Quotients of Calabi–Yau varieties

János Kollár, Michael Larsen

Research output: Chapter in Book/Report/Conference proceedingChapter

31 Scopus citations


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
Number of pages33
StatePublished - 2009

Publication series

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

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


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


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