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Title
L2 approaches in several complex variables [electronic resource] : development of Oka-Cartan Theory by L2 estimates for the ? ̄operator / Takeo Ōsawa.
ISBN
9784431557470 electronic book
4431557474 electronic book
9784431557463
4431557466
Published
Tokyo : Springer, 2015.
Language
English
Description
1 online resource (196 pages)
Item Number
10.1007/978-4-431-55747-0 doi
Call Number
QA331.7
Dewey Decimal Classification
515.9/4
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.
Bibliography, etc. Note
Includes bibliographical references.
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Access limited to authorized users.
Source of Description
Description based on print version record.
Series
Springer monographs in mathematics.
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.
