Fundamental groups of links of isolated singularities

Kapovich, Michael; Kollár, János

doi:10.1090/S0894-0347-2014-00807-9

Fundamental groups of links of isolated singularities
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by Michael Kapovich and János Kollár

J. Amer. Math. Soc. 27 (2014), 929-952

DOI: https://doi.org/10.1090/S0894-0347-2014-00807-9

Published electronically: May 22, 2014

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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 ${\mathbb Q}$-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.

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