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

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.

Autoren/Hrsg.

Tschinkel, Yuri

Zarhin, Yuri

Inhaltsverzeichnis

1;Preface;8
2;Contents
;10
3;Contents of Volume I;12
4;Potential Automorphy of Odd-Dimensional Symmetric Powers of Elliptic Curves and Applications;14
4.1;1 Reciprocity for n-dimensional Galois representations;16
4.2;2 Potential modularity of a Galois representation;21
4.3;3. A lemma about certain residual representations;25
4.4;4. Removing t;27
4.5;5. Applications and generalizations;30
4.6;6. Concluding remarks;33
4.7;References;33
5;Cyclic Homology with Coefficients;35
5.1;1 Recollection on cyclic homology.;37
5.2;2 Cyclic bimodules.;40
5.3;3 Gauss–Manin connection.;45
5.4;4 Categorical Approach.;48
5.5;5 Discussion;56
5.6;References;58
6;Noncommutative Geometry and Path Integrals;60
6.1;1 Noncommutative Monomials and Lattice Paths;62
6.2;2 Noncommutative exponential functions.;65
6.3;3 Generalities on the Noncommutative Fourier Transform;75
6.4;4 Noncommutative Gaussian and the Wiener Measure;79
6.5;5 Futher Examples of NCFT;87
6.6;6 Fourier Transform of Noncommutative Measures;88
6.7;7 Toward the Inverse Noncommutative Fourier Transform;93
7;Another Look at the Dwork Family;99
7.1;1 Introduction and a bit of history;99
7.2;2 The situation to be studied: generalities;101
7.3;3 The particular situation to be Studied: details;103
7.4;4 Interlude: Hypergeometric sheaves;107
7.5;5 Statement of the main theorem;110
7.6;6 Proof of the main theorem: the strategy;111
7.7;7 Proof of Theorem 6.1;114
7.8;8 Appendix I: The transcendental approach;121
7.9;9 Appendix II: The situation in characteristic p, when p divides some wi;129
7.10;10 Appendix III: Interesting pieces in the original Dwork family;132
7.11;References;134
8;Graphs, Strings, and Actions;137
8.1;1 Graphs, Spaces of Graphs, and Cell Models;140
8.1.1;1.1 Classes of Graphs;140
8.1.1.1;Graphs;140
8.1.1.2;Ribbon Graphs;140
8.1.1.3;The genus of a ribbon graph and its surface;141
8.1.1.4;Treelike, normalized Marked ribbon graphs;141
8.1.1.5;The intersection tree of an almost treelike ribbon graph;142
8.1.1.6;Dual b/w tree of a Marked ribbon graph;142
8.1.1.7;Spineless marked ribbon graphs;142
8.1.2;1.2 Operations on graphs;143
8.1.2.1;Contracting Edges;143
8.1.3;1.3 Spaces of Graphs with Metrics;144
8.1.3.1;Graphs with a Metric;144
8.1.3.2;Projective Metrics;144
8.1.3.3;The Space of Metric Ribbon Graphs;145
8.1.3.4;Cacti and Spineless Cacti and Thickened Cacti;145
8.1.3.5;Marked Ribbon Graphs with Metric and Maps of Circles;145
8.1.3.6;Cactus Terminology;146
8.1.3.7;Normalized Treelike and Almost Treelike Ribbon Graphs and Their Cell Complexes;146
8.1.3.8;Details of the Bicrossed Product Structure for Cacti;148
8.2;2 The Tree Level: Cell Models for (Framed) Little Discs and Their Operations;149
8.2.1;2.1 A First Cell Model for the Little Discs: Cact1;149
8.2.2;2.2 A CW Decomposition for Cacti1 and a Cellular Chain Model for the Framed Little Discs;150
8.2.3;2.3 The GBV Structure;153
