Fundamental groups of links of isolated singularities

Abstract

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G G there is a complex projective surface S S with simple normal crossing singularities only, so that the fundamental group of S S is isomorphic to G G . We use this to construct 3-dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to G G . Lastly, we prove that a finitely-presented group G G is Q {\mathbb Q} -superperfect (has vanishing rational homology in dimensions 1 and 2) if and only if G G is isomorphic to the fundamental group of the link of a rational 6-dimensional complex singularity.