Let X be a toric Fano manifold and denote by Crit(f X ) ⊂ (C * ) n the solution scheme of the corresponding Landau-Ginzburg system of equations. For toric Del-Pezzo surfaces and various toric Fano threefolds we define a map L : Crit(f X ) → P ic(X) such that E L (X) := L(Crit(f X )) ⊂ P ic(X) is a full strongly exceptional collection of line bundles. We observe the existence of a natural monodromy mapwhere L(X) is the space of all Laurent polynomials whose Newton polytope is equal to the Newton polytope of f X , the Landau-Ginzburg potential of X, and R X ⊂ L(X) is the space of all elements whose corresponding solution scheme is reduced. We show that monodromies of Crit(f X ) admit non-trivial relations to quiver representations of the exceptional collection E L (X). We refer to this property as the M -aligned property of the maps L : Crit(f X ) → P ic(X). We discuss possible applications of the existence of such M -aligned exceptional maps to various aspects of mirror symmetry of toric Fano manifolds.
Let X be a toric Del-Pezzo surface and let Crit(W ) ⊂ (C * ) n be the solution scheme of the Landau-Ginzburg system of equations. Denote by X • the polar variety of X. Our aim in this work is to describe a map L : Crit(W ) → F uk trop (X • ) whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.
Let X = P(O P s ⊕ r i=1 O P s (a i )) be a Fano projective bundle over P s and denote by Crit(X) ⊂ (C * ) n the solution scheme of the Landau-Ginzburg system of equations of X. We describe a map E : Crit(X) → P ic(X) whose image E = {E(z)|z ∈ Crit(X)} is the full strongly exceptional collection described by Costa and Miró-Roig in [15]. We further show that Hom(E(z), E(w)) for z, w ∈ Crit(X) can be described in terms of a monodromy group acting on Crit(X).
For a toric Fano manifold X denote by Crit(X) ⊂ (C * ) n the solution scheme of the Landau-Ginzburg system of equations of X. Examples of toric Fano manifolds with rk(P ic(X)) ≤ 3 which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map E : Crit(X) → P ic(X) whose image E = {E(z)|z ∈ Crit(X)} is a full strongly exceptional collection satisfying the M-aligned property. That is, under this map, the groups Hom(E(z), E(w)) for z, w ∈ Crit(X) are naturally related to the structure of the monodromy group acting on Crit(X).
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