L 2 and intersection cohomologies for a polarizable variation of Hodge structure

Abstract

We consider a polarized variation of Hodge structure (V, Vz, S, F) of weight k, over a complex manifold X 1-16]. Here V denotes a locally constant sheaf of finite dimensional complex vector spaces, V z a sheaf of lattices in V, and F a decreasing filtration of Ox | by locally free sheaves of Ox-modules F p. By assumption, the filtration F induces Hodge structure of weight k on the stalks of V, and satisfies the Riemann bilinear relations, as welt as the transversality relation, relative to the fiat bilinear form S. Typically variations of Hodge structure arise from the cohomology of the fibres in a family of smooth projective varieties.Henceforth the base X will be assumed to lie as a Zariski open subset in a compact K/ihler manifold X, such that X-X is a divisor with normal crossings. In other words, every x~X lies in some neighborhood U, with (1.1) U_~A", Uc~X~--(A*)mXA n-m(A=unit disc in C, A*=A-{0}), for some m=m(x) between 0 and n=dimX.Any smooth quasiprojective variety X has a projective completion )? of this sort. We equip X with a complete K/ihler metric g, whose restriction to any neighborhood U as in (1.1) is asymptotic to a product of metrics on the one dimensional factors: Euclidean on the n-m factors A, and of constant negative curvature, complete at the punctures, on the m factors A*; such a metric g always exists I-9]. Whether a V-valued form on X is square integrable over compact subsets of X', relative to the metric g on X and the Hodge metrics I-6, 18] on the stalks of V, depends only on the asymptotic behavior of g, not on the particular choice of g, nor on the choice of polarization. The complex of sheaves A~2)(V ) on X, defined by the assignment * Partially supported by NSF Grant DMS-8501949 ** Partially supported by NSF Grant DMS-8317436 218 E. Cattani et al.(1.2) U~--,complex of V-valued forms ~o on U n X, with measurable coefficients and measurable exterior derivative, such that 09 and d~o have finite L 2 norm on K n X, for every compact K c U, is therefore canonically attached to the completion X'. Because of the special nature of the metric g -on neighborhoods U as in (I.I) it splits into a product of metrics on the one dimensional factors, up to quasi-isometry -the sheaves A~z)(V ) are fine, so the complex of global sections computes the hypercohomo- The proof of Theorem (1.5) is a local problem -it suffices to identify the L 2 cohomology and the intersection cohomology of any neighborhood U~-A n as in (1.1). Both cohomologies have weight filtrations, which are induced by the local monodromy. The nature of the metric g and of the Hodge metrics on V enables us to compare the L 2 cohomology groups of A" to those of A "-{0}, via a Mayer-Vietoris spectral sequence: the L 2 cohomology of A n agrees with that of A n-{0} below a certain critical weight, and vanishes above it; the contribution of the critical weight itself is isomorphic to the tensor product of a L 2 and intersection cohomologies 219 certain infinite dimensional space with the L 2 cohomology of A"-{0}, also at the critical leve...

Mixed Hodge Modules

“…Moreover it is an abelian full subcategory of MHW(L7), because we can take X such that X\U is a locally principal divisor and use 7, or j^ for the extension. Similarly it is stable by \// g9 (/) gjl for a function g on (7 for a general X and for the compatibility with its definition for a quasi-projective variety, it is enough to check (4.2.6) for X quasi-projective. We take a projective completion X such that X\X is a locally principal divisor, and take a refinement of the covering X = U U t such that X\U t is a w.l.p.d.…”

“…the filiations W (i} on M^ are constructed functorially by induction on i using W (i} on the other M 7 and FFon M r In fact this is just (3.23.13) if n = 1, and we can prove it by induction on n using (3.23.13). Clearly it is enough to show the assertion for a = 0.…”

Asymptotic Behaviour of Tame Nilpotent Harmonic Bundles with Trivial Parabolic Structure

“…We know that the degree of e in R V with respect to the filtration W (N (j) ∧ R ) is b = b1 − h1 for any j. Here R and b are given as follows: (See (7). We use h − 1 instead of h.)…”

Untitled

“…The Poincaré intersection form on H * (L) is nondegenerate, as usual, and also because of the Hodge-Riemann bilinear relations (38) on E.…”