8.2.4;2.4 Cells for the Araki–Kudo–Cohen, Dyer–Lashof Operations;155
8.2.5;2.5 A Smooth Cellular Model for the Framed Little Discs: Cacti;156
8.2.5.1;The Relevant Trees;156
8.2.6;2.6 The KW Cell Model for the Little Discs;159
8.2.6.1;Trees;159
8.2.6.2;The Minimal A Complex;160
8.2.7;2.7 A Finer Cell Model, the Generalized Boardman–Vogt Decomposition;160
8.2.7.1;Decomposing the Stasheff Polytope;160
8.2.7.2;Decomposing the Cyclohedra;161
8.2.7.3;Trees and Their Cell Complex;161
8.2.7.4;The Homotopy from KS to Cact1;162
8.2.7.5;The Cell Level: Maps and i;162
8.2.7.6;The Versions for the Framed Little Discs;163
8.3;3 Operations of the Cell Models on Hochschild Complexes;164
8.3.1;3.1 The Cyclic Deligne Conjecture;164
8.3.1.1;Assumption;164
8.3.1.2;Notation;164
8.3.1.3;Assumption;165
8.3.1.4;Correlators from Decorated Trees;165
8.3.1.5;The Foliage Operator;166
8.3.1.6;Signs;166
8.3.1.7;Examples;167
8.3.2;3.2 The Araki–Kudo–Cohen, Dyer–Lashof Operations on the Hochschild Complex;169
8.3.3;3.3 The A-Deligne Conjecture;169
8.3.4;3.4 The Cyclic A Case;170
8.4;4 The Moduli Space vs. the Sullivan PROP;170
8.4.1;4.1 Ribbon Graphs and Arc Graphs;170
8.4.1.1;A Short Introduction to the Arc Operad;170
8.4.2;4.2 Spaces of Graphs on Surfaces;170
8.4.2.1;Embedded Graphs;171
8.4.2.2;A Linear Order on Arcs;171
8.4.2.3;The Poset Structure;172
8.4.2.4;CW Structure of Ag,rs;172
8.4.2.5;Open-Cell Cell Complex;173
8.4.2.6;Relative Cells;173
8.4.2.7;Elements of the Ag,rs as Projectively Weighted Graphs;173
8.4.3;4.3 Topological Operad Structure;175
8.4.3.1;The Spaces Arc(n);175
8.4.3.2;Topological Description of the Glueing ;175
8.4.3.3;The Dual Graph;176
8.4.4;4.4 DArc;176
8.4.4.1;The Relation to Moduli Space;176
8.4.5;4.5 Cells;176
8.4.6;4.6 Digraphs and Sullivan Chord Diagrams;177
8.4.6.1;Ribbon Digraphs;177
8.4.6.2;Sullivan Chord and Ribbon Diagrams;177
8.4.7;4.7 Graph Actions, Feynman Rules, and Correlation Functions;178
8.4.7.1;Operadic Correlation Functions;178
8.4.8;4.8 Operadic Correlation Functions with Values in a Twisted Hom Operad;179
8.4.8.1;Signs;179
8.4.9;4.9 Arc Correlation Functions;179
8.4.9.1;Correlation Functions on the Tensor Algebra of an Algebra;180
8.4.9.2;Correlators for the Hochschild Cochains of a Frobenius Algebra ;181
8.4.9.3;The Sullivan–Chord Diagram Case;183
8.4.10;4.10 Correlators for A;183
8.4.11;4.11 Application to String Topology;184
8.5;5 Stabilization and Outlook;184
8.6;References;185
9;Quotients of Calabi–Yau Varieties;189
9.1;1 Uniruled quotients;193
9.2;2 Maps of Calabi–Yau Varieties;196
9.3;3 Basic Non-Reid–Tai Pairs;204
9.4;4 Quotients of Abelian Varieties;210
9.5;5 Examples;217
9.6;References;219
10;Notes on Motives in Finite Characteristic;222
10.1;0.1 An explicit example;224
10.2;1 First proposal: algebraic dynamics;226
10.2.1;1.1 The case of GL(1);226
10.2.2;1.2 Moduli of local systems on surfaces;227
10.2.2.1;Example: SL(2)-local systems on the sphere with three punctures;228
10.2.3;1.3 Equivariant bundles and Ruelle-type zeta functions;229
10.2.3.1;Reminder: Trace formula and Ruelle-type zeta function;230
