The Cheeger–Goresky–MacPherson conjecture for complex projective varieties with non-isolated singularities

For a complex projective variety $Z$, the Cheeger–Goresky–MacPherson (CGM) conjecture predicts that the reduced $L^2$-de Rham cohomology of the regular locus $Z^*=\mathrm{Reg}\,Z$, equipped with the restriction of the Fubini–Study metric, is naturally isomorphic to the middle-perversity intersection cohomology of $Z$. The conjecture was proved by Ohsawa for varieties with isolated singularities (1991), and by Saper and Pardon–Stern in companion forms in the same setting, but it remains open when the singular set has positive dimension — the surviving open core, highlighted explicitly in Ruppenthal's work on weighted Dolbeault complexes (2009). The problem asks for a proof or an explicit refutation of this comparison in the non-isolated case.

Background

Let $Z\subset\mathbb{CP}^{N}$ be a complex projective variety and let $Z^{*}:=\mathrm{Reg}\,Z$ be its regular locus, a complex manifold carrying the (generally incomplete) Riemannian metric obtained by restricting the Fubini–Study metric — the homogeneous Kähler metric, i.e. a Hermitian metric whose associated 2-form is closed — of the ambient projective space. The reduced $L^{2}$-de Rham cohomology $H^{k}_{(2)}(Z^{*})$ is the cohomology of the complex of differential $k$-forms on $Z^{*}$ that are square-integrable together with their exterior derivative, taken modulo the closure of the image of $d$ so that the groups are Hausdorff; on complete manifolds it is represented by square-integrable harmonic forms, but on the incomplete manifold $Z^{*}$ even this harmonic representation is a subtle analytic statement. Intersection cohomology $IH^{*}(Z)$, introduced by Goresky and MacPherson, is a topological invariant of a stratified space that agrees with ordinary cohomology for nonsingular varieties and restores Poincaré duality for singular ones; the middle-perversity version — the grading condition adapted to complex varieties — is meant throughout, and when $Z$ admits a small resolution (a resolution of singularities whose fibers all have dimension strictly less than half of $\dim Z$), $IH^{*}(Z)$ agrees with the ordinary cohomology of that resolution.

In 1982 Cheeger, Goresky and MacPherson conjectured that for every complex projective variety there is a natural isomorphism $H^{k}_{(2)}(Z^{*})\cong IH^{k}(Z)$: an identification of the two most canonical cohomology theories of a singular variety, one analytic and one topological. The original motivation was Hodge-theoretic — such an isomorphism, combined with a Hodge decomposition of the $L^{2}$-cohomology of the Kähler manifold $Z^{*}$, would carry a pure Hodge structure onto $IH^{*}(Z)$, giving analytic access to the intersection cohomology of singular varieties. Two neighbouring results support the expected picture: Cheeger's theorem identifies $L^{2}$-cohomology with intersection cohomology for complete metrics that are locally cone-like along the strata (near each stratum the space is, metrically, approximately a product of the stratum with a cone over a cross-section), and the Zucker conjecture, proved by Looijenga and by Saper–Stern, identifies the $L^{2}$-cohomology of the regular part of a Shimura variety — a locally symmetric complex algebraic variety arising in the theory of automorphic forms — with the intersection cohomology of its natural compactification. The difficulty is that the Fubini–Study-induced metric on $Z^{*}$ is incomplete near the singular set: at a point of a singular stratum of positive dimension it behaves asymptotically like a product of a transverse cone singularity with a smooth factor, but with distortions that the complete-metric theory does not control, and in the borderline middle degrees the $L^{2}$-condition becomes fragile.

The state of the art is as follows. Ohsawa proved the conjecture for varieties with isolated singularities; Saper obtained the corresponding identification for Kähler varieties with isolated singularities; Pardon and Stern computed the $L^{2}$-$\bar\partial$-cohomology of complex projective varieties with isolated singularities and later endowed the $L^{2}$-cohomology of such varieties with a pure Hodge structure (here $H^{p,q}_{(2)}(Z^{*})$ denotes the reduced $L^{2}$-cohomology in bidegree $(p,q)$ of the $\bar\partial$-operator on the Dolbeault complex of $Z^{*}$, i.e. square-integrable $(p,q)$-forms with square-integrable $\bar\partial$, modulo closure of the $\bar\partial$-image). Beyond the isolated case only partial information exists: Diederich, Vassiliadou and Fornæss computed local $L^{2}$ $\bar\partial$-groups on surfaces with singular curves, and Ruppenthal constructed weighted $L^{p}$ Dolbeault complexes and local $\bar\partial$-solution operators near singularities with normal crossings (locally the union of coordinate hyperplanes), together with an $L^{2}$-representation program for cohomology on resolutions of singularities — a resolution is a proper holomorphic map from a complex manifold onto $Z$ that is an isomorphism away from the singular set — which however so far covers top-degree forms in general and all degrees only for isolated singularities. No comparison theorem with $IH^{*}(Z)$ is known for the induced metric when the singular set has positive dimension.

Problem Statement

Let $Z\subset\mathbb{CP}^{N}$ be a complex projective variety whose singular set $\mathrm{Sing}(Z)$ has positive dimension — the case left open after Ohsawa's theorem for isolated singularities — and let $Z^{*}=\mathrm{Reg}\,Z$ carry the restriction of the Fubini–Study metric. Let $H^{k}_{(2)}(Z^{*})$ denote the reduced $L^{2}$-de Rham cohomology of $Z^{*}$ and $IH^{k}(Z)$ the middle-perversity intersection cohomology of $Z$ with real coefficients, so that both sides of the comparison are real vector spaces; complexifying yields the corresponding statement for $IH^{k}(Z;\mathbb{C})$. The problem is to settle the Cheeger–Goresky–MacPherson conjecture in this remaining generality, in one of two ways:

1. Prove that there is a natural (choice-free) isomorphism $H^{k}_{(2)}(Z^{*})\cong IH^{k}(Z)$ for all degrees $k$ and all complex projective varieties $Z$ with positive-dimensional singular set; or

2. Refute the conjecture by an explicit projective variety $Z$ with a positive-dimensional singular set for which some $H^{k}_{(2)}(Z^{*})$ and $IH^{k}(Z)$ fail to be isomorphic, together with a rigorous computation of both groups under the induced metric.

Both branches address the same scientific objective: the CGM comparison between the analytic $L^{2}$-cohomology of the regular locus (for the natural induced metric) and the intersection-theoretic cohomology of singular projective varieties with non-isolated singularities. The Hodge-theoretic refinement — identifying the reduced $L^{2}$-$\bar\partial$-cohomology groups $H^{p,q}_{(2)}(Z^{*})$ with the $(p,q)$-Hodge pieces of $IH^{*}(Z;\mathbb{C})$ and putting a pure Hodge structure on $IH^{*}(Z)$ — was the original motivation for the conjecture and was achieved by Pardon–Stern in the isolated case, but it rests on an $L^{2}$-Hodge decomposition on the incomplete Kähler manifold $Z^{*}$ beyond the isolated case and is a separate theorem rather than part of the comparison sought here. No restriction is placed on method; the middle-degree and borderline-parameter behaviour that drives the known $\epsilon$-losses in local weighted estimates reflects the intrinsic analytic difficulty of the comparison near the singular stratum.

References

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