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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.
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| ==Definition through trivialization==
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| Specifically, one requires that the trivialization maps
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| :<math>\phi_U\colon \pi^{-1}(U) \to U\times\mathbb C^k</math>
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| are [[biholomorphic map]]s. This is equivalent to requiring that the [[transition map|transition function]]s
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| :<math>t_{UV}\colon U\cap V \to \mathrm{GL}_k\mathbb C</math>
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| 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.
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| ==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''' 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.
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| ==The sheaves of forms with values in a holomorphic vector bundle==
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| 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]].
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| 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.
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| ==Cohomology of holomorphic vector bundles==
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| 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.
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| The cohomology of the vector bundle is then defined as the [[sheaf cohomology]] of <math>\mathcal O(E)</math>.
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| 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]].
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| ==The Picard group==
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| 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.
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| 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.
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| ==References==
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| *{{Springer|id=v/v096400|title=Vector bundle, analytic}}
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| [[Category:Vector bundles]]
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| [[Category:Complex manifolds]]
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