L2 approaches in several complex variables : development of Oka-Cartan Theory by L2 estimates for the? ̄operator /

L2 approaches in several complex variables : development of Oka-Cartan Theory by L2 estimates for the? ̄operator /

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Bibliographic Details
Author / Creator:Ōsawa, Takeo, 1951- author.
Imprint:Tokyo : Springer, 2015.
Description:1 online resource (196 pages).
Language:English
Series:Springer monographs in mathematics
Springer monographs in mathematics.
Subject:
Functions of several complex variables.
Functions of several complex variables.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/11096366
Hidden Bibliographic Details
ISBN:9784431557470
4431557474
9784431557463
4431557466
Notes:Includes bibliographical references.
Print version record.
Summary:The purpose of this monograph is to present 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. Highlighted are the new precise results on the L² extension of holomorphic functions. 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 by Q.-A. Guan and X.-Y. Zhou separately. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani-Yamaguchi, Berndtsson, and Guan-Zhou. 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 these 15 years.
Other form:Print version: Ōsawa, Takeo, 1951- L2 approaches in several complex variables 9784431557463
Standard no.:10.1007/978-4-431-55747-0
Table of Contents:
  • Part I Holomorphic Functions and Complex Spaces
  • Convexity Notions
  • Complex Manifolds
  • Classical Questions of Several Complex Variables
  • Part II The Method of L² Estimates
  • Basics of Hilb ert Space Theory
  • Harmonic Forms
  • Vanishing Theorems
  • Finiteness Theorems
  • Notes on Complete Kahler Domains (= CKDs)
  • Part III L² Variant of Oka-Cartan Theory
  • Extension Theorems
  • Division Theorems
  • Multiplier Ideals
  • Part IV Bergman Kernels
  • The Bergman Kernel and Metric
  • Bergman Spaces and Associated Kernels
  • Sequences of Bergman Kernels
  • Parameter Dependence
  • Part V L² Approaches to Holomorphic Foliations
  • Holomorphic Foliation and Stable Sets
  • L² Method Applied to Levi Flat Hypersurfaces
  • LFHs in Tori and Hopf Surfaces.

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