Fock-Goncharov dual cluster varieties and Gross-Siebert mirrors
arXiv:2206.10584 · doi:10.1515/crelle-2023-0043
Abstract
Cluster varieties come in pairs: for any cluster variety there is an associated Fock-Goncharov dual cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi-Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross-Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross-Siebert mirror symmetry. Particularly, we show that the mirror to the cluster variety is a degeneration of the Fock-Goncharov dual cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross-Hacking-Keel-Kontsevich compares with the canonical scattering diagram defined by Gross-Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi-Yau varieties obtained as blow-ups of toric varieties.
51 pages, revised version published in Journal für die reine und angewandte Mathematik (Crelles Journal)