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Curve-counting invariants and crepant resolutions of Calabi-Yau threefolds Steinberg, David Christopher


The Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi-Yau threefold ought to be equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but Bryan, Cadman, and Young have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold χ. The crepant resolution conjecture of Bryan-Cadman-Young gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland.

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