Q:

Uniformization of Kodaira fibered surfaces

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 $\mathbb{C}^{2}$. 

A:

Here are the arguments to exclude polydisk and the ball (there are no other complex 2-dimensional bounded symmetric domains: In fact, one can do without this and argue that any domain other than the ball would have rank $\ge 2$ and, hence, Margulis superrigidity theorem would apply). 

Kefeng Liu ("Geometric height inequalities", Math. Research Letters, 3 (1996), 693–702) proved that a compact complex-hyperbolic surface cannot admit a  holomorphic submersion to a Riemann surface.  This excludes the complex ball. 
Consider the 2-dimensional polydisk $D^2$ and a group $\Gamma$ acting discretely, holomorphically and cocompactly on $D^2$. Then $\Gamma$ is a lattice in $Isom(D^2)$ and by Margulis' superrigidity theorem either $\Gamma$ is reducible or it is superrigid, and , ehnce, does not admit an epimorphism to a surface group. The second is impossible in the case of fibrations over Riemann surfaces, the first contradicts non-isotriviality assumption in the question.

On the other hand, the universal cover of your complex manifold $X$ is a certain bounded domain in ${\mathbb C}^2$: This result could be found in 
P.A. Griffiths, Complex-analytic properties of certain Zariski open sets on algebraic varieties. Ann. of Math. (2) 94 (1971), 21–51.  
who proves it using Bers' simultaneous uniformization. 

A:

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.

