论文标题
曲线模量空间上的全态1形
Holomorphic 1-forms on the moduli space of curves
论文作者
论文摘要
由于六十年代，众所周知，Moduli Space $ \ Mathcal {M} _g $ g> 2 $的光滑投射曲线的$ \ Mathcal {m} _g $上没有非平凡的封闭全态$ 1 $。在本文中，我们加强了这样的结果，证明对于$ g \ geq 5 $，没有非平凡的全态$ 1 $ forms。以此目的，我们证明了针对投影型品种$ x $的本地免费滑轮$ \ MATHCAL {F} $的部分的扩展结果。更准确地说，我们给出了限制映射的过滤性$ρ_d：h^0（\ Mathcal {f}）\的表征。然后，我们将其应用于hodge类$ \ Mathcal {m} _g $的hodge class给出的线束$ \ mathcal {l} $。
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 $\mathcal{M}_g$.
