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-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.

95 pages, 10 figures. To appear in Geometry & Topology, accepted version

References in corpus (1)

• Local mirror symmetry in the tropics

Cited by in corpus (5)

• The canonical wall structure and intrinsic mirror symmetry
• Punctured logarithmic maps
• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• Polystable log Calabi-Yau varieties and Gravitational instantons
• Equations of mirrors to log Calabi--Yau pairs via the heart of canonical wall structures