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.
Contents
- Definition through trivialization
- The sheaf of holomorphic sections
- The sheaves of forms with values in a holomorphic vector bundle
- Cohomology of holomorphic vector bundles
- The Picard group
- Hermitian metrics on a holomorphic vector bundle
- References
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 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.
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
The sheaves of forms with values in a holomorphic vector bundle
If
These sheaves are fine, meaning that they have partitions of the unity.
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
the space of global holomorphic sections of E. We also have that
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
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,Indeed, if u = (e1, …, en) is a holomorphic frame, then let
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
Let
The curvature Ω appears prominently in the vanishing theorem for higher cohomology of holomorphic vector bundles; e.g., Kodaira's vanishing theorem and Nakano's vanishing theorem.