Tschinkel / Zarhin Algebra, Arithmetic, and Geometry

Volume II: In Honor of Yu. I. Manin
1. Auflage 2010
ISBN: 978-0-8176-4747-6
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 704 Seiten, Web PDF

Reihe: Mathematics and Statistics (R0)

The two volumes of are composed of invited expository articles and extensions detailing Manin’s contributions to the subjects, and are in celebration of his 70 birthday.

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"