Holomorphic vector bundle: Difference between revisions
AHusain314 (talk | contribs) m Undid revision 592980202 by 138.51.31.39 (talk) It would be called "holomorphic" this seems to imply it is called "holomorphic everywhere on U" |
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==The sheaf of holomorphic sections== |
==The sheaf of holomorphic sections== |
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Let ''E'' be a holomorphic vector bundle. A ''local section'' <math>s : U \to E_{|U}</math> is said to be '''holomorphic''' '''everywhere''' on ''U'', |
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. |
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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. |
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. |
Revision as of 09:11, 31 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
- "Vector bundle, analytic", Encyclopedia of Mathematics, EMS Press, 2001 [1994]