ON THE UNIVERSAL COVERINGS OF ALGEBRAIC-SURFACES

In this paper we use some recent developments in Nonabelian Hedge theo ry to study the existence of holomorphic functions on the universal co verings of algebraic surfaces. In particular we prove that if the fund amental 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 s mooth projective variety is holomorphically convex. (C) Elsevier, Pari s.