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Holomorphic vector bundle : ウィキペディア英語版
Holomorphic vector bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space 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.
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
:\phi_U : \pi^(U) \to U \times \mathbf^k
are biholomorphic maps. This is equivalent to requiring that the transition functions
:t_ : U\cap V \to \mathrm_k(\mathbf)
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.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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