Algebra, Arithmetic, and Geometry (eBook)

Preface 8
Contents 
10 
Contents of Volume I 12
Potential Automorphy of Odd-Dimensional Symmetric Powers of Elliptic Curves and Applications 14
1 Reciprocity for n-dimensional Galois representations 16
2 Potential modularity of a Galois representation 21
3. A lemma about certain residual representations 25
4. Removing t 27
5. Applications and generalizations 30
6. Concluding remarks 33
References 33
Cyclic Homology with Coefficients 35
1 Recollection on cyclic homology. 37
2 Cyclic bimodules. 40
3 Gauss–Manin connection. 45
4 Categorical Approach. 48
5 Discussion 56
References 58
Noncommutative Geometry and Path Integrals 60
1 Noncommutative Monomials and Lattice Paths 62
2 Noncommutative exponential functions. 65
3 Generalities on the Noncommutative Fourier Transform 75
4 Noncommutative Gaussian and the Wiener Measure 79
5 Futher Examples of NCFT 87
6 Fourier Transform of Noncommutative Measures 88
7 Toward the Inverse Noncommutative Fourier Transform 93
Another Look at the Dwork Family 99
1 Introduction and a bit of history 99
2 The situation to be studied: generalities 101
3 The particular situation to be Studied: details 103
4 Interlude: Hypergeometric sheaves 107
5 Statement of the main theorem 110
6 Proof of the main theorem: the strategy 111
7 Proof of Theorem 6.1 114
8 Appendix I: The transcendental approach 121
9 Appendix II: The situation in characteristic p, when p divides some wi 129
10 Appendix III: Interesting pieces in the original Dwork family 132
References 134
Graphs, Strings, and Actions 137
1 Graphs, Spaces of Graphs, and Cell Models 140
1.1 Classes of Graphs 140
Graphs 140
Ribbon Graphs 140
The genus of a ribbon graph and its surface 141
Treelike, normalized Marked ribbon graphs 141
The intersection tree of an almost treelike ribbon graph 142
Dual b/w tree of a Marked ribbon graph 142
Spineless marked ribbon graphs 142
1.2 Operations on graphs 143
Contracting Edges 143
1.3 Spaces of Graphs with Metrics 144
Graphs with a Metric 144
Projective Metrics 144
The Space of Metric Ribbon Graphs 145
Cacti and Spineless Cacti and Thickened Cacti 145
Marked Ribbon Graphs with Metric and Maps of Circles 145
Cactus Terminology 146
Normalized Treelike and Almost Treelike Ribbon Graphs and Their Cell Complexes 146
Details of the Bicrossed Product Structure for Cacti 148
2 The Tree Level: Cell Models for (Framed) Little Discs and Their Operations 149
2.1 A First Cell Model for the Little Discs: Cact1 149
2.2 A CW Decomposition for Cacti1 and a Cellular Chain Model for the Framed Little Discs 150
2.3 The GBV Structure 153
2.4 Cells for the Araki–Kudo–Cohen, Dyer–Lashof Operations 155
2.5 A Smooth Cellular Model for the Framed Little Discs: Cacti 156
The Relevant Trees 156
2.6 The KW Cell Model for the Little Discs 159
Trees 159
The Minimal A Complex 160
2.7 A Finer Cell Model, the Generalized Boardman–Vogt Decomposition 160
Decomposing the Stasheff Polytope 160
Decomposing the Cyclohedra 161
Trees and Their Cell Complex 161
The Homotopy from KS to Cact1 162
The Cell Level: Maps and i 162
The Versions for the Framed Little Discs 163
3 Operations of the Cell Models on Hochschild Complexes 164
3.1 The Cyclic Deligne Conjecture 164
Assumption 164
Notation 164
Assumption 165
Correlators from Decorated Trees 165
The Foliage Operator 166
Signs 166
Examples 167
3.2 The Araki–Kudo–Cohen, Dyer–Lashof Operations on the Hochschild Complex 169
3.3 The A-Deligne Conjecture 169
3.4 The Cyclic A Case 170
4 The Moduli Space vs. the Sullivan PROP 170
4.1 Ribbon Graphs and Arc Graphs 170
A Short Introduction to the Arc Operad 170
4.2 Spaces of Graphs on Surfaces 170
Embedded Graphs 171
A Linear Order on Arcs 171
The Poset Structure 172
CW Structure of Ag,rs 172
Open-Cell Cell Complex 173
Relative Cells 173
Elements of the Ag,rs as Projectively Weighted Graphs 173
4.3 Topological Operad Structure 175
The Spaces Arc(n) 175
Topological Description of the Glueing 175
The Dual Graph 176
