Two definitions of holomorphic vector bundle

I am stuck in the task of understanding of the following. I am trying to learn about holomorphic vector bundle. So far, I have found two definitions.

Definition 1: One starts with the complex (smooth or topological, does not matter) vector bundle $\pi: E \rightarrow M$ over the complex manifold $M$, assuming there is a local trivialization of $E$ whose transition maps are holomorphic maps.

Definition 2: One again starts with complex vector bundle $\pi: E \rightarrow M$ over the complex manifold $M$, only assuming that there is a complex structure on $E$ making $\pi: E \rightarrow M$ into a holomorphic map of complex manifolds.

The implication D1 $\Rightarrow$ D2 is quite clear, as one can use the local trivialization together with holomorphic atlas on $M$ to define the holomorphic atlas on $E$, such that $\pi$ is locally just a projection.

The definition D1 is quite common. For example, I use the Lectures on Kähler Geometry. On the other hand, in lecture notes by Sean Pohorence, they use D2. In the first reference, they give the equivalence D1 $\Leftrightarrow$ D2 as an excercise.

I am completely stuck in the D2 $\Rightarrow$ D1 part. Excercise 2.0.7. in the second reference hints one to show that based on D2, every local trivialization map $\phi: U \times \mathbb{C}^{k} \rightarrow \pi^{-1}(U)$ is automatically holomorphic. Clearly, this would prove D1. Can someone hint me on this one? Some good reference (at least stating this precisely) would be fine.

Interestingly, there is already some discussion in this topic here, where people suggest that this implication may not be true, which confuses me even more...

With kind regards, Jan Vysoký

• In D2, $E$ is a complex manifold: by definition of complex manifold, $\phi:U\times\mathbb{C}^k\to\pi^{-1}(U)$ is a holomorphic map. – Armando j18eos Jan 14 '18 at 11:51
• I understand that E is a complex manifold, but $\phi$ is by assumption just a diffeomorphism. Why is it a holomorphism? – Jan Vysoky Jan 15 '18 at 8:26