  We classify up to diffeomorphism all smooth manifolds homeomorphic to the
complex projective m-space $\mathbb{C}P^{m}$ for $m = 5, 6, 7$ and $8$. As an
application, for $m = 7$ and $8$, we compute the smooth tangential structure
set of $\mathbb{C}P^{m}$ and obtain a bound on the number of smooth homotopy
complex projective m-spaces with given Pontryagin classes up to
orientation-preserving diffeomorphism. We also show that there exists a smooth
manifold which is tangentially homotopy equivalent but not homeomorphic to
$\mathbb{C}P^{8}$.
