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.
Definition through trivialization
Specifically, one requires that the trivialization maps
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 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.
This condition is local, meaning that holomorphic sections form a sheaf on X. This sheaf is sometimes denoted . 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.
The sheaves of forms with values in a holomorphic vector bundle
If denotes the sheaf of C∞ differential forms of type (p, q), then the sheaf of type (p, q) forms with values in E can be defined as the tensor product
A fundamental distinction between smooth and holomorphic vector bundles is that in the latter, there is a canonical differential operator called the Dolbeault operator:
It is obtained by taking antiholomorphic derivatives in local coordinates.
Cohomology of holomorphic vector bundles
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.
The Picard group
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.
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; that is, ∇ is a connection such that
- (1) For any smooth sections s of E, where p takes the (0, 1)-component of an E-valued 1-form.
- (2) For any smooth sections s, t of E and a vector field X on M,
Indeed, if u = (e1, …, en) 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 (1, 0)-form, the (0, 1)-component of is .
Let be the curvature form of ∇. Since squares to zero, Ω has no (0, 2)-component and since Ω is easily shown to be skew-hermitian, it also has no (2, 0)-component. Consequently, Ω is a (1, 1)-form given by
- For example, the existence of a Hermitian metric on E means the structure group of the frame bundle can be reduced to the unitary group and Ω has values in the Lie algebra of this unitary group, which consists of skew-hermitian metrices.
- Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523
- Hazewinkel, Michiel, ed. (2001), "Vector bundle, analytic", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4