Log BPS numbers of log Calabi-Yau surfaces

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

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Let (S,E) be a log Calabi-Yau surface pair with E a smooth divisor. We define new conjecturally integer-valued counts of A1-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

Original languageEnglish
Pages (from-to)687-732
JournalTransactions of the American Mathematical Society
Issue number1
Early online date3 Nov 2020
Publication statusPublished - Jan 2021

Bibliographical note

Funding Information:
Received by the editors February 28, 2019, and, in revised form, May 6, 2020, and June 23, 2020. 2020 Mathematics Subject Classification. Primary 14N35; Secondary 14J33. 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.

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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