Abstract
In this paper we use some recent developments in Nonabelian Hodge theory to study the existence of holomorphic functions on the universal coverings of algebraic surfaces. In particular we prove that if the fundamental group of an algebraic surface is reductive then its universal covering is holomorphically convex. This is a partial verification of the Shafarevich conjecture claiming that the universal covering of a smooth projective variety is holomorphically convex.
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