On the Fukaya categories of projective hypersurfaces of general type
Kazushi Ueda
TL;DR
The paper proves homological mirror symmetry for projective hypersurfaces of sufficiently high degree by relating the wrapped Fukaya category of the affine complement $U$ to the Fukaya category of the boundary $X$ via a Lagrangian correspondence $C$, inducing an $A_\infty$-functor $\\Phi: \\mathcal{W}(U) \\to \\mathcal{F}(X)$ and a descended functor on the stable category. It shows unobstructedness with $\\mathfrak{m}_0(1)=0$ and identifies the target with a matrix-factorization category $\\operatorname{mf}([\\mathbb{A}^{n+2}/G], w-x_1\\cdots x_{n+2})$, using Hochschild cohomology to establish fullness and essential surjectivity. Consequently, the Fukaya category of $X$ is equivalent to the mirror matrix-factorization category, yielding HMS in the high-degree regime. The work ties together wrapped/Fukaya categories, Milnor fiber Picard-Lefschetz theory, and Jacobian-ring data in the Brieskorn–Pham setting, expanding HMS to new geometric contexts.
Abstract
We prove homological mirror symmetry for projective hypersurfaces of sufficiently high degree using a functor from the wrapped Fukaya category of an affine hypersurface to the Fukaya category of its boundary at infinity.