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 : EX 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.

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 : UE|U is said to be holomorphic if, in a neighborhood of each point of U, it is holomorphic in some (equivalently any) trivialization.

By an application of the Newlander-Nirenberg theorem, one obtains a converse to the construction of the Dolbeault operator of a holomorphic bundle:[1]

Dolbeault operator has local inverse in terms of homotopy operator.[2]

These sheaves are fine, meaning that they admit partitions of unity. A fundamental distinction between smooth and holomorphic vector bundles is that in the latter, there is a canonical differential operator, given by the Dolbeault operator defined above:

Let E be a holomorphic vector bundle on a complex manifold M and suppose there is a hermitian metric on E; that is, fibers Ex are equipped with inner products <·,·> that vary smoothly. Then there exists a unique connection ∇ on E that is compatible with both complex structure and metric structure, called the Chern connection; that is, ∇ is a connection such that

If u' = ug is another frame with a holomorphic change of basis g, then

The curvature Ω appears prominently in the vanishing theorems for higher cohomology of holomorphic vector bundles; e.g., Kodaira's vanishing theorem and Nakano's vanishing theorem.