论文标题
当地拓扑障碍
Local topological obstruction for divisors
论文作者
论文摘要
鉴于平滑，投影的品种$ x $和有效的除数$ d \，\ subseteq \，x $，众所周知，（拓扑）阻碍了基本类别的$ d $作为hodge类的变形，是$ h^2（\ h^2（\ mathcal {o} _x）$。在本文中，我们用$ h^2_d（\ Mathcal {o} _x）$替换$ h^2（\ mathcal {o} _x）$，并提供类似的拓扑阻塞理论。我们将最终的局部拓扑阻塞理论与几何阻塞理论进行了比较（即，$ d $的变形的障碍物是一阶的无限变形$ x $的有效卡地亚除数）。我们将其应用于研究线性系统和Noether-Lefschetz基因座的跳跃基因座。最后，我们给出了一阶变形的示例$ x_t $ x $ $ x $的$ x $，同级$ [d] $变形为hodge类，但$ d $却不是作为有效的卡地亚除数$ x_t $。
Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in $H^2(\mathcal{O}_X)$. In this article, we replace $H^2(\mathcal{O}_X)$ by $H^2_D(\mathcal{O}_X)$ and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of $D$ as an effective Cartier divisor of a first order infinitesimal deformations of $X$). We apply this to study the jumping locus of families of linear systems and the Noether-Lefschetz locus. Finally, we give examples of first order deformations $X_t$ of $X$ for which the cohomology class $[D]$ deforms as a Hodge class but $D$ does not lift as an effective Cartier divisor of $X_t$.