10.2.3.2;Rationality conjecture for motivic local systems;231
10.3;2 Second proposal: formalism of motivic function spaces and higher-dimensional Langlands correspondence;233
10.3.1;2.1 Motivic functions and the tensor category Ck;233
10.3.1.1;Fiber functors for finite fields;234
10.3.1.2;Extensions and variants;234
10.3.1.3;Example: motivic Radon transform;235
10.3.2;2.2 Commutative algebras in Ck;236
10.3.2.1;Elementary examples of algebras;236
10.3.2.2;Categorification;237
10.3.3;2.3 Algebras parameterizing motivic local systems;237
10.3.3.1;Preparations on ramification and motivic local systems;237
10.3.3.2;Conjecture on algebras parameterizing motivic local systems;238
10.3.3.3;Arguments in favor, and extensions;239
10.3.4;2.4 Toward integrable systems over local fields;240
10.4;3 Third proposal: lattice models;242
10.4.1;3.1 Traces depending on two indices;242
10.4.2;3.2 Two-dimensional translation invariant lattice models;243
10.4.2.1;Transfer matrices;244
10.4.3;3.3 Two-dimensional Weil conjecture;245
10.4.4;3.4 Higher-dimensional lattice models and a higher-dimensional Weil conjecture;246
10.4.4.1;Evidence: p-adic Banach lattice models;247
10.4.5;3.5 Tensor category A and the Master Conjecture;248
10.4.5.1;Machine modelling finite fields;251
10.4.6;3.6 Corollaries of the Master Conjecture;251
10.4.6.1;Good sign: Bombieri–Dwork bound;251
10.4.6.2;Bad sign: cohomology theories for motives over finite fields;252
10.5;4 Categorical afterthoughts;252
10.5.1;4.1 Decategorifications of 2-categories;252
10.5.1.1;Noncommutative stable homotopy theory;253
10.5.1.2;Elementary algebraic model of bivariant K-theory;253
10.5.1.3;Noncommutative pure and mixed motives;253
10.5.1.4;Motivic integral operators;254
10.5.1.5;Correspondences for free algebras;254
10.5.2;4.2 Trace of an exchange morphism;254
10.6;References;255
11;PROPped-Up Graph Cohomology;257
11.1;1.1 PROPs, Dioperads, and 12PROPs;259
11.2;1.2 Free PROPs;263
11.3;1.3 From 12PROPs to PROPs;264
11.4;1.4 Quadratic Duality and Koszulness for 12PROPs;267
11.5;1.5 Perturbation Techniques for Graph Cohomology;273
11.6;1.6 Minimal Models of PROPs;279
11.7;1.7 Classical Graph Cohomology;285
11.8;References;288
12;Symboles de Manin et valeurs de fonctions L;290
12.1;1 Introduction;290
12.1.1;1.1 Les symboles de Manin;290
12.1.2;1.2 Analyse de Fourier multiplicative;291
12.1.3;1.3 Interprétation arithmétique;293
12.1.4;1.4 Perspectives;294
12.2;2 Formulaire préliminaire;295
12.2.1;2.1 Suppression des facteurs d'Euler;295
12.2.2;2.2 Opérateurs d'Atkin–Lehner;296
12.2.3;2.3 Torsion des formes modulaires par des caractères quelconques;297
12.2.4;2.4 La torsion des formes modulaires par des caractères de niveaux divisant N;298
12.2.5;2.5 La torsion des formes modulaires par des caractères additifs;299
12.2.6;2.6 Invariants locaux des tordues de formes modulaires, première analyse;299
12.2.7;2.7 Invariants locaux des tordues de formes modulaires, cas de série principale;300
12.2.8;2.8 Invariants locaux des tordues de formes modulaires, cas supercuspidal;301
