Configurations in abelian categories. III. Stability conditions and identities

Abstract

This is the third 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 the theory of 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 . This paper introduces ( weak) stability conditions ( τ , T , ⩽ ) on A . We show the moduli spaces Obj ss α , Obj si α , Obj st α ( τ ) of τ-semistable, indecomposable τ-semistable and τ-stable objects in class α are constructible sets in Obj A , and some associated configuration moduli spaces M ss , M si , M st , M ss b , M si b , M st b ( I , ≼ , κ , τ ) A constructible in M ( I , ≼ ) A , so their characteristic functions δ ss α , δ si α , δ st α ( τ ) and δ ss , … , δ st b ( I , ≼ , κ , τ ) are constructible. We prove many identities relating these constructible functions, and their stack function analogues, under pushforwards. We introduce interesting algebras H τ pa , H τ to , H ¯ τ pa , H ¯ τ to of constructible and stack functions, and study their structure. In the fourth paper we show H τ pa , … , H ¯ τ to are independent of ( τ , T , ⩽ ) , and construct invariants of A , ( τ , T , ⩽ ) .