Let X be a toric Fano manifold given by a Fano polytope Δ and let Δ^{0} be the corresponding polar polytope. The polytope Δ^{0} determines the space L(Δ^{0}) of all Laurent polynomials of the form f(z)=∑_{}a_{n}z^{n}, where the sum runs over all n∈Δ^{0}(0). To any element f∈L(Δ^{0}(0)) we associate the corresponding Landau-Ginzburg system z_{i}·(∂f/∂z_{i})=0 for i=1,...,n. We denote by Crit(f)⊂(ℂ∖0)^{n} the solution scheme of the system. Landau-Ginzburg potentials were first introduced by Batyrev in the description of quantum cohomology of toric Fano manifolds in [Ba1] and in the description of the middle homology of mirrors of Calabi-Yau hypersurfaces, in [Ba2]. In the context of HMS of toric Fano manifolds Landau-Ginzburg systems play a fundamental role and the mirror of X is given by a pair ((ℂ∖0)^{n},f). Let us mention the following two fundamental works:

If f∈L(Δ^{0}) is a generic element it has the structure of a Lefschetz fibration. Concretely, f is an algebraic fibration with fiber W_{z}=f^{-1}(z)⊂(ℂ∖0)^{n} for z∈ℂ. A fiber is singular exactly when Crit(f)∩W_{z }is not empty. If f is generic all the singularities are isolated and quadratic (which is the Lefschetz condition). A fundamental property of Lefschetz fibrations, from the symplectic view-point, is that the vanishing cycles of the fibration in H^{n-1}(W_{z}), for a smooth fiber, can be represented by Lagranagian spheres, see [Ar,Se1]. An important feature is that the vanishing Lagrangian spheres are **not** uniquely determined by their corresponding vanishing cycles. In fact, a vanishing Lagrangian sphere L(z;γ) is determined by a curve γ⊂f(Crit(f)) connecting a singular value w_{i}_{}=f(z_{i}_{ }) where Crit(f)={z_{1},...,z_{n}} to the regular value w=f(z). Seidel's remarkable view-point (see, for instance [Se2,Se3,Se4,Se5]) is that a collection of vanishing Lagrangian spheres determined by a choice of paths {γ_{1},...,γ_{N}} has properties analog to that of exceptional collections in D^{b}^{}(X). In fact, the resulting collection {L_{1},...,L_{N}} is an exceptional collection in the corresponding Fukaya-Seidel category FS((ℂ∖0)^{n},f)associated to the Lefschetz fibration f, we refer the reader to the book [Se5]. In particular, one obtains a homological mirror symmetry functor by finding an equivalent (full strongly) exceptional collection {E1,...,EN}⊂D^{b}(X) analog to a collection of Lagrangian vanishing spheres {L_{1},...,L_{N}} in the mirror. This was done in various cases, for instance, Auroux, Katzarkov and Orlov describe such a functor in the case of Del-Pezzo surfaces, in [AKO]. In [Se3] the case of the quartic surface is studied in detail. It should be noted that the exceptional objects in D^{}^{b}^{}(X) corresponding to vanishing cycles are not necessarily given in terms of vector bundles.

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**__Abouzaid's homological mirror symmetry for toric Fano manifolds__

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Abouzaid's results on homological mirror symmetry for toric Fano manifolds were developed in [Ab1,Ab2]. On the algebraic side, it is noted in [Ab2] that the derived category of coherent sheaves D^{b}(X) of a toric manifold X is generated as a triangulated category by the elements of Pic(X). On the other hand, Abouzaid describes a class of Lagrangian submanifolds in ((ℂ∖0)^{n},Φ) which are mirror analogs of line bundles in Pic(X), called tropical Lagrangian sections (Abouzaid works with a certain mild deformation Φ of a Laurent polynomial f). This correspondence is the basis of Abouzaid's homological mirror symmetry functor. More concretely, a *Lagrangian brane *L⊂(ℂ∖0)^{n }is an embedded compact graded Lagrangian submanifold, which is spin and exact. A Lagrangian brane is said to be admisible if ∂L⊂M, where M:={f=0}, and there exists a small neighborhood of ∂L in L which agrees with the parallel transport of ∂L along a segment in ℂ. Based on ideas of Kontsevich, the admissible Lagrangians are taken to be the objects of the Fukaya

A_{∞}-pre-category Fuk((ℂ∖0)^{n},Φ).

One has the following two maps (T-fibrations): the moment map μ:(ℂ∖0)^{n}→ℝ^{n} and the logarithm map Log:(ℂ∖0)^{n}→ℝ. Recall that the image of the moment map coincides with the polytope, μ((ℂ∖0)^{n})=Δ. On the other hand, it is noted by Abouzaid that there exists a component of the complement of the amobea of M under the map Log which coincides with (a -smoothning of) Δ. A *tropical Lagrangian section* is an admissible Lagrangian brane which is a section of Log restricted to (a smoothning of) Δ. We denote by Fuk_{trop}(Φ)⊂Fuk(Φ) the collection of tropical Lagrangian sections. It is shown by Abouzaid in [Ab2] that up to Hamiltonian isotopy torpical Lagrangian sections are in one to one correspondence with elements of Pic(X).

[Ab1] M. Abouzaid. Homogeneous coordinate rings and mirror symmetry for toric varieties. Geometry and Topology 10 (2006) 1097-1156.

[Ab2] M. Abouzaid. Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. Selecta Mathematica August 2009, Volume 15, Issue 2, pp 189-270.

[AKO] D. Auroux, L. Katzarkov, D. Orlov. Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves. Invent. Math. 166 (2006), no. 3, 537-582.

[Ar] V. I. Arnold, Some remarks on symplectic monodromy of Milnor fibrations, Progress in mathematics, volume 133, the Floer memorial volume, 99-103.

[Ba1] V. Batyrev, Quantum cohomology rings of toric manifolds, Astérisque 218 (1993) 9–34 Journées de Géométrie Algébrique d'Orsay.

[Ba2] V. Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3 (1994), no. 3, 493-535.

[HV] K. Hori, C. Vafa. Mirror Symmetry. arXiv:hep-th/0002222

[K] M. Kontsevich. Homological algebra of mirror symmetry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), 120-139, Birkhauser, Basel, 1995.

[K2] M. Kontsevich. Course at ENS. 1998.

[Se1] P. Seidel, Floer homology and the symplectic isotopy problem, thesis, Oxford, 1997.

[Se2] P, Seidel, Vanishing cycles and mutation, European congress of mathematics , volume 202 of the series Progress in Mathematics, 65-85.

[Se3] P. Seidel. More about vanishing cycles and mutation. Symplectic geometry and mirror symmetry (Seoul, 2000), 429-465, World Sci. Publ., River Edge, NJ, 2001.

[Se4] P. Seidel, Homological mirror symmetry for the quartic surface, Memoirs of the AMS, volume 236, 2015.

[Se5] P. Seidel, Fukaya categories and Picad-Lefschetz theory, Zurich lectures in advanced mathematics , European mathemathical society, 2008.