Algebraic Geometry over the Complex Numbers

by Arapura, Donu

• ISBN13: 9781461418085
• ISBN10: 1461418089
• Format: Paperback
• Copyright: 2012-02-10
• Publisher: Springer Verlag

Summary

This is a relatively fast paced graduate level introduction to complex algebraic geometry, from the basics to the frontier of the subject. It covers sheaf theory, cohomology, some Hodge theory, as well as some of the more algebraic aspects of algebraic geometry.The author frequently refers the reader if the treatment of a certain topic is readily available elsewhere but goes into considerable detail on topics for which his treatment puts a twist or a more transparent viewpoint. His cases of exploration and are chosen very carefully and deliberately. The textbook achieves its purpose of taking new students of complex algebraic geometry through this a deep yet broad introduction to a vast subject, eventually bringing them to the forefront of the topic via a non-intimidating style.

Table of Contents

Preface	p. vii
Introduction through Examples
Plane Curves	p. 3
Conics	p. 3
Singularities	p. 5
Bézout's Theorem	p. 7
Cubics	p. 9
Genus 2 and 3	p. 11
Hyperelliptic Curves	p. 14
Sheaves and Geometry
Manifolds and Varieties via Sheaves	p. 21
Sheaves of Functions	p. 22
Manifolds	p. 24
Affine Varieties	p. 28
Algebraic Varieties	p. 32
Stalks and Tangent Spaces	p. 35
1-Forms, Vector Fields, and Bundles	p. 41
Compact Complex Manifolds and Varieties	p. 45
More Sheaf Theory	p. 49
The Category of Sheaves	p. 49
Exact Sequences	p. 53
Affine Schemes	p. 58
Schemes and Gluing	p. 62
Sheaves of Modules	p. 66
Line Bundles on Projective Space	p. 70
Direct and Inverse Images	p. 72
Differentials	p. 76
Sheaf Cohomology	p. 79
Flasque Sheaves	p. 79
Cohomology	p. 81
Soft Sheaves	p. 86
C∞-Modules Are Soft	p. 89
Mayer-Vietoris Sequence	p. 90
Products*	p. 93
De Rham Cohomology of Manifolds	p. 97
Acyclic Resolutions	p. 97
De Rham's Theorem	p. 100
Künneth's Formula	p. 102
Poincaré Duality	p. 105
Gysin Maps	p. 108
Projections	p. 109
Inclusions	p. 110
Fundamental Class	p. 111
Lefschetz Trace Formula	p. 113
Riemann Surfaces	p. 117
Genus	p. 117
∂-Cohomology	p. 122
Projective Embeddings	p. 126
Function Fields and Automorphisms	p. 130
Modular Forms and Curves	p. 133
Simphicial Methods	p. 137
Simplicial and Singular Cohomology	p. 137
Cohomology of Projective Space	p. 142
Cech Cohomology	p. 144
Cech Versus Sheaf Cohomology	p. 147
First Chern Class	p. 150
Hodge Theory
The Hodge Theorem for Riemannian Manifolds	p. 157
Hodge Theory on a Simplicial Complex	p. 157
Harmonic Forms	p. 159
The Heat Equation*	p. 163
Toward Hodge Theory for Complex Manifolds	p. 169
Riemann Surfaces Revisited	p. 169
Dolbeault's Theorem	p. 172
Complex Tori	p. 173
Kähler Manifolds	p. 179
Kähler Metrics	p. 179
The Hodge Decomposition	p. 183
Picard Groups	p. 187
A Little Algebraic Surface Theory	p. 189
Examples	p. 189
The Neron-Severi Group	p. 193
Adjunction and Riemann-Roch	p. 195
The Hodge Index Theorem	p. 198
Fibered Surfaces*	p. 200
Hodge Structures and Homological Methods	p. 203
Pure Hodge Structures	p. 203
Canonical Hodge Decomposition	p. 205
Hodge Decomposition for Moishezon Manifolds	p. 210
Hypercohomology*	p. 212
Holomorphic de Rham Complex*	p. 216
The Deligne-Hodge Decomposition*	p. 217
Topology of Families	p. 223
Topology of Families of Elliptic Curves	p. 223
Local Systems	p. 228
Higher Direct Images*	p. 230
First Betti Number of a Fibered Variety*	p. 235
The Hard Lefschetz Theorem	p. 237
Hard Lefschetz	p. 237
Proof of Hard Lefschetz	p. 239
Weak Lefschetz and Barth's Theorem	p. 241
Lefschetz Pencils*	p. 242
Cohomology of Smooth Projective Maps*	p. 247
Coherent Cohomology
Coherent Sheaves	p. 255
Coherence on Ringed Spaces	p. 255
Coherent Sheaves on Affine Schemes	p. 257
Coherent Sheaves on Pn	p. 259
GAGA, Part I	p. 263
Cohomology of Coherent Sheaves	p. 265
Cohomology of Affine Schemes	p. 265
Cohomology of Coherent Sheaves on Pn	p. 267
Cohomology of Analytic Sheaves	p. 272
GAGA, Part II	p. 274
Computation of Some Hodge Numbers	p. 279
Hodge Numbers of Pn	p. 279
Hodge Numbers of a Hypersurface	p. 282
Hodge Numbers of a Hypersurface II	p. 285
Double Covers	p. 287
Griffiths Residues*	p. 289
Deformations and Hodge Theory	p. 293
Families of Varieties via Schemes	p. 293
Semicontinuity of Coherent Cohomology	p. 297
Deformation Invariance of Hodge Numbers	p. 300
Noether-Lefschetz*	p. 302
Analogies and Conjectures*
Analogies and Conjectures	p. 307
Counting Points and Euler Characteristics	p. 307
The Weil Conjectures	p. 309
A Transcendental Analogue of Weil's Conjecture	p. 312
Conjectures of Grothendieck and Hodge	p. 313
Problem of Computability	p. 317
Hodge Theory without Analysis	p. 319
References	p. 321
Index	p. 327
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