Smooth Projective Toric Variety Representatives in Complex Cobordism

Abstract

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.