**Graduate Studies in Mathematics**

Volume: 125;
2011;
236 pp;
Hardcover

MSC: Primary 30; 31; 32;

Print ISBN: 978-0-8218-5369-6

Product Code: GSM/125

List Price: $67.00

Individual Member Price: $53.60

**Electronic ISBN: 978-1-4704-1186-2
Product Code: GSM/125.E**

List Price: $67.00

Individual Member Price: $53.60

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#### Supplemental Materials

# Riemann Surfaces by Way of Complex Analytic Geometry

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*Dror Varolin*

This book establishes the basic function theory and complex geometry of Riemann surfaces, both open and compact. Many of the methods used in the book are adaptations and simplifications of methods from the theories of several complex variables and complex analytic geometry and would serve as excellent training for mathematicians wanting to work in complex analytic geometry.

After three introductory chapters, the book embarks on its central, and certainly most novel, goal of studying Hermitian holomorphic line bundles and their sections. Among other things, finite-dimensionality of spaces of sections of holomorphic line bundles of compact Riemann surfaces and the triviality of holomorphic line bundles over Riemann surfaces are proved, with various applications. Perhaps the main result of the book is Hörmander's Theorem on the square-integrable solution of the Cauchy-Riemann equations. The crowning application is the proof of the Kodaira and Narasimhan Embedding Theorems for compact and open Riemann surfaces.

The intended reader has had first courses in real and complex analysis, as well as advanced calculus and basic differential topology (though the latter subject is not crucial). As such, the book should appeal to a broad portion of the mathematical and scientific community.

*This book is the first to give a textbook
exposition of Riemann surface theory from the viewpoint of positive
Hermitian line bundles and Hörmander \(\bar \partial\)
estimates. It is more analytical and PDE oriented than prior texts in
the field, and is an excellent introduction to the methods used
currently in complex geometry, as exemplified in J. P. Demailly's
online but otherwise unpublished book “Complex analytic and
differential geometry.” I used it for a one quarter course on
Riemann surfaces and found it to be clearly written and
self-contained. It not only fills a significant gap in the large
textbook literature on Riemann surfaces but is also rather
indispensible for those who would like to teach the subject from a
differential geometric and PDE viewpoint.*

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#### Table of Contents

# Table of Contents

## Riemann Surfaces by Way of Complex Analytic Geometry

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Complex analysis 120 free
- Riemann surfaces 2140
- Functions on Riemann surfaces 3756
- Complex line bundles 6180
- Complex differential forms 87106
- Calculus on line bundles 101120
- Potential theory 115134
- Solving \overline{∂} with smooth data 133152
- Harmonic forms 145164
- Uniformization 165184
- Hörmander’s Theorem 177196
- Embedding Riemann surfaces 197216
- The Riemann-Roch Theorem 211230
- Abel’s Theorem 223242
- Bibliography 233252
- Index 235254 free
- Back Cover Back Cover1258

#### Readership

Graduate students and research mathematicians interested in complex analysis and geometry and in PDE on complex spaces.

#### Reviews

...the text will be very helpful for those who want to study Riemann surfaces from a differential geometric and PDE viewpoint.

-- Montash Math