Quotients of affine spaces for actions of reductive groups

Details Quotients of affine spaces for actions of reductive groups Quotients of affine spaces for actions of reductive groups

2004-12-14

arxiv.org/abs/math/0412278v1

Mathematics

Algebraic Geometry

37 pages

Scientific paper

In this paper we study actions of reductive groups on affine spaces. We prove that there is a fan structure on the space of characters of the group, which parameterizes the possible invariant quotients. In the second half of the paper we study various geometrical properties of the quotients: we compute their Chow ring and the cohomology ring (in case we work over the complex numbers), both with rational coefficients. Actually, the cycle map turns out to be an isomorphism. Further, we construct families of such quotients over reduced base schemes, and compute again the Chow and the cohomology rings. Finally, we prove the vanishing of the higher cohomology groups of nef line bundles on the geometric quotients that we obtain.

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