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

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

2004-10-11

arxiv.org/abs/math/0410268v5

Mathematics

Algebraic Geometry

83 pages, LaTeX. (v5) Minor changes, new references

Scientific paper

This is the fourth in a series of papers math.AG/0312190, math.AG/0503029, math.AG/0410267 on configurations in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration is a finite collection of objects and morphisms in A satisfying some axioms. Configurations describe how an object X in A decomposes into subobjects. The first paper math.AG/0312190 defined configurations and studied moduli spaces Obj_A, M(I,<)_A of objects and (I,<)-configurations in A, using the theory of Artin stacks. The second math.AG/0503029 considered algebras of constructible functions and "stack functions" on Obj_A, using the theories developed in math.AG/0403305, math.AG/0509722. The third math.AG/0410267 introduced stability conditions (t,T,<) on A, and showed the moduli space Obj_{ss}^a(t) of t-semistable objects in class a in A is a constructible set in Obj_A, so its characteristic function d_{ss}^a(t) is constructible. It proved many identities on constructible and stack functions such as d_{ss}^a(t). This paper first studies how Obj_{ss}^a(t) changes as we vary the stability condition (t,T,<) to (t',T',<), by writing d_{ss}^a(t') as a sum of products of d_{ss}^b(t). Then we discuss invariants I_{ss}^a(t) or I_{ss}(I,<,k,t) 'counting' t-semistable objects and configurations in A, satisfying identities and transformation laws from (t,T,<) to (t',T',<). We compute the invariants when A is a category mod-KQ of representations of a quiver Q or coh(P) of coherent sheaves on a smooth projective curve P. We find special properties of the invariants when A=coh(P) for P a surface with K_P^{-1} nef, or P a Calabi-Yau 3-fold.

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