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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Log BPS numbers of log Calabi-Yau surfaces


Authors: Jinwon Choi, Michel van Garrel, Sheldon Katz and Nobuyoshi Takahashi
Journal: Trans. Amer. Math. Soc.
MSC (2020): Primary 14N35; Secondary 14J33
DOI: https://doi.org/10.1090/tran/8234
Published electronically: November 3, 2020
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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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Additional Information

Jinwon Choi
Affiliation: Department of Mathematics & Research Institute of Natural Sciences, Sookmyung Women’s University, Cheongpa-ro 47-gil 100, Youngsan-gu, Seoul 04310, Republic of Korea
Email: jwchoi@sookmyung.ac.kr

Michel van Garrel
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: michel.van-garrel@warwick.ac.uk

Sheldon Katz
Affiliation: Department of Mathematics, MC-382, University of Illinois at Urbana-Champaign, Urbana, Illinois
Email: katz@math.uiuc.edu

Nobuyoshi Takahashi
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan
Email: tkhsnbys@hiroshima-u.ac.jp

DOI: https://doi.org/10.1090/tran/8234
Received by editor(s): February 28, 2019
Received by editor(s) in revised form: May 6, 2020, and June 23, 2020
Published electronically: November 3, 2020
Additional Notes: The first author was supported by the Korea NRF grant NRF-2018R1C1B6005600
The second author was supported by the German Research Foundation DFG-RTG-1670 and the European Commission Research Executive Agency MSCA-IF-746554
The third author was supported in part by NSF grant DMS-1502170 and NSF grant DMS-1802242, as well as by NSF grant DMS-1440140 while in residence at MSRI in Spring, 2018
The fourth author was supported by JSPS KAKENHI Grant Number JP17K05204. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 746554
Article copyright: © Copyright 2020 American Mathematical Society