Hodge theory and complex algebraic geometry: 2
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Cambridge
Cambridge University Press
2005
|
| Edition: | 1. publ., reprint. |
| Series: | Cambridge studies in advanced mathematics
77 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 351 S. Ill., graph. Darst. |
| ISBN: | 0521802830 |
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MARC
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| 100 | 1 | |a Voisin, Claire |d 1962- |e Verfasser |0 (DE-588)1075027810 |4 aut | |
| 245 | 1 | 0 | |a Hodge theory and complex algebraic geometry |n 2 |c Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) |
| 250 | |a 1. publ., reprint. | ||
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2005 | |
| 300 | |a IX, 351 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 77 | |
| 490 | 0 | |a Cambridge studies in advanced mathematics | |
| 650 | 0 | 7 | |a Hodge-Theorie |0 (DE-588)4135967-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Algebraische Geometrie |0 (DE-588)4001161-6 |2 gnd |9 rswk-swf |
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| _version_ | 1804135558150094848 |
|---|---|
| adam_text | Contents
0
Introduction page
1
1 The Topology of Algebraic Varieties
17
1
The Lefschetz Theorem on
Hyperplane
Sections
19
1.1
Morse theory
20
1.1.1
Morse s lemma
20
1.1.2
Local study of the level set
23
1.1.3
Globalisation
27
1.2
Application to
affine
varieties
28
1.2.1
Index of the square of the distance function
28
1.2.2
Lefschetz theorem on
hyperplane
sections
31
1.2.3
Applications
34
1.3
Vanishing theorems and Lefschetz theorem
36
Exercises
39
2
Lefschetz Pencils
41
2.1
Lefschetz pencils
42
2.1.1
Existence
42
2.1.2
The holomorphic Morse lemma
46
2.2
Lefschetz degeneration
47
2.2.1
Vanishing spheres
47
2.2.2
An application of Morse theory
48
2.3
Application to Lefschetz pencils
53
2.3.1
Blowup of the base locus
53
2.3.2
The Lefschetz theorem
54
2.3.3
Vanishing cohomology and primitive cohomology
57
2.3.4
Cones over vanishing cycles
60
Exercises
62
vi
Contents
Monodromy
67
3.1
The monodromy
action
69
3.1.1
Local systems and representations of
π
69
3.1.2
Local systems associated to a fibration
73
3.1.3
Monodromy and variation of Hodge structure
74
3.2
The case of Lefschetz pencils
77
3.2.1
The Picard-Lefschetz formula
77
3.2.2
Zariski s theorem
85
3.2.3
Irreducibility of the monodromy action
87
3.3
Application: the Noether-Lefschetz theorem
89
3.3.1
The Noether-Lefschetz locus
89
3.3.2
The Noether-Lefschetz theorem
93
Exercises
94
The Leray Spectral Sequence
98
4.1
Definition of the spectral sequence
100
4.1.1
The hypercohomology spectral sequence
100
4.1.2
Spectral sequence of a composed functor
107
4.1.3
The Leray spectral sequence
109
4.2
Deligne s theorem
113
4.2.1
The cup-product and spectral sequences
113
4.2.2
The relative Lefschetz decomposition
115
4.2.3
Degeneration of the spectral sequence
117
4.3
The invariant cycles theorem
118
4.3.1
Application of the degeneracy of the Leray-spectral
sequence
118
4.3.2
Some background on mixed Hodge theory
119
4.3.3
The global invariant cycles theorem
123
Exercises
124
II Variations of Hodge Structure
127
5
Transversality and Applications
129
5.1
Complexes associated to IVHS
130
5.1.1
The
de Rham
complex of a flat bundle
130
5.1.2
Transversality
133
5.1.3
Construction of the complexes mathcatKi r
137
5.2
The holomorphic Leray spectral sequence
138
5.2.1
The Leray filtration on
Ω£
and the complexes Kp
,,
138
5.2.2
Infinitesimal invariants
141
5.3
Local study of Hodge loci
143
5.3.1
General properties
143
5.3.2
Infinitesimal study
