Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics

Cover......Page 1
London Mathematical Society Lecture Note Series 338......Page 2
Surveys in Geometry and Number Theory: Reports on contemporary Russian mathematics......Page 4
9780521691826......Page 5
Contents......Page 6
Preface......Page 8
Introduction......Page 10
1 Remarkable classes of affine embeddings......Page 14
2 Connections with Hilbert’s 14th Problem......Page 23
3 Some properties of affine embeddings......Page 28
4 Embeddings with a finite number of orbits......Page 35
5 Application One: Invariant algebras on homogeneous spaces of compact Lie groups......Page 44
6 Application Two: G-algebras with finitely generated invariant subalgebras......Page 51
Bibliography......Page 57
Introduction......Page 61
1 Basic definitions......Page 63
3 Explicit reciprocity formulas......Page 68
4 The matrix method......Page 74
5 Fractional part invariant: classification up to a strict isogeny......Page 77
6 Invariant Cartier-Dieudonne modules; the module invariant of a formal group......Page 81
7 Classification for small ramification index......Page 85
8 Some other classification results......Page 89
9 Connection of formal groups with associated Galois modules......Page 90
Bibliography......Page 95
Introduction......Page 97
1 Conjectures on mirror symmetry for Fano varieties......Page 98
2 The Iskovskikh classification and D3 equations.......Page 105
3 (N, d)-modular variation......Page 110
4 (N, d)-modular D3 equations: the necessary condition.......Page 114
5 (N, d)-modular D3 equations: a sufficient condition.......Page 116
6 A conjecture on counting matrices. The Iskovskikh classification revisited......Page 123
7 What next?......Page 126
Bibliography......Page 128
Preliminaries......Page 131
1 The rigidity problem for del Pezzo fibrations......Page 137
2 Projective models of del Pezzo fibrations......Page 141
3 Fibre-to-fibre transformations......Page 150
4 Mori structures on del Pezzo fibrations: the case d = 1......Page 157
5 Mori structures on del Pezzo fibrations: the case d = 2......Page 160
Bibliography......Page 164
1 Best Diophantine approximations in the sense of linear forms......Page 167
2 Best simultaneous Diophantine approximations......Page 176
Bibliography......Page 190
Introduction......Page 192
1 Projectively dual varieties......Page 193
3 The contact locus and its normal bundle......Page 195
4 Fibrations and divisors......Page 196
5 Dual varieties and jet bundles......Page 197
6 The Kac-Kleiman-Holme formula......Page 199
8 Dual varieties and Mori theory......Page 200
9 The class formula......Page 201
10 Flag varieties......Page 202
11 Adjoint varieties......Page 204
12 The Pyasetskii pairing......Page 206
13 The multisegment duality......Page 207
14 Kashin’s diagrams......Page 208
15 Parabolic subgroups with Abelian unipotent radical......Page 210
16 Tangents and secants......Page 212
18 Degrees of hyperdeterminants......Page 215
19 Parametric formulas......Page 216
20 Polarized flag varieties with positive defect......Page 218
21 Varieties with small codegree......Page 219
22 The Matsumura-Monsky theorem......Page 220
23 Quasiderivations of commutative algebras......Page 221
24 Around the Hartshorne conjecture......Page 223
25 Self-dual nilpotent orbits......Page 224
26 Discriminants of anticommutative algebras......Page 225
Bibliography......Page 229
Introduction......Page 235
1 Complexity and rank......Page 240
2 Embedding theory......Page 257
3 Spherical varieties......Page 263
4 Spaces of complexity one......Page 280
Bibliography......Page 284
Introduction......Page 288
1 Geometric quantization and its geometric formulation......Page 293
2 The correspondence principle in algebraic Lagrangian geometry......Page 300
3 The dynamic correspondence in algebraic Lagrangian geometry......Page 312
4 Reduction of ALG(a)-quantization......Page 319
Bibliography......Page 326