Resumen de Configurations in abelian categories. III. Stability conditions and identities

Dominic Joyce

• This is the third in a series on configurations in an abelian category . Given a finite poset (I,), an (I,)-configuration (s,?,p) is a finite collection of objects s(J) and morphisms ?(J,K) or in satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in decomposes into subobjects.

  The first paper defined configurations and studied moduli spaces of configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category coh(P) of coherent sheaves on a projective scheme P, or of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on .

  This paper introduces (weak) stability conditions (t,T,) on . We show the moduli spaces , , of t-semistable, indecomposable t-semistable and t-stable objects in class a are constructible sets in , and some associated configuration moduli spaces constructible in , so their characteristic functions and are constructible.

  We prove many identities relating these constructible functions, and their stack function analogues, under pushforwards. We introduce interesting algebras of constructible and stack functions, and study their structure. In the fourth paper we show are independent of (t,T,), and construct invariants of .