Holomorphic vector bundle

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map $\pi:E\to X$ is holomorphic.

Specifically, one requires that the trivialization maps

$\phi_U\colon \pi^{-1}(U) \to U\times\mathbb C^k$

are biholomorphic maps. This is equivalent to requiring that the transition functions

$t_{UV}\colon U\cap V \to \mathrm{GL}_k\mathbb C$

are holomorphic maps.

A holomorphic line bundle is a rank one holomorphic vector bundle.

This geometry-related article is a stub. You can help Wikipedia by expanding it.