Holomorphic 1–forms on the moduli space of curves

Favale, Filippo Francesco; Pirola, Gian Pietro; Torelli, Sara

doi:10.2140/gt.2024.28.3001

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Holomorphic $1$–forms on the moduli space of curves

Filippo Francesco Favale, Gian Pietro Pirola and Sara Torelli

Geometry & Topology 28 (2024) 3001–3022

DOI: 10.2140/gt.2024.28.3001

Abstract

Since the 1960s it has been well known that there are no nontrivial closed holomorphic $1$ –forms on the moduli space $ℳ_{g}$ of smooth projective curves of genus $g > 2$ . We strengthen this result, proving that for $g \geq 5$ there are no nontrivial holomorphic $1$ –forms. With this aim, we prove an extension result for sections of locally free sheaves $ℱ$ on a projective variety $X$ . More precisely, we give a characterization for the surjectivity of the restriction map $ρ_{D} : H^{0} (ℱ) \to H^{0} (ℱ |_{D})$ for divisors $D$ in the linear system of a sufficiently large multiple of a big and semiample line bundle $ℒ$ . Then we apply this to the line bundle $ℒ$ given by the Hodge class on the Deligne–Mumford compactification of $ℳ_{g}$ .

Keywords

moduli space, 1–forms, extension of sections, positivity

Mathematical Subject Classification

Primary: 14H15

Secondary: 32L10

References

Publication

Received: 2 November 2020

Revised: 10 March 2023

Accepted: 16 April 2023

Published: 25 November 2024

Proposed: Mark Gross

Seconded: Richard P Thomas, Jim Bryan

Authors

Filippo Francesco Favale
Dipartimento di Matematica Felice Casorati Università degli Studi di Pavia Pavia Italy

Gian Pietro Pirola
Dipartimento di Matematica Felice Casorati Università degli Studi di Pavia Pavia Italy

Sara Torelli
Institut für Algebraische Geometrie Leibniz Universität Hannover Hannover Germany

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Volume 28, issue 7 (2024)