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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 $variety\phantom{\rule{1.em}{0ex}}X$ with a holomorphic vector field V having nontrivial zero $set\phantom{\rule{1.em}{0ex}}Z·$ 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 ${i}^{*}{H}^{•}\left(X;ℂ\right)$, $i:Y\to X$ being the inclusion, is the image under a $ℂ$-algebra homomorphism $\psi$ of the graded algebra associated to a certain filtration of ${H}^{0}\left(Y\cap Z;ℂ\right)$. 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 $𝔥$ 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 $h\in 𝔥$, consider the regular orbit $W·h\subset 𝔥$ as a finite reduced subvariety of $𝔥$ with ring of regular functions $A\left(W·h\right)=A\left(𝔥\right)/I\left(W·h\right)$, the ring of complex polynomials on $𝔥$ modulo those vanishing on $W·h·$

The ascending filtration on A($𝔥\right)$ coming from the degree of a polynomial gives an ascending filtration F of A(W$·h\right)$ whose associated graded ring Gr A(W$·h\right)$ is isomorphic with ${H}^{•}\left(X;ℂ\right)$. The upshot of our result on torus action is that if ${X}_{w}={\cup }_{v\le w}BvB/B$ is the generalized Schubert variety in X determined by $w\in W$, then ${H}^{•}\left({X}_{w};ℂ\right)\cong GrA\left(\left[e,w\right]·h\right)$, where $\left[e,w\right]·h=\left\{v·h|\phantom{\rule{1.em}{0ex}}v\le w\right\}$ and the $ℂ$-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 ${i}^{*}:{H}^{•}\left(X;ℂ\right)\to {H}^{•}\left({X}_{w};ℂ\right)$ is precisely the restriction ${j}_{h}^{*}:Gr\left(A\left(W·h\right)\right)\to Gr\left(A\left(\left[e,w\right]·h\right)\right)$ where ${j}_{h}:\left[e,w\right]·h\to W·h$ is the inclusion.

Reviewer: D.Laksov

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