Flags, Schubert polynomials, degeneracy loci, and determinantal formulas
The author proves a formula for degeneracy loci of a map of flagged vector bundles. Let \(h:E\to F\) be a map of vector bundles on a variety \(X\) and consider flags \(E_ 1\subset E_ 2\subset\cdots\subset E_ s=E\) (resp., \(F=F_ t\twoheadrightarrow F_{t- 1}\twoheadrightarrow\cdots\twoheadrightarrow F_ 1)\) of subbundles (resp., quotient bundles) of \(E\) (resp., \(F)\). One can consider the degeneracy loci: \(\Omega_ r(h):=\{x\in X|\text{rk}(E_ p(x)\to F_ q(x))\leq r(p,q)\), for all \(p,q\}\), where \(r\) is a collection of rank numbers satisfying certain conditions, which guarantee that, for generic \(h\), \(\Omega_ r(h)\) is irreducible, reduced, Cohen-Macaulay. The author gives a formula for the class \([\Omega_ r(h)]\) of this locus in the Chow ring of \(X\), as a polynomial in the Chern classes of the vector bundles. When expressed in terms of Chern roots, these polynomials are the ``double Schubert polynomials'' introduced and studied by Lascoux and Schützenberger. Special cases of this formula recover the Kempf-Laksov determinantal formula, the Giambelli-Thom-Porteous formula, as well as a formula of \textit{P. Pragacz} [Ann. Sci. Éc. Norm Supér. IV. Ser. 21, No. 3, 413- 454 (1988; Zbl 0687.14043)]. When specialized to the flag manifold of flags in an \(n\)-dimensional vector space, the formula implies that of \textit{I. N. Bernstein}, \textit{M. Gel'fand} and \textit{S. T. Gel'fand} [Russ. Math. Surveys, 28, No. 3, 1-26 (1973; Zbl 0289.57024)] and \textit{M. Demazure} [Ann. sci. Ec. Norm. Super., IV. Ser. 7, 53-88 (1974; Zbl 0312.14009)]. Doing the general case makes the proof easier. The simplicity of the proof arises from the realization of the operators considered in the papers cited above as correspondences (a fact noticed by several people, and, as the author asserts, communicated to him by R. MacPherson).
scientific article; zbMATH DE number 62989
scientific article