Equivariant Cohomology in Algebraic Geometry Lecture Nine: Flag Varieties
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Remark 0.1. We saw that the study of cohomology and equivariant cohomology of Grassmannians leads to interesting symmetric polynomials, namely, the Schur polynomials sλ(x) and sλ(x|t). These arise in contexts other than intersection theory and representation theory. For example, Griffiths asked which polynomials P in c1(E), . . . , cn(E) are positive whenever E is an ample vector bundle on an n-dimensional variety. (That is, ∫
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Equivariant Cohomology in Algebraic Geometry Lecture Seven: Equivariant Cohomology of Grassmannians
1.2. Schubert basis. To get more information, we must restrict to a torus. Take V = Cn, and let T be the subgroup of diagonal matrices in GLnC. We have the same description of H∗ TX, where X = Gr(k, n), but now Λ = ΛT = Z[t1, . . . , tn] and c(E) = ∏n i=1(1+ ti). Taking a T -invariant flag F•, we have T -invariant Schubert varieties Ωλ(F•). (In this section, we always assume a partition λ is co...
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تاریخ انتشار 2007