The analytic Hodge-theory program begun by Cheeger, Goresky and MacPherson in 1982 asks whether the intersection cohomology of a singular projective variety can be realized as the L2-cohomology of a complete Kähler metric on the variety's regular locus. For varieties with isolated singularities this was established by Saper (1985, 1992, via complete Kähler metrics) and Ohsawa (1991, via hermitian methods). For varieties with arbitrary, non-isolated singularities, no construction of a complete Kähler metric whose L2-cohomology is isomorphic to middle-perversity intersection cohomology in all degrees has appeared in the peer-reviewed literature.

Background

A complex projective variety VV is in general singular; its regular locus reg⁡(V)\operatorname{reg}(V), the complement of the singular set, is a smooth complex manifold. A Kähler metric on it is a Hermitian metric whose associated (1,1)(1,1)-form is closed; such a metric is complete if every geodesic runs for all time. The L2L^2-cohomology of a Riemannian metric gg on reg⁡(V)\operatorname{reg}(V) is the cohomology H(2)∗(reg⁡(V),g)H^*_{(2)}(\operatorname{reg}(V),g) of the complex of smooth differential forms α\alpha for which both α\alpha and dαd\alpha are square-integrable; it is a purely analytic object that can behave badly for incomplete metrics, while completeness makes the Laplacian self-adjoint and supports the Kähler identities needed for a Hodge decomposition. Intersection cohomology IH∗(V;C)IH^*(V;\mathbb{C}), introduced by Goresky and MacPherson, is a topological invariant of a stratified singular space, defined by sheaf complexes (the intersection complex) that are insensitive to strata of sufficiently small dimension; with the middle perversity — the standard choice of the growth conditions that a sheaf complex must satisfy along the strata — it restores Poincaré duality for singular varieties, and for projective varieties it carries a canonical pure Hodge structure, the direct analogue of the Hodge decomposition of smooth projective varieties; this is a theorem of the theory of Hodge modules, Saito's framework extending Hodge theory to singular spaces and to coefficients varying in families. The guiding question of Cheeger, Goresky and MacPherson (1982) was whether this topological invariant admits an analytic incarnation: they conjectured that the L2L^2-cohomology of reg⁡(V)\operatorname{reg}(V) with respect to the Fubini–Study metric (the standard Kähler metric on complex projective space) induced by a projective embedding is naturally isomorphic to IH∗(V;C)IH^*(V;\mathbb{C}). That induced metric is generally incomplete, which motivated the complete-metric variant studied ever since: seek a complete (ideally Kähler, to retain Hodge-theoretic structure) metric on reg⁡(V)\operatorname{reg}(V) whose L2L^2-cohomology computes intersection cohomology. Cheeger had already shown that with purely Riemannian conical metrics on a stratified space the L2L^2-cohomology does compute intersection cohomology in all degrees, but conical metrics are not Kähler in general, and the Kähler condition is the crux. Saper later constructed, for varieties with isolated singularities, complete Kähler metrics (now called Saper metrics) whose full L2L^2-cohomology is IH∗(V;C)IH^*(V;\mathbb{C}), with the L2L^2-Hodge structure matching the algebraic one; Ohsawa settled the isolated-singularity case of the original Fubini–Study conjecture, and Hsiang and Pati had earlier handled normal surfaces. Related but distinct realization theorems identify L2L^2-cohomology with intersection cohomology for Poincaré-type metrics (complete metrics with cusp-like blow-up near a divisor) on complements of normal-crossing divisors (singular divisors locally resembling unions of coordinate hyperplanes) — Zucker for curves, Cattani–Kaplan–Schmid in general — and for the canonical metrics on locally symmetric varieties (quotients of bounded symmetric domains by arithmetic groups), the latter the content of the Zucker conjecture proved by Looijenga and by Saper–Stern. What none of these settings covers is a variety with arbitrary, non-isolated singularities, where the singular locus is itself singular and the transverse geometry iterates: no complete Kähler metric with full-degree L2L^2-cohomology equal to intersection cohomology is known in the published literature for that case.

