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

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29/11/2018##### Author

Beentjes, Sjoerd Viktor

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##### Abstract

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.