  For a complex projective space the inertia group, the homotopy inertia group
and the concordance inertia group are isomorphic. In complex dimension 4n+1,
these groups are related to computations in stable cohomotopy. Using stable
homotopy theory, we make explicit computations to show that the inertia group
is non-trivial in many cases. In complex dimension 9, we deduce some results on
geometric structures on homotopy complex projective spaces and complex
hyperbolic manifolds.
