Moduli of spherical tori with one conical point
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2020
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| _version_ | 1866910383238807552 |
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| author | Eremenko, Alexandre Mondello, Gabriele Panov, Dmitri |
| author_facet | Eremenko, Alexandre Mondello, Gabriele Panov, Dmitri |
| contents | 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 $m>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_02772 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Moduli of spherical tori with one conical point Eremenko, Alexandre Mondello, Gabriele Panov, Dmitri Differential Geometry Geometric Topology 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 $m>1$. |
| title | Moduli of spherical tori with one conical point |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2008.02772 |