Weak Form of Holzapfel’s Conjecture

Abstract

Let $\mathbb{B} \subset \mathbb{C}^{2}$ be the unit ball and $\Gamma$ be a lattice of $\mathrm{SU}(2,1)$. 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 noncompact 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.