The Cheeger–Goresky–MacPherson conjecture: L²-cohomology of the smooth part of a complex projective variety in the Fubini–Study metric versus intersection cohomology

The Cheeger–Goresky–MacPherson conjecture (1982) predicts that for a complex projective variety $X\subset\mathbb{P}^N$, the $L^2$-cohomology of the smooth locus $X_{\mathrm{reg}}$ equipped with the induced Fubini–Study metric is naturally isomorphic to the middle-perversity intersection cohomology of $X$. It is proved only in special classes (conical metrics, normal surfaces, isolated singularities, arithmetic/locally symmetric varieties), and a 2021 metric-existence variant of Shentu–Zhao does not cover the prescribed Fubini–Study metric.

Background

On a closed smooth manifold, de Rham's theorem and Hodge theory identify topological cohomology with spaces of harmonic differential forms. For singular spaces this picture breaks down, and two substitute theories are relevant here. First, Goresky and MacPherson's intersection homology (1980–83) modifies singular homology by imposing allowability conditions on chains relative to a stratification of the singular set; with the middle perversity (the allowability cutoff placed halfway along the dimension range of strata) it restores Poincaré duality on any complex projective variety, and its cohomology $IH^*(X;\mathbb{C})$ carries the Kähler–Hodge package — Poincaré duality, the hard Lefschetz theorem (wedging with a hyperplane class induces isomorphisms on cohomology up to the middle degree), and a Hodge decomposition — by work of Beĭlinson–Bernstein–Deligne–Gabber and Saito. Second, Cheeger's $L^2$-cohomology: given a (typically incomplete) Riemannian metric on the smooth part of a singular space, one forms the de Rham complex of square-integrable forms $\omega$ whose differential $d\omega$ is also square-integrable, and takes its cohomology (classically the reduced version, quotienting by the closure of the image of $d$, so that the result agrees with the space of $L^2$ harmonic forms). Cheeger (1980) proved that for spaces with conical singularities and a conical metric, this $L^2$-cohomology computes intersection cohomology, giving harmonic representatives for intersection cohomology classes. In their 1982 paper, Cheeger, Goresky, and MacPherson conjectured the corresponding statement for the metric that comes for free in algebraic geometry: if $X\subset\mathbb{P}^N$ is a complex projective variety and $ds^2_{\mathrm{FS}}$ is the Fubini–Study metric (the standard homogeneous Kähler metric on projective space, restricted to the smooth locus $X_{\mathrm{reg}}$; the induced metric is generally NOT conical near singular strata), then $H^*_{(2)}(X_{\mathrm{reg}},ds^2_{\mathrm{FS}})\simeq IH^*(X;\mathbb{C})$. An affirmative answer would give a genuine $L^2$-Hodge theory — harmonic forms, heat kernels, hard Lefschetz by analytic means — for arbitrary projective varieties, paralleling Zucker's conjecture, which identified the $L^2$-cohomology of a locally symmetric variety with the intersection cohomology of its Baily–Borel compactification (the minimal projective compactification of such a variety) and was proved by Looijenga and by Saper–Stern. The conjecture is confirmed in: conical-metric settings (Cheeger; Nagase for stratified spaces), normal algebraic surfaces (Hsiang–Pati; Pati; Nagase), varieties with isolated singularities (Saper; Ohsawa; with pure Hodge structure by Pardon–Stern), and arithmetic varieties, where the Saper–Stern theorem on $L^2$-cohomology of arithmetic varieties establishes the parallel Zucker statement for the locally symmetric metric rather than for the prescribed Fubini–Study metric. The essential difficulty is that near a singular stratum of codimension at least 2 the Fubini–Study metric degenerates in a non-product, non-conical fashion governed by the projective embedding; monomial coordinates taming it (Hsiang–Pati coordinates) are only known up to dimension 3 (Belotto da Silva–Bierstone–Grandjean–Milman). Shentu and Zhao (2021) proved a CGM-type statement for an existentially constructed complete Kähler 'distinguished' metric on any compact complex space, explicitly leaving the original prescribed-metric conjecture open.

Problem Statement

Let $X\subset\mathbb{P}^N$ be a complex projective variety (with arbitrary, possibly non-isolated singularities), let $X_{\mathrm{reg}}$ be its smooth locus equipped with the Fubini–Study metric $ds^2_{\mathrm{FS}}$ induced by the embedding, and let $IH^*(X;\mathbb{C})$ denote middle-perversity intersection cohomology. Prove or refute the Cheeger–Goresky–MacPherson conjecture (1982): there is a natural isomorphism $H^*_{(2)}(X_{\mathrm{reg}},ds^2_{\mathrm{FS}})\;\simeq\;IH^*(X;\mathbb{C})$ in all degrees, where the left side is the (reduced) $L^2$-cohomology of the de Rham complex of square-integrable complex-valued differential forms $\omega$ on $X_{\mathrm{reg}}$ with $d\omega$ square-integrable. The conjecture is already known for normal surfaces, for varieties with isolated singularities, for conical metrics, and for arithmetic/locally symmetric varieties (the latter via the locally symmetric metric of Zucker's conjecture, not the prescribed Fubini–Study metric); the open content is the general case of arbitrary singular strata with the specifically prescribed Fubini–Study metric.

References

1. J. Cheeger, M. Goresky, R. MacPherson, L²-cohomology and intersection homology of singular algebraic varieties, in: Seminar on Differential Geometry (AM-102), Annals of Mathematics Studies 102, Princeton University Press, 1982, pp. 303–340. DOI: 10.1515/9781400881918-018, https://doi.org/10.1515/9781400881918-018
2. J. Cheeger, On the Hodge theory of Riemannian pseudomanifolds, Proceedings of Symposia in Pure Mathematics 36 (1980) 91–146. DOI: 10.1090/pspum/036/573430, https://doi.org/10.1090/pspum/036/573430
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6. Masayoshi Nagase, Remarks on the L²-cohomology of singular algebraic surfaces, Journal of the Mathematical Society of Japan 41 (1989) 97–116. DOI: 10.2969/jmsj/04110097, https://doi.org/10.2969/jmsj/04110097
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12. Junchao Shentu, Chen Zhao, L²-representation of Hodge Modules, arXiv:2103.04030 (2021). https://arxiv.org/abs/2103.04030
13. Junchao Shentu, Chen Zhao, L²-Dolbeault resolution of the lowest Hodge piece of a Hodge module, Advances in Mathematics 442 (2024) 109547. DOI: 10.1016/j.aim.2024.109547, https://doi.org/10.1016/j.aim.2024.109547
14. A. Belotto da Silva, E. Bierstone, V. Grandjean, P. D. Milman, Resolution of singularities of the cotangent sheaf of a singular variety, arXiv:1504.07280 (2015). https://arxiv.org/abs/1504.07280
15. Jiaming Luo, Shirong Li, Stratified Interpretation for De Rham Cohomology and Non-Witt Spaces, arXiv:2505.00320 (2025). https://arxiv.org/abs/2505.00320
16. Jiaming Luo, Shirong Li, Derived Stratifications and Arithmetic Intersection Theory for Varieties with Isolated Singularities, arXiv:2508.01679 (2025). https://arxiv.org/abs/2508.01679