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The Craighero–Gattazzo surface is simply connected

Part of: Surfaces and higher-dimensional varieties Families, fibrations

Published online by Cambridge University Press:  28 February 2017

Julie Rana ,
Jenia Tevelev  and
Giancarlo Urzúa
Julie Rana
Affiliation:
School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455, USA email jrana@umn.edu
Jenia Tevelev
Affiliation:
Department of Mathematics and Statistics, Lederle Graduate Research Tower, 1623D, University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA email tevelev@math.umass.edu
Giancarlo Urzúa
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile email urzua@mat.uc.cl
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Abstract

We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod $p$ technique and deformation theory.

Keywords

Godeaux surfacesfundamental groupdeformation theorymoduli space

MSC classification

Primary: 14J10: Families, moduli, classification 14J29: Surfaces of general type
Secondary: 14J25: Special surfaces 14D06: Fibrations, degenerations
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 3 , March 2017 , pp. 557 - 585
Copyright
© The Authors 2017 

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