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Algebraic approximation of Kähler threefolds of Kodaira dimension zero

Authors
  • Graf, Patrick1
  • 1 Universität Bayreuth, Lehrstuhl für Mathematik I, Bayreuth, 95440, Germany , Bayreuth (Germany)
Type
Published Article
Journal
Mathematische Annalen
Publisher
Springer Berlin Heidelberg
Publication Date
Aug 03, 2017
Volume
371
Issue
1-2
Pages
487–516
Identifiers
DOI: 10.1007/s00208-017-1577-4
Source
Springer Nature
Keywords
License
Yellow

Abstract

We prove that for a compact Kähler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every Kähler threefold of Kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modification. As a corollary, we see that the fundamental group of any Kähler threefold is a quotient of an extension of fundamental groups of projective manifolds, up to subgroups of finite index. In the course of the proof, we show that for a canonical threefold with c1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1 = 0$$\end{document}, the Albanese map decomposes as a product after a finite étale base change. This generalizes a result of Kawamata, valid in all dimensions, to the Kähler case. Furthermore we generalize a Hodge-theoretic criterion for algebraic approximability, due to Green and Voisin, to quotients of a manifold by a finite group.

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