Complex Hyperbolic Structures on Disc Bundles over Surfaces

Details Complex Hyperbolic Structures on Disc Bundles over Surfaces Complex Hyperbolic Structures on Disc Bundles over Surfaces

2005-11-30

arxiv.org/abs/math/0511741v6

International Mathematics Research Notices (2011) 2011 (19): 4295-4375

Mathematics

Geometric Topology

52 pages, 12 pictures, 10 tables, 20 references. Changes: final version

Scientific paper

10.1093/imrn/rnq161

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/2PU(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.

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