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Sheridan, Nick

On the Fukaya category of a Fano hypersurface in projective space. (English) Zbl 1453.53079

Publ. Math., Inst. Hautes Étud. Sci. 124, 165-317 (2016).
Summary: This paper is about the Fukaya category of a Fano hypersurface \(X \subset \mathbf {CP}^{n}\). 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.

Cited in 1 Review
Cited in 39 Documents

MSC:

53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14J33 Mirror symmetry (algebro-geometric aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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