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Complex Hyperbolic Structures on Disc Bundles over Surfaces

Sasha Anan'in·Carlos H. Grossi·Nikolay Gusevskii
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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 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.

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Sasha Anan'in, Carlos H. Grossi, Nikolay Gusevskii. 2011-07-26. Complex Hyperbolic Structures on Disc Bundles over Surfaces. https://doi.org/10.1093/imrn%2Frnq161

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