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Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. (English) Zbl 0788.14044
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 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 I. N. Bernstein, M. Gel’fand and S. T. Gel’fand [Russ. Math. Surveys, 28, No. 3, 1-26 (1973; Zbl 0289.57024)] and 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).

MSC:
14M12 Determinantal varieties
14M15 Grassmannians, Schubert varieties, flag manifolds
13C40 Linkage, complete intersections and determinantal ideals
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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[1] I. N. Bernšteĭ n, I. M. Gel’fand, and S. I. Gel’fand, Schubert cells, and the cohomology of the spaces \(G/P\) , Uspehi Mat. Nauk 28 (1973), no. 3(171), 3-26. · Zbl 0289.57024
[2] R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces , Amer. J. Math. 80 (1958), 964-1029. JSTOR: · Zbl 0101.39702
[3] P. Bressler, M. Finkelberg, and V. Lunts, Vanishing cycles on Grassmannians , Duke Math. J. 61 (1990), no. 3, 763-777. · Zbl 0727.14027
[4] Séminaire C. Chevalley, 1956-1958. Classification des groupes de Lie algébriques , 2 vols, Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1958. · Zbl 0092.26301
[5] C. Chevalley, Séminaire C. Chevalley; 2e année: 1958. Anneaux de Chow et applications , Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1958. · Zbl 0098.13101
[6] M. Demazure, Désingularisation des variétés de Schubert généralisées , Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88. · Zbl 0312.14009
[7] C. Ehresmann, Sur la topologie de certains espaces homogènes , Ann. of Math. (2) 35 (1934), 396-443. JSTOR: · Zbl 0009.32903
[8] W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[9] G. Giambelli, Ordine de una varietà piu ampia di quella rappresentata coll’ annullare tutti i minori di dato ordine estratti da una data matrice generica di forme , Mem. R. Istituto Lombardo (3) 11 (1904), 101-125.
[10] G. Giambelli, La teoria delle formole d’incidenza e di posizione speciale e le forme binarie , Atti della R. Accad. delle Scienze di Torino 40 (1904), 1041-1062. · JFM 36.0604.02
[11] G. Giambelli, Risoluzione del problema generale numerativo per gli spazi plurisecanti di una curva algebrica , Mem. Acad. Sci. Torino (2) 59 (1909), 433-508. · JFM 40.0612.04
[12] G. Kempf, Linear systems on homogeneous spaces , Ann. of Math. (2) 103 (1976), no. 3, 557-591. JSTOR: · Zbl 0327.14016
[13] G. Kempf and D. Laksov, The determinantal formula of Schubert calculus , Acta Math. 132 (1974), 153-162. · Zbl 0295.14023
[14] A. Lascoux, Puissances extérieures, déterminants et cycles de Schubert , Bull. Soc. Math. France 102 (1974), 161-179. · Zbl 0295.14024
[15] A. Lascoux, Classes de Chern des variétés de drapeaux , C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 5, 393-398. · Zbl 0495.14032
[16] A. Lascoux, Anneau de Grothendieck de la variété des drapeaux , preprint, 1988.
[17] A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert , C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447-450. · Zbl 0495.14031
[18] A. Lascoux and M.-P. Schützenberger, Symmetry and flag manifolds , Invariant theory (Montecatini, 1982) ed. F. Gherardelli, Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 118-144. · Zbl 0542.14031
[19] A. Lascoux and M.-P. Schützenberger, Schubert polynomials and the Littlewood-Richardson rule , Lett. Math. Phys. 10 (1985), no. 2-3, 111-124. · Zbl 0586.20007
[20] I. G. Macdonald, Notes on Schubert Polynomials , Départment de mathématiques et d’informatique, Université du Québec, Montréal, 1991.
[21] D. Monk, The geometry of flag manifolds , Proc. London Math. Soc. (3) 9 (1959), 253-286. · Zbl 0096.36201
[22] C. Musili and C. S. Seshadri, Schubert varieties and the variety of complexes , Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, Papers Dedicated to I. R. Shafarevich on the Occasion of his Sixtieth Birthday, pp. 329-359. · Zbl 0567.14030
[23] P. Pragacz, Enumerative geometry of degeneracy loci , Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 413-454. · Zbl 0687.14043
[24] A. Ramanathan, Schubert varieties are arithmetically Cohen-Macaulay , Invent. Math. 80 (1985), no. 2, 283-294. · Zbl 0541.14039
[25] T. A. Springer, Quelques applications de la cohomologie d’intersection , Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92, Soc. Math. France, Paris, 1982, pp. 249-273. · Zbl 0526.22014
[26] M. L. Wachs, Flagged Schur functions, Schubert polynomials, and symmetrizing operators , J. Combin. Theory Ser. A 40 (1985), no. 2, 276-289. · Zbl 0579.05001
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