Quotient categories, stability conditions, and birational geometry

TL;DR

This paper studies quotient categories of coherent sheaves on smooth projective varieties, revealing their homological dimension and describing stability conditions and equivalences, with applications to birational geometry.

Contribution

It introduces a new perspective on quotient categories of coherent sheaves and characterizes their stability conditions and equivalences, linking to birational geometry classification.

Findings

Homological dimension of quotient categories is c.

Describes the space of stability conditions for c=1.

Classifies exact equivalences between quotient categories.

Abstract

This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has homological dimension c. As an application of this, we will describe the space of stability conditions on its derived category in the case c=1. Moreover, we describe all exact equivalences between these quotient categories in this particular case which is closely related to classification problems in birational geometry.