Basic algebraic geometry 2 Schemes and complex manifolds
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2013
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| Ausgabe: | 3. ed. |
| Online-Zugang: | Inhaltsverzeichnis |
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| 020 | |a 9783642380099 |q Hardcover |9 978-3-642-38009-9 | ||
| 020 | |a 9783662514016 |q Softcover |9 978-3-662-51401-6 | ||
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| 100 | 1 | |a Šafarevič, Igorʹ R. |d 1923-2017 |e Verfasser |0 (DE-588)119280337 |4 aut | |
| 240 | 1 | 0 | |a Osnovj algebraičeskoj geometrii |
| 245 | 1 | 0 | |a Basic algebraic geometry |n 2 |p Schemes and complex manifolds |c Igor R. Shafarevich |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2013 | |
| 300 | |a XIV, 262 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
| DE-19_call_number | 1601/SK 240 S128 B31-2(3)+2 1601/SK 240 S128 B31-2(3) |
|---|---|
| DE-19_location | 95 |
| DE-BY-TUM_call_number | 0102 MAT 140 2001 A 29535 |
| DE-BY-TUM_katkey | 2623231 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040009673769 |
| DE-BY-UBM_katkey | 4948094 |
| DE-BY-UBM_media_number | 41623350120016 41623350130018 |
| _version_ | 1823055417343737856 |
| adam_text | Titel: Bd. 2. Basic algebraic geometry. Schemes and complex manifolds
Autor: Šafarevič, Igorʹ R
Jahr: 2013
Contents
Book 2: Schemes and Varieties
5 Schemes 3
1 The Spec of a Ring 5
1.1 Definition of Spec A 5
1.2 Properties of Points of Spec A 7
1.3 The Zariski Topology of Spec A 9
1.4 Irreducibility, Dimension 11
1.5 Exercises to Section 1 14
2 Sheaves 15
2.1 Presheaves 15
2.2 The Structure Presheaf 17
2.3 Sheaves 19
2.4 Stalks of a Sheaf 23
2.5 Exercises to Section 2 24
3 Schemes 25
3.1 Definition of a Scheme 25
3.2 Glueing Schemes 30
3.3 Closed Subschemes 32
3.4 Reduced Schemes and Nilpotents 35
3.5 Finiteness Conditions 36
3.6 Exercises to Section 3 38
4 Products of Schemes 40
4.1 Definition of Product 40
4.2 Group Schemes 42
4.3 Separatedness 43
4.4 Exercises to Section 4 46
6 Varieties 49
1 Definitions and Examples 49
1.1 Definitions 49
VII
http://d-nb.info/1033092460
VIII Contents
1.2 Vector Bundles 53
1.3 Vector Bundles and Sheaves 56
1.4 Divisors and Line Bundles 63
1.5 Exercises to Section 1 67
2 Abstract and Quasiprojective Varieties 68
2.1 Chow s Lemma 68
2.2 Blowup Along a Subvariety 70
2.3 Example of Non-quasiprojective Variety 74
2.4 Criterions for Projectivity 79
2.5 Exercises to Section 2 81
3 Coherent Sheaves 81
3.1 Sheaves of Ox-Modules 81
3.2 Coherent Sheaves 85
3.3 Devissage of Coherent Sheaves 88
3.4 The Finiteness Theorem 92
3.5 Exercises to Section 3 93
4 Classification of Geometric Objects and Universal Schemes .... 94
4.1 Schemes and Functors 94
4.2 The Hilbert Polynomial 100
4.3 Flat Families 103
4.4 The Hilbert Scheme 107
4.5 Exercises to Section 4 110
Book 3: Complex Algebraic Varieties and Complex Manifolds
7 The Topology of Algebraic Varieties 115
1 The Complex Topology 115
1.1 Definitions 115
1.2 Algebraic Varieties as Differentiable Manifolds;
Orientation 117
1.3 Homology of Nonsingular Projective Varieties 118
1.4 Exercises to Section 1 121
2 Connectedness 121
2.1 Preliminary Lemmas 121
2.2 The First Proof of the Main Theorem 122
2.3 The Second Proof 124
2.4 Analytic Lemmas 126
2.5 Connectedness of Fibres 127
2.6 Exercises to Section 2 128
3 The Topology of Algebraic Curves 129
3.1 Local Structure of Morphisms 129
3.2 Triangulation of Curves 131
3.3 Topological Classification of Curves 133
3.4 Combinatorial Classification of Surfaces 137
3.5 The Topology of Singularities of Plane Curves 140
3.6 Exercises to Section 3 142
Contents IX
4 Real Algebraic Curves 142
4.1 Complex Conjugation 143
4.2 Proof of Harnack s Theorem 144
4.3 Ovals of Real Curves 146
4.4 Exercises to Section 4 147
8 Complex Manifolds 149
1 Definitions and Examples 149
1.1 Definition 149
