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

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

2005-12-17

arxiv.org/abs/math/0512406v3

Mathematics

Geometric Topology

19 pages, 6 figures, 9 references. Changes: revised exposition

Scientific paper

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 $\Sigma$ (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 $2(\chi+e)=3\tau$ which is necessary for the existence of a holomorphic section of the bundle, where $\chi=\chi\Sigma$ stands for the Euler characteristic of $\Sigma$, e=eM, for the Euler number of the bundle, and $\tau$, 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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