The Du Bois complex of a hypersurface and the minimal exponent
The Du Bois complex of a hypersurface and the minimal exponent
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DOI：
10.1215/00127094-2022-0074
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发表时间：
2021-05
影响因子：
2.5
通讯作者：
M. Mustaţă;S. Olano;M. Popa;J. Witaszek
中科院分区：
文献类型：
--
作者：
M. Mustaţă;S. Olano;M. Popa;J. Witaszek
我们研究了光滑复代数变量中超曲面$Z$的Du Bois复合体$\underline{\Omega}_Z^\bullet$的最小指数$\widetilde{\alpha}(Z)$。后者是奇异不变量，定义为$Z$的简化Bernstein-Sato多项式的最大根的负数，并改进了对数正则阈值。我们表明，如果$\widetilde{\alpha}(Z)\geq p+1$，那么规范态射$\Omega_Z^p\to \underline{\Omega}_Z^p$是一个同构，其中$\underline{\Omega}_Z^p$是关于Hodge过滤的Du Bois复合物的$p$ -相关梯度块。另一方面，如果$Z$是奇异且$\widetilde{\alpha}(Z)>p\geq 2$，我们得到了$\underline{\Omega}_Z^{n-p}$的一些高上同调的不灭结果。
We study the Du Bois complex $\underline{\Omega}_Z^\bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $\widetilde{\alpha}(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein-Sato polynomial of $Z$, and refining the log canonical threshold. We show that if $\widetilde{\alpha}(Z)\geq p+1$, then the canonical morphism $\Omega_Z^p\to \underline{\Omega}_Z^p$ is an isomorphism, where $\underline{\Omega}_Z^p$ is the $p$-th associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if $Z$ is singular and $\widetilde{\alpha}(Z)>p\geq 2$, we obtain non-vanishing results for some of the higher cohomologies of $\underline{\Omega}_Z^{n-p}$.