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I'm trying to understand how to view complex vector bundles as real vector bundles.

I am confused about what is going on in the wikipedia page for complex vector bundles. Here is a screenshot:

enter image description here In particular I am confused about the sentence:

If $E$ is a complex vector bundle, then the complex structure $J$ can be defined by setting $J_x$ to be the scalar multiplication by $i$.

Here $E$ is a complex vector bundle, but $J$ is defined on real vector bundles. What should be the domain and target of $J_x$ in this case?

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  • $\begingroup$ What does $\Re(E_x)$ mean? $\endgroup$
    – Eric Wofsey
    Commented Sep 7, 2019 at 22:20
  • $\begingroup$ I intended it to mean the real part of $E_x$. I should have been more precise. I hope it is clear that $E_x$ is the fibre of $E$ at $x$ $\endgroup$
    – T. Stark
    Commented Sep 7, 2019 at 22:23
  • $\begingroup$ What does "real part" mean? $\endgroup$
    – Eric Wofsey
    Commented Sep 7, 2019 at 22:23
  • $\begingroup$ Opps, I assumed we could take the real part of a complex vector space similarly to how we can take the real part of a complex number, I realise now that that does not work. $\endgroup$
    – T. Stark
    Commented Sep 7, 2019 at 22:27

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Any complex vector space is also a real vector space, by just restricting the scalar multiplication operation to just real scalars. In the same way, any complex vector bundle is also a real vector bundle, by considering each fiber as a real vector space. (To see that this bundle of real vector spaces is locally trivial, note that any local trivialization of the complex vector bundle also gives a local trivialization as a real vector bundle by identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ in the obvious way.)

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  • $\begingroup$ Where does the complex structure mentioned in the wikipedia article come into play here? If we only consider real scalars then we are losing some of the structure of the complex vector bundle. $\endgroup$
    – T. Stark
    Commented Sep 7, 2019 at 22:31
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    $\begingroup$ The complex structure is exactly the additional data you can use to recover the original complex vector bundle from the real vector bundle, since it tells you how to multiply by $i$. $\endgroup$
    – Eric Wofsey
    Commented Sep 7, 2019 at 22:32
  • $\begingroup$ I am still a little confused on how this all fits together. Is this correct. Given a complex vector bundle $E$, let $E_{\mathbb{R}}$ denote the real vector space obtained by restricting the scalar multiplication operation to real scalars. Then we define the complex structure as $J:E_{\mathbb{R}}\rightarrow E_{\mathbb{R}}$? $\endgroup$
    – T. Stark
    Commented Sep 7, 2019 at 22:42
  • $\begingroup$ $E_\mathbb{R}$ is a real vector bundle, not just a real vector space. But yes, $J:E_\mathbb{R}\to E_\mathbb{R}$. You will hardly ever see anyone bother to explicitly distinguish $E$ and $E_\mathbb{R}$ though. $\endgroup$
    – Eric Wofsey
    Commented Sep 7, 2019 at 22:45
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    $\begingroup$ There is nothing at all mysterious about this map $J$: it's just the map which is scalar multiplication by $i$ in each fiber (using the original complex vector space structure on each fiber). $\endgroup$
    – Eric Wofsey
    Commented Sep 7, 2019 at 22:46

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