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.


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)