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arXiv · 2210.07125

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

Filippo Francesco Favale·Juan Carlos Naranjo·Gian Pietro Pirola·Sara Torelli
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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.

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Filippo Francesco Favale, Juan Carlos Naranjo, Gian Pietro Pirola, Sara Torelli. 2022-10-13. Holomorphic 1-forms on some coverings of the moduli space of curves. https://arxiv.org/abs/2210.07125

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