Homogeneous Koszul manifolds in \(\mathbb{C}^ n\)
Suppose \(X\) is a complex manifold which is homogeneous under the real Lie group \(G\) acting by holomorphic automorphisms and assume \(X\) has a \(G\)- invariant volume form. The theory of such manifolds was developed by \textit{J. L. Koszul} [Can. J. Math. 7, 562-576 (1955; Zbl 0066.161)] and the case when the Koszul structure is Kähler has been studied in detail, see the work of \textit{M. Dorfmeister} and \textit{K. Nakajima} [Acta Math. 161, No. 1/2, 23-70 (1988; Zbl 0662.32025)] and the references given there. In this paper homogeneous pseudo-Kähler manifolds for which the group contains an exponential solvable group acting transitively on the given space are studied. Such manifolds are referred to as type \(E\). Using a theorem of \textit{J.-I. Hano} [Scripta Math. 29, 125-139 (1971; Zbl 0259.32009)] it is proved that all type \(E\) manifolds may be realized as Siegel \(N - P\) domains, a class of homogeneous domains in \(\mathbb{C}^ n\) introduced by the author [Ann. Math., II. Ser. 126, 389-414 (1987; Zbl 0655.32027)], and that the identity component of the holomorphic isometry group has a rational action. A notion of rank is defined, is shown to be independent of the group acting and attention is then restricted to manifolds of type \(E\) and rank one which are called type I. Koszul domains of type I are shown to be biholomorphic to homogeneous nil-balls. It is conjectured that a contractable homogeneous domain with analytic, Levi nondegenerate boundary is a type \(I\) domain. As a step towards proving a result of this kind it is proved that a contractable domain in \(\mathbb{C}^ n\) which is homogeneous under a rationally homogeneous action of \(G\) and whose boundary is real analytic, codimension-one, Levi nondegenerate, homogeneous under an analytic extension of the action of \(G\) and has an attractive fixed point is type \(I\). By means of a result of \textit{S. M. Webster} [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)] it is shown that for a domain of type \(I\) which is not pseudoconvex its nil-ball realization has a group which acts polynomially. Now one can work with the Chern-Moser normal form [\textit{S. S. Chern} and \textit{J. K. Moser}, Acta Math. 113(1974), 219-271 (1974; Zbl 0302.32015)]. There is a generalized Cartan type result: Given a biholomorphism of two nonpseudoconvex type \(I\) domains which preserves the origin, then the corresponding local biholomorphism of the Chern- Moser normal forms is linear. The equivalence problem reduces to checking whether the normal forms are linearly equivalent. There are some results when the dilation degree is small. Considerable progress has been made for a special class of domains, the holomorphically abelian, on solving the problem of producing all polynomials \(r\) such that the domain \(\tau>r(z)\) is the normal form of a type I domain. A characterization of holomorphically abelian Siegel \(N - P\) pairs in terms of homogeneous duality algebras is given and the Chern- Moser normal form is computed. Finally there are a couple of examples, one being a holomorphically abelian domain in \(\mathbb{C}^ 4\) whose full automorphism group is solvable.
scientific article; zbMATH DE number 146366