Lectures on Kähler geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2007
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society student texts
69 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 171 S. |
| ISBN: | 9780521688970 9780521868914 |
Staff View
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| 245 | 1 | 0 | |a Lectures on Kähler geometry |c Andrei Moroianu |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 2007 | |
| 300 | |a IX, 171 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a London Mathematical Society student texts |v 69 | |
| 650 | 4 | |a Kählerian manifolds | |
| 650 | 0 | 7 | |a Kähler-Geometrie |0 (DE-588)4631472-6 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804136400044425216 |
|---|---|
| adam_text | Contents
Introduction
ix
Part
1. Basics
of differential geometry
1
Chapter
1.
Smooth manifolds
3
1.1.
Introduction
3
1.2.
The tangent space
4
J.3. Vector fields
б
1.4.
Exercises
9
Chapter
2.
Tensor ftelds on smooth manifolds
13
2.1.
Exterior and tensor algebras
13
2.2.
Tensor fields
15
2.3.
Lie derivative of tensors
17
2.4.
Exercises
19
Chapter
3.
The exterior derivative
21
3.1.
Exterior forms
21
3.2.
The exterior derivative
21
3.3.
The
Cartari
formula
23
3.4.
Integration
24
3.5.
Exercises
26
Chapter
4.
Principal and vector bundles
29
4.1.
Lie groups
29
4.2.
Principal bundles
31
4.3.
Vector bundles
33
4.4.
Correspondence between principal and vector bundles
33
4.5.
Exercises
35
Chapter
5.
Connections
37
5.1.
Covariant derivatives on vector bundles
37
5.2.
Connections on principal bundles
39
5.3.
Linear connections
41
5.4.
Pull-back of bundles
41
5.5.
Parallel transport
42
5.G. Holonomy
43
5.7.
Reduction of connections
44
vi
Contents
5.8.
Exercises
45
Chapter
6.
Riemaniiian manifolds
47
6.1.
Ricinannian
metrics
47
6.2.
Tlie
Levi- Civita
connection
48
6.3.
The curvature tensor
49
6.4.
Killing vector fields
51
6.5.
Exercises
52
Part
2.
Complex and Hermitian geometry
55
Chapter
7.
Complex structures and holomorphic maps
57
7.1.
Preliminaries
57
7.2.
Holomorphic functions
59
7.3.
Complex manifolds
59
7.4.
The complexified tangent bundle
61
7.5.
Exercises
62
Chapter
8.
Holomorphic forms and vector fields
65
8.1.
Decomposition of the (complexified) exterior bundle
65
8.2.
Holomorphic objects on complex manifolds
67
8.3.
Exercises
68
Chapter
9.
Complex and holomorphic vector bundles
71
9.1.
Holomorphic vector bundles
71
9.2.
Holomorphic structures
72
9.3.
The canonical bundle of CPW
74
9.4.
Exercises
75
Chapter
10.
Hermitian bundles
77
10.1.
The curvature operator of a connection
77
10.2.
Hermitian structures and connections
78
10.3.
Exercises
80
Chapter
11.
Hermit
ian
and
Kahler
metrics
81
11.1.
Hermitian metrics
81
11.2. Kahler
metrics
82
11.3.
Characterization of
Kahler
metrics
83
11.4.
Comparison of the
Levi
Civita and Chcrn connections
85
11.5.
Exercises
86
Chapter
12.
The curvature tensor of
Kahler
manifolds
87
12.1.
The
Kählerian
curvature tensor
87
12.2.
The curvature tensor in local coordinates
88
12.3.
Exercises
91
Chapter
13.
Examples of
Kahler
metrics
93
Contents
vii
13.1.
The flat metric on
С
93
13.2.
Tlie
Fubini- Study metric on the complex projective space
93
13.3.
Geometrical properties of the Fubini-Study metric
95
13.4.
Exercises
97
Chapter
14.
Natural operators on R.iemannian and
Kahler
manifolds
99
14.1.
The formal adjoint of a linear differential operator
99
14.2.
The Laplace operator on Riemannian manifolds
100
14.3.
The Laplace operator on
Kahler
manifolds
101
14.4.
Exercises
104
Chapter
15.
Hodge and Dolbeault theories
105
15.1.
Hodge theory
105
15.2.
Dolbeault theory
107
15.3.
Exercises
109
Part
3.
Topics on compact
Kahler
manifolds 111
Chapter
16.
Cliern classes
113
16.1.
Chern-Weil theory
113
16.2.
Properties of the
hrst
Chern class
116
16.3.
Exercises
118
Chapter
17.
The
Ricci form
of
Kahler
manifolds
119
17.1. Kahler
metrics as geometric
Um-structures
119
17.2.
