Abstract
Soit M une variété holomorphe compacte admettant pour espace de revêtement universel un domaine borné homogène Ω de CN, par exemple une surface de Riemann compacte de genre > 1. Soit A le groupe des automorphismes holomorphes de Ω. Puisque Ω est supposé homogène, le groupe A opère transitivement dans Ω. Muni de la topologie “compacts-ouverts”, A est un groupe de Lie; le groupe des automorphismes du revêtement Ω → M est un sous-groupe discret Г de A. On sait d'autre part que A opère proprement dans Ω. Choisissons un point z°Є Ω. Son stabilisateur est un sous-groupe compact K de A. L'application canonique de Г\A sur Г\A/K est donc propre. Or Ω s'identifie à A/K et Г\A/K s'identifie à M. Ceci montre que Г\A est compact, autrement dit que Г est sous-groupe uniforme de A. On retrouve les mêmes circonstances dans la composante connexe neutre A° de A. Celle ci opère encore transitivement dans Ω et Г° = Г ∩ A° est un sous-groupe discret uniforme de A°. Or un groupe de Lie connexe contenant un sous-groupe discret uniforme est unimodulaire. D'autre part, on sait qu un domaine borné admettant un groupe unimodulaire transitif de transformations holomorphes est un domaine borné symétrique (Théorème de Hano): tout point de Ω est donc point fixe isolé (et unique) d'un automorphisme involutif holomorphe de Ω.
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Koszul, J.L. (2011). Formes Harmoniques Vectorielles Sur Les Espaces Localement Symetriques. In: Vesentini, E. (eds) Geometry of Homogeneous Bounded Domains. C.I.M.E. Summer Schools, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11060-3_5
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