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, Indranil ; Florentino, Carlos
Annales de l'Institut Fourier, Tome 65 (2015), p. 2601-2612 / Harvested from Numdam

Soit G un groupe algébrique affine réductif complexe connexe, et soit KG 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)/KHom(Γ,G)//G. L’action naturelle de * sur l’espace des modules de G-fibrés de Higgs sur X s’étend à une action de . Ceci produit la rétraction par déformation mentionnée ci-dessus.

Let G be a connected complex reductive affine algebraic group, and let KG 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)/KHom(Γ,G)//G. The natural action of * on the moduli space of G–Higgs bundles over X extends to an action of . This produces the above mentioned deformation retraction.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2997
Classification:  20G20,  14J60
Mots clés: Groupes de Kähler, variété des caractères, G-fibrés de Higgs, groupe virtuellement nilpotent
@article{AIF_2015__65_6_2601_0,
     author = {Biswas, Indranil and Florentino, Carlos},
     title = {Character varieties of virtually nilpotent K\"ahler groups and $G$--Higgs bundles},
     journal = {Annales de l'Institut Fourier},
     volume = {65},
     year = {2015},
     pages = {2601-2612},
     doi = {10.5802/aif.2997},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2015__65_6_2601_0}
}
Biswas, Indranil; Florentino, Carlos. Character varieties of virtually nilpotent Kähler groups and $G$–Higgs bundles. Annales de l'Institut Fourier, Tome 65 (2015) pp. 2601-2612. doi : 10.5802/aif.2997. http://gdmltest.u-ga.fr/item/AIF_2015__65_6_2601_0/

[1] Amorós, J.; Burger, M.; Corlette, K.; Kotschick, D.; Toledo, D. Fundamental groups of compact Kähler manifolds, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 44 (1996), pp. xii+140 | 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, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 590 (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), pp. xiv+247 (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