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

Résumé

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 ${\mathbb{C}}^*$ sur l’espace des modules de $G$-fibrés de Higgs sur $X$ s’étend à une action de $\mathbb{C}$. Ceci produit la rétraction par déformation mentionnée ci-dessus.

Abstract

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 ${\mathbb{C}}^*$ on the moduli space of $G$–Higgs bundles over $X$ extends to an action of $\mathbb{C}$. This produces the above mentioned deformation retraction.

Détail

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

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