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Let be a smooth, affine complex variety, which, considered as a complex manifold, has the singular -cohomology of a point. Suppose that is a complex algebraic group acting algebraically on . Our main results are the following: if is semi-simple, then the generic fiber of the quotient map contains a dense orbit. If is connected and reductive, then the action has fixed points if .
We give a classification of finite group actions on a surface giving rise to quotients, from the point of view of their fixed points. It is shown that except two cases, each such group gives rise to a unique type of fixed point set.
We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford’s geometric invariant theory and tame stacks.
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