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On a characterization of unbounded homogeneous domains with boundaries of light cone type. (English) Zbl 1388.32015
The authors consider the two homogeneous \((n+1)\)-dimensional unbounded Reinhardt domains \[ C^{n,1}=\big\{(z_0,z_1,\ldots,z_n)\in {\mathbb C}^{n+1}~\big |~-|z_0|^2+|z_1|^2+\ldots+|z_n|^2<0\big\}, \] and \[ D^{n,1}=\big\{(z_0,z_1,\ldots,z_n)\in {\mathbb C}^{n+1}~\big |~-|z_0|^2+|z_1|^2+\ldots+|z_n|^2>0\big\}. \] The main results of the paper are the determination of the holomorphic automorphism group of \( D^{n,1} \) by \[ \mathrm{Aut}(D^{n,1})=GU(n,1), \] where \(GU(n,1):=\{A\in \mathrm{GL}(n+1,{\mathbf C})~|~ A^*JA=\nu(A) J\}\), with \(\nu(A)\) a non-zero real number depending on \(A\), and the characterization of \(D^{n,1}\) by \(\mathrm{Aut}(D^{n,1})\) in a class of complex manifolds containing the Stein manifolds:
Theorem. Let \(M\) be a connected complex manifold of dimension \(n+1\) that is holomorphically separable and admits a smooth envelope of holomorphy. Assume that \(\mathrm{Aut}(M)\) is isomorphic to \(\mathrm{Aut}(D^{n,1})\) as a topological group. Then \(M\) is biholomorphic to \(D^{n,1}\).
Important ingredients in the proof of the theorem are the determination of the holomorphic automorphism group of \( C^{n,1} \), and the existence in \(\mathrm{Aut}(M)\) of an effective \((n+1)\)-dimensional torus \(T\) acting effectively on \(M\). The latter fact implies that \(M\) is \(T\)-equivariantly biholomorphic to a Reinhardt domain in \({\mathbb C}^{n+1}\).
The above characterization is generally false for unbounded homogeneous domains in \({\mathbb C}^n\), for \(n\geq 5\). Counterexamples are provided by the domains \[ C^{p,q}=\big\{(Z,W)\in {\mathbb C}^{p+q}~\big|~\| Z\|^2-\| W\|^2<0\big\}\text{ and }D^{p,q}=\big\{(Z,W)\in {\mathbb C}^{p+q}~\big|~\| Z\|^2-\| W\|^2>0\big\}, \] for \(p,q>1\), \(p\not=q\).

32M10 Homogeneous complex manifolds
32M05 Complex Lie groups, group actions on complex spaces
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