Lie Group Actions in Complex Analysis

Diese Monographie beschäftigt sich mit der Rolle von Lie-Gruppen in der Komplexen Analysis. Einerseits treten Lie-Gruppen als Automorphismengruppen von komplexen Räumen auf, daher sind sie eine wichtige Invariante dieser Räume. Andererseits lassen sich mit Lie-Gruppen häufig allgemeine Problemstellungen explizit lösen, zum Beispiel lassen sich globale analytische Eigenschaften homogener Mannigfaltigkeiten auf diese Weise in eine algebraische Sprache übertragen. Das Buch richtet sich an Diplomanden, Doktoranden und Dozenten der Mathematik, die in dem aktuellen Gebiet der Lie-Gruppen arbeiten.

Autorentext

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

Klappentext

The main topic of this book is the sudy of the interaction between two major subjects of modern mathematics, namely, the theory of Lie groups with its specific methods and ways of thinking on the one hand and complex analysis with all its analytic, algebraic and geometric aspects. More specifically, the author concentrates on the double role of Lie groups in complex analysis, namely, as groups of biholomorphic self-made of certain complex analytic objects on the one hand and as a special class of complex manifolds with an additional strong structure on the other hand. The book starts from the basics of this subject and introduces the reader into many fields of recent research.

Inhalt
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.