Periods and Nori motives:
Saved in:
| Main Authors: | , |
|---|---|
| Other Authors: | , |
| Format: | Book |
| Language: | English |
| Published: |
Cham, Switzerland
Springer
[2017]
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge
volume 65 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxiii, 372 Seiten |
| ISBN: | 9783319509259 9783319845241 |
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MARC
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Record in the Search Index
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| adam_text | Contents
Part I Background Material
1 General Set-Up................................................... 3
1.1 Varieties .................................................... 3
1.1.1 Linearising the Category of Varieties.................. 3
1.1.2 Divisors with Normal Crossings......................... 4
1.2 Complex Analytic Spaces ...................................... 5
1.2.1 Analytification ...................................... 5
1.3 Complexes..................................................... 6
1.3.1 Basic Definitions ..................................... 6
1.3.2 Filtrations............................................ 7
1.3.3 Total Complexes and Signs ............................. 8
1.4 Hypercohomology............................................... 9
1.4.1 Definition............................................ 10
1.4.2 Godement Resolutions.................................. 11
1.4.3 Čech Cohomology ...................................... 13
1.5 Simplicial Objects........................................... 15
1.6 Grothendieck Topologies...................................... 20
1.7 Torsors ..................................................... 22
1.7.1 Sheaf-Theoretic Definition ........................... 23
1.7.2 Torsors in the Category of Sets ...................... 24
1.7.3 Torsors in the Category of Schemes
(Without Groups)..................................... 27
2 Singular Cohomology.............................................. 31
2.1 Relative Cohomology ......................................... 31
2.2 Singular (Co)homology........................................ 34
2.3 Simplicial Cohomology........................................ 36
2.4 The Künneth Formula and Poincaré Duality .................... 41
2.5 The Basic Lemma............................................. 45
v
v* Contents
2.5.1 Formulations of the Basic Lemma........................ 45
2.5.2 Direct Proof of Basic Lemma I ......................... 47
2.5.3 Nori’s Proof of Basic Lemma II......................... 49
2.5.4 Beilinson’s Proof of Basic Lemma II.................... 52
2.5.5 Perverse Sheaves and Artin Vanishing................... 55
2.6 Triangulation of Algebraic Varieties ....................... 59
2.6.1 Semi-algebraic Sets ................................... 60
2.6.2 Semi-algebraic Singular Chains ........................ 66
2.7 Singular Cohomology via the h -Topology ....................... 70
3 Algebraic de Rham Cohomology...................................... 73
3.1 The Smooth Case ............................................ 73
3.1.1 Definition............................................. 73
3.1.2 Functoriality . . ..................................... 76
3.1.3 Cup Product............................................ 77
3.1.4 Change of Base Field................................... 79
3.1.5 Étale Topology......................................... 80
3.1.6 Differentials with Log Poles........................... 81
3.2 The General Case: Via the h-Topology....................... 83
3.3 The General Case: Alternative Approaches...................... 87
3.3.1 Deligne’s Method....................................... 87
3.3.2 Hartshome’s Method .................................... 90
3.3.3 Using Geometric Motives............................. 91
3.3.4 The Case of Divisors with Normal Crossings ........... 94
4 Hoiomorphic de Rham Cohomology.................................... 97
4.1 Hoiomorphic de Rham Cohomology................................. 97
4.1.1 Definition............................................. 97
4.1.2 Hoiomorphic Differentials with Log Poles............... 99
4.1.3 GAGA................................................ 100
4.2 Hoiomorphic de Rham Cohomology via the h -Topology.......... 102
4.2.1 h -Differentials ..................................... 102
4.2.2 Hoiomorphic de Rham Cohomology ....................... 103
4.2.3 GAGA.................................................. 104
5 The Period Isomorphism............................................ 107
5.1 The Category(k, Q)—Vect..................................... 107
5.2 A Triangulated Category.................................... 108
5.3 The Period Isomorphism in the Smooth Case .................. 109
5.4 The General Case (via the h -Topology) ..................... Ill
