Quotients of Calabi-Yau varieties

János Kollár; Michael Larsen

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Quotients of Calabi-Yau varieties

János Kollár·Michael Larsen

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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 $-\infty$ or equivalently (as we show) to be uniruled. These stabilizers are closely related to unitary reflection groups.

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János Kollár, Michael Larsen. 2007-01-21. Quotients of Calabi-Yau varieties. https://arxiv.org/abs/math/0701466

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