Configurations in abelian categories. III. Stability conditions and identities

Abstract

This is the third in a series math.AG/0312190, math.AG/0503029, math.AG/0410268 on configurations in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration (\sigma,\iota,\pi) is a finite collection of objects \sigma(J) and morphisms \iota(J,K) or \pi(J,K) : \sigma(J) --> \sigma(K) in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects. The first paper math.AG/0312190 defined configurations and studied moduli spaces Obj_A, M(I,<)_A of objects and (I,<)-configurations in A, using the theory of Artin stacks. The second math.AG/0503029 considered algebras of constructible functions and "stack functions" on Obj_A, using the theories developed in math.AG/0403305, math.AG/0509722. This paper introduces (weak) stability conditions (t,T,<) on A. We show the moduli spaces Obj_{ss}^a(t),Obj_{st}^a(t) of t-(semi)stable objects in class a in K(A) are constructible sets in the stack Obj_A, and some configuration moduli spaces M_{ss},...,M_{st}^b(I,<,k,t)_A are constructible in M(I,<)_A. So their characteristic functions d_{ss}^a(t),... and d_{ss}(I,<,k,t),... are constructible functions on Obj_A and M(I,<)_A. We prove many identities relating pushforwards of these functions under 1-morphisms between moduli stacks. These encode facts about, for example, the Euler characteristic of the family of ways of decomposing a t-semistable object into t-stable factors, and constitute a kind of "universal algebra of t-(semi)stability". Using these we define interesting (Lie) algebras of constructible functions H^{pa}_t,H^{to}_t and L^{pa}_t,L^{to}_t on Obj_A. All this is generalized to "stack functions".