What are some general classes of compact complex manifolds whose universal covers are bounded domains? One class I know are the Kodaira fibered surfaces.

1$\begingroup$ Compact Shimura varieties for instance (associated to torsionfree arithmetic groups). But also any smooth proper curve of genus at least two. $\endgroup$– Ariyan JavanpeykarDec 20, 2016 at 8:52

6$\begingroup$ Such a manifold must be Kobayashi hyperbolic. Conjecturally they should be all projective and with ample canonical bundle. Moreover, again conjecturally, they should have only subvariety of general type. This gives (again conjecturally) some necessary conditions (far from being sufficient). They also should have of course infinite fundamental group. Of course in dimension one this is also sufficient, so that compact complex curves admit a bounded domain as universal cover iff they are of genus greater than or equal to two. $\endgroup$– diveriettiDec 20, 2016 at 9:11

$\begingroup$ @ diverietti Thanks! Are there are no good sufficient conditions? $\endgroup$– JaikrishnanDec 20, 2016 at 11:02

$\begingroup$ You're welcome. Not at my knowledge. But the paper cited in T. Amdeberhan's answer is definitely a good place to look at! $\endgroup$– diveriettiDec 21, 2016 at 14:18
1 Answer
I'm not aware of a sufficient condition, although many classes have been recognized as carrying the "bounded domain" covers. Here is one interesting paper to look into.