Abstract
This paper is about the Fukaya category of a Fano hypersurface X ⊂ 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 splitgeneration 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 nontrivial information about Gromov–Witten invariants of X from its Fukaya category.
Original language  English 

Pages (fromto)  165317 
Number of pages  153 
Journal  Publications mathématiques de l'IHÉS 
Volume  124 
Issue number  1 
Early online date  15 Feb 2016 
DOIs  
Publication status  Published  30 Nov 2016 
Fingerprint
Dive into the research topics of 'On the Fukaya category of a Fano hypersurface in projective space'. Together they form a unique fingerprint.Profiles

Nick Sheridan
 School of Mathematics  Royal Society University Research Fellow
Person: Academic: Research Active