Lagrangian constant cycle subvarieties in Lagrangian fibrations
Lagrangian constant cycle subvarieties in Lagrangian fibrations
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拉格朗日纤维中的拉格朗日恒定循环子类型
DOI：
10.1093/imrn/rnx334
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Hsueh
中科院分区：
文献类型：
--
作者：
Hsueh
我们证明了不可约紧致Calabi-Yau流形$X$的显性亚纯映射的像是有理连通的，它的一般纤维的维度严格介于$0$和$\dim X$之间。利用这一结果，我们对任何允许拉格朗日纤维的超Kahler流形$X$构造了一个拉格朗日常数循环子簇$\Sigma_H$，它依赖于一个因子类$H$，它对一些光滑的拉格朗日纤维的限制是充分的。如果$\dim X=4$，我们还证明了直到标量倍数，$\mathm{CH}_0(X)$中的$\Sigma_H$所支持的零圈的类既不依赖于$H$，也不依赖于拉格朗日纤维(提供了$b_2(X)\ge 8$)。
We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold $X$ whose general fiber is of dimension strictly between $0$ and $\dim X$ is rationally connected. Using this result, we construct for any hyper-Kahler manifold $X$ admitting a Lagrangian fibration a Lagrangian constant cycle subvariety $\Sigma_H$ in $X$ which depends on a divisor class $H$ whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on $\Sigma_H$ in $\mathrm{CH}_0(X)$ depend neither on $H$ nor on the Lagrangian fibration (provided $b_2(X) \ge 8$).