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RANK TWO TOPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI OF QUASI-PROJECTIVE MANIFOLDS

  • Stefan Papadima (a1) and Alexander I. Suciu (a2)

Abstract

We study the germs at the origin of $G$ -representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\text{SL}_{2}(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\text{SL}_{n}(\mathbb{C})$ for some $n\geqslant 3$ , or the depth is greater than 1, then certain natural inclusions of germs are strict.

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The first author’s work was partially supported by the Romanian Ministry of Research and Innovation, CNCS–UEFISCDI, grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III. The second author was partially supported by the Simons Foundation collaboration grant for mathematicians 354156.

Deceased 10 January 2018.

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1.Arapura, D., Geometry of cohomology support loci for local systems I, J. Algebraic Geom. 6(3) (1997), 563597; MR 1487227.
2.Berceanu, B., Măcinic, A., Papadima, S. and Popescu, R., On the geometry and topology of partial configuration spaces of Riemann surfaces, Algebr. Geom. Topol. 17(2) (2017), 11631188; MR 3623686.
3.Budur, N. and Wang, B., Cohomology jump loci of differential graded Lie algebras, Compos. Math. 151(8) (2015), 14991528; MR 3383165.
4.Campana, F., Claudon, B. and Eyssidieux, P., Représentations linéaires des groupes kählériens et de leurs analogues projectifs, J. Éc. Polytech. Math. 1 (2014), 331342; MR 3322791.
5.Campana, F., Claudon, B. and Eyssidieux, P., Représentations linéaires des groupes kählériens: factorisations et conjecture de Shafarevich linéaire, Compos. Math. 151(2) (2015), 351376; MR 3314830.
6.Corlette, K. and Simpson, C., On the classification of rank-two representations of quasiprojective fundamental groups, Compos. Math. 144(5) (2008), 12711331; MR 2457528.
7.Deligne, P., Griffiths, P., Morgan, J. W. and Sullivan, D., Real homotopy theory of Kähler manifolds, Invent. Math. 29(3) (1975), 245274; MR 0382702.
8.Dimca, A. and Papadima, S., Non-abelian cohomology jump loci from an analytic viewpoint, Commun. Contemp. Math. 16(4) (2014), 1350025, 47 pp, MR 3231055.
9.Dimca, A., Papadima, S. and Suciu, A. I., Alexander polynomials: essential variables and multiplicities, Int. Math. Res. Not. IMRN 2008 (2008), Art. ID rnm119, 36 pp;MR 2416998.
10.Dimca, A., Papadima, S. and Suciu, A. I., Topology and geometry of cohomology jump loci, Duke Math. J. 148(3) (2009), 405457; MR 2527322.
11.Dupont, C., The Orlik–Solomon model for hypersurface arrangements, Ann. Inst. Fourier (Grenoble) 65(6) (2015), 25072545; MR 3449588.
12.Falk, M., Arrangements and cohomology, Ann. Combin. 1(2) (1997), 135157; MR 1629681.
13.Goldman, W. and Millson, J., The deformation theory of representations of fundamental groups of compact Kähler manifolds, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 4396; MR 0972343.
14.Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Volume 9 (Springer, New York, 1972); MR 0323842.
15.Humphreys, J. E., Linear Algebraic Groups, Graduate Texts in Mathematics, Volume 21 (Springer, New York-Heidelberg, 1975); MR 0396773.
16.Kapovich, M. and Millson, J., On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Publ. Math. Inst. Hautes Études Sci. 88 (1998), 595; MR 1733326.
17.Libgober, A. and Yuzvinsky, S., Cohomology of the Orlik–Solomon algebras and local systems, Compos. Math. 121(3) (2000), 337361; MR 1761630.
18.Loray, F., Pereira, J. V. and Touzet, F., Representations of quasiprojective groups, flat connections and transversely projective foliations, J. Éc. Polytech. Math. 3 (2016), 263308; MR 3522824.
19.Măcinic, A., Papadima, S., Popescu, R. and Suciu, A. I., Flat connections and resonance varieties: from rank one to higher ranks, Trans. Amer. Math. Soc. 369(2) (2017), 13091343; MR 3572275.
20.Morgan, J. W., The algebraic topology of smooth algebraic varieties, Publ. Math. Inst. Hautes Études Sci. 48 (1978), 137204; MR 0516917.
21.Papadima, S. and Suciu, A. I., The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy, Proc. Lond. Math. Soc. 114(6) (2017), 9611004; MR 3661343.
22.Papadima, S. and Suciu, A. I., The topology of compact Lie group actions through the lens of finite models, Int. Math. Res. Notices IMRN. Published electronically athttp://doi.org/10.1093/imrn/rnx294 (2018).
23.Papadima, S. and Suciu, A. I., Naturality properties and comparison results for topological and infinitesimal embedded jump loci, Preprint, 2016, arXiv:1609.02768v2.
24.Sullivan, D., Infinitesimal computations in topology, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 269331; MR 0646078.
25.Tougeron, J.-Cl., Idéaux de fonctions différentiables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 71 (Springer, Berlin-New York, 1972); MR 0440598.
26.Whitehead, G. W., Elements of Homotopy Theory, Graduate Texts in Mathematics, Volume 61, (Springer, New York-Berlin, 1978); MR 0516508.
27.Zuo, K., Representations of Fundamental Groups of Algebraic Varieties, Lecture Notes in Mathematics, Volume 1708 (Springer, Berlin, 1999); MR 1738433.
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RANK TWO TOPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI OF QUASI-PROJECTIVE MANIFOLDS

  • Stefan Papadima (a1) and Alexander I. Suciu (a2)

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