Isoparametric foliations on complex projective spaces

TL;DR

This paper classifies irreducible isoparametric foliations on complex projective spaces, revealing their properties, examples, and conditions for homogeneity, and introduces a new method based on generalized Vogan diagrams for studying such foliations.

Contribution

It provides a comprehensive classification of irreducible isoparametric foliations on complex projective spaces and develops a novel graph-based method for analyzing singular Riemannian foliations.

Findings

Classification of foliations except for n=15, q=1

Existence of noncongruent foliations pulling back to spheres

Every irreducible foliation is homogeneous iff n+1 is prime

Abstract

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.