1.2 Quotient Spaces 152
1.3 Commutative Algebraic Groups as Quotient Spaces .... 155
1.4 Examples of Compact Complex Manifolds not Isomorphic
to Algebraic Varieties 157
1.5 Complex Spaces 163
1.6 Exercises to Section 1 165
2 Divisors and Meromorphic Functions 166
2.1 Divisors 166
2.2 Meromorphic Functions 169
2.3 The Structure of the Field A4(X) 171
2.4 Exercises to Section 2 174
3 Algebraic Varieties and Complex Manifolds 175
3.1 Comparison Theorems 175
3.2 Example of Nonisomorphic Algebraic Varieties that Are
Isomorphic as Complex Manifolds 178
3.3 Example of a Nonalgebraic Compact Complex Manifold
with Maximal Number of Independent Meromorphic
Functions 181
3.4 The Classification of Compact Complex Surfaces 183
3.5 Exercises to Section 3 185
4 Kahler Manifolds 185
4.1 Kahler Metric 186
4.2 Examples 188
4.3 Other Characterisations of Kahler Metrics 190
4.4 Applications of Kahler Metrics 193
4.5 Hodge Theory 196
4.6 Exercises to Section 4 198
9 Uniformisation 201
1 The Universal Cover 201
1.1 The Universal Cover of a Complex Manifold 201
1.2 Universal Covers of Algebraic Curves 203
1.3 Projective Embedding of Quotient Spaces 205
1.4 Exercises to Section 1 206
2 Curves of Parabolic Type 207
2.1 Theta Functions 207
2.2 Projective Embedding 209
X Contents
2.3 Elliptic Functions, Elliptic Curves and Elliptic Integrals . . 210
2.4 Exercises to Section 2 213
3 Curves of Hyperbolic Type 213
3.1 Poincare Series 213
3.2 Projective Embedding 216
3.3 Algebraic Curves and Automorphic Functions 218
3.4 Exercises to Section 3 221
4 Uniformising Higher Dimensional Varieties 221
4.1 Complete Intersections are Simply Connected 221
4.2 Example of Manifold with tt a Given Finite Group .... 222
4.3 Remarks 226
4.4 Exercises to Section 4 227
Historical Sketch 229
1 Elliptic Integrals 229
2 Elliptic Functions 231
3 Abelian Integrals 233
4 Riemann Surfaces 235
5 The Inversion of Abelian Integrals 237
6 The Geometry of Algebraic Curves 239
7 Higher Dimensional Geometry 241
8 The Analytic Theory of Complex Manifolds 243
9 Algebraic Varieties over Arbitrary Fields and Schemes 244
References 247
References for the Historical Sketch 250
Index 253
|
| any_adam_object | 1 |
| author | Šafarevič, Igorʹ R. 1923-2017 |
| author_GND | (DE-588)119280337 |
| author_facet | Šafarevič, Igorʹ R. 1923-2017 |
| author_role | aut |
| author_sort | Šafarevič, Igorʹ R. 1923-2017 |
| author_variant | i r š ir irš |
| building | Verbundindex |
| bvnumber | BV041270382 |
| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)935012533 (DE-599)BVBBV041270382 |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV041270382 |
| illustrated | Illustrated |
| indexdate | 2025-02-03T17:41:49Z |
| institution | BVB |
| isbn | 9783642380099 9783662514016 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026244005 |
| oclc_num | 935012533 |
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| physical | XIV, 262 S. Ill., graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| spellingShingle | Šafarevič, Igorʹ R. 1923-2017 Basic algebraic geometry |
| title | Basic algebraic geometry |
| title_alt | Osnovj algebraičeskoj geometrii |
| title_auth | Basic algebraic geometry |
| title_exact_search | Basic algebraic geometry |
| title_full | Basic algebraic geometry 2 Schemes and complex manifolds Igor R. Shafarevich |
| title_fullStr | Basic algebraic geometry 2 Schemes and complex manifolds Igor R. Shafarevich |
| title_full_unstemmed | Basic algebraic geometry 2 Schemes and complex manifolds Igor R. Shafarevich |
| title_short | Basic algebraic geometry |
| title_sort | basic algebraic geometry schemes and complex manifolds |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026244005&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009785411 |
| work_keys_str_mv | AT safarevicigorʹr osnovjalgebraiceskojgeometrii AT safarevicigorʹr basicalgebraicgeometry2 |