Does anyone know an example of a smooth hyperbolic surface bundle over a hyperbolic surface (surface = compact two-manifold) which does *not* have a complex structure? Is there any decision procedure to tell, given such a bundle, whether it has a complex structure, or is it more of a black art?



A four-dimensional counterexample?
This [paper by Hillmann](http://www.ams.org/mathscinet-getitem?mr=1733043) addresses this question. He proves that a surface bundle over a surface which is a complex surface has a holomorphic fibration over the base, for some choice of complex structure on the base. He uses this to prove that when the fiber has genus 2, the bundle must be finitely covered by a product (see the last Corollary in the paper; this follows from a [result of Kas](http://www.ams.org/mathscinet-getitem?mr=242199)). So for any fiber bundle with genus 2 fiber $S\_2\to X\to S$, if the image of $\pi\_1(S)\to Mod(S\_2)$ induced by the fibration is infinite, then $X$ will not admit a complex structure. 


