Automatic split-generation for the Fukaya category

We prove a structural result in mirror symmetry for projective Calabi--Yau (CY) manifolds. Let $X$ be a connected symplectic CY manifold, whose Fukaya category $\mathcal{F}(X)$ is defined over some suitable Novikov field $\mathbb{K}$; its mirror is assumed to be some smooth projective scheme $Y$ over $\mathbb{K}$ with `maximally unipotent monodromy'. Suppose that some split-generating subcategory of (a $\mathsf{dg}$ enhancement of) $D^bCoh( Y)$ embeds into $\mathcal{F}(X)$: we call this hypothesis `core homological mirror symmetry'. We prove that the embedding extends to an equivalence of categories, $D^bCoh(Y) \cong D^\pi( \mathcal{F}(X))$, using Abouzaid's split-generation criterion. Our results are not sensitive to the details of how the Fukaya category is set up. In work-in-preparation [PS], we establish the necessary foundational tools in the setting of the `relative Fukaya category', which is defined using classical transversality theory.

我们证明了射影卡拉比 - 丘（CY）流形的镜像对称的一个结构结果。设\(X\)是一个连通的辛CY流形，其富卡亚范畴\(\mathcal{F}(X)\)是在某个合适的诺维科夫域\(\mathbb{K}\)上定义的；假定它的镜像是\(\mathbb{K}\)上某个具有“极大幂幺单值性”的光滑射影概型\(Y\)。假设（\(D^bCoh(Y)\)的一个\(\mathsf{dg}\)强化的）某个分裂生成子范畴嵌入到\(\mathcal{F}(X)\)中：我们将这个假设称为“核心同调镜像对称”。我们利用阿布扎伊德的分裂生成准则证明该嵌入可扩展为范畴的等价，\(D^bCoh(Y)\cong D^\pi(\mathcal{F}(X))\)。我们的结果对富卡亚范畴的构建细节不敏感。在准备中的工作[PS]中，我们在“相对富卡亚范畴”的设定下建立了必要的基础工具，它是使用经典的横截性理论定义的。