When we say something to the effect of, "Consider a holomorphic bundle $V$ on a complex manifold $X$...", we are saying that the transition functions of $V$ are holomorphic functions with respect to the specific complex structure on $X$.

But I think this idea is somewhat taken for granted, in courses and books. I would like to see a specific, sufficiently non-trivial, example of a specific vector bundle with specific transition functions that are provably holomorphic with respect to a specific complex structure on a specific manifold, with few if any details spared. (Apologies for the over-use of "specific".)

Can anyone help out? I actually have some familiarity with some of the basic examples, such as the tautological bundle and tangent bundle on projective space, but are there more interesting examples where holomorphic-ness can be checked explicitly?

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    $\begingroup$ I suggest you start by working out the transition functions of the tautological line bundle on $\Bbb CP^n$ (start with $n=1$). $\endgroup$ – Ted Shifrin Dec 28 '14 at 19:14
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    $\begingroup$ I don; t understand what you mean by «taken for granted»; no one is insinuating anything: that's just the definition and you can't be more explicit that that! Just pick any complex manifold and look at the transition functions of its tangent bundle, for example, tensor products of tangent bundles, symmtric or alternating powers, and so on. My guess is that almost every sensible vector bundle you know on a complex manifold is actually holomophic in a canonical way: just go through the examples you know and check this yourself $\endgroup$ – Mariano Suárez-Álvarez Dec 28 '14 at 19:29
  • $\begingroup$ Ah, you edited. In the original version wasn't super clear to me at first that you had worked things out for $\operatorname{Pic}(\mathbb{P}^n)$, $\Omega_X$, $T_X$ yet — sorry to speak as if you hadn't. $\endgroup$ – Hoot Dec 31 '14 at 6:24
  • $\begingroup$ No worries -- I'm glad the question is clearer now. $\endgroup$ – MathsByTheSea Dec 31 '14 at 15:43

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