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L(2) Approaches in Several Complex Variables - Towards the Oka-Cartan Theory with Precise Bounds (Hardcover, 2nd ed. 2018)
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L(2) Approaches in Several Complex Variables - Towards the Oka-Cartan Theory with Precise Bounds (Hardcover, 2nd ed. 2018)
Series: Springer Monographs in Mathematics
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Total price: R3,312
Discovery Miles: 33 120
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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
L(2) 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 L(2) 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
L(2) 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 L(2) 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 L(2) method
obtained during the past 15 years.
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