Question

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?

Answer 1

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.

1. Projective manifold: closed submanifold of some complex projective space.
2. Moishezon manifold: compact complex manifold such that the field of meromorphic functions on it has transcendence degree equal to its complex dimension.
3. (Compact) Kähler manifold: compact complex manifold carrying a Kähler form, that is a closed positive smooth (1,1)-form.
4. 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:

1. 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.
2. 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.
3. 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).
4. 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.