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

  • Duc-Manh Nguyen

    Université de Bordeaux I, Talence, France


The space H(2)\mathcal{H}(2) consists of pairs (M,ω)(M,\omega), where MM is a Riemann surface of genus two, and ω\omega is a holomorphic 1-form which has only one zero of order two. There exists a natural action of C\mathbb{C}^* on H(2)\mathcal{H}(2) by multiplication to the holomorphic 1-form. In this paper, we single out a proper subgroup Γ\Gamma of Sp(4,Z)\mathrm{Sp}(4,\mathbb{Z}) generated by three elements, and show that the space H(2)/C\mathcal{H}(2)/\mathbb{C}^* can be identified with the quotient Γ\J2\Gamma\backslash\mathcal{J}_2, where J2\mathcal{J}_2 is the Jacobian locus in the Siegel upper half space H2\mathfrak{H}_2. A direct consequence of this result is that [Sp(4,Z):Γ]=6[\mathrm{Sp}(4,\mathbb{Z}):\Gamma]=6. The group Γ\Gamma can also be interpreted as the image of the fundamental group of H(2)/C\mathcal{H}(2)/\mathbb{C}^* in the symplectic group Sp(4,Z)\mathrm{Sp}(4,\mathbb{Z}).

Cite this article

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

DOI 10.4171/GGD/237