*(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 $set\phantom{\rule{1.em}{0ex}}Z\xb7$ 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 ${i}^{*}{H}^{\u2022}(X;\u2102)$, $i:Y\to X$ being the inclusion, is the image under a $\u2102$-algebra homomorphism $\psi $ of the graded algebra associated to a certain filtration of ${H}^{0}(Y\cap Z;\u2102)$. 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 $\U0001d525$ 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 $h\in \U0001d525$, consider the regular orbit $W\xb7h\subset \U0001d525$ as a finite reduced subvariety of $\U0001d525$ with ring of regular functions $A(W\xb7h)=A\left(\U0001d525\right)/I(W\xb7h)$, the ring of complex polynomials on $\U0001d525$ modulo those vanishing on $W\xb7h\xb7$

The ascending filtration on A($\U0001d525)$ coming from the degree of a polynomial gives an ascending filtration F of A(W$\xb7h)$ whose associated graded ring Gr A(W$\xb7h)$ is isomorphic with ${H}^{\u2022}(X;\u2102)$. 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 ${H}^{\u2022}({X}_{w};\u2102)\cong GrA([e,w]\xb7h)$, where $[e,w]\xb7h=\{v\xb7h|\phantom{\rule{1.em}{0ex}}v\le w\}$ and the $\u2102$-algebra on the right is the graded algebra associated to the ring of regular functions on the subvariety [e,w]$\xb7h$ of $W\xb7h$ with natural ascending filtration defined as above. In addition, the natural map ${i}^{*}:{H}^{\u2022}(X;\u2102)\to {H}^{\u2022}({X}_{w};\u2102)$ is precisely the restriction ${j}_{h}^{*}:Gr\left(A(W\xb7h)\right)\to Gr\left(A([e,w]\xb7h)\right)$ where ${j}_{h}:[e,w]\xb7h\to W\xb7h$ is the inclusion.

##### 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 |