AI 中文总结
本文针对光滑复仿射对数卡拉比-丘曲面，通过构造近 toric 模型并结合辛托雷利定理，利用同调镜像对称定理建立了相关范畴的等价关系，完成了其同调镜像对称的相关研究。
AI 中文摘要
设$U$为光滑复仿射对数卡拉比-丘曲面，$\boxed{\boldsymbol{\textit{Ũ}}}$为其具有自然分次结构的完备有限型刘维尔流形。本文给出一种算法，构造有限型拟射影$\boxed{\boldsymbol{\textit{Z}}}$概型$U^\text{vee}$，使得对任意域$\boxed{\boldsymbol{\textit{k}}}$，有$D^\text{π}(\boxed{\boldsymbol{\textit{W}}}(\boxed{\boldsymbol{\textit{Ũ}}},\boxed{\boldsymbol{\textit{k}}})) \boxed{\boldsymbol{\textit{≅}}} D^\text{b}\text{Coh}(U^\text{vee}_{\boxed{\boldsymbol{\textit{k}}}})$。核心贡献是构造由精确特征射线图编码的近 toric 模型，利用辛对数卡拉比-丘对的辛托雷利定理证明该模型是保分次强精确辛同构于$\boxed{\boldsymbol{\textit{Ũ}}}$，结果由 Hacking–Keating 的同调镜像对称定理导出。
英文摘要
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 \[ D^π\bigl(\mathcal{W}(\widehat{U},\Bbbk)\bigr) \cong D^b\operatorname{Coh}(U^\vee_{\Bbbk}) \] for every field $\Bbbk$. 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.
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