On the upper-semicontinuity of automorphism groups of model domains in almost complex manifolds
This work deals with the upper semicontinuity of automorphism groups of almost complex manifolds. A typical result for the upper semicontinuity of holomorphic automorphism groups is the Greene-Krantz Theorem [\textit{R. E. Greene} and \textit{S. G. Krantz}, Math. Ann. 261, 425--446 (1982; Zbl 0531.32016); Math. Z. 190, 455--467 (1985; Zbl 0584.58014)], which deals with the automorphism groups in case of deformation of complex structures on bounded pseudo convex domains. It says that if \(\Omega\) is a \(\mathbb C^{\infty} \) bounded pseudoconvex domain in \(\mathbb C^{n} \) with the standard complex structure \( J_{\mathrm{st}} \), and if \(( J_{\nu} ) \) is a sequence of complex structures on \(\bar {\Omega} \) which converge to \(J_{\mathrm{st}} \) in the \( \mathbb C^{\infty} \) topology of \(\bar {\Omega} \), then \(\text{Aut} ( \Omega, J_{\nu}) \) is isomorphic to a subgroup of \(\text{Aut} (\Omega , J_{\mathrm{st}}) \), for sufficiently large \(\nu \).NEWLINENEWLINEFor general (even unbounded) strongly pseudo convex domains in \( \mathbb C^{n} \), a local version of [\textit{J.-P. Rosay}, Ann. Inst. Fourier 29, No. 4, 91--97 (1979; Zbl 0402.32001)] (see also [\textit{H. Gaussier} et al., Complex Variables, Theory Appl. 47, No. 9, 761--768 (2002; Zbl 1044.32019)]) which characterizes the unit ball, \(\mathbb B^{n}\), in \(\mathbb C^{n} \) is used. However, this characterization fails in case of almost complex manifolds [\textit{H. Gaussier} and \textit{A. Sukhov}, Math. Z. 254, No. 3, 567--589 (2006; Zbl 1107.32009)]. There are many strongly pseudoconvex homogeneous domains in almost complex manifolds, which are biholomorphically distinct from each other. Such a domain is called a model domain in the paper under review.NEWLINENEWLINELet \( J_{\mathrm{st}}\) denote the standard complex structure on \(\mathbb C^{n} \). More precisely, a model domain is a pair \( ( \mathbb H^{n+1} , J_{ \mathfrak B } ) \), where \(\mathbb H^{n+1} \) is the Siegel half plane in \(\mathbb C^{n+1} \) (i.e., the usual unbounded representation of the unit ball, \((\mathbb H^{n+1} , J_{\mathrm{st}} ) \simeq (\mathbb B^{n+1}, J_{\mathrm{st}})\) ) and \( J_{\mathfrak B} \) is an almost complex structure on \(\mathbb C^{n+1} \) given by an \( n \times n \) skew-symmetric matrix \( \mathfrak{B}\).NEWLINENEWLINE The automorphism group of such a model domain, \(\text{Aut} (\mathbb{H}^{n+1} , J_{\mathfrak{B}})\), is not compact. The paper examines the upper semicontinuity of automorphism groups for this deformation. The main result in the paper is:NEWLINENEWLINETheorem. For every model domain \((\mathbb{H}^{n+1} , J_{\mathfrak{B}}) \), its automorphism group is conjugate-isomorphic to a subgroup of the automorphism group of the unit ball \(\mathbb B^{n+1} \) with the standard complex structure.NEWLINENEWLINEHere, a conjugate isomorphism means an isomorphism given by \( \varphi \mapsto F^{-1} \circ \varphi \circ F \), for some diffeomorphism \( F \colon \mathbb{H}^{n+1} \rightarrow \mathbb B^{n+1} \).NEWLINENEWLINETo prove the main theorem, the authors show that there exists an integrable almost complex structure on \((\mathbb{H}^{n+1} , J_{\mathfrak{B} }) \), which is invariant under the action of \(\text{Aut} (\mathbb{H}^{n+1}, J_{\mathfrak{B}}) \), and for which \( \mathbb{H}^{n+1} \) is strongly pseudoconvex. Then, using the Wong-Rosay theorem from [Zbl 1044.32019], they get a conjugate isomorphism from \(\text{Aut} (\mathbb{H}^{n+1} , J_{\mathfrak{B}}) \) into \( \text{Aut} (\mathbb B^{n+1} , J_{\mathrm{st}}) \), which is induced by a diffeomorphism \(\mathbb{H}^{n+1} \rightarrow \mathbb B^{n+1} \).
scientific article; zbMATH DE number 6334524