On the crepant resolution conjecture for Donaldson-Thomas invariants

Abstract

We prove a comparison formula for curve-counting invariants in the setting of the McKay correspondence, related to the crepant resolution conjecture for Donaldson-Thomas invariants. The conjecture is concerned with comparing the invariants of a (hard Lefschetz) Calabi-Yau orbifold of dimension three with those of a specific crepant resolution of its coarse moduli space. We prove the conjecture for point classes and give a conditional proof for general curve classes. We also prove a variant of the formula for curve classes. Along the way we identify the image of the standard heart of the orbifold under the Bridgeland-King-Reid equivalence.

Curve counting invariants for crepant resolutions

“…It would be very interesting to find the precise relationship between the two approaches. An even more recent preprint [18] of Calabrese proves the DT crepant resolution conjecture, utilizing his earlier paper [17].…”

Weyl symmetry for curve counting invariants via spherical twists

“…Proof of theorem 47. We give the proof of theorem 47 based on the application of Iqbal's formula (9). We first remark that it's enough to prove the result when β = jC for some j ≥ 0.…”

“…The notion of perverse stable pairs on Y coincides with the image of stable pairs on X . The results of Theorems 1, 2 and 3 are the rationality and functional equation of PT(X ) and the wall-crossing between Φ −1 PT(X ) and Bryan-Steinberg pairs of Y → X [1,7,9]. The nef class is given by the pullback of an ample class on X and the derived anti-equivalence ρ corresponds to the derived dual of X…”

Erratum to Donaldson–Thomas invariants and flops (J. reine angew. Math. 716 (2016), 103–145)

“…Nonetheless, the actual perverse/ordinary comparison formula is of interest beyond the context of flops (for example in the setting of the crepant resolution conjecture for DonaldsonThomas invariants [2,3]). …”