of the regular locus $X_{\rm reg}$ with respect to the (generally incomplete) Fubini–Study metric computes $IH^*(X)$; $L^2$-cohomology $H^*_{(2)}$ is the cohomology of square-integrable differential forms $\alpha$ whose differentials $d\alpha$ are also square-integrable, i.e. the de Rham cohomology with an analytic boundary condition at the singular set. This Cheeger–Goresky–MacPherson (CGM) conjecture is known only in special cases (normal surfaces by Hsiang–Pati; isolated singularities by Ohsawa); the general case with the Fubini–Study metric remains open.

Independently, Saito's theory of pure Hodge modules - analytic/algebraic objects combining filtered $D$-modules (sheaves with a compatible system of differential operators) and perverse sheaves - endows $IH^*(X,\mathbb{V})$, the intersection cohomology with coefficients in a polarized variation of Hodge structure (VHS) $\mathbb{V}$, with a canonical pure Hodge structure. A variation of Hodge structure is a local system carrying a Hodge filtration by holomorphic subbundles; "polarized" means it carries a flat pairing making each fiber a polarized Hodge structure, and "quasi-unipotent local monodromies" means the monodromy around components of the complement $X\setminus X^o$ has eigenvalues that are roots of unity after a finite power - the standard condition ensuring a well-behaved local study.

Saito's construction is algebraic, but there has long been a competing analytic route: when the metric is complete and Kähler, the classical $L^2$ Hodge theory of Kashiwara–Kawai and Zucker identifies $H^k_{(2)}$ with the space of $L^2$-harmonic forms and the total bi-degree decomposition of forms induces a $(p,q)$-decomposition, hence a pure Hodge structure of the expected weight. This was established for curves by Zucker and for normal-crossing divisors by Cattani–Kaplan–Schmid and Kashiwara–Kawai, and Zucker proved that for varieties with isolated singularities the resulting $L^2$ Hodge structure on intersection homology agrees with the algebraically defined one.

Shentu and Zhao (arXiv:2103.04030, 2021) advanced this program decisively. They define, for a pure-dimensional complex space $X$ and a dense Zariski open $X^o\subset X_{\rm reg}$, a family of "distinguished" hermitian metrics on $X^o$ (their Definition 4.12), constructed locally from bounded potential functions along an equisingular stratification; such metrics are complete and locally of finite volume, exist for compactable pairs and for germs, and can be chosen Kähler when $X$ is a compact Kähler space. Their main theorem (Theorem 1.4) gives a canonical quasi-isomorphism between the complex of locally $L^2$ $\mathbb{V}$-valued forms and the intersection complex $IC_X(\mathbb{V})$ - with $(\mathbb{V},h)$ a "0-tame harmonic bundle", a pluriharmonic bundle with controlled asymptotics that includes every $\mathbb{R}$-polarized VHS with quasi-unipotent local monodromies - so that $H^k_{(2)}(X^o,\mathbb{V};ds^2,h)\simeq IH^k(X,\mathbb{V})$ for every $k$. This solves the CGM problem for distinguished metrics (though not for the Fubini–Study metric). Via completeness and the Kähler identities, the $(p,q)$-decomposition of $L^2$-forms then yields a pure Hodge structure on these groups (the "$L^2$-pure Hodge structure"), together with the Kähler package: Poincaré duality, hard Lefschetz, and Lefschetz decomposition, proved by harmonic representatives.

The remaining question, posed as Conjecture 8.5 of that paper, is whether this analytically defined Hodge structure is the same as Saito's algebraically defined pure Hodge structure on $IH^*(X,\mathbb{V})$. The question has a precedent in the Harris–Zucker conjecture (Mém. SMF 85, 2001), which predicts the same compatibility for the $L^2$ Hodge structure of a Shimura variety with respect to its Satake–Baily–Borel compactification, where the relevant $L^2$-intersection-cohomology isomorphism was established in the late 1980s; Shentu–Zhao's conjecture is the general-singularity analogue of that compatibility question.

Problem Statement

Let $X$ be a compact Kähler space of pure dimension $n$ and $X^o\subset X_{\rm reg}$ a dense Zariski open subset. Let $(\mathbb{V},\nabla,F^\bullet,h)$ be an $\mathbb{R}$-polarized variation of Hodge structure of weight $r$ on $X^o$ with quasi-unipotent local monodromies (more generally, let $(\mathcal{V},h)$ be a 0-tame harmonic bundle on $(X,X^o)$ in the sense of Shentu–Zhao, Definition 4.20), and let $ds^2$ be a Kähler distinguished metric associated to $(X,X^o)$ in the sense of Shentu–Zhao, Definition 4.12. By Shentu–Zhao's Theorem 1.4 there is a canonical isomorphism

