  We introduce Chern classes in $U(m)$-equivariant homotopical bordism that
refine the Conner-Floyd-Chern classes in the $MU$-cohomology of $B U(m)$. For
products of unitary groups, our Chern classes form regular sequences that
generate the augmentation ideal of the equivariant bordism rings. Consequently,
the Greenlees-May local homology spectral sequence collapses for products of
unitary groups. We use the Chern classes to reprove the $MU$-completion theorem
of Greenlees-May and La Vecchia.
