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The Newlander-Nirenberg theorem states that any Integrable Almost Complex manifold is a complex manifold. I am looking for natural examples of complex structures that are not integrable.

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    I changed the title by adding "almost", because *nonintegrable complex structure* is an oxymoron. Also, I added some relevant tags.2011-06-23

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The sphere $S^6$ naturally sits inside of the imaginary octonians $\operatorname{Im}\mathbb{O}$. At the point $p\in S^6$, multiplication by $p$ on $ T_p S^6 = p^\bot \subseteq \operatorname{Im}\mathbb{O}$ defines an almost complex structure.

This almost complex structure is not integrable, due to the nonassociativity of the octonians.

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    @Michael: It's probably also worth adding that my only interaction with (almost) comple$x$ manifolds has been through a few cursory google searches - I've probably spent a grand total of a couple of days of my life thinking about them in any kind of detail. Sorry to be useless!2015-01-21