Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space

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We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi–Yau hypersurface in projective space, for any d \ge 3 (for example, d=3 is the quintic threefold). 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.
Original languageEnglish
Pages (from-to)1-186
Number of pages186
JournalInventiones mathematicae
Issue number1
Early online date25 Feb 2014
Publication statusPublished - 31 Jan 2015


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