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

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    $\begingroup$ Compact Shimura varieties for instance (associated to torsion-free arithmetic groups). But also any smooth proper curve of genus at least two. $\endgroup$ – Ariyan Javanpeykar Dec 20 '16 at 8:52
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    $\begingroup$ 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. $\endgroup$ – diverietti Dec 20 '16 at 9:11
  • $\begingroup$ @ diverietti Thanks! Are there are no good sufficient conditions? $\endgroup$ – Jaikrishnan Dec 20 '16 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! $\endgroup$ – diverietti Dec 21 '16 at 14:18
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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.

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