An analogue of the Novikov Conjecture in complex algebraic geometry

Abstract

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.