We prove Homological Mirror Symmetry for a smooth (Formula presented.) (for example, (Formula presented.) is the quintic threefold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the ‘(Formula presented.)-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.
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