The Cheeger–Goresky–MacPherson conjecture: $L^2$-de Rham cohomology of the Fubini–Study metric and intersection cohomology of singular projective varieties

For a complex projective variety $X\subset\mathbb{CP}^N$, the Cheeger–Goresky–MacPherson (CGM) conjecture predicts that the $L^2$-de Rham cohomology of the smooth locus $X_{\mathrm{reg}}$, computed with the restriction of the Fubini–Study metric and the exterior derivative in the sense of distributions, is naturally isomorphic to the middle-perversity intersection cohomology $IH^*(X)$. The conjecture is proved for isolated singularities (Ohsawa, building on Hsiang–Pati, Saper, and Cheeger's conical-singularity theory) and for arithmetic varieties (Saper–Stern), but for singular loci of positive dimension it remains open.

Background

Intersection homology was introduced by Goresky and MacPherson as a homology theory for singular spaces that restores key features of manifolds, notably Poincaré duality: one computes simplicial chains subject to allowability conditions, governed by a perversity parameter, and the middle perversity is the choice adapted to complex varieties. Cheeger, Goresky, and MacPherson conjectured in 1982 that this topological invariant has a direct analytic realization. Let $X\subset\mathbb{CP}^N$ be a (reduced, pure-dimensional) projective variety with smooth locus $X_{\mathrm{reg}}=X\setminus\mathrm{Sing}\,X$. The Fubini–Study metric is the standard Kähler (i.e., compatible with the complex structure and closed) metric on $\mathbb{CP}^N$ induced by the Hermitian inner product on $\mathbb{C}^{N+1}$; its restriction $ds^2_{\mathrm{FS}}$ to $X_{\mathrm{reg}}$ is generally incomplete because the singular points lie at finite distance. One then forms the $L^2$-de Rham complex: square-integrable differential forms $\alpha$ on $X_{\mathrm{reg}}$ whose distributional exterior derivative $d\alpha$ is also square-integrable (the maximal, or distributional, closed extension of $d$). Its cohomology $H^k_{(2)}(X_{\mathrm{reg}},ds^2_{\mathrm{FS}})$ depends on the metric, unlike ordinary de Rham cohomology. The CGM conjecture states that these groups compute intersection cohomology $IH^k(X)$ in middle perversity. The original motivation was Hodge-theoretic: such an identification, combined with $L^2$-Hodge decomposition, would put a pure Hodge structure — a canonical decomposition of cohomology into summands of Hodge type $(p,q)$, as on the cohomology of smooth projective varieties — on $IH^*(X)$ analytically (a structure now also available algebraically, e.g., through Saito's theory of polarizable Hodge modules). Known cases: Cheeger established the identification for spaces with conical singularities, where the metric is an exact cone near the singularity; Hsiang–Pati proved it for normal algebraic surfaces (normality forces the singular locus of a surface to be zero-dimensional, so this is again an isolated-singularity case); Ohsawa settled the full isolated-singularity case for projective varieties with the actual projective (Kähler) metric; Saper constructed complete Kähler metrics on the regular part of varieties with isolated singularities whose $L^2$-cohomology is $IH^*$, and Pardon–Stern established the pure Hodge structure on the $L^2$-cohomology in the isolated case; Saper–Stern computed the $L^2$-cohomology of arithmetic varieties — quotients of symmetric complex domains by arithmetic groups, carrying their natural incomplete locally symmetric metric — confirming Zucker's conjecture, which identified this $L^2$-cohomology with the intersection cohomology of the variety's Baily–Borel compactification (its natural projective compactification). In all of these, either the singular set is zero-dimensional or the geometry is highly special. On the analytic side, Ruppenthal developed an $L^2$-theory for the $\bar\partial$-operator on singular complex spaces with arbitrary singularities (via resolutions of singularities and weighted Dolbeault complexes), which clarifies the $L^2$-Dolbeault groups but does not settle the de Rham statement for the Fubini–Study metric when the singular locus has positive dimension. Shentu and Zhao later constructed, for arbitrary compact complex spaces, some complete (Kähler, when the space is Kähler) metric on the regular locus whose $L^2$-cohomology is $IH^*$ — solving a CGM-type problem for an existentially defined metric, while explicitly noting that the original Fubini–Study statement remains unknown in general.

Problem Statement

Prove or refute the Cheeger–Goresky–MacPherson conjecture in its remaining generality. Let $X\subset\mathbb{CP}^N$ be a reduced pure-dimensional complex projective variety, let $X_{\mathrm{reg}}=X\setminus\mathrm{Sing}\,X$ be its smooth locus, and let $ds^2_{\mathrm{FS}}$ be the restriction of the Fubini–Study metric to $X_{\mathrm{reg}}$. Let $H^k_{(2)}(X_{\mathrm{reg}},ds^2_{\mathrm{FS}})$ be the $L^2$-de Rham cohomology: the cohomology of the complex of square-integrable differential forms on $X_{\mathrm{reg}}$ whose exterior derivative, taken in the sense of distributions, is also square-integrable. The conjecture asserts that there is a natural isomorphism $H^k_{(2)}\left(X_{\mathrm{reg}},ds^2_{\mathrm{FS}}\right)\;\cong\;IH^k(X)$ for all $k$, where $IH^k(X)$ is the middle-perversity intersection cohomology of $X$ with complex coefficients. This is known for isolated singularities (Ohsawa, with the normal-surface case earlier by Hsiang–Pati), for conical singularities (Cheeger), and for arithmetic varieties with their natural locally symmetric metric (Saper–Stern, in the sense of Zucker's conjecture). The open core is the case of an arbitrary singular locus of positive dimension, $\dim\mathrm{Sing}\,X\geq 1$: establish or disprove the natural isomorphism in that generality, without restricting the class of singularities or the dimension, and with the metric fixed to $ds^2_{\mathrm{FS}}$ (no substitution of another metric).

References

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12. Junchao Shentu, Chen Zhao, 'L²-representation of Hodge Modules', arXiv:2103.04030 (2021), https://arxiv.org/abs/2103.04030