Schur Q-functions and Degeneracy Locus Formulas for Morphisms with Symmetries

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle and is based on a push-forward formula for these polynomials in a Grassmann bundle, established in [P4]. “Something which is not testable is not scientific.” Poper’s criterium Introduction The goal of the present paper is to state and prove new closed-form formulas for the fundamental classes of some degeneracy loci. We will be here interested in degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices that are symmetric (resp. skewsymmetric) w.r.t. the main diagonal. To be more precise, let α : E ։ F be a surjection of two vector bundles of respective ranks e and f on a variety X . We denote by E ∨ F (resp. E ∧ F ) the kernel of the surjection E ⊗ F α⊗1 −−−−→ F ⊗ F ։ ∧F (resp. E ⊗ F α⊗1 −−−−→ F ⊗ F ։ SF ). In the present paper, we will call a morphism φ : E → F symmetric (resp. skewsymmetric) provided it is induced by a section of the subbundle E∨F (resp. E∧F ) of E ⊗ F . (Note that for E = F these notions coincide with the usual notions of “symmetric” and “skew-symmetric” morphisms.) 1991 Mathematics Subject Classification. 14M12, 14C17, 14M15. This research was supported by the grant No.5031 of French-Polish cooperation C.N.R.S. – P.A.N., and the KBN grant No.2P03A 05112.