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Schubert varieties in flag bundles for the classical groups. (English) Zbl 0862.14032

Teicher, Mina (ed.), Proceedings of the Hirzebruch 65 conference on algebraic geometry, Bar-Ilan University, Ramat Gan, Israel, May 2-7, 1993. Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 9, 241-262 (1996).
Let \(V\) be a vector bundle on an algebraic variety \(X\), \(L\) a line bundle on \(X\) and \(Q: V \otimes V \to L\) a nondegenerate skew-symmetric or symmetric bilinear form. Let \({\mathcal F}\) be the bundle of complete flags of isotropic subbundles of \(V\) with respect to \(Q\) and let \(F_\bullet\) be such a flag. For each permutation \(w\) in the corresponding Weyl group \(W\) one can define, using \(F_\bullet\), a Schubert variety \({\mathcal X}_w \subset {\mathcal F}\). In this paper, the author gives a formula for the class of \({\mathcal X}_w\) in the Chow ring of \({\mathcal F}\), thus completing his results from Duke Math. J. 65, No. 3, 381-420 (1992; Zbl 0788.14044) where he considered the case of the Schubert subvarieties of the bundle of complete flags of subbundles of \(V\). The formulas found by the author imply formulas for the corresponding degeneracy loci whenever one has a bundle with a symplectic or orthogonal form and two flags of isotropic subbundles. This was worked out in detail by the author [J. Differ. Geom. 43, No. 2, 276-290 (1996)]. The author uses some splitting principles to reduce the proof of the formulas to the case where \(V\) is a direct sum of lines bundles and the symplectic or orthogonal forms are diagonal. Then, he proves the formulas for \(w=\) identity and deduces the general case by relating one locus to the next by a \(\mathbb{P}^1\)-bundle (or conic bundle) correspondence.
For the entire collection see [Zbl 0828.00035].

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14C05 Parametrization (Chow and Hilbert schemes)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L35 Classical groups (algebro-geometric aspects)

Citations:

Zbl 0788.14044
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