6
$\begingroup$

After doing a bit of playing around (I think) I was able to show that the map $\operatorname{id}\otimes\ \psi : \Omega^{p,q}(X, E) \to \Omega^{p,q}(X, \bar{E})$, where $\psi $ is the conjugation map $E \to \bar{E}$, is well-defined for a complex vector bundle $E$ provided that $E$ has real transition functions. The trivial complex bundle has this property, but are there others? If so, what can we say about them?


Added later: Alex has given some examples in the comments below so I am left with the following question.

Can we classify complex vector bundles with real transition functions?

Classify may be to strong a requirement. What I am after is either a name for this class of bundles or some properties which demonstrate how few there are. Compare this to the constant transition functions scenario in which case the bundles are flat and we can extend the exterior derivative to $\Omega^{\bullet}(X, E)$ by $d\otimes\operatorname{id}_E$.

  • 0
    Complex vector bundles with real transition functions are common. Take any real vector bundle and tensor it with the trivial complex line bundle, so that the fibers are complex vector spaces with the same basis as the real vector space.2014-05-10

0 Answers 0