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 $D_{\mathrm{coh}}^b(\mathrm{Tot}(U))$ into several exceptional objects and several copies of $D_{\mathrm{coh}}^b({\mathbb{A}}^n)$. 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 $k=1$.

18 pages