论文标题
Koszul复合物和与谎言代数相关的光谱序列
Koszul complexes and spectral sequences associated with Lie algebroids
论文作者
论文摘要
我们研究了一些与本地免费的$ \ Mathcal O_x $ -Module $ \ Mathcal A $相关的光谱序列，该序列具有Lie代数结构。在这里，$ x $是一个复杂的歧管，或者是代数封闭的字段$ k $的常规方案。一个光谱序列可以通过选择$ \ Mathcal a $的全局$ v $ $ \ Mathcal a $与$ \ Mathcal A $相关联，并考虑一个由$ v $给出的具有差异差异的Koszul综合体。通过使用Deligne的退化标准，该光谱序列显示在第二页上退化。 我们研究的另一个光谱序列在考虑复杂歧管上的Holomolorphic Vector Bundle $ e $的Atiyah algebroid $ \ Mathcal d_e $时会产生。如果$ v $是带有标量符号的$ e $上的差异操作员，即$ \ Mathcal d_e $的全局部分，我们将与Pair $（E，V）$ a Twisted Koszul Complect相关联。已知与该复合物相关的第一频谱序列在未拨打的情况下的第一页（$ e = 0 $）案例中退化
We study some spectral sequences associated with a locally free $\mathcal O_X$-module $\mathcal A$ which has a Lie algebroid structure. Here $X$ is either a complex manifold or a regular scheme over an algebraically closed field $k$. One spectral sequence can be associated with $\mathcal A$ by choosing a global section $V$ of $\mathcal A$, and considering a Koszul complex with a differential given by inner product by $V$. This spectral sequence is shown to degenerate at the second page by using Deligne's degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid $\mathcal D_E$ of a holomolorphic vector bundle $E$ on a complex manifold. If $V$ is a differential operator on $E$ with scalar symbol, i.e, a global section of $\mathcal D_E$, we associate with the pair $(E,V)$ a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ($E=0$) case
