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Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. http://eom.springer.de/s/s084990.htmSiegel domain and references therein (those references probably answer question 2 as well). Another useful link is http://eom.springer.de/h/h047630.htmHomogeneous 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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Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. http://eom.springer.de/s/s084990.htm and references therein (those references probably answer question 2 as well). Another useful link is http://eom.springer.de/h/h047630.htm

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.
show/hide this revision's text 1

Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. http://eom.springer.de/s/s084990.htm and references therein (those references probably answer question 2 as well). Another useful link is http://eom.springer.de/h/h047630.htm