Holomorphic vector bundle

Complex vector bundle on a complex manifold
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.

Let E be a holomorphic vector bundle. A local section 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 , or abusively by 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 then this sheaf coincides with the structure sheaf of the complex manifold X.

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 .

If E is a holomorphic vector bundle, the cohomology of E is defined to be the sheaf cohomology of In particular, we have

the space of global holomorphic sections of E. We also have that parametrizes the group of extensions of the trivial line bundle of X by E, that is, exact sequences of holomorphic vector bundles 0 → E → F → X × 'C' → 0. For the group structure, see also Baer sum as well as sheaf extension. By Dolbeault's theorem, this sheaf cohomology can alternatively be described as the cohomology of the chain complex defined by the sheaves of forms with values in the holomorphic bundle Namely we have

In the context of complex differential geometry, the Picard group Pic(X) of the complex manifold X is the group of isomorphism classes of holomorphic line bundles with group law given by tensor product and inversion given by dualization. It can be equivalently defined as the first cohomology group of the sheaf of non vanishing holomorphic functions.