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 ≥ 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 H^0(F)->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.
Holomorphic 1-forms on the moduli space of curves
Filippo Favale
;Gian Pietro Pirola;
2024-01-01
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 ≥ 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 H^0(F)->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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
