Holomorphic 1–forms on the moduli space of curves

Favale, Filippo Francesco¹ ; Pirola, Gian Pietro¹ ; Torelli, Sara²

¹ Dipartimento di Matematica Felice Casorati, Università degli Studi di Pavia, Pavia, Italy
² Institut für Algebraische Geometrie, Leibniz Universität Hannover, Hannover, Germany

Geometry & topology, Tome 28 (2024) no. 7, pp. 3001-3022.

Résumé

Since the 1960s it has been well known that there are no nontrivial closed holomorphic $1$–forms on the moduli space ${\mathcal{ℳ}}_g$ of smooth projective curves of genus $g>2$. We strengthen this result, proving that for $g\ge 5$ there are no nontrivial 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 ${\mathcal{ℳ}}_g$.

Détail

DOI : 10.2140/gt.2024.28.3001

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

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