Quadratic-linear duality and rational homotopy theory of chordal arrangements

Bibby, Christin; Hilburn, Justin

doi:10.2140/agt.2016.16.2637

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Quadratic-linear duality and rational homotopy theory of chordal arrangements

Christin Bibby and Justin Hilburn

Algebraic & Geometric Topology 16 (2016) 2637–2661

DOI: 10.2140/agt.2016.16.2637

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 $K (π, 1)$ .

Keywords

hyperplane arrangement, toric arrangement, elliptic arrangement, Koszul duality, rational homotopy theory

Mathematical Subject Classification 2010

Primary: 16S37, 52C35, 55P62

References

Publication

Received: 17 October 2014

Revised: 21 July 2015

Accepted: 29 January 2016

Published: 7 November 2016

Authors

Christin Bibby
Department of Mathematics University of Western Ontario London, ON N6A 5B7 Canada

Justin Hilburn
Department of Mathematics University of Oregon 1380 Lawrence #2 Eugene, OR 97403 United States

Volume 16, issue 5 (2016)