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Riemann Surfaces by Way of Complex Analytic Geometry

Dror Varolin Stony Brook University, Stony Brook, NY
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Hardcover ISBN: 978-0-8218-5369-6
Product Code: GSM/125
List Price: $72.00 MAA Member Price:$64.80
AMS Member Price: $57.60 Electronic ISBN: 978-1-4704-1186-2 Product Code: GSM/125.E List Price:$67.00
MAA Member Price: $60.30 AMS Member Price:$53.60
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List Price: $108.00 MAA Member Price:$97.20
AMS Member Price: $86.40 Click above image for expanded view Riemann Surfaces by Way of Complex Analytic Geometry Dror Varolin Stony Brook University, Stony Brook, NY Available Formats:  Hardcover ISBN: 978-0-8218-5369-6 Product Code: GSM/125  List Price:$72.00 MAA Member Price: $64.80 AMS Member Price:$57.60
 Electronic ISBN: 978-1-4704-1186-2 Product Code: GSM/125.E
 List Price: $67.00 MAA Member Price:$60.30 AMS Member Price: $53.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$108.00 MAA Member Price: $97.20 AMS Member Price:$86.40
• Book Details

Volume: 1252011; 236 pp
MSC: Primary 30; 31; 32;

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

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

• Chapters
• Chapter 1. Complex analysis
• Chapter 2. Riemann surfaces
• Chapter 3. Functions on Riemann surfaces
• Chapter 4. Complex line bundles
• Chapter 5. Complex differential forms
• Chapter 6. Calculus on line bundles
• Chapter 7. Potential theory
• Chapter 8. Solving $\overline {\partial }$ with smooth data
• Chapter 9. Harmonic forms
• Chapter 10. Uniformization
• Chapter 11. Hörmander’s Theorem
• Chapter 12. Embedding Riemann surfaces
• Chapter 13. The Riemann-Roch Theorem
• Chapter 14. Abel’s Theorem

• Reviews

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

Montash Math
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1252011; 236 pp
MSC: Primary 30; 31; 32;

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

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

• Chapters
• Chapter 1. Complex analysis
• Chapter 2. Riemann surfaces
• Chapter 3. Functions on Riemann surfaces
• Chapter 4. Complex line bundles
• Chapter 5. Complex differential forms
• Chapter 6. Calculus on line bundles
• Chapter 7. Potential theory
• Chapter 8. Solving $\overline {\partial }$ with smooth data
• Chapter 9. Harmonic forms
• Chapter 10. Uniformization
• Chapter 11. Hörmander’s Theorem
• Chapter 12. Embedding Riemann surfaces
• Chapter 13. The Riemann-Roch Theorem
• Chapter 14. Abel’s Theorem
• ...the text will be very helpful for those who want to study Riemann surfaces from a differential geometric and PDE viewpoint.

Montash Math
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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