# ​Overview

Let X be a projective Fano toric manifold given by a polytope Δ and let Δ​0  be the corresponding polar polytope. My current research involves the study of the relationship between two classes of objects associated to X:
​homological mirror symmetry
​here.
​Motivated by ideas from
(a) Full strongly excpetional collections of line bundles in Pic(X)

# (I) E-maps and M-algined property

Let f∈L(Δ​0​) be a Laurent polynomial (potential) and let Crit(f)⊂(ℂ​\0)​n be the solution of the corresponding Landau-Ginzburg system. We refer to a map E:Crit(f)→Pic(X) as an exceptional map (or E-map) if the image ℰ=E(Crit(f))⊂Pic(X)  is a full strongly exceptional collection. Let us show an example:

Example ​(Projective space):​ Let X=ℙ​s be s-dimensional projective space. Let f(z​1,...,z​s)=z​1+...+z​s+1/z​1·...·z​s be the Landau-Ginzburg potential. The system of equations is given by z​i-1/z​1·...·z​s=0, for i=1,...,s, whose solutions are z​k=(μ​k,...,μ​k) for k=0,...,s where μ=e2πi/s+1. Define the map E:Crit(f)→Pic(ℙ​s) by E(z):=[∑​nArg(z​n)H​n] (notations explained below). The map gives E(z​k)=[(k/s+1)·(H​0+...+H​k)]=k ·O(1)=O(k), for k=0,...,s, which is Beilinson's collection. In particular, it is an E-map.

In general, recall that one has 0→ℤ​n→Div​T(X)→Pic(X)→0 and that in the Fano caseDiv​T(X)={∑a​FD​F:F∈Δ(n-1)}={∑​na​nD​n:n∈Δ​0(0)}. Define the map E​f(z) =[∑​nArg(z​n)D​n]∈Pic(X). We have:

​Theorem ​ ​A (J.): Let be a toric Fano manifold of one of the following classes:
(1) dimX≤3.
(2) Projective Fano bundles over projective space (rk(Pic(X))=2).
(3) Blow up of ℙ​sXℙ​r along ℙ​s-1Xℙ​r-1 for 0≤s,r.
(4) Blow up of ℙ(O​n-1(b)⊕O​n-1) along for b
Then there exists an E-map of the form E​f for some f∈L(Δ​0).

In the space L(Δ​0) one has the hypersurface R={f:Crit(f) is non-reduced}. Consider the monodromy map M:π​1(L(Δ​0)\R,f)→Aut(Crit(f)). In the cases (1)-(4) we described an element γ​D∈π​1(L(Δ)\R,f) for any D∈Div​T(X). The E-maps of Theorem A have the following additional geometric property to which we refer as the ​M-aligned property (M stands for monodromy):

​Theorem B ( ​J.):​ Let be a toric Fano manifold of class (1)-(4) and as above. For any two z​1,z​2∈Crit(f) one has M(γ​D)(z​1)=z​2 if
[D]=E​f(z​2)-E​f(z​1).
​We present various examples of these monodromies
(b) Solutions of Landau-Ginzburg systems determined by Δ​0
​and
​here

# (II) E-maps and homological mirror symmetry for toric Fano manifolds

Recall that Div​T(X)=⊕F∈Δ(n-1) ℤ·DF has a dual description in terms of piece-wise linear Δ-functions. In particular, any T-invariant divisor D∈Div​T(X) and a vertex m∈Δ(0) is associated with a weight w​X(D,m)∈ℤ, viewed as the linear functional obtained by restricting the piece-wise linear function corresponding to D to the cone determined by m.

On the other hand, for any two solutions z,z’∈Crit(f) consider G(z,z’)={γ:M(γ)(z)=z’}⊂π​1(L(Δ​0)\R,f). In the examples above (Theorem A) one has a distinguished solution z​0 such that E(z​0​)=O​X. Note that γ∈G(z,z​0) can be naturally associated with a weight W(γ)∈ℤ​n by lifting the image of M(γ) under the argument map to the universal cover ℝ​n of T​n=ℝ​n/ℤ​n. In a recent work we observed that any z∈Crit(f)  and m∈Δ(0) can be associated with an element γ(z,m)∈G(z,z​0) (determined uniquely by the combinatorics of the polytope Δ). In particular, the pair (z,m) is associated with a “mirror weight” given by
W​f(z,m)=W(γ(z,m)). In a recent work, we proved the following result, which we view as a manifestation of homological mirror symmetry:

​Theorem C ​(J.):​ Let X be a toric Fano and let z∈Crit(f) be any solution. Then W​f(z,m)=W​X​(D​z​,m) for any m∈Δ(0), where
D​z∈Div​T(X) is a (unique) toric divisor such that E(z)=[D​z]∈Pic(X).

In fact, Abouzaid's homological mirror symmetry functor HMS gives a correspondence between Div​T​(X) and the group of tropical Lagrangian sections Fuk​trop​(X​0​). Theorem C is used to construct a map L:Crit(f)→Fuk​trop(X​0) such that
HMS(D​z)=L(z). The corresponding Lagrangian tropical sections are given as embeddings i​z:Δ→(ℂ∖0) with , defined in a non-trivial way. We refer the reader here for examples of the mirror weights.

​References could be found
​here.
​Abouzaid's homological mirror symmetry functor