The
Ricci form
as curvature form on the canonical bundle
119
17.3.
Ricci-fìat
Kahler
manifolds
121
17.4.
Exercises
122
Chapter
18.
The Calabi-Yau theorem
125
18.1.
An overview
125
18.2.
Exercises
127
Chapter
19. Kahler-
Einstein metrics
129
19.1.
The
Aubin
■
Yau theorem
129
19.2.
Holomorphic vector fields on
Kühler
Einstein manifolds
131
19.3.
Exercises
133
Chapter
20. Wcitzenböck
techniques
135
20.1.
The
Wcitzenböck
formula
135
20.2.
Vanishing results on
Kahler
manifolds
137
20.3.
Exercises
139
Chapter
21.
The Hirzebruch-Riemann-Roch formula
141
21.1.
Positive line bundles
141
21.2.
The Hirzebruch-Riemami
......Roch
formula
142
21.3.
Exercises
145
viii
Contents
Chapter
22.
Further vanishing results
. 147
22.1.
The Lichnerowicz formula for
Kahler
manifolds
147
22.2.
The Kodaira vanishing theorem
149
22.3.
Exercises
151
Chapter
23.
Ricci-flat
Kahler
metrics
153
23.1. Hyperkähler
manifolds
153
23.2.
Projective
manifolds
155
23.3.
Exercises
156
Chapter
24.
Explicit examples of Calabi-Yau manifolds
159
24.1.
Divisors
159
24.2.
Line bundles and divisors
161
24.3.
Adjunction formulas
162
24.4.
Exercises
165
Bibliography
167
Index
169
|
| adam_txt |
Contents
Introduction
ix
Part
1. Basics
of differential geometry
1
Chapter
1.
Smooth manifolds
3
1.1.
Introduction
3
1.2.
The tangent space
4
J.3. Vector fields
б
1.4.
Exercises
9
Chapter
2.
Tensor ftelds on smooth manifolds
13
2.1.
Exterior and tensor algebras
13
2.2.
Tensor fields
15
2.3.
Lie derivative of tensors
17
2.4.
Exercises
19
Chapter
3.
The exterior derivative
21
3.1.
Exterior forms
21
3.2.
The exterior derivative
21
3.3.
The
Cartari
formula
23
3.4.
Integration
24
3.5.
Exercises
26
Chapter
4.
Principal and vector bundles
29
4.1.
Lie groups
29
4.2.
Principal bundles
31
4.3.
Vector bundles
33
4.4.
Correspondence between principal and vector bundles
33
4.5.
Exercises
35
Chapter
5.
Connections
37
5.1.
Covariant derivatives on vector bundles
37
5.2.
Connections on principal bundles
39
5.3.
Linear connections
41
5.4.
Pull-back of bundles
41
5.5.
Parallel transport
42
5.G. Holonomy
43
5.7.
Reduction of connections
44
vi
Contents
5.8.
Exercises
45
Chapter
6.
Riemaniiian manifolds
47
6.1.
Ricinannian
metrics
47
6.2.
Tlie
Levi- Civita
connection
48
6.3.
The curvature tensor
49
6.4.
Killing vector fields
51
6.5.
Exercises
52
Part
2.
Complex and Hermitian geometry
55
Chapter
7.
Complex structures and holomorphic maps
57
7.1.
Preliminaries
57
7.2.
Holomorphic functions
59
7.3.
Complex manifolds
59
7.4.
The complexified tangent bundle
61
7.5.
Exercises
62
Chapter
8.
Holomorphic forms and vector fields
65
8.1.
Decomposition of the (complexified) exterior bundle
65
8.2.
Holomorphic objects on complex manifolds
67
8.3.
Exercises
68
Chapter
9.
Complex and holomorphic vector bundles
71
9.1.
Holomorphic vector bundles
71
9.2.
Holomorphic structures
72
9.3.
The canonical bundle of CPW
74
9.4.
Exercises
75
Chapter
10.
Hermitian bundles
77
10.1.
The curvature operator of a connection
77
10.2.
Hermitian structures and connections
78
10.3.
Exercises
80
Chapter
11.
Hermit
ian
and
Kahler
metrics
81
11.1.
Hermitian metrics
81
11.2. Kahler
metrics
82
11.3.
Characterization of
Kahler
metrics
83
11.4.
Comparison of the
Levi
Civita and Chcrn connections
85
11.5.
Exercises
86
Chapter
12.
The curvature tensor of
Kahler
manifolds
87
12.1.
The
Kählerian
curvature tensor
87
12.2.
The curvature tensor in local coordinates
88
12.3.
