Overview

This book was planned as an introduction to a vast area, where many contri­ butions have been made in recent years. The choice of material is based on my understanding of the role of Lie groups in complex analysis. On the one hand, they appear as the automorphism groups of certain complex spaces, e. g. , bounded domains in en or compact spaces, and are therefore important as being one of their invariants. On the other hand, complex Lie groups and, more generally, homoge­ neous complex manifolds, serve as a proving ground, where it is often possible to accomplish a task and get an explicit answer. One good example of this kind is the theory of homogeneous vector bundles over flag manifolds. Another example is the way the global analytic properties of homogeneous manifolds are translated into algebraic language. It is my pleasant duty to thank A. L. Onishchik, who first introduced me to the theory of Lie groups more than 25 years ago. I am greatly indebted to him and to E. B. Vinberg forthe help and advice they have given me for years. I would like to express my gratitude to M. Brion, B. GilIigan, P. Heinzner, A. Hu­ kleberry, and E. Oeljeklaus for valuable discussions of various subjects treated here. A part of this book was written during my stay at the Ruhr-Universitat Bochum in 1993. I thank the Deutsche Forschungsgemeinschaft for its research support and the colleagues in Bochum for their hospitality.

Full Product Details

Author:   Dimitrij Akhiezer
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag
Edition:   Softcover reprint of the original 1st ed. 1995
Volume:   27
Dimensions:   Width: 16.20cm , Height: 1.20cm , Length: 22.90cm
Weight:   0.342kg
ISBN:  

9783322802699

ISBN 10:   3322802698
Pages:   204
Publication Date:   16 December 2011
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.
Language:   English

Table of Contents

1 Lie theory.- 1.1 Complex spaces.- 1.2 Lie group actions.- 1.3 One-parameter transformation groups.- 1.4 Vector fields.- 1 5 Infinitesimal transformations.- 1.6 Analyticity of Lie group actions.- 1.7 Lie homomorphism.- 1.8 Global actions.- 2 Automorphism groups.- 2.1 Topology in Hol(X, Y).- 2.2 Local linearization of a compact group with a fixed point.- 2.3 The automorphism group of a compact complex space.- 2.4 Automorphisms of fiber bundles.- 2.5 Proper actions.- 2.6 The automorphism group of a bounded domain.- 2.7 The automorphism groups of the polydisk and the ball.- 2.8 A characterization of the ball.- 2.9 Bounded domains with compact quotient D/Aut(D).- 3 Compact homogeneous manifolds.- 3.1 Flag manifolds.- 3.2 Equivariant projective embeddings.- 3.3 Automorphism groups of flag manifolds.- 3.4 Parallelizable manifolds.- 3.5 Tits fibration.- 3.6 Manifolds fibered by tori.- 3.7 The role of the fundamental group.- 3.8 An estimate of the dimension of Aut(X).- 3.9 Compact homogeneousKähler manifolds.- 4 Homogeneous vector bundles.- 4.1 Coherent analytic G-sheaves.- 4.2 Holomorphic vector G-bundles.- 4.3 Theorem of R.Bott. Proof of the Borel-Weil theorem.- 4.4 Application of the Leray spectral sequence.- 4.5 Proof of the theorem of R.Bott.- 4.6 Invertible sheaves on G/P for P maximal parabolic.- 4.7 Computations in root systems.- 4.8 Cohomology of the tangent sheaf.- 5 Function theory on homogeneous manifolds.- 5.1 Representations of compact Lie groups on Fréchet spaces.- 5.2 Differentiable vectors and Fourier series in O(X).- 5.3 Reductive complex Lie groups.- 5.4 Quasi-affine homogeneous varieties.- 5.5 Holomorphically separable homogeneous manifolds.- 5.6 Stein homogeneous manifolds.- 5.7 Observable subgroups.- 5.8 Invariant plurisubharmonic functions and geodesic convexity.- Concluding remarks.- Index of notations.- Index of terminology.

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Prof. Dr. Dimitri Akhiezer lehrt Mathematik an einer Moskauer Universität.

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