Quadratic-linear duality and rational homotopy theory of chordal arrangements

TL;DR

This paper explores the rational homotopy theory of hypercurve arrangements associated with graphs and smooth algebraic curves, focusing on chordal graphs and using quadratic-linear duality to analyze their topological properties.

Contribution

It introduces a novel application of quadratic-linear duality to compute the Malcev Lie algebra and minimal model for chordal arrangements, establishing their rational $K(\pi,1)$ property.

Findings

Computed Malcev Lie algebra for chordal arrangements

Established the minimal model of the complement

Proved the complement is rationally $K(\pi,1)$

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 ($C=\mathbb{C}$), there is considerable literature on the rational homotopy theory of $X$, and the trigonometric case ($C = \mathbb{C}^\times$) is similar in flavor. The case of 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 $K(\pi,1)$.