142,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: ISTE Invoiced
- Sprache: Englisch
Complex Manifolds and Geometric Algebraic Analysis is intended for graduate students in mathematics, physics, and beyond.
The book is divided into ten chapters. Chapter 1 deals with the properties of holomorphic functions of several complex variables. Chapter 2 introduces tools for studying complex manifolds and analytic varieties, whilst Chapter 3 covers the foundational material from sheaves and cohomology. Chapter 4 concerns the study of divisors and line bundles on complex manifolds, and Chapter 5 is devoted to some fundamental theorems. Chapter 6 covers definitions and examples of abelian varieties, whilst Chapter 7 studies theta functions on complex projective tori. Lastly, the aim of Chapter 8 is to discuss an interesting interaction between complex algebraic geometry and dynamical systems.
This book is supplemented with two appendices, one on Riemann surfaces and algebraic curves and the other covering elliptic functions and elliptic integrals. Additionally, various examples, exercises, and problems with solutions are provided throughout the book.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 400
Veröffentlichungsjahr: 2025
Ähnliche
Table of Contents
Cover
Table of Contents
Title Page
Copyright Page
Preface
1 Holomorphic Functions of Several Complex Variables
1.1. Notations, definitions and properties
1.2. Cauchy integral formula and applications
1.3. Power series in several variables
1.4. Various fundamental properties
1.5. Inverse mapping and implicit function theorems
1.6. Exercises
2 Complex Manifolds and Analytic Varieties
2.1. Preliminaries
2.2. Examples of complex manifolds
2.3. Tangent spaces and tangent bundles
2.4. Constant rank theorem
2.5. Submanifolds, subvarieties and examples
2.6. Exercises
3 Sheaves and Cohomology
3.1. Sheaves
3.2. Cech cohomology
3.3. De Rham cohomology
3.4. Dolbeault cohomology
3.5. Connections and curvature
3.6. Curvature form, first Chern class of line bundles
3.7. The Poincaré dual
3.8. Exercises
4 Divisors and Line Bundles
4.1. Divisors
4.2. Line bundles
4.3. Sections of line bundles
4.4. Tori and Riemann form
4.5. Line bundles on complex tori
4.6. Exercises
5 Some Fundamental Theorems
5.1. Preliminaries and various notions
5.2. The Kodaira–Nakano vanishing theorem
5.3. The Lefschetz theorem on hyperplane sections
5.4. The Lefschetz theorem on (1, 1)-classes
5.5. The Kodaira embedding theorem
5.6. Exercises
6 Abelian Varieties
6.1. Definitions and examples
6.2. Riemann conditions, polarization and Riemann form
6.3. Isogenies and reducible Abelian varieties
6.4. Dual Abelian varieties
6.5. Prym varieties
6.6. Projectively normal embedding
6.7. Number of even and odd sections of a line bundle
6.8. Exercises
7 Theta Functions and Complex Projective Tori
7.1. Meromorphic functions and theta functions
7.2. Lefschetz theorem
7.3. Exercises
8 Algebraically Completely Integrable Systems
8.1. Preliminaries
8.2. The Hénon–Heiles system
8.3. Kowalewski’s spinning top
8.4. Kirchhoff’s equations of motion of a solid in an ideal fluid
8.5. Exercises
9 Appendix: Riemann Surfaces and Algebraic Curves
10 Appendix: Elliptic Functions and Elliptic Integrals
10.1. Elliptic functions
10.2. Weierstrass functions
10.3. Elliptic integrals and Jacobi elliptic functions
10.4. Application: simple pendulum
References
Index
Other titles from iSTE in Mathematics and Statistics
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1.
Domain of convergence
Figure 1.2.
Laurent series
Chapter 2
Figure 2.1.
Holomorphic map
Figure 2.2.
Elliptic and hyperelliptic curve
Figure 2.3.
Stereographic projection
Figure 2.4.
Torus
Chapter 7
Figure 7.1.
Normal representation (genus two)
Chapter 8
Figure 8.1.
