Grauert-Remmert is probably irrelevant for this simple case.

Thanks a lot Steven for the link. So do you know of a simple proof that there exists only one real analytic structure on $\mathbb{R}$ which is compatible with its smooth structure?

So know very little about deformation theory but in my setting do we have the existence of a miniversal deformation $g:Y\rightarrow FormalSpec(R)$ which reduces modulo the maximal ideal of $R$ to $X_0\rightarrow Spec(\overline{\mathbb{Q}})$ Because if we have such a miniversal deformation then I think that this is good enough. So in otherwords, there is no need to have a moduli space!

No, I used the word geometry in a more naive sense, i.e., understanding enough about the action of the Weyl group on the root system $\Delta$ in order to deduce the Weyl integral formula.

Thnaks a lot for your nice set of notes :)

Thanks Tom, yes if you don't take the involution then this is the classical example of an Hopf surface!

Yes, I just worked it out for myself. I had forgotten about the universal coefficient theorem for chain complexes. I also just amused myself to construct explicitly all the $n$ extensions in $Ext(\mathbb{Z}/n,\mathbb{Z})$.