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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Simply transitive groups and Kähler structures on homogeneous Siegel domains

Author: Josef Dorfmeister
Journal: Trans. Amer. Math. Soc. 288 (1985), 293-305
MSC: Primary 32M10; Secondary 53C55
MathSciNet review: 773062
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Abstract: We determine the Lie algebras of all simply transitive groups of automorphisms of a homogeneous Siegel domain $ D$ as modifications of standard normal $ j$-algebras. We show that the Lie algebra of all automorphisms of $ D$ is a "complete isometry algebra in standard position". This implies that $ D$ carries a riemannian metric $ \tilde g$ with nonpositive sectional curvature satisfying Lie $ \operatorname{Iso}(D,\tilde g) = \operatorname{Lie}\; \operatorname{Aut}\,$   D. We determine all Kähler metrics $ f$ on $ D$ for which the group $ \operatorname{Aut}(D,f)$ of holomorphic isometries acts transitively. We prove that in this case $ \operatorname{Aut}(D,f)$ contains a simply transitive split solvable subgroup.

The results of this paper are used to prove the fundamental conjecture for homogeneous Kähler manifolds admitting a solvable transitive group of automorphisms.

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Keywords: Homogeneous bounded domain, Kähler metric, solvable transitive group of automorphisms
Article copyright: © Copyright 1985 American Mathematical Society