In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map 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. Definition through trivialization 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. The sheaf of holomorphic sections Let be a holomorphic vector bundle. A local section is said to be holomorphic if, in a neighborhood of each point of , it is holomorphic in some (equivalently any) trivialization. This condition is local, meaning that holomorphic sections form a sheaf on . This sheaf is sometimes denoted , or abusively by . Such a sheaf is always locally free of the same rank as the rank of the vector bundle. If is the trivial line bundle then this sheaf coincides with the structure sheaf of the complex manifold . Basic Examples There are line bundles over whose global sections correspond to homogeneous polynomials of degree (for a positive integer). In particular, corresponds to the trivial line bundle. If we take the covering then we can find charts defined byWe can construct transition functions defined byNow, if we consider the trivial bundle we can form induced transition functions . If we use the coordinate on the fiber, then we can form transition functionsfor any integer . Each of these are associated with a line bundle . Since vector bundles necessarily pull back, any holomorphic submanifold has an associated line bundle , sometimes denoted . Dolbeault operators Suppose is a holomorphic vector bundle. Then there is a distinguished operator defined as follows. In a local trivialisation of , with local frame , any section may be written for some smooth functions . Define an operator locally by where is the regular