Title:

Crepant resolution conjecture for DonaldsonThomas invariants via wallcrossing

Let Y be a smooth complex projective Calabi{Yau threefold. DonaldsonThomas invariants [Tho00] are integer invariants that virtually enumerate curves on Y. They are organised in a generating series DT(Y) that is interesting from a variety of perspectives. For example, wellknown series in mathematics and physics appear in explicit computations. Furthermore, closer to the topic of this thesis, the generating series of birational CalabiYau threefolds determine one another [Cal16a]. The crepant resolution conjecture for DonaldsonThomas invariants [BCY12] conjectures another such comparison result. It relates the Donaldson{Thomas generating series of a certain type of threedimensional CalabiYau orbifold to that of a particular resolution of singularities of its coarse moduli space. The conjectured relation is an equality of generating series. In this thesis, I first provide a counterexample showing that this conjecture cannot hold as an equality of generating series. I then verify that both generating series are the Laurent expansion about different points of the same rational function. This suggests a reinterpretation of the crepant resolution conjecture as an equality of rational functions. Second, following a strategy of Bridgeland [Bri11] and Toda [Tod10a, Tod13, Tod16a], I prove a wallcrossing formula in a motivic Hall algebra relating the Hilbert scheme of curves on the orbifold to that on the resolution. I introduce the notion of pair object associated to a torsion pair, putting ideal sheaves and stable pairs on the same footing, and generalise the wallcrossing formula to this setting, essentially breaking the former in many pieces. Pairs, and their wallcrossing formula, are fundamentally objects of the bounded derived category of the CalabiYau orbifold. Finally, I present joint work with J. Calabrese and J. Rennemo [BCR] in which we use the wallcrossing formula and Joyce's integration map to prove the crepant resolution conjecture for DonaldsonThomas invariants as an equality of rational functions. A crucial ingredient is a result of J. Rennemo that detects when two generating functions related by a wallcrossing are expansions of the same rational function.
