  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.
