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?