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Quasimaps and stable pairs

Part of: Cycles and subschemes Projective and enumerative geometry

Published online by Cambridge University Press:  12 April 2021

Henry Liu Open the ORCID record for Henry Liu [Opens in a new window]
Henry Liu*
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York NY 10027, USA; E-mail: hliu@math.columbia.edu

Abstract

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We prove an equivalence between the Bryan-Steinberg theory of $\pi $-stable pairs on $Y = \mathcal {A}_{m-1} \times \mathbb {C}$ and the theory of quasimaps to $X = \text{Hilb}(\mathcal {A}_{m-1})$, in the form of an equality of K-theoretic equivariant vertices. In particular, the combinatorics of both vertices are described explicitly via box counting. Then we apply the equivalence to study the implications for sheaf-counting theories on Y arising from 3D mirror symmetry for quasimaps to X, including the Donaldson-Thomas crepant resolution conjecture.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14C05: Parametrization (Chow and Hilbert schemes)

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e32
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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