Genus-three curve with a canonical basis of cycles
Guide
Cover Page
Table of Contents
Title Page
Copyright Page
Preface
Begin Reading
References
Index
Other titles from iSTE in Mathematics and Statistics
Wiley End User License Agreement
Pages
iii
iv
ix
x
xi
xii
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
251
252
253
254
255
257
258
259
260
261
262
263
264
256
Complex Manifolds and Geometric Algebraic Analysis
Ahmed Lesfari
First published 2026 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2026
The rights of Ahmed Lesfari to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2025945943
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-83669-091-7
The manufacturer’s authorized representative according to the EU General Product Safety Regulation is Wiley-VCH GmbH, Boschstr. 12, 69469 Weinheim, Germany, e-mail: [email protected].
Preface
This essential book is intended for graduate students in mathematics, physics and beyond. It may also be of interest to doctoral students wishing to acquire a broad spectrum of mathematical knowledge and combine it with other disciplines. It aims to present several important aspects of complex manifolds with various properties in geometric algebraic analysis. The topics covered in this book are objects of extraordinary richness that appear in many scientific disciplines. The presentation is clear and well organized, and many examples, exercises and problems with solutions are provided throughout the text. This book is divided into 10 chapters, each containing several sections.
Chapter 1 deals with some basic properties of holomorphic functions of several complex variables. In this chapter, we cover fundamental concepts in the theory of holomorphic functions of several complex variables, including complex differentiation, the Cauchy integral formula and its applications, power series expansions, expansion in Reinhardt domains, various fundamental properties, inverse mapping and implicit function theorems.
In Chapter 2, we introduce tools for studying complex manifolds and analytic varieties by illustrating them with numerous examples and applications. Several fundamental exercises and problems are studied in order to elucidate the specificities of these manifolds. We explore in particular open subsets, complex algebraic curves or compact Riemann surfaces, spheres, projective spaces, grassmannians and tori, among others. We also study tangent spaces and tangent bundles in the case of complex manifolds. It is well known that we can treat a complex manifold M of dimension n as a real manifold of dimension 2n. We can therefore define tangent vectors by identifying them with the derivations of the algebra of real differential functions on M with values in R. We can also define tangent vectors by identifying them with the complex-valued derivations of the C-algebra of holomorphic functions. We prove the constant rank theorem, which states, up to a change of coordinates, that an application whose rank is locally constant is equivalent to a linear application. Next, we present submanifolds in different ways, starting by reviewing some definitions of a submanifold and then demonstrating some results that bring together the different ways of locally describing submanifolds: by equations, by the graph of an application and by a parameterization. Each of these characterizations has its own interest and often comes up when solving exercises, and each plays a crucial role.
In Chapter 3, we sketch the foundational material from sheaves and cohomology. First, we recall the definition of sheaves and we examine many examples. Then, we give an overview of Cech, de Rham and Dolbeault cohomologies, connections, curvature form, the first Chern class of line bundles and the Poincaré dual.
Chapter 4 concerns the study of divisors and line bundles on complex manifolds. We are interested in divisors, line bundles, sections of lines bundles, tori, Riemann form and line bundles on complex tori.
Chapter 5 is devoted to some fundamental theorems. We first introduce various notions concerning projective variety, tangent and cotangent bundles (Dolbeault), cohomology groups, the first Chern class, Hodge forms, hermitian metrics, harmonic space, the Hodge star operator and Hodge theorem, the Kähler metric, the Kähler form, the Kähler manifold and Moishezon variety. Next, we prove the Kodaira–Nakano vanishing theorem, the Lefschetz vanishing theorem on hyperplane sections and the Lefschetz vanishing theorem for classes of type (1,1). Particular attention is devoted to the study of the Kodaira embedding theorem and other versions of this theorem. This asserts that a compact complex analytic variety is algebraically projective if and only if it is a Hodge variety, in other words if and only if it admits a closed, entire and positive two-differential form of type (1,1), or again if and only if it admits a holomorphic line bundle whose first Chern class is positive. The connection with the theory of Kähler manifolds will be explained. A Kähler manifold is a complex manifold equipped with a hermitian metric whose imaginary part, which is a two-form of type (1,1) relative to the complex structure, is closed. Such a metric is called a Kähler metric, and the form is called a Kähler form. In the language of Kähler varieties, the Kodaira embedding theorem means that a compact complex variety is projective if and only if it admits a Kähler form whose cohomology class is entire. We discuss another interesting result concerning Kähler varieties obtained by Moishezon. A Moishezon variety is a compact complex analytic variety that becomes projective after a finite number of blow-ups of smooth centers and has the maximum number of algebraically independent meromorphic functions. More precisely, a compact Kähler variety of dimension n is projective if and only if it has n algebraically independent meromorphic functions.
