  We introduce an analogue of the Novikov Conjecture on higher signatures in
the context of the algebraic geometry of (nonsingular) complex projective
varieties. This conjecture asserts that certain "higher Todd genera" are
birational invariants. This implies birational invariance of certain extra
combinations of Chern classes (beyond just the classical Todd genus) in the
case of varieties with large fundamental group (in the topological sense). We
prove the conjecture under the assumption of the "strong Novikov Conjecture"
for the fundamental group, which is known to be correct for many groups of
geometric interest. We also show that, in a certain sense, our conjecture is
best possible.
