Inventiones mathematicae

, Volume 199, Issue 1, pp 1–186 | Cite as

Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space



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.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Princeton University and the Institute for Advanced StudyPrincetonUSA

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