Complex hyperbolic geometry on disc bundles over surfaces

Contact for this talk: Grigory Avramidi (gavramidi @ mpim-bonn.mpg.de)

This talk explores 4-manifolds with complex hyperbolic structures, i.e., the model geometry is the complex hyperbolic plane: the unit ball in $\mathbb{C}^2$ with its group of holomorphic automorphisms $\mathrm{PU}(2,1)$. More precisely, we focus on oriented disc bundles over oriented closed hyperbolic surfaces.

Several examples are discussed with attention to three distinguished invariants: the Euler number of the disc bundle, the Euler characteristic of the surface, and the Toledo invariant. Empirical data suggests that the interplay of these invariants is related to the existence of complex hyperbolic structures and holomorphic sections.

Additionally, we present the joint work "Quotients of the holomorphic 2-ball and the turnover" with Carlos Grossi, which analyzes disc orbibundles over hyperbolic 2-spheres with three conic points, the simplest among closed hyperbolic 2-orbifolds. The examples of complex hyperbolic disc orbibundles emerge from faithful representations of von Dyck groups into $\mathrm{PU}(2,1)$. Additionally, our description of the character varieties of such representations leads to the discovery of the first non-rigid complex hyperbolic disc orbibundles, meaning their complex hyperbolic structures have non-trivial moduli spaces.