Configurations in abelian categories. IV. Invariants and changing stability conditions

Abstract

This is the fourth in a series of papers math.AG/0312190, math.AG/0503029, math.AG/0410267 on configurations in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration is a finite collection of objects and morphisms in A satisfying some axioms. 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. The third math.AG/0410267 introduced stability conditions (t,T,<) on A, and showed the moduli space Obj_{ss}^a(t) of t-semistable objects in class a in A is a constructible set in Obj_A, so its characteristic function d_{ss}^a(t) is constructible. It proved many identities on constructible and stack functions such as d_{ss}^a(t). This paper first studies how Obj_{ss}^a(t) changes as we vary the stability condition (t,T,<) to (t',T',<), by writing d_{ss}^a(t') as a sum of products of d_{ss}^b(t). Then we discuss invariants I_{ss}^a(t) or I_{ss}(I,<,k,t) 'counting' t-semistable objects and configurations in A, satisfying identities and transformation laws from (t,T,<) to (t',T',<). We compute the invariants when A is a category mod-KQ of representations of a quiver Q or coh(P) of coherent sheaves on a smooth projective curve P. We find special properties of the invariants when A=coh(P) for P a surface with K_P^{-1} nef, or P a Calabi-Yau 3-fold.