论文标题
本地完整交叉点的最小指数和$ k $ - 理性
The minimal exponent and $k$-rationality for local complete intersections
论文作者
论文摘要
我们表明，如果$ z $是光滑复杂品种$ x $的本地完整交点子，则纯codimension $ r $，那么$ z $具有$ k $ - 理性的奇异性，并且仅当$ \widetildeα（z）> k+r $，$ \ \ \ \widetildeα（z）$是$ z $的最小值。我们还根据$ z $的交叉路口共同体霍奇模块的Hodge过滤来表征这种情况。此外，我们表明，如果$ z $具有$ k $ - 合理的奇异性，那么对本地协同学的hodge过滤$ \ mathcal {h}^r_z（\ nathcal {o} _x _x _x）$是在级别上生成的，以$ \ dim（x） - \ lceil \ wideteLdel $ kel $ \ z）生成1 $和$ z $是奇异的，尺寸$ d $，$ \ mathcal {h}^k（\undeslineΩ_z^{d-k} {d-k}）\ neq 0 $。所有这些结果都以光滑品种中的超曲面而闻名。
We show that if $Z$ is a local complete intersection subvariety of a smooth complex variety $X$, of pure codimension $r$, then $Z$ has $k$-rational singularities if and only if $\widetildeα(Z)>k+r$, where $\widetildeα(Z)$ is the minimal exponent of $Z$. We also characterize this condition in terms of the Hodge filtration on the intersection cohomology Hodge module of $Z$. Furthermore, we show that if $Z$ has $k$-rational singularities, then the Hodge filtration on the local cohomology sheaf $\mathcal{H}^r_Z(\mathcal{O}_X)$ is generated at level $\dim(X)-\lceil \widetildeα(Z)\rceil-1$ and, assuming that $k\geq 1$ and $Z$ is singular, of dimension $d$, that $\mathcal{H}^k(\underlineΩ_Z^{d-k})\neq 0$. All these results have been known for hypersurfaces in smooth varieties.
