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Abelian duality and propagation of resonance

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Abstract

We explore the relationship between a certain “abelian duality” property of spaces and the propagation properties of their cohomology jump loci. To that end, we develop the analogy between abelian duality spaces and those spaces which possess what we call the “EPY property”. The same underlying homological algebra allows us to deduce the propagation of jump loci: in the former case, characteristic varieties propagate, and in the latter, the resonance varieties. We apply the general theory to arrangements of linear and elliptic hyperplanes, as well as toric complexes, right-angled Artin groups, and Bestvina–Brady groups. Our approach brings to the fore the relevance of the Cohen–Macaulay condition in this combinatorial context.

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Authors and Affiliations

  1. Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada

    Graham Denham

  2. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Alexander I. Suciu

  3. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Sergey Yuzvinsky

Authors
  1. Graham Denham
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  2. Alexander I. Suciu
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  3. Sergey Yuzvinsky
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Corresponding author

Correspondence to Sergey Yuzvinsky.

Additional information

G. Denham: Partially supported by an NSERC Discovery Grant (Canada).

A. I. Suciu: Partially supported by NSF Grant DMS-1010298, NSA Grant H98230-13-1-0225, and Simons Foundation collaborative Grant 354156.

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Denham, G., Suciu, A.I. & Yuzvinsky, S. Abelian duality and propagation of resonance. Sel. Math. New Ser. 23, 2331–2367 (2017). https://doi.org/10.1007/s00029-017-0343-5

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