Gross–Hacking–Keel mirror construction conjecture for maximal intersection log Calabi–Yau pairs

Let $(Y,D_Y)$ be a simple normal crossings maximal intersection Calabi–Yau pair such that $D_Y$ supports an ample divisor. Set $U=Y\setminus D_Y$, let $R=k[\mathrm{Pic}(Y)^\times]$ and let $\Omega$ be the canonical volume form on $U$. Define

$$U^{\mathrm{trop}}(\mathbb Z)=\Big\{\text{divisorial valuations }v\colon k(U)\setminus\{0\}\to\mathbb Z\text{ with }v(\Omega)<0\Big\}\cup\{0\}.$$

Let $K=\mathrm{Ker}\{\mathrm{Pic}(Y)\to\mathrm{Pic}(U)\}$.

Gross–Hacking–Keel mirror construction conjecture. The free $R$-module $V$ with basis $U^{\mathrm{trop}}(\mathbb Z)$ has a natural finitely generated $R$-algebra structure whose structure constants are non-negative integers determined by counts of rational curves on $U$. The fibration

$$p\colon\operatorname{Spec}(V)\to\operatorname{Spec}(R)=T_{\mathrm{Pic}(Y)}$$

is a $T_K$-equivariant flat family of affine maximal intersection log Calabi–Yau varieties, and the quotient

$$\operatorname{Spec}(V)/T_K\to T_{\mathrm{Pic}(U)}$$

only depends on $U$ and is the mirror family of $U$.

This conjectural construction extends mirror symmetry from Calabi–Yau varieties to maximal intersection log Calabi–Yau pairs. It is motivated by the role of toric models and toric degenerations in known mirror constructions; the source presents the characterization of pairs admitting toric models as an open and difficult problem.