Cohomology of complex projective Stiefel manifolds

The cohomology algebra mod p of the complex projective Stiefel manifolds is determined for all primes p. When p = 2 we also determine the action of th e Steenrod algebra and apply this to the problem of existence of trivial su bbundles of multiples of the canonical line bundle over a lens space with 2 -torsion, obtaining optimal results in many cases.