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Pure Hodge structure on the L2 -cohomology of varieties with isolated singularities

W. Pardon, M. Stern

2001
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Journal für die Reine und Angewandte Mathematik
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Introduction. Cheeger, Goresky, and MacPherson conjectured in [CGM] an L 2 -de Rham theorem: that the intersection cohomology of a projective variety V is naturally isomorphic to the L 2 -cohomology of the incomplete manifold V −Sing V , with metric induced by a projective embedding. The early interest in this conjecture was motivated in large part by the hope that one could then put a pure Hodge structure on the intersection cohomology of V and even extend the rest of the "Kähler package"
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... ähler package" ([CGM]) to this context. Saito ([S1,S2]) eventually established the Kähler package for intersection cohomology without recourse to L 2 -cohomology techniques. However, interest in L 2 -cohomology did not disappear with this result, since, among other things, L 2 -cohomology provides intrinsic geometric invariants of an arbitrary complex projective variety which are not apparent from the point of view of D-modules. For instance, L 2 −∂-coholomology groups depend on boundary conditions ([PS]), which, as we show here, must be treated carefully in order to give the correct Hodge components for the L 2 -cohomology of a singular variety. A related fact is that for incomplete manifolds the pure Hodge structure and Lefschetz decompositions are not direct consequences of the Kähler condition as they are in the compact case. Indeed, the primary obstruction to obtaining a Hodge structure on the L 2 -cohomology is the following apparent technicality: on an incomplete Kähler manifold there are several potentially distinct definitions of a square integrable harmonic form. For example, a form h might be considered harmonic if dh = 0 = δh, or if∂h = 0 = ϑh, or simply if ∆h = 0. Moreover there are further domain considerations: one imposes boundary conditions, which turn out to have no effect on cohomology in the case of d, but are crucial for∂-cohomology. On a compact, or even complete manifold all these definitions of harmonics coincide, and one obtains the pure Hodge structure by decomposing harmonic forms into their (p, q) components. The (p, q) components are harmonic in the weakest sense -they are in the kernel of ∆. The equality of the different notions of harmonic then allows one to realize these (p, q) components as spaces of both∂ and d cohomology classes. The equivalence of the different definitions of harmonic is also required in order to obtain the Lefschetz decomposition. A local computation shows that interior product with the Kähler form preserves the kernel of ∆, but one requires the equivalence to see that this also induces an endomorphism on the L 2 -cohomology. Partially supported by NSF grants DMS 95-04900 (Pardon) and DMS 9505040 (Stern) 1

doi:10.1515/crll.2001.026
fatcat:4shwf3a4zvf7fpogkwpcrsyteq