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


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
Full Text: Numdam EuDML


[1] E. Akyildiz : Vector fields and cohomology of G/P , Lecture Notes in Math., Vol. 956 (1981). · Zbl 0498.14025
[2] E. Akyildiz , J.B. Carrell , D.I. Lieberman and A.J. Sommese : Graded rings associated to holomorphic vector fields with a unique zero . Proc. of Symp. in Pure Mathematics: Singularities, Vol. 40 (1983) 55. · Zbl 0523.57031
[3] E. Akyildiz , J.B. Carrell , D.I. Lieberman and A.J. Sommese : Graded rings associated to holomorphic vector fields with a unique zero . Preprint. · Zbl 0523.57031
[4] I.N. Bernstein , I.M. Gel’Fand and S.I. Gel’Fand : Schubert cells and cohomology of the space G/P , Russian Math. Surveys 28 (1973) 1-26. · Zbl 0289.57024
[5] J.B. Carrell : Vector fields and the cohomology of G/B. Manifolds and Lie groups, Papers in honor of Y. Matsushima . Progress in Mathematics, Vol. 14 Birkhauser, Boston (1981). · Zbl 0482.57017
[6] J.B. Carrell : Orbits of the Weyl group and a theorem of DeConcini and Procesi . Preprint. · Zbl 0613.20029
[7] J.B. Carrell and D.I. Lieberman : Holomorphic vector fields and compact Kaehler manifolds , Inventiones Math. 21 (1973) 303-309. · Zbl 0253.32017
[8] J.B. Carrell and D.I. Lieberman : Vector fields and Chern numbers , Math. Annalen, 225 (1977) 263-273. · Zbl 0365.32020
[9] J.B. Carrell and D.I. Lieberman : Vector fields, Chern classes and cohomology , Proc. Symp. Pure Math., Vol. 30 (1977) 251-254. · Zbl 0365.32019
[10] J.B. Carrell and A.J. Sommese , Some topological aspects of C*-actions on compact Kaehler manifolds , Comment. Math. Helv. 54 (1979) 567-582. · Zbl 0466.32015
[11] C. Deconcini and C. Procesi : Symmetric functions, conjugacy classes, and the flag variety , Invent. Math., 64 (1981) 203-219. · Zbl 0475.14041
[12] P. Deligne : ThĂ©orie de Hodge, III . Inst. Hautes Etudes Sci. Publ. Math., Vol. 44 (1972) 5-77. · Zbl 0237.14003
[13] A. Fujiki : Fixed points of the actions on compact Kaehler manifolds , Publ. RIMS, Kyoto Univ. 15 (1979) 1-45. · Zbl 0446.53021
[14] P. Griffiths and J. Harris : Principles of Algebraic Geometry , John Wiley and Sons, New York (1978). · Zbl 0408.14001
[15] H. Hironaka : Bimeromorphic smoothing of a complex analytic space , Math. Inst. Warwick Univ., England (1971). · Zbl 0407.32006
[16] S. Kumar : Geometry of Schubert cells and cohomology of Kac-Moody Lie algebras , J. of Diff. Geometry 20 (1984) 389-432. · Zbl 0565.17007
[17] S. Kumar : Cohomology algebra of Schubert varieties associated to Kac-Moody groups, to appear in proceedings of the CMS Seminar in Algebraic Geometry , Vancouver (1984). · Zbl 0591.14037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.