Toric Polynomial Generators of Complex Cobordism

TL;DR

This paper constructs explicit polynomial generators for the complex cobordism ring using smooth projective toric varieties, providing a combinatorial approach to understanding its structure in various dimensions.

Contribution

It introduces a method to explicitly construct complex cobordism generators in many dimensions via toric varieties and torus-equivariant blow-ups.

Findings

Generators constructed in all odd dimensions and dimensions one less than a prime power

Toric varieties serve as convenient algebraic models for cobordism generators

Evidence suggests possible extension to all remaining dimensions

Abstract

Although it is well-known that the complex cobordism ring is a polynomial ring $\Omega_{*}^{U}\cong\mathbb{Z}\left[\alpha_{1},\alpha_{2},\ldots\right]$, an explicit description for convenient generators $\alpha_{1},\alpha_{2},\ldots$ has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blow-ups to a special class of smooth projective toric varieties, such generators can be constructed in every complex dimension that is odd or one less than a prime power. A large amount of evidence suggests that smooth projective toric varieties can serve as polynomial generators in the remaining dimensions as well.