146
Contents
vii
5.3.3
The Noether-Lefschetz locus
148
5.3.4
A density criterion
151
Exercises
153
Hodge Filtration of Hypersurfaces
156
6.1
Filtration by the order of the pole
158
6.1.1
Logarithmic complexes
158
6.1.2
Hodge filtration and filtration by the order of
the pole
160
6.1.3
The case of hypersurfaces of P
163
6.2
IVHS of hypersurfaces
167
6.2.1
Computation of V
167
6.2.2
Macaulay s theorem
171
6.2.3
The symmetriser lemma
175
6.3
First applications
177
6.3.1
Hodge loci for families of hypersurfaces
177
6.3.2
The generic
Torelli
theorem
179
Exercises
184
Normal Functions and Infinitesimal Invariants
188
7.1
The Jacobian fibration
189
7.1.1
Holomorphic structure
189
7.1.2
Normal functions
191
7.1.3
Infinitesimal invariants
192
7.2
The Abel-Jacobi map
193
7.2.1
General properties
193
7.2.2
Geometric interpretation of the infinitesimal
invariant
197
7.3
The case of hypersurfaces of high degree in P
205
7.3.1
Application of the symmetriser lemma
205
7.3.2
Generic triviality of the Abel-Jacobi map
207
Exercises
212
Nori
s Work
215
8.1
The connectivity theorem
217
8.1.1
Statement of the theorem
217
8.1.2
Algebraic translation
218
8.1.3
The case of hypersurfaces of
projective
space
223
8.2
Algebraic equivalence
228
8.2.1
General properties
228
8.2.2
The Hodge class of a normal function
229
8.2.3
Griffiths theorem
233
viu
Contents
8.3
Application
of the connectivity theorem
235
8.3.1
The
Nori
equivalence
235
8.3.2
Nori s theorem
237
Exercises
240
III Algebraic Cycles
243
9
Chow Groups
245
9.1
Construction
247
9.1.1
Rational equivalence
247
9.1.2
Functoriality
:
proper morphisms and flat
morphisms
248
9.1.3
Localisation
254
9.2
Intersection and cycle classes
256
9.2.1
Intersection
256
9.2.2
Correspondences
259
9.2.3
Cycle classes
261
9.2.4
Compatibilities
263
9.3
Examples
269
9.3.1
Chow groups of curves
269
9.3.2
Chow groups of
projective
bundles
269
9.3.3
Chow groups of blowups
271
9.3.4
Chow groups of hypersurfaces of small degree
273
Exercises
275
10
Mumford s Theorem and its Generalisations
278
10.1
Varieties with representable CHo
280
10.1.1
Representability
280
10.1.2
Roitman s theorem
284
10.1.3
Statement of Mumford s theorem
289
10.2
The Bloch-Srinivas construction
291
10.2.1
Decomposition of the diagonal
291
10.2.2
Proof of Mumford s theorem
294
10.2.3
Other applications
298
10.3
Generalisation
301
10.3.1
Generalised decomposition of the diagonal
301
10.3.2
An application
303
Exercises
304
11
The Bloch Conjecture and its Generalisations
307
11.1
Surfaces with pg
= 0
308
11.1.1
Statement of the conjecture
308
11.1.2
Classification
310
Contents ix
313
315
322
322
324
327
328
328
329
336
339
340
343
348
11.1.3
Bloch s conjecture for surfaces which are not
of general type
11.1.4
Godeaux surfaces
11.2 Filtrations
on Chow groups
1.2.1
The generalised Bloch conjecture
1.2.2
Conjectural filtration on the Chow groups
1.2.3
The
Saito
filtration
11.3
Τ
lie case of
abel
ian
varieties
1.3.1
The Pontryagin product
1.3.2
Results of Bloch
1.3.3
Fourier transform
1.3.4
Results of Beauville
Exercises
References
Index
|
| adam_txt |
Contents
0
Introduction page
1
1 The Topology of Algebraic Varieties
17
1
The Lefschetz Theorem on
Hyperplane
Sections
19
1.1
Morse theory
20
1.1.1
Morse's lemma
20
1.1.2
Local study of the level set
23
1.1.3
Globalisation
27
1.2
Application to
affine
varieties
28
1.2.1
Index of the square of the distance function
28
1.2.2
Lefschetz theorem on
hyperplane
sections
31
1.2.3
Applications
34
1.3
Vanishing theorems and Lefschetz' theorem
36
Exercises
39
2
Lefschetz Pencils
41
2.1
Lefschetz pencils
42
2.1.1
Existence
42
2.1.2
