Morse inequalities for covering manifolds

Abstract

We study the existence of $L^2$ holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of $L^2$ holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.