Comparison of non-archimedean and logarithmic mirror constructions via the Frobenius structure theorem

TL;DR

This work bridges two mirror-geometry frameworks for log Calabi–Yau pairs by proving that, under a semipositivity/favorable degeneracy setup, the Gross–Siebert intrinsic mirror and Keel–Yu non-archimedean mirror coincide. Central to the result is showing that the structure constants defining the mirror algebra are governed by naive curve counts $\eta$ rather than virtual counts, leading to a Frobenius-structure-type determination of the full product from its 2- and 3-point data. The authors derive this via a careful tropical analysis of punctured log curves, broken-line types, and balancing conditions, tying log Gromov–Witten invariants to theta-function combinatorics. Consequences include the equality of the two mirror constructions in cases with a Zariski-dense torus, the birational invariance of the mirror under log étale modifications, and a resolution of Mandel’s mirror conjecture for a broad class of Fano pairs, as well as a classical–quantum period equivalence $igl(\hat{G}_X = \pi_W\bigr)$ for these scenarios. The results deepen the understanding of mirror symmetry for open Calabi–Yau geometries and provide concrete computational tools via tropical and log-GW techniques.

Abstract

For a log Calabi Yau pair (X,D) with X\D smooth affine, satisfying either assumption 1.1 of "The canonical wall structure and intrinsic mirror symmetry" or contains a Zariski dense torus, we prove under the condition that D is the support of a nef divisor, that the structure constants defining a trace form on the mirror algebra constructed by Gross-Siebert are given by the naive curve counts defined by Keel-Yu in definition 1.1 of "The Frobenius structure theorem for log Calabi-Yau varieties containing a torus". As a corollary, we deduce the equality of the mirror algebras constructed by Gross-Siebert and Keel-Yu in the case X\D contains a Zariski dense torus. In addition, we use this result to prove a mirror conjecture proposed by Mandel in "Fano mirror periods from the Frobenius structure conjecture" for Fano pairs satisfying assumption 1.1.

Figures (3)

• Figure 1: An example of a tropical curve contributing to a a structure constant, colored in red. The legs with infinite extent correspond to the "input" contact orders $p$ and $q$, and the bounded leg is the "output".
• Figure 2: An example of a broken line type in the tropicalization of a blow up of $\mathbb{P}^1\times\mathbb{P}^1$ at a point contained in the interior of a boundary divisor. The $1$-dimensional cone contained in the lower half plane is identified with the dashed line to give the integral affine structure. The spine of the type is depicted in red, and a wall type produced by forgetting the spine is depicted in green, in this case corresponding to the exceptional divisor. Note that a tropical balancing condition holds at all vertices in the spine. Moreover, a broken line can only bend along the line spanned by this red ray.
• Figure 3: An example of behavior disallowed by Step $1$. The tail without legs is disallowed by the semipositivity assumption.

Theorems & Definitions (50)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Example 1.4
• Theorem 1.5: int_mirror Conjecture $9.2$
• Corollary 1.6
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 3.2