Holomorphic line bundle

Summary

A holomorphic line bundle is a holomorphic vector bundle of rank one. Its transition functions are nowhere vanishing holomorphic functions. So the isomorphism classes of holomorphic line bundles on a complex manifold $X$ form the Picard group $H^1(X, \mathcal{O}_X^*)$ under tensor product. Line bundles encode divisors, maps to projective space and positivity, as with the canonical bundle and the bundles $\mathcal{O}(k)$ on $\mathbb{P}^n$.