Holomorphic Functions of Exponential Growth on Abelian Coverings of a Projective Manifold

Abstract

Let M be a projective manifold, p:M_{G} --> M a regular covering over M with a free abelian transformation group G. We describe holomorphic functions on M_{G} of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary we obtain for such functions Cartwright and Liouville type theorems. Our approach brings together L_{2} cohomology technique for holomorphic vector bundles on complete K\"{a}hler manifolds and geometric properties of projective manifolds.