A gadget defined for complex vector bundles. The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The ith Chern class is an obstruction to the existence of (n - i + 1) everywhere complex linearly independent vector fields on that vector bundle. The ith Chern class is in the (2i)th cohomology group of the base space.

Chern number | obstruction | Pontryagin class | Stiefel-Whitney class