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Isoparametric foliations on complex projective spaces


Author: Miguel Domínguez-Vázquez
Journal: Trans. Amer. Math. Soc. 368 (2016), 1211-1249
MSC (2010): Primary 53C40; Secondary 53C12, 53C35
DOI: https://doi.org/10.1090/S0002-9947-2014-06415-5
Published electronically: September 23, 2014
MathSciNet review: 3430362
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Abstract: Irreducible isoparametric foliations of arbitrary codimension $ q$ on complex projective spaces $ \mathbb{C} P^n$ are classified, for $ (q,n)\neq (1,15)$. 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 $ \mathbb{C} P^n$ 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 a certain graph that generalizes extended Vogan diagrams of inner symmetric spaces.


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Additional Information

Miguel Domínguez-Vázquez
Affiliation: Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, Brazil
Email: mvazquez@impa.br

DOI: https://doi.org/10.1090/S0002-9947-2014-06415-5
Keywords: Isoparametric foliation, polar action, inhomogeneous isoparametric foliation, FKM-foliation, extended Vogan diagram, inner symmetric space, complex projective space
Received by editor(s): December 13, 2012
Received by editor(s) in revised form: January 26, 2014
Published electronically: September 23, 2014
Additional Notes: The author was supported by the FPU programme of the Spanish Government, by a Marie-Curie European Reintegration Grant (PERG04-GA-2008-239162), and projects MTM2009-07756 and INCITE09207151PR (Spain).
Article copyright: © Copyright 2014 American Mathematical Society