Fan cohomology and equivariant Chow rings of toric varieties

Mu-wan Huang, University of Nebraska - Lincoln

Abstract

Toric varieties are varieties equipped with a torus action and constructed from cones and fans. In the joint work with Suanne Au and Mark E. Walker, we prove that the equivariant K -theory of an affine toric variety constructed from a cone can be identified with a group ring determined by the cone. When a toric variety X (Δ) is smooth, we interpret equivariant K -groups as presheaves on the associated fan space Δ. Relating the sheaf cohomology groups to equivariant K -groups via a spectral sequence, we provide another proof of a theorem of Vezzosi and Vistoli: equivariant K -theory is formed by patching equivariant K -theory of equivariant affine toric subvarieties. This dissertation studies the sheaf cohomology groups for the equivariant K -groups tensored with [Special characters omitted.] and completed, and how they relate to the equivariant K -groups of non-smooth and non-affine toric varieties. The equivariant K -groups tensored with [Special characters omitted.] and completed coincide with the equivariant Chow rings for affine toric varieties. For a three-dimensional complete fan , we calculate the dimensions of the sheaf cohomology groups for the symmetric algebra sheaf. When the fan is given by a convex polytope, this information computes the equivariant K -groups tensored with [Special characters omitted.] and completed as extensions of sheaf cohomology groups.

Recommended Citation

Mu-wan Huang, "Fan cohomology and equivariant Chow rings of toric varieties" (January 1, 2009). ETD collection for University of Nebraska - Lincoln. Paper AAI3360497.
http://digitalcommons.unl.edu/dissertations/AAI3360497