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A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In particular, not every compact Kähler manifold can admit a positive line bundle. What about in the non-compact case? That is:

Are there any restrictions as to which non-compact Kähler manifolds can admit a positive line bundle?

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I'm not sure if this answers your question. There are positive line bundles on non-compact manifolds which are not ample, see for instance the paper of T. Ohsawa "A counter example of ampleness of positive line bundles", Proc. Japan Acad. Ser. A Math. Sci. Volume 55, Number 5 (1979), 193-194. On the positive side, a theorem of S. Takayama asserts that a weakly 1-complete manifold, which carries a positive line bundle can be embedded (of course, not properly) into a projective space. However, the embedding is given by a twisting of the positive line bundle with the canonical line bundle. –  user20497 Sep 5 '12 at 9:03

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Take a K3, or a general $n$-dimensional complex torus $M$, $n>1$, without any integer (1,1)-classes, and remove a point $x$. You will obtain a non-compact Kahler manifold without any non-trivial line bundles (because its second cohomology stays the same). For such a manifold it's not hard to show that no positive bundles exist. Indeed, if there is a positive line bundle, it must be trivial. Then you would have a positive form $\eta$ which is exact. By Sibony's lemma (see e.g. arXiv:0712.4036, Theorem 5.1), $\eta$ is locally integrable around $x$. Then, by Skoda-El Mir theorem, the trivial extension of $\eta$ to $M$ is a closed, positive and hence exact current. This is impossible, because $M$ is compact and Kahler.

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