no. 6
Character varieties of virtually nilpotent Kähler groups and G–Higgs bundles
[Variétés de caractères des groupes de Kähler virtuellement nilpotents et G-fibrés de Higgs]
Biswas, Indranil1 ; Florentino, Carlos2
1 School of Mathematics Tata Institute of Fundamental Research, Homi Bhabha Road Bombay 400005 (India)
2 Departamento Matemática Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisbon (Portugal)
Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2601-2612.

Soit G un groupe algébrique affine réductif complexe connexe, et soit K⊂G un sous-groupe compact maximal. Soit X une variété Kählerienne compacte connexe dont le groupe fondamental Γ est virtuellement nilpotent. Nous montrons que la variété de caractères Hom(Γ,G)//G admet une rétraction par déformation forte naturelle sur le sous-ensemble Hom(Γ,K)/K⊂Hom(Γ,G)//G. L’action naturelle de ℂ * sur l’espace des modules de G-fibrés de Higgs sur X s’étend à une action de ℂ. Ceci produit la rétraction par déformation mentionnée ci-dessus.

Let G be a connected complex reductive affine algebraic group, and let K⊂G be a maximal compact subgroup. Let X be a compact connected Kähler manifold whose fundamental group Γ is virtually nilpotent. We prove that the character variety Hom(Γ,G)//G admits a natural strong deformation retraction to the subset Hom(Γ,K)/K⊂Hom(Γ,G)//G. The natural action of ℂ * on the moduli space of G–Higgs bundles over X extends to an action of ℂ. This produces the above mentioned deformation retraction.

DOI : 10.5802/aif.2997
Classification : 20G20, 14J60
Keywords: Kähler group, character variety, $G$–Higgs bundle, virtually nilpotent group
Mot clés : Groupes de Kähler, variété des caractères, $G$-fibrés de Higgs, groupe virtuellement nilpotent

Biswas, Indranil 1 ; Florentino, Carlos 2

1 School of Mathematics Tata Institute of Fundamental Research, Homi Bhabha Road Bombay 400005 (India)
2 Departamento Matemática Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisbon (Portugal) 
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Biswas, Indranil; Florentino, Carlos. Character varieties of virtually nilpotent Kähler groups and $G$–Higgs bundles. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2601-2612. doi : 10.5802/aif.2997. https://aif.centre-mersenne.org/articles/10.5802/aif.2997/

[1] Amorós, J.; Burger, M.; Corlette, K.; Kotschick, D.; Toledo, D. Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, 44, American Mathematical Society, Providence, RI, 1996, pp. xii+140 | DOI | MR | Zbl

[2] Anchouche, Boudjemaa; Biswas, Indranil Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold, Amer. J. Math., Volume 123 (2001) no. 2, pp. 207-228 | MR | Zbl

[3] Bergeron, Maxime The topology of nilpotent representations in reductive groups and their maximal compact subgroups, Geom. Topol., Volume 19 (2015) no. 3, pp. 1383-1407 | DOI | MR

[4] Biswas, Indranil; Florentino, Carlos The topology of moduli spaces of group representations: the case of compact surface, Bull. Sci. Math., Volume 135 (2011) no. 4, pp. 395-399 | DOI | MR | Zbl

[5] Biswas, Indranil; Florentino, Carlos Commuting elements in reductive groups and Higgs bundles on abelian varieties, J. Algebra, Volume 388 (2013), pp. 194-202 | DOI | MR | Zbl

[6] Biswas, Indranil; Gómez, Tomás L. Connections and Higgs fields on a principal bundle, Ann. Global Anal. Geom., Volume 33 (2008) no. 1, pp. 19-46 | DOI | MR | Zbl

[7] Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2), Volume 117 (1983) no. 1, pp. 109-146 | DOI | MR | Zbl

[8] Delzant, Thomas L’invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes, Math. Ann., Volume 348 (2010) no. 1, pp. 119-125 | DOI | MR | Zbl

[9] Donaldson, S. K. Infinite determinants, stable bundles and curvature, Duke Math. J., Volume 54 (1987) no. 1, pp. 231-247 | DOI | MR | Zbl

[10] Florentino, Carlos; Lawton, Sean The topology of moduli spaces of free group representations, Math. Ann., Volume 345 (2009) no. 2, pp. 453-489 | DOI | MR | Zbl

[11] Florentino, Carlos; Lawton, Sean Character varieties and moduli of quiver representations, In the tradition of Ahlfors-Bers. VI (Contemp. Math.), Volume 590, Amer. Math. Soc., Providence, RI, 2013, pp. 9-38 | DOI | MR

[12] Florentino, Carlos; Lawton, Sean Topology of character varieties of Abelian groups, Topology Appl., Volume 173 (2014), pp. 32-58 | DOI | MR | Zbl

[13] Franco, Emilio; Garcia-Prada, Oscar; Newstead, P. E. Higgs bundles over elliptic curves for complex reductive Lie groups (http://arxiv.org/abs/1310.2168) | MR

[14] Gukov, Sergei Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial, Comm. Math. Phys., Volume 255 (2005) no. 3, pp. 577-627 | DOI | MR | Zbl

[15] Hitchin, N. J. The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Volume 55 (1987) no. 1, pp. 59-126 | DOI | MR | Zbl

[16] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, pp. xiv+247 (Graduate Texts in Mathematics, No. 21) | MR | Zbl

[17] Narasimhan, M. S.; Seshadri, C. S. Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2), Volume 82 (1965), pp. 540-567 | MR | Zbl

[18] Pettet, Alexandra; Souto, Juan Commuting tuples in reductive groups and their maximal compact subgroups, Geom. Topol., Volume 17 (2013) no. 5, pp. 2513-2593 | DOI | MR

[19] Ramanathan, A.; Subramanian, S. Einstein-Hermitian connections on principal bundles and stability, J. Reine Angew. Math., Volume 390 (1988), pp. 21-31 | MR | Zbl

[20] Simpson, Carlos T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 5-95 | Numdam | MR | Zbl

[21] Uhlenbeck, K.; Yau, S.-T. On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math., Volume 39 (1986) no. S, suppl., p. S257-S293 Frontiers of the mathematical sciences: 1985 (New York, 1985) | DOI | MR | Zbl

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