**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: $72.00

AMS Member Price: $57.60

MAA Member Price: $64.80

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

List Price: $67.00

AMS Member Price: $53.60

MAA Member Price: $60.30

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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introduction to the methods used currently in complex geometry, as exemplified in J. P. Demailly's online but otherwise unpublished book “Complex analytic and differential geometry.”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.

—Steven Zelditch

#### Readership

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

#### Reviews & Endorsements

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

-- Montash Math

#### 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