Stabilization of the cohomology of thickenings
Stabilization of the cohomology of thickenings
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DOI：
10.1353/ajm.2019.0013
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发表时间：
2016-05
影响因子：
1.7
通讯作者：
B. Bhatt;Manuel Blickle;G. Lyubeznik;Anurag Singh;Wenliang Zhang
中科院分区：
文献类型：
--
作者：
B. Bhatt;Manuel Blickle;G. Lyubeznik;Anurag Singh;Wenliang Zhang
摘要：对于特征为零的域上${\Bbb P}^n$中的局部完全交子簇$X=V({\cal I})$，证明了在小于$X$奇异轨迹余维数的上同调度下，在$X$的形式补全上~${\Bbb P}^n$的向量束沿$X$的上同调可以有效地计算为任意充分高增厚~$X_t=V({\cal I}^t)$上的上同调；这里的主要成分是~$X$的正常束的正结果。进一步，我们证明了Kodaira消失定理在相同的上同次范围内对所有加厚~$X_t$成立；这扩展了已知的Kodaira在X$上消失的版本，主要的新成分是X$上的Kodaira- akizuki - nakano消失定理的一个版本，用余切复合体来表述。
Abstract:For a local complete intersection subvariety $X=V({\cal I})$ in ${\Bbb P}^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $X$, the cohomology of vector bundles on the formal completion of~${\Bbb P}^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening~$X_t=V({\cal I}^t)$; the main ingredient here is a positivity result for the normal bundle of~$X$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings~$X_t$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $X$, and the main new ingredient is a version of the Kodaira-Akizuki-Nakano vanishing theorem for $X$, formulated in terms of the cotangent complex.