The structure of rational homogeneous domains in \({\mathbb{C}}^ n\)
Bounded homogeneous domains in \({\mathbb{C}}^ n\) have been investigated from many different angles by numerous authors since Eli Cartan started the subject in 1935. One of the highlights in this field is the work of Piatetski-Shapiro who showed that bounded homogeneous domains are biholomorphically equivalent with homogeneous ''Siegel domains''. A simple consequence of his work is that such a Siegel domain \(\Omega\) is (i) contractible, (ii) homogeneous under a connected Lie group G, (iii) ``rational'', i.e., for all \(g\in G\) the map \(z\to g\cdot z\) of \(\Omega\) onto \(\Omega\) is rational and (iv) G acts effectively. The purpose of this paper is to study ``rational, homogeneous domains'', i.e., domains in \({\mathbb{C}}^ n\) satisfying the four properties above. The first main result of this paper is Theorem (0.1). Let \(\Omega\) be a domain in \({\mathbb{C}}^ n\). Let G be a connected Lie group which acts rationally on \(\Omega\). Then there exists a real, algebraic group \(G_ R\) such that (i) G is a Zariski-dense, analytic subgroup of \(G_ R.\) (ii) The action \(\mu\) : \(G\times \Omega \to \Omega\) extends to a rational mapping \({\bar \mu}\): \(G_ R\times {\mathbb{C}}^ n\to {\mathbb{C}}^ n\) (which may have singularities). Furthermore, if G acts transitively on \(\Omega\), then \(G_ R\) may be chosen so that \(\Omega\) is homogeneous under the connected component \((G_ R)_ 0.\) The second main result of the paper is a description of the rationally homogeneous domains in \({\mathbb{C}}^ n\) as ``Siegel domains of type N-P''. ``To define this concept, let N be a connected, simply connected nilpotent Lie group with Lie algebra \({\mathcal N}\). Let \({\mathcal N}_ c\) be the complexification of \({\mathcal N}\) and let \(N_ c\) be the corresponding complex Lie group. Let \(P\subset N_ c\) be a complex, connected subgroup. Any automorphism of N extends holomorphically to an automorphism of \(N_ c\). Let \(A_ p\) be the group of automorphisms of N which leave P invariant. Note that \(A_ p\) is a real algebraic group. Let T be a maximal \({\mathfrak R}\)-split torus in \(A_ p\). Then T acts on the double quotient space \(Y=N\setminus N_ c/P\). We shall say that (N,P) is a Siegel pair if \(T_ 0\) has an open orbit in Y in the quotient topology. \((T_ 0\) is the identity component of T.) This is equivalent to saying that the group \(S=T_ 0\times_ SN\) has an open orbit in \(N_ c/P\). The space \(N_ c/P\) is identifiable with \({\mathbb{C}}^ n\) and each open orbit can be seen to be a contractible, rationally homogeneous domain. It is also true that there are only a finite number of open orbits. These orbits are referred to as the Siegel domains of type N-P associated with N and P.'' With this notation the second main result is Theorem (0.2). Any contractible, rationally homogeneous domain \(\Omega\) is biholomorphic with a Siegel domain of type N-P. Specifically, there is a Siegel pair N- P, complex-Zariski open subsets \({\mathcal O}\subset N_ c/P\) and \({\mathcal W}\subset {\mathbb{C}}^ n\) and a one-to-one, non-singular, open, rational mapping \(\Psi\) of \({\mathcal O}\) onto \({\mathcal W}\) such that each of the Siegel domains associated with N-P is contained in \({\mathcal O}\) and \(\Psi\) maps one of these domains onto \(\Omega\). The rest of the paper is primarily concerned with the investigation of special rationally homogeneous domains, so-called ``generically non- degenerate Siegel N-P domains.'' The main results here are Theorem (0.3). Any generically non-degenerate, regular Siegel domain of type N-P is biholomorphic with a co-adjoint orbit of a completely solvable algebraic group corresponding to a totally complex polarization. Corollary (0.4). Generically non-degenerate, regular, Siegel N-P domains are pseudo-Kählerian in the sense that they carry a pseudo-Kählerian form which is invariant under a transitive group of bi-holomorphisms. This is a remarkable paper. It should open new avenues for research in a classical field.
rationally homogeneous domains in \({\mathbb{C}}^ n\)
scientific article; zbMATH DE number 4069316