Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry: Volume 1

Cover......Page 1
Title......Page 5
Copyright......Page 6
Contents......Page 7
List of contributors......Page 12
1 Model theory......Page 15
2 Non-archimedean geometry......Page 18
3 Motives and the Grothendieck ring of varieties......Page 20
4.2 Geometric nature of p-adic integrals......Page 23
5 Motivic versus p-adic integration......Page 24
5.1 p-adic integration......Page 25
5.2 The geometric approach to motivic integration......Page 26
5.3 The model theoretic approach to motivic integration......Page 28
5.4 Specializing motivic integrals to p-adic integrals......Page 31
6.1 Approaches based on geometry......Page 32
6.2 Approaches based on model theory......Page 36
7 Connections between motivic integration, model theory and non-archimedean geometry......Page 41
8.1 Motivic integration......Page 42
8.3 Non-archimedean geometry......Page 44
9 About this book......Page 45
References......Page 46
1.1 Languages.......Page 49
1.3 Substructures.......Page 50
1.6 Examples of languages, structures, and substructures.......Page 51
1.8 Terms.......Page 55
1.9 Formulas.......Page 56
1.10 Warning.......Page 57
1.11 Adding constant symbols, diagrams.......Page 58
1.13 Satisfaction.......Page 60
1.14 Parameters, definable sets.......Page 61
1.17 Interpretability of a structure in another one......Page 62
2.1 Theories, models of theories, etc........Page 63
2.3 Comments.......Page 64
2.6 Decidability.......Page 65
2.10 Craig’s interpolation theorem.......Page 66
2.12 Saturated models.......Page 67
2.14 Definable and algebraic closures......Page 68
2.17 Principal and non-principal ultrafilters, Fréchet filter.......Page 69
2.19 Reduced products of L-structures.......Page 70
2.24 Application 1: another proof of the compactness theorem.......Page 71
2.26......Page 72
2.28 Preservation theorems.......Page 73
2.30 Divisible ordered abelian groups.......Page 74
2.31 Ordered Z-groups.......Page 75
2.34 Algebraically closed valued fields.......Page 76
2.35 Definition.......Page 78
2.37 Elimination of imaginaries.......Page 79
2.39 Examples......Page 80
3 The results of Ax, Kochen, and Er?sov......Page 81
3.3 Other results: the AKE-principle.......Page 82
4 More results on valued fields......Page 83
4.4......Page 84
4.5 Definition.......Page 85
4.9 Application to power series.......Page 86
Analytic structures.......Page 87
Main omissions.......Page 88
References......Page 89
3 On the definition of rigid analytic spaces......Page 94
1 Affinoid spaces......Page 96
2 Rigid analytic spaces via Grothendieck topologies......Page 101
3 Rigid analytic spaces via Tate’s h-structures......Page 105
4 Grothendieck topologies versus h-structures......Page 110
References......Page 116
1 Introduction......Page 117
Notation......Page 118
Raynaud’s vewpoint......Page 119
Comparison of topology and formal birational patching......Page 120
The classical Zariski-Riemann space......Page 122
Quasi-compactness......Page 123
Points and the structure sheaf......Page 124
Zariski’s argument......Page 126
2.3 Fresh start......Page 127
2.4 Visualization......Page 129
Points......Page 131
Affinoids......Page 132
3.1 Adic topology......Page 134
3.2 Adhesive rings......Page 135
3.3 Some examples......Page 138
3.4 Artin-Rees type property......Page 139
3.5 Some useful properties......Page 141
4.1 T.u.a. rings and universally adhesive formal schemes......Page 142
4.2 Complete valuation rings......Page 143
GFGA comparison theorem......Page 146
Finiteness theorem......Page 148
Admissible blow-ups......Page 149
Admissible topology......Page 150
5.2 General rigid spaces......Page 151
5.3 Visualization......Page 152
GAGA functor......Page 153
GAGA comparison theorem......Page 155
GAGA existence theorem......Page 156
References......Page 157
1 Introduction......Page 159
Notations, conventions......Page 160
2.1 The Grothendieck group of an additive category......Page 161
2.2 The Grothendieck group of an exact category......Page 163
3.1 Definition and functoriality......Page 167
3.2 Scissor relations and piecewise isomorphisms.......Page 168
