Suppose that we have a compact Kaehler manifold $X$ with big and nef canonical class $c\_1(K\_{X})$, does it imply that $X$ is projective? By the base point free theorem, big and nef implies semi ample but it is for projective algebraic manifolds. So it seems to suggest that big and nef does not necessarily imply projectivity. But I have seen in literature that people claim that big and nef does imply projectivity.



Does big and nef imply projectivity?
If $X$ has a big line bundle $L$ then for an appropriate natural number $m$, sections of $L^m$ define a meromorphic map $\varphi: X \dashrightarrow \mathbf P^N$ which is bimeromorphic onto its image. Therefore $X$ is bimeromorphic to the projective variety $\overline{\varphi(X)}$, hence it is Moishezon. But Moishezon plus Kähler equals projective.


