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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 \(i^*H^{\bullet}(X;{\mathbb{C}})\), \(i: Y\to X\) 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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