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?

  • 4
    $\begingroup$ 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. $\endgroup$ – user20497 Sep 5 '12 at 9:03

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.