Moduli of spherical tori with one conical point

TL;DR

This paper characterizes the topology of the moduli space of genus one spherical surfaces with one conical point, revealing its connectivity, orbifold Euler characteristic, and complex structure depending on the cone angle parameter.

Contribution

It provides a detailed topological classification of the moduli space based on the conical angle, including its connectedness, orbifold Euler characteristic, and complex structure for even angles.

Findings

Connectedness of the moduli space varies with the cone angle.

Orbifold Euler characteristic is computed as -m^2/12 for certain angles.

For even angles, the moduli space has a natural complex structure and is biholomorphic to a quotient of hyperbolic space.

Abstract

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 $2\pi\vartheta$. 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 $m>1$.