We study the existence of $L^2$ holomorphic sections of invariant line bundles over Galois coverings of Zariski open sets in Moishezon manilolds. We show that the von Neuman dimension of the space of $L^2$ holomorphic sections is bounded below under reasonable curvature conditions. We also give criteria for a a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. Their coverings are then studied with the same methods. As applications we give weak Lefschetz theorems using the Napier--Ramachandran proof of the Nori theorem.