Nonalgebraizable real analytic tubes in C n

Abstract

We give necessary conditions for certain