## Relatório de pesquisa 31/04

A. Anan´in, Carlos H. Grossi and Nikolay Gusevskii, Complex Hyperbolic Structures on Disc Bundles Over Surfaces

Abstract: We study oriented disc bundles $M$ over a closed orientable surface $\Sigma$ that arise from certain discrete subgroups in $\PU(2,1)$ generated by reflections in ultraparallel complex lines in the complex hyperbolic plane $\Bbb H_{\Bbb C}^2$. The results obtained allow us to construct the first examples of

$\bullet$ Disc bundles $M$ over $\Sigma$ that satisfy the equality $2(\chi+e)=3\tau$,

$\bullet$ Disc bundles $M$ over $\Sigma$ that satisfy the inequality $\frac{1}{2}\chi<e$,

$\bullet$ Disc bundles $M$ over $\Sigma$ that admit both real hyperbolic and complex hyperbolic structures,

$\bullet$ Discrete and faithful representations $\varrho:\pi_1\Sigma\to\PU(2,1)$ with fractional Toledo invariant, and

$\bullet$ Nonhomeomorphic disc bundles $M$ over the same $\Sigma$ and with the same $\tau$, where $\chi$ stands for the Euler characteristic $\chi(\Sigma)$ of $\Sigma$, $e$, for the Euler number $e(M)$ of $M$, and $\tau$, 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.

If you are interested in obtaining a copy of this report please contact the authors either via e-mail or via snail mail. Their postal address is the following (the first two authors):

IMECC, UNICAMP, Cx. P. 6065

13083-970 Campinas, SP, BRAZIL

June 30, 2004