Hodge theory and complex algebraic geometry: 1
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| Main Author: | |
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| Format: | Book |
| Language: | English French |
| Published: |
Cambridge
Cambridge University Press
2007
|
| Edition: | 1. publ., reprint., transferred to digital print. |
| Series: | Cambridge studies in advanced mathematics
76 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 322 S. |
| ISBN: | 0521802601 9780521802604 9780521718011 |
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| 245 | 1 | 0 | |a Hodge theory and complex algebraic geometry |n 1 |c Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) |
| 250 | |a 1. publ., reprint., transferred to digital print. | ||
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2007 | |
| 300 | |a IX, 322 S. | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 76 | |
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Record in the Search Index
| _version_ | 1804137256871526400 |
|---|---|
| adam_text | Contents
0
Introduction page
1
1 Preliminaries
19
1
Holomorphic
Functions of Many
Variables
21
1.1
Holomorphic functions of one variable
22
1.1.1
Definition and basic properties
22
1.1.2
Background on Stokes formula
24
1.1.3
Cauchy s formula
27
1.2
Holomorphic functions of several variables
28
1.2.1
Cauchy s formula and
analyticky
28
1.2.2
Applications of Cauchy
s
formula
30
1.3
The equation
§§ = ƒ 35
Exercises
37
2
Complex Manifolds
38
2.1
Manifolds and vector bundles
39
2.1.1
Definitions
39
2.1.2
The tangent bundle
41
2.1.3
Complex manifolds
43
2.2
Integrability of almost complex structures
44
2.2.1
Tangent bundle of a complex manifold
44
2.2.2
The Frobenius theorem
46
2.2.3
The Newlander-Nirenberg theorem
50
2.3
The operators
9
and
õ
53
2.3.1
Definition
53
2.3.2
Local exactness
55
2.3.3
Dolbeault complex of a holomorphic bundle
57
2.4
Examples of complex manifolds
59
Exercises
61
vi
Contents
3 Kahler
Metrics
63
3.1
Definition and basic properties
64
3.1.1
Hermitian geometry
64
3.1.2
Hermitian and
Kahler
metrics
66
3.1.3
Basic properties
67
3.2
Characterisations of
Kahler
metrics
69
3.2.1
Background on connections
69
3.2.2 Kahler
metrics and connections
71
3.3
Examples of
Kahler
manifolds
75
3.3.1
Chern form of line bundles
75
3.3.2
Fubini-Study metric
76
3.3.3
Blowups
78
Exercises
82
4
Sheaves and Cohomology
83
4.1
Sheaves
85
4.1.1
Definitions, examples
85
4.1.2
Stalks, kernels, images
89
4.1.3
Resolutions
91
4.2
Functors and derived functors
95
4.2.1
Abelian categories
95
4.2.2
Injective resolutions
96
4.2.3
Derived functors
99
4.3
Sheaf cohomology
102
4.3.1
Acyclic resolutions
103
4.3.2
The
de Rham
theorems
108
4.3.3
Interpretations of the group
H
110
Exercises
113
II The Hodge Decomposition
115
5
Harmonic Forms and Cohomology
117
5.1
Laplacians
119
5.1.1
The L2 metric
119
5.1.2
Formal adjoint operators
121
5.1.3
Adjoints
of the operators
9 121
5.1.4
Laplacians
124
5.2
Elliptic differential operators
125
5.2.1
Symbols of differential operators
125
5.2.2
Symbol of the Laplacian
126
5.2.3
The fundamental theorem
128
5.3
Applications
129
5.3.1
Cohomology and harmonic forms
129
Contents
vii
5.3.2
Duality theorems
130
Exercises
136
The Case of
Kahler
Manifolds
137
6.1
The Hodge decomposition
139
6.1.1 Kahler
identities
139
6.1.2
Comparison of the Laplacians
141
6.1.3
Other applications
142
6.2
Lefschetz decomposition
144
6.2.1
Commutators
144
6.2.2
Lefschetz decomposition on forms
146
6.2.3
Lefschetz decomposition on the cohomology
148
6.3
The Hodge index theorem
150
6.3.1
Other Hermitian identities
150
6.3.2
The Hodge index theorem
152
Exercises
154
Hodge Structures and Polarisations
156
7.1
Definitions, basic properties
157
7.1.1
Hodge structure
157
7.1.2
Polarisation
160
7.1.3
Polarised varieties
