On the interplay between discrete invariants of complex hyperbolic disc bundles over surfaces

TL;DR

This paper explores the relationships between key invariants of complex hyperbolic disc orbibundles over hyperbolic 2-orbifolds, proposing and proving conjectures about their interdependence.

Contribution

It introduces a conjecture linking the Toledo invariant, Euler number, and Euler characteristic, and proves specific cases where this relation holds.

Findings

Proved that for orbibundles over quadrangles of bisectors, 3τ = 2e + 2χ.

Established that -3|τ| = 2e + 2χ when a section without complex tangent planes exists.

Conjectured a universal relation -3|τ| = 2e + 2χ for all such orbibundles.

Abstract

We investigate the relationship between three natural invariants of complex hyperbolic disc orbibundles over oriented and closed hyperbolic $2$-orbifolds. These invariants are the Euler characteristic $\chi$ of the $2$-orbifold, the Euler number $e$ of the disc orbibundle, and the Toledo invariant $\tau$ of a faithful representation of the surface group into $\mathrm{PU}(2,1)$ attached to the complex hyperbolic structure of the disc orbibundle. Based on previous examples, we conjecture that $-3|\tau| = 2e+2\chi$ always holds. For complex hyperbolic disc orbibundles over $2$-orbifolds derived from quadrangles of bisectors via tessellation, we prove that $3\tau = 2e+2\chi$. Furthermore, we demonstrate that $-3|\tau| = 2e+2\chi$ holds when a section with no complex tangent planes is present.