Derived categories of small toric Calabi-Yau 3-folds and curve counting invariants

We first construct a derived equivalence between a small crepant resolution of an affine toric Calabi-Yau 3-fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wallcrossing formula for the generating function of the counting invariants of perverse coherent sheaves. As an application we provide some equations on Donaldson-Thomas, Pandeharipande-Thomas and Szendroi’s invariants. Finally, we show that moduli spaces associated with a quiver given by successive mutations are realized as the moduli spaces associated the original quiver by changing the stability conditions. Introduction This is a subsequent paper of [NN]. We study variants of Donaldson-Thomas (DT in short) invariants on small crepant resolutions of affine toric Calabi-Yau varieties. The original Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are defined by virtual counting of moduli spaces of ideal sheaves IZ of 1-dimensional closed subschemes Z ⊂ Y ([Tho00], [Beh]). These are conjecturally equivalent to Gromov-Witten invariants after normalizing the contribution of 0-dimensional sheaves ([MNOP06]). A variant has been introduced Pandharipande and Thomas (PT in short) as virtual counting of moduli spaces of stable coherent systems ([PTa]). They conjectured these invariants also coincide with DT invariants after suitable normalization and mentioned that the coincidence should be recognized as a wallcrossing phenomenon. Here, a coherent system is a pair of a coherent sheaf and a morphism to it from the structure sheaf, which is first introduced by Le Potier in his study on moduli problems ([LP93]). Note that an ideal sheaf IZ is the kernel of the canonical surjections from the structure sheaf OY to the structure sheaf OZ . So in this sense DT invariants also count coherent systems. On the other hand, a variety sometimes has a derived equivalence with a noncommutative algebra. A typical example is a noncommutative crepant resolution of a Calabi-Yau 3-fold introduced by Michel Van den Bergh ([VdB04], [VdB]). In the case of [VdB04], the Abelian category of modules of the noncommutative crepant resolution corresponds to the Abelian category of perverse coherent sheaves in the sense of Tom Bridgeland ([Bri02]). Recently, Balazs Szendroi proposed to study counting invariants of ideals of such noncommutative algebras