On the partial algebraicity of holomorphic mappings between real algebraic sets

Details On the partial algebraicity of holomorphic mappings between real algebraic sets On the partial algebraicity of holomorphic mappings between real algebraic sets

1999-06-09

arxiv.org/abs/math/9906057v2

Bull. Soc. Math. France 129 (2001), no. 4, 547--591

Mathematics

Complex Variables

Scientific paper

In this paper, we consider local holomorphic mappings f: M\to M' between real algebraic CR generic manifolds (or more generally, real algebraic sets with singularities) in the complex euclidean spaces of different dimensions and we search necessary and sufficient conditions for f to be algebraic. These conditions appear to exclude two certain flatness of M and of M'. From the point of view of CR geometry, a real analytic CR manifold M can be flat in essentially to ways, being biholomorphic to a product M_1\times \Delta^k, k\geq 1, by a polydisc (algebraic degeneracy), or to a product M_1\times I^l,l\geq 1 by a real cube (transversal degeneracy), in a neighborhod of a Zariski generic point. We also require that the CR manifold M is minimal in the sense of Tumanov at a generic point. Our first result provides a characterization of mappings with positive transcendence degree k. Such maps have the property that near a Zariski generic point f(p) in M', there exists a k-algebraically degenerate real algebraic set X'' which contains f(M) and which is contained in M'. This solves the algebraic mapping problem for a minimal source M completely. Our second main result is the construction of canonical foliations by Segre surfaces of the extrinsic complexification of M. We prove in particular that a holomorphic function defined in a neighborhood of a minimal M is algebraic if and only if its restriction to each Segre surface of M is algebraic. We also show by an example that the double reflection foliation in the spirit of tangential CR derivations does not yield a characterization of positivity of transcendence degree of holomorphic mappings.

Affiliated with

Also associated with

No associations

Rating

On the partial algebraicity of holomorphic mappings between real algebraic sets does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the partial algebraicity of holomorphic mappings between real algebraic sets, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the partial algebraicity of holomorphic mappings between real algebraic sets will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-199579