no. 6
Character varieties of virtually nilpotent Kähler groups and G–Higgs bundles
Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2601-2612.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2997
Classification: 20G20,  14J60
Keywords: Kähler group, character variety, G–Higgs bundle, virtually nilpotent group 
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     title = {Character varieties of virtually nilpotent {K\"ahler} groups and $G${{\textendash}Higgs} bundles},
     journal = {Annales de l'Institut Fourier},
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Biswas, Indranil; Florentino, Carlos. Character varieties of virtually nilpotent Kähler groups and $G$–Higgs bundles. Annales de l'Institut Fourier, Volume 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, Tome 44, American Mathematical Society, Providence, RI, 1996, xii+140 pages | Article | MR: 1379330 | Zbl: 0849.32006

[2] Anchouche, Boudjemaa; Biswas, Indranil Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold, Amer. J. Math., Tome 123 (2001) no. 2, pp. 207-228 | MR: 1828221 | Zbl: 1007.53026

[3] Bergeron, Maxime The topology of nilpotent representations in reductive groups and their maximal compact subgroups, Geom. Topol., Tome 19 (2015) no. 3, pp. 1383-1407 | Article | MR: 3352239

[4] Biswas, Indranil; Florentino, Carlos The topology of moduli spaces of group representations: the case of compact surface, Bull. Sci. Math., Tome 135 (2011) no. 4, pp. 395-399 | Article | MR: 2799816 | Zbl: 1225.14011

[5] Biswas, Indranil; Florentino, Carlos Commuting elements in reductive groups and Higgs bundles on abelian varieties, J. Algebra, Tome 388 (2013), pp. 194-202 | Article | MR: 3061684 | Zbl: 1285.14045

[6] Biswas, Indranil; Gómez, Tomás L. Connections and Higgs fields on a principal bundle, Ann. Global Anal. Geom., Tome 33 (2008) no. 1, pp. 19-46 | Article | MR: 2369185 | Zbl: 1185.14032

[7] Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2), Tome 117 (1983) no. 1, pp. 109-146 | Article | MR: 683804 | Zbl: 0529.57005

[8] Delzant, Thomas L’invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes, Math. Ann., Tome 348 (2010) no. 1, pp. 119-125 | Article | MR: 2657436 | Zbl: 1293.32030

[9] Donaldson, S. K. Infinite determinants, stable bundles and curvature, Duke Math. J., Tome 54 (1987) no. 1, pp. 231-247 | Article | MR: 885784 | Zbl: 0627.53052

[10] Florentino, Carlos; Lawton, Sean The topology of moduli spaces of free group representations, Math. Ann., Tome 345 (2009) no. 2, pp. 453-489 | Article | MR: 2529483 | Zbl: 1200.14093

[11] Florentino, Carlos; Lawton, Sean Character varieties and moduli of quiver representations, In the tradition of Ahlfors-Bers. VI (Contemp. Math.) Tome 590, Amer. Math. Soc., Providence, RI, 2013, pp. 9-38 | Article | MR: 3087924

[12] Florentino, Carlos; Lawton, Sean Topology of character varieties of Abelian groups, Topology Appl., Tome 173 (2014), pp. 32-58 | Article | MR: 3227204 | Zbl: 1300.14045

[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: 3331841

[14] Gukov, Sergei Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial, Comm. Math. Phys., Tome 255 (2005) no. 3, pp. 577-627 | Article | MR: 2134725 | Zbl: 1115.57009

[15] Hitchin, N. J. The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Tome 55 (1987) no. 1, pp. 59-126 | Article | MR: 887284 | Zbl: 0634.53045

[16] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, xiv+247 pages (Graduate Texts in Mathematics, No. 21) | MR: 396773 | Zbl: 0471.20029

[17] Narasimhan, M. S.; Seshadri, C. S. Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2), Tome 82 (1965), pp. 540-567 | MR: 184252 | Zbl: 0171.04803

[18] Pettet, Alexandra; Souto, Juan Commuting tuples in reductive groups and their maximal compact subgroups, Geom. Topol., Tome 17 (2013) no. 5, pp. 2513-2593 | Article | MR: 3190294

[19] Ramanathan, A.; Subramanian, S. Einstein-Hermitian connections on principal bundles and stability, J. Reine Angew. Math., Tome 390 (1988), pp. 21-31 | MR: 953674 | Zbl: 0648.53017

[20] Simpson, Carlos T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 5-95 | Numdam | MR: 1179076 | Zbl: 0814.32003

[21] Uhlenbeck, K.; Yau, S.-T. On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math., Tome 39 (1986) no. S, suppl., p. S257-S293 (Frontiers of the mathematical sciences: 1985 (New York, 1985)) | Article | MR: 861491 | Zbl: 0615.58045

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