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Quotient dimensions and the fixed point problem for reductive groups. (English) Zbl 0834.14027
Summary: 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 \(\dim X/ / G\leq 3\).

MSC:
14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
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