The Cheeger–Goresky–MacPherson conjecture for the Fubini–Study metric: does the $L^2$-cohomology of the regular part of a singular projective variety compute its intersection cohomology?

For a complex projective variety $X\subseteq\mathbb{P}^N$, Cheeger, Goresky, and MacPherson (1982) conjectured that the $L^2$-cohomology of the smooth locus $X_{\mathrm{reg}}$, equipped with the restriction of the Fubini–Study metric, is naturally isomorphic to the middle-perversity intersection cohomology of $X$. The conjecture is proven for normal surfaces and for varieties with isolated singularities, and a 2021 result of Shentu–Zhao settles an existential-metric variant (a specially constructed complete Kähler metric), but the general case — arbitrary singularities, with the metric fixed a priori — remains open.

Background

Let $X\subseteq\mathbb{P}^N$ be a complex projective variety, possibly singular, and let $X_{\mathrm{reg}}$ be its smooth locus. The Fubini–Study metric $ds_{\mathrm{FS}}^2$ — the Kähler metric on projective space induced by the standard Hermitian inner product on $\mathbb{C}^{N+1}$, a Kähler metric being one compatible with the complex structure whose associated $2$-form is closed — restricts to a Kähler metric on $X_{\mathrm{reg}}$. The $L^2$-cohomology $H_{(2)}^*(X_{\mathrm{reg}},ds_{\mathrm{FS}}^2)$ is the cohomology of the complex of square-integrable complex differential forms $\alpha$ on $X_{\mathrm{reg}}$ such that the exterior derivative $d\alpha$ is also square-integrable (in the established convention one passes to the reduced $L^2$-cohomology, i.e. one quotients by the closure of exact forms, so that the groups are Hausdorff topological vector spaces); it is an analytic invariant of the metric completion of the smooth part. Intersection homology was introduced by Goresky and MacPherson: a perversity is a sequence controlling how far cycles may deviate from the singular strata, and middle perversity is the choice that restores Poincaré duality for suitably behaved singular spaces; the resulting intersection cohomology $IH^*(X,\mathbb{C})$ — the hypercohomology of the intersection cohomology complex $IC_X$, i.e. the cohomology obtained by replacing the complex of sheaves $IC_X$ by a resolution by suitably soft sheaves and taking global sections — is a topological invariant of the stratified space. Cheeger's Hodge theory on conical singularities showed that $L^2$-cohomology often admits harmonic representatives (forms annihilated by the Laplace operator), so the conjectured identification would place a pure Hodge structure, hard Lefschetz (the isomorphisms given by iterated cup product with the hyperplane class), and harmonic representatives on the analytic side. In their 1982 Annals-of-Mathematics-Studies chapter, Cheeger, Goresky, and MacPherson conjectured that for every projective variety these two invariants agree: $H_{(2)}^*(X_{\mathrm{reg}},ds_{\mathrm{FS}}^2)\cong IH^*(X,\mathbb{C})$. Known cases: Hsiang–Pati proved it for normal surfaces (1985); Saper (1985, 1992) and then Ohsawa (1991) settled isolated singularities — Saper via adapted complete Kähler metrics he constructed, Ohsawa for the given hermitian metric. Saper–Stern (1990) proved the analogous statement for arithmetic varieties, confirming Zucker's conjecture (which identifies the $L^2$-cohomology of locally symmetric spaces with their intersection cohomology); there the metric is the locally symmetric metric rather than the Fubini–Study metric. Pardon–Stern (2001) established the pure Hodge structure on the $L^2$-cohomology for isolated singularities. Shentu–Zhao (2021) resolved a variant: for any compact complex space there exists a complete ('distinguished') Kähler metric with $H_{(2)}^*\cong IH^*$, but the metric is existentially constructed, not the fixed Fubini–Study metric. The remaining difficulties have a precise analytic shape. Near a singular stratum $Y$ of positive dimension, the Fubini–Study metric is only asymptotic to an iterated cone-edge metric built from the link $L_Y$ (the cross-section of a local conical model $\mathbb{C}^m\times c(L_Y)$, where $c(L_Y)$ denotes the cone on $L_Y$); the relevant elliptic theory is the weighted Sobolev Fredholm theory of cone and edge operators, which is Fredholm only for weights avoiding a discrete exceptional set $\Lambda_{\mathrm{ex}}$ of indicial (critical) exponents determined by the spectrum of the link Laplacians — resonance cases where Fredholm alternatives fail. A stratified space satisfies the Witt condition when the middle-dimensional cohomology of every even-codimensional link vanishes, which frees middle-perversity intersection cohomology of extra choices; on non-Witt strata, where that middle link cohomology does not vanish, the middle-perversity self-duality condition requires choices of Lagrangian subspaces of $H^{\mathrm{mid}}(L_Y)$, so even the topological side needs extra structure (neighboring theories exist: Hodge theory on Cheeger spaces — stratified spaces with iterated cone-edge metrics satisfying a spectral tameness hypothesis — by Albin–Leichtnam–Mazzeo–Piazza, and refined intersection homology on non-Witt spaces by Albin–Banagl–Leichtnam–Mazzeo–Piazza), and matching an analytic $L^2$ condition to those choices is unresolved. Unlike Saper's and Shentu–Zhao's settings, the Fubini–Study metric is fixed a priori and cannot be adapted to the stratification, which is the crux of the open problem.

