## Weak form of Holzapfel’s conjecture.(English)Zbl 1210.14027

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

### MSC:

 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties 11F23 Relations with algebraic geometry and topology 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)