Automatic Split-Generation for the Fukaya Category

Abstract

Abstract 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 ${\mathbb {K}}$-variety $Y$ that is “maximally degenerating”. Suppose that some split-generating subcategory of (a $\textsf {dg}$ 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, $D^{b}Coh(Y) \simeq D^{\pi }( \mathcal {F}(X))$, using Abouzaid’s split-generation criterion. The results only depend on certain formal properties of the Fukaya category, which have been established in certain cases but not in complete generality. The appendix, which can be read on its own, proves a result about Hochschild cohomology for schemes: the compatibility of the global Hochschild–Kostant–Rosenberg isomorphism with Kodaira–Spencer deformation theory.