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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?

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    $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented Sep 11, 2011 at 3:40

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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.

The Gromov-Phillips h-principle for closed manifolds is false as well, immersions are the only special case which applies to closed manifolds I am aware of. All other versions (e.g. submersions, symplectic structures, positively or negatively curved metrics) fail, and each of them fails in a fairly spectacular manner.

There are some exceptions to the rule, but in general I would say that one needs noncompactness to push away all possible obstructions to infinity.

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