Weak Form of Holzapfel’s Conjecture

Abstract

Let ${\mathbb B} \subset {\mathbb C}^2$ be the unit ball and $\Gamma$ be a lattice of ${\rm SU} (2,1)$. Bearing in mind that all compact Riemann surfaces are discrete quotients of the unit disc $\Delta \subset {\mathbb C}$, Holzapfel conjectures that the discrete ball quotients ${\mathbb B} / \Gamma$ and their compactifications are widely spread among the smooth projective surfaces. There are known ball quotients ${\mathbb B} / \Gamma$ of general type, as well as rational, abelian, K3 and elliptic ones. The present note constructs three non-compact ball quotients, which are birational, respectively, to a hyperelliptic, Enriques or a ruled surface with an elliptic base. As a result, we establish that the ball quotient surfaces have representatives in any of the eight Enriques classification classes of smooth projective surfaces.