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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]] <math>\pi:E\to X</math> 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
:<math>\phi_U\colon \pi^{-1}(U) \to U\times\mathbb C^k</math>
are [[biholomorphic map]]s. This is equivalent to requiring that the [[transition map|transition function]]s
:<math>t_{UV}\colon U\cap V \to \mathrm{GL}_k\mathbb C</math>
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'' <math>s : U \to E_{|U}</math> 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 (mathematics)|sheaf]] on ''X'', sometimes denoted <math>\mathcal O(E)</math>. If ''E'' is the trivial line bundle <math>\underline{\mathbb C}</math>, then this sheaf coincides with the [[structure sheaf]] <math>\mathcal O_X</math> of the complex manifold X.
 
==The sheaves of forms with values in a holomorphic vector bundle==
 
If <math>\mathcal E_X^{p,q}</math> denotes the sheaf of <math>\mathcal C^\infty</math> differential forms of type ''(p,q)'', then the sheaf <math>\mathcal E^{p,q}(E)</math> of type ''(p,q)'' forms with values in ''E'' can be defined as the [[tensor product]] <math>\mathcal E_X^{p,q}\otimes E</math>. These sheaves are [[fine sheaf|fine]], which means that it has [[partition of unity|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]]''' <math>\overline \partial : \mathcal E^{p,q}(E) \to \mathcal E^{p,q+1}(E) </math> obtained in trivializations.
 
==Cohomology of holomorphic vector bundles==
 
If <math>E</math> is a holomorphic vector bundle of rank <math>r</math> over <math>X</math>, one denotes <math>\mathcal O(E)</math> the [[sheaf (mathematics)|sheaf]] of holomorphic sections of <math>E</math>. Recall that it is a [[locally free sheaf]] of rank <math>r</math> over the [[structure sheaf]] <math>\mathcal O_X</math> of its base.
 
The cohomology of the vector bundle is then defined as the [[sheaf cohomology]] of <math>\mathcal O(E)</math>.
 
We have <math>H^0(X, \mathcal O(E)) = \Gamma (X, \mathcal O(E))</math>, the space of global holomorphic sections of ''E'', whereas <math>H^1(X, \mathcal O(E))</math> parametrizes the group of extensions of the trivial line bundle of ''X'' by ''E'', that is [[exact sequences]] of holomorphic vector bundles <math>0 \to E \to F \to X \times \mathbb C \to 0</math>. 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 <math>H^1(X, \mathcal O_X^*)</math> of the bundle of non-locally zero holomorphic functions.
 
==References==
*{{Springer|id=v/v096400|title=Vector bundle, analytic}}
 
[[Category:Vector bundles]]
[[Category:Complex manifolds]]

Revision as of 09:37, 28 January 2014

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 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

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 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 . If E is the trivial line bundle , then this sheaf coincides with the structure sheaf of the complex manifold X.

The sheaves of forms with values in a holomorphic vector bundle

If denotes the sheaf of 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 . 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 obtained in trivializations.

Cohomology of holomorphic vector bundles

If is a holomorphic vector bundle of rank over , one denotes the sheaf of holomorphic sections of . Recall that it is a locally free sheaf of rank over the structure sheaf of its base.

The cohomology of the vector bundle is then defined as the sheaf cohomology of .

We have , the space of global holomorphic sections of E, whereas parametrizes the group of extensions of the trivial line bundle of X by E, that is exact sequences of holomorphic vector bundles . 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 of the bundle of non-locally zero holomorphic functions.

References

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