The higher Todd genus birational-invariance conjecture

Let $V$ be a nonsingular complex projective variety of complex dimension $n$, viewed as a complex manifold, with fundamental class $[V]\in H_{2n}(V,\mathbb Z)$ and total Todd class $\mathcal T(V)\in H^*(V,\mathbb Q)$. Let $\pi$ be a discrete group, let $x\in H^*(B\pi,\mathbb Q)$, and let $u\colon V\to B\pi$ be a map to the classifying space of $\pi$. The higher Todd genus is $\langle \mathcal T(V)\cup u^*(x),[V]\rangle$. Higher Todd genus conjecture. The higher Todd genus is a birational invariant within the class of nonsingular complex projective varieties. This is the paper's algebraic-geometric analogue of Novikov's conjecture, replacing higher signatures and homotopy invariance by higher Todd genera and birational invariance; the supplied text states that the paper proves it under the strong Novikov conjecture for the fundamental group.