A complex vector bundle is a vector bundle π:E->M whose fiber bundles π^(-1)(m) are a copy of C^k. π is a holomorphic vector bundle if it is a holomorphic map between complex manifolds and its transition functions are holomorphic. The simplest example is a holomorphic line bundle, where the fiber is simply a copy of C.

complex manifold | Hermitian metric | holomorphic function | holomorphic line bundle | holomorphic tangent bundle | vector bundle