We study log Calabi–Yau varieties obtained as a blow-up of a toric variety along
hypersurfaces in its toric boundary. Mirrors to such varieties are constructed by
Gross and Siebert from a canonical scattering diagram built by using punctured
Gromov–Witten invariants of Abramovich, Chen, Gross and Siebert. We
show that there is a piecewise-linear isomorphism between the canonical
scattering diagram and a scattering diagram defined algorithmically, following a
higher-dimensional generalization of the Kontsevich–Soibelman construction. We
deduce that the punctured Gromov–Witten invariants of the log Calabi–Yau
variety can be captured from this algorithmic construction. This generalizes
previous results of Gross, Pandharipande and Siebert on “the tropical vertex” to
higher dimensions. As a particular example, we compute these invariants
for a nontoric blow-up of the three-dimensional projective space along two
lines.
Keywords
mirror symmetry, tropical geometry, Gromov–Witten theory