Semiorthogonal decompositions on total spaces of tautological bundles

Abstract

Let U be the tautological subbundle on the Grassmannian $\mathrm{Gr}(k, n)$. There is a natural morphism $\mathrm{Tot}(U) \to \mathbb{A}^n$. Using it, we give a semiorthogonal decomposition for the bounded derived category $D^b_{\mathrm{coh}}(\mathrm{Tot}(U))$ into several exceptional objects and several copies of $D^b_{\mathrm{coh}}(\mathbb{A}^n)$. 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 $\mathrm{Gr}(k, E)$ of those subspaces which 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 $k = 1$.