Semiorthogonal decompositions on total spaces of tautological bundles

TL;DR

The paper constructs semiorthogonal decompositions for derived categories of total spaces of tautological bundles on Grassmannians, generalizing Orlov's blow-up formula to broader settings involving vector bundles and their sections.

Contribution

It introduces a new semiorthogonal decomposition framework for total spaces of tautological bundles, extending classical blow-up formulas to more general vector bundle contexts.

Findings

Derived categories decompose into exceptional objects and copies of the base.

Global decompositions relate subvarieties of Grassmannians to zero loci of sections.

Generalizes Orlov's blow-up formula to higher rank 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 $k = 1$.