Character varieties of virtually nilpotent Kähler groups and $G$–Higgs bundles

Biswas, Indranil; Florentino, Carlos

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2601-2612.

Soit $G$ un groupe algébrique affine réductif complexe connexe, et soit $K\subset G$ un sous-groupe compact maximal. Soit $X$ une variété Kählerienne compacte connexe dont le groupe fondamental $\Gamma$ est virtuellement nilpotent. Nous montrons que la variété de caractères $\text{Hom}(\Gamma ,G)//G$ admet une rétraction par déformation forte naturelle sur le sous-ensemble $\text{Hom}(\Gamma ,K)/K\subset \text{Hom}(\Gamma ,G)//G$. 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 $K\subset G$ be a maximal compact subgroup. Let $X$ be a compact connected Kähler manifold whose fundamental group $\Gamma$ is virtually nilpotent. We prove that the character variety $\text{Hom}(\Gamma ,G)//G$ admits a natural strong deformation retraction to the subset $\text{Hom}(\Gamma ,K)/K\subset \text{Hom}(\Gamma ,G)//G$. 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.

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

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