  Irreducible isoparametric foliations of arbitrary codimension q on complex
projective spaces CP^n are classified, except if n=15 and q=1. Remarkably,
there are noncongruent examples that pull back under the Hopf map to congruent
foliations on the sphere. Moreover, there exist many inhomogeneous
isoparametric foliations, even of higher codimension. In fact, every
irreducible isoparametric foliation on the complex projective n-space is
homogeneous if and only if n+1 is prime.
  The main tool developed in this work is a method to study singular Riemannian
foliations with closed leaves on complex projective spaces. This method is
based on certain graph that generalizes extended Vogan diagrams of inner
symmetric spaces.
