Cohomogeneity one Einstein metrics on complex projective spaces

We study Einstein metrics on complex projective spaces that are invariant under cohomogeneity one actions of compact connected Lie groups, under the assumption that the singular orbits are totally geodesic. These actions were classified by Takagi into five models. For each of them, we write the Einstein equation for diagonal invariant metrics and determine the corresponding smoothness conditions at the singular orbits. Our main result is the nonexistence of smooth globally defined invariant Einstein metrics in four of the five models and a necessary condition for global existence in the remaining one.