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Q:
Embedding of schemes in non-algebraic varieties
Let $k$ be an algebraically closed field and let $X$ be a non-algebraic $k$-scheme.
Suppose that we have found a closed subscheme $S \subset X$ such that the underlying topological space of $S$ is a manifold.
My question is:
Can we always find an open subscheme $U \subset S$ and a closed subscheme $T \subset U$ such that $S \setminus T$ is a $\mathbb{Z}$-motive of Kähler type over $k$?
More precisely:
Let $C$ be a complex manifold.
A $\mathbb{Z}$-motive of Kähler type is a closed subscheme $Z \subset C$ such that the underlying space of $Z$ is $\mathbb{Z}$-torsion.
So, if $Z$ is of Kähler type, we can always find a Zariski open subscheme $U \subset Z$ and a closed subscheme $T \subset U$ such that $Z \setminus T$ is a $\mathbb{Z}$-motive.
My question is: can we always find an open subscheme $U \subset S$ and a closed subscheme $T \subset U$ such that $S \setminus T$ is a $\mathbb{Z}$-motive?
Any suggestion will be appreciated.
A:
Let $X$ be a smooth variety. Suppose $H^1(X_{\mathrm{red}}, \mathbb Z)$ has infinitely generated dual. Then any non-empty closed subscheme of $X$ is a $\mathbb Z$-motive. Indeed, note that if $Y$ is a $\mathbb Z$-submotive of $X$ and $V$ is a dense open subscheme of $Y$, then $Y\setminus V$ is $\mathbb Z
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