5.5 The General Case (Deligne’s Method)......................... 113
Contents
vii
6 Categories of (Mixed) Motives ...................................... 117
6.1 Pure Motives.................................................. 117
6.2 Geometric Motives . .......................................... 119
6.3 Absolute Hodge Motives........................................ 124
6.4 Mixed Tate Motives ......................................... 129
Part II Nori Motives
7 Nori’s Diagram Category ............................................ 137
7.1 Main Results................................................ 137
7.1.1 Diagrams and Representations ........................ 137
7.1.2 Explicit Construction of the Diagram Category ....... 139
7.1.3 Universal Property: Statement ....................... 140
7.1.4 Discussion of the Tannakian Case .................... 144
7.2 First Properties of the Diagram Category ..................... 145
7.3 The Diagram Category of an Abelian Category................... 149
7.3.1 A Calculus of Tensors................................ 150
7.3.2 Construction of the Equivalence...................... 156
7.3.3 Examples and Applications ........................... 164
7.4 Universal Property of the Diagram Category.................... 165
7.5 The Diagram Category as a Category of Comodules............... 168
7.5.1 Preliminary Discussion............................... 168
7.5.2 Coalgebras and Comodules ............................ 169
8 More on Diagrams.................................................... 177
8.1 Multiplicative Structure...................................... 177
8.2 Localisation.................................................. 188
8.3 Nori’s Rigidity Criterion.................................... 191
8.4 Comparing Fibre Functors...................................... 195
8.4.1 The Space of Comparison Maps ........................ 196
8.4.2 Some Examples ....................................... 201
8.4.3 The Description as Formal Periods.................... 204
9 Nori Motives....................................................... 207
9.1 Essentials of Nori Motives ................................... 207
9.1.1 Definition........................................... 207
9.1.2 Main Results ........................................ 209
9.2 Yoga of Good Pairs............................................ 212
9.2.1 Good Pairs and Good Filtrations ..................... 212
9.2.2 Cech Complexes ...................................... 213
9.2.3 Putting Things Together.............................. 216
9.2.4 Comparing Diagram Categories ........................ 218
yiü Contents
9.3 Tensor Structure .......................................... 220
9.3.1 Collection of Proofs............................... 225
9.4 Artin Motives ............................................. 226
9.5 Change of Fields........................................... 228
10 Weights and Pure Nori Motives .................................... 233
10.1 Comparison Functors.......................................... 233
10.2 Weights and Nori Motives .................................... 236
10.2.1 Andre’s Motives.................................... 237
10.2.2 Weights ........................................... 238
10.3 Tate Motives............................................... 241
Part III Periods
11 Periods of Varieties ............................................... 247
11.1 First Definition ............................................ 247
11.2 Periods for the Category(k, Q)֊֊Vect ........................ 250
11.3 Periods of Algebraic Varieties............................... 253
11.3.1 Definition........................................... 253
11.3.2 First Properties..................................... 255
11.4 The Comparison Theorem....................................... 256
11.5 Periods of Motives .......................................... 258
12 Kontsevich-Zagier Periods .......................................... 261
12.1 Definition .................................................. 261
12.2 Comparison of Definitions of Periods ........................ 265
13 Formal Periods and the Period Conjecture ........................... 273
13.1 Formal Periods and Nori Motives ............................. 273
13.2 The Period Conjecture...................................... 277
13.2.1 Formulation in the Number Field Case ................ 278
13.2.2 Consequences......................................... 279
13.2.3 Special Cases and the Older Literature............... 282
13.2.4 The Function Field Case ............................. 284
13.3 The Case of 0-Dimensional Varieties ......................... 287
Part IV Examples 289
14 Elementary Examples .............................................. 291
14.1 Logarithms .................................................. 291
14.2 More Logarithms............................................ 293
