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(Probably some basic question, but I've never worked in the real world.)

Let $X\subset\mathbb{P}^n_\mathbb{C}$ be a complex variety with the complex conjugation $\tau:X\to X$. 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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    $\begingroup$ You need to assume that $X$ is defined over the reals, and likewise for $F$. Otherwise, there's no way to get a conjugation on $X$ or $F$. $\endgroup$
    – Donu Arapura
    Aug 16, 2010 at 20:22

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Perhaps my comment was a bit too cryptic, so let me expand it slighty. If $X\subset \mathbb{P}^n_\mathbb{C}=\mathbb{P}$ is defined by real polynomials, then conjugation $\tau:[x_0,\ldots, x_n]\mapsto [\bar x_0,\ldots, \bar x_n]$ induces an action on $X$. The story for a general coherent sheaf $F$ is similar. It can always be given as the cokernel of a matrix $\oplus \mathcal{O}_{\mathbb{P}}(a_i)\to \oplus \mathcal{O}_{\mathbb{P}}(b_j)$. In order to get a natural action of $\tau$ on $F$, which is an isomorphism $\tau^*F\cong F$ with "square" equaling the identity, it would be enough to assume that some presentation matrix is given by real polynomials. To put it more canonically, the pair $(X,F)$ should be obtained by base change from a pair defined over $Spec\mathbb{R}$.

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  • $\begingroup$ Thanks for the reply. Of course I meant X is stabilized by $\tau$. (It implies that X is defined over the reals, right?) If a presentation is given one can define the complex conjugation. But the conjugation can be defined in various ways. Apparently they are all compatible, but probably this should be proved? Could you recommend some short intro/summary? $\endgroup$
    – Dmitry Kerner
    Aug 17, 2010 at 0:45
  • $\begingroup$ I wish I could recommend a reference, but I'm not sure of one. Perhaps someone else can help? $\endgroup$
    – Donu Arapura
    Aug 17, 2010 at 1:01
  • $\begingroup$ F. Catanese wrote some papers on this subject. For instance "Moduli spaces of surfaces and real structures", Ann. Math. 158. But I do not remember whether he considers also the case of coherent sheaves. One can find all of them on arXiv. $\endgroup$
    – Francesco Polizzi
    Aug 17, 2010 at 8:27
  • $\begingroup$ Does this answer imply, by choosing $\bigoplus \mathcal{O}_{\mathbb{P}}(a_i)=0$, that on $\mathbb{P}^1$ we have $\tau^* \mathcal{O}(2) \simeq \mathcal{O}(2)$, where $\tau$ is complex conjugation? That doesn't seem right, because $d \tau$ defines an isomorphism $T \mathbb{P}^1 = \overline{T \mathbb{P}^1}=(T \mathbb{P}^1)^*$, i.e. an isomorphism from $\mathcal{O}(2)$ to $\mathcal{O}(-2)$. Also, $\tau$ does not preserve $c_1(\mathcal{O}(2))$. $\endgroup$
    – user505117
    Feb 11, 2022 at 11:39

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