Geometry & Topology 13 (2009) 2389–2425

DOI: 10.2140/gt.2009.13.2389

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.

Keywords

stability condition, derived category, counting invariant, wall crossing, Donaldson–Thomas invariant

Mathematical Subject Classification

Primary: 14F05, 18E30

Secondary: 14D20, 14J32, 14N35

References

Publication

Received: 28 March 2009
Accepted: 8 May 2009
Published: 12 June 2009
Proposed: Jim Bryan
Seconded: Simon Donaldson, Frances Kirwan

Authors