Configurations in abelian categories. IV. Invariants and changing stability conditions - Joyce, D (Elsevier, 2008)
This is the last in a series on configurations in an abelian category A. Given a finite poset (I, and#10927;), an (I, and#10927;)-configuration (σ, ι, π) is a finite collection of objects σ(J) and morphisms ι(J, K) or andpi;(J, K) : σ (J) andrightarrow; σ (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 defined configurations and studied moduli spaces of configurations in A, using Artin stacks. It showed well-behaved moduli stacks ObjA, M(I, and#10927;)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, and#8804;) on A, and showed the moduli space Objandalpha;ss(τ) of τ-semistable objects in class α is a constructible subset in ObjA, so its characteristic function anddelta;andalpha;ss(τ) is a constructible function. It formed algebras Hpaτ, Htoτ, H-paτ, H-toτ of constructible and stack functions on ObjA, and proved many identities in them. In this paper, if (τ, T, and#8804;) and (andtilde;andtau;, andtilde;T, and#8804;) are stability conditions on A we write anddelta;andalpha;ss(andtilde;andtau;) in terms of the anddelta;andbeta;ss(τ), and deduce the algebras Hpaτ,..., H-toτ are independent of (τ, T, and#8804;). We study invariants Iandalpha;ss(τ) or Iss(I, and#10927;, 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 K-1P nef, or a Calabi–Yau 3-fold.