Overview
- Presents quite recent research works, all of very high standard, in the field of several complex variables
- Selects only extremely important materials from the conventional basic theory of complex analysis and manifold theory
- Requires no more than a one-semester introductory course in complex analysis as a prerequisite for understanding
- Makes the content more informative with the addition of new materials and sections to each chapter
- Proves Andreotti–Grauert's finiteness theorems by the method of Andreotti –Vesentini
- Proves optimal L² extensions on the basis of new methods invented in the past 5 years
- Describes ongoing development on Levi flat domains
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (5 chapters)
Keywords
About this book
In Chapter 1, the classical questions of several complex variables motivating the development of this field are reviewed after necessary preparations from the basic notions of those variables and of complex manifolds such as holomorphic functions, pseudoconvexity, differential forms, and cohomology. In Chapter 2, the L² method of solving the d-bar equation is presented emphasizing its differential geometric aspect. In Chapter 3, a refinement of the Oka–Cartan theory is given by this method. The L² extension theorem with an optimal constant is included, obtained recently by Z. Błocki and separately by Q.-A. Guan and X.-Y. Zhou. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani–Yamaguchi, Berndtsson, Guan–Zhou, and Berndtsson–Lempert. Most of these results are obtained by the L² method. In the last chapter, rather specific results are discussed on the existence and classification of certain holomorphic foliations and Levi flat hypersurfaces as their stables sets. These are also applications of the L² method obtained during the past 15 years.
Authors and Affiliations
Bibliographic Information
Book Title: L² Approaches in Several Complex Variables
Book Subtitle: Towards the Oka–Cartan Theory with Precise Bounds
Authors: Takeo Ohsawa
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-4-431-56852-0
Publisher: Springer Tokyo
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Japan KK, part of Springer Nature 2018
Hardcover ISBN: 978-4-431-56851-3Published: 10 December 2018
eBook ISBN: 978-4-431-56852-0Published: 28 November 2018
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 2
Number of Pages: XI, 258
Number of Illustrations: 5 b/w illustrations
Topics: Several Complex Variables and Analytic Spaces, Algebraic Geometry, Differential Geometry, Functional Analysis