Problem Statement

Let $X\subseteq\mathbb{P}^N$ be an arbitrary complex projective variety (no assumption on dimension, number, or codimension of singular strata), let $ds_{\mathrm{FS}}^2$ be the Fubini–Study metric restricted to the smooth locus $X_{\mathrm{reg}}$, and let $IH^*(X,\mathbb{C})$ denote middle-perversity intersection cohomology. The Cheeger–Goresky–MacPherson conjecture asks: is there a natural isomorphism

$$H_{(2)}^*(X_{\mathrm{reg}},ds_{\mathrm{FS}}^2)\;\cong\;IH^*(X,\mathbb{C}),$$

where $H_{(2)}^*$ is the reduced $L^2$-cohomology of the square-integrable de Rham complex on $X_{\mathrm{reg}}$? The problem is open precisely beyond the settled cases (normal surfaces, settled by Hsiang–Pati, and isolated singularities, settled by Saper and by Ohsawa); the proof of Zucker's conjecture for arithmetic varieties (Saper–Stern) concerns the locally symmetric metric and is an analogous but distinct statement. The general case may involve singular strata of positive dimension, where the metric is only asymptotically an iterated cone-edge metric; non-Witt links, where the middle-perversity complex requires choices of self-dual subspaces of middle link cohomology, so that in that regime the precise form of the isomorphism — including how the $L^2$ complex singles out the corresponding polarized intersection complex — is itself part of what a proof must supply; and resonant cases, where critical exponents $\lambda_Y$ determined by the spectra of link Laplacians lie in the exceptional set $\Lambda_{\mathrm{ex}}$ of the weighted elliptic theory and the Fredholm alternative fails. These features describe the ambient problem class.

References

1. Jeff Cheeger, Mark Goresky, and Robert 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. W. C. Hsiang and V. Pati, 'L²-cohomology of normal algebraic surfaces. I', Inventiones Mathematicae 81 (1985), 395–412. DOI: 10.1007/bf01388578, https://doi.org/10.1007/bf01388578
3. Leslie Saper, 'L²-cohomology and intersection homology of certain algebraic varieties with isolated singularities', Inventiones Mathematicae 82 (1985), 207–255. DOI: 10.1007/bf01388801, https://doi.org/10.1007/bf01388801
4. Leslie Saper and Mark Stern, 'L²-Cohomology of Arithmetic Varieties', Annals of Mathematics 132 (1990). DOI: 10.2307/1971500, https://doi.org/10.2307/1971500
5. Takeo Ohsawa, 'Cheeger-Goreski-MacPherson's conjecture for the varieties with isolated singularities', Mathematische Zeitschrift 206 (1991), 219–224. DOI: 10.1007/bf02571337, https://doi.org/10.1007/bf02571337
6. Rafe Mazzeo, 'Elliptic theory of differential edge operators I', Communications in Partial Differential Equations 16 (1991), 1615–1664. DOI: 10.1080/03605309108820815, https://doi.org/10.1080/03605309108820815
7. Leslie Saper, 'L₂-cohomology of Kähler varieties with isolated singularities', Journal of Differential Geometry 36 (1992). DOI: 10.4310/jdg/1214448444, https://doi.org/10.4310/jdg/1214448444
8. Walter Pardon and Mark Stern, 'Pure Hodge structure on the L²-cohomology of varieties with isolated singularities', Journal für die reine und angewandte Mathematik 533 (2001). DOI: 10.1515/crll.2001.026, https://doi.org/10.1515/crll.2001.026
9. Pierre Albin, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, 'Hodge theory on Cheeger spaces', Journal für die reine und angewandte Mathematik 744 (2018), 29–102. DOI: 10.1515/crelle-2015-0095, https://doi.org/10.1515/crelle-2015-0095
10. Pierre Albin, Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, 'Refined intersection homology on non-Witt spaces', Journal of Topology and Analysis 7 (2015), 105–133. DOI: 10.1142/s1793525315500065, https://doi.org/10.1142/s1793525315500065
11. Junchao Shentu and Chen Zhao, 'L²-representation of Hodge Modules', arXiv:2103.04030 (2021), https://arxiv.org/abs/2103.04030