Exercises
91
Chapter
13.
Examples of
Kahler
metrics
93
Contents
vii
13.1.
The flat metric on
С"'
93
13.2.
Tlie
Fubini- Study metric on the complex projective space
93
13.3.
Geometrical properties of the Fubini-Study metric
95
13.4.
Exercises
97
Chapter
14.
Natural operators on R.iemannian and
Kahler
manifolds
99
14.1.
The formal adjoint of a linear differential operator
99
14.2.
The Laplace operator on Riemannian manifolds
100
14.3.
The Laplace operator on
Kahler
manifolds
101
14.4.
Exercises
104
Chapter
15.
Hodge and Dolbeault theories
105
15.1.
Hodge theory
105
15.2.
Dolbeault theory
107
15.3.
Exercises
109
Part
3.
Topics on compact
Kahler
manifolds 111
Chapter
16.
Cliern classes
113
16.1.
Chern-Weil theory
113
16.2.
Properties of the
hrst
Chern class
116
16.3.
Exercises
118
Chapter
17.
The
Ricci form
of
Kahler
manifolds
119
17.1. Kahler
metrics as geometric
Um-structures
119
17.2.
The
Ricci form
as curvature form on the canonical bundle
119
17.3.
Ricci-fìat
Kahler
manifolds
121
17.4.
Exercises
122
Chapter
18.
The Calabi-Yau theorem
125
18.1.
An overview
125
18.2.
Exercises
127
Chapter
19. Kahler-
Einstein metrics
129
19.1.
The
Aubin
■
Yau theorem
129
19.2.
Holomorphic vector fields on
Kühler
Einstein manifolds
131
19.3.
Exercises
133
Chapter
20. Wcitzenböck
techniques
135
20.1.
The
Wcitzenböck
formula
135
20.2.
Vanishing results on
Kahler
manifolds
137
20.3.
Exercises
139
Chapter
21.
The Hirzebruch-Riemann-Roch formula
141
21.1.
Positive line bundles
141
21.2.
The Hirzebruch-Riemami
.Roch
formula
142
21.3.
Exercises
145
viii
Contents
Chapter
22.
Further vanishing results
. 147
22.1.
The Lichnerowicz formula for
Kahler
manifolds
147
22.2.
The Kodaira vanishing theorem
149
22.3.
Exercises
151
Chapter
23.
Ricci-flat
Kahler
metrics
153
23.1. Hyperkähler
manifolds
153
23.2.
Projective
manifolds
155
23.3.
Exercises
156
Chapter
24.
Explicit examples of Calabi-Yau manifolds
159
24.1.
Divisors
159
24.2.
Line bundles and divisors
161
24.3.
Adjunction formulas
162
24.4.
Exercises
165
Bibliography
167
Index "
169 |
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| id | DE-604.BV022354259 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:00:54Z |
| indexdate | 2024-07-09T20:55:47Z |
| institution | BVB |
| isbn | 9780521688970 9780521868914 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015563697 |
| oclc_num | 77012505 |
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| physical | IX, 171 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | London Mathematical Society student texts |
| series2 | London Mathematical Society student texts |
| spelling | Moroianu, Andrei 1971- Verfasser (DE-588)114940053 aut Lectures on Kähler geometry Andrei Moroianu 1. publ. Cambridge Cambridge Univ. Press 2007 IX, 171 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society student texts 69 Kählerian manifolds Kähler-Geometrie (DE-588)4631472-6 gnd rswk-swf Kähler-Geometrie (DE-588)4631472-6 s DE-604 London Mathematical Society student texts 69 (DE-604)BV000841726 69 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015563697&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Moroianu, Andrei 1971- Lectures on Kähler geometry London Mathematical Society student texts Kählerian manifolds Kähler-Geometrie (DE-588)4631472-6 gnd |
| subject_GND | (DE-588)4631472-6 |
| title | Lectures on Kähler geometry |
| title_auth | Lectures on Kähler geometry |
| title_exact_search | Lectures on Kähler geometry |
| title_exact_search_txtP | Lectures on Kähler geometry |
| title_full | Lectures on Kähler geometry Andrei Moroianu |
| title_fullStr | Lectures on Kähler geometry Andrei Moroianu |
| title_full_unstemmed | Lectures on Kähler geometry Andrei Moroianu |
| title_short | Lectures on Kähler geometry |
| title_sort | lectures on kahler geometry |
| topic | Kählerian manifolds Kähler-Geometrie (DE-588)4631472-6 gnd |
| topic_facet | Kählerian manifolds Kähler-Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015563697&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000841726 |
| work_keys_str_mv | AT moroianuandrei lecturesonkahlergeometry |