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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)
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