Tschinkel / Zarhin Algebra, Arithmetic, and Geometry

Volume II: In Honor of Yu. I. Manin
2009
ISBN: 978-0-8176-4747-6
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 270, 704 Seiten, eBook

Reihe: Progress in Mathematics

Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin consists of
invited expository and research articles on new developments arising from
Manin’s outstanding contributions to mathematics.

Zielgruppe

Research

Autoren/Hrsg.

Tschinkel, Yuri

Zarhin, Yuri

Inhaltsverzeichnis

Potential Automorphy of Odd-Dimensional Symmetric Powers of Elliptic Curves and Applications.- Cyclic Homology with Coefficients.- Noncommutative Geometry and Path Integrals.- Another Look at the Dwork Family.- Graphs, Strings, and Actions.- Quotients of Calabi–Yau Varieties.- Notes on Motives in Finite Characteristic.- PROPped-Up Graph Cohomology.- Symboles de Manin et valeurs de fonctions L.- Graph Complexes with Loops and Wheels.- Yang–Mills Theory and a Superquadric.- A Generalization of the Capelli Identity.- Hidden Symmetries in the Theory of Complex Multiplication.- Self-Correspondences of K3 Surfaces via Moduli of Sheaves.- Foliations in Moduli Spaces of Abelian Varieties and Dimension of Leaves.- Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities.- Rankin’s Lemma of Higher Genus and Explicit Formulas for Hecke Operators.- Rank-2 Vector Bundles on ind-Grassmannians.- Massey Products on Cycles of Projective Lines and Trigonometric Solutions of the Yang–Baxter Equations.- On Linnik and Selberg’s Conjecture About Sums of Kloosterman Sums.- Une Algèbre Quadratique Liée à la Suite de Sturm.- Fields of u-Invariant 2r + 1.- Cubic Surfaces and Cubic Threefolds, Jacobians and Intermediate Jacobians.- De Jong-Oort Purity for p-Divisible Groups.

Leseproben

"Quotients of Calabi–Yau Varieties(p. 179-180)

J´anos Koll´ar and Michael Larsen

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 ?nite 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 classi?cation 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 ?xed points, then ?(X/G) = ?(X) = 0; thus the interesting case is when G has ?xed points. We answer the above questions in terms of the action of the stabilizer subgroups near the ?xed points. We give a rough classi?cation of possible stabilizer groups which cause X/G to have Kodaira dimension -8 or equivalently (as we show) to be uniruled. These stabilizers are closely related to unitary re?ection groups.

Key words: Calabi–Yau, uniruled, rationally connected, re?ection group

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

Let X be a Calabi–Yau variety over C, that is, a projective variety with canonical singularities whose canonical class is numericaly trivial. Let G be a ?nite 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 classi?cation 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 ?xed points, then ?(X/G) = ?(X) = 0; thus the interesting case is that in which G has ?xed points. We answer the above questions in terms of the action of the stabilizer subgroups near the ?xed points. The answer is especially nice if X is smooth. In the introduction we concentrate on this case. The precise general results are formulated later. Definition 1. Let V be a complex vector space and g ? GL(V ) an element of ?nite order. Its eigenvalues (with multiplicity) can be written as"