## A generalization of Mumford's geometric invariant theory

### Summary

Summary: 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.

14L24, 14L30

### Keywords/Phrases

geometric invariant theory, good quotients, reductive group actions