The new edition of this celebrated and long-unavailable book preserves much of the content and structure of the original, which is still unrivaled in its presentation of a universal method for the resolution of a class of singularities in algebraic geometry. At the same time, the book has been completely retypeset, errors have been eliminated, proofs have been streamlined, the notation has been made consistent and uniform, an index has been added, and a guide to recent literature has been added. The authors begin by reviewing key results in the theory of toroidal embeddings and by explaining examples that illustrate the theory. Chapter II develops the theory of open self-adjoint homogeneous cones and their polyhedral reduction theory. Chapter III is devoted to basic facts on hermitian symmetric domains and culminates in the construction of toroidal compactifications of their quotients by an arithmetic group. The final chapter considers several applications of the general results. The book brings together ideas from algebraic geometry, differential geometry, representation theory and number theory, and will continue to prove of value for researchers and graduate students in these areas.
Author(s): Avner Ash, David Mumford, Michael Rapoport, Yung-sheng Tai
Series: Cambridge Mathematical Library
Edition: 2
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 242
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the second edition......Page 9
Preface to the first edition......Page 11
References......Page 12
1 Torus embeddings over the complex numbers......Page 13
2 The functor of a torus embedding......Page 19
3 Toroidal embeddings over the complex numbers......Page 21
4 Compactification of the universal elliptic curve......Page 26
5 Hirzebruch’s theory of the Hilbert modular group......Page 37
References......Page 47
II Polyhedral reduction theory in self-adjoint cones......Page 49
1.2......Page 50
1.3......Page 53
2 Jordan algebras......Page 55
3.1......Page 63
3.2......Page 64
3.3......Page 66
3.5......Page 68
3.6......Page 69
3.7......Page 70
3.8......Page 71
3.9......Page 75
3.10......Page 76
4.1......Page 79
4.2......Page 81
4.3......Page 83
5.1......Page 87
5.2......Page 93
5.3......Page 96
5.4......Page 99
6 Positive-definite forms in low dimensions......Page 102
References......Page 106
1 Tube domains and compactification of their cusps......Page 109
Appendix: Groups of Q-rank 1 acting on tube domains......Page 115
2.1......Page 117
2.2......Page 120
2.3......Page 123
2.4......Page 131
2.5......Page 134
3.1......Page 135
3.2......Page 141
3.3......Page 146
3.4......Page 149
3.5......Page 152
4.1......Page 154
4.2......Page 157
4.3......Page 161
4.4......Page 166
Appendix: Connected components......Page 169
5 Statement of the Main Theorem......Page 171
6.1......Page 176
6.2......Page 182
6.3......Page 185
7 An intrinsic form of the Main Theorem......Page 188
References......Page 198
1.1......Page 201
1.2......Page 205
1.3......Page 206
2.1......Page 211
2.2......Page 213
2.3......Page 220
References......Page 226
Survey Papers and General Expositions......Page 227
Geometric applications and classification problems......Page 228
Cohomological applications......Page 230
Papers with an arithmetic flavor or functor descriptions......Page 232
Comparison with other compactifications......Page 234
Explicit resolutions......Page 235
Higher weight Hodge structures......Page 236
Reduction theory......Page 237
Index......Page 241
