The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

Abstract: Let $U$ be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebra equipped with a compatible multilinear form. This proves a variant of the Frobenius structure conjecture by Gross-Hacking-Keel in mirror symmetry, and the spectrum of this algebra is supposed to give the hypothetical mirror family. Although the statement of our theorem involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of $U$. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when $U$ is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram constructed from counts of infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK, as well as the positivity in the Laurent phenomenon, follow readily from the geometric description.