Algebra, Arithmetic, and Geometry (eBook)

Preface 7
Contents 8
Curriculum Vitae Yuri Ivanovich Manin 10
List of Publications 15
Gerstenhaber and Batalin–Vilkovisky Structures on Lagrangian Intersections 33
Introduction 33
1 Algebra 41
2 Symplectic geometry 50
3 Derived Lagrangian intersections on polarized symplectic manifolds 53
4 The Gerstenhaber structure on Tor and the Batalin–Vilkovisky structure on Ext 69
5 Further remarks 76
References 79
A Non-Archimedean Interpretation of the Weight Zero Subspaces of Limit Mixed Hodge Structures 80
Introduction 80
1 Topological spaces associated with algebraic varieties over a commutative Banach ring 83
2 The case of the Banach ring (C,) 85
3 Topological spaces associated with algebraic varietiesover the ring OC,0 87
4 The main result 89
5 An interpretation of the weight zero subspaces 96
References 97
Analytic Curves in Algebraic Varieties over Number Fields 99
1 Introduction 99
2 Preliminary: the geometric case 102
3 A-analyticity of formal curves 106
4 Analytic curves in algebraic varieties over local fields and canonical seminorms 113
5 Capacitary metrics on p-adic curves 120
6 An algebraicity criterion for A-analytic curves 135
7 Rationality criteria 139
Appendix 148
References 151
Riemann–Roch for Real Varieties 155
1 Introduction 155
2 Background on Lie algebroids, groupoids and gerbes 159
3 Background on homology of differential operators 166
4 Characteristic classes from Lie algebra cohomology 172
5 The real Riemann–Roch 182
6 Comparison with the gerbe picture 188
References 193
Universal KZB Equations: The Elliptic Case 195
Introduction 195
1 Bundles with flat connections on (reduced) configuration spaces 197
2 Formality of pure braid groups on the torus 202
3 Bundles with flat connection on M1,n and M1,[n] 207
4 The monodromy morphisms G1,[n] . Gn Sn 225
5 Construction of morphisms G1,[n] . Gn Sn 243
6 The KZB connection as a realization of the universal KZB connection 260
7 The universal KZB connection and representations of Cherednik algebras 270
8 Explicit realizations of certain highest weight representations of the rational Cherednik algebra of type An 1 272
9 Equivariant D-modules and representations of the rational Cherednik algebra 278
References 294
Exact Category of Modules of Constant Jordan Type 297
1 Introduction 297
2 The exact category C(kG) 298
3 The Grothendieck group K0(C(kG)) 302
4 Realization of Jordan types 310
5 Stratification by C(kG) 316
References 319
Del Pezzo Moduli via Root Systems 321
Introduction 321
1 Moduli spaces for marked Del Pezzo surfaces 323
2 Coble’s covariants 331
3 Anticanonical divisors with a cusp 338
4 Coble’s representations 346
5 The Coble linear system 356
References 366
The Weil Proof and the Geometry of the Adèles Class Space 368
1 Introduction 369
2 A look at the Weil proof 371
3 Quantum statistical mechanics and arithmetic 380
4 The adèles class space 385
5 Primitive cohomology 395
6 A cohomological Lefschetz trace formula 397
7 Correspondences 400
8 Thermodynamics and geometry of the primes 404
9 Functoriality of the adèles class space 421
10 Vanishing cycles: an analogy 427
References 432
Elliptic Curves with Large Analytic Order of X(E) 435
Introduction 435
1 Examples of elliptic curves with large |X(E)| 440
2 Values of the Goldfeld–Szpiro ratio GS(E) 444
3 Large and small (nonzero) values of L(E, 1) 444
4 Remarks on Conjecture 3 446
References 447
p-adic Entropy and a p-adic Fuglede–Kadison Determinant 450
1 Introduction 450
2 Preliminaries 453
3 The Frobenius group determinant and a proof of Theorem 1 457
4 The logarithm on the 1-units of a p-adic Banach algebra 459
5 A p-adic logarithmic Fuglede–Kadison determinant and its relation to p-adic entropy 464
References 469
Finite Subgroups of the Plane Cremona Group 470
1 Introduction 470
2 First examples 473
3 Rational G-surfaces 478
4 Automorphisms of minimal ruled surfaces 486
5 Automorphisms of conic bundles 495
6 Automorphisms of Del Pezzo surfaces 511
7 Elementary links and factorization theorem 552
8 Birational classes of minimal G-surfaces 561
9 What is left? 566
10 Tables 567
References 573
Lie Algebra Theory without Algebra 576
1 Introduction 576
2 More general setting 578
3 Lengths of vectors 579
4 Riemannian geometry argument 582
5 Discussion 588
6 Appendix 591
References 592
Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties 594
1 Introduction 594
2 Combinatorics 599
3 The Flag Space 612
4 Strata on the Flag Space 616
5 Extension to the boundary 625
6 Existence of boundary components 627
7 Superspecial fibers 629
8 Local structure of strata 632
9 Punctual flag spaces 636
10 Pieri formulas 640
11 Irreducibility properties 645
12 The Cycle Classes 649
13 Tautological rings 656
14 Comparison with S(g, p) 659
15 Appendix 660
References 661
Experiments with General Cubic Surfaces 664
1 Introduction 664
2 Background 667
3 Computation of the Galois group 668
4 Computation of Peyre’s constant 670
5 Numerical Data 672
6 A concrete example 676
References 679
Cluster Ensembles, Quantization and the Dilogarithm II: The Intertwiner 681
1 Cluster ensembles 683
2 Motivation: *-quantization of cluster X-varieties 687
3 The intertwiner 691
4 The quantum logarithm and dilogarithm functions 697
References 699
Operads Revisited 700
1 Modular operads as symmetric monoidal functors 703
2 Patterns 708
3 The simplicial pattern Cob 712
Appendix. Enriched categories 716
References 722