This paper is a contribution to the topics of mirror symmetry and Gromov-Witten (GW) theory. From the GW point of view, it gives a large class of examples in which the newly defined punctured log GW invariants are computed. From the mirror symmetry point of view, it gives a large class of examples of dimension \(\geq 3\) of the Gross-Siebert approach to mirror symmetry based on curve counting invariants.
Let \((X, D)\) be a log Calabi-Yau variety which admits a toric model, i.e. which is obtained by blowing up a toric variety along hypersurfaces of the toric boundary divisors. Gross and Siebert have defined the canonical scattering diagram of \((X, D)\), which is an object encoding a class of enumerative invariants of \((X, D)\) called punctured log GW invariants. The main result of the paper is that the canonical scattering diagram can be computed in an entirely algorithmic manner, without reference to enumerative geometry. In particular, the punctured log GW invariants can be efficiently computed.
The main result is a generalization in arbitrary dimension of the paper “The tropical vertex” by Gross-Pandharipande-Siebert in dimension 2. Whereas the 2-dimensional case uses log GW theory, i.e. virtual counts of \(X\) with presribed tangency conditions along \(D\), the higher-dimensional case requires the more general punctured log GW theory developped by Abramovich-Chen-Gross-Siebert, in which negative contact orders with \(D\) are allowed.
The paper also includes concrete non-trivial examples in dimension 3 such as the blow-up of \(\mathbb{P}^3\) along two disjoint lines.