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 $\pi:E\to X$ 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.

1 Definition through trivialization
2 The sheaf of holomorphic sections
3 The sheaves of forms with values in a holomorphic vector bundles
4 Cohomology of holomorphic vector bundles
5 The Picard group
6 References

Definition through trivialization

Specifically, one requires that the trivialization maps

$\phi_U\colon \pi^{-1}(U) \to U\times\mathbb C^k$

are biholomorphic maps. This is equivalent to requiring that the transition functions

$t_{UV}\colon U\cap V \to \mathrm{GL}_k\mathbb C$

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 $s�: U \to E_{|U}$ is said to be holomorphic if everywhere on U, it is holomorphic in some (equivalently any) trivialization.

This condition is local, so that holomorphic sections form a sheaf on X, sometimes denoted $\mathcal O(E)$ . If E is the trivial line bundle $\underline{\mathbb C}$ , then this sheaf coincides with the structure sheaf $\mathcal O_X$ of the complex manifold X.

The sheaves of forms with values in a holomorphic vector bundles

If $\mathcal E_X^{p,q}$ denotes the sheaf of $\mathcal C^\infty$ differential forms of type (p,q), then the sheaf $\mathcal E^{p,q}(E)$ of type (p,q) forms with values in E can be defined as the tensor product $\mathcal E_X^{p,q}\otimes E$ . These sheaves are fine, which means that it has partitions of the unity.

A fundamental distinction between smooth and holomorphic vector bundles is that in the latter, there is a canonical differential operator : the Dolbeault operator $\overline \partial�: \mathcal E^{p,q}(E) \to \mathcal E^{p,q%2B1}(E)$ obtained in trivializations.

Cohomology of holomorphic vector bundles

If $E$ is a holomorphic vector bundle of rank $r$ over $X$ , one denotes $\mathcal O(E)$ the sheaf of holomorphic sections of $E$ . Recall that it is a locally free sheaf of rank $r$ over the structure sheaf $\mathcal O_X$ of its base.

The cohomology of the vector bundle is then defined as the sheaf cohomology of $\mathcal O(E)$ .

We have $H^0(X, \mathcal O(E)) = \Gamma (X, \mathcal O(E))$ , the space of global holomorphic sections of E, whereas $H^1(X, \mathcal O(E))$ parametrizes the group of extensions of the trivial line bundle of X by E, that is exact sequences of holomorphic vector bundles $0 \to E \to F \to X \times \mathbb C \to 0$ . For the group structure, see also Baer sum.

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 law the tensor product and inversion given by dualization.

It can be equivalently defined as the first cohomology group $H^1(X, \mathcal O_X^*)$ of the bundle of non-locally zero holomorphic functions.

References

Hazewinkel, Michiel, ed. (2001), "Vector bundle, analytic", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=v/v096400