Fixed points for reductive group actions on acyclic varieties

Fankhauser, Martin

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

Let $X$ be a smooth, affine complex variety, which, considered as a complex manifold, has the singular $\mathbb{Z}$-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 $\pi :X\to X//G$ contains a dense orbit. If $G$ is connected and reductive, then the action has fixed points if $\dim X//G\le 3$.

Soit $X$ une variété complexe, affine et lisse, qui, considérée comme variété analytique, a la $\mathbb{Z}$-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 $\pi :X\to X//G$ contient une orbite dense. Si $G$ est connexe et réductif, l’action a des points fixes si $\dim X//G\le 3$.

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