Positive line bundle

Summary

A holomorphic line bundle is positive when it has a Hermitian metric whose Chern connection has positive curvature. Then $i$ times the curvature is a Kähler form, so a compact complex manifold with a positive line bundle is a Kähler manifold. By the Kodaira embedding theorem, on a compact complex manifold the positive line bundles are exactly the ample line bundles. Positivity is the analytic hypothesis of the Kodaira and Kodaira–Nakano vanishing theorems.