Handbook of Geometry and Topology of Singularities V: Foliations by Felipe Cano & José Luis Cisneros-Molina & Lê Dũng Tráng & José Seade ISBN 9783031524806, 9783031524813, 3031524802, 3031524810 instant download

Handbook of Geometry and Topology of Singularities V: Foliations by Felipe Cano & José Luis Cisneros-Molina & Lê Dũng Tráng & José Seade ISBN 9783031524806, 9783031524813, 3031524802, 3031524810 instant download

The geometric theory of complex differential equations was originated in the early 1950s by Petrovsky and Landis. They introduced complex limit cycles and the analogue of the real Poincaré map, and proved that a generic planar polynomial vector field has no algebraic orbits, a fact known in the West as the Jouanolou theorem. They stated a persistence conjecture for complex limit cycles; this conjecture stays open even now. There are two approaches to the theory of planar polynomial foliations. One may consider a class of polynomial vector fields that have degree no greater than n in a fixed affine neighborhood of the complex projective plane. These foliations generically have an invariant line at infinity, which contains in general n+1 singular points. They give rise to a so called monodromy group at infinity that determines very specific properties of the foliation. The class of these foliations is denoted by An. Another class is the class of vector fields that have degree not greater than n in any affine neighborhood of CP2. This class is denoted by Bn. Generic foliations of this class have no invariant complex lines at all. Khudai-Verenov, a student of Landis, proved that a generic planar polynomial vector field of class An has the minimality property: all its orbits except for the singular points and the line at infinity are dense. This was the origin of the topological theory of complex foliations. For a while this theory attracted the attention of the leading young mathematicians of the 1960s: Anosov, Arnold, Novikov, Vinogradov and others. A conjecture that a generic planar polynomial vector field is structurally stable was discussed. Being an undergraduate student, I found a gap in the published proof of the Petrovsky-Landis persistence theorem. Since then I had a dream to find a correct proof: a goal that is not yet achieved. At the same time, complex differential equations became my main subject.