A Gadget defined for Complex Vector Bundles. The Chern classesof a Complex Manifold are the Chern classes of its Tangent Bundle. The th Chernclass is an Obstruction to the existence of everywhere Complex linearly independentVector Fields on that Vector Bundle. The th Chern class is in the thcohomology group of the base Space.
See also Obstruction, Pontryagin Class, Stiefel-Whitney Class