Flat pushforwards of Chern classes and the smoothability of cycles below the middle dimension

We prove in this paper the smoothability of cycles modulo rational equivalence below the middle dimension, 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 below the middle dimension. 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.