Ypatia 2022 - June 8-10, 2022

Session

Abstract: The moduli space ${\mathcal{M}}_g$ of smooth projective curves of genus $g$ is a quasi-projective variety, singular on loci of dimension at most $2g-1$. Let ${\mathcal{M}}_g^0$ denote its smooth locus. Not much is known about the cohomology $H^i({\mathcal{M}}_g^0,\mathbb{C})$ and even less about the spaces of holomorphic forms $H^i({\mathrm{\Omega }}_{{\mathcal{M}}_g^0}^j)$. Notice that ${\mathcal{M}}_g$ is not compact, so in particular it doesn't carry a Hodge decomposition and thus $H^i({\mathrm{\Omega }}_{{\mathcal{M}}_g^0}^j)$ can't be recovered from $H^i({\mathcal{M}}_g^0,\mathbb{C})$ just using Hodge theory. In the talk I will present the result for $i=1,j=0$, namely that ${\mathcal{M}}_g$ do not admit holomorphic 1-forms, and I will briefly discuss its generalization to other moduli spaces realized as finite coverings of ${\mathcal{M}}_g$ (e.q. spin curves). The techniques comes from Hodge theory on the Deligne-Mumford compactification and intersection theory on the Satake compactification of ${\mathcal{M}}_g$. The work is joint with F.F. Favale and G.P.Pirola.)

8 juin 2022, 15:15