Descent theory for semiorthogonal decompositions

TL;DR

This paper develops a descent method for constructing semiorthogonal decompositions of equivariant derived categories, enabling new decompositions for projective bundles, blow-ups, and varieties with full exceptional collections.

Contribution

It introduces a descent theory approach to semiorthogonal decompositions, extending their construction to equivariant categories under group actions.

Findings

Semiorthogonal decompositions for equivariant categories of projective bundles.

Decomposition results for blow-ups with smooth centers.

Applications to varieties with full exceptional collections.

Abstract

In this paper a method of constructing a semiorthogonal decomposition of the derived category of $G$-equivariant sheaves on a variety $X$ is described, provided that the derived category of sheaves on $X$ admits a semiorthogonal decomposition, whose components are preserved by the action of the group $G$ on $X$. Using this method, semiorthogonal decompositions of equivariant derived categories were obtained for projective bundles and for blow-ups with a smooth center, and also for varieties with a full exceptional collection, preserved by the action of the group. As a main technical instrument, descent theory for derived categories is used.