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So let $D\subseteq \mathbb{C}^n$ be a bounded connected open set with a transitive action of its group of biholomorphisms (which we denote by $Hol(D)$). Note that I'm not assuming that $D$ is symmetric. We thus have that $D$ is "homeomorphic" to $Hol(D)/K$ where $K=Stab(d_0)$ for some $d_0\in D$.

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.

Under my assumptions:

(1) Is $Hol(D)$ always a Lie group?

(2) Is $K$ always a maximal compact?

(3) In general is $D$ always contractible (or simply connected)?

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Dear Hugo -- re "is Hol(D) always a Lie group": no, take $D=\mathbb{C}^2$; it has automorphisms of the form $(x,y)\mapsto (x,y+f(x))$ where $f$ is any holomorphic function $\mathbb{C}\to\mathbb{C}$. –  algori Sep 8 '11 at 22:40
Maybe I am missing something: isn't $D=\mathbb{C}^\times$ an example where $D$ is not contractible? –  M P Sep 8 '11 at 22:40
Yes you are right $\mathbb{C}^{\times}$ is a counter-example, so I'll redit my question –  Hugo Chapdelaine Sep 8 '11 at 23:09
I forgot put that $D$ was bounded –  Hugo Chapdelaine Sep 8 '11 at 23:21
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2 Answers

up vote 2 down vote accepted

Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. Siegel domain and references therein (those references probably answer question 2 as well). Another useful link is Homogeneous bounded domain.

upd: Another Google search gave the following references:

"Homogeneous Bounded Domains and Siegel Domains" by Soji Kaneyuki, Springer LNM 241.

"Theory of complex homogeneous bounded domains" by Yichao Xu, Mathematics and its applications 569.

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Thanks a lot @algori for the reference –  Hugo Chapdelaine Sep 9 '11 at 1:02
I fixed the links. –  George Lowther Dec 2 '11 at 1:28
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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.

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thanks @Gjergji –  Hugo Chapdelaine Sep 9 '11 at 1:04
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