Chern connection

Summary

The Chern connection is a canonical connection on a holomorphic vector bundle with a Hermitian metric. It is the unique connection that preserves the metric and whose $(0,1)$-part is the $\bar\partial$-operator of the bundle. It turns the pair (holomorphic structure, metric) into curvature, a $(1,1)$-form with values in endomorphisms. Its curvature is the basis of positivity notions, of vanishing theorems such as the Kodaira vanishing theorem, and of Chern–Weil representatives of Chern classes.