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Springer Monographs in Mathematics

L² Approaches in Several Complex Variables

Towards the Oka–Cartan Theory with Precise Bounds

Authors: Ohsawa, Takeo

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  • 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
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  • ISBN 978-4-431-56851-3
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About this book

This monograph presents the current status of a rapidly developing part of several complex variables, motivated by the applicability of effective results to algebraic geometry and differential geometry. Special emphasis is put on the new precise results on the extension of holomorphic functions in the past 5 years.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 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 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 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 method obtained during the past 15 years.

Table of contents (5 chapters)

Table of contents (5 chapters)

Buy this book

eBook $54.99
$84.99 (listprice)
price for USA in USD (gross)
valid through July 31, 2020
  • ISBN 978-4-431-56852-0
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $69.99
$109.99 (listprice)
price for USA in USD
valid through July 31, 2020
  • ISBN 978-4-431-56851-3
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
L² Approaches in Several Complex Variables
Book Subtitle
Towards the Oka–Cartan Theory with Precise Bounds
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2018
Publisher
Springer Japan
Copyright Holder
Springer Japan KK, part of Springer Nature
eBook ISBN
978-4-431-56852-0
DOI
10.1007/978-4-431-56852-0
Hardcover ISBN
978-4-431-56851-3
Series ISSN
1439-7382
Edition Number
2
Number of Pages
XI, 258
Number of Illustrations
5 b/w illustrations
Topics

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