Log BPS numbers of log Calabi-Yau surfaces

Choi, Jinwon; van Garrel, Michel; Katz, Sheldon; Takahashi, Nobuyoshi

doi:10.1090/tran/8234

Log BPS numbers of log Calabi-Yau surfaces

Authors: Jinwon Choi, Michel van Garrel, Sheldon Katz and Nobuyoshi Takahashi
Journal: Trans. Amer. Math. Soc. 374 (2021), 687-732
MSC (2020): Primary 14N35; Secondary 14J33
DOI: https://doi.org/10.1090/tran/8234
Published electronically: November 3, 2020
MathSciNet review: 4188197
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Abstract: Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We define new conjecturally integer-valued counts of $\mathbb {A}^1$-curves in $(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $E$ via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.

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