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

TL;DR

This paper constructs a map linking solutions of Landau-Ginzburg equations to exceptional line bundle collections on certain toric Fano manifolds, revealing a deep connection between algebraic geometry and monodromy actions.

Contribution

It introduces a new map from Landau-Ginzburg critical points to Picard groups, establishing a link with exceptional collections and monodromy for toric Fano manifolds with small Picard rank.

Findings

Constructed a map from Crit(X) to Pic(X) for specific toric Fano manifolds.

Established that the image forms a full strongly exceptional collection.

Linked Hom groups to monodromy group actions.

Abstract

For a toric Fano manifold $X$ denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. Examples of toric Fano manifolds with $rk(Pic(X)) \leq 3$ which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map $E : Crit(X) \rightarrow Pic(X)$ whose image $\mathcal{E}=\left \{ E(z) \vert z \in Crit(X) \right \}$ is a full strongly exceptional collection satisfying the M-aligned property. That is, under this map, the groups $Hom(E(z),E(w))$ for $z,w \in Crit(X)$ are naturally related to the structure of the monodromy group acting on $Crit(X)$.