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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 $$p,q>1$$, $$p\not=q$$.

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