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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.


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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


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