Complex Hyperbolic Structures on Disc Bundles over Surfaces

Abstract

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->Σ that: admit both real and complex hyperbolic structures; satisfy the equality 2(χ+e)=3τ; satisfy the inequality χ/2<e; and induce discrete and faithful representations π_1Σ->PU(2,1) with fractional Toledo invariant; where χ is the Euler characteristic of Σ, e denotes the Euler number of M, and τ stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(χ+e)=3τ, 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.