4.4 DArc 176
The Relation to Moduli Space 176
4.5 Cells 176
4.6 Digraphs and Sullivan Chord Diagrams 177
Ribbon Digraphs 177
Sullivan Chord and Ribbon Diagrams 177
4.7 Graph Actions, Feynman Rules, and Correlation Functions 178
Operadic Correlation Functions 178
4.8 Operadic Correlation Functions with Values in a Twisted Hom Operad 179
Signs 179
4.9 Arc Correlation Functions 179
Correlation Functions on the Tensor Algebra of an Algebra 180
Correlators for the Hochschild Cochains of a Frobenius Algebra 181
The Sullivan–Chord Diagram Case 183
4.10 Correlators for A 183
4.11 Application to String Topology 184
5 Stabilization and Outlook 184
References 185
Quotients of Calabi–Yau Varieties 189
1 Uniruled quotients 193
2 Maps of Calabi–Yau Varieties 196
3 Basic Non-Reid–Tai Pairs 204
4 Quotients of Abelian Varieties 210
5 Examples 217
References 219
Notes on Motives in Finite Characteristic 222
0.1 An explicit example 224
1 First proposal: algebraic dynamics 226
1.1 The case of GL(1) 226
1.2 Moduli of local systems on surfaces 227
Example: SL(2)-local systems on the sphere with three punctures 228
1.3 Equivariant bundles and Ruelle-type zeta functions 229
Reminder: Trace formula and Ruelle-type zeta function 230
Rationality conjecture for motivic local systems 231
2 Second proposal: formalism of motivic function spaces and higher-dimensional Langlands correspondence 233
2.1 Motivic functions and the tensor category Ck 233
Fiber functors for finite fields 234
Extensions and variants 234
Example: motivic Radon transform 235
2.2 Commutative algebras in Ck 236
Elementary examples of algebras 236
Categorification 237
2.3 Algebras parameterizing motivic local systems 237
Preparations on ramification and motivic local systems 237
Conjecture on algebras parameterizing motivic local systems 238
Arguments in favor, and extensions 239
2.4 Toward integrable systems over local fields 240
3 Third proposal: lattice models 242
3.1 Traces depending on two indices 242
3.2 Two-dimensional translation invariant lattice models 243
Transfer matrices 244
3.3 Two-dimensional Weil conjecture 245
3.4 Higher-dimensional lattice models and a higher-dimensional Weil conjecture 246
Evidence: p-adic Banach lattice models 247
3.5 Tensor category A and the Master Conjecture 248
Machine modelling finite fields 251
3.6 Corollaries of the Master Conjecture 251
Good sign: Bombieri–Dwork bound 251
Bad sign: cohomology theories for motives over finite fields 252
4 Categorical afterthoughts 252
4.1 Decategorifications of 2-categories 252
Noncommutative stable homotopy theory 253
Elementary algebraic model of bivariant K-theory 253
Noncommutative pure and mixed motives 253
Motivic integral operators 254
Correspondences for free algebras 254
4.2 Trace of an exchange morphism 254
References 255
PROPped-Up Graph Cohomology 257
1.1 PROPs, Dioperads, and 12PROPs 259
1.2 Free PROPs 263
1.3 From 12PROPs to PROPs 264
1.4 Quadratic Duality and Koszulness for 12PROPs 267
1.5 Perturbation Techniques for Graph Cohomology 273
1.6 Minimal Models of PROPs 279
1.7 Classical Graph Cohomology 285
References 288
Symboles de Manin et valeurs de fonctions L 290
1 Introduction 290
1.1 Les symboles de Manin 290
1.2 Analyse de Fourier multiplicative 291
1.3 Interprétation arithmétique 293
1.4 Perspectives 294
2 Formulaire préliminaire 295
2.1 Suppression des facteurs d'Euler 295
2.2 Opérateurs d'Atkin–Lehner 296
2.3 Torsion des formes modulaires par des caractères quelconques 297
2.4 La torsion des formes modulaires par des caractères de niveaux divisant N 298
2.5 La torsion des formes modulaires par des caractères additifs 299
2.6 Invariants locaux des tordues de formes modulaires, première analyse 299
2.7 Invariants locaux des tordues de formes modulaires, cas de série principale 300
2.8 Invariants locaux des tordues de formes modulaires, cas supercuspidal 301
2.9 Invariants locaux des tordues de formes primitives par torsion, conclusion 301
3 Le théorème 1 et ses corollaires 301
