## 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