Algebraic geometry over the complex numbers:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2012
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | XII, 329 S. graph. Darst. |
| ISBN: | 9781461418085 |
Staff View
MARC
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| 245 | 1 | 0 | |a Algebraic geometry over the complex numbers |c Donu Arapura |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2012 | |
| 300 | |a XII, 329 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
| _version_ | 1804148903964049408 |
|---|---|
| adam_text | Contents
Preface
............................................................
vii
Part I Introduction through Examples
1
Plane Curves
.................................................. 3
1.1
Conies
.................................................... 3
1.2
Singularities
............................................... 5
1.3
Bézouťs
Theorem
.......................................... 7
1.4
Cubics
.................................................... 9
1.5
Genus
2
and
3 ............................................. 11
1.6
Hyperelliptic Curves
........................................ 14
Part
Π
Sheaves and Geometry
2
Manifolds and Varieties via Sheaves
.............................. 21
2.1
Sheaves of Functions
....................................... 22
2.2
Manifolds
................................................. 24
2.3
Affine
Varieties
............................................ 28
2.4
Algebraic Varieties
......................................... 32
2.5
Stalks and Tangent Spaces
................................... 35
2.6 1
-Forms, Vector Fields, and Bundles
.......................... 41
2.7
Compact Complex Manifolds and Varieties
..................... 45
3
More Sheaf Theory
............................................. 49
3.1
The Category of Sheaves
.................................... 49
3.2
Exact Sequences
........................................... 53
3.3
Affine
Schemes
............................................ 58
3.4
Schemes and Gluing
........................................ 62
3.5
Sheaves of Modules
........................................ 66
3.6
Line Bundles on Projective Space
............................. 70
3.7
Direct and Inverse Images
................................... 72
3.8
Differentials
............................................... 76
IX
x
Contents
4
Sheaf Cohomology
............................................. 79
4.1
Flasque
Sheaves
............................................ 79
4.2
Cohomology
.............................................. 81
4.3
Soft Sheaves
............................................... 86
4.4
C-Modules Are Soft
....................................... 89
4.5
Mayer-Vietoris Sequence
.................................... 90
4.6
Products*
................................................. 93
5
De Rham
Cohomology of Manifolds
.............................. 97
5.1
Acyclic Resolutions
........................................ 97
5.2
De Rham s
Theorem
........................................100
5.3 Künneth s
Formula
.........................................102
5.4
Poincaré
Duality
...........................................105
5.5
Gysin Maps
...............................................108
5.5.1
Projections
..........................................109
5.5.2
Inclusions
..........................................110
5.6
Fundamental Class
.........................................
Ill
5.7
Lefschetz Trace Formula
....................................113
6
Riemann Surfaces
..............................................117
6.1
Genus
....................................................117
6.2
d-Cohomology
............................................122
6.3
Projective
Embeddings
......................................126
6.4
Function Fields and Automorphisms
..........................130
6.5
Modular Forms and Curves
..................................133
7
Simplicial Methods
.............................................137
7.1
Simplicial and Singular Cohomology
..........................137
7.2
Cohomology of
Projective
Space
..............................142
7.3
Čech
Cohomology
..........................................144
7.4
Čech
Versus Sheaf Cohomology
..............................147
7.5
First
Chem
Class
...........................................150
Partili
Hodge Theory
8
The Hodge Theorem for Riemannian Manifolds
...................157
8.1
Hodge Theory on a Simplicial Complex
........................
I57
8.2
Harmonic Forms
...........................................159
8.3
The Heat Equation*
........................................163
9
Toward Hodge Theory for Complex Manifolds
....................
I69
9.1
Riemann Surfaces Revisited
..................................169
9.2
Dolbeault s Theorem
........................................172
9.3
Complex Tori
...................... ......................173
Contents xi
10 Kahler
Manifolds..............................................
179
10.1 Kahler
Metrics.............................................
