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Fundamental groups of links of isolated singularities


Authors: Michael Kapovich and János Kollár
Journal: J. Amer. Math. Soc. 27 (2014), 929-952
MSC (2010): Primary 14B05, 14J17, 14F35; Secondary 20F05, 53C55
DOI: https://doi.org/10.1090/S0894-0347-2014-00807-9
Published electronically: May 22, 2014
MathSciNet review: 3230815
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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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Additional Information

Michael Kapovich
Affiliation: Department of Mathematics, University of California, Davis, California 95616
Email: kapovich@math.ucdavis.edu

János Kollár
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
Email: kollar@math.princeton.edu

DOI: https://doi.org/10.1090/S0894-0347-2014-00807-9
Received by editor(s): January 9, 2012
Received by editor(s) in revised form: September 19, 2012, February 19, 2013, and February 20, 2013
Published electronically: May 22, 2014
Article copyright: © Copyright 2014 American Mathematical Society