ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 32, 34 Scárdua, Bruno, Instituto de Matemática Pura e Aplicada, Brazil Integration of Complex Differential Equations Let F be an algebraic foliation admitting a Liouvillian first integral on ℂP (2). Then F is transversely affine outside some algebraic invariant curve S ⊂ ℂP (2). If moreover for some irreducible component S_{0} ⊂ S the singularities q ∈ SingF ∩S are nondicritical and generic then either F is given by a closed rational 1-form or it is a rational pull-back from a Riccati foliation R: p(x)dy − (y^{2}a(x) + yb(x))dx = 0 on \overline{ℂ} × \overline{ℂ}. The result applies on the study of foliations with algebraic limit sets on ℂP (n), on the classification polynomial complete vector fields over ℂ^{2}, and topological rigidity of foliations on ℂP (2). Reference: B. Scárdua: Transversely affine and transversely projective holomorphic foliations. Ann. Sc. E. N. S., t.30 (1997), 169-204.