8
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Let $K\subset L$ be a finite galois extension of fields (the case I have in mind is $K=\mathbb{R},L=\mathbb{C}$). Given a scheme $X$ over $K$ by pulling back to $L$ we get a scheme $Y=X\times _K L$ over $L$ with a $K$ - action of $G=\mathrm{Gal}(L/K)$. It is easy to see that this action satisfies the following: There exists an affine open cover $X=\bigcup_{i\in I}U_i $ such that any $\sigma \in G$ takes each $U_i$ to itself. Now let $Y$ be an arbitrary scheme over $L$ with a $K$ - action of $G$ that satisfies the above property (for all of this you can assume whatever finiteness and niceness conditions you want on everything, the main case I have in mind is that of finite type separated schemes over the base fields, not necessarily reduced or affine and if possible not necessarily quasi projective). My questions are the following:

1) What other necessary conditions on the action of $G$ on $Y$ (other then the one I stated) must I assume to have a chance of $Y$ coming from a scheme $X$ over $K$?

2) Are there any simple sufficient conditions for $Y$ to come from such an $X$?

3)What are important examples of schemes $Y$ over $\mathbb{C}$ with a $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$ action not coming from a scheme over $\mathbb{R}$ I should keep in mind? If possible, I want all actions to be anti-holomorphic i.e. inducing anti-holomorphic homomorphisms on the stalks of the analytification. Interesting examples for analytic spaces which don't come from schemes will also be very nice.

Edit: Assume throughout that the structural map $Y\rightarrow \mathrm{Spec(L)}$ is $G$-equivariant to avoid redundancies, thanks to Julian Rosen for pointing this out.

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    $\begingroup$ A necessary condition is that the structure map $Y\to\mathrm{Spec}(L)$ is $\mathrm{Gal}(L/K)$-equivariant. $\endgroup$ – Julian Rosen Aug 4 '17 at 22:00
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    $\begingroup$ Equivariance is also sufficient in the case that $Y$ is quasi-affine. $\endgroup$ – Julian Rosen Aug 4 '17 at 23:01
  • $\begingroup$ I was hoping for a more general answer but I guess this will do, do you have a reference for this? $\endgroup$ – Anonymous Coward Aug 6 '17 at 8:10
  • $\begingroup$ I believe Weil's paper "The field of definition of a variety" is the original reference, and applies in the case $Y$ is quasi-projective (Theorem 2). Another reference is Tag 0246 on the Stacks Project, which states the result for quasi-affine varieties and fpqc coverings ($\mathrm{Spec}(L)\to\mathrm{Spec}(K)$ is an fpqc covering). I also found these seminar notes, which claim that... $\endgroup$ – Julian Rosen Aug 6 '17 at 14:41
  • $\begingroup$ ...(cont.) equivariant structure map + invariant affine open cover is sufficient (p. 8). The page hosting these notes says "Given the nature of the seminar, the topics are rarely given a 'complete' treatmeant and instead presented impressionistically---and often times incorrectly. Use at your own risk." So I am a bit skeptical. $\endgroup$ – Julian Rosen Aug 6 '17 at 14:41

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