The higher-dimensional tropical vertex

Abstract

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.