  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. 
Inertia groups of high dimensional complex projective spaces