Formulas for Lagrangian and orthogonal degeneracy loci; the Q-polynomials approach

Abstract

Let V be a vector bundle on a scheme X endowed with a nondegenerate symplectic or orthogonal form. Let G be a Grassmannian bundle parametrizing maximal isotropic subbundles of V. The main goal of the paper is to give formulas for the classes of the loci in G parametrizing those subbundles which intersect successive members of of a fixed flag of isotropic subbundles of V in dimensions greater than 1,2,3,... . The formulas which we give are quadratic expressions in Q-polynomials of the tautological bundle on G and flag S-polynomials of the members of the flag. These formulas globalize to Lagrangian and orthogonal degeneracy loci. They have especially nice form for the loci of points in X where two maximal isotropic subbundles intersect in dimension exceeding a given number. (A problem for finding formulas for these loci was posed by J. Harris several years ago.) In this case, our formulas are simple quadratic expressions of Q-polynomials applied to E and F. The work generalizes formulas given in [P.Pragacz, Algebro- geometric applications of Schur S- and Q-polynomials, SLN 1478 (1991), 130-191]. One of its applications is computation of the classes of Brill-Noether loci in Pryms in [De Concini, P. Pragacz, On the class of Brill-Noether loci for Prym varieties, Math. Ann. 302 (1995), 687-697].