Number of connected components of an Automorphism group Let $X$ be a smooth quasi-projective irreducible variety over the field of complex numbers $\mathbb{C}$. We denote by $\mathrm{Aut}(X)$ the group of algebraic automorphisms of $X$. Moreover, for a variety $V$, we call a map $V \to \mathrm{Aut}(X)$ a morphism, if the induced map $V \times X \to X$
is a morphism of varieties. The connected component of $\mathrm{Aut}(X)$
of the neutral element $e \in \mathrm{Aut}(X)$ we define by
$$
\mathrm{Aut}(X)^\circ = \left\{ g \in \mathrm{Aut}(X) \Big|
\begin{array}{l} 
\, \textrm{$\exists$
an irreducible variety $V$ and a morphism} \\
\textrm{$V \to \mathrm{Aut}(X)$ s.t. the image contains $g$ and $e$}
\end{array}  \right\} \, .
$$
This notion goes back to Ramanujam, see [Ram64]. Clearly, $\mathrm{Aut}(X)^\circ$ is a normal subgroup of $\mathrm{Aut}(X)$.
I am interested in the size of the group $Q(X) = \mathrm{Aut}(X) / \mathrm{Aut}(X)^\circ$. In case $X$ is projective, then $Q(X)$ is countable. Also in case $X$ is affine,
$Q(X)$ is countable. My question is, whether this is true in general, i.e. whether for all smooth irreducible quasi-projective variety $X$, the group $Q(X)$
is countable. Every proof, counter-example or textbook reference would be perfect.
[Ram64] Ramanujam, C.P., A note on automorphism groups of algebraic varieties, Math. Ann. 156, 25-33 (1964). ZBL0121.16103.
 A: Let $i:X\to \overline{X}$ be any dense open immersion of $X$ in a projective scheme.  For every morphism $\phi:X\to X$, denote by $\overline{\Gamma}_\phi\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ the closure of the graph of $\phi$.  For every choice of ample invertible sheaf $\mathcal{L}$ on $\overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$, there is an associated Hilbert polynomial $P(t)\in \mathbb{Q}[t]$ of $\overline{\Gamma}_\phi$ with respect to $\mathcal{L}$.  For every choice of $P(t)$, denote by $\text{Hilb}^{P(t)}$ the associated Hilbert scheme parameterizing closed subschemes $Z\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ with Hilbert polynomial $P(t)$.  There is an open subscheme $U$ that parameterizes those $Z$ such that both projections $Z \cap (X\times_{\text{Spec}(\mathbb{C})} X) \to X$ are isomorphisms.  As an open subscheme of a projective scheme, $U$ has only finitely many irreducible components.  Also there are only countably many possible numerical polynomials $P(t)$.  Thus, the group $Q(X)$ is finite or countably infinite.
