Holomorphic 1-forms on the moduli space of curves

TL;DR

This paper proves the non-existence of non-trivial holomorphic 1-forms on the moduli space of curves for genus g≥5, extending previous results and providing new extension theorems for sections of sheaves.

Contribution

It establishes a new non-existence result for holomorphic 1-forms on moduli spaces for higher genus and introduces an extension theorem for sheaf sections on projective varieties.

Findings

No non-trivial holomorphic 1-forms on e4gb0 for ga8 5.

Extension theorem characterizing surjectivity of restriction maps for sheaf sections.

Application of the extension theorem to the Hodge line bundle on e4gb0 moduli space.

Abstract

Since the sixties it is well known that there are no non-trivial closed holomorphic $1$-forms on the moduli space $\mathcal{M}_g$ of smooth projective curves of genus $g>2$. In this paper, we strengthen such result proving that for $g\geq 5$ there are no non-trivial holomorphic $1$-forms. With this aim, we prove an extension result for sections of locally free sheaves $\mathcal{F}$ on a projective variety $X$. More precisely, we give a characterization for the surjectivity of the restriction map $\rho_D:H^0(\mathcal{F})\to H^0(\mathcal{F}|_{D})$ for divisors $D$ in the linear system of a sufficiently large multiple of a big and semiample line bundle $\mathcal{L}$. Then, we apply this to the line bundle $\mathcal{L}$ given by the Hodge class on the Deligne Mumford compactification of $\mathcal{M}_g$.