The Higher Dimensional Tropical Vertex

TL;DR

This paper extends the tropical vertex concept to higher dimensions, linking log Gromov-Witten invariants with scattering diagrams and providing explicit computations for complex blow-up examples.

Contribution

It introduces a higher-dimensional generalization of the tropical vertex and establishes a piecewise linear isomorphism between scattering diagrams, connecting geometric invariants with algorithmic constructions.

Findings

Established a higher-dimensional Kontsevich-Soibelman type construction.

Proved the equivalence of scattering diagrams from geometric and algebraic perspectives.

Computed log Gromov-Witten invariants for a non-toric blow-up of P^3.

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-Siebert from a canonical scattering diagram built by using punctured log Gromov-Witten invariants of Abramovich-Chen-Gross-Siebert. We show that there is a piecewise linear isomorphism between the canonical scattering diagram and a scattering diagram defined algortihmically, following a higher dimensional generalisation of the Kontsevich-Soibelman construction. We deduce that the punctured log Gromov-Witten invariants of the log Calabi-Yau variety can be captured from this algorithmic construction. As a particular example, we compute these invariants for a non-toric blow-up of the three dimensional projective space along two lines. This generalizes previous results of Gross-Pandharipande-Siebert on "The Tropical Vertex" to higher dimensions.