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 $Crit(X)\subset ({\mathbb{C}}^{\ast }{)}^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. Examples of toric Fano manifolds with $rk(Pic(X))\le 3$ which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map $E:Crit(X)\to Pic(X)$ whose image $\mathcal{E}=\left\{E(z)|z\in Crit(X)\right\}$ 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 $Crit(X)$.

The following is an extensively extended version of arXiv:1412.2687

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Cited by in corpus (1)

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