A Generalization of Mumford's Geometric Invariant Theory

We generalize Mumford's construction of good quotients for reductive group actions. Replacing a single linearized invertible sheaf with a certain group of sheaves, we obtain a Geometric Invariant Theory producing not only the quasiprojective quotient spaces, but more generally all divisorial ones. As an application, we characterize in terms of the Weyl group of a maximal torus, when a proper reductive group action on a smooth complex variety admits an algebraic variety as orbit space.

2000 Mathematics Subject Classification: 14L24, 14L30

Keywords and Phrases: Geometric Invariant Theory, good quotients, reductive group actions

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