Projective toric generators in the unitary cobordism ring

Abstract

By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: \(\Omega^U_*\simeq \mathbb Z[a_1,a_2,\dots]\), \({\rm deg}(a_i)=2i\). In this paper we solve a well-known problem of constructing geometric representatives for \(a_i\) 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 smooth complex manifold \(X\) of (complex) dimension \(n\) (\(n\geq 2\), \(k=0,\dots,n-2\)). 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.