Lie Group Actions in Complex Analysis by Dmitri Akhiezer

By Dmitri Akhiezer

This publication used to be deliberate as an advent to an enormous sector, the place many contri­ butions were made in recent times. the alternative of fabric relies on my realizing of the function of Lie teams in advanced research. at the one hand, they seem because the automorphism teams of convinced complicated areas, e. g. , bounded domain names in en or compact areas, and are for that reason very important as being certainly one of their invariants. nevertheless, complicated Lie teams and, extra as a rule, homoge­ neous complicated manifolds, function a proving floor, the place it's always attainable to complete a role and get an particular solution. One solid instance of this sort is the speculation of homogeneous vector bundles over flag manifolds. one other instance is the way in which the worldwide analytic homes of homogeneous manifolds are translated into algebraic language. it really is my friendly accountability to thank A. L. Onishchik, who first brought me to the speculation of Lie teams greater than 25 years in the past. i'm enormously indebted to him and to E. B. Vinberg for the assistance and suggestion they've got given me for years. i want to specific my gratitude to M. Brion, B. GilIigan, P. Heinzner, A. Hu­ kleberry, and E. Oeljeklaus for useful discussions of varied matters handled the following. part of this booklet was once written in the course of my remain on the Ruhr-Universitat Bochum in 1993. I thank the Deutsche Forschungsgemeinschaft for its study help and the colleagues in Bochum for his or her hospitality.

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Since the intersection of all Hj is trivial, each of these groups equals {e}. 3 The automorphism group of a compact complex space In order to prove (*) take an hE Hd and write mod where fi are the same as in the proof of the theorem and qi,x E m~+l. More precisely, ' "1,x····· fan qi,x - " ~f Ci,Q: n,x' where a = (al, ... ,an ) E zn, ai ~ 0, 2:ai = d+ 1, and Ci,a E C. By iterating we obtain (fi 0 h-m)x == Ax + mqi,x mod m~+2. Since h m E H d +l for some m ~ 1, it follows that qi,x E m~+2. 3 The automorphism group of a compact complex space Let X be a compact complex space.

Montgomery. The proof makes use of powerful theorems from the general theory of topological groups. They are stated here without proof, but the reduction to the topological setting is complete. This reduction requires, among other things, the identity theorem for compact groups of holomorphic transformations with a fixed point. The identity theorem, in its turn, is deduced from the local linearization theorem for such groups. In addition to general results, we consider several examples of compact complex spaces, for which Aut(X) is explicitly determined.

In this section we prove that Aut(X) is a complex Lie group and that the action of Aut(X) on X is holomorphic. Montgomery, see [BM3]. Kerner [Ker] extended the result to reduced complex spaces. Kaup, see [Ka1]. The proof is based on the general theory of topological groups. For the convenience of the reader we state all necessary results and give the corresponding references. All proofs can be also found in [MZ]. A topological group G is called a group without small subgroups, if there exists a neighborhood of e E G which contains no subgroups of G except {e}.

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