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Holomorphic $1$-forms on the moduli space of curves | | Filippo Francesco Favale
; Gian Pietro Pirola
; Sara Torelli
; | | Date: |
22 Sep 2020 | | 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 $ggeq 5$ 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 $
ho_D:H^0(mathcal{F}) o
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: | arXiv, 2009.10490 | | Services: | Forum | Review | PDF | Favorites | |
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