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The word "rank" is used in several places in the article, but it is neither defined nor linked to its definition. I hope someone knowledgeable on this...

complex vector bundles (i.e. ( 0 , 1 ) {\displaystyle (0,1)} -connections, which are intimately related to holomorphic vector bundles), as well as potentially...

the top exterior power of the dual of the holomorphic tangent space). Notice that this is a complex line bundle, hence each fiber is a 2-dimensional real...

to call TM+ a holomorphic vector bundle when M is only an almost complex manifold. In what sense in the projection map TM+ → M holomorphic? Is this naming...

Ser. A 58, 158-162 (1982; Zbl 0538.32021)] that stability of a holomorphic vector bundle should be a necessary and sufficient condition for the existence...

01:53, 5 May 2016 (UTC) "Algebraic vector bundle" redirects here. BTotaro (talk) 17:24, 20 May 2016 (UTC) "Holomorphic sheaf" redirects here. BTotaro (talk)...

connection with complex geometry (e.g. holomorphic maps on a Riemann surface, holomorphic sections of a hermitian vector bundle, etc.). One could also make a comment...

G-torsors (principal bundles) on X. Easy example: isomorphism classes of (holomorphic) C ∗ {\displaystyle \mathbb {C} ^{*}} -principal bundles on a complex manifold...

For the reason to describe holomorphic bundles it may be something unavoidable　to describe the definitions of line bundles and divisors. However, the...

More generally, the result is that topologically trivial vector bundles are holomorphically trivial. — Preceding unsigned comment added by...

"Coherent cohomology", "holomorphic Euler characteristic", and "Serre's vanishing theorem" redirect here. BTotaro (talk) 17:19, 20 May 2016 (UTC) This...

holomorphic) contrasting them with a non-example (such as bounded functions). As for vector fields, I would present the example with a general vector...

detailed should this page be? for instance tangent bundle is given a cursory definition, but tangent vectors are not defined as far as I can tell. I will continue...

smooth line bundles. A holomorphic line bundle can have vanishing first Chern class (i.e. it is smoothly trivial) without being holomorphically trivial....

tensors on a general vector space, but these are promptly then confined to tangent spaces at a point (usually followed by the tangent bundle, sometimes by its...

whether it's stated for coherent sheaves on proper schemes or, say, holomorphic vector bundles on complex projective manifolds. This type of definition is used...

conformal maps which are not linear). So we shouldn't expect to get a vector bundle at all. Thank you, that cleared up some of these things a bit for me...

affine space (which admits a vector space model) and an affine bundleis is confusing; just as calling a vector bundle a vector space is confusing, even if...

Hamiltonian mechanics. A complex manifold is a manifold modeled on Cn with holomorphic transition functions on chart overlaps. These manifolds are the basic...

the coordinate transformation – the distinction is significant for non-holomorphic bases) pushes one into the concept of densities as defined in this article...