# 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* → *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*.

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* → *E*|_{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 *E*_{x} 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.