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 F(X) is defined over some suitable Novikov field K; its mirror is assumed to be some smooth projective scheme Y over K with `maximally unipotent monodromy'. Suppose that some split-generating subcategory of (a dg enhancement of) DbCoh(Y) embeds into F(X): we call this hypothesis `core homological mirror symmetry'. We prove that the embedding extends to an equivalence of categories, DbCoh(Y)≅Dπ(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.