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.
© Walter de Gruyter Berlin · New York 2009