Some mixed Hodge structure on l^2-cohomology of covering of K\"ahler manifolds

We give methods to compute l^2-cohomology groups of a covering manifolds obtained by removing pullback of a (normal crossing) divisor to a covering of a compact K\"ahler manifold. We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by expected Gysin maps.Comment: 40 pages. This revised version will be published in Mathematische Annale

Un théorème du type d'Oka–Levi pour les domaines étalés au dessus de variétés projectives

AbstractIn this article, we study spread domains Π:U→V over a projective manifold V such that Π to be a Stein morphism, e.g., hull of meromorphy. We prove, such a domain is an existence domain of some holomorphic section s∈H0(U,El), where E=Π∗(H), H an ample line bundle on V. This is done by proving some line bundle convexity theorem for U. We deduce various results, e.g., a Lelong–Bremermann theorem for almost plurisubharmonic functions and a general Levi type theorem: Let U→V a locally pseudoconvex spread domain over a projective manifold, then U is an almost domain of meromorphy, that is Ũ\U=H some hypersurface in Ũ, the hull of meromorphy of U. Hence, if W is a general spread domain over V then its pseudoconvex hull is obtained from its meromorphic hull minus some hypersurface