TY - JOUR

T1 - On the Fukaya category of a Fano hypersurface in projective space

AU - Sheridan, Nick

N1 - Funding Information:
This work was partially supported by the IAS and the National Science Foundation under agreement number DMS-1128155, and by the National Science Foundation through Grant number DMS-1310604. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Publisher Copyright:
© 2016, IHES and Springer-Verlag Berlin Heidelberg.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - This paper is about the Fukaya category of a Fano hypersurface X⊂ CPn. 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.

AB - This paper is about the Fukaya category of a Fano hypersurface X⊂ CPn. 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.

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U2 - 10.1007/s10240-016-0082-8

DO - 10.1007/s10240-016-0082-8

M3 - Article

AN - SCOPUS:84958253635

SN - 0073-8301

VL - 124

SP - 165

EP - 317

JO - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques

JF - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques

IS - 1

ER -