The holomorphic Morse lemma
46
2.2
Lefschetz degeneration
47
2.2.1
Vanishing spheres
47
2.2.2
An application of Morse theory
48
2.3
Application to Lefschetz pencils
53
2.3.1
Blowup of the base locus
53
2.3.2
The Lefschetz theorem
54
2.3.3
Vanishing cohomology and primitive cohomology
57
2.3.4
Cones over vanishing cycles
60
Exercises
62
vi
Contents
Monodromy
67
3.1
The monodromy
action
69
3.1.1
Local systems and representations of
π\
69
3.1.2
Local systems associated to a fibration
73
3.1.3
Monodromy and variation of Hodge structure
74
3.2
The case of Lefschetz pencils
77
3.2.1
The Picard-Lefschetz formula
77
3.2.2
Zariski's theorem
85
3.2.3
Irreducibility of the monodromy action
87
3.3
Application: the Noether-Lefschetz theorem
89
3.3.1
The Noether-Lefschetz locus
89
3.3.2
The Noether-Lefschetz theorem
93
Exercises
94
The Leray Spectral Sequence
98
4.1
Definition of the spectral sequence
100
4.1.1
The hypercohomology spectral sequence
100
4.1.2
Spectral sequence of a composed functor
107
4.1.3
The Leray spectral sequence
109
4.2
Deligne's theorem
113
4.2.1
The cup-product and spectral sequences
113
4.2.2
The relative Lefschetz decomposition
115
4.2.3
Degeneration of the spectral sequence
117
4.3
The invariant cycles theorem
118
4.3.1
Application of the degeneracy of the Leray-spectral
sequence
118
4.3.2
Some background on mixed Hodge theory
119
4.3.3
The global invariant cycles theorem
123
Exercises
124
II Variations of Hodge Structure
127
5
Transversality and Applications
129
5.1
Complexes associated to IVHS
130
5.1.1
The
de Rham
complex of a flat bundle
130
5.1.2
Transversality
133
5.1.3
Construction of the complexes mathcatKi r
137
5.2
The holomorphic Leray spectral sequence
138
5.2.1
The Leray filtration on
Ω£
and the complexes Kp
,,
138
5.2.2
Infinitesimal invariants
141
5.3
Local study of Hodge loci
143
5.3.1
General properties
143
5.3.2
Infinitesimal study
146
Contents
vii
5.3.3
The Noether-Lefschetz locus
148
5.3.4
A density criterion
151
Exercises
153
Hodge Filtration of Hypersurfaces
156
6.1
Filtration by the order of the pole
158
6.1.1
Logarithmic complexes
158
6.1.2
Hodge filtration and filtration by the order of
the pole
160
6.1.3
The case of hypersurfaces of P"
163
6.2
IVHS of hypersurfaces
167
6.2.1
Computation of V
167
6.2.2
Macaulay's theorem
171
6.2.3
The symmetriser lemma
175
6.3
First applications
177
6.3.1
Hodge loci for families of hypersurfaces
177
6.3.2
The generic
Torelli
theorem
179
Exercises
184
Normal Functions and Infinitesimal Invariants
188
7.1
The Jacobian fibration
189
7.1.1
Holomorphic structure
189
7.1.2
Normal functions
191
7.1.3
Infinitesimal invariants
192
7.2
The Abel-Jacobi map
193
7.2.1
General properties
193
7.2.2
Geometric interpretation of the infinitesimal
invariant
197
7.3
The case of hypersurfaces of high degree in P"
205
7.3.1
Application of the symmetriser lemma
205
7.3.2
Generic triviality of the Abel-Jacobi map
207
Exercises
212
Nori
's Work
215
8.1
The connectivity theorem
217
8.1.1
Statement of the theorem
217
8.1.2
Algebraic translation
218
8.1.3
The case of hypersurfaces of
projective
space
223
8.2
Algebraic equivalence
228
8.2.1
General properties
228
8.2.2
The Hodge class of a normal function
229
8.2.3
Griffiths' theorem
233
viu
Contents
8.3
Application
of the connectivity theorem
235
8.3.1
The
Nori
equivalence
235
8.3.2
Nori's theorem
237
Exercises
240
III Algebraic Cycles
243
9
Chow Groups
245
9.1
Construction
247
9.1.1
Rational equivalence
247
9.1.2
Functoriality
:
proper morphisms and flat
morphisms
248
9.1.3
Localisation
254
9.2
Intersection and cycle classes
256
9.2.1
Intersection
