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 OkaCartan theory is given by this method. The L² extension theorem with an optimal constant is included, obtained recently by Z. Bocki 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 MaitaniYamaguchi, Berndtsson, GuanZhou, and BerndtssonLempert. 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.

Presents quite recent research works, all of very high standard, in the field of several complex variables Selects only extremely important materials from the conventional basic theory of complex analysis and manifold theory Requires no more than a one-semester introductory course in complex analysis as a prerequisite for understanding Makes the content more informative with the addition of new materials and sections to each chapter Proves AndreottiGrauert's finiteness theorems by the method of Andreotti Vesentini Proves optimal L² extensions on the basis of new methods invented in the past 5 years Describes ongoing development on Levi flat domains

Inhalt
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 Hilbert 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. <p