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On the Local Donaldson-Thomas theory of curves Song, Yinan

Abstract

In this thesis, we study the Donaldson-Thomas theory of local curves. The motivation is the Gromov-Witten/Donaldson-Thomas correspondence. First, we review the gauge theory motivation of the original construction and the history of the Donaldson-Thomas theory. Then we review the construction of Gieseker-Maruyama-Simpson moduli spaces and their relation with Hilbert schemes of threefolds. We also review the concept of a perfect obstruction theory and its relation with the virtual fundamental classes. Then we describe the Gromov-Witten/Donaldson-Thomas correspondence and the equivariant generalization. We study the equivariant Donaldson-Thomas theory of two types of threefolds. First, we consider the total space of P²-bundles over smooth curves of genus g with (C*)³ action along the fiber, and we consider the curve class of a section plus the class of a line in the fiber. We compute the Donaldson-Thomas partition function up to the first order and we find that it agrees with the Gromov-Witten prediction. The second case we consider is the local curve case. The threefold is the total space of a direct sum of two line bundles of opposite degree with an anti-diagonal action along the fiber. Based on the calculation from Gromov-Witten theory, the only non-trivial contributions to the Donaldson-Thomas partition function come from components that parametrize subschemes of pure dimension. We compute these contributions and find that they agree with the Gromov-Witten prediction. We are also able to verify the vanishing of contributions from the so-called product-type components.

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