Holomorphic vector bundle

Summary

A holomorphic vector bundle is a complex vector bundle over a complex manifold. Its total space is a complex manifold, and its local trivializations are holomorphic. Equivalently, its transition functions are holomorphic maps into $GL_r(\C)$. Holomorphic vector bundles carry sheaves of holomorphic sections, and their cohomology is a main tool of complex geometry. Examples are the holomorphic tangent bundle, the bundles of holomorphic forms, and the line bundles $\mathcal{O}(k)$ on projective space.