Holomorphic vector bundle

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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 π : EX 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

${\displaystyle \phi _{U}:\pi ^{-1}(U)\to U\times \mathbf {C} ^{k}}$

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

${\displaystyle t_{UV}:U\cap V\to \mathrm {GL} _{k}(\mathbf {C} )}$

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 : UE|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 ${\displaystyle {\mathcal {O}}(E).}$ 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 ${\displaystyle {\underline {\mathbf {C} }},}$ then this sheaf coincides with the structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$ of the complex manifold X.

The sheaves of forms with values in a holomorphic vector bundle

If ${\displaystyle {\mathcal {E}}_{X}^{p,q}}$ 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

${\displaystyle {\mathcal {E}}^{p,q}(E)\triangleq {\mathcal {E}}_{X}^{p,q}\otimes E.}$

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:

${\displaystyle {\overline {\partial }}:{\mathcal {E}}^{p,q}(E)\to {\mathcal {E}}^{p,q+1}(E).}$

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 ${\displaystyle {\mathcal {O}}(E)}$. In particular, we have

${\displaystyle H^{0}(X,{\mathcal {O}}(E))=\Gamma (X,{\mathcal {O}}(E)),}$

the space of global holomorphic sections of E. We also have that ${\displaystyle H^{1}(X,{\mathcal {O}}(E))}$ parametrizes the group of extensions of the trivial line bundle of X by E, that is, exact sequences of holomorphic vector bundles 0 → EFX × 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 ${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})}$ 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, ${\displaystyle p\nabla s={\bar {\partial }}s}$ 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,
${\displaystyle X\cdot \langle s,t\rangle =\langle \nabla _{X}s,t\rangle +\langle s,\nabla _{X}t\rangle }$
where we wrote ${\displaystyle \nabla _{X}s}$ for the contraction of ${\displaystyle \nabla s}$ by X. (This is equivalent to saying that the parallel transport by ∇ preserves the metric <·,·>.)

Indeed, if u = (e1, …, en) is a holomorphic frame, then let ${\displaystyle h_{ij}=\langle e_{i},e_{j}\rangle }$ and define ωu by the equation ${\displaystyle \sum h_{ik}\,{(\omega _{u})}_{j}^{k}=\partial h_{ij}}$, which we write more simply as:

${\displaystyle \omega _{u}=h^{-1}\partial h.}$

If u' = ug is another frame with a holomorphic change of basis g, then

${\displaystyle \omega _{u'}=g^{-1}dg+g\omega _{u}g^{-1},}$

and so ω is indeed a connection form, giving rise to ∇ by ∇s = ds + ω · s. Now, since ${\displaystyle {\overline {\omega }}^{T}={\overline {\partial }}h\cdot h^{-1}}$,

${\displaystyle d\langle e_{i},e_{j}\rangle =\partial h_{ij}+{\overline {\partial }}h_{ij}=\langle {\omega }_{i}^{k}e_{k},e_{j}\rangle +\langle e_{i},{\omega }_{j}^{k}e_{k}\rangle =\langle \nabla e_{i},e_{j}\rangle +\langle e_{i},\nabla e_{j}\rangle .}$

That is, ∇ is compatible with metric structure. Finally, since ω is a (1, 0)-form, the (0, 1)-component of ${\displaystyle \nabla s}$ is ${\displaystyle {\bar {\partial }}s}$.

Let ${\displaystyle \Omega =d\omega +\omega \wedge \omega }$ be the curvature form of ∇. Since ${\displaystyle p\nabla ={\bar {\partial }}}$ squares to zero, Ω has no (0, 2)-component and since Ω is easily shown to be skew-hermitian,[1] it also has no (2, 0)-component. Consequently, Ω is a (1, 1)-form given by

${\displaystyle \Omega ={\bar {\partial }}\omega .}$

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.

Notes

1. ^ 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.