3.1 La démonstration du théorème 1 301
3.2 Réciproque du corollaire 2 et observations algorithmiques sur les aspects locaux 307
a. Les invariants de f en termes de la fonction f 307
b. Torsion de f par des caractères tels que N=N 307
c. Les invariants locaux des tordues de f par des caractères tels que N=N 308
d. Les invariants locaux des tordues de f pour caractère quelconque 308
e. Les nombres (f,1) pour caractère de niveau divisant N 308
f. Que faire lorsque f n'est pas primitive par torsion ? 308
3.3 Équations fonctionnelles et relations de Manin 309
4 Produit de formes modulaires 310
4.1 Le produit scalaire de Petersson 310
4.2 La fonction L du carré tensoriel 313
Références 315
Graph Complexes with Loops and Wheels 317
1 Introduction 317
2 Dg Props Versus Sheaves of dg Lie Algebras 319
3 Directed Graph Complexes with Loops and Wheels 335
4 Examples 348
5 Wheeled Cyclic Complex 357
References 359
Yang–Mills Theory and a Superquadric 361
1 Introduction 361
2 Infinitesimal constructions 365
2.1 Real structure on the Lie algebra gl(4|3) 366
2.2 Symmetries of the ambitwistor space 368
3 Reduced theory 371
3.1 The manifold F 371
3.2 Properties of the manifold F 372
4 Nonreduced theory 378
4.1 Construction of the algebra A(Z) 378
4.2 Proof of the equivalence 379
4.3 Relation between a CR structure on Z and an algebra A(Z) 381
5 Appendix 381
5.1 On the definition of a graded real superspace 381
5.2 On homogeneous CR-structures 383
5.3 General facts about CR structures on supermanifolds 384
6 Acknowledgments 387
References 387
A Generalization of the Capelli Identity 389
1 Introduction 389
2 Identities 392
2.1 The main identity 392
2.2 A presentation as a row determinant of size M+N 393
2.3 A Relation Between Determinants of Sizes M and N 394
2.4 A relation to the Capelli identity 395
2.5 Proof of Theorem 1 396
3 The (glM,glN) Duality and the Bethe Subalgebras 398
3.1 Bethe subalgebra 398
3.2 The (glM,glN) Duality 400
3.3 Scalar Differential Operators 402
3.4 The Simple Joint Spectrum of the Bethe Subalgebra 403
References 403
Hidden Symmetries in the Theory of Complex Multiplication 405
0 Introduction 405
0.1 405
0.2 405
0.3 406
0.5 407
0.7 Idle speculation 407
0.8 408
0.10 409
0.12 410
1 Background material 410
1.1 Wreath products and Galois theory 411
1.1.1 Notation 411
1.1.2 Basic construction 411
1.1.5 414
1.2 Class Field Theory 416
1.2.1 416
1.2.2 416
1.2.3 417
1.2.4 417
1.3 CM fields 417
1.3.1 Complex conjugations 418
1.3.2 Transfer maps 418
1.3.3 419
1.4 Tate's construction 421
1.4.1 Tate's half-transfer 421
1.4.2 The Taniyama element 422
1.4.3 422
1.5 The Serre torus 423
1.5.1 423
1.5.2 423
1.5.3 424
1.5.4 424
1.6 Universal Taniyama elements , 425
1.6.1 425
1.6.4 426
1.7 The Taniyama group , , 426
1.7.1 427
1.7.2 427
1.7.3 427
1.7.4 428
1.7.6 429
2 Hidden symmetries in the CM theory 429
2.1 Generalised half-transfer 429
2.1.1 429
2.1.2 Rewriting Tate's Half-Transfer in Terms of s 430
2.1.5 Change of s 431
2.1.8 Galois functoriality of F"0365F 433
2.2 Generalised Taniyama elements 434
2.2.1 434
2.2.5 Action of AutF-alg(FQ)0 on CM points of Hilbert modular varieties 437
2.3 Generalised universal Taniyama elements 439
2.4 Generalised Taniyama group 440
2.4.1 440
2.4.2 441
2.4.3 441
2.4.4 442
2.4.5 442
References 442
Self-Correspondences of K3 Surfaces via Moduli of Sheaves 444
1 Introduction 444
1.1 Preliminary Notation for Lattices 447
2 Isomorphisms Between MX (v) and X for a General K3 Surface X and a Primitive Isotropic Mukai Vector v 448
3 Isomorphisms Between MX (v) and X for X a General K3 Surface with (X)=2 453
4 Isomorphisms Between MX (v) and X for a General K3 Surface X with (X)3 460
5 Composing Self-Correspondences of a K3 Surface via Moduli of Sheaves and the General Classification Problem 464
5.1 General Problem of Classifying Self-Correspondences of a K3 Surface via Moduli of Sheaves 466
References 468
Foliations in Moduli Spaces of Abelian Varieties and Dimension of Leaves 470
