local kählerforms on complex manifold hallo,
Let $M$ be a complex manifold. Assume we have a covering of $M$ by complex charts $\{U_{i}\}$. Furthermore assume that we have on each $U_{i}$ a Kählerform $\omega_{i}$ (i.e. $d\omega_{i} = 0$). Is it possible to construct from these forms a global Kählerform $\omega$ on $M$? Or under wat circumstances is this possible ? Is there any book or paper where this issue is discussed ? Thanks in advance.
gary
 A: (I decided to repost multiple comments as an answer. It's partly an answer if I understand the OP correctly.)
I think the OP does not necessarily want to conclude that these "local" forms are the restrictions of a global form. Rather, he wants to "glue" them together in the sense of "glueing" constructions used to solve elliptic PDE's, such as the Kummer construction of Calabi-Yau metrics on K3. That is, the OP (I think) wants to interpolate between these forms, and on the interpolation region the new globally defined form need not restrict to the original local pieces.
Gary, is this indeed what you meant? If so, it sounds very difficult in general. If you have only a few charts with topologically simple intersections (like annuli), then this is often possible with cutoff functions, but topological obstructions can and do arise. Ideally, one has to be able to match the cohomology classes of your local Kahler forms $[\omega_i]$ on the intersection regions. If they do not match already, "corrections" are sometimes possible. It's rather nontrivial in general. Before saying more, I would need more details about the specific situation. 
