TITLE: When is the tangent bundle of a manifold naturally a complex manifold?
QUESTION [8 upvotes]: 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?

REPLY [12 votes]: $TM$ 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.