A proof of N. Takahashi’s conjecture for (P2,E) and a refined sheaves/Gromov–Witten correspondence

Journal article

We prove N.\ Takahashi’s conjecture determining the contribution of each contact point in genus-0 maximal contact Gromov–Witten theory of $\mathbb{P}^2$ 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 $\mathbb{P}^2$.
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 $\mathbb{P}^2$, or equivalently refined genus-0 Gopakumar–Vafa invariants of local $\mathbb{P}^2$, with higher-genus maximal contact Gromov–Witten theory of $(\mathbb{P}^2, E)$. The correspondence involves the non-trivial change of variables $y = e^{i\hbar}$, where $y$ is the refined/cohomological variable on the sheaf side, and $\hbar$ is the genus variable on the Gromov–Witten side. We explain how this correspondence can be heuristically motivated

Authors: Bousseau, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Duke University Press
• Journal: Duke Mathematical Journal
• Volume: 172
• Issue: 15
• Pages: 2895-2955
• Publication date: 2023-10-15
• DOI: 10.1215/00127094-2022-0095
• EISSN: 1547-7398
• ISSN: 0012-7094
• Language: English
• Keywords: Gromov–Witten invariants; coherent sheaves; Donaldson–Thomas invariants