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
- Donu Arapura, Geometry of cohomology support loci for local systems. I, J. Algebraic Geom. 6 (1997), no. 3, 563–597. MR 1487227
- Richard Body, Mamoru Mimura, Hiroo Shiga, and Dennis Sullivan, $p$-universal spaces and rational homotopy types, Comment. Math. Helv. 73 (1998), no. 3, 427–442. MR 1633367, DOI 10.1007/s000140050063
- Kuo-Tsai Chen, Extension of $C^{\infty }$ function algebra by integrals and Malcev completion of $\pi _{1}$, Advances in Math. 23 (1977), no. 2, 181–210. MR 458461, DOI 10.1016/0001-8708(77)90120-7
- Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MR 24908, DOI 10.1090/S0002-9947-1948-0024908-8
- Kevin Corlette and Carlos Simpson, On the classification of rank-two representations of quasiprojective fundamental groups, Compos. Math. 144 (2008), no. 5, 1271–1331. MR 2457528, DOI 10.1112/S0010437X08003618
- Alexandru Dimca and Ştefan Papadima, Non-abelian cohomology jump loci from an analytic viewpoint, Commun. Contemp. Math. 16 (2014), no. 4, 1350025, 47. MR 3231055, DOI 10.1142/S0219199713500259
- Alexandru Dimca, Ştefan Papadima, and Alexander I. Suciu, Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009), no. 3, 405–457. MR 2527322, DOI 10.1215/00127094-2009-030
- J. Dixmier, Cohomologie des algèbres de Lie nilpotentes, Acta Sci. Math. (Szeged) 16 (1955), 246–250 (French). MR 74780
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- William M. Goldman and John J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43–96. MR 972343
- Peter John Hilton and Urs Stammbach, A course in homological algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York-Berlin, 1971. MR 0346025
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- Michael Kapovich and John J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 5–95 (1999). MR 1733326
- A. Libgober, First order deformations for rank one local systems with a non-vanishing cohomology, Topology Appl. 118 (2002), no. 1-2, 159–168. Arrangements in Boston: a Conference on Hyperplane Arrangements (1999). MR 1877722, DOI 10.1016/S0166-8641(01)00048-7
- John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372
- John W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204. MR 516917
- Stefan Papadima and Alexander I. Suciu, Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006), no. 3, 533–555. MR 2207874, DOI 10.1007/s00208-005-0704-9
- Stefan Papadima and Alexander I. Suciu, Bieri-Neumann-Strebel-Renz invariants and homology jumping loci, Proc. Lond. Math. Soc. (3) 100 (2010), no. 3, 795–834. MR 2640291, DOI 10.1112/plms/pdp045
- Stefan Papadima and Alexander I. Suciu, Vanishing resonance and representations of Lie algebras, J. Reine Angew. Math. 706 (2015), 83–101. MR 3393364, DOI 10.1515/crelle-2013-0073
- Stefan Papadima and Alexander I. Suciu, Jump loci in the equivariant spectral sequence, Math. Res. Lett. 21 (2014), no. 4, 863–883. MR 3275650, DOI 10.4310/MRL.2014.v21.n4.a13
- Ştefan Papadima and Alexander I. Suciu, Non-abelian resonance: product and coproduct formulas, Bridging algebra, geometry, and topology, Springer Proc. Math. Stat., vol. 96, Springer, Cham, 2014, pp. 269–280. MR 3297121, DOI 10.1007/978-3-319-09186-0_{1}7
- S. Papadima and A. Suciu, The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy, preprint http://arxiv.org/abs/1401.0868v2.
- Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
- Alexander I. Suciu, Characteristic varieties and Betti numbers of free abelian covers, Int. Math. Res. Not. IMRN 4 (2014), 1063–1124. MR 3168402, DOI 10.1093/imrn/rns246
- Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078
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