Polynomial Bridgeland stability conditions and the large volume limit

TL;DR

This paper introduces polynomial stability conditions as a generalization of Bridgeland stability, constructing a family that includes large volume limits and connecting wall-crossing phenomena to enumerative invariants.

Contribution

It defines polynomial stability conditions, constructs a broad family on any normal projective variety, and relates wall-crossing to important enumerative invariants like PT/DT correspondence.

Findings

Polynomial stability conditions generalize Bridgeland stability.

Constructed a family of stability conditions including large volume limits.

Connected wall-crossing in stability conditions to PT/DT invariants.

Abstract

We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpson-stability, and large volume limits of Bridgeland stability conditions. We show that the PT/DT-correspondence relating stable pairs to Donaldson-Thomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wall-crossing in our family of polynomial stability conditions. Similarly, we show that the relation between stable pairs and invariants of one-dimensional torsion sheaves (proven recently by the same authors) is a wall-crossing formula.