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:

Motivated by ideas from

(a)
__Full strongly excpetional collections of line bundles in Pic(X)__

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:

In general, recall that one has 0→ℤ^{n}→Div^{}_{T}(X)→Pic(X)→0 and that in the Fano case Div_{T}(X)={∑a_{F}D_{F}:F∈Δ(n-1)}={∑_{n}_{}_{}a_{n}D_{n}:n∈Δ^{0}(0)}. Define the map E_{f}(z) =[∑_{n}Arg(z_{n})D_{n}]∈Pic(X). We have:

(1) dimX≤3.

(2) Projective Fano bundles over projective space (rk(Pic(X))=2).

(3) Blow up of ℙ^{s}Xℙ_{}^{r}_{} along ℙ^{s-1}^{}Xℙ^{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):

[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

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