Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space

TL;DR

This paper proves Homological Mirror Symmetry for smooth Calabi-Yau hypersurfaces in projective space of dimension greater than two, using advanced techniques in symplectic geometry and category theory.

Contribution

It introduces the relative Fukaya category, constructs an immersed Lagrangian sphere, and establishes mirror symmetry for Calabi-Yau hypersurfaces in projective space.

Findings

Homological Mirror Symmetry is proven for all smooth Calabi-Yau hypersurfaces in projective space.

Development of the relative Fukaya category and its grading structure.

Explicit computations enabled by a Morse-Bott model and graded matrix factorizations.

Abstract

We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.