On the nonexistence of certain morphisms from Grassmannian to   Grassmannian in characteristic 0
  Here, we utilize facts about the big Chern classes discovered by M. Kapranov and independently by M. V. Nori to prove certain results about the nonexistence of certain morphisms from Grassmannian to Grassmannian in characteristic 0. In particular, we show that the filtration F_. on CH^l(X) where F_rCH^l(X) is the linear span of ch_l(V) for all vector bundles V of rank <=r is nontrivial as a theory by showing that for a fixed l>=2, for infinitely many r, for a Grassmannian of r dimensional quotient spaces in an n dimensional vector space (n sufficiently large) over a field of charactersitic 0, ch_l(Q) is in F_rCH^l(G(r,n)) but not in F^(r-1)CH^l(G(r,n)) where Q is the universal quotient bundle of the Grassmannian G(r,n). Separately but using similar methods, we show that for r>=2, if Q denotes the universal quotient bundle of the Grassmannian of r dimensional quotients G(r,n), n>=2r+1, then the class [\psi^pQ] can never be equal to the class of a genuine vector bundle in K(G). In parallel, we obtain a formula for the bid Chern classes in terms of the ch_l. This leads to the identification of proper subfunctors of the Hodge functors H^q(X, \Omega^p) in characteristic 0. 
Ajay C. Ramadoss