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Here is a question that I have thought about recently but I lack the required intuition and knowledge of theorems to see where to even start with this.

  1. $\mathbb{C}$ is a field extension of $\mathbb{R}$ and thus for a scheme X of finite type over $\mathbb R$ we have a diagram $Spec \mathbb C \rightarrow Spec \mathbb R \leftarrow X $ whose pullback yields a scheme Y of finite type (sanity check?) over $\mathbb C$.

By Serre's GAGA this gives us a complex manifold $Y_{an}$.

Is there a nice description of what complex manifolds can be obtained in this way?

Noel Lundström
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  • You may find this related question useful. – jgon Jun 09 '19 at 23:33
  • You are sane, morphisms of finite type are closed under base change so the scheme $Y \longrightarrow Spec \mathbb C$ (which is often denoted $X_{\mathbb C}$ or $\overline X$ as $\mathbb C$ is the algebraic closure of $\mathbb R$) is of finite type. Here's a reference from the stacks project: https://stacks.math.columbia.edu/tag/01T4 – paul blart math cop Jun 10 '19 at 01:59
  • It's just a guess, but I think you'll end up with complexifications of real analytic manifolds. Not all such manifolds will come from this process, just like not all complex manifolds come from schemes over $\mathbb C$. – Gunnar Þór Magnússon Jun 12 '19 at 11:13

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