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Fixed points for reductive group actions on acyclic varieties

Martin Fankhauser — 1995

Annales de l'institut Fourier

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 $\dim X / / G \leq 3$ .

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