Two remarks on the topology of projective surfaces

Abstract

We will prove two results about the topology of complex projective surfaces. The first result says that if the Shafarevich Conjecture has an affirmative answer in dimension two then the second homotopy group of a smooth projective surface is a torsion-free abelian group. The second result is that for any 2-dimensional function field K/C there is a normal projective simply-connected surface with function field K.