Specific line bundle over complex manifold implies Kähler? 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 Kähler?
 A: 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.


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*Projective manifold: closed submanifold of some complex projective space.

*Moishezon manifold: compact complex manifold such that the field of meromorphic functions on it has transcendence degree equal to its complex dimension.

*(Compact) Kähler manifold: compact complex manifold carrying a Kähler form, that is a closed positive smooth (1,1)-form.

*Manifold in the Fujiki class ($\mathcal C$): compact complex manifold bimeromorphic to a compact Kähler manifold.


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 projective-Moishezon and Kähler-Fujiki). Here is the characterization for you:


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*A compact complex manifold is projective if and only if it carries a (1,1) rational cohomology class which can be represented by a positive (1,1)-form (or equivalently if it carries a positive hermitian line bundle). This is the content of Kodaira's embedding theorem.

*A compact complex manifold is Kähler if and only if it carries a (1,1) real cohomology class which can be represented by a positive (1,1)-form. This is almost the definition.

*A compact complex manifold is Moishezon if and only if it carries a (1,1) rational cohomology class which can be represented by a (1,1) Kähler current, that is a (1,1)-closed positive current which is bounded from below by a (non necessarily closed) smooth positive (1,1)-form (or equivalently if it carries a big line bundle).

*A compact complex manifold is in the Fujiki class ($\mathcal C$) if and only if it carries a (1,1) real cohomology class which can be represented by a (1,1) Kähler current. This is the content of a theorem by Demailly-Paun. 

