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Zero of holomorphic vector fields on singular spaces and intersection rings of Schubert varieties. (English) Zbl 0613.14035

From the introduction: The purpose of this paper is to initiate a study of the cohomology rings of invariant subvarieties of a smooth projective varietyX with a holomorphic vector field V having nontrivial zero setZ· 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 YZ is finite, then the graded ring i * H (X;), i:YX being the inclusion, is the image under a -algebra homomorphism ψ of the graded algebra associated to a certain filtration of H 0 (YZ;). In certain cases, for example when Z is finite and i * surjective, ψ is an isomorphism.

Applying this to the vector fields on flag varieties X=G/B 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 X=G/B the associated flag variety. Let 𝔥 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 h𝔥, consider the regular orbit W·h𝔥 as a finite reduced subvariety of 𝔥 with ring of regular functions A(W·h)=A(𝔥)/I(W·h), the ring of complex polynomials on 𝔥 modulo those vanishing on W·h·

The ascending filtration on A(𝔥) coming from the degree of a polynomial gives an ascending filtration F of A(W·h) whose associated graded ring Gr A(W·h) is isomorphic with H (X;). The upshot of our result on torus action is that if X w = vw BvB/B is the generalized Schubert variety in X determined by wW, then H (X w ;)GrA([e,w]·h), where [e,w]·h={v·h|vw} and the -algebra on the right is the graded algebra associated to the ring of regular functions on the subvariety [e,w]·h of W·h with natural ascending filtration defined as above. In addition, the natural map i * :H (X;)H (X w ;) is precisely the restriction j h * :Gr(A(W·h))Gr(A([e,w]·h)) where j h :[e,w]·hW·h is the inclusion.

Reviewer: D.Laksov

MSC:
14L30Group actions on varieties or schemes (quotients)
14F05Sheaves, derived categories of sheaves, etc.
14M17Homogeneous spaces and generalizations
14L24Geometric invariant theory of group schemes