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The local Donaldson-Thomas theory of curves.(English)Zbl 1205.14067

The authors carry out calculations of the local Donaldson-Thomas theory of curves, which refers to all Donaldson-Thomas invariants on the total space of a rank two bundle over a curve relative to surfaces determined by the fibers over fixed points in the curve. By comparing these results to those obtained byJ. Bryan andR. Pandharipande [J. Am. Math. Soc. 21, No. 1, 101–136 (2008;Zbl 1126.14062)] on the local Gromov-Witten theory of curves, they establish the Gromov-Witten/Donaldson-Thomas correspondence for the local theory of curves. This provides a rich class of new examples where the GW/DT correspondence is verified, and these techniques are likely to play a basic role in the general proof of the GW/DT correspondence for 3-folds. The calculations are achieved by localization and degeneration methods, and the results relate both the Gromov-Witten and Donaldson-Thomas invariants to the quantum cohomology of the Hilbert scheme of points in the plane.

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

Citations:

Zbl 1126.14062

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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