Cheeger–Goresky–MacPherson question: complete Kähler metrics on projective varieties whose $L^2$-cohomology computes intersection cohomology

For a complex projective variety $V$ with singular locus, does there exist a complete Kähler metric on the regular part $\operatorname{reg}(V)$ whose full $L^2$-cohomology is isomorphic to the middle-perversity intersection cohomology $IH^*(V)$ in all degrees? The question goes back to Cheeger, Goresky, and MacPherson (1982). It is settled affirmatively for isolated singularities by Saper's complete Kähler metrics, and Saper-type complete Kähler metrics exist on the regular part of every projective variety by work of Grant Melles and Milman, but no construction, and no obstruction, is known for arbitrary singular loci: the recent identifications of Bei and Piazza cover only top-holomorphic bidegrees. No published resolution is known as of August 2026.

Background

Let $V$ be a complex projective variety of complex dimension $v$ embedded in some complex projective space, and let $\operatorname{reg}(V)$ and $\operatorname{sing}(V)$ denote its sets of smooth and singular points. The regular part is a complex manifold of dimension $v$, and one can study it with the tools of Riemannian and Kähler geometry; a Kähler metric on $\operatorname{reg}(V)$ is called complete if every geodesic runs for all time, equivalently if every point of $\operatorname{sing}(V)$ lies at infinite metric distance.

Given a complete Riemannian metric, the $L^2$-cohomology is the cohomology of the complex of square-integrable differential forms with square-integrable differentials, closed modulo exact such forms (one often passes to the reduced quotient, which divides additionally by the closure of the exact classes; the distinction matters when the range of the differential is not closed). On a closed manifold, Hodge theory identifies ordinary cohomology with spaces of harmonic forms, and the guiding question for singular varieties is whether an analogous analytic representative exists for the correct topological invariant of $V$.

The correct invariant is intersection cohomology, introduced by Goresky and MacPherson: it is the cohomology of a complex of chains allowed to meet the singular strata only with controlled dimension, the control being encoded by a perversity function; the middle perversity is the symmetric choice that restores Poincaré duality and supports a Kähler-type Hodge theory. In their 1982 study of $L^2$-cohomology and intersection homology of singular algebraic varieties, Cheeger, Goresky, and MacPherson verified the matching between the two theories for metrics that are conical near the singular strata, and raised the general question — the subject of this problem — of whether every complex projective variety carries a complete Kähler metric on its regular part whose $L^2$-cohomology (in all degrees) recovers the middle-perversity intersection cohomology of $V$.

Three lines of later work delimit what is known. First, Saper (1985, 1992) settled the case of isolated singularities affirmatively: he constructed on $\operatorname{reg}(V)$ a complete Kähler metric — now called a Saper metric, obtained by blowing up the singularity and modifying the pullback of the ambient metric by iterated logarithmic potentials — whose full $L^2$-cohomology computes the middle-perversity intersection cohomology. Second, Grant Melles and Milman (1995, 2006), inspired by Saper's construction, built Saper-type complete Kähler metrics on the regular part of an arbitrary analytic subvariety of a compact Kähler manifold, with no restriction on the singular set; these are the natural candidate metrics, but their $L^2$-cohomology was not computed, and the matching with intersection cohomology for non-isolated singularities has remained open since. Third, in the complementary direction of the metric induced from the ambient space (which is incomplete at the singularities), Pardon and Stern (1991), completing earlier work of Pardon and Haskell on curves and surfaces, identified the top-holomorphic-bidegree $L^2$-Dolbeault groups of $\operatorname{reg}(V)$ with the Dolbeault cohomology of a resolution of $V$; and Saper and Stern (1990) proved the Zucker conjecture, identifying the $L^2$-cohomology of arithmetic varieties (quotients of Hermitian symmetric domains by arithmetic groups, equipped with their canonical locally symmetric metric, which is complete) with the intersection cohomology of their Baily–Borel compactification, the minimal projective compactification of such a quotient — a celebrated special case with non-isolated singularities, but for one specific complete metric on one specific family of varieties rather than by a general construction.

