An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

By: Martin Schlichenmaier

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Hardcover | 21 August 2007 | Edition Number 2

At a Glance

Hardcover
236 Pages

Edition Type
Revised

Dimensions(cm)
23.39 x 15.6 x 1.42

Hardcover

$163.69

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This book gives an introduction to modern geometry. Starting from an elementary level the author develops deep geometrical concepts, playing an important role today 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 - the modern language and modern view of Algebraic Geometry, and Mirror Symmetry. The book grew out of lecture courses. The presentation style is therefore similar to a lecture.

Industry Reviews

From the reviews of the second edition:

"As the title suggests, this book is an introduction to Riemann surfaces, with the target audience being students of string theory. ... an excellent book to use to become familiar with these concepts, and as a result the book is able to touch on a wide variety of concepts which are not broached by more traditional treatments of the subject. ... I would certainly recommend the book for anyone who wants an enjoyable conceptual introduction to what can be a highly technical subject." (Mark Gross, Mathematical Reviews, Issue 2008 k)

"This book is an introduction to the language of modern algebraic geometry, designed mainly for Physics students who are interested in string theory. ... Overall the book is very readable and it fulfills its goal remarkably well ... that are useful to physicists interested in string theory, with all the necessary references for further reading. This book will be an excellent addition to the bookshelf of any physics student or researcher who wants to learn about the mathematical aspects of string theory." (Valentino Tosatti, Zentrablatt MATH, Vol. 1153, 2009)

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You Can Find This Book In

Non-Fiction Mathematics Algebra Calculus & Mathematical Analysis Geometry Algebraic Geometry Differential & Riemannian Geometry Topology Science Physics Nuclear Physics Mathematical Physics

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Paperback

Published: 30th November 2010

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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
Table of Contents provided by Ingram. All Rights Reserved.

An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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