Geometry, Integrability and Quantization Weak Form of Holzapfel's Conjecture *
Geometry, Integrability and Quantization Weak Form of Holzapfel's Conjecture *
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Holzapfel 猜想的几何、可积性和量化弱形式 *
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通讯作者：
Boris Kotzev
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作者：
A. Kasparian;Boris Kotzev
设B ∈ C2是单位球，Γ是SU（2，1）的格.考虑到所有紧致黎曼曲面都是单位圆盘球面C的离散球面，Holzapfel证明了离散球面球面B/Γ及其紧化在光滑射影曲面中广泛分布。已知有一般类型的球幂函数B/Γ，以及有理的、阿贝尔的、K3的和椭圆的球幂函数。本文构造了三个非紧球曲面，它们分别是超椭圆曲面、Enriques曲面和以椭圆为底的直纹曲面的双有理曲面。结果表明，球商曲面在光滑射影曲面的八个Enriques分类类中均有代表.
Let B ⊂ C 2 be the unit ball and Γ be a lattice of SU(2, 1). Bearing in mind that all compact Riemann surfaces are discrete quotients of the unit disc ∆ ⊂ C, Holzapfel conjectures that the discrete ball quotients B/Γ and their compactifications are widely spread among the smooth projective surfaces. There are known ball quotients B/Γ 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 En-riques classification classes of smooth projective surfaces.