Vector field construction of Segre sets

A CR generic real analytic CR manifold M carries two families of Segre varieties and conjugate Segre varieties. We observe in this article that their complexifications give rise to two families of foliations of the complexification of M which coincide with the flow foliations induced by the complexified CR (1,0) and (0,1) vector fields tangent to M. As an application, we derive a new proof of the characterization of finite type in terms of Segre sets.Comment: Completely rewritten version, 42 pages Sujclass: Complex Variable

On transfer of biholomorphisms across nonminimal loci

A connected real analytic hypersurface M in C^(n+1) whose Levi form is nondegenerate in at least one point - hence at every point of some Zariski-open subset - is locally biholomorphic to the model Heisenberg quadric pseudosphere of signature (k, n-k) in one point if and only if, at every other Levi nondegenerate point, it is also locally biholomorphic to some Heisenberg pseudosphere, possibly having different signature (l, n-l).Comment: 12 pages, reproving and generalizing a bit a recent theorem of Kossovskiy-Shafikov according to a suggestion of Beloshapk