Abstract:
According to Milnor and Novikov's classical result, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: $\Omega^U_*\simeq \mathbb{Z}[a_1,a_2,\dots]$, $\deg(a_i)=2i$. In this paper we solve the well-known problem of constructing geometric representatives for the $a_i$ among smooth projective toric varieties, $a_n=[X^{n}]$, $\dim_\mathbb{C} X^{n}=n$. Our proof uses a family of equivariant modifications (birational isomorphisms) $B_k(X)\to X$ of an arbitrary complex manifold $X$ of complex dimension $n$ ($n\geqslant 2$, $k=0,\dots,n-2$). The key fact is that the change of the Milnor number under these modifications depends only on the dimension $n$ and the number $k$ and does not depend on the manifold $X$ itself.
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