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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

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

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 bundle

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 ${\overline {\partial }}:{\mathcal {E}}^{p,q}(E)\to {\mathcal {E}}^{p,q+1}(E)$ 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 $H^{0}(X,{\mathcal {O}}(E))=\Gamma (X,{\mathcal {O}}(E))$, 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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