# Holomorphic h-principle for compact manifolds

The Oka principle for Stein manifolds says (roughly) that the only obstructions for "things" are topological obstructions (for instance every smooth complex vector bundle admits a holomorphic structure, etc). Is there a similar principle (atleast in some cases) for compact complex manifolds? Or atleast some version of a h-principle for compact manifolds?

-
Hi Vamsi, As Johannes states, of course the answer is "no" in general. However, for fiber bundles of very special type, and up to replacing holomorphic section by meromorphic section, there are some such results in the compact case. The basic example is Tsen's theorem: a $\mathbb{P}^1$-bundle over a Riemann surface always has a holomorphic section. –  Jason Starr Sep 11 '11 at 3:40

I don't think you get an $h$-principle for compact complex manifolds.
Example: Given a complex line bundle $L \to M$, it admits a holomorphic structure iff the image of its Chern class in $H^2 (M;\mathcal{O})$ is zero. Similarly, the group of holomorphic line bundles which are topologically trivial is the cokernel of the homomorphism $H^1 (M; \mathbb{Z}) \to H^1 (M;\mathcal{O})$. Complex tori show that Oka's priniciple fails for compact complex manifolds.