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