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 $π : E \to X$ is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.

By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety X (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e., locally free sheaves of finite rank) on X.

Specifically, one requires that the trivialization maps

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

are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guaranteed by the remark that the derivative (in the appropriate sense) of a vector-valued holomorphic function is itself holomorphic.

Let $E$ be a holomorphic vector bundle. A local section $s : U \to E | U$ is said to be holomorphic if, in a neighborhood of each point of $U$ , it is holomorphic in some (equivalently any) trivialization.

This condition is local, meaning that holomorphic sections form a sheaf on $X$ . This sheaf is sometimes denoted ${\mathcal {O}}(E).$ Such a sheaf is always locally free of the same rank as the rank of the vector bundle. If $E$ is the trivial line bundle ${\underline {\mathbf {C} }},$ then this sheaf coincides with the structure sheaf ${\mathcal {O}}_{X}$ of the complex manifold $X$ .

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