On the Fukaya category of a Fano hypersurface in projective space

By Nick Sheridan. Publications mathématiques (Bures-sur-Yvette), 2013

123 citations recorded by the source corpus.

This paper is about the Fukaya category of a Fano hypersurface X⊂CPn$X \subset \mathbf {CP}^{n}$. 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$X$: we construct a configuration of monotone Lagrangian spheres in X$X$, and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of X$X$ (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$X$. We also explain how to extract non-trivial information about Gromov–Witten invariants of X$X$ from its Fukaya category.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ed3302ce9d517cbe283dbcdfa078deecb3faf40d

How researchers see this paper

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In Applied Mathematics

1. P. Seidel
2. Yankı Lekili
3. J. Evans
4. James Pascaleff
5. Dmitry Tonkonog
6. Timothy Perutz
7. Nick Sheridan

In Geometry and Topology

1. Yankı Lekili
2. P. Seidel
3. D. Auroux
4. Nick Sheridan
5. Timothy Perutz
6. Sheel Ganatra
7. J. Evans
8. I. Smith

Citations in the corpus

• Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “ussion of Assumption 2.1, this gives rise to a class in SH1(E) red, and by applying red, to a class (2.25) s2SH0(E): The last-mentioned class has an alternative and more direct description, following [38, 2]. With that description in mind, we call sthe Borman-Sheridan class. Remark 2.6. From the per”
• The wall-crossing formula and Lagrangian mutations (2017, background): “An important case where these hypotheses are satisfied is the case of Kähler pairs and anchored Lagrangian submanifolds as studied by Sheridan [32].”
• Mirror symmetry: from categories to curve counts (2015, background): “There is an action of the character group G ∗ on X n , and H A ( X n ) G ∗ is precisely the Hodge part of the cohomology, i.e., the part generated by the K¨ahler class [ ω ] (see [She13, Lemma 7.5] for a proof of this fact).”
• Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, background): “Finding a split-generator is also the first step in many modern proofs of homological mirror symmetry (for example [7, 57, 59]).”