(Probably some basic question, but I've never worked in the real world.)
Let $Xsubsetmathbb{P}^n_mathbb{C}$ be a complex variety with the complex conjugation $tau:Xto X$. So $tau$ acts on $mathcal{O}_X(k)$ too.
Suppose $F$ is a sheaf of modules with prescribed embedding of modules of its local sections: $F(U)subset mathcal{O}^{oplus d}(U)$.
The complex conjugation acts on $mathcal{O}^{oplus d}(U)$, hence the images of $F(U)$ are defined. Hence the image of $F$ too.
Now should check that this is compatible with exact sequences etc...
Other ways to define the action of complex conjugation?
References?
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