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

Abstract

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 π ( J , K ) : σ ( J ) → 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 Obj A , 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- K Q of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on Obj A . The third introduced stability conditions ( τ , T , ⩽ ) on A , and showed the moduli space Obj ss α ( τ ) of τ-semistable objects in class α is a constructible subset in Obj A , so its characteristic function δ ss α ( τ ) is a constructible function. It formed algebras H τ pa , H τ to , H ¯ τ pa , H ¯ τ to of constructible and stack functions on Obj A , and proved many identities in them. In this paper, if ( τ , T , ⩽ ) and ( τ ˜ , T ˜ , ⩽ ) are stability conditions on A we write δ ss α ( τ ˜ ) in terms of the δ ss β ( τ ) , and deduce the algebras H τ pa , … , H ¯ τ to are independent of ( τ , T , ⩽ ) . We study invariants I ss α ( τ ) or I ss ( I , ≼ , κ , τ ) ‘counting’ τ-semistable objects or configurations in A , which satisfy additive and multiplicative identities. We compute them completely when A = mod- K Q 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 P −1 nef, or a Calabi–Yau 3-fold.