L² Approaches in Several Complex Variables : Development of Oka–Cartan Theory by L² Estimates for the d-bar Operator / by Takeo Ohsawa.

Format:

Book

Author/Creator:

Ohsawa, Takeo., Author.

Series:

Springer Monographs in Mathematics, 2196-9922

Language:

English

Subjects (All):

Functions of complex variables.

Geometry, Algebraic.

Geometry, Differential.

Functional analysis.

Several Complex Variables and Analytic Spaces.

Algebraic Geometry.

Differential Geometry.

Functional Analysis.

Local Subjects:

Several Complex Variables and Analytic Spaces.

Algebraic Geometry.

Differential Geometry.

Functional Analysis.

Physical Description:

1 online resource (202 p.)

Edition:

1st ed. 2015.

Place of Publication:

Tokyo : Springer Japan : Imprint: Springer, 2015.

Language Note:

English

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.

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.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

4-431-55747-4