161
7.2
Examples
167
7.2.1
Projective
space
167
7.2.2
Hodge structures of weight
1
and abelian varieties
168
7.2.3
Hodge structures of weight
2 170
7.3
Functoriality
174
7.3.1
Morphisms of Hodge structures
174
7.3.2
The pullback and the Gysin morphism
176
7.3.3
Hodge structure of a blowup
180
Exercises
182
Holomorphic
de Rham
Complexes and Spectral Sequences
184
8.1
Hypercohomology
186
8.1.1
Resolutions of complexes
186
8.1.2
Derived functors
189
8.1.3
Composed functors
194
8.2
Holomorphic
de Rham
complexes
196
8.2.1
Holomorphic
de
Rham resolutions
196
8.2.2
The logarithmic case
197
8.2.3
Cohomology of the logarithmic complex
198
8.3
Filtrations and spectral sequences
200
8.3.1
Filtered complexes
200
viii Contents
8.3.2
Spectral
sequences
201
8.3.3
The Frölicher
spectral sequence
204
8.4
Hodge theory of open manifolds
207
8.4.1
Filtratíons
on the logarithmic complex
207
8.4.2
First terms of the spectral sequence
208
8.4.3
Deligne s theorem
213
Exercises
214
III Variations of Hodge Structure
217
9
Families and Deformations
219
9.1
Families of manifolds
220
9.1.1
Trivialisations
220
9.1.2
The Kodaira-Spencer map
223
9.2
The Gauss-Manin connection
228
9.2.1
Local systems and flat connections
228
9.2.2
The Cartan-Lie formula
231
9.3
The
Kahler
case
232
9.3.1
Semicontinuity theorems
232
9.3.2
The Hodge numbers are constant
235
9.3.3
Stability of
Kahler
manifolds
236
10
Variations of Hodge Structure
239
10.1
Period domain and period map
240
10.1.1
Grassmannians
240
10.1.2
The period map
243
10.1.3
The period domain
246
10.2
Variations of Hodge structure
249
10.2.1
Hodge bundles
249
10.2.2
Transversality
250
10.2.3
Computation of the differential
251
10.3
Applications
254
10.3.1
Curves
254
10.3.2
Calabi-Yau manifolds
258
Exercises
259
IV Cycles and Cycle Classes
261
11
Hodge Classes
263
11.1
Cycle class
264
11.1.1
Analytic subsets
264
11.1.2
Cohomology class
269
11.1.3
The
Kahler
case
273
11.1.4
Other approaches
275
Contents ix
12
11.2 Chern
classes
276
11.2.1
Construction
276
11.2.2
The
Kahler
case
279
11.3
Hodge classes
279
11.3.1
Definitions and examples
279
11.3.2
The Hodge conjecture
284
11.3.3
Correspondences
285
Exercises
287
Deligne-Beilinson Cohomology and the Abel-Jacobi Map
290
12.1
The Abel-Jacobi map
291
12.1.1
Intermediate Jacobians
291
12.1.2
The Abel-Jacobi map
292
12.1.3
Picard
and
Albanese
varieties
296
12.2
Properties
300
12.2.1
Correspondences
300
12.2.2
Some results
302
12.3
Deligne cohomology
304
12.3.1
The Deligne complex
304
12.3.2
Differential characters
306
12.3.3
Cycle class
310
Exercises
313
Bibliography
315
Index
319
|
| adam_txt |
Contents
0
Introduction page
1
1 Preliminaries
19
1
Holomorphic
Functions of Many
Variables
21
1.1
Holomorphic functions of one variable
22
1.1.1
Definition and basic properties
22
1.1.2
Background on Stokes' formula
24
1.1.3
Cauchy's formula
27
1.2
Holomorphic functions of several variables
28
1.2.1
Cauchy's formula and
analyticky
28
1.2.2
Applications of Cauchy
's
formula
30
1.3
The equation
§§ = ƒ 35
Exercises
37
2
Complex Manifolds
38
2.1
Manifolds and vector bundles
39
2.1.1
Definitions
39
2.1.2
The tangent bundle
41
2.1.3
Complex manifolds
43
2.2
Integrability of almost complex structures
44
2.2.1
Tangent bundle of a complex manifold
44
2.2.2
The Frobenius theorem
46
2.2.3
The Newlander-Nirenberg theorem
50
2.3
The operators
9
and
õ
53
2.3.1
Definition
53
2.3.2
Local exactness
55
2.3.3
Dolbeault complex of a holomorphic bundle
57
2.4
Examples of complex manifolds
59
Exercises
61
vi
Contents
3 Kahler
Metrics
63
3.1
Definition and basic properties
64
3.1.1
Hermitian geometry
64
3.1.2
Hermitian and
Kahler
metrics
66
3.1.3
Basic properties
67
3.2
Characterisations of
Kahler
metrics
69
3.2.1
Background on connections
69
3.2.2 Kahler
metrics and connections
71
3.3
Examples of
