Homological Mirror Symmetry for Affine Log Calabi-Yau Surfaces

Let $U$ be a smooth complex affine log Calabi–Yau surface, and let $\widehat{U}$ be its complete finite-type Liouville manifold, equipped with its natural grading structure. We give an algorithm that constructs a finite-type quasi-projective $\mathbb{Z}$-scheme $U^{\vee }$ such that $$D^{\pi }(\mathcal{W}(\widehat{U},\mathbb{k}))\cong D^b\mathrm{Coh}(U_{\mathbb{k}}^{\vee })$$ for every field $\mathbb{k}$. Our main contribution is to construct an almost toric model encoded by an exact eigenray diagram and, using the symplectic Torelli theorem for symplectic log Calabi–Yau pairs, to prove that this model is grading-preserving strongly exact symplectomorphic to $\widehat{U}$. The result then follows from the homological mirror symmetry theorem of Hacking–Keating.