Complex Hyperbolic Structures on Disc Bundles over Surfaces. II. Example of a Trivial Bundle

Sasha Anan'in; Nikolay Gusevskii

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Complex Hyperbolic Structures on Disc Bundles over Surfaces. II. Example of a Trivial Bundle

Sasha Anan'in·Nikolay Gusevskii

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Abstract

This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface $Σ$ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open Question 8.1] if a trivial circle bundle over a closed surface of genus >1 admits a holomorphically fillable contact structure. The constructed example M satisfies the relation $2(χ+e)=3τ$ which is necessary for the existence of a holomorphic section of the bundle, where $χ=χΣ$ stands for the Euler characteristic of $Σ$, e=eM, for the Euler number of the bundle, and $τ$, for the Toledo invariant. (The relation is also valid for the series of examples constructed in [AGG].) Open question: Does there exist a holomorphic section of the bundle M?

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BibTeX RIS

Sasha Anan'in, Nikolay Gusevskii. 2007-02-23. Complex Hyperbolic Structures on Disc Bundles over Surfaces. II. Example of a Trivial Bundle. https://arxiv.org/abs/math/0512406

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