Projective toric generators in the unitary cobordism ring

TL;DR

This paper constructs explicit geometric representatives for the generators of the unitary cobordism ring using smooth projective toric varieties, employing equivariant modifications to control Milnor numbers.

Contribution

It provides a method to realize the generators of the unitary cobordism ring as smooth projective toric varieties, solving a longstanding problem in geometric cobordism theory.

Findings

Constructed geometric representatives for all generators of the cobordism ring.

Developed equivariant modifications that alter Milnor numbers predictably.

Showed that Milnor number changes depend only on dimension and modification index.

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.