Higher Chern numbers of toric varieties

It is a well-known fact that the unitary bordism ring is isomorphic to a polynomial ring with countably many multiplicative generators: $\Omega^{U}_{*}\simeq \mathbb{Z}[a_{1},a_{2}\dots]$, ${\rm deg}(a_{i})=2i$. Subject of my talk is a proof of the following fact: there exists a sequence of smooth projective toric varieties giving polynomial generators of the ring $\Omega^{U}_{*}$, $a_{n}=[X^{n}]$, $\dim_{\mathbb{C}} X^{n}=n$. Method of the proof is based on considering a family of equivariant modifications (birational isomorphisms) $B_{k}(X)\to X$ of an arbtrary smooth complex manifold $X$ of complex dimension $n$ ($n\geq 2$, $k=0,\dots,n-2$). These modifications change the Chern number $s_{n}$ in a way depending only on the dimension $n$ and a value of the parameter $k$. In particular, the change does not depend on the manifold $X$. The result is complementary to the results of A. Wilfong '13. The talk is based on a joint work "Projective toric generators in the unitary cobordism ring", '16, with Y. Ustinovsky.