Graphs and Equivariant Cohomology

Abstract

Let X be a T-skeletal variety, that is, a complex algebraic variety where a complex torus T acts with only nitely many xed points and invariant curves. By a result of Goresky, Kottwtiz and MacPherson, the equivariant cohomology of X can be read off from the associated graph of xed points and invariant curves. The purpose of this paper is to compute explicitly and combinatorially the equivariant cohomology of certain projective toric surfaces and projective homogeneous spaces. In all these cases the equivariant cohomology is known to be a free module over a polynomial ring, and we provide explicit combinatorial and geometric bases for such modules. Furthermore, we exhibit an e cient algorithm to obtain such bases from a suitable order relation on the associated graph.