Riemann surfaces by way of complex analytic geometry
著者
書誌事項
Riemann surfaces by way of complex analytic geometry
（Graduate studies in mathematics, v. 125）
American Mathematical Society, c2011
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注記
Includes bibliographical references (p. 233) and index
内容説明・目次
内容説明
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 Hoermander'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.
目次
Preface
Complex analysis
Riemann surfaces
Functions on Riemann surfaces
Complex line bundles
Complex differential forms
Calculus on line bundles
Potential theory
Solving \overline{ } with smooth data
Harmonic forms
Uniformization
Hoermander's Theorem
Embedding Riemann surfaces
The Riemann-Roch Theorem
Abel's Theorem
Bibliography
Index
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