### 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 language | English (US) |
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Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

Pages | 179-211 |

Number of pages | 33 |

DOIs | |

State | Published - Jan 1 2009 |

### Publication series

Name | Progress in Mathematics |
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Volume | 270 |

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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## Cite this

*Progress in Mathematics*(pp. 179-211). (Progress in Mathematics; Vol. 270). Springer Basel. https://doi.org/10.1007/978-0-8176-4747-6_6