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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 $variety\quad X$ 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 $i\sp*H\sp{\bullet}(X;{\bbfC})$, $i: Y\to X$ being the inclusion, is the image under a ${\bbfC}$-algebra homomorphism $\psi$ of the graded algebra associated to a certain filtration of $H\sp 0(Y\cap Z;{\bbfC})$. In certain cases, for example when Z is finite and $i\sp*$ 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 ${\frak h}$ 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 $h\in {\frak h}$, consider the regular orbit $W\cdot h\subset {\frak h}$ as a finite reduced subvariety of ${\frak h}$ with ring of regular functions $A(W\cdot h)=A({\frak h})/I(W\cdot h)$, the ring of complex polynomials on ${\frak h}$ modulo those vanishing on $W\cdot h.$ The ascending filtration on A(${\frak h})$ 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 $H\sp{\bullet}(X;{\bbfC})$. The upshot of our result on torus action is that if $X\sb w=\cup\sb{v\le w}BvB/B$ is the generalized Schubert variety in X determined by $w\in W$, then $H\sp{\bullet}(X\sb w;{\bbfC})\cong Gr A([e,w]\cdot h)$, where $[e,w]\cdot h=\{v\cdot h\vert \quad v\le w\}$ and the ${\bbfC}$-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 $i\sp*: H\sp{\bullet}(X;{\bbfC})\to H\sp{\bullet}(X\sb w;{\bbfC})$ is precisely the restriction $j\sp*\sb h: Gr(A(W\cdot h))\to Gr(A([e,w]\cdot h))$ where $j\sb h: [e,w]\cdot h\to W\cdot h$ is the inclusion.
Reviewer: D.Laksov

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