On Landau–Ginzburg systems, co-tropical geometry, and Db(X) of various toric Fano manifolds

Abstract

Let X be a toric Fano manifold given by a reflexive polytope Δ and let Crit(f)⊂(C*)s be the solution scheme of the Landau–Ginzburg system associated with a Laurent polynomial fu(z)≔∑n∈Δ◦(0)∩Zsunzn∈L(Δ◦). Motivated by mirror symmetry, we construct a map E : Crit(fu) → Pic(X) and show various cases of (X, fu) for which its image Eu(X)≔E(z)|z∈Crit(fu)⊂Pic(X) is a full strongly exceptional collection of line bundles. Moreover, we study relations between Hom(E(z), E(w)) for z, w ∈ Crit(fu) and the structure of the monodromy group acting on Crit(fu). The construction relies on a study of the tropical and co-tropical properties of Crit(fu) as log|u| → ±∞.