$H^k_{(2)}(X^o,\mathbb{V};ds^2,h)\simeq IH^k(X,\mathbb{V})$

for every $k$, where the left side is $L^2$-cohomology and the right side is intersection cohomology $H^k(X,IC_X(\mathbb{V}))$. The $(p,q)$-decomposition of $L^2$-forms with respect to the complete Kähler metric $ds^2$ endows $H^k_{(2)}(X^o,\mathbb{V};ds^2,h)$ with a pure Hodge structure of weight $k+r$, and Saito's theory of polarizable Hodge modules endows $IH^k(X,\mathbb{V})$ with a pure Hodge structure of weight $k+r$. Prove or refute Shentu–Zhao's Conjecture 8.5: these two pure Hodge structures are compatible via the canonical isomorphism - that is, for every $k$ and every $(p,q)$ with $p+q=k+r$, the isomorphism identifies the $(p,q)$-summand $H^{p,q}_{(2)}(X^o,\mathbb{V})$ of the $L^2$ Hodge decomposition with the $(p,q)$-summand $IH^{p,q}(X,\mathbb{V})$ of Saito's Hodge decomposition. The statement is required in this generality: arbitrary compact Kähler spaces with arbitrary singularities, arbitrary polarized variations of Hodge structure (more generally, 0-tame harmonic bundles) with quasi-unipotent local monodromies, and arbitrary Kähler distinguished metrics; the known cases (isolated singularities with trivial coefficients; the top Hodge piece) do not cover the conjecture.

References

1. Junchao Shentu and Chen Zhao, "$L^2$-representation of Hodge Modules", preprint, arXiv:2103.04030 (2021). https://arxiv.org/abs/2103.04030 (Conjecture 8.5 appears on p. 55 of v1)
2. Junchao Shentu and Chen Zhao, "$L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module", Advances in Mathematics 442 (2024), article 109547. DOI 10.1016/j.aim.2024.109547, https://doi.org/10.1016/j.aim.2024.109547 (preprint arXiv:2104.04905)
3. Jeff Cheeger, Mark Goresky and Robert MacPherson, "$L^2$-cohomology and intersection homology of singular algebraic varieties", in: Seminar on Differential Geometry, Annals of Mathematics Studies 102, Princeton University Press (1982), pp. 303–340. DOI 10.1515/9781400881918-018, https://doi.org/10.1515/9781400881918-018
4. Morihiko Saito, "Modules de Hodge polarisables", Publications of the Research Institute for Mathematical Sciences 24 (1988), no. 6, 849–995. DOI 10.2977/prims/1195173930, https://doi.org/10.2977/prims/1195173930
5. Takeo Ohsawa, "Cheeger-Goreski-MacPherson's conjecture for the varieties with isolated singularities", Mathematische Zeitschrift 206 (1991), no. 2, 219–224. DOI 10.1007/bf02571337, https://doi.org/10.1007/bf02571337
6. W. C. Hsiang and V. Pati, "$L^2$-cohomology of normal algebraic surfaces. I", Inventiones Mathematicae 81 (1985), no. 3, 395–412. DOI 10.1007/bf01388578, https://doi.org/10.1007/bf01388578
7. Masaki Kashiwara and Takahiro Kawai, "The Poincaré lemma for variations of polarized Hodge structure", Publications of the Research Institute for Mathematical Sciences 23 (1987), no. 2, 345–407. DOI 10.2977/prims/1195176545, https://doi.org/10.2977/prims/1195176545
8. Steven Zucker, "The Hodge structures on the intersection homology of varieties with isolated singularities", Duke Mathematical Journal 55 (1987), no. 3, 603–616. DOI 10.1215/s0012-7094-87-05530-x, https://doi.org/10.1215/s0012-7094-87-05530-x
9. Michael Harris and Steven Zucker, "Boundary cohomology of Shimura varieties. III. Coherent cohomology on higher-rank boundary strata and applications to Hodge theory", Mémoires de la Société Mathématique de France, Nouvelle Série 85 (2001), vi+116 pp. DOI 10.24033/msmf.398, https://doi.org/10.24033/msmf.398
10. Junchao Shentu and Chen Zhao, "$L^2$-extension of adjoint bundles and Kollár's conjecture", Mathematische Annalen 386 (2023), no. 3–4, 1429–1462. DOI 10.1007/s00208-022-02432-6, https://doi.org/10.1007/s00208-022-02432-6
11. Mads Bach Villadsen, "Hodge modules and Kähler morphisms", preprint, arXiv:2401.09544 (2024). https://arxiv.org/abs/2401.09544
12. Chen Zhao, "$L^2$ representation of Simpson-Mochizuki's prolongation of Higgs bundles and the Kawamata-Viehweg vanishing theorem for semistable parabolic Higgs bundles", preprint, arXiv:2210.04130 (2022). https://arxiv.org/abs/2210.04130