Stability conditions for derived categories

Abstract

The notion of stability condition on a triangulated category has been introduced by Bridgeland in [65], following ideas from physics by Douglas [104] on π-stability for D-branes. A stability condition on a triangulated category ℑ is given by abstracting the usual properties of μ-stability for sheaves on complex projective varieties; one introduces the notion of slope, using a group homomorphism from the Grothendieck group K(ℑ) of ℑ to ℂ, and requires that a stability condition has generalized Harder-Narasimhan filtrations and is compatible with the shift functor. The main property is that there exists a parameter space Stab(ℑ) for stability conditions, endowed with a natural topology, which is a (possibly infinitedimensional) complex manifold. The space of stability conditions Stab(ℑ) thus yields a geometric invariant naturally attached to a triangulated category ℑ.