Homological Mirror Symmetry for Affine Log Calabi-Yau Surfaces

Abstract

Let $U$ be a smooth complex affine log Calabi--Yau surface, and let $\hat{U}$ 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 \[ D^π\bigl(\mathcal{W}(\widehat{U},\Bbbk)\bigr) \cong D^b\operatorname{Coh}(U^\vee_{\Bbbk}) \] 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 $\hat{U}$. The result then follows from the homological mirror symmetry theorem of Hacking--Keating.

Comments welcome