Mathematical Reviews MR2951762

'A theory of generalized Donaldson-Thomas invariants', by Dominic Joyce and Yinan Song

Algebraic Donaldson-Thomas invariants of a complex Calabi-Yau threefold X were first introduced in [R. P. Thomas, J. Differential Geom. 54 (2000), no. 2, 367–438; MR1818182]. They are defined by means of the virtual fundamental cycle of the moduli spaces of Gieseker stable sheaves over X of a specific K-group class α, denoted by ℳαst. This is a 0-cycle and obtained by studying the deformation-obstruction of sheaves. The corresponding Donaldson-Thomas invariant, denoted by DTα, can be defined when ℳαst is proper, which is the case when there are no strictly semistable sheaves on X in the class α. In this sense DTα is an integer giving a virtual count of the stable sheaves on X in the class α. DTα is unchanged by deformations of X. Another remarkable property of DTα proven in [K. A. Behrend, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338; MR2600874] is that DTα can always be expressed as the Euler characteristics of ℳαst weighted by Behrend's constructible function.
   The paper under review makes a breakthrough by defining the generalized Donaldson-Thomas invariants DTα in the presence of strictly semistable sheaves in the class α. These invariants generalizing DTα are in general rational numbers and remarkably they have both important properties of the Donaldson-Thomas invariants mentioned above. They are obtained by taking into account the complicated ℚ-valued contributions made by the strictly semistable sheaves in the class α.
   This is a continuation of the project "Configurations in abelian categories'' by the first author of the paper under review [Adv. Math. 203 (2006), no. 1, 194–255; MR2231046; Adv. Math. 210 (2007), no. 2, 635–706; MR2303235; Adv. Math. 215 (2007), no. 1, 153–219; MR2354988; Adv. Math. 217 (2008), no. 1, 125–204; MR2357325]. The generalized Donaldson-Thomas invariants are defined more generally for any 3-Calabi-Yau abelian categories with stability conditions. The abelian category of coherent sheaves on X and the abelian category of the representations of quivers with superpotentials are the examples. The generalized Donaldson-Thomas invariants then depend on the choice of the stability. The relation between DTα(τ) and DTα(τ̃) for two stability conditions τ and τ̃is then given in terms of explicit wall crossing formulas.
   Importantly, the generalized Donaldson-Thomas invariants DTα(τ) are expressed in terms of other invariants ĎTα(τ) which are conjecturally integer-valued. This conjecture is proven in some special cases. This is analogous to the conjectural definition of the integer-valued BPS invariants in terms of the ℚ-valued Gromov-Witten invariants and the famous Gopakumar integrality conjecture.

Reviewed by Amin Gholampour. Back