TITLE: What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler?
QUESTION [5 upvotes]: I was wondering if this implication is true. I read a few places that
$$\text{nonprojective} \Longrightarrow \text{nonKähler}$$
but I think I maybe have misunderstood. Equivalently, this is of course asking if $$\text{Kähler} \Longrightarrow \text{projective ?}$$

REPLY [2 votes]: I give the Campana-Moishezon criterion here which is less known.
Definition: A complex manifold $X$ is algebraically connected if for any general pair of points $x, y \in X,$
there exists a proper, connected and not necessarily irreducible curve in $X$ containing $x$ and $y$. 
Campana-Moishezon criterion: A compact complex manifold is projective if and only if it is Kähler and algebraically connected.
Note that by another result from Campana et al,
A compact hyper- Kahler manifold $X$ is projective if and only if the Néron-Severi group $NS(X)$ is hyperbolic, I.e., negative semi definite but not negative definite with respect to Beauville-Bogomolov-Fujiki's form
F. Campana. Coréduction algébrique d’un espace analytique faiblement kählérien compact. Invent. Math., 63(2):187–223, 1981.
https://projecteuclid.org/euclid.jdg/1292940689