# On the topology of $\mathcal{H}(2)$

### Duc-Manh Nguyen

Université de Bordeaux I, Talence, France

## Abstract

The space $\mathcal{H}(2)$ consists of pairs $(M,\omega)$, where $M$ is a Riemann surface of genus two, and $\omega$ is a holomorphic 1-form which has only one zero of order two. There exists a natural action of $\mathbb{C}^*$ on $\mathcal{H}(2)$ by multiplication to the holomorphic 1-form. In this paper, we single out a proper subgroup $\Gamma$ of $\mathrm{Sp}(4,\mathbb{Z})$ generated by three elements, and show that the space $\mathcal{H}(2)/\mathbb{C}^*$ can be identified with the quotient $\Gamma\backslash\mathcal{J}_2$, where $\mathcal{J}_2$ is the Jacobian locus in the Siegel upper half space $\mathfrak{H}_2$. A direct consequence of this result is that $[\mathrm{Sp}(4,\mathbb{Z}):\Gamma]=6$. The group $\Gamma$ can also be interpreted as the image of the fundamental group of $\mathcal{H}(2)/\mathbb{C}^*$ in the symplectic group $\mathrm{Sp}(4,\mathbb{Z})$.

## Cite this article

Duc-Manh Nguyen, On the topology of $\mathcal{H}(2)$. Groups Geom. Dyn. 8 (2014), no. 2, pp. 513–551

DOI 10.4171/GGD/237