Quadratic-linear duality and rational homotopy theory of chordal arrangements

Abstract

To any graph and smooth algebraic curve $C$, one may associate a “hypercurve” arrangement, and one can study the rational homotopy theory of the complement $X$. In the rational case ($C=ℂ$), there is considerable literature on the rational homotopy theory of $X$, and the trigonometric case ($C={ℂ}^{\times }$) is similar in flavor. The case when $C$ is a smooth projective curve of positive genus is more complicated due to the lack of formality of the complement. When the graph is chordal, we use quadratic-linear duality to compute the Malcev Lie algebra and the minimal model of $X$, and we prove that $X$ is rationally $K\left(\pi ,1\right)$.