. The determinant is a sum of forms of different
orders: 
where 
is a 2j-form.
with 
structure group is GL(k,C). 
total chern class = c(F)= . The determinant is a sum of forms of different
orders: where is a 2j-form.
G compact means F is anti-hermitian real differential form.
is hermitian so can be diagonalized by a unitary transformation.
this means 
where the 
's are 2-forms.
This means 
For the second churn class, you get:
properties:
and 
over same base M, do whitney sum, 
, then
