Ohsawa | L² Approaches in Several Complex Variables | Buch | 978-4-431-56851-3 | sack.de

Buch, Englisch, 258 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 576 g

Reihe: Springer Monographs in Mathematics

Ohsawa

L² Approaches in Several Complex Variables

Towards the Oka-Cartan Theory with Precise Bounds
2. Auflage 2018
ISBN: 978-4-431-56851-3
Verlag: Springer Japan

Towards the Oka-Cartan Theory with Precise Bounds

Buch, Englisch, 258 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 576 g

Reihe: Springer Monographs in Mathematics

ISBN: 978-4-431-56851-3
Verlag: Springer Japan


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. Blocki 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.
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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.



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