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

This is the last in a series on configurations in an abelian category A. Given a finite poset (I,≼), an (I,≼)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or 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 defined configurations and studied moduli spaces of configurations in A, using Artin stacks. It showed well-behaved moduli stacks ObjA,M(I,≼)A of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA.The third introduced stability conditions(τ,T,⩽) on A, and showed the moduli space of τ-semistable objects in class α is a constructible subset in ObjA, so its characteristic function is a constructible function. It formed algebras , , , of constructible and stack functions on ObjA, and proved many identities in them.In this paper, if (τ,T,⩽) and are stability conditions on A we write in terms of the , and deduce the algebras are independent of (τ,T,⩽). We study invariants or Iss(I,≼,κ,τ) ‘counting’τ-semistable objects or configurations in A, which satisfy additive and multiplicative identities. We compute them completely when A=mod-KQ or A=coh(P) for P a smooth curve. We also find invariants with special properties when A=coh(P) for P a smooth surface with nef, or a Calabi–Yau 3-fold.