12.2.9;2.9 Invariants locaux des tordues de formes primitives par torsion, conclusion;301
12.3;3 Le théorème 1 et ses corollaires;301
12.3.1;3.1 La démonstration du théorème 1;301
12.3.2;3.2 Réciproque du corollaire 2 et observations algorithmiques sur les aspects locaux;307
12.3.2.1;a. Les invariants de f en termes de la fonction f;307
12.3.2.2;b. Torsion de f par des caractères tels que N=N;307
12.3.2.3;c. Les invariants locaux des tordues de f par des caractères tels que N=N;308
12.3.2.4;d. Les invariants locaux des tordues de f pour caractère quelconque;308
12.3.2.5;e. Les nombres (f,1) pour caractère de niveau divisant N;308
12.3.2.6;f. Que faire lorsque f n'est pas primitive par torsion ?;308
12.3.3;3.3 Équations fonctionnelles et relations de Manin;309
12.4;4 Produit de formes modulaires;310
12.4.1;4.1 Le produit scalaire de Petersson;310
12.4.2;4.2 La fonction L du carré tensoriel;313
12.5;Références;315
13;Graph Complexes with Loops and Wheels;317
13.1;1 Introduction;317
13.2;2 Dg Props Versus Sheaves of dg Lie Algebras;319
13.3;3 Directed Graph Complexes with Loops and Wheels;335
13.4;4 Examples;348
13.5;5 Wheeled Cyclic Complex;357
13.6;References;359
14;Yang–Mills Theory and a Superquadric;361
14.1;1 Introduction;361
14.2;2 Infinitesimal constructions;365
14.2.1;2.1 Real structure on the Lie algebra gl(4|3);366
14.2.2;2.2 Symmetries of the ambitwistor space;368
14.3;3 Reduced theory;371
14.3.1;3.1 The manifold F;371
14.3.2;3.2 Properties of the manifold F;372
14.4;4 Nonreduced theory;378
14.4.1;4.1 Construction of the algebra A(Z);378
14.4.2;4.2 Proof of the equivalence;379
14.4.3;4.3 Relation between a CR structure on Z and an algebra A(Z);381
14.5;5 Appendix;381
14.5.1;5.1 On the definition of a graded real superspace;381
14.5.2;5.2 On homogeneous CR-structures;383
14.5.3;5.3 General facts about CR structures on supermanifolds;384
14.6;6 Acknowledgments;387
14.7;References;387
15;A Generalization of the Capelli Identity;389
15.1;1 Introduction;389
15.2;2 Identities;392
15.2.1;2.1 The main identity;392
15.2.2;2.2 A presentation as a row determinant of size M+N;393
15.2.3;2.3 A Relation Between Determinants of Sizes M and N;394
15.2.4;2.4 A relation to the Capelli identity;395
15.2.5;2.5 Proof of Theorem 1;396
15.3;3 The (glM,glN) Duality and the Bethe Subalgebras;398
15.3.1;3.1 Bethe subalgebra;398
15.3.2;3.2 The (glM,glN) Duality;400
15.3.3;3.3 Scalar Differential Operators;402
15.3.4;3.4 The Simple Joint Spectrum of the Bethe Subalgebra;403
15.4;References;403
16;Hidden Symmetries in the Theory of Complex Multiplication;405
16.1;0 Introduction;405
16.1.1;0.1 ;405
16.1.2;0.2 ;405
16.1.3;0.3 ;406
16.1.4;0.5 ;407
16.1.5;0.7 Idle speculation;407
16.1.6;0.8 ;408
16.1.7;0.10 ;409
16.1.8;0.12 ;410
16.2;1 Background material;410
16.2.1;1.1 Wreath products and Galois theory;411
16.2.1.1;1.1.1 Notation;411
16.2.1.2;1.1.2 Basic construction;411
16.2.1.3;1.1.5 ;414
16.2.2;1.2 Class Field Theory;416
16.2.2.1;1.2.1 ;416
16.2.2.2;1.2.2 ;416
16.2.2.3;1.2.3 ;417
16.2.2.4;1.2.4 ;417
16.2.3;1.3 CM fields;417
