Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds:
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| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2004
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXII, 150 S. graph. Darst. |
| ISBN: | 9812388516 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
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| 001 | BV021241795 | ||
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| 100 | 1 | |a Nieper-Wißkirchen, Marc Arnold |e Verfasser |0 (DE-588)124200486 |4 aut | |
| 245 | 1 | 0 | |a Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds |c Marc Nieper-Wißkirchen |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2004 | |
| 300 | |a XXII, 150 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Kähler-Mannigfaltigkeit |0 (DE-588)4162978-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Chern-Zahl |0 (DE-588)4322052-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Invariante |0 (DE-588)4128781-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Kähler-Mannigfaltigkeit |0 (DE-588)4162978-4 |D s |
| 689 | 0 | 1 | |a Chern-Zahl |0 (DE-588)4322052-6 |D s |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-014284486 | ||
Datensatz im Suchindex
| _version_ | 1804134689590476800 |
|---|---|
| adam_text | Contents
Preface vii
Introduction
ix
Notation
xiii
1.
Compact
hyper-Kähler
manifolds and holomorphic sym-
plectic manifolds
1
1.1
Basics on compact
hyper-Kähler
manifolds
.......... 1
1.1.1
Hokmomy of Riemannian manifolds
.......... 1
1.1.2
Definition of a compact
hyper-Kähler
manifold
... 3
1.1.3
Holomorphic symplectic manifolds
.......... 4
1.1.4
Deformations of compact complex manifolds
..... 7
1.2
Examples
............................ 8
1.2.1
The
КЗ
surface
..................... 8
1.2.2
The Hubert scheme of points on a surface
...... 11
1.2.3
Construction of line bundles and classes in H2 on the
Hubert schemes of points on surfaces
......... 13
1.2.4
Hubert schemes of points on
КЗ
surfaces
....... 15
1.2.5
Generalised
Kummer
varieties
............. 16
1.2.6
Further examples
.................... 19
1.3
Characteristic classes
...................... 20
1.3.1
Symplectic sheaves
................... 20
1.3.2
Characteristic classes of symplectic sheaves
..... 21
1.3.3
Chern numbers of holomorphic symplectic manifolds
24
1.4
The Atiyah class
........................ 26
xx
Chern Numbers and RW-Invariants of Compact
Hyper-Kăhler
Manifolds
1.4.1
Definition
........................ 26
1.4.2
Description in terms of
Čech
cohomology
....... 28
1.4.3
The
Bianchi
identity
.................. 29
1.4.4
Torsion and the Atiyah class of the tangent bundle
. 31
1.4.5
The Atiyah class of symplectic sheaves
........ 32
1.4.6
Chern-Weil theory
................... 34
1.5
On the second cohomology group of
a hyper-Kähler
manifold
36
1.5.1
The period map
..................... 36
1.5.2
A vanishing result for polynomials on H2
....... 37
2.
Graph homology
39
2.1
The space of graph homology
................. 39
2.1.1
Jacobi diagrams
..................... 39
2.1.2
Chains of Jacobi diagrams
............... 41
2.1.3
Glueing legs and product of Jacobi diagrams
..... 42
2.1.4
Subspaces and ideals
.................. 44
2.1.5
The graph homology spaces
.............. 45
2.2
Symmetric monoidal categories
................ 47
2.2.1
Definition
........................ 47
2.2.2
fc-linear categories
.................... 49
2.2.3
Global sections
..................... 50
2.2.4
External tensor and symmetric algebras
....... 51
2.3
Metric Lie algebra objects
................... 52
2.3.1
Definition
........................ 52
2.3.2
Examples from the category of vector spaces
..... 54
2.3.3
Morphisms between tensor powers of metric Lie alge¬
bra objects
....................... 55
2.3.4
The PROP of metric Lie algebras
........... 59
2.3.5
The universality of the PROP of metric Lie algebras
61
2.4
Weight systems
......................... 64
2.4.1
Definition
........................ 65
2.4.2
Constructions of weight systems
............ 65
2.4.3
Modules of metric Lie algebra objects
......... 66
2.5
Operation with graphs and special graphs
.......... 67
2.5.1
Special graphs
...................... 67
2.5.2
Operations
........................ 68
2.5.3
Closed and connected graphs
............. 72
2.5.4
Polywheels
........................ 75
Contents xxi
2.5.5 The Hopf
algebra structure on the space of graph ho-
mology
.......................... 76
2.6
The Wheeling Theorem
.................... 79
2.6.1
The wheeling element
Ω
................ 79
2.6.2
Wheeling and the Wheeling Theorem
......... 80
3.
