The cotangent bundle of a manifold has a canonical symplectic form and if we choose a riemannian metric on $M$, we can give it an almost complex structure.

Is this structure integrable, and if it isn't in general, what are the conditions on the manifold for it to be integrable? Can we give $T^*M$ a complex structure in some other way?

Furthermore, what are the conditions on $M$ to ensure that the symplectic form is a Kaehler form?

I read this thread: Kähler structure on cotangent bundle?, but I honestly didn't understand, how far it answers these questions.

M by Eliashberg's result. It is not Kahler, in general. A Kahler metric exists on TM = TM in a neighborhood of the zero section ("Grauert tube"). $\endgroup$ – Eric Zaslow Jan 5 '12 at 15:06