The minimal exponent and $k$-rationality for local complete intersections
The minimal exponent and $k$-rationality for local complete intersections
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发表时间：
2022-12
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通讯作者：
Qianyu Chen;Bradley Dirks;Mircea Mustactua
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作者：
Qianyu Chen;Bradley Dirks;Mircea Mustactua
我们证明了如果$Z$是纯余维数$r$的光滑复变$X$的局部完全交子变，则$Z$有$k$ -有理奇点当且仅当$\widetilde{\alpha}(Z)>k+r$，其中$\widetilde{\alpha}(Z)$是$Z$的最小指数。我们还用$Z$的交上同调Hodge模上的Hodge滤波来描述这种情况。进一步证明，如果$Z$具有$k$ -有理奇点，则在$\dim(X)-\lceil \widetilde{\alpha}(Z)\rceil-1$层次上产生局部上同轴$\mathcal{H}^r_Z(\mathcal{O}_X)$上的Hodge过滤，假设$k\geq 1$和$Z$是奇点，则在$d$维数上，$\mathcal{H}^k(\underline{\Omega}_Z^{d-k})\neq 0$。所有这些结果都是已知的光滑品种的超曲面。
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 $\widetilde{\alpha}(Z)>k+r$, where $\widetilde{\alpha}(Z)$ 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 $\dim(X)-\lceil \widetilde{\alpha}(Z)\rceil-1$ and, assuming that $k\geq 1$ and $Z$ is singular, of dimension $d$, that $\mathcal{H}^k(\underline{\Omega}_Z^{d-k})\neq 0$. All these results have been known for hypersurfaces in smooth varieties.