TITLE: Uniformization of Kodaira fibered surfaces
QUESTION [10 upvotes]: Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the fibers (which are complex curves) are at least $2$. By abuse of notation I call $X$ a Kodaira fibered surface. What is known about the structure of the universal cover of $X$? In particular, when is it a bounded symmetric domain? I know that $X$ is a minimal algebraic surface of general type and that the universal cover of $X$ can never be a ball in $\mathbb{C}^{2}$.

REPLY [6 votes]: If an algebraic surface has the bidisk as universal cover, then, by Hirzebruch's proportionality theorem, its topological index is 0 (the index is 1/3 (c_1^2 - 2 c_2)).
But Kodaira proved that for Kodaira  fibrations the index is strictly positive.
To exclude the case that the universal covering is the ball, where 
c_1^2 = 3 c_2, again by Hirzebruch's theorem, is harder and was done by Kefeng Liu. You may look in my article with Rollenske for more results on Kodaira fibrations. 
Regards, Fabrizio Catanese.