Rozansky-Witten theory
83
3.1
The Rozansky-Witten weight system
............. 83
3.1.1
The derived category
.................. 83
3.1.2
A metric Lie algebra object in the derived category
. 85
3.1.3
Rozansky-Witten weight systems
........... 88
3.1.4
Properties of the Rozansky-Witten weight system
. . 90
3.1.5
An inner product on the cohomology of a holomorphic
symplectic manifold
................... 92
3.1.6
Rozansky-Witten invariants
.............. 94
3.1.7
Complex genera and Rozansky-Witten invariants
. . 95
3.2
Some applications
....................... 99
3.2.1
Chebyshev polynomials
................. 99
3.2.2
An application of the Wheeling Theorem
....... 100
3.2.3
On the genus td5(a)td5(/?) of an irreducible holo¬
morphic symplectic manifold
.............. 101
3.2.4
The
Ьг-погт
of the Riemannian curvature tensor of
a compact
hyper-Kähler
manifold
........... 105
3.2.5
The Beauville-Bogomolov form
............ 105
3.2.6
A Hirzebruch-Riemann-Roch formula
........ 107
4.
Calculations for the example series
109
4.1
More on the geometry of the Hubert schemes of points on
surfaces
............................. 109
4.1.1
The universal family
.................. 109
4.1.2
The incidence variety
XÍn n+^
............. 110
4.1.3
Calculations in various K-groups
........... 112
4.1.4
Chem
numbers of the Hubert schemes
........ 113
4.2
Genera of Hubert schemes of points on surfaces
....... 116
4.2.1
Two decomposition results
............... 116
4.2.2
A structural result on genera of Hubert schemes of
points on surfaces
....................
H
4.2.3
Genera of the generalised
Kummer
varieties
..... 121
xxii
Chern Numbers and RW-Invariants of Compact
Hyper-Kähler
Manifolds
4.3
Calculations of the power series
Аф, Вф, С
ψ
and
D
φ
.... 123
4.3.1
Bott s residue formula
................. 123
4.3.2
How to calculate
Сф
and
Ώφ
.............. 125
4.3.3
The calculation of
Αφ
and
Вф
............. 126
4.3.4
Chern numbers for the example series
........ 128
4.4
Calculations of Rozansky-Witten invariants
......... 128
4.4.1
A lemma from
umbral
calculus
............ 129
4.4.2
More on Rozansky-Witten invariants of closed graph
homology classes
.................... 131
4.4.3
A structural result on the Rozansky-Witten weights
of closed connected graphs on the example series
. . 134
4.4.4
Explicit calculation
................... 136
Bibliography
141
Index
143
|
| adam_txt |
Contents
Preface vii
Introduction
ix
Notation
xiii
1.
Compact
hyper-Kähler
manifolds and holomorphic sym-
plectic manifolds
1
1.1
Basics on compact
hyper-Kähler
manifolds
. 1
1.1.1
Hokmomy of Riemannian manifolds
. 1
1.1.2
Definition of a compact
hyper-Kähler
manifold
. 3
1.1.3
Holomorphic symplectic manifolds
. 4
1.1.4
Deformations of compact complex manifolds
. 7
1.2
Examples
. 8
1.2.1
The
КЗ
surface
. 8
1.2.2
The Hubert scheme of points on a surface
. 11
1.2.3
Construction of line bundles and classes in H2 on the
Hubert schemes of points on surfaces
. 13
1.2.4
Hubert schemes of points on
КЗ
surfaces
. 15
1.2.5
Generalised
Kummer
varieties
. 16
1.2.6
Further examples
. 19
1.3
Characteristic classes
. 20
1.3.1
Symplectic sheaves
. 20
1.3.2
Characteristic classes of symplectic sheaves
. 21
1.3.3
Chern numbers of holomorphic symplectic manifolds
24
1.4
The Atiyah class
. 26
xx
Chern Numbers and RW-Invariants of Compact
Hyper-Kăhler
Manifolds
1.4.1
Definition
. 26
1.4.2
Description in terms of
Čech
cohomology
. 28
1.4.3
The
Bianchi
identity
. 29
1.4.4
Torsion and the Atiyah class of the tangent bundle
. 31
1.4.5
The Atiyah class of symplectic sheaves
. 32
1.4.6
Chern-Weil theory
. 34
1.5
On the second cohomology group of
a hyper-Kähler
manifold
36
1.5.1
The period map
. 36
1.5.2
A vanishing result for polynomials on H2
. 37
2.
Graph homology
39
2.1
The space of graph homology
. 39
2.1.1
Jacobi diagrams
. 39
2.1.2
Chains of Jacobi diagrams
. 41
2.1.3
Glueing legs and product of Jacobi diagrams
. 42
2.1.4
Subspaces and ideals
. 44
2.1.5
The graph homology spaces
. 45
2.2
Symmetric monoidal categories
. 47
2.2.1
Definition
. 47
2.2.2
fc-linear categories
. 49
2.2.3
Global sections
. 50
2.2.4
External tensor and symmetric algebras
. 51
2.3
Metric Lie algebra objects
. 52
2.3.1
Definition
. 52
2.3.2
Examples from the category of vector spaces
. 54
2.3.3
Morphisms between tensor powers of metric Lie alge¬
bra objects
. 55
2.3.4
The PROP of metric Lie algebras
. 59
2.3.5
The universality of the PROP of metric Lie algebras
61
2.4
Weight systems
. 64
2.4.1
Definition
. 65
2.4.2
Constructions of weight systems
. 65
2.4.3
Modules of metric Lie algebra objects
. 66
2.5
Operation with graphs and special graphs
. 67
2.5.1
Special graphs
. 67
2.5.2
Operations
. 68
2.5.3
Closed and connected graphs
. 72
2.5.4
Polywheels
. 75
Contents xxi
2.5.5 The Hopf
algebra structure on the space of graph ho-
mology
. 76
2.6
The Wheeling Theorem
. 79
2.6.1
The wheeling element
Ω
. 79
2.6.2
Wheeling and the Wheeling Theorem
. 80
3.
