An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

by Schlichenmaier, Martin

• ISBN13: 9783540711742
• ISBN10: 3540711740
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-10-01
• Publisher: SPRINGER

Summary

This book gives an introduction to modern geometry. Starting from an elementary level the author develops deep geometrical concepts, playing an important role nowadays in contemporary theoretical physics. He presents various techniques and viewpoints, thereby showing the relations between the alternative approaches. At the end of each chapter suggestions for further reading are given to allow the reader to study the touched topics in greater detail. This second edition of the book contains two additional more advanced geometric techniques: (1) The modern language and modern view of Algebraic Geometry and (2) Mirror Symmetry. The book grew out of lecture courses. The presentation style is therefore similar to a lecture. Graduate students of theoretical and mathematical physics will appreciate this book as textbook. Students of mathematics who are looking for a short introduction to the various aspects of modern geometry and their interplay will also find it useful. Researchers will esteem the book as reliable reference.

Author Biography

Martin Schlichenmaier is full professor for mathematics at the University of  Luxemburg. He has held several teaching and research positions in the mathematics department of the University of Mannheim.

Table of Contents

Introduction from a Physicist's Viewpoint	p. 1
Manifolds	p. 7
Generalities	p. 7
Complex Manifolds	p. 9
The Classification Problem	p. 13
Hints for Further Reading	p. 14
Topology of Riemann Surfaces	p. 17
Fundamental Group	p. 17
Simplicial Homology	p. 21
Universal Covering Space	p. 28
Hints for Further Reading	p. 29
Analytic Structure	p. 31
Holomorphic and Meromorphic Functions	p. 31
Divisors and the Theorem of Riemann-Roch	p. 35
Meromorphic Functions on the Torus	p. 38
Hints for Further Reading	p. 41
Differentials and Integration	p. 43
Tangent Space and Differentials	p. 43
Differential Forms of Second Order	p. 48
Integration	p. 50
Hints for Further Reading	p. 52
Tori and Jacobians	p. 53
Higher Dimensional Tori	p. 53
Jacobians	p. 55
Hints for Further Reading	p. 59
Projective Varieties	p. 61
Generalities	p. 61
Embedding of One-Dimensional Tori	p. 65
Theta Functions	p. 67
Hints for Further Reading	p. 69
Moduli Spaces of Curves	p. 71
The Definition	p. 71
Methods of Construction	p. 74
The Geometry of the Moduli Space and Its Compactification	p. 78
Hints for Further Reading	p. 85
Vector Bundles, Sheaves and Cohomology	p. 87
Vector Bundles	p. 87
Sheaves	p. 91
Cohomology	p. 95
Hints for Further Reading	p. 100
The Theorem of Riemann-Roch for Line Bundles	p. 103
Divisors and Line Bundles	p. 103
An Application: The Krichever-Novikov Algebra	p. 109
Hints for Further Reading	p. 117
The Mumford Isomorphism on the Moduli Space	p. 119
The Mumford Isomorphism	p. 119
The Grothendieck-Riemann-Roch Theorem	p. 125
Hints for Further Reading	p. 131
Modern Algebraic Geometry	p. 133
Varieties	p. 133
The Spectrum of a Ring	p. 139
Homomorphisms	p. 146
Noncommutative Spaces	p. 149
Hints for Further Reading	p. 153
Schemes	p. 155
Affine Schemes	p. 155
General Schemes	p. 159
The Structure Sheaf O[subscript R]	p. 162
Examples of Schemes	p. 164
Hints for Further Reading	p. 167
Hodge Decomposition and Kahler Manifold	p. 169
Some Introductory Remarks on Mirror Symmetry	p. 169
Compact Complex Manifolds and Hodge Decomposition	p. 171
Kahler Manifolds	p. 177
Hodge Numbers of the Projective Space	p. 181
Hints for Further Reading	p. 182
Calabi-Yau Manifolds and Mirror Symmetry	p. 183
Calabi-Yau Manifolds	p. 183
K3 Surfaces, Hypersurfaces and Complete Intersections	p. 187
Geometric Mirror Symmetry	p. 192
Example of a Calabi-Yau Three-fold and Its Mirror: Results of Givental	p. 196
Hints for Further Reading	p. 200
p-adic Numbers	p. 203
Index	p. 213
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