Complex Hyperbolic Structures on Disc Bundles over Surfaces
  We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyperbolic disc bundles M->{\Sigma} that: admit both real and complex hyperbolic structures; satisfy the equality 2(\chi+e)=3\tau; satisfy the inequality \chi/2<e; and induce discrete and faithful representations \pi_1\Sigma->PU(2,1) with fractional Toledo invariant; where {\chi} is the Euler characteristic of \Sigma, e denotes the Euler number of M, and {\tau} stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(\chi+e)=3\tau, we conjecture that there exists a holomorphic section in all our examples.   In order to reduce the amount of calculations, we systematically explore coordinate-free methods. 
Sasha Anan'in, Carlos H. Grossi, Nikolay Gusevskii