Although it is well known that the complex cobordism ring
is isomorphic to the
polynomial ring
,
an explicit description for convenient generators
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.
