Vector fields on singular varieties:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2009
|
| Series: | Lecture notes in mathematics
1987 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018899446&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Literaturangaben |
| Physical Description: | XX, 225 S. 24 cm |
| ISBN: | 9783642052040 |
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Record in the Search Index
| _version_ | 1819359904480100352 |
|---|---|
| adam_text | Titel: Vector fields on singular varieties
Autor: Brasselet, Jean-Paul
Jahr: 2009
Contents
The Case of Manifolds.............................................. 1
1.1 Poincare?Hopf Index Theorem ................................. 1
1.1.1 Poincare?Hopf Index at Isolated Points.............. 1
1.1.2 Poincare-Hopf Index at Nonisolated Points.......... 3
1.2 Poincare and Alexander Dualities.............................. 5
1.3 Chern Classes via Obstruction Theory......................... 6
1.3.1 Chern Classes of Almost Complex Manifolds........ 6
1.3.2 Relative Chern Classes................................. 8
1.4 Chern-Weil Theory of Characteristic Classes................. 11
1.5 Cech-de Rham Cohomology..................................... 15
1.5.1 Integration on the Cech-de Rham Cohomology...... 16
1.5.2 Relative Cech-de Rham Cohomology -
Alexander Duality...................................... 17
1.6 Localization of Chern Classes................................... 19
1.6.1 Characteristic Classes in the Cech-de Rham
Cohomology............................................. 19
1.6.2 Localization of Characteristic Classes
of Complex Vector Bundles............................ 20
1.6.3 Localization of the Top Chern Class.................. 22
1.6.4 Hyperplane Bundle..................................... 23
1.6.5 Grothendieck Residues................................. 25
1.6.6 Residues at an Isolated Zero........................... 26
1.6.7 Examples................................................ 28
The Schwartz Index................................................. 31
2.1 Isolated Singularity Case........................................ 31
2.2 Whitney Stratifications.......................................... 33
2.3 Radial Extension of Vector Fields.............................. 34
2.4 The Schwartz Index on a Stratified Variety................... 37
2.4.1 Case of Vector Fields with an Isolated Singularity... 37
2.4.2 Case of Vector Fields with Nonisolated Singularity.. 38
Contents
The GSV Index...................................................... 43
3.1 Vector Fields Tangent to a Hypersurface...................... 43
3.2 The Index for Vector Fields on ICIS........................... 46
3.3 Some Applications and Examples.............................. 49
3.4 The Case of Isolated Smoothable Singularities................ 52
3.5 Nonisolated Singularities........................................ 53
3.5.1 The Strict Thom Condition for Complex
Analytic Maps.......................................... 54
3.5.2 The Hypersurface Case ................................ 56
3.5.3 The Complete Intersection Case....................... 57
3.6 The Proportionality Theorem.................................. 58
3.7 Geometric Applications......................................... 61
3.7.1 Topological Invariance of the Milnor Number........ 61
3.7.2 The Canonical Contact Structure on the Link....... 62
3.7.3 On the Normal Bundle of Holomorphic
Singular Foliations...................................... 68
Indices of Vector Fields on Real Analytic Varieties......... 71
4.1 The Schwartz Index on Real Analytic Varieties............... 71
4.2 The GSV Index on Real Analytic Varieties.................... 73
4.3 A Geometric Interpretation of the GSV Index................ 77
4.4 Topological Invariants and Gurvatura Integra................. 78
4.5 Relation with the Milnor Number for Real Singularities..... 81
The Virtual Index................................................... 85
5.1 The Virtual Tangent Bundle of a Local Complete Intersection 85
5.2 Chern-Weil Theory for Virtual Bundles....................... 86
5.3 Characteristic Numbers on Singular Varieties................. 88
5.4 The Virtual Index............................................... 91
5.5 Identification with GSV Index When Singularities
are Isolated....................................................... 92
5.6 A Generalization of the Adjunction Formula.................. 93
5.7 An Integral Formula for the Virtual Index.................... 95
The Case of Holomorphic Vector Fields........................ 97
6.1 Baum Bott Residues of Holomorphic Vector Fields.......... 98
6.2 One-Dimensional Singular Foliations...........................101
6.3 Residues of Holomorphic Vector Fields on Singular Varieties 104
6.3.1 Grothendieck Residues Relative to a Subvariety.....104
6.3.2 Residues for the Ambient Tangent Bundle
(Generalized Variation)................................105
6.3.3 Residues for the Normal Bundle (Residues
of Type Camacho-Sad) ................................107
6.3.4 Residues for the Virtual Tangent Bundle
(Singular Bauni-Bott) .................................110
Contents xi
7 The Homological Index and Algebraic Formulas.............115
7.1 The Homological Index..........................................116
7.2 The Hypersurface Case..........................................119
7.3 The Index of Real Analytic Vector Fields.....................123
7.3.1 The Signature Formula of Eisenbud-
Levine-Khimshiashvili .................................124
7.3.2 The Index on Real Hypersurface Singularities.......126
