Holomorphic 1-forms on the moduli space of curves

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 $g\ge 5$ 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 ${\mathcal{M}}_g$.

Several improvements were made. The last version has been accepted for publication in "Geometry & Topology"

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