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

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}) .