L^2 Cohomology on coverings of a compact K\"ahler manifold

TL;DR

This paper develops a framework for L^2 cohomology on coverings of compact Kähler manifolds, extending classical theorems and introducing new tools for geometric analysis of such coverings.

Contribution

It defines canonical L^2 cohomology groups for unramified coverings of analytic varieties, preserving key properties and extending index and vanishing theorems.

Findings

L^2 cohomology shares properties with standard sheaf cohomology.

Finite b3-dimension of cohomology groups under Galois coverings.

Extension of Atiyah's L^2 index theorem to arbitrary coherent sheaves.

Abstract

Andreotti-Vesentini, Ohsawa, Gromov, Koll\'ar, among others, have observed that Hodge theory could be extended to (non compact) K\"ahler complete manifolds, within the L^2 framework. Also, many vanishing theorems on projective or K\"ahler manifolds rely on the Kodaira-Bochner-Nakano identity, and thus possess natural L^2 versions. Our goal is to define canonical L^2 cohomology groups on any unramified covering of an analytic variety X, with values in a coherent analytic sheaf on X. This cohomology shares all usual properties of standard coherent sheaf cohomology (especially, exact sequences, spectral sequences, vanishing theorems,...). These properties are obtained by incorporating the information provided by L^2 estimates in the standard proofs, with suitable adaptations. L^2 cohomology should provide a comfortable and efficient framework for the study the geometry of coverings, by providing a relevant fonctorial formalism. When the base variety is compact and the covering is a Galois covering of group $\Gamma$, there is a $\Gamma$ action on cohomology groups and a related concept of $\Gamma$-dimension. In that case, we prove that the $\Gamma$-dimension of cohomology groups is finite, and we extend Atiyah's L^2 index theorem to the case of arbitrary coherent sheaves. Finally, if the base manifold is projective, L^2 analogues of the usual vanishing theorems (Kodaira-Serre, Kawamata-Viehweg,...) are valid. Similar constructions and results have also been studied by P. Eyssidieux.