### 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 G if is isomorphic to the fundamental group of the link of a rational 6-dimensional complex singularity.

Original language | English (US) |
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Pages (from-to) | 929-952 |

Number of pages | 24 |

Journal | Journal of the American Mathematical Society |

Volume | 27 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1 2014 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal of the American Mathematical Society*,

*27*(4), 929-952. https://doi.org/10.1090/S0894-0347-2014-00807-9