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Quotients of Calabi–Yau Varieties

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Algebra, Arithmetic, and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 270))

Summary

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.

2000 Mathematics Subject Classifications: 14J32, 14K05, 20E99 (Primary) 14M20, 14E05, 20F55 (Secondary)

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Author information

Authors and Affiliations

  1. Department of Mathematics, Princeton University, Fine Hall, Washington Road Princeton, NJ, 08544-1000, U.S.A.

    János Kollár

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, U.S.A.

    Michael Larsen

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  1. János Kollár
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  2. Michael Larsen
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Correspondence to János Kollár .

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Editors and Affiliations

  1. Courant Inst. Mathematical, New York University, Mercer St. 251, New York, 10012, USA

    Yuri Tschinkel

  2. Eberly College of Science, Pennsylvania State University, University Park, 16802, USA

    Yuri Zarhin

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Kollár, J., Larsen, M. (2009). Quotients of Calabi–Yau Varieties. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 270. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4747-6_6

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