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 1/2 chi 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 obtain 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.194295437