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I'm aware that similar questions have been asked here and here but neither of these seems to have settled my issue. I have the following three definitions of a holomorphic vector bundle (in all cases $M$ is a complex manifold and a complex vector bundle means a smooth vector bundle whose fibres are complex vector spaces):

A holomorphic vector bundle $\pi:E\rightarrow M$ is a complex vector bundle together with the structure of a complex manifold on $E$ for which there exist trivialisations \begin{equation} \phi_U:\pi^{-1}(U)\rightarrow U\times\mathbb{C}^k \end{equation} which are biholomorphic maps of complex manifolds.


A holomorphic vector bundle $\pi:E\rightarrow M$ is a complex vector bundle for which the transition functions $g_{\alpha\beta}:U_\alpha\cap U_\beta \rightarrow GL_k(\mathbb{C})$ are holomorphic maps of complex manifolds.


A holomorphic vector bundle $\pi:E\rightarrow M$ is a complex vector bundle together with the structure of a complex manifold on $E$ for which the projection $\pi:E\rightarrow M$ is a holomorphic map of complex manifolds.


The wikipedia article here seems to merge all three definitions into one! Whereas Griffiths' and Harris' book goes with the first definition. I am unsure how to see whether these definitions are equivalent, or whether this is actually the case.

Moreover, I am actually trying to prove that the holomorphic tangent bundle $T^{1,0}M$ is indeed a holomorphic vector bundle - although I have no idea where to start with this, with any of the three definitions... any help on either of these problems would be much appreciated!

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