Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes

314 Pages·1994·9.884 MB·English

by  I. R. Shafarevich (auth.), I. R. Shafarevich (eds.)

Encyclopaedia of
Mathematical Sciences
Volume 23
Editor-in-Chief: R. V. Gamkrelidze
1. R. Shafarevich (Ed.)
Algebraic Geometry 1
Algebraic Curves
Algebraic Manifolds and Schemes
With 49 Figures
Springer-Verlag Berlin Heidelberg GmbH
Consulting Editors of the Series:
A.A. Agrachev, A.A. Gonchar, E. F. Mishchenko,
N. M. Ostianu, V. P. Sakharova, A. B. Zhishchenko
Title of the Russian edition:
ltogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 23, Algebraicheskaya geometriya 1
Publisher VINITI, Moscow 1988
Mathematics Subject Classification (1991):
14Axx, 14Cxx, 14Exx, 14Fxx, 14Hxx, 14Kxx, 14Mxx, 14Nxx, 30Fxx
ISBN 978-3-540-63705-9
Library of Congress Cataloging-in-Publication Data
Aigebraicheskaia geometriia 1. English.
Aigebraic geometry 1: algebraic curves. algebraic manifolds and schemes 1 1. R. Shafarevich (ed.).
p. cm. - (Encyclopaedia of mathematical sciences; v. 23)
Includes bibliographical references and index.
ISBN 978-3-540-63705-9 ISBN 978-3-642-57878-6 (eBook)
DOI 10.1007/978-3-642-57878-6
1. Curves. Algebraic. 2. Manifolds (Mathematics) 3. Schemes (Algebraic geometry)
1. Shafarevich. 1. R. (1gor' RostislavovichJ. 1923 - . II. Title. Il!. Series.
QA56S.A4413 1994 SI6.3'S-dc20 93-4995
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned,
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© Springer-Verlag Berlin Heidelberg 1994
Springer-Verlag Berlin Heidelberg New York in 1994
Softcover reprint ofthe hardcover Ist edition 1994
Typesetting: Springer TEX in-hause system
41/3140 -5 4 3 210 -Printed an acid·free paper
List of Editors, Authors and Translators
Editor-in-Chief
R.V Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute,
ul. Vavilova 42, 117966 Moscow, Institute for Scientific Information (VINITI),
ul. U sievicha 20a, 125219 Moscow, Russia
Consulting Editor
I. R. Shafarevich, Steklov Mathematical Institute, ul. Vavilova 42,
117966 Moscow, Russia
Authors
V. I. Danilov, Central Economic and Mathematical Institute of the Russian
Academy of Sciences, ul. Krasikova 32, 117418 Moscow, Russia
V. V. Shokurov, Department of Mathematics, The Johns Hopkins University,
Baltimore, Maryland 21218, USA
Translators
D. Coray, Section de Mathematiques, Universite de Geneve, 2-4, rue du Lievre,
Case postale 240, 1211 Geneve 24. Switzerland
V. N. Shokurov t
Contents
I. Riemann Surfaces and Algebraic Curves
V. V. Shokurov
1
II. Algebraic Varieties and Schemes
V. I. Danilov
167
Author Index
299
Subject Index
301
I. Riemann Surfaces and
Algebraic Curves
V. V. Shokurov
Translated from the Russian
by V. N. Shokurov
Contents
Introduction by 1. R. Shafarevich 5
Chapter 1. Riemann Surfaces 16
§ 1. Basic Notions ............... 16
1.1. Complex Chart; Complex Coordinates 16
1.2. Complex Analytic Atlas . . . . 17
1.3. Complex Analytic Manifolds 17
1.4. Mappings of Complex Manifolds 19
1.5. Dimension of a Complex Manifold 20
1.6. Riemann Surfaces .. . . 20
1. 7. Differentiable Manifolds . . . . . 22
§ 2. Mappings of Riemann Surfaces 23
2.1. Nonconstant Mappings of Riemann Surfaces are Discrete 23
2.2. Meromorphic Functions on a Riemann Surface ..... 23
2.3. Meromorphic Functions with Prescribed Behaviour at Poles 25
2.4. Multiplicity of a Mapping; Order of a Function ..... 26
2.5. Topological Properties of Mappings of Riemann Surfaces 27
2.6. Divisors on Riemann Surfaces . . . . . . . 27
2.7. Finite Mappings of Riemann Surfaces . . . 29
2.8. Unramified Coverings of Riemann Surfaces 30
2.9. The Universal Covering .......... 30
2.10. Continuation of Mappings . . . . . . . . . 31
2.11. The Riemann Surface of an Algebraic Function 32
2 V. V. Shokurov
§ 3. Topology of Riemann Surfaces 35
3.1. Orient ability . . . . . . 35
3.2. Triangulability..... 36
3.3. Development; Topological Genus 37
3.4. Structure of the Fundamental Group 38
3.5. The Euler Characteristic ...... 39
3.6. The Hurwitz Formulae ...... 39
3.7. Homology and Cohomology; Betti Numbers 41
3.8. Intersection Product; Poincare Duality 42
§ 4. Calculus on Riemann Surfaces . . . . . 44
4.1. Tangent Vectors; Differentiations 44
4.2. Differential Forms . . . . . . . . 45
