Semiorthogonal decomposition via categorical action

TL;DR

This paper introduces a method using categorical actions of shifted affine algebras to construct semiorthogonal decompositions, recovering known exceptional collections and extending to Grassmannians of sheaves with low homological dimension.

Contribution

It develops a new categorical approach to semiorthogonal decompositions, unifying and extending previous constructions to broader classes of varieties.

Findings

Recovers Kapranov's exceptional collections for Grassmannians and flag varieties.

Constructs semiorthogonal decompositions for Grassmannians of sheaves with homological dimension ≤ 1.

Provides a new framework linking categorical actions to geometric decompositions.

Abstract

We show that the categorical action of the shifted $q=0$ affine algebra can be used to construct semiorthogonal decomposition on the weight categories. In particular, this construction recovers Kapranov's exceptional collection when the weight categories are the derived categories of coherent sheaves on Grassmannians and $n$-step partial flag varieties. Finally, as an application, we use this result to construct a semiorthogonal decomposition on the derived categories of coherent sheaves on Grassmannians of a coherent sheaf with homological dimension $\leq 1$ over a smooth projective variety $X$.