Kahler
manifolds
75
3.3.1
Chern form of line bundles
75
3.3.2
Fubini-Study metric
76
3.3.3
Blowups
78
Exercises
82
4
Sheaves and Cohomology
83
4.1
Sheaves
85
4.1.1
Definitions, examples
85
4.1.2
Stalks, kernels, images
89
4.1.3
Resolutions
91
4.2
Functors and derived functors
95
4.2.1
Abelian categories
95
4.2.2
Injective resolutions
96
4.2.3
Derived functors
99
4.3
Sheaf cohomology
102
4.3.1
Acyclic resolutions
103
4.3.2
The
de Rham
theorems
108
4.3.3
Interpretations of the group
H
' 110
Exercises
113
II The Hodge Decomposition
115
5
Harmonic Forms and Cohomology
117
5.1
Laplacians
119
5.1.1
The L2 metric
119
5.1.2
Formal adjoint operators
121
5.1.3
Adjoints
of the operators
9 121
5.1.4
Laplacians
124
5.2
Elliptic differential operators
125
5.2.1
Symbols of differential operators
125
5.2.2
Symbol of the Laplacian
126
5.2.3
The fundamental theorem
128
5.3
Applications
129
5.3.1
Cohomology and harmonic forms
129
Contents
vii
5.3.2
Duality theorems
130
Exercises
136
The Case of
Kahler
Manifolds
137
6.1
The Hodge decomposition
139
6.1.1 Kahler
identities
139
6.1.2
Comparison of the Laplacians
141
6.1.3
Other applications
142
6.2
Lefschetz decomposition
144
6.2.1
Commutators
144
6.2.2
Lefschetz decomposition on forms
146
6.2.3
Lefschetz decomposition on the cohomology
148
6.3
The Hodge index theorem
150
6.3.1
Other Hermitian identities
150
6.3.2
The Hodge index theorem
152
Exercises
154
Hodge Structures and Polarisations
156
7.1
Definitions, basic properties
157
7.1.1
Hodge structure
157
7.1.2
Polarisation
160
7.1.3
Polarised varieties
161
7.2
Examples
167
7.2.1
Projective
space
167
7.2.2
Hodge structures of weight
1
and abelian varieties
168
7.2.3
Hodge structures of weight
2 170
7.3
Functoriality
174
7.3.1
Morphisms of Hodge structures
174
7.3.2
The pullback and the Gysin morphism
176
7.3.3
Hodge structure of a blowup
180
Exercises
182
Holomorphic
de Rham
Complexes and Spectral Sequences
184
8.1
Hypercohomology
186
8.1.1
Resolutions of complexes
186
8.1.2
Derived functors
189
8.1.3
Composed functors
194
8.2
Holomorphic
de Rham
complexes
196
8.2.1
Holomorphic
de
Rham resolutions
196
8.2.2
The logarithmic case
197
8.2.3
Cohomology of the logarithmic complex
198
8.3
Filtrations and spectral sequences
200
8.3.1
Filtered complexes
200
viii Contents
8.3.2
Spectral
sequences
201
8.3.3
The Frölicher
spectral sequence
204
8.4
Hodge theory of open manifolds
207
8.4.1
Filtratíons
on the logarithmic complex
207
8.4.2
First terms of the spectral sequence
208
8.4.3
Deligne's theorem
213
Exercises
214
III Variations of Hodge Structure
217
9
Families and Deformations
219
9.1
Families of manifolds
220
9.1.1
Trivialisations
220
9.1.2
The Kodaira-Spencer map
223
9.2
The Gauss-Manin connection
228
9.2.1
Local systems and flat connections
228
9.2.2
The Cartan-Lie formula
231
9.3
The
Kahler
case
232
9.3.1
Semicontinuity theorems
232
9.3.2
The Hodge numbers are constant
235
9.3.3
Stability of
Kahler
manifolds
236
10
Variations of Hodge Structure
239
10.1
Period domain and period map
240
10.1.1
Grassmannians
240
10.1.2
The period map
243
10.1.3
The period domain
246
10.2
Variations of Hodge structure
249
10.2.1
Hodge bundles
249
10.2.2
Transversality
250
10.2.3
Computation of the differential
251
10.3
Applications
254
10.3.1
Curves
254
10.3.2
Calabi-Yau manifolds
258
Exercises
259
IV Cycles and Cycle Classes
261
11
Hodge Classes
263
11.1
Cycle class
264
11.1.1
Analytic subsets
264
11.1.2
Cohomology class
269
11.1.3
The
Kahler
case
273
11.1.4
Other approaches
275
Contents ix
12
11.2 Chern
classes
276
11.2.1
Construction
276
11.2.2
The
Kahler
case
279
11.3
Hodge classes
279
11.3.1
Definitions and examples
279
11.3.2
The Hodge conjecture
284
11.3.3
Correspondences
285
Exercises
287
Deligne-Beilinson Cohomology and the Abel-Jacobi Map
290
12.1
