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

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 2-level structures, of spin curves and of Prym curves. In particular, we obtain that there are no non-trivial holomorphic 1-forms on the smooth open set of the Prym locus.