On the nonexistence of certain morphisms from Grassmannian to Grassmannian in characteristic $0$

Abstract

This paper proves some properties of the big Chern classes of a vector bundle on a smooth scheme over a field of characteristic 0. These properties together with the explicit computation of the big Chern classes of universal quotient bundles of Grassmannians are used to prove the main Theorems (Theorems 1,2 and 3) of this paper. The nonexistence certain morphisms between Grassmannians over a field of characteristic 0 follows directly from these theorems. One of our theorems, for instance, states that the higher Adams operations applied to the class of a universal quotient bundle of a Grassmannian that is not a line bundle yield elements in the K-ring of the Grassmannian that are not representable as classes of genuine vector bundles. This is not true for Grassmannians over a field of characteristic $p$.