Inequalities for semi-stable fibrations on surfaces, and their relation to the Coleman-Oort conjecture

For semi-stable fibrations in curves over a curve Arakelov established a basic inequality which can be interpreted Hodge theoretically as an estimate for the degree of the Deligne extended Hodge bundle. When equality holds, it is well known that the period map embeds the base of the fibration as a Shimura curve in the relevant quotient of the Siegel upper half space. Going back to the situation of curve fibrations, one can ask if for those the Arakelov bound is sharp. This turns out not to be the case except if the genus of the fiber is “small.” This result is due to X. Lu and K. Zuo. In this note, I derive a simplified proof, using detailed surface theory. I also explain the relation with Shimura curves and the Coleman–Oort conjecture.