For Q-factorial projective toric varieties whose toric boundary divisors are nef, the genus-zero log and local Gromov-Witten invariants with point and descendant insertions agree after the log-local normalization, and both are computed in closed form.
Log BPS numbers of log Calabi-Yau surfaces
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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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2019 1verdicts
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On the log-local principle for the toric boundary
For Q-factorial projective toric varieties whose toric boundary divisors are nef, the genus-zero log and local Gromov-Witten invariants with point and descendant insertions agree after the log-local normalization, and both are computed in closed form.