Pure Hodge Structure on the L2-cohomology of Varieties with Isolated Singularities

Cheeger, Goresky, and MacPherson conjectured [CGM] an L2-de Rham theorem: that the intersection cohomology of a projective variety V is naturally isomorphic to the L2cohomology 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 Hodge structure on the intersection cohomology of V and even extend the rest of the “Kähler package” ([CGM]) to this context. Saito [S1,S2] eventually established the Kahler package for intersection cohomology without recourse to L2-cohomology techniques. Interest in L2-cohomology did not disappear with this result, since, among other things, L2-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, L2 − ∂̄-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 L2-cohomology of a singular variety. It was quickly realized, however, that for incomplete manifolds the Hodge and Lefschetz decompositions are not direct consequences of the Kähler condition as they are in the complete case. The primary obstruction to obtaining a Hodge structure on the L2-cohomology is an 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 complete manifold all these definitions of harmonics coincide, and one obtains the Hodge decomposition 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 representing both ∂̄ and d cohomology classes. The equivalence of the different definitions of harmonic is also required in order to obtain the Lefschetz decomposition. Interior product with the Kahler form preserves the kernel of ∆ by local computation. One requires the equivalence to see that it also preserves the kernel of d.