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In Hartshorne's book 《Algebraic Geometry》 p.443, the author introduces a construction of a non-projective complex manifold from a projective one. His method can be summarized as following:
Let $X$ be a projective complex threefold, take two non-singular curves $c,d\subseteq X$ which meet transversally at two points $P,Q$, and nowhere else. On $X-Q$, he first blow up the curve $c$, then blow up the strict transform of the curve $d$, we denote the result of these two blow-ups by $X_1$, on $X-P$, first blow up the curve $d$, then blow up the strict transform of the curve $c$, we denote the result of these two blow-ups on $X-P$ by $X_2$. On $X-P-Q$ it doesn't matter in which order we blow up the curves $c$ and $d$, so we can glue $X_1$ and $X_2$ along the inverse images of $X-P-Q$. The result is a non-projective compact complex manifold $\tilde X$.

My question is: where the is condition projective used? what if we change the condition projective manifold to any compact Kähler manifold (or maybe any compact complex manifold), do we still have these blow-up operations? is the result a manifold in Fujiki class $\mathcal C$? I guess the main point is to guarantee there exist two non-singular curves meet transversally at two points, then what's the most loose condition to ensure it? Any comment is welcome!

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    $\begingroup$ If this construction is meant to be a proof of the existence of non-projective compact Kähler manifolds, one has to start with something that is known to exists, i.e. a projective threefold. You can also define the blow-up of a complex manifold along a submanifold, see here. Since blowing-up is a biholomorphic operation, if you start with something in $\mathcal C$, the result would be in $\mathcal C$ as well? $\endgroup$
    – red_trumpet
    Dec 14, 2021 at 9:50
  • $\begingroup$ Also to show that $\tilde X$ is not projective, one uses that $X$ is projective, but the morphism $\tilde X \to X$ is not projective. $\endgroup$
    – red_trumpet
    Dec 14, 2021 at 9:51
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    $\begingroup$ Projectivity is used implicitly to find the curves; by embedding $X \subset \mathbb P^r$ and intersecting with a generic linearly embedded $\mathbb P^2 \subset \mathbb P^r$ we obtain (by Bertini's theorem) a nonsingular curve $c \subset X$. By choosing a second general plane in $\mathbb P^r$ which intersects the first in a sufficiently general line, we obtain $d$ (nonsingular by generality of the second plane) meeting $c$ transversely (by generality of the line) in some number of points. Now replace $X$ with its blowup along all but two points of $c\cap d$ and you have the needed setup. $\endgroup$
    – Tabes Bridges
    Dec 14, 2021 at 9:55
  • $\begingroup$ So in other words, projectivity is not to ensure the existence of blowups but of curves. $\endgroup$
    – Tabes Bridges
    Dec 14, 2021 at 9:57
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    $\begingroup$ There are even compact Kahler manifolds having no subvarieties $\endgroup$
    – Nick L
    Dec 14, 2021 at 11:26

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