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Flat connections and resonance varieties: From rank one to higher ranks


Authors: Daniela Anca Măcinic, Ştefan Papadima, Clement Radu Popescu and Alexander I. Suciu
Journal: Trans. Amer. Math. Soc. 369 (2017), 1309-1343
MSC (2010): Primary 55N25, 55P62; Secondary 14F35, 20F36, 20J05
DOI: https://doi.org/10.1090/tran/6799
Published electronically: April 8, 2016
MathSciNet review: 3572275
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Abstract: Given a finitely generated group $ \pi $ and a linear algebraic group $ G$, the representation variety $ \mathrm {Hom}(\pi ,G)$ has a natural filtration by the characteristic varieties associated to a rational representation of $ G$. Its algebraic counterpart, the space of $ \mathfrak{g}$-valued flat connections on a commutative, differential graded algebra $ (A,d)$, admits a filtration by the resonance varieties associated to a representation of $ \mathfrak{g}$. We establish here a number of results concerning the structure and qualitative properties of these embedded resonance varieties, with particular attention to the case when the rank $ 1$ resonance variety decomposes as a finite union of linear subspaces. The general theory is illustrated in detail in the case when $ \pi $ is either an Artin group or the fundamental group of a smooth, quasi-projective variety.


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Additional Information

Daniela Anca Măcinic
Affiliation: Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: Anca.Macinic@imar.ro

Ştefan Papadima
Affiliation: Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: Stefan.Papadima@imar.ro

Clement Radu Popescu
Affiliation: Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: Radu.Popescu@imar.ro

Alexander I. Suciu
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: a.suciu@neu.edu

DOI: https://doi.org/10.1090/tran/6799
Keywords: Resonance variety, characteristic variety, differential graded algebra, Lie algebra, flat connection, quasi-projective manifold, Artin group.
Received by editor(s): March 2, 2014
Received by editor(s) in revised form: March 16, 2015, and June 14, 2015
Published electronically: April 8, 2016
Additional Notes: The first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-RU-PD-2011-3-0149
The second author was partially supported by the Romanian Ministry of National Education, CNCS-UEFISCDI, grant PNII-ID-PCE-2012-4-0156
The third author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2012-3-0492
The fourth author was partially supported by NSF grant DMS–1010298 and NSA grant H98230-13-1-0225
Article copyright: © Copyright 2016 American Mathematical Society