The Gross–Siebert algebra and homological mirror symmetry for log Calabi–Yau surfaces

Let $Y$ be a smooth projective surface, let $D$ be an anticanonical divisor supporting an ample divisor, let $R=\Bbbk[\operatorname{NE}(Y)]$, and let $A=\bigoplus_{q\in B(\mathbb Z)}R\cdot\vartheta_q$ be the free $R$-module with basis $\{\vartheta_q\}$. Define the symmetric $s$-point functions $\langle a_1,\ldots,a_s\rangle$ as in the construction from curve-counting invariants, with $\vartheta_0$ satisfying $\langle\vartheta_0\rangle=1$ and $\langle\vartheta_q\rangle=0$ for $q\in B_0(\mathbb Z)$. Let $T_Y=\operatorname{Hom}(A_1(Y),\Bbb G_m)$ be the big torus of $S=\operatorname{Spec}(R)$. The Gross–Siebert mirror-family conjecture. There is a unique finitely generated commutative and associative $R$-algebra structure on $A$, with $\vartheta_0=1$, such that

$$\langle a_1,a_2,\ldots,a_s\rangle=\langle a_1\cdot a_2\cdots a_s\rangle\in R.$$

The induced morphism $X=\operatorname{Spec}(A)\to S$ is a flat family of affine varieties with trivial relative dualizing sheaf and semi log canonical singularities; its fibers over $T_Y$ are irreducible and log canonical. If $\Bbbk=\mathbb C$, $A$ is the symplectic cohomology ring $SH^0(U)$, where $U=Y\setminus D$. For a complexified Kähler form $\mathbf B+i\omega$ and $s=\exp(2\pi i(\mathbf B+i\omega))\in T_Y$, if the fiber $X_s$ is smooth, then $X_s$ is homologically mirror to $(U,(\mathbf B+i\omega)|_U)$, meaning that its bounded derived category is equivalent to the wrapped Fukaya category. This conjecture packages the proposed mirror family and its expected categorical and singularity-theoretic properties; the source gives no resolution status for the full assertion.