Review of 'A theory of generalized Donaldson-Thomas invariants'

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 ℳ^α_st. This is a 0-cycle and obtained by studying the deformation-obstruction of sheaves. The corresponding Donaldson-Thomas invariant, denoted by DT^α, can be defined when ℳ^α_st is proper, which is the case when there are no strictly semistable sheaves on X in the class α. In this sense DT^α is an integer giving a virtual count of the stable sheaves on X in the class α. DT^α is unchanged by deformations of X. Another remarkable property of DT^α proven in [K. A. Behrend, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338; MR2600874] is that DT^α can always be expressed as the Euler characteristics of ℳ^α_st weighted by Behrend's constructible function.
   The paper under review makes a breakthrough by defining the generalized Donaldson-Thomas invariants DT^α in the presence of strictly semistable sheaves in the class α. These invariants generalizing DT^α 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 DT^α(τ) and DT^α(τ̃) for two stability conditions τ and τ̃is then given in terms of explicit wall crossing formulas.
   Importantly, the generalized Donaldson-Thomas invariants DT^α(τ) are expressed in terms of other invariants ĎT^α(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