Let $M$ be a smooth manifold and $I:TM\to TM$ an integrable almost complex structure. Then, the cotangent bundle $T^\*M$ admits a canonical complex structure, which can be built from holomorphic charts on $M$.


If $I$ is not necessarily integrable, is there always an almost complex structure on $T^\*M$ such that $T^\*M\to M$ is holomorphic? Is it canonical? If not, can we construct it using some additional choice on $M$?



Almost complex structures on cotangent bundles of almost complex manifolds
There is one due to Satô, *"Almost analytic vector fields in almost complex manifolds"*. He constructs a $J'$ and $J''$ on $T^\ast M$ using a $J$ on $M$, such that $J''$ is always an almost complex structure but $J'$ is only an almost complex structure when $J$ is integrable (in which case $J'$ is also integrable), and $J''$ is a modified version of $J'$ using the nontrivial Nijenhuis tensor of $J$. This is elaborated on in Yano--Patterson's paper *"Vertical and complete lifts from a manifold to its cotangent bundle"* (which might be easier to digest).


There is also a way using connections on $M$ (which generalizes Satô's work) due to Yano--Patterson, *"Horizontal lifts from a manifold to its cotangent bundle"*. Then there is a unifying generalization of both works, see Bertrand's paper *"Almost complex structures on the cotangent bundle"* (<https://arxiv.org/abs/math/0507068>); here Theorem 3.1 asserts holomorphicity of the projection $\pi$.