Problem Statement

Let VV be a complex projective variety with regular locus reg⁡(V)\operatorname{reg}(V). Does there exist a complete Kähler metric gg on reg⁡(V)\operatorname{reg}(V) such that, for every degree i≥0i \geq 0, the L2L^2-cohomology H(2)i(reg⁡(V),g)H^i_{(2)}(\operatorname{reg}(V),g) — the ii-th cohomology of the complex of smooth complex differential forms α\alpha on reg⁡(V)\operatorname{reg}(V) with both α\alpha and dαd\alpha square-integrable with respect to gg — is canonically isomorphic to the middle-perversity intersection cohomology IHi(V;C)IH^i(V;\mathbb{C})? Here 'canonically' means that the isomorphism is the one induced by a quasi-isomorphism of the global sheaf complex obtained by gluing the local L2L^2 complexes on reg⁡(V)\operatorname{reg}(V) with the intersection complex on VV, rather than an incidental identification of vector spaces. The case where VV has only isolated singularities is settled affirmatively by Saper's construction; the question is open for varieties whose singular locus may itself be positive-dimensional and singular. The equivalent formulation for compact Kähler spaces, and the extension where the constant coefficients C\mathbb{C} are replaced by a polarized variation of Hodge structure (a holomorphic family of Hodge structures over a dense Zariski-open subset satisfying Griffiths transversality and a polarization condition), are natural companion questions of the same problem; the core target is the trivial-coefficient statement above for arbitrary singularities.

References

  1. Jeff Cheeger, Mark Goresky, Robert MacPherson: L2-cohomology and intersection homology of singular algebraic varieties (1982), in Seminar on Differential Geometry, Annals of Mathematics Studies 102, Princeton University Press, pp. 303–340. DOI: 10.1515/9781400881918-018, https://doi.org/10.1515/9781400881918-018
  2. Jeff Cheeger: On the Hodge theory of Riemannian pseudomanifolds (1980), in Geometry of the Laplace Operator, Proc. Sympos. Pure Math. 36, pp. 91–146. DOI: 10.1090/pspum/036/573430, https://doi.org/10.1090/pspum/036/573430
  3. W. C. Hsiang, V. Pati: L2-cohomology of normal algebraic surfaces. I (1985), Inventiones Mathematicae 81, 395–412. DOI: 10.1007/bf01388578, https://doi.org/10.1007/bf01388578
  4. Leslie Saper: L2-cohomology and intersection homology of certain algebraic varieties with isolated singularities (1985), Inventiones Mathematicae 82, 207–255. DOI: 10.1007/bf01388801, https://doi.org/10.1007/bf01388801
  5. Takeo Ohsawa: Cheeger-Goreski-MacPherson's conjecture for the varieties with isolated singularities (1991), Mathematische Zeitschrift 206, 219–224. DOI: 10.1007/bf02571337, https://doi.org/10.1007/bf02571337
  6. Leslie Saper: L2-cohomology of Kähler varieties with isolated singularities (1992), Journal of Differential Geometry 36, no. 1, 89–161. DOI: 10.4310/jdg/1214448444, https://doi.org/10.4310/jdg/1214448444
  7. Steven Zucker: Hodge theory with degenerating coefficients: L2 cohomology in the Poincaré metric (1979), Annals of Mathematics 109, 415–476. DOI: 10.2307/1971221, https://doi.org/10.2307/1971221
  8. Eduardo Cattani, Aroldo Kaplan, Wilfried Schmid: L2 and intersection cohomologies for a polarizable variation of Hodge structure (1987), Inventiones Mathematicae 87, 217–252. DOI: 10.1007/bf01389415, https://doi.org/10.1007/bf01389415
  9. Leslie Saper, Mark Stern: L2-cohomology of arithmetic varieties (1990), Annals of Mathematics 132, 1–69. DOI: 10.2307/1971500, https://doi.org/10.2307/1971500
  10. W. Pardon, M. Stern: Pure Hodge structure on the L2-cohomology of varieties with isolated singularities (2001), Journal für die reine und angewandte Mathematik 533, 55–80. DOI: 10.1515/crll.2001.026, https://doi.org/10.1515/crll.2001.026