Let $L$ be a holomorphic line bundle on complex manifold $X$, such that it admits a hermitian structure whose Chern connection has positive curvature. Is $X$ then Kahler?
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As Dima said, it is much more: in fact it is projective. But let me give you some more insights on this kind of questions. I shall give you the definition of four different classes of compact complex manifolds.
A Moishezon manifold can be shown to be bimeromorphic to a projective manifold, so that in some sense Moishezon manifolds are with respect to projective manifolds as manifolds in the Fujiki class ($\mathcal C$) are with respect to Kähler manifolds. It turns out, that one can characterize these four classes in terms of cohomological properties (these characterizations reflect again this relation between projectiveMoishezon and KählerFujiki). Here is the characterization for you:


