Holomorphic 1–forms on the moduli space of curves
Year: 2024
Type: article
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 $ρ_D:H^0(\mathcal{F})\to H^0(\mathcal{F}|_{D})$ 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 $\mathcal{M}_g$.
Source: Geometry & Topology
Institutions: University of Pavia, Leibniz University Hannover
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funders: Alexander von Humboldt-Stiftung, Istituto Nazionale di Alta Matematica "Francesco Severi", Università degli Studi di Milano, Dipartimenti di Eccellenza, Università degli Studi di Milano-Bicocca, Ministero dell’Istruzione, dell’Università e della Ricerca, Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni
OpenAlex ID: W3088471773