I am writing because I am extremely confused with the structure of complex vector bundles. Ok first of all I understand that a complex vector bundle is a just a vector bundle $\pi:E\to X$ such that each fiber $E_x$ is a complex vector space. Also, the transition functions $\tau_{ij}: (U_i\cap U_j)\times \mathbb{C}^n\to (U_i\cap U_j)\times \mathbb{C}^n$ are elements of $GL(n, \mathbb{C})$. Moreover, if the transition functions are holomorphic we say that $\pi: E\to X$ is a holomorphic vector bundle over $X$.

Now, for any complex vector bundle of rank $n$ is equivalent to a real vector bundle of rank $2n$ with an endomorphism $I$ such that $I^2=-1$. That's all cool.

My question is this, say we have $E$ a real vector bundle of dimension $2n$ and $J:E\to E$ an almost complex structure. Does that make $E$ a complex vector bundle of dimension $n$? If yes, then how can I multiply with $i$?

Also, if we tensor $E\otimes\mathbb C$ then we have a $2n$ dimensional complex vector bundle, and of course we can multiply by $i$. What is the relation between multiplication by $i$ and $J$??

What do all these have to do with holomorphic vector bundles?!


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