15 October 2023 A proof of N. Takahashi’s conjecture for (P2,E) and a refined sheaves/Gromov–Witten correspondence
Pierrick Bousseau
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Duke Math. J. 172(15): 2895-2955 (15 October 2023). DOI: 10.1215/00127094-2022-0095

Abstract

We prove N. Takahashi’s conjecture determining the contribution of each contact point in genus-0 maximal contact Gromov–Witten theory of P2 relative to a smooth cubic E. This is a new example of a question in Gromov–Witten theory that can be fully solved despite the presence of contracted components and multiple covers. The proof relies on a tropical computation of the Gromov–Witten invariants and on the interpretation of the tropical picture as describing wall-crossing in the derived category of coherent sheaves on P2.

The same techniques allow us to prove a new sheaves/Gromov–Witten correspondence, relating Betti numbers of moduli spaces of one-dimensional Gieseker semistable sheaves on P2, or equivalently, refined genus-0 Gopakumar–Vafa invariants of local P2, with higher-genus maximal contact Gromov–Witten theory of (P2,E). The correspondence involves the nontrivial change of variables y=ei, where y is the refined/cohomological variable on the sheaf side, and is the genus variable on the Gromov–Witten side. We explain how this correspondence can be heuristically motivated by a combination of mirror symmetry and hyper-Kähler rotation.

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Pierrick Bousseau. "A proof of N. Takahashi’s conjecture for (P2,E) and a refined sheaves/Gromov–Witten correspondence." Duke Math. J. 172 (15) 2895 - 2955, 15 October 2023. https://doi.org/10.1215/00127094-2022-0095

Information

Received: 8 December 2021; Revised: 23 August 2022; Published: 15 October 2023
First available in Project Euclid: 7 December 2023

Digital Object Identifier: 10.1215/00127094-2022-0095

Subjects:
Primary: 14N35

Keywords: coherent sheaves , Donaldson–Thomas invariants , Gromov–Witten Invariants

Rights: Copyright © 2023 Duke University Press

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Vol.172 • No. 15 • 15 October 2023
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