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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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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