L² Approaches in Several Complex Variables

Development of Oka–Cartan Theory by L² Estimates for the d-bar Operator

  • Takeo Ohsawa

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Takeo Ohsawa
    Pages 1-40
  3. Takeo Ohsawa
    Pages 93-125
  4. Takeo Ohsawa
    Pages 127-151
  5. Takeo Ohsawa
    Pages 153-176
  6. Back Matter
    Pages 177-196

About this book


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.


Bergman kernel Levi flat hypersurfaces L² extension of holomorphic functions Multiplier ideals Vanishing and finiteness theorems

Authors and affiliations

  • Takeo Ohsawa
    • 1
  1. 1.Graduate School of Mathematics Nagoya UniversityNagoyaJapan

Bibliographic information