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

TL;DR

This paper characterizes holomorphic functions of exponential growth on abelian coverings of projective manifolds, deriving Liouville-type theorems using L2 cohomology and geometric methods.

Contribution

It introduces a new approach combining L2 cohomology and geometry to describe exponential growth holomorphic functions on abelian covers of projective manifolds.

Findings

Holomorphic functions of exponential growth are characterized on abelian coverings.

Liouville-type theorems are established for these functions.

The approach integrates L2 cohomology techniques with geometric properties.

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.