Holomorphic vector bundle


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 is equivalent to the category of algebraic vector bundles 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 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 trivialization.
This condition is local, meaning that holomorphic sections form a sheaf on. This sheaf is sometimes denoted 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 . In particular, corresponds to the trivial line bundle. If we take the covering then we can find charts defined by
We can construct transition functions defined by
Now, 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 functions
for 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 distinguised 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 Cauchy-Riemann operator of the base manifold. This operator is well-defined on all of because on an overlap of two trivialisations with holomorphic transition function, if where is a local frame for on, then, and so
because the transition functions are holomorphic. This leads to the following definition: A [|Dolbeault operator] on a smooth complex vector bundle is an -linear operator
such that
By an application of the Newlander-Nirenberg theorem, one obtains a converse to the construction of the Dolbeault operator of a holomorphic bundle:
Theorem: Given a Dolbeault operator on a smooth complex vector bundle, there is a unique holomorphic structure on such that is the associated Dolbeault operator as constructed above.

With respect to the holomorphic structure induced by a Dolbeault operator, a smooth section is holomorphic if and only if. This is similar morally to the definition of a smooth or complex manifold as a ringed space. Namely, it is enough to specify which functions on a topological manifold are smooth or complex, in order to imbue it with a smooth or complex structure.
Dolbeault operator has local inverse in terms of homotopy operator.

The sheaves of forms with values in a holomorphic vector bundle

If denotes the sheaf of differential forms of type, then the sheaf of type forms with values in can be defined as the tensor product
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:

Cohomology of holomorphic vector bundles

If is a holomorphic vector bundle, the cohomology of is defined to be the sheaf cohomology of. In particular, we have
the space of global holomorphic sections of. We also have that parametrizes the group of extensions of the trivial line bundle of by, that is, exact sequences of holomorphic vector bundles. 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

The Picard group

In the context of complex differential geometry, the Picard group of the complex manifold 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.

Hermitian metrics on a holomorphic vector bundle

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
Indeed, if u = is a holomorphic frame, then let and define ωu by the equation, which we write more simply as:
If u' = ug is another frame with a holomorphic change of basis g, then
and so ω is indeed a connection form, giving rise to ∇ by ∇s = ds + ω · s. Now, since,
That is, ∇ is compatible with metric structure. Finally, since ω is a -form, the -component of is.
Let be the curvature form of ∇. Since squares to zero by the definition of a Dolbeault operator, Ω has no -component and since Ω is easily shown to be skew-hermitian, it also has no -component. Consequently, Ω is a -form given by
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.