  We prove in this paper the smoothability of cycles modulo rational
equivalence in the Whitney range, that is, when the dimension is strictly
smaller than the codimension. We introduce and study the class of cycles
obtained as ``flat pushforwards of Chern classes" (or equivalently, flat
pushforwards of products of divisors) and prove that they are smoothable in the
Whitney range. Our main result is that all cycles (of any dimension) on a
smooth projective variety are flat pushforwards of Chern classes. In the case
of abelian varieties, one can even restrict to smooth pushforwards of Chern
classes.
