On Landau-Ginzburg systems and $\mathcal{D}^b(X)$ of various toric Fano manifolds with small picard group

Abstract

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).

Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

“…Remark 3. 34. In what follows, we will use results in [2] that are stated for manifolds with boundary instead of corners.…”

“…However, a transitive action on critical points is not sufficient as we would also need to guarantee that the monodromy can produce thimbles for all critical points that correspond to disjoint paths. Jerby has used a related method to produce full strong exceptional collections of line bundles in D b Coh(X) for some lowdimensional examples in [32,33,34] (cf. Section 1.1).…”