## 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 $$variety\quad X$$ with a holomorphic vector field V having nontrivial zero $$set\quad Z.$$ 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 $$Y\cap Z$$ is finite, then the graded ring $$i^*H^{\bullet}(X;{\mathbb{C}})$$, $$i: Y\to X$$ being the inclusion, is the image under a $${\mathbb{C}}$$-algebra homomorphism $$\psi$$ of the graded algebra associated to a certain filtration of $$H^ 0(Y\cap Z;{\mathbb{C}})$$. In certain cases, for example when Z is finite and $$i^*$$ surjective, $$\psi$$ 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 $${\mathfrak h}$$ 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 $$h\in {\mathfrak h}$$, consider the regular orbit $$W\cdot h\subset {\mathfrak h}$$ as a finite reduced subvariety of $${\mathfrak h}$$ with ring of regular functions $$A(W\cdot h)=A({\mathfrak h})/I(W\cdot h)$$, the ring of complex polynomials on $${\mathfrak h}$$ modulo those vanishing on $$W\cdot h.$$
The ascending filtration on A($${\mathfrak h})$$ coming from the degree of a polynomial gives an ascending filtration F of A(W$$\cdot h)$$ whose associated graded ring Gr A(W$$\cdot h)$$ is isomorphic with $$H^{\bullet}(X;{\mathbb{C}})$$. The upshot of our result on torus action is that if $$X_ w=\cup_{v\leq w}BvB/B$$ is the generalized Schubert variety in X determined by $$w\in W$$, then $$H^{\bullet}(X_ w;{\mathbb{C}})\cong Gr A([e,w]\cdot h)$$, where $$[e,w]\cdot h=\{v\cdot h| \quad v\leq w\}$$ and the $${\mathbb{C}}$$-algebra on the right is the graded algebra associated to the ring of regular functions on the subvariety [e,w]$$\cdot h$$ of $$W\cdot h$$ with natural ascending filtration defined as above. In addition, the natural map $$i^*: H^{\bullet}(X;{\mathbb{C}})\to H^{\bullet}(X_ w;{\mathbb{C}})$$ is precisely the restriction $$j^*_ h: Gr(A(W\cdot h))\to Gr(A([e,w]\cdot h))$$ where $$j_ h: [e,w]\cdot h\to W\cdot h$$ is the inclusion.
Reviewer: D.Laksov

### MSC:

 14L30 Group actions on varieties or schemes (quotients) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14M17 Homogeneous spaces and generalizations 14L24 Geometric invariant theory
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### References:

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