L² Approaches in Several Complex Variables

Towards the Oka–Cartan Theory with Precise Bounds

  • Takeo Ohsawa

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

Table of contents

  1. Front Matter
    Pages i-xi
  2. Takeo Ohsawa
    Pages 1-46
  3. Takeo Ohsawa
    Pages 115-164
  4. Takeo Ohsawa
    Pages 165-204
  5. Takeo Ohsawa
    Pages 205-237
  6. Back Matter
    Pages 239-258

About this book


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  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  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  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  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  method obtained during the past 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.Professor Emeritus, Nagoya UniversityNagoyaJapan

Bibliographic information