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.



I would like to thank my advisor, Paul Seidel, for all of his help with this work. These results owe a great deal to him, both because his previous work [48, 49, 50, 51] laid the foundations for them, and because he gave me a lot of guidance and made many useful suggestions along the way. I would also like to thank Mohammed Abouzaid for many helpful discussions and suggestions, and for showing me a preliminary version of [1]. I would also like to thank Grisha Mikhalkin for helping me to find the construction of the immersed Lagrangian sphere in the pair of pants, on which this work is based.


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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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