Chern class

Summary

Chern classes are characteristic classes of complex vector bundles, with values in cohomology. In algebraic geometry they are defined for algebraic vector bundles, with values in the Chow ring. A vector bundle $E$ of rank $r$ has classes $c_0(E) = 1, c_1(E), \ldots, c_r(E)$, with $c_i$ in degree $i$. They measure how far $E$ is from being trivial, and they enter the Riemann–Roch theorems through the Todd class. For a line bundle, $c_1$ is the class of the corresponding divisor.