Arakelov Geometry over Adelic Curves

Contents......Page 6
Introduction......Page 10
1.1 Norms and seminorms......Page 20
1.1.1 Topology......Page 22
1.1.2 Operator seminorm......Page 24
1.1.3 Quotient seminorm......Page 25
1.1.4 Topology of normed vector spaces of finite dimension......Page 26
1.1.5 Dual norm......Page 31
1.1.6 Seminorm of the dual operator......Page 33
1.1.7 Lattices and norms......Page 34
1.1.8 Trivial valuation case......Page 38
1.1.9 Metric on the space of norms......Page 41
1.1.10 Direct sums......Page 43
1.1.11 Tensor product seminorms......Page 49
1.1.12 Exterior power seminorm......Page 55
1.1.13 Determinant seminorm......Page 56
1.1.14 Seminormed graded algebra......Page 58
1.1.15 Norm of polynomial......Page 60
1.2.1 Inner product......Page 61
1.2.3 Orthogonality in general cases......Page 63
1.2.4 Orthogonality and lattice norms......Page 74
1.2.5 Orthogonality and Hadamard property......Page 75
1.2.6 Ultrametric Gram-Schimdt process......Page 78
1.2.7 Dual determinant norm......Page 86
1.2.8 Ellipsoid of John and Löwner......Page 92
1.2.9 Hilbert-Schmidt tensor norm......Page 96
1.3 Extension of scalars......Page 100
1.3.1 Basic properties......Page 101
1.3.2 Direct sums......Page 106
1.3.3 Orthogonality......Page 108
1.3.4 Extension of scalars in the real case......Page 119
2.1.1 Berkovich space associated with a scheme......Page 126
2.1.2 Metric on a vector bundle......Page 130
2.1.3 Base change......Page 134
2.2.1 Dual metric and tensor product metric......Page 135
2.2.2 Distance between metrics......Page 138
2.2.3 Fubini-Study metric......Page 139
2.3.1 Definition and basic properties......Page 146
2.3.2 Model metrics......Page 150
2.3.4 Extension property......Page 156
2.3.4.1 A generalisation of a result in [155] and [111]......Page 160
2.3.4.2 Extension property over an Archimedean field......Page 164
2.3.4.3 Extension property over a non-Archimedean field......Page 165
2.4.1 Reminder on Cartier divisors......Page 171
2.4.2 Linear system of a divisor......Page 173
2.4.3 Q-Cartier and R-Cartier divisors......Page 175
2.5.1 Green functions of Cartier divisors......Page 179
2.5.2 Green functions for Q-Cartier and R-Cartier divisors......Page 181
2.5.3 Canonical Green functions with respect to endmorphisms......Page 183
3 Adelic curves......Page 185
3.1 Definition of Adelic curves......Page 186
3.2.2 Number fields......Page 188
3.2.4 Polarised varieties......Page 189
3.2.5 Function field over Q......Page 190
3.2.6 Polarised arithmetic variety......Page 192
3.2.9 Restriction of adelic structure to a measurable subset......Page 195
3.3 Finite separable extensions......Page 196
3.3.1 Integration along fibres......Page 197
3.3.2 Measurability of fibre integrals......Page 198
3.3.3 Construction of the measure......Page 202
3.4.1 Finite extension......Page 206
3.4.2 General algebraic extensions......Page 209
3.5 Height function and Northcott property......Page 217
3.6 Measurability of automorphism actions......Page 220
3.7 Morphisms of adelic curves......Page 221
4.1.1 Definition and algebraic constructions......Page 223
4.1.1.3 Direct sums......Page 224
4.1.1.7 Determinant......Page 225
4.1.1.10 Local distance......Page 226
4.1.2 Dominated norm families......Page 227
4.1.3 Measurability of norm families......Page 241
4.1.4 Adelic vector bundles......Page 254
4.1.5 Examples......Page 257
4.2 Adelic divisors......Page 259
4.3.1 Arakelov degree of adelic line bundles......Page 260
4.3.2 Arakelov degree of adelic vector bundles......Page 262
4.3.3 Arakelov degree of tensor adelic vector bundles......Page 267
4.3.4 Positive degree......Page 268
4.3.5 Riemann-Roch formula......Page 270
4.3.6.1 Function field case......Page 271
4.3.6.2 Number field case......Page 272
4.3.7 Slopes and slope inequalities......Page 273
4.3.8 Finiteness of slopes......Page 275
4.3.9 Some slope estimates......Page 278
4.3.10 Harder-Narasimhan filtration: Hermitian case......Page 282
4.3.11 Harder-Narasimhan filtration: general case......Page 288
4.3.12 Absolute positive degree and absolute maximal slope......Page 300
4.3.13 Successive minima......Page 301
4.3.14 Minkowski property......Page 303
4.4 Adelic vector bundles over number fields......Page 307
4.4.1 Coherency for a norm family......Page 308
4.4.2 Finite generation of a dominated vector bundle over S......Page 309
4.4.3 Invariants  and......Page 312
5.1 Reminder on R-filtrations......Page 315
5.2 Reminder on geometric invariant theory......Page 320
5.3 Estimate for the minimal slope under semi-stability assumption......Page 327
5.4 An interpretation of the geometric semistability......Page 330
5.5 Lifting and refinement of filtrations......Page 335
5.6 Estimation in general case......Page 337
6.1 Metrised line bundles on an arithmetic variety......Page 345
6.1.1 Quotient metric families......Page 346
6.1.2 Dominated metric families......Page 347
6.1.3 Universally dense point families......Page 355
6.1.4 Measurable metric families......Page 360
6.2.1 Height function......Page 363
6.2.2 Essential minimum......Page 365
6.2.3 Adelic divisors......Page 368
6.2.4 The canonical compatifications of Cartier divisors with respect to endomorphisms......Page 374
6.3.1 Reminder on some facts about convex sets......Page 376
6.3.2 Graded semigroups......Page 377
6.3.3 Concave transform......Page 383
6.3.4 Applications to the study of graded algebras......Page 394
6.3.5 Applications to the study of the volume function......Page 398
6.4.1 Asymptotic maximal slope......Page 402
6.4.2 Arithmetic volume function......Page 407
6.4.3 Volume of adelic R-Cartier divisors......Page 409
7.1 Graded algebra of adelic vector bundles......Page 421
7.2 Fundamental estimations......Page 424
7.3 A consequence of the extension property of semipositive metrics......Page 431
7.4 Nakai-Moishezon\'s criterion in a general settings......Page 433
7.5 Nakai-Moishezon\'s criterion over a number field......Page 436
7.5.1 Invariants  and  for a graded algebra of adelic vector bundles......Page 437
7.5.2 Dominancy and coherency of generically pure metric......Page 439
7.5.3 Fine metric family......Page 440
7.5.4 A generalization of Nakai-Moishezon\'s criterion......Page 443
A.1 Monotone class theorems......Page 445
A.3 Vague convergence and weak convergence of measures......Page 447
A.4 Upper and lower integral......Page 449
A.5 L1 space......Page 455
References......Page 456
Index......Page 463