On the Fukaya category of a Fano hypersurface in projective space

Sheridan, Nick

doi:10.1007/s10240-016-0082-8

Geodesic

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On the Fukaya category of a Fano hypersurface in projective space

Sheridan, Nick ¹

¹ Department of Mathematics, Princeton University, Fine Hall Washington Road 08544-1000 Princeton NJ USA

Publications Mathématiques de l'IHÉS, Tome 124 (2016), pp. 165-317

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Résumé

This paper is about the Fukaya category of a Fano hypersurface $X \subset {CP}^{n}$ . Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak proper Calabi–Yau structure, Abouzaid’s split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface $X$ : we construct a configuration of monotone Lagrangian spheres in $X$ , and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of $X$ (up to a constant shift in the Fano index 1 case). As a consequence, we give a proof of Kontsevich’s homological mirror symmetry conjecture for $X$ . We also explain how to extract non-trivial information about Gromov–Witten invariants of $X$ from its Fukaya category.

MR Zbl

DOI : 10.1007/s10240-016-0082-8

Keywords: Modulus Space, Maslov Index, Quantum Cohomology, Hochschild Cohomology, Homological Mirror Symmetry

Affiliations des auteurs :

Sheridan, Nick ¹

¹ Department of Mathematics, Princeton University, Fine Hall Washington Road 08544-1000 Princeton NJ USA

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     title = {On the {Fukaya} category of a {Fano} hypersurface in projective space},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {165--317},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {124},
     year = {2016},
     doi = {10.1007/s10240-016-0082-8},
     mrnumber = {3578916},
     zbl = {1453.53079},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-016-0082-8/}
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Sheridan, Nick. On the Fukaya category of a Fano hypersurface in projective space. Publications Mathématiques de l'IHÉS, Tome 124 (2016), pp. 165-317. doi: 10.1007/s10240-016-0082-8

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