Rozansky-Witten theory
83
3.1
The Rozansky-Witten weight system
. 83
3.1.1
The derived category
. 83
3.1.2
A metric Lie algebra object in the derived category
. 85
3.1.3
Rozansky-Witten weight systems
. 88
3.1.4
Properties of the Rozansky-Witten weight system
. . 90
3.1.5
An inner product on the cohomology of a holomorphic
symplectic manifold
. 92
3.1.6
Rozansky-Witten invariants
. 94
3.1.7
Complex genera and Rozansky-Witten invariants
. . 95
3.2
Some applications
. 99
3.2.1
Chebyshev polynomials
. 99
3.2.2
An application of the Wheeling Theorem
. 100
3.2.3
On the genus td5(a)td5(/?) of an irreducible holo¬
morphic symplectic manifold
. 101
3.2.4
The
Ьг-погт
of the Riemannian curvature tensor of
a compact
hyper-Kähler
manifold
. 105
3.2.5
The Beauville-Bogomolov form
. 105
3.2.6
A Hirzebruch-Riemann-Roch formula
. 107
4.
Calculations for the example series
109
4.1
More on the geometry of the Hubert schemes of points on
surfaces
. 109
4.1.1
The universal family
. 109
4.1.2
The incidence variety
XÍn'n+^
. 110
4.1.3
Calculations in various K-groups
. 112
4.1.4
Chem
numbers of the Hubert schemes
. 113
4.2
Genera of Hubert schemes of points on surfaces
. 116
4.2.1
Two decomposition results
. 116
4.2.2
A structural result on genera of Hubert schemes of
points on surfaces
.
H"
4.2.3
Genera of the generalised
Kummer
varieties
. 121
xxii
Chern Numbers and RW-Invariants of Compact
Hyper-Kähler
Manifolds
4.3
Calculations of the power series
Аф, Вф, С
ψ
and
D
φ
. 123
4.3.1
Bott's residue formula
. 123
4.3.2
How to calculate
Сф
and
Ώφ
. 125
4.3.3
The calculation of
Αφ
and
Вф
. 126
4.3.4
Chern numbers for the example series
. 128
4.4
Calculations of Rozansky-Witten invariants
. 128
4.4.1
A lemma from
umbral
calculus
. 129
4.4.2
More on Rozansky-Witten invariants of closed graph
homology classes
. 131
4.4.3
A structural result on the Rozansky-Witten weights
of closed connected graphs on the example series
. . 134
4.4.4
Explicit calculation
. 136
Bibliography
141
Index
143 |
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| author | Nieper-Wißkirchen, Marc Arnold |
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| id | DE-604.BV021241795 |
| illustrated | Illustrated |
| index_date | 2024-07-02T13:31:26Z |
| indexdate | 2024-07-09T20:28:36Z |
| institution | BVB |
| isbn | 9812388516 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014284486 |
| oclc_num | 255389988 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-11 |
| physical | XXII, 150 S. graph. Darst. |
| publishDate | 2004 |
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| publisher | World Scientific |
| record_format | marc |
| spelling | Nieper-Wißkirchen, Marc Arnold Verfasser (DE-588)124200486 aut Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds Marc Nieper-Wißkirchen New Jersey [u.a.] World Scientific 2004 XXII, 150 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd rswk-swf Chern-Zahl (DE-588)4322052-6 gnd rswk-swf Invariante (DE-588)4128781-2 gnd rswk-swf Kähler-Mannigfaltigkeit (DE-588)4162978-4 s Chern-Zahl (DE-588)4322052-6 s Invariante (DE-588)4128781-2 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014284486&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nieper-Wißkirchen, Marc Arnold Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd Chern-Zahl (DE-588)4322052-6 gnd Invariante (DE-588)4128781-2 gnd |
| subject_GND | (DE-588)4162978-4 (DE-588)4322052-6 (DE-588)4128781-2 |
| title | Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds |
| title_auth | Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds |
| title_exact_search | Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds |
| title_exact_search_txtP | Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds |
| title_full | Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds Marc Nieper-Wißkirchen |
| title_fullStr | Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds Marc Nieper-Wißkirchen |
| title_full_unstemmed | Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds Marc Nieper-Wißkirchen |
| title_short | Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds |
| title_sort | chern numbers and rozansky witten invariants of compact hyper kahler manifolds |
| topic | Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd Chern-Zahl (DE-588)4322052-6 gnd Invariante (DE-588)4128781-2 gnd |
| topic_facet | Kähler-Mannigfaltigkeit Chern-Zahl Invariante |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014284486&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT nieperwißkirchenmarcarnold chernnumbersandrozanskywitteninvariantsofcompacthyperkahlermanifolds |