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

In this paper, we consider unramified coverings of the moduli space 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 M-g. This applies to several moduli spaces, as the moduli space of curves with 2-level 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.