179
10.2
The Hodge Decomposition
...................................183
10.3
Picard
Groups
.............................................187
11
A Little Algebraic Surface Theory
...............................189
11.1
Examples
.................................................189
11.2
The Neron-Severi Group
....................................193
11.3
Adjunction and Riemann-Roch
...............................195
11.4
The Hodge Index Theorem
...................................198
11.5
Fibered Surfaces*
..........................................200
12
Hodge Structures and Homological Methods
......................203
12.1
Pure Hodge Structures
......................................203
12.2
Canonical Hodge Decomposition
.............................205
12.3
Hodge Decomposition for Moishezon Manifolds
................210
12.4
Hypercohomology*
.........................................212
12.5
Holomorphic
de Rham
Complex*
.............................216
12.6
The Deligne-Hodge Decomposition*
..........................217
13
Topology of Families
............................................223
13.1
Topology of Families of Elliptic Curves
........................223
13.2
Local Systems
.............................................228
13.3
Higher Direct Images*
......................................230
13.4
First
Betti
Number of a Fibered Variety*
.......................235
14
The Hard Lefschetz Theorem
....................................237
14.1
Hard Lefschetz
.............................................237
14.2
Proof of Hard Lefschetz
.....................................239
14.3
Weak Lefschetz and Barth s Theorem
..........................241
14.4
Lefschetz Pencils*
..........................................242
14.5
Cohomology of Smooth
Projective
Maps*
......................247
Part IV Coherent Cohomology
15
Coherent Sheaves
..............................................255
15.1
Coherence on Ringed Spaces
.................................255
15.2
Coherent Sheaves on
Affine
Schemes
..........................257
15.3
Coherent Sheaves on F1
.....................................259
15.4
GAGA, Part I
..............................................263
16
Cohomology of Coherent Sheaves
................................265
16.1
Cohomology of
Affine
Schemes
..............................265
16.2
Cohomology of Coherent Sheaves on P
.......................267
16.3
Cohomology of Analytic Sheaves
.............................272
16.4
GAGA, Part II
.............................................274
xii
Contents
17
Computation of Some Hodge Numbers
...........................279
17.
í
Hodge Numbers of
Рл
.......................................279
17.2
Hodge Numbers of a Hypersurface
............................282
17.3
Hodge Numbers of a Hypersurface II
..........................285
17.4
Double Covers
.............................................287
17.5
Griffiths Residues*
.........................................289
18
Deformations and Hodge Theory
................................293
18.1
Families of Varieties via Schemes
.............................293
18.2
Semicontinuity of Coherent Cohomology
......................297
18.3
Deformation
Invariance
of Hodge Numbers
.....................300
18.4
Noether-Lefschetz*
........................................302
Part V Analogies and Conjectures*
19
Analogies and Conjectures
......................................307
19.1
Counting Points and
Euler
Characteristics
......................307
19.2
The Weil Conjectures
.......................................309
19.3
A Transcendental Analogue of Weil s Conjecture
................312
19.4
Conjectures of Grothendieck and Hodge
.......................313
19.5
Problem of Computability
...................................317
19.6
Hodge Theory without Analysis
..............................319
References
.........................................................321
Index
...................................,........................327
Universitext
Donu Arapura
Algebraic Geometry over the Complex Numbers
Algebraic Geometry over the Complex Numbers is a strong addition to existing introductory
literature on algebraic geometry. The authors treatment combines the study of algebraic
geometry with differential and complex geometry and unifies these subjects using sheaf-
theoretic ideas. It is an ideal text for showing students the connections between algebraic
geometry, complex geometry, and topology, and brings the reader close to the forefront
of research in Hodge theory and related fields.
Unique features of Algebraic Geometry over the Complex Numbers:
•
Contains a rapid introduction to complex algebraic geometry
•
Includes background material on topology, manifold theory and sheaf theory
•
Analytic and algebraic approaches are developed somewhat in parallel
The presentation is easy going, elementary, and well illustrated with examples. This
textbook is intended for graduate level courses in algebraic geometry and related
fields. It can be used as a main text for a second semester graduate course in algebraic
geometry with emphasis on sheaf theoretical methods or a more advanced graduate
course on algebraic geometry and Hodge Theory.
Mathematics
ISBN
978-1-4614-1808-5
9 781461 418085
springer.com
|
| any_adam_object | 1 |
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| author_GND | (DE-588)1020210079 |
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| ctrlnum | (OCoLC)785837239 (DE-599)BVBBV039937071 |
| discipline | Mathematik |
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| id | DE-604.BV039937071 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:14:32Z |
| institution | BVB |
| isbn | 9781461418085 |
| language | English |
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| spelling | Arapura, Donu 1958- Verfasser (DE-588)1020210079 aut Algebraic geometry over the complex numbers Donu Arapura New York [u.a.] Springer 2012 XII, 329 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Komplexe Geometrie (DE-588)4164898-5 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s Komplexe Geometrie (DE-588)4164898-5 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4614-1809-2 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024795212&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024795212&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arapura, Donu 1958- Algebraic geometry over the complex numbers Komplexe Geometrie (DE-588)4164898-5 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4164898-5 (DE-588)4001161-6 |
| title | Algebraic geometry over the complex numbers |
| title_auth | Algebraic geometry over the complex numbers |
| title_exact_search | Algebraic geometry over the complex numbers |
| title_full | Algebraic geometry over the complex numbers Donu Arapura |
| title_fullStr | Algebraic geometry over the complex numbers Donu Arapura |
| title_full_unstemmed | Algebraic geometry over the complex numbers Donu Arapura |
| title_short | Algebraic geometry over the complex numbers |
| title_sort | algebraic geometry over the complex numbers |
| topic | Komplexe Geometrie (DE-588)4164898-5 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Komplexe Geometrie Algebraische Geometrie |
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