Completely integrable vector fields

Friday, April 5, 2019 - 14:20
Javier Ribon
Completely integrable vector fields

We study local holomorphic vector fields in dimension 3 with a maximum number of holomorphic first integrals. There are no such examples with a finite number of separatrices and isolated singularity as was proved in a joint work with Felipe Cano (UVA) and Marianna Ravara Vago (UFSC). As a consequence, it is natural to allow non-isolated singularities. Anyway, this is still a very hard problem and so we introduce a hypothesis that forces the dynamics of the foliation to be somehow manageable. This tameness condition has a simple formulation: there exists a holomorphic first integral that is non-constant in every of the irreducible components of the singular set. In this setting we will show that the leaf space is a germ of regular surface and the ring of holomorphic first integrals is a ring of complex power series in two variables. Given a transversal, we consider the finite group of diffeomorphisms whose orbits are contained in leaves of the foliation. Surprisingly, even if the leaf space is simple, we can construct explicit examples where the aforementioned group is non-solvable. This is a joint work with Rudy Rosas (PUCPE).




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