Open Access
2010 Weak Form of Holzapfel’s Conjecture
Azniv Kasparian, Boris Kotzev
J. Geom. Symmetry Phys. 19: 29-42 (2010). DOI: 10.7546/jgsp-19-2010-29-42


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.


Download Citation

Azniv Kasparian. Boris Kotzev. "Weak Form of Holzapfel’s Conjecture." J. Geom. Symmetry Phys. 19 29 - 42, 2010.


Published: 2010
First available in Project Euclid: 25 May 2017

zbMATH: 1210.14027
MathSciNet: MR2674960
Digital Object Identifier: 10.7546/jgsp-19-2010-29-42

Rights: Copyright © 2010 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

Back to Top