16.2.3.1;1.3.1 Complex conjugations;418
16.2.3.2;1.3.2 Transfer maps;418
16.2.3.3;1.3.3 ;419
16.2.4;1.4 Tate's construction;421
16.2.4.1;1.4.1 Tate's half-transfer;421
16.2.4.2;1.4.2 The Taniyama element;422
16.2.4.3;1.4.3 ;422
16.2.5;1.5 The Serre torus;423
16.2.5.1;1.5.1 ;423
16.2.5.2;1.5.2 ;423
16.2.5.3;1.5.3 ;424
16.2.5.4;1.5.4 ;424
16.2.6;1.6 Universal Taniyama elements , ;425
16.2.6.1;1.6.1 ;425
16.2.6.2;1.6.4 ;426
16.2.7;1.7 The Taniyama group , , ;426
16.2.7.1;1.7.1 ;427
16.2.7.2;1.7.2 ;427
16.2.7.3;1.7.3 ;427
16.2.7.4;1.7.4 ;428
16.2.7.5;1.7.6 ;429
16.3;2 Hidden symmetries in the CM theory;429
16.3.1;2.1 Generalised half-transfer;429
16.3.1.1;2.1.1 ;429
16.3.1.2;2.1.2 Rewriting Tate's Half-Transfer in Terms of s;430
16.3.1.3;2.1.5 Change of s;431
16.3.1.4;2.1.8 Galois functoriality of F"0365F;433
16.3.2;2.2 Generalised Taniyama elements;434
16.3.2.1;2.2.1 ;434
16.3.2.2;2.2.5 Action of AutF-alg(FQ)0 on CM points of Hilbert modular varieties;437
16.3.3;2.3 Generalised universal Taniyama elements;439
16.3.4;2.4 Generalised Taniyama group;440
16.3.4.1;2.4.1 ;440
16.3.4.2;2.4.2 ;441
16.3.4.3;2.4.3 ;441
16.3.4.4;2.4.4 ;442
16.3.4.5;2.4.5 ;442
16.4;References;442
17;Self-Correspondences of K3 Surfaces via Moduli of Sheaves;444
17.1;1 Introduction;444
17.1.1;1.1 Preliminary Notation for Lattices;447
17.2;2 Isomorphisms Between MX (v) and X for a General K3 Surface X and a Primitive Isotropic Mukai Vector v;448
17.3;3 Isomorphisms Between MX (v) and X for X a General K3 Surface with (X)=2;453
17.4;4 Isomorphisms Between MX (v) and X for a General K3 Surface X with (X)3;460
17.5;5 Composing Self-Correspondences of a K3 Surface via Moduli of Sheaves and the General Classification Problem;464
17.5.1;5.1 General Problem of Classifying Self-Correspondences of a K3 Surface via Moduli of Sheaves;466
17.6;References;468
18;Foliations in Moduli Spaces of Abelian Varieties and Dimension of Leaves;470
18.1;1 Notations;472
18.2;2 Computation of the dimension of automorphism schemes;478
18.3;3 Serre-Tate coordinates;481
18.4;4 The dimension of central leaves, the unpolarized case;483
18.5;5 The dimension of central leaves, the polarized case;487
18.6;6 The dimension of Newton polygon strata;493
18.7;7 Some results used in the proofs;496
18.8;8 Some questions and some remarks;502
18.9;References;504
19;Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities;507
19.1;1 Triangulated Categories of Singularities for Graded Algebras;509
19.1.1;1.1 Localization in Triangulated Categories and Semiorthogonal Decomposition;509
19.1.2;1.2 Triangulated Categories of Singularities for Algebras;511
19.1.3;1.3 Morphisms in Categories of Singularities;513
19.2;2 Categories of Coherent Sheaves and Categories of Singularities;515
19.2.1;2.1 Quotient Categories of Graded Modules;515
19.2.2;2.2 Triangulated Categories of Singularities for Gorenstein Algebras;516
19.2.3;2.3 Categories of Coherent Sheaves for Gorenstein Schemes;522
19.3;3 Categories of Graded D-branes of Type B in Landau–Ginzburg Models;526
19.3.1;3.1 Categories of Graded Pairs;526
