Three 'equivalent' definitions of a holomorphic vector bundle

Question

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.

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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.

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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.

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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!

Answer 1

Consider $E$'s transition functions as a smooth manifold, induced by its transition functions as a complex vector bundle:

$$\tau_{UV}(p, v) = \phi_V \circ \phi_U^{-1}(p, v) = (p, g_{UV}(p) \cdot v)$$

where $p \in U \cap V$ and $v \in \mathbb C^k$. By definition,

• $\pi : E \to M$ is a holomorphic vector bundle in the first sense if the $\phi_U$'s are biholomorphisms.

• $\pi : E \to M$ is a holomorphic vector bundle in the second sense if the $g_{UV}$'s are holomorphic maps.

• $\pi : E \to M$ is a holomorphic vector bundle in the third sense if the $\tau_{UV}$'s are biholomorphisms.

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Let $\mathscr U$ be the open cover of $M$ on which the local trivializations are defined. (WLOG we may assume that each distinguished open $U \in \mathscr U$ is biholomorphic to an open subset of $\mathbb C^n$.) Since $\mathscr F = \{ \phi_U \}_{U \in \mathscr U}$ is a smooth atlas on $M$, the following statements are equivalent:

• $\mathscr F$ is a holomorphic atlas on $M$.

• $\mathscr F$'s charts $\phi_U$ are biholomorphisms.

• $\mathscr F$'s transition functions $\tau_{UV}$ are biholomorphisms.

Hence the first and third definitions are equivalent.

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Since $\tau_{UV}$ is already a diffeomorphism, the following statements are equivalent

• $\tau_{UV}$ is a biholomorphism.

• $\tau_{UV}$ is a holomorphic map.

• $g_{UV}$ is a holomorphic map.

Hence the second and third definitions are equivalent.