We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d ≥ 3 (for example, d = 3 is the quintic three-fold).

Abstract. The n-dimensional pair of pants is defined to be the complement of n+2 generic hyperplanes in CP n . We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism A ∞ algebra in the Fukaya category. On the level of cohomology, it is an exterior algebra with n + 2 generators. It is not formal, and we compute certain higher products in order to determine it up to quasi-isomorphism. This allows us to give some evidence for the Homological Mirror Symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model (C n+2 , W ), where W = z 1 ...z n+2 . We show that the endomorphism A ∞ algebra of our Lagrangian is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.

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 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.

We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests how the Gamma class should arise from the Strominger-Yau-Zaslow conjecture. We use it to give a new proof of (a version of) the Gamma Conjecture for Batyrev pairs of mirror Calabi-Yau hypersurfaces.