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Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex variety $X^\sigma$ in the following way:

1. $X$ and $X^\sigma$ are equal as schemes.
2. If $p:X\longrightarrow\textrm{Spec}\,\mathbb C$ is the structural morphism of $X$, then the structural morphism of $X^\sigma$ is given by $\left(\textrm{Spec}\,\sigma\right)\circ p$.

In this way, the group $\textrm{Aut}(\mathbb C)$ acts on the set of all complex varieties, but $X$ and $X^\sigma$ are in general not isomorphic (is some cases not homeomorphic).

Now suppose that $X$ is a complex surface, and consider the orbit set $$\Omega_X=\{X^\sigma\,:\sigma\in\textrm{Aut}(\mathbb C)\}$$ For which type of $X$ the set $\Omega_X$ contains only a countable number of isomorphism classes? Moreover, can you give me any reference/result that analyzes the orbit set $\Omega_X$?

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex variety $X^\sigma$ in the following way:

1. $X$ and $X^\sigma$ are equal as schemes.
2. If $p:X\longrightarrow\textrm{Spec}\,\mathbb C$ is the structural morphism of $X$, then the structural morphism of $X^\sigma$ is given by $\left(\textrm{Spec}\,\sigma\right)\circ p$.

In this way, the group $\textrm{Aut}(\mathbb C)$ acts on the set of all complex varieties, but $X$ and $X^\sigma$ are in general not isomorphic (is some cases not homeomorphic).

Now suppose that $X$ is a complex surface, and consider the orbit set $$\Omega_X=\{X^\sigma\,:\sigma\in\textrm{Aut}(\mathbb C)\}$$ For which type of $X$ the set $\Omega_X$ contains only a countable number of isomorphism classes? Moreover, can you give me any reference/result that analyzes the orbit set $\Omega_X$?

Thanks in advance.

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex variety $X^\sigma$ in the following way:

1. $X$ and $X^\sigma$ are equal as schemes.
2. If $p:X\longrightarrow\textrm{Spec}\,\mathbb C$ is the structural morphism of $X$, then the structural morphism of $X^\sigma$ is given by $\left(\textrm{Spec}\,\sigma\right)\circ p$.

In this way the group $\textrm{Aut}(\mathbb C)$ acts on the set of all complex varieties, but $X$ and $X^\sigma$ are in general not isomorphic (is some cases not homeomorphic).

Now suppose that $X$ is a complex surface, and consider the orbit set $$\Omega_X=\{X^\sigma\,:\sigma\in\textrm{Aut}(\mathbb C)\}$$ For which type of $X$ the set $\Omega_X$ contains only a countable number of isomorphism classes? Moreover, can you give me any reference/result that analyzes the set $\Omega_X$?

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex variety $X^\sigma$ in the following way:

1. $X$ and $X^\sigma$ are equal as schemes.
2. If $p:X\longrightarrow\textrm{Spec}\,\mathbb C$ is the structural morphism of $X$, then the structural morphism of $X^\sigma$ is given by $\left(\textrm{Spec}\,\sigma\right)\circ p$.

In this way, the group $\textrm{Aut}(\mathbb C)$ acts on the set of all complex varieties, but $X$ and $X^\sigma$ are in general not isomorphic (is some cases not homeomorphic).

Now suppose that $X$ is a complex surface, and consider the orbit set $$\Omega_X=\{X^\sigma\,:\sigma\in\textrm{Aut}(\mathbb C)\}$$ For which type of $X$ the set $\Omega_X$ contains only a countable number of isomorphism classes? Moreover, can you give me any reference/result that analyzes the orbit set $\Omega_X$?