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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Flat connections and resonance varieties: From rank one to higher ranks
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by Daniela Anca Măcinic, Ştefan Papadima, Clement Radu Popescu and Alexander I. Suciu PDF
Trans. Amer. Math. Soc. 369 (2017), 1309-1343 Request permission

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
  • MR Author ID: 863641
  • 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
  • MR Author ID: 832600
  • Email: Radu.Popescu@imar.ro
  • Alexander I. Suciu
  • Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
  • MR Author ID: 168600
  • ORCID: 0000-0002-5060-7754
  • Email: a.suciu@neu.edu
  • 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
  • © Copyright 2016 American Mathematical Society
  • 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
  • MathSciNet review: 3572275