## 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.## References

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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