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 : E → X 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.
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.
Let E be a holomorphic vector bundle. A local section s : U → EU 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 , or abusively by 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 then this sheaf coincides with the structure sheaf of the complex manifold X.
There are line bundles over whose global sections correspond to homogeneous polynomials of degree (for a positive integer). In particular, corresponds to the trivial line bundle. If we take the covering then we can find charts defined byWe can construct transition functions defined byNow, if we consider the trivial bundle we can form induced transition functions . If we use the coordinate on the fiber, then we can form transition functionsfor any integer . Each of these are associated with a line bundle .
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This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex
In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. Suppose is a complex manifold of complex dimension and let denote the ring of holomorphic functions on We call a Stein manifold if the following conditions hold: is holomorphically convex, i.
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