Configurations in abelian categories. III. Stability conditions and identities

TL;DR: In this article, generalized Donaldson-Thomas invariants are defined for all classes of coherent sheaves, and they are equal to $DT^\alpha(\tau)$ when it is defined.

TL;DR: In this article, the authors present the new ideas coming out of the interactions of string theory and algebraic geometry in a coherent logical context, including the Strominger-Yau-Zaslow conjecture, the theory of stability conditions on triangulated categories, and a physical basis for the McKay correspondence.

TL;DR: In this paper, the second part of a series on configurations in an abelian category A is presented, where the authors define an associative multiplication ∗ on CF (Obj A ) using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that CF ( Obj A ) is a Q -algebra.

TL;DR: In this article, the Euler characteristic version of DT/PT correspondence, using the notion of weak stability conditions and the wall-crossing formula, was shown, and it was shown to be equivalent to the stable pairs of the Donaldson-Thomas invariant.

TL;DR: In this article, the authors studied moduli spaces of (I, <)-configurations in an abelian category A, using the theory of Artin stacks, and proved well-behaved moduli stacks of objects and configurations in A, M(I,<)_A, exist when A is the coh(P) of coherent sheaves on a projective K-scheme P, or mod-KQ of representations of a quiver Q.

TL;DR: In this article, the moduli stack of vector budles has been studied in the context of algebraic stacks, and a detailed comparison with geometric invariant moduli schemes has been made.

TL;DR: In this article, the authors studied moduli spaces of (I, <)-configurations in an abelian category A, using the theory of Artin stacks, and showed that well-behaved moduli stacks exist when A is an ABELIAN category of coherent sheaves or vector bundles on a projective K-scheme P, or of representations of a quiver Q.

TL;DR: In this article, the notions of Euler characteristic for constructible sets in algebraically closed fields and pushforwards and pullbacks of constructible functions with functorial behaviour were defined.