14.3 Quadratic Forms ............................................ 294
14.4 Elliptic Curves............................................. 297
14.5 Periods of 1-Forms on Arbitrary Curves....................... 301
Contents ix
15 Multiple Zeta Values .............................................. 307
15.1 A C՜value, the Basic Example................................ 307
15.2 Definition of Multiple Zeta Values ......................... 310
15.3 Kontsevich’s Integral Representation........................ 312
15.4 Relations Among Multiple Zeta Values........................ 314
15.5 Multiple Zeta Values and Moduli Space
of Marked Curves ........................................... 320
15.6 Multiple Polylogarithms .................................. 321
15.6.1 The Configuration .................................. 322
15.6.2 Singular Homology .................................. 323
15.6.3 Smooth Singular Homology............................ 326
15.6.4 Algebraic de Rham Cohomology and the Period
Matrix of (.X, D)................................... 327
15.6.5 Varying the Parameters a and b...................... 331
16 Miscellaneous Periods: An Outlook ................................. 337
16.1 Special Values of L-Functions............................... 337
16.2 Feynman Periods............................................. 341
16.3 Algebraic Cycles and Periods ............................... 343
16.4 Periods of Homotopy Groups ................................. 347
16.5 Exponential Periods......................................... 349
16.6 Non-periods................................................. 350
Glossary............................................................... 355
References............................................................. 359
Index.................................................................. 369
|
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| ctrlnum | (OCoLC)991544150 (DE-599)BSZ484618180 |
| discipline | Mathematik |
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| id | DE-604.BV044238576 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:47:26Z |
| institution | BVB |
| isbn | 9783319509259 9783319845241 |
| language | English |
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| physical | xxiii, 372 Seiten |
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| series | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge |
| spelling | Huber, Annette 1967- (DE-588)172146569 aut Periods and Nori motives Annette Huber, Stefan Müller-Stach ; with contributions by Benjamin Friedrich and Jonas von Wangenheim Cham, Switzerland Springer [2017] © 2017 xxiii, 372 Seiten txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge volume 65 Kohomologie (DE-588)4031700-6 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Periode (DE-588)4197595-9 gnd rswk-swf Motiv Mathematik (DE-588)4197596-0 gnd rswk-swf Periode (DE-588)4197595-9 s Zahlentheorie (DE-588)4067277-3 s Motiv Mathematik (DE-588)4197596-0 s Kohomologie (DE-588)4031700-6 s DE-604 Müller-Stach, Stefan 1962- (DE-588)111714141 aut Friedrich, Benjamin 1980- (DE-588)130563307 ctb Wangenheim, Jonas von (DE-588)1165244942 ctb Erscheint auch als Online-Ausgabe 978-3-319-50926-6 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge volume 65 (DE-604)BV000899194 65 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029643981&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Huber, Annette 1967- Müller-Stach, Stefan 1962- Periods and Nori motives Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge Kohomologie (DE-588)4031700-6 gnd Zahlentheorie (DE-588)4067277-3 gnd Periode (DE-588)4197595-9 gnd Motiv Mathematik (DE-588)4197596-0 gnd |
| subject_GND | (DE-588)4031700-6 (DE-588)4067277-3 (DE-588)4197595-9 (DE-588)4197596-0 |
| title | Periods and Nori motives |
| title_auth | Periods and Nori motives |
| title_exact_search | Periods and Nori motives |
| title_full | Periods and Nori motives Annette Huber, Stefan Müller-Stach ; with contributions by Benjamin Friedrich and Jonas von Wangenheim |
| title_fullStr | Periods and Nori motives Annette Huber, Stefan Müller-Stach ; with contributions by Benjamin Friedrich and Jonas von Wangenheim |
| title_full_unstemmed | Periods and Nori motives Annette Huber, Stefan Müller-Stach ; with contributions by Benjamin Friedrich and Jonas von Wangenheim |
| title_short | Periods and Nori motives |
| title_sort | periods and nori motives |
| topic | Kohomologie (DE-588)4031700-6 gnd Zahlentheorie (DE-588)4067277-3 gnd Periode (DE-588)4197595-9 gnd Motiv Mathematik (DE-588)4197596-0 gnd |
| topic_facet | Kohomologie Zahlentheorie Periode Motiv Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029643981&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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