L² approaches in several complex variables : towards the Oka-Cartan theory with precise bounds / Takeo Ohsawa.

Format:

Book

Author/Creator:

Ōsawa, Takeo, 1951- author.

Series:

Springer monographs in mathematics

Springer monographs in mathematics, 1439-7382

Language:

English

Subjects (All):

Functions of several complex variables.

Holomorphic functions.

Analytic spaces.

Physical Description:

xi, 258 pages ; 24 cm.

Edition:

Second edition.

Place of Publication:

Tokyo : Springer Japan KK, [2018]

Summary:

"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 L² 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 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."-Publisher.

Contents:

Basic notions and classical results

Analyzing the L² ∂ ̄-cohomology

L² Oka-Cartan theory

Bergman kernels

L² approaches to holomorphic foliations.

Notes:

First edition published as "L² approaches in several complex variables: development of Oka-Cartan theory by L² estimates for the ∂̄ operator" in 2015.

Includes bibliographical references (pages 239-254) and index.

ISBN:

4431568514

9784431568513

OCLC:

1051682266