On bounded homogeneous connected domains of C^n - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:51:26Z http://mathoverflow.net/feeds/question/74944 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74944/on-bounded-homogeneous-connected-domains-of-cn On bounded homogeneous connected domains of C^n Hugo Chapdelaine 2011-09-08T22:30:56Z 2011-12-02T01:26:01Z <p>So let $D\subseteq \mathbb{C}^n$ be a <strong>bounded</strong> connected open set with a transitive action of its group of biholomorphisms (which we denote by $Hol(D)$). Note that <strong>I'm not</strong> assuming that $D$ is symmetric. We thus have that $D$ is "homeomorphic" to $Hol(D)/K$ where $K=Stab(d_0)$ for some $d_0\in D$.</p> <p>In the special case where $Hol(D)$ is a real Lie group and that $K$ is a maximal compact of $Hol(D)$ then by a theorem of Elie Cartan we have that $Hol(D)/K$ is homeomoprphic to $\mathbb{R}^m$ and thus contractible. </p> <p>Under my assumptions: </p> <p>(1) Is $Hol(D)$ always a Lie group?</p> <p>(2) Is $K$ always a maximal compact?</p> <p>(3) In general is $D$ always contractible (or simply connected)?</p> http://mathoverflow.net/questions/74944/on-bounded-homogeneous-connected-domains-of-cn/74952#74952 Answer by Gjergji Zaimi for On bounded homogeneous connected domains of C^n Gjergji Zaimi 2011-09-08T23:46:28Z 2011-09-08T23:46:28Z <p>It is a theorem of H. Cartan that $Hol(D)$ for any bounded such $D\subset \mathbb C^n$ is a finite dimensional real Lie group. See for example chapter 9 of "Several complex variables" by R. Narasimhan.</p> http://mathoverflow.net/questions/74944/on-bounded-homogeneous-connected-domains-of-cn/74959#74959 Answer by algori for On bounded homogeneous connected domains of C^n algori 2011-09-09T00:40:26Z 2011-12-02T01:26:01Z <p>Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. <a href="http://www.encyclopediaofmath.org/index.php?title=Siegel_domain" rel="nofollow">Siegel domain</a> and references therein (those references probably answer question 2 as well). Another useful link is <a href="http://www.encyclopediaofmath.org/index.php?title=Homogeneous_bounded_domain" rel="nofollow">Homogeneous bounded domain</a>.</p> <p>upd: Another Google search gave the following references:</p> <p>"Homogeneous Bounded Domains and Siegel Domains" by Soji Kaneyuki, Springer LNM 241.</p> <p>"Theory of complex homogeneous bounded domains" by Yichao Xu, Mathematics and its applications 569.</p>