Semi-orthogonal decomposition and smoothing

TL;DR

This paper studies how semi-orthogonal decompositions of derived categories of singular varieties behave under deformations to smooth varieties, focusing on the deformation of pretilting sheaves into exceptional vector bundles.

Contribution

It demonstrates that a pretilting sheaf associated with a non-commutative deformation deforms into a sum of mutually orthogonal exceptional vector bundles during smoothing.

Findings

Pretilting sheaves deform into exceptional vector bundles.

Semi-orthogonal components are preserved under deformation.

Explicit construction for Q-Gorenstein smoothings of surfaces.

Abstract

We investigate the behavior of semi-orthogonal decompositions of bounded derived categories of singular varieties under flat deformations to smooth varieties. We consider a Q-Gorenstein smoothing of a surface with a quotient singularity, and prove that a pretilting sheaf, which is constructed from a non-commutative deformation of a divisorial sheaf and generates a semi-orthogonal component of a bounded derived category, deforms to a direct sum of exceptional vector bundles which are mutually totally orthogonal.