Complex line bundle ($L\to X$)

A complex line bundle is a vector bundle whose fibers are one-dimensional complex vector spaces and whose transition maps are complex linear. Its underlying real bundle has rank two and a canonical orientation, and its Euler class equals its First Chern class.

Every smooth complex line bundle on ${\mathbb{CP}}^1$ is isomorphic to exactly one twisting bundle $\mathcal{O}(k)$. Trivializing on the two standard affine charts leaves a transition function on ${\mathbb{C}}^{\ast }$; its winding number determines $k$, and a partition of unity removes the zero-winding factor.