256
9.2.2
Correspondences
259
9.2.3
Cycle classes
261
9.2.4
Compatibilities
263
9.3
Examples
269
9.3.1
Chow groups of curves
269
9.3.2
Chow groups of
projective
bundles
269
9.3.3
Chow groups of blowups
271
9.3.4
Chow groups of hypersurfaces of small degree
273
Exercises
275
10
Mumford's Theorem and its Generalisations
278
10.1
Varieties with representable CHo
280
10.1.1
Representability
280
10.1.2
Roitman's theorem
284
10.1.3
Statement of Mumford's theorem
289
10.2
The Bloch-Srinivas construction
291
10.2.1
Decomposition of the diagonal
291
10.2.2
Proof of Mumford's theorem
294
10.2.3
Other applications
298
10.3
Generalisation
301
10.3.1
Generalised decomposition of the diagonal
301
10.3.2
An application
303
Exercises
304
11
The Bloch Conjecture and its Generalisations
307
11.1
Surfaces with pg
= 0
308
11.1.1
Statement of the conjecture
308
11.1.2
Classification
310
Contents ix
313
315
322
322
324
327
328
328
329
336
339
340
343
348
11.1.3
Bloch's conjecture for surfaces which are not
of general type
11.1.4
Godeaux surfaces
11.2 Filtrations
on Chow groups
1.2.1
The generalised Bloch conjecture
1.2.2
Conjectural filtration on the Chow groups
1.2.3
The
Saito
filtration
11.3
Τ
lie case of
abel
ian
varieties
1.3.1
The Pontryagin product
1.3.2
Results of Bloch
1.3.3
Fourier transform
1.3.4
Results of Beauville
Exercises
References
Index |
| 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 | BV021717951 |
| 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)179934460 (DE-599)BVBBV021717951 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. publ., reprint. |
| format | Book |
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| id | DE-604.BV021717951 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:22:30Z |
| indexdate | 2024-07-09T20:42:25Z |
| institution | BVB |
| isbn | 0521802830 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014931664 |
| oclc_num | 179934460 |
| open_access_boolean | |
| owner | DE-384 DE-19 DE-BY-UBM DE-29T DE-824 DE-355 DE-BY-UBR DE-83 DE-188 DE-92 |
| owner_facet | DE-384 DE-19 DE-BY-UBM DE-29T DE-824 DE-355 DE-BY-UBR DE-83 DE-188 DE-92 |
| physical | IX, 351 S. Ill., graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| 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 2 Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) 1. publ., reprint. Cambridge Cambridge University Press 2005 IX, 351 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 77 Cambridge studies in advanced mathematics Hodge-Theorie (DE-588)4135967-7 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Hodge-Theorie (DE-588)4135967-7 s Algebraische Geometrie (DE-588)4001161-6 s DE-604 (DE-604)BV016433308 2 Cambridge studies in advanced mathematics 77 (DE-604)BV000003678 77 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014931664&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 Hodge-Theorie (DE-588)4135967-7 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4135967-7 (DE-588)4001161-6 |
| 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 2 Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) |
| title_fullStr | Hodge theory and complex algebraic geometry 2 Claire Voisin (CNRS, Institut de Mathématiques de Jussieu) |
| title_full_unstemmed | Hodge theory and complex algebraic geometry 2 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 | Hodge-Theorie (DE-588)4135967-7 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Hodge-Theorie Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014931664&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 hodgetheoryandcomplexalgebraicgeometry2 |