From the introduction: The purpose of this paper is to initiate a study of the cohomology rings of invariant subvarieties of a smooth projective with a holomorphic vector field V having nontrivial zero We will first consider the case in which V is generated by a torus action on X, showing that if V is tangent to the set of smooth points of a closed subvariety Y of X such that is finite, then the graded ring , being the inclusion, is the image under a -algebra homomorphism of the graded algebra associated to a certain filtration of . In certain cases, for example when Z is finite and surjective, is an isomorphism.
Applying this to the vector fields on flag varieties gives a surprising description of the cohomology algebra of a Schubert variety which is now explained. Suppose G is a semi-simple complex Lie group, B a Borel subgroup and the associated flag variety. Let be a Cartan subalgebra of Lie(G) and Lie(B), and let W be the associated partially ordered Weyl group of G. For any regular element , consider the regular orbit as a finite reduced subvariety of with ring of regular functions , the ring of complex polynomials on modulo those vanishing on
The ascending filtration on A( coming from the degree of a polynomial gives an ascending filtration F of A(W whose associated graded ring Gr A(W is isomorphic with . The upshot of our result on torus action is that if is the generalized Schubert variety in X determined by , then , where and the -algebra on the right is the graded algebra associated to the ring of regular functions on the subvariety [e,w] of with natural ascending filtration defined as above. In addition, the natural map is precisely the restriction where is the inclusion.