Holomorphic 1-forms on some coverings of the Moduli space of curves

Abstract

In this paper, we consider unramified coverings of the moduli space $\mathcal{M}_g$ of smooth projective complex curves of genus $g$. Under some hypothesis on the branch locus of the finite extended map to the Deligne-Mumford compactification, we prove the vanishing of the vector space of holomorphic 1 -forms on the preimage of the smooth locus of $\mathcal{M}_g$. This applies to several moduli spaces, as the moduli space of curves with 2level structures, of spin curves and of Prym curves. In particular, we obtain that there are no nontrivial holomorphic 1 -forms on the smooth open set of the Prym locus.