3.3 Bittner’s presentation.......Page 169
3.4 Spreading out.......Page 170
3.5 The λ-structure and Kapranov’s zeta function.......Page 172
3.6 The Grothendieck ring of definable sets......Page 173
3.7 The Grothendieck ring of a theory......Page 174
3.8 The modified Grothendieck ring of varieties and the theory of algebraically closed fields......Page 175
3.9 Fibrations......Page 179
4.1 The notion of additive invariant......Page 181
4.2 Modified Grothendieck ring of varieties and realization morphisms......Page 188
4.3 Direct consequences for the geometry of varieties......Page 190
5.1 What we know about the Grothendieck ring of varieties......Page 191
5.2 What we don’t know about the Grothendieck ring of varieties......Page 193
6.2 The question of Larsen and Lunts......Page 195
6.3 The work of Liu and Sebag......Page 196
6.4 Some consequences......Page 198
6.5 Related problems......Page 199
References......Page 200
1 The invention of motivic integration......Page 203
2 Geometric motivic integration......Page 206
2.1 The value ring of the motivic measure......Page 207
Dimension......Page 208
2.2 The arc space J∞(X)......Page 210
2.3 An algebra of measurable sets......Page 213
2.4 The measurable function associated to a subscheme......Page 214
2.5 Definition and computation of the motivic integral......Page 216
The induced map on the arc space......Page 219
3.1 Images of cylinders under birational maps.......Page 221
3.2 Proof of transformation rule using Weak Factorization......Page 225
4.1 Properties of the motivic measure......Page 227
Comparison with Lebesgue integra......Page 230
4.2 Motivic integration on singular varieties......Page 231
5 Birational invariants via motivic integration......Page 233
5.1 Notation from birational geometry......Page 234
5.2 Proof of threshold formula......Page 236
5.3 Bounds for the log canonical threshold......Page 240
5.4 Inversion of adjunction......Page 242
5.5 Geometry of arc spaces without explicit motivic integration.......Page 244
A.1 The relative canonical divisor and differentials......Page 249
A.2 Proof of Theorem 3.5......Page 250
B Solutions to some exercises......Page 253
Appendix References......Page 255
1 Introduction......Page 258
Notation......Page 260
2.1 Formal schemes and rigid varieties......Page 261
2.2 Greenberg schemes......Page 264
2.3 Motivic integration on smooth formal schemes......Page 271
2.4 The change of variables formula......Page 275
2.5 Néron smoothening and dilatation......Page 281
2.6 Motivic integration on special formal schemes and rigid varieties......Page 286
3.1 The motivic Serre invariant for rigid varieties......Page 289
3.2 The motivic Serre invariant for algebraic varieties......Page 292
3.3 The trace formula......Page 294
4.1 Analytic manifolds and p-adic integration......Page 298
4.2 The p-adic Serre invariant......Page 299
5.1 The Milnor fibration......Page 302
5.2 The motivic zeta function......Page 303
6.1 Construction of the analytic Milnor fiber......Page 305
6.2 Étale cohomology of the analytic Milnor fiber......Page 306
6.3 Points of the analytic Milnor fiber and arc spaces......Page 307
6.4 Motivic Weil generating series......Page 309
6.5 The motivic zeta function as a Weil generating series......Page 311
6.6 Non-archimedean geometry and complex singularities......Page 312
6.7 Singular cohomology of the analytic Milnor fiber......Page 313
7.2 The motivic zeta function of an abelian variety......Page 314
References......Page 315
1 Introduction......Page 319
2.1......Page 324
3.1 A list of theories......Page 325
3.2 The axiomatic set-up......Page 326
4.1......Page 330
4.2 Dimension......Page 331
5 Summation over the value group......Page 332
6 Integration over the residue rings......Page 334
7 Putting P+ and Q+ together to form C+......Page 335
7.2 Integration over residue rings and value group......Page 336
8 Integration over one valued field variable......Page 338
9 General integration......Page 339
10 Further properties......Page 340
11 Direct image formalism......Page 342
11.1 Comparison with [19]......Page 345
References......Page 346