Abstract

Let $U$ be the tautological subbundle on the Grassmannian $\operatorname{Gr}(k, n)$. There is a natural morphism $\textrm{Tot}(U) \to{\mathbb{A}}^n$. Using it, we give a semiorthogonal decomposition for the bounded derived category $D^b_{\!\textrm{coh}}(\textrm{Tot}(U))$ into several exceptional objects and several copies of $D^b_{\!\textrm{coh}}({\mathbb{A}}^n)$. We also prove a global version of this result: given a vector bundle $E$ with a regular section $s$, consider a subvariety of the relative Grassmannian $\operatorname{Gr}(k, E)$ of those subspaces that contain the value of $s$. The derived category of this subvariety admits a similar decomposition into copies of the base and the zero locus of $s$. This may be viewed as a generalization of the blow-up formula of Orlov, which is the case $k = 1$.

中文翻译：

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重言丛总空间的半正交分解

摘要

令$U$ 是Grassmannian $\operatorname{Gr}(k, n)$ 上的重言式子丛。存在一个自然态射 $\textrm{Tot}(U)\to{\mathbb{A}}^n$。使用它，我们将有界派生类别 $D^b_{\!\textrm{coh}}(\textrm{Tot}(U))$ 半正交分解为几个特殊对象和 $D^b_{ 的几个副本\!\textrm{coh}}({\mathbb{A}}^n)$. 我们还证明了这个结果的一个全局版本：给定一个带有正则部分 $s$ 的向量束 $E$，考虑那些包含$s$ 的价值。该子变体的派生类别允许类似的分解为基数和 $s$ 的零轨迹的副本。这可以看作是对 Orlov 的爆破公式的推广，即 $k = 1$。