# Three 'equivalent' definitions of a holomorphic vector bundle

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 $$\phi_U:\pi^{-1}(U)\rightarrow U\times\mathbb{C}^k$$ 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!

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.

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.

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.