On the moduli space of certain smooth codimension-one foliations of the 5-sphere by complex surfaces

Laurent Meersseman 1  and Alberto Verjovsky 2
  • 1 I.M.B., Université de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France. e-mail: laurent.meersseman@u-bourgogne.fr
  • 2 Instituto de Matemáticas de la Universidad Nacional Autonóma de México, Unidad Cuernavaca, Apartado Postal 273-3, Admon. de correos No. 3, Cuernavaca, Morelos, México. e-mail: alberto@matcuer.unam.mx


In this paper we first determine the set of all possible integrable almost CR-structures on the smooth foliation of 𝕊5 constructed in [Meersseman, Verjovsky, Ann. Math. 156: 915–930, 2002]. We give a specific concrete model of each of these structures. We show that this set can be naturally identified with ℂ × ℂ × ℂ. We then adapt the classical notions of coarse and fine moduli space to the case of a foliation by complex manifolds. We prove that the previous set, identified with ℂ3, defines a coarse moduli space for the foliation of [Meersseman, Verjovsky, Ann. Math. 156: 915–930, 2002], but that it does not have a fine moduli space. Finally, using the same ideas we prove that the standard Lawson foliation on the 5-sphere can be endowed with almost CR-structures but none of these is integrable. This is a foliated analogue to the examples of almost complex manifolds without complex structure.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.