Beyond Liouvillian transcendence

Abstract

To a codimension one foliation F defined by a meromorphic 1-form ω, one may associate a Godbillon-Vey sequence (ωj),j =0 , 1 ,... , of meromorphic 1-forms ωj with ω0 = ω. The sequence is said to have finite length k if ωk � and ωj = 0 for j>k . The case k =0 , 1 or 2 corresponds, respectively, to the case where the foliation F has additive, affine or projective transverse structure and k ≤ 1 is equivalent to the existence of a Liouvillian first integral. These are the only possible cases where the transverse structures come from an action of a Lie group on C and a non-trivial model for these foliations is the Riccati differential equation. We propose to go beyond the Lie group transverse structure by studying the case of general k and, for this case, we determine a model differential equation, which generalizes the Riccati equation. We also discuss some other related topics.