The Abel-Jacobi map
291
12.1.1
Intermediate Jacobians
291
12.1.2
The Abel-Jacobi map
292
12.1.3
Picard
and
Albanese
varieties
296
12.2
Properties
300
12.2.1
Correspondences
300
12.2.2
Some results
302
12.3
Deligne cohomology
304
12.3.1
The Deligne complex
304
12.3.2
Differential characters
306
12.3.3
Cycle class
310
Exercises
313
Bibliography
315
Index
319 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Voisin, Claire 1962- |
| author_GND | (DE-588)1075027810 |
| author_facet | Voisin, Claire 1962- |
| author_role | aut |
| author_sort | Voisin, Claire 1962- |
| author_variant | c v cv |
| building | Verbundindex |
| bvnumber | BV023031893 |
| callnumber-first | Q - Science |
| callnumber-label | QA564 |
| callnumber-raw | QA564 |
| callnumber-search | QA564 |
| callnumber-sort | QA 3564 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)612965182 (DE-599)BVBBV023031893 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. publ., reprint., transferred to digital print. |
| format | Book |
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| id | DE-604.BV023031893 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:17:21Z |
| indexdate | 2024-07-09T21:09:25Z |
| institution | BVB |
| isbn | 0521802601 9780521802604 9780521718011 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016235760 |
| oclc_num | 612965182 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-29T DE-824 DE-355 DE-BY-UBR DE-83 DE-92 DE-188 |
| owner_facet | DE-19 DE-BY-UBM DE-29T DE-824 DE-355 DE-BY-UBR DE-83 DE-92 DE-188 |
| physical | IX, 322 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Cambridge University Press |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Voisin, Claire 1962- Verfasser (DE-588)1075027810 aut Hodge theory and complex algebraic geometry 1 Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) 1. publ., reprint., transferred to digital print. Cambridge Cambridge University Press 2007 IX, 322 S. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 76 Cambridge studies in advanced mathematics Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Hodge-Theorie (DE-588)4135967-7 gnd rswk-swf Hodge-Theorie (DE-588)4135967-7 s Algebraische Geometrie (DE-588)4001161-6 s DE-604 (DE-604)BV016433308 1 Cambridge studies in advanced mathematics 76 (DE-604)BV000003678 76 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016235760&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Voisin, Claire 1962- Hodge theory and complex algebraic geometry Cambridge studies in advanced mathematics Algebraische Geometrie (DE-588)4001161-6 gnd Hodge-Theorie (DE-588)4135967-7 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4135967-7 |
| title | Hodge theory and complex algebraic geometry |
| title_auth | Hodge theory and complex algebraic geometry |
| title_exact_search | Hodge theory and complex algebraic geometry |
| title_exact_search_txtP | Hodge theory and complex algebraic geometry |
| title_full | Hodge theory and complex algebraic geometry 1 Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) |
| title_fullStr | Hodge theory and complex algebraic geometry 1 Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) |
| title_full_unstemmed | Hodge theory and complex algebraic geometry 1 Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) |
| title_short | Hodge theory and complex algebraic geometry |
| title_sort | hodge theory and complex algebraic geometry |
| topic | Algebraische Geometrie (DE-588)4001161-6 gnd Hodge-Theorie (DE-588)4135967-7 gnd |
| topic_facet | Algebraische Geometrie Hodge-Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016235760&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV016433308 (DE-604)BV000003678 |
| work_keys_str_mv | AT voisinclaire hodgetheoryandcomplexalgebraicgeometry1 |