Holomorphic h-principle for compact manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:55:51Z http://mathoverflow.net/feeds/question/75069 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75069/holomorphic-h-principle-for-compact-manifolds Holomorphic h-principle for compact manifolds Vamsi 2011-09-10T06:29:43Z 2011-09-10T10:11:00Z <p>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?</p> http://mathoverflow.net/questions/75069/holomorphic-h-principle-for-compact-manifolds/75082#75082 Answer by Johannes Ebert for Holomorphic h-principle for compact manifolds Johannes Ebert 2011-09-10T10:11:00Z 2011-09-10T10:11:00Z <p>I don't think you get an $h$-principle for compact complex manifolds. </p> <p>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.</p> <p>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.</p> <p>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.</p>