Highest weight categories via pairs of dual exceptional sequences
Agnieszka Bodzenta, Alexey Bondal
TL;DR
The paper develops a framework linking highest weight abelian categories to dual pairs of exceptional sequences in derived categories, providing concrete, checkable criteria for the heart of a glued t-structure to be highest weight. It shows Ringel duality corresponds to exchanging a sequence with its left dual and furnishes restricted and equivalence results that tie HW structures to dual exceptional data. The authors apply the theory to geometry, giving new proofs and explicit HW descriptions for perverse sheaves with middle perversity, the derived category of Grassmannians, and abelian null categories arising from birational morphisms of regular surfaces, including explicit standard, costandard, and tilting objects. They also discuss algebraic examples, including Kalck’s counterexample and directed algebras, to illuminate the scope and limitations of HW characterizations via exceptional sequences. Overall, the work unifies algebraic and geometric perspectives on highest weight structures and provides practical tools for constructing and recognizing HW hearts from triangulated data.
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.