论文标题
一个圆锥形的球形托里模量
Moduli of spherical tori with one conical point
论文作者
论文摘要
在本文中，我们确定模量空间$ \ MATHCAL {MS} _ {1,1}（\ vartheta）的拓扑，该属属的表面具有恒定曲率$ 1 $的riemannian指标，一个圆锥形的角度$2π\ vartheta $。特别是，对于$ \ vartheta \ in（2m-1,2m+1）$ non-odd，$ \ nathcal {ms} _ {1,1}（\ vartheta）$已连接，具有Orbifold Euler特征$ -M^2/12 $，其拓扑及其拓扑取决于integer $ m m m oste $ m m m m m> 0 $ m> 0 $ m> 0 $ m> 0 $> 0 $> 0> 0。对于$ \ vartheta = 2M+1 $奇数，$ \ MATHCAL {MS} _ {1,1}（2M+1）$具有$ \ lceil {m（m+1）/6} \ rceil $ connected组件。对于$ \ vartheta = 2m $，$ \ nathcal {ms} _ {1,1}（2m）$具有自然复杂的结构，并且对于某个子g_m $ g_m $ g_m $ g_m $ g_m $ g_m $ \ mathrm的$ \ mathrm $ \ mathrm {sl} sl}（2，2，2，2，2，2，k_m g_m $ biholomorphic to biholomorphic to biholomorphic to $ \ Mathbb {h}^2/g_m $ $ m^2 $，对于$ m> 1 $，这是非正常的。
In this paper we determine the topology of the moduli space $\mathcal{MS}_{1,1}(\vartheta)$ of surfaces of genus one with a Riemannian metric of constant curvature $1$ and one conical point of angle $2π\vartheta$. In particular, for $\vartheta\in (2m-1,2m+1)$ non-odd, $\mathcal{MS}_{1,1}(\vartheta)$ is connected, has orbifold Euler characteristic $-m^2/12$, and its topology depends on the integer $m>0$ only. For $\vartheta=2m+1$ odd, $\mathcal{MS}_{1,1}(2m+1)$ has $\lceil{m(m+1)/6}\rceil$ connected components. For $\vartheta=2m$ even, $\mathcal{MS}_{1,1}(2m)$ has a natural complex structure and it is biholomorphic to $\mathbb{H}^2/G_m$ for a certain subgroup $G_m$ of $\mathrm{SL}(2,\mathbb{Z})$ of index $m^2$, which is non-normal for $m>1$.
