  A general problem in complex cobordism theory is to find useful
representatives for cobordism classes. One particularly convenient class of
complex manifolds consists of smooth projective toric varieties. The bijective
correspondence between these varieties and smooth polytopes allows us to
examine which complex cobordism classes contain a smooth projective toric
variety by studying the combinatorics of polytopes. These combinatorial
properties determine obstructions to a complex cobordism class containing a
smooth projective toric variety. However, the obstructions are only necessary
conditions, and the actual distribution of smooth projective toric varieties in
complex cobordism appears to be quite complicated. The techniques used here
provide descriptions of smooth projective toric varieties in low-dimensional
cobordism.