19.3.2;3.2 Categories of Graded Pairs and Categories of Singularities;528
19.3.3;3.3 Graded D-branes of Type B and Coherent Sheaves;531
19.4;References;533
20;Rankin's Lemma of Higher Genus and Explicit Formulas for Hecke Operators;536
20.1;1 Introduction: Generating Series for the Hecke Operators;536
20.2;2 Results;538
20.2.1;2.1 Preparation: A Formula for the Total Hecke Operator T(p) of Genus 2;538
20.2.2;2.2 Rankin's Generating Series in Genus 2;539
20.2.3;2.3 Symmetric Square Generating Series in Genus 2;541
20.2.4;2.4 Cubic Generating Series in Genus 2;542
20.3;3 Proofs: Formulas for the Hecke Operators of Spg;542
20.3.1;3.1 Satake's Spherical Map ;542
20.3.2;3.2 Use of Andrianov's Generating Series in Genus 2;543
20.3.3;3.3 Rankin's Lemma of Genus 2 (Compare with [Jia96]);543
20.4;4 Relations with L-Functions and Motives for Spn ;545
20.5;5 A Holomorphic Lifting from GSp2 GSp2 to GSp4: A Conjecture;547
20.6;References;555
21;Rank-2 Vector Bundles on ind-Grassmannians;558
21.1;1 Introduction;558
21.2;2 Notation and Conventions;559
21.3;3 The Linear Case;561
21.4;4 Auxiliary Results;564
21.5;5 The Case rkE=2;565
21.6;References;575
22;Massey Products on Cycles of Projective Lines and Trigonometric Solutions of the Yang–Baxter Equations;576
22.1;1 The AYBE and the QYBE;581
22.2;2 Solutions of the AYBE Associated with Simple Vector Bundles on Degenerations of Elliptic Curves;590
22.3;3 Simple Vector Bundles on Cycles of Projective Lines;592
22.4;4 Computation of the Associative r-Matrix Arising as a Massey Product;594
22.5;5 Associative Belavin–Drinfeld Triples Associated with Simple Vector Bundles;598
22.6;6 Solutions of the AYBE and Associative BD-Structures;602
22.7;7 Meromorphic Continuation;610
22.8;8 Classification of Trigonometric Solutions of the AYBE;615
22.9;References;620
23;On Linnik and Selberg's Conjecture About Sums of Kloosterman Sums;621
23.1;1 Statements;621
23.2;2 Proofs;625
23.3;References;636
24;Une Algèbre Quadratique Liée à la Suite de Sturm;638
24.1;§ 1 Introduction;638
24.2;§ 2 Algèbre B;642
24.3;§ 3 Début de la démonstration du théorème 1.5;646
24.4;§ 4 Formule (A);648
24.5;§ 5 Formule (B);649
24.6;§ 1 Nombres (j)i;653
24.7;§ 2 Polynômes d'Euler et fonction hypergéométrique;655
24.8;§ 3 Asymptotiques;657
24.9;Bibliography;660
25;Fields of u-Invariant 2r+1;661
25.1;1 Introduction;661
25.2;2 Elementary Discrete Invariant;662
25.3;3 Generic Points of Quadrics and Chow Groups;669
25.3.1;3.1 Algebraic Cobordisms;670
25.3.2;3.2 Beyond Theorem 3.1;671
25.3.3;3.3 Some Auxiliary Facts;676
25.4;4 Even u-invariants;678
25.5;5 Odd u-invariants;679
25.6;References;684
26;Cubic Surfaces and Cubic Threefolds, Jacobians and Intermediate Jacobians;686
26.1;1 Principally Polarized Abelian Varieties That Admit an Automorphism of Order 3;686
26.2;2 Cubic Threefolds;689
26.3;3 Intermediate Jacobians;690
26.4;References;690
27;De Jong-Oort Purity for p-Divisible Groups;691
27.1;1 Introduction;691
27.2;2 Frobenius Modules;692
27.3;3 Proof of Purity;696
27.4;References;699

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"