It is well-known that the cotangent bundle of a manifold is naturally a symplectic manifold. Inspired by mirror symmetry, when is the tangent bundle $TM$ of a manifold $M$ naturally a complex manifold? There is always canonical almost complex structure as the tangent bundle of $TM$ is point wise $TM\times TM$, i,e. $J(x,y)=J(-y,x)$. When is this integrable?
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$\begingroup$ In the mirror symmetry story, this kind of argument appears when $M$ as a extra structure: the structure of an affine manifold. Then it is true that the tangent bundle $TM$ of an affine manifold $M$ is naturally a complex manifold (where "naturally" means preserved by diffeomorphisms coming from diffeomorphisms of $M$ preserving the affine structure) (and of course not from any diffeomorphism of $M$ as indicated in Robert Bryant answer). $\endgroup$– user25309Sep 20, 2015 at 13:04
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$TM$ is not ever `naturally' a complex manifold, in the sense that there is no complex structure on $TM$ that is preserved by all of the diffeomorphisms $f':TM\to TM$ where $f:M\to M$ ranges over all diffeomorphisms of $M$ with itself.
In spite of what the OP wrote, there is no canonical (i.e., natural) almost complex structure on $TM$ either. This is a misunderstanding about $TTM$, which is not $TM\times TM$ 'pointwise' is any natural sense.
answered Sep 20, 2015 at 11:31