4.3. Exterior Differentiations; de Rham Cohomology 46
4.4. Kahler and Riemann Metrics ......... 47
4.5. Integration of Exterior Differentials; Green's Formula 48
4.6. Periods; de Rham Isomorphism ....... 51
4.7. Holomorphic Differentials; Geometric Genus;
Riemann's Bilinear Relations ........ 52
4.8. Meromorphic Differentials; Canonical Divisors 54
4.9. Meromorphic Differentials with Prescribed Behaviour
at Poles; Residues . . . . . . . . . . 56
4.10. Periods of Meromorphic Differentials ....... 57
4.11. Harmonic Differentials . . . . . . . . . . . . . . . 58
4.12. Hilbert Space of Differentials; Harmonic Projection 59
4.13. Hodge Decomposition . . . . . . . . . . . . . . . 61
4.14. Existence of Meromorphic Differentials and Functions 62
4.15. Dirichlet's Principle . . . . 65
§ 5. Classification of Riemann Surfaces 65
5.1. Canonical Regions . . . . . 66
5.2. Uniformization ...... 66
5.3. Types of Riemann Surfaces 67
5.4. Automorphisms of Canonical Regions 68
5.5. Riemann Surfaces of Elliptic Type 69
5.6. Riemann Surfaces of Parabolic Type 69
5.7. Riemann Surfaces of Hyperbolic Type 71
5.8. Automorphic Forms; Poincare Series 74
5.9. Quotient Riemann Surfaces; the Absolute Invariant 75
5.10. Moduli of Riemann Surfaces . . . . . . . . . . . . 76
§ 6. Algebraic Nature of Compact Riemann Surfaces 79
6.1. Function Spaces and Mappings Associated with Divisors 79
6.2. Riemann-Roch Formula; Reciprocity Law for Differentials
of the First and Second Kind .............. 82
6.3. Applications of the Riemann-Roch Formula to Problems
of Existence of Meromorphic Functions and Differentials 84
6.4. Compact Riemann Surfaces are Projective ....... 85
1. Riemann Surfaces and Algehraic Curves 3
6.5. Algebraic Nature of Projective Models;
Arithmetic Riemann Surfaces ..... 86
6.6. Models of Riemann Surfaces of Genus 1 87
Chapter 2. Algebraic Curves 89
§ 1. Basic Notions ..... 89
1.1. Algebraic Varieties; Zariski Topology 89
1.2. Regular Functions and Mappings 90
1.3. The Image of a Projective Variety is Closed 93
1.4. Irreducibility; Dimension ........ . 93
1.5. Algebraic Curves . . . . . . . . . 94
1.6. Singular and Nonsingular Points on Varieties 94
1. 7. Rational Functions, Mappings and Varieties 96
1.8. Differentials . . . . . 102
1.9. Comparison Theorems 104
1.10. Lefschetz Principle 105
§ 2. Riemann-Roch Formula .. 106
2.1. Multiplicity of a Mapping; Ramification 106
2.2. Divisors ......... . 107
2.3. Intersection of Plane Curves . . . . . . 109
2.4. The Hurwitz Formulae ....... . 111
2.5. Function Spaces and Spaces of Differentials Associated
with Divisors ........... 112
2.6. Comparison Theorems (Continued) 112
2.7. Riemann-Roch Formula 113
2.8. Approaches to the Proof 113
2.9. First Applications . . . 113
2.10. Riemann Count 117
§ 3. Geometry of Projective Curves 118
3.1. Linear Systems ..... 118
3.2. Mappings of Curves into Ipm 120
3.3. Generic Hyperplane Sections 121
3.4. Geometrical Interpretation of the Riemann-Roch Formula 123
3.5. Clifford's Inequality . . 124
3.6. Castelnuovo's Inequality 126
3.7. Space Curves ..... 127
3.8. Projective Normality 128
3.9. The Ideal of a Curve; Intersections of Quadrics 129
3.10. Complete Intersections ...... 132
3.11. The Simplest Singularities of Curves 134
3.12. The Clebsch Formula 135
3.13. Dual Curves . . . . . . . . . . . . 135
3.14. Plucker Formula for the Class ... 137
3.15. Correspondence of Branches; Dual Formulae 137
4 V. V. Shokurov
Chapter 3. Jacobians and Abelian Varieties 139
§ 1. Abelian Varieties 139
1.1. Algebraic Groups 139
1.2. Abelian Varieties 140
1.3. Algebraic Complex Tori; Polarized Tori 140
1.4. Theta Function and Riemann Theta Divisor 145
1.5. Principally Polarized Abelian Varieties 147
1.6. Points of Finite Order on Abelian Varieties 148
1. 7. Elliptic Curves ........... . 150
§ 2. Jacobians of Curves and of Riemann Surfaces 154
2.1. Principal Divisors on Riemann Surfaces 154
2.2. Inversion Problem .......... . 155
2.3. Picard Group ............ . 156
2.4. Picard Varieties and their Universal Property 156
2.5. Polarization Divisor of the Jacobian of a Curve;
Poincare Formulae . . . . . . . 158
2.6. Jacobian of a Curve of Genus 1 161
References . . . . . . . . . . . . . . . . 163