1 Notations 472
2 Computation of the dimension of automorphism schemes 478
3 Serre-Tate coordinates 481
4 The dimension of central leaves, the unpolarized case 483
5 The dimension of central leaves, the polarized case 487
6 The dimension of Newton polygon strata 493
7 Some results used in the proofs 496
8 Some questions and some remarks 502
References 504
Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities 507
1 Triangulated Categories of Singularities for Graded Algebras 509
1.1 Localization in Triangulated Categories and Semiorthogonal Decomposition 509
1.2 Triangulated Categories of Singularities for Algebras 511
1.3 Morphisms in Categories of Singularities 513
2 Categories of Coherent Sheaves and Categories of Singularities 515
2.1 Quotient Categories of Graded Modules 515
2.2 Triangulated Categories of Singularities for Gorenstein Algebras 516
2.3 Categories of Coherent Sheaves for Gorenstein Schemes 522
3 Categories of Graded D-branes of Type B in Landau–Ginzburg Models 526
3.1 Categories of Graded Pairs 526
3.2 Categories of Graded Pairs and Categories of Singularities 528
3.3 Graded D-branes of Type B and Coherent Sheaves 531
References 533
Rankin's Lemma of Higher Genus and Explicit Formulas for Hecke Operators 536
1 Introduction: Generating Series for the Hecke Operators 536
2 Results 538
2.1 Preparation: A Formula for the Total Hecke Operator T(p) of Genus 2 538
2.2 Rankin's Generating Series in Genus 2 539
2.3 Symmetric Square Generating Series in Genus 2 541
2.4 Cubic Generating Series in Genus 2 542
3 Proofs: Formulas for the Hecke Operators of Spg 542
3.1 Satake's Spherical Map 542
3.2 Use of Andrianov's Generating Series in Genus 2 543
3.3 Rankin's Lemma of Genus 2 (Compare with [Jia96]) 543
4 Relations with L-Functions and Motives for Spn 545
5 A Holomorphic Lifting from GSp2 GSp2 to GSp4: A Conjecture 547
References 555
Rank-2 Vector Bundles on ind-Grassmannians 558
1 Introduction 558
2 Notation and Conventions 559
3 The Linear Case 561
4 Auxiliary Results 564
5 The Case rkE=2 565
References 575
Massey Products on Cycles of Projective Lines and Trigonometric Solutions of the Yang–Baxter Equations 576
1 The AYBE and the QYBE 581
2 Solutions of the AYBE Associated with Simple Vector Bundles on Degenerations of Elliptic Curves 590
3 Simple Vector Bundles on Cycles of Projective Lines 592
4 Computation of the Associative r-Matrix Arising as a Massey Product 594
5 Associative Belavin–Drinfeld Triples Associated with Simple Vector Bundles 598
6 Solutions of the AYBE and Associative BD-Structures 602
7 Meromorphic Continuation 610
8 Classification of Trigonometric Solutions of the AYBE 615
References 620
On Linnik and Selberg's Conjecture About Sums of Kloosterman Sums 621
1 Statements 621
2 Proofs 625
References 636
Une Algèbre Quadratique Liée à la Suite de Sturm 638
§ 1 Introduction 638
§ 2 Algèbre B 642
§ 3 Début de la démonstration du théorème 1.5 646
§ 4 Formule (A) 648
§ 5 Formule (B) 649
§ 1 Nombres (j)i 653
§ 2 Polynômes d'Euler et fonction hypergéométrique 655
§ 3 Asymptotiques 657
Bibliography 660
Fields of u-Invariant 2r+1 661
1 Introduction 661
2 Elementary Discrete Invariant 662
3 Generic Points of Quadrics and Chow Groups 669
3.1 Algebraic Cobordisms 670
3.2 Beyond Theorem 3.1 671
3.3 Some Auxiliary Facts 676
4 Even u-invariants 678
5 Odd u-invariants 679
References 684
Cubic Surfaces and Cubic Threefolds, Jacobians and Intermediate Jacobians 686
1 Principally Polarized Abelian Varieties That Admit an Automorphism of Order 3 686
2 Cubic Threefolds 689
3 Intermediate Jacobians 690
References 690
De Jong-Oort Purity for p-Divisible Groups 691
1 Introduction 691
2 Frobenius Modules 692
3 Proof of Purity 696
References 699

"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 κ(X/G) = κ(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 −∞ 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 κ(X/G) = κ(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"