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

TL;DR

This paper proves the vanishing of holomorphic 1-forms on certain coverings of the moduli space of curves, including spaces with level structures, spin, and Prym curves, under specific hypotheses.

Contribution

It establishes new vanishing results for holomorphic 1-forms on various coverings of the moduli space of curves, extending previous knowledge.

Findings

No non-trivial holomorphic 1-forms on Prym locus

Vanishing results for coverings with specific branch locus hypotheses

Applicable to moduli spaces with level, spin, and Prym structures

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 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.