The paper by Bei and Piazza (2015, published 2018) returns to the complete-metric question with a sheaf-theoretic method. For a compact irreducible complex space whose regular part carries a complete Hermitian metric $h$, they prove, under a finiteness hypothesis (finiteness of the pushforwards of certain sheaves of maximal domains of the $\bar\partial$-operator on a resolution — the maximal domain being the largest domain of the weakly defined $\bar\partial$-operator on which it is a closed operator) and a local quasi-isometry hypothesis (near each singular point the metric must be comparable to a Kähler metric admitting a bounded potential), that the top-holomorphic-bidegree groups satisfy $H^{v,q}_{2,\bar\partial}(\operatorname{reg}(V),h)\cong H^{v,q}_{\bar\partial}(M)$ for a resolution $\pi:M\to V$, and they verify the hypotheses for Saper-type metrics and for complete Kähler metrics of finite volume with pinched negative sectional curvature, which by the Siu–Yau compactification theorem are biholomorphic to regular parts of projective varieties with isolated singularities. These identifications are consistent with, but far short of, the full matching: the Dolbeault cohomology of a resolution agrees with intersection cohomology only in restricted degree ranges, and nothing is obtained for the middle-degree de Rham groups where the heart of the intersection cohomology of $V$ lives.

Problem Statement

Let $V$ be a complex projective variety of complex dimension $v$ with regular part $\operatorname{reg}(V)$ and arbitrary singular locus. Does there exist a complete Kähler metric $g$ on $\operatorname{reg}(V)$ such that the $L^2$-de Rham cohomology of $(\operatorname{reg}(V),g)$ is isomorphic in every degree to the middle-perversity intersection cohomology $IH^*(V)$ — as asked by Cheeger, Goresky, and MacPherson? (The Saper-type metrics of Grant Melles and Milman described in the Background are constructed on analytic subvariety of arbitrary compact Kähler manifolds, a class strictly larger than the class of projective varieties, since subvarieties of a general compact Kähler manifold need not be projective; the question of Cheeger–Goresky–MacPherson restated here is the projective one.) A negative answer must exhibit a complex projective variety for which no complete Kähler metric on its regular part has $L^2$-cohomology isomorphic to $IH^*(V)$ in all degrees. The question is open for general varieties whose singular locus is not discrete; the isolated-singularity case was settled affirmatively by Saper, and for Baily–Borel compactifications of arithmetic varieties the matching follows from the Zucker conjecture theorem of Saper and Stern, whose locally symmetric metric on the regular part is complete.

References

1. J. Cheeger, M. Goresky, R. MacPherson, $L^2$-cohomology and intersection homology of singular algebraic varieties, in: Seminar on Differential Geometry, Annals of Mathematics Studies 102, Princeton University Press, Princeton (1982), pp. 303–340. DOI: 10.1515/9781400881918-018, https://doi.org/10.1515/9781400881918-018
2. L. Saper, $L^2$-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
3. L. Saper, $L_2$-cohomology of Kähler varieties with isolated singularities, Journal of Differential Geometry 36 (1992), no. 1, 89–161. DOI: 10.4310/jdg/1214448444, https://doi.org/10.4310/jdg/1214448444
4. C. Grant and P. Milman, Metrics for singular analytic spaces, Pacific Journal of Mathematics 168 (1995), no. 1, 61–156. DOI: 10.2140/pjm.1995.168.61, https://doi.org/10.2140/pjm.1995.168.61
5. C. Grant Melles and P. Milman, Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization, Annales de la Faculté des Sciences de Toulouse, Mathématiques (6) 15 (2006), no. 4, 689–771. DOI: 10.5802/afst.1134, https://doi.org/10.5802/afst.1134
6. L. Saper and M. Stern, $L_2$-cohomology of arithmetic varieties, Annals of Mathematics (2) 132 (1990), no. 1, 1–69. DOI: 10.2307/1971500, https://doi.org/10.2307/1971500
7. W. L. Pardon and M. A. Stern, $L^2$–$\overline\partial$-cohomology of complex projective varieties, Journal of the American Mathematical Society 4 (1991), no. 3, 603–621. DOI: 10.1090/S0894-0347-1991-1102582-6, https://doi.org/10.1090/S0894-0347-1991-1102582-6
8. Y. T. Siu and S. T. Yau, Compactification of negatively curved complete Kähler manifolds of finite volume, in: Seminar on Differential Geometry, Annals of Mathematics Studies 102, Princeton University Press, Princeton (1982), pp. 363–380. DOI: 10.1515/9781400881918-021, https://doi.org/10.1515/9781400881918-021
9. F. Bei and P. Piazza, On the $L^2$-$\overline\partial$-cohomology of certain complete Kähler metrics, Mathematische Zeitschrift 290 (2018), no. 1-2, 521–537. DOI: 10.1007/s00209-017-2029-2; arXiv:1509.06174, https://arxiv.org/abs/1509.06174