Quotients of Calabi–Yau varieties

János Kollár, Michael Larsen

Research output: Chapter in Book/Report/Conference proceedingChapter

20 Scopus citations

Abstract

Let X be a complex Calabi–Yau variety, that is, a complex projective variety with canonical singularities whose canonical class is numerically trivial. Let G be a finite group acting on X and consider the quotient variety X/G. The aim of this paper is to determine the place of X/G in the birational classification of varieties. That is, we determine the Kodaira dimension of X/G and decide when it is uniruled or rationally connected. If G acts without fixed points, then κ(X/G) = κ(X) = 0; thus the interesting case is when G has fixed points. We answer the above questions in terms of the action of the stabilizer subgroups near the fixed points. We give a rough classification of possible stabilizer groups which cause X/G to have Kodaira dimension −∞ or equivalently (as we show) to be uniruled. These stabilizers are closely related to unitary reflection groups.

Original languageEnglish (US)
Title of host publicationProgress in Mathematics
PublisherSpringer Basel
Pages179-211
Number of pages33
DOIs
StatePublished - Jan 1 2009

Publication series

NameProgress in Mathematics
Volume270
ISSN (Print)0743-1643
ISSN (Electronic)2296-505X

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

Keywords

  • Calabi–Yau
  • Rationally connected
  • Reflection group
  • Uniruled

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