Crepant resolution conjecture for Donaldson-Thomas invariants via wall-crossing

Let Y be a smooth complex projective Calabi{Yau threefold. Donaldson-Thomas 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, well-known series in mathematics and physics appear in explicit computations. Furthermore, closer to the topic of this thesis, the generating series of birational Calabi-Yau threefolds determine one another [Cal16a]. The crepant resolution conjecture for Donaldson-Thomas invariants [BCY12] conjectures another such comparison result. It relates the Donaldson{Thomas generating series of a certain type of three-dimensional Calabi-Yau 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 wall-crossing 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 wall-crossing formula to this setting, essentially breaking the former in many pieces. Pairs, and their wall-crossing formula, are fundamentally objects of the bounded derived category of the Calabi-Yau orbifold. Finally, I present joint work with J. Calabrese and J. Rennemo [BCR] in which we use the wall-crossing formula and Joyce's integration map to prove the crepant resolution conjecture for Donaldson-Thomas 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 wall-crossing are expansions of the same rational function