Highest weight categories via pairs of dual exceptional sequences

TL;DR

This paper establishes criteria using dual exceptional sequences to identify highest weight categories, and applies these to geometric contexts like perverse sheaves, Grassmannians, and null categories of surfaces, providing new proofs and descriptions.

Contribution

It introduces criteria based on dual exceptional sequences for recognizing highest weight categories and applies them to various geometric categories, offering new proofs and geometric insights.

Findings

Perverse sheaves of middle perversity are highest weight under certain conditions.

Derived categories of Grassmannians have a highest weight t-structure.

Null categories of proper birational morphisms of surfaces are highest weight.

Abstract

In this paper we present criteria in terms of dual pairs of exceptional sequences for an abelian category to be highest weight. The criteria are applied in three situations of geometric origin. We give new proofs for the facts that the category of perverse sheaves of middle perversity on complex-analytic manifolds with suitable conditions on the stratification is highest weight and that the derived coherent category of any Grassmannian has a $t$-structure with highest weight heart. Also we show that the abelian null category of any proper birational morphism of regular surfaces is highest weight. For this null category, we give a geometric description of some special objects related to the highest weight structure, such as standard, costandard and characteristic tilting objects.