8 The Local Euler Obstruction......................................129
8.1 Definition of the Euler Obstruction. The Nash
Blow Up..........................................................129
8.1.1 Proportionality Theorem for Vector Fields...........131
8.2 Euler Obstruction and Hyperplane Sections...................133
8.3 The Local Euler Obstruction of a Function ...................136
8.4 The Euler Obstruction and the Euler Defect..................137
8.5 The Euler Defect at General Points............................139
8.6 The Euler Obstruction via Morse Theory.....................140
9 Indices for 1-Forms..................................................143
9.1 Some Basic Facts About 1-Forms..............................143
9.2 Radial Extension and the Schwartz Index.....................147
9.3 Local Euler Obstruction of a 1-Form
and the Proportionality Theorem..............................149
9.4 The Radial Index................................................151
9.5 The GSV Index..................................................153
9.5.1 Isolated Singularity Case...............................154
9.5.2 Nonisolated Singularity Case..........................155
9.6 The Homological Index..........................................158
9.7 On the Milnor Number of an Isolated Singularity............159
9.8 Indices for Collections of 1-Forms..............................161
9.8.1 The GSV Index for Collections of 1-Forms...........165
9.8.2 Local Chern Obstructions..............................165
10 The Schwartz Classes...............................................167
10.1 The Local Schwartz Index of a Frame.........................167
10.2 Proportionality Theorem........................................171
10.3 The Schwartz Classes............................................ 175
10.4 Alexander and Other Homomorphisms........................176
10.5 Localization of the Schwartz Classes...........................178
10.5.1 The Topological Viewpoint............................179
10.5.2 The Differential Geometric Viewpoint................180
10.6 MacPherson and Mather Classes...............................182
xii Contents
11 The Virtual Classes.................................................185
11.1 Virtual Classes...................................................185
11.2 Lifting a Frame to the Milnor Fiber ...........................187
11.3 The Fulton-Johnson Classes....................................189
11.4 Localization of the Virtual Classes.............................191
12 Milnor Number and Milnor Classes.............................193
12.1 Milnor Classes ...................................................194
12.2 Localization of Milnor Classes..................................195
12.3 Differential Geometric Point of View...........................196
12.4 Generalized Milnor Number ....................................200
13 Characteristic Classes of Coherent Sheaves
on Singular Varieties................................................201
13.1 Local Chern Classes and Characters
in the Cech-de Rham Cohomology.............................201
13.2 Thorn Class......................................................206
13.3 Riemann-Roch Theorem for Embeddings......................207
13.4 Homology Chern Characters and Classes......................210
13.5 Characteristic Classes of the Tangent Sheaf...................212
References...................................................................215
Index.........................................................................223
|
| any_adam_object | 1 |
| author | Brasselet, Jean-Paul Seade, José 1954- Suwa, Tatsuo |
| author_GND | (DE-588)132349078 (DE-588)130563072 (DE-588)129584029 |
| author_facet | Brasselet, Jean-Paul Seade, José 1954- Suwa, Tatsuo |
| author_role | aut aut aut |
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| dewey-full | 515.94 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.94 |
| dewey-search | 515.94 |
| dewey-sort | 3515.94 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036006824 |
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| indexdate | 2024-12-20T14:04:48Z |
| institution | BVB |
| isbn | 9783642052040 |
| language | English |
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| physical | XX, 225 S. 24 cm |
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| publisher | Springer |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spellingShingle | Brasselet, Jean-Paul Seade, José 1954- Suwa, Tatsuo Vector fields on singular varieties Lecture notes in mathematics Index (DE-588)4161483-5 gnd Singularität Mathematik (DE-588)4077459-4 gnd Vektorfeld (DE-588)4139571-2 gnd |
| subject_GND | (DE-588)4161483-5 (DE-588)4077459-4 (DE-588)4139571-2 |
| title | Vector fields on singular varieties |
| title_auth | Vector fields on singular varieties |
| title_exact_search | Vector fields on singular varieties |
| title_full | Vector fields on singular varieties Jean-Paul Brasselet ; José Seade ; Tatsuo Suwa |
| title_fullStr | Vector fields on singular varieties Jean-Paul Brasselet ; José Seade ; Tatsuo Suwa |
| title_full_unstemmed | Vector fields on singular varieties Jean-Paul Brasselet ; José Seade ; Tatsuo Suwa |
| title_short | Vector fields on singular varieties |
| title_sort | vector fields on singular varieties |
| topic | Index (DE-588)4161483-5 gnd Singularität Mathematik (DE-588)4077459-4 gnd Vektorfeld (DE-588)4139571-2 gnd |
| topic_facet | Index Singularität Mathematik Vektorfeld |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018899446&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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