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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.

Definition through trivialization

Specifically, one requires that the trivialization maps

${\varphi }_U:{\pi }^{-1}(U)\to U\times {\mathbb{ℂ}}^k$

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

$t_{UV}:U\cap V\to {\mathrm{G}\mathrm{L}}_k\mathbb{ℂ}$

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 $\underset{\_}{\mathbb{ℂ}}$, 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 bundle

If ${\mathcal{ℰ}}_X^{p,q}$ denotes the sheaf of ${\mathcal{𝒞}}^{\mathrm{\infty }}$ 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 $\bar{\partial }:{\mathcal{ℰ}}^{p,q}(E)\to {\mathcal{ℰ}}^{p,q+1}(E)$ 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 $H^0(X,\mathcal{𝒪}(E))=\mathrm{\Gamma }(X,\mathcal{𝒪}(E))$, 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

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