Tschinkel / Zarhin Algebra, Arithmetic, and Geometry

Volume I: In Honor of Yu. I. Manin
2009
ISBN: 978-0-8176-4745-2
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 269, 698 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;7
2;Contents;8
3;Curriculum Vitae Yuri Ivanovich Manin;10
4;List of Publications;15
5;Gerstenhaber and Batalin–Vilkovisky Structures on Lagrangian Intersections;33
5.1;Introduction;33
5.2;1 Algebra;41
5.3;2 Symplectic geometry;50
5.4;3 Derived Lagrangian intersections on polarized symplectic manifolds;53
5.5;4 The Gerstenhaber structure on Tor and the Batalin–Vilkovisky structure on Ext;69
5.6;5 Further remarks;76
5.7;References;79
6;A Non-Archimedean Interpretation of the Weight Zero Subspaces of Limit Mixed Hodge Structures;80
6.1;Introduction;80
6.2;1 Topological spaces associated with algebraic varieties over a commutative Banach ring;83
6.3;2 The case of the Banach ring (C,);85
6.4;3 Topological spaces associated with algebraic varietiesover the ring OC,0;87
6.5;4 The main result;89
6.6;5 An interpretation of the weight zero subspaces;96
6.7;References;97
7;Analytic Curves in Algebraic Varieties over Number Fields;99
7.1;1 Introduction;99
7.2;2 Preliminary: the geometric case;102
7.3;3 A-analyticity of formal curves;106
7.4;4 Analytic curves in algebraic varieties over local fields and canonical seminorms;113
7.5;5 Capacitary metrics on p-adic curves;120
7.6;6 An algebraicity criterion for A-analytic curves;135
7.7;7 Rationality criteria;139
7.8;Appendix;148
7.9;References;151
8;Riemann–Roch for Real Varieties;155
8.1;1 Introduction;155
8.2;2 Background on Lie algebroids, groupoids and gerbes;159
8.3;3 Background on homology of differential operators;166
8.4;4 Characteristic classes from Lie algebra cohomology;172
8.5;5 The real Riemann–Roch;182
8.6;6 Comparison with the gerbe picture;188
8.7;References;193
9;Universal KZB Equations: The Elliptic Case;195
9.1;Introduction;195
9.2;1 Bundles with flat connections on (reduced) configuration spaces;197
9.3;2 Formality of pure braid groups on the torus;202
9.4;3 Bundles with flat connection on M1,n and M1,[n];207
9.5;4 The monodromy morphisms G1,[n] . Gn Sn;225
9.6;5 Construction of morphisms G1,[n] . Gn Sn;243
9.7;6 The KZB connection as a realization of the universal KZB connection;260
9.8;7 The universal KZB connection and representations of Cherednik algebras;270
9.9;8 Explicit realizations of certain highest weight representations of the rational Cherednik algebra of type An 1;272
9.10;9 Equivariant D-modules and representations of the rational Cherednik algebra;278
9.11;References;294
10;Exact Category of Modules of Constant Jordan Type;297
10.1;1 Introduction;297
10.2;2 The exact category C(kG);298
10.3;3 The Grothendieck group K0(C(kG));302
10.4;4 Realization of Jordan types;310
10.5;5 Stratification by C(kG);316
10.6;References;319
11;Del Pezzo Moduli via Root Systems;321
11.1;Introduction;321
11.2;1 Moduli spaces for marked Del Pezzo surfaces;323
11.3;2 Coble’s covariants;331
11.4;3 Anticanonical divisors with a cusp;338
11.5;4 Coble’s representations;346
11.6;5 The Coble linear system;356
11.7;References;366
12;The Weil Proof and the Geometry of the Adèles Class Space;368
12.1;1 Introduction;369
12.2;2 A look at the Weil proof;371
12.3;3 Quantum statistical mechanics and arithmetic;380
12.4;4 The adèles class space;385
12.5;5 Primitive cohomology;395
12.6;6 A cohomological Lefschetz trace formula;397
12.7;7 Correspondences;400
12.8;8 Thermodynamics and geometry of the primes;404
12.9;9 Functoriality of the adèles class space;421
12.10;10 Vanishing cycles: an analogy;427
12.11;References;432
13;Elliptic Curves with Large Analytic Order of X(E);435
13.1;Introduction;435
13.2;1 Examples of elliptic curves with large |X(E)|;440
13.3;2 Values of the Goldfeld–Szpiro ratio GS(E);444
13.4;3 Large and small (nonzero) values of L(E, 1);444
13.5;4 Remarks on Conjecture 3;446
13.6;References;447
14;p-adic Entropy and a p-adic Fuglede–Kadison Determinant;450
14.1;1 Introduction;450
14.2;2 Preliminaries;453
14.3;3 The Frobenius group determinant and a proof of Theorem 1;457
14.4;4 The logarithm on the 1-units of a p-adic Banach algebra;459
14.5;5 A p-adic logarithmic Fuglede–Kadison determinant and its relation to p-adic entropy;464
14.6;References;469
15;Finite Subgroups of the Plane Cremona Group;470
15.1;1 Introduction;470
15.2;2 First examples;473
15.3;3 Rational G-surfaces;478
15.4;4 Automorphisms of minimal ruled surfaces;486
15.5;5 Automorphisms of conic bundles;495
15.6;6 Automorphisms of Del Pezzo surfaces;511
15.7;7 Elementary links and factorization theorem;552
15.8;8 Birational classes of minimal G-surfaces;561
15.9;9 What is left?;566
15.10;10 Tables;567
15.11;References;573
16;Lie Algebra Theory without Algebra;576
16.1;1 Introduction;576
16.2;2 More general setting;578
16.3;3 Lengths of vectors;579
16.4;4 Riemannian geometry argument;582
16.5;5 Discussion;588
16.6;6 Appendix;591
16.7;References;592
17;Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties;594
17.1;1 Introduction;594
17.2;2 Combinatorics;599
17.3;3 The Flag Space;612
17.4;4 Strata on the Flag Space;616
17.5;5 Extension to the boundary;625
17.6;6 Existence of boundary components;627
17.7;7 Superspecial fibers;629
17.8;8 Local structure of strata;632
17.9;9 Punctual flag spaces;636
17.10;10 Pieri formulas;640
17.11;11 Irreducibility properties;645
17.12;12 The Cycle Classes;649
17.13;13 Tautological rings;656
17.14;14 Comparison with S(g, p);659
17.15;15 Appendix;660
17.16;References;661
18;Experiments with General Cubic Surfaces;664
18.1;1 Introduction;664
18.2;2 Background;667
18.3;3 Computation of the Galois group;668
18.4;4 Computation of Peyre’s constant;670
18.5;5 Numerical Data;672
18.6;6 A concrete example;676
18.7;References;679
19;Cluster Ensembles, Quantization and the Dilogarithm II: The Intertwiner;681
19.1;1 Cluster ensembles;683
19.2;2 Motivation: *-quantization of cluster X-varieties;687
19.3;3 The intertwiner;691
19.4;4 The quantum logarithm and dilogarithm functions;697
19.5;References;699
20;Operads Revisited;700
20.1;1 Modular operads as symmetric monoidal functors;703
20.2;2 Patterns;708
20.3;3 The simplicial pattern Cob;712
20.4;Appendix. Enriched categories;716
20.5;References;722