In Chapter 6, we first deal with some definitions and examples of Abelian varieties. We also discuss Chow’s theorem, which states that any closed analytic subvariety of the projective space PN (C) is a projective variety. He states that such a variety is defined by homogeneous polynomial equations and can therefore be studied by either analytic or algebraic methods. Then, one of the objectives is to see how to characterize complex tori that have an embedding in a complex projective space and are therefore considered as Abelian varieties. The so-called Riemann conditions or Riemann bilinear relations, as well as the Riemann form, play a crucial role in this study. We also introduce isogenies, reducible Abelian varieties, dual Abelian varieties and projectively normal embedding. We provide sufficient conditions, which guarantee that a complex n-dimensional manifold is analytically isomorphic to an n-dimensional complex torus and a Kähler manifold. We discuss the relation with the Hodge theory and an immediate consequence is that a complex manifold will complete to Abelian variety by adjoining some divisors. Several examples are given. Moreover, the classical definition of Prym varieties deals with the unramified covers of curves. The aim here is to give explicit algebraic descriptions of the Prym varieties associated with ramified double covers of algebraic curves. In addition, we provide two explicit formulas for determining the number of even and odd sections of a line bundle.
Chapter 7 deals with theta functions and complex projective tori. We begin by reviewing some results concerning theta functions which play a major role in much current research and explore some of the numerous consequences, a subject of renewed interest in recent years. The purpose of this chapter is to provide a short and quick exposition of some important aspects of meromorphic theta functions for compact Riemann surfaces. The study of theta functions is done via an analytical approach using meromorphic functions in the framework of Mumford. Some interesting examples are given. We show how the meromorphic functions on the Riemann surfaces of an arbitrary genus can be constructed explicitly in terms of the multi-dimensional theta functions. We discuss the important role of the zeros of theta functions and the Jacobi inversion problem, which askes whether we can find a divisor that is the preimage for an arbitrary point in the Jacobian. The Lefschetz theorem on projective embeddings over the complex numbers is of utmost importance in the complex geometric theory of compact manifolds. The next step is to provide an analytic proof of this theorem on the projectivity of complex tori using theta functions. This theorem shows how these theta functions provide an embedding of the complex torus into a projective space. We explain how positive line bundles on Abelian varieties can be explicitly described in terms of multipliers and how their sections can be described by theta functions.
The aim of Chapter 8 is to discuss an interesting interaction between complex algebraic geometry and dynamical systems. We briefly explain how Abelian varieties and in particular Prym varieties are involved in the modern study and active area of algebraically completely integrable hamiltonian dynamical systems. We first consider some notions and properties concerning these systems and explain how the solutions are expressed in the form of a Laurent series of nonlinear differential equations in the complex domain. Such meromorphic solutions depending on a sufficient number of free parameters play a crucial role in the study of so-called algebraically integrable differential equations. This means that we require that the invariants of the differential system be polynomial (in adequate coordinates) and that, moreover, the complex varieties obtained by equating these polynomial invariants to generic constants form the affine part of an algebraic complex torus (Abelian variety) in such a way that the complex flows generated by the invariants are linear on these complex tori. These ideas are applied to the study of three important algebraically completely integrable hamiltonian systems, namely the Hénon–Heiles differential system, Kowalewski’s spinning top and Kirchhoff’s equations of motion of a solid in an ideal fluid.
The book is supplemented with two appendixes (Chapters 8 and 9), one on Riemann surfaces and algebraic curves and the other on elliptic functions and elliptic integrals. In addition, numerous examples, exercises and problems with solutions are scattered throughout the text.
I have included a bibliography of some key works and a detailed index allowing the reader to quickly locate any item discussed in the book. We sincerely thank ISTE and John Wiley & Sons, Inc. for the quality of their work, their thoroughness and their professionalism. I dedicate this book to all those who have kindly contributed to it in one way or another.
A. LESFARI
October 2025
