Fundamental groups of links of isolated singularities

TL;DR

This paper demonstrates that any finitely-presented group can be realized as the fundamental group of the link of a 6-dimensional complex singularity, linking algebraic properties to geometric structures.

Contribution

It constructs complex projective surfaces with prescribed fundamental groups and extends this to 3D isolated singularities, establishing a correspondence with Q-superperfect groups.

Findings

Any finitely-presented group G can be realized as the fundamental group of a link of a 6D complex singularity.

Constructed projective surfaces with simple normal crossing singularities matching any finitely-presented group.

Characterized Q-superperfect groups as those arising from links of rational 6D complex 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 there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. We use this to construct 3-dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to G. Lastly, we prove that a finitely-presented group G is Q-superperfect (has vanishing rational homology in dimensions 1 and 2) if and only if G is isomorphic to the fundamental group of the link of a rational 6-dimensional complex singularity.