Fixed points for reductive group actions on acyclic varieties

Fankhauser, Martin

doi:10.5802/aif.1495

Fankhauser, Martin

Annales de l'Institut Fourier, Tome 45 (1995) no. 5, pp. 1249-1281.

Résumé
Abstract

Soit $X$ une variété complexe, affine et lisse, qui, considérée comme variété analytique, a la $ℤ$ -cohomologie singulière d’un point. Supposons que $G$ soit un groupe complexe algébrique agissant algébriquement sur $X$ . Nos résultats principaux sont les suivants : Si $G$ est semisimple, la fibre générique de l’application quotient $π : X \to X / / G$ contient une orbite dense. Si $G$ est connexe et réductif, l’action a des points fixes si $\dim X / / G \leq 3$ .

Let $X$ be a smooth, affine complex variety, which, considered as a complex manifold, has the singular $ℤ$ -cohomology of a point. Suppose that $G$ is a complex algebraic group acting algebraically on $X$ . Our main results are the following: if $G$ is semi-simple, then the generic fiber of the quotient map $π : X \to X / / G$ contains a dense orbit. If $G$ is connected and reductive, then the action has fixed points if $\dim X / / G \leq 3$ .

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1495

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     author = {Fankhauser, Martin},
     title = {Fixed points for reductive group actions on acyclic varieties},
     journal = {Annales de l'Institut Fourier},
     pages = {1249--1281},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {5},
     year = {1995},
     doi = {10.5802/aif.1495},
     mrnumber = {97a:14047},
     zbl = {0834.14027},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1495/}
}

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Fankhauser, Martin. Fixed points for reductive group actions on acyclic varieties. Annales de l'Institut Fourier, Tome 45 (1995) no. 5, pp. 1249-1281. doi : 10.5802/aif.1495. http://archive.numdam.org/articles/10.5802/aif.1495/

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