This question is somewhat related to this other question of mine.

I was wondering which are the known examples of bounded domains $\Omega$ in $\mathbb C^n$ admitting a compact free quotient.

By a theorem of Siegel, such a domain must be holomorphically convex. Moreover, if the boundary is sufficiently regular, say $C^2$ (even if, by a recent theorem of A. Zimmer, $C^{1,1}$ suffices), by the classical theorem of Wong-Rosay, then $\Omega$ must be biholomorphic to the unit ball.

Of course all bounded symmetric domains give such examples, by a classical theorem of E. Borel. But I am interested in more "exotic" examples, specifically **non symmetric** examples.

The only I am aware of live in $\mathbb C^2$ and are the universal covers of Kodaira fibrations (see this question for more details).

- Is it possibile for instance to construct higher dimensional analogous of the universal cover of a Kodaira fibration?

For dimension $n\ge 4$, there exists (many in fact) homogeneous bounded domains in $\mathbb C^n$ which are non symmetric. In 1979 J. E. D’Atri proved that there exists bounded homogeneous non-symmetric domains, for $n\ge 6$, whose Bergman metric has positive holomorphic sectional curvature somewhere.

Unfortunately, they can **never** cover a compact manifold. Indeed, it was shown by J. Hano in 1957 that if a homogeneous bounded domain covers a manifold of finite volume, *i.e.* if its automorphism group admits a lattice so that it's unimodular, than the domain is in fact symmetric.

Remark that I am really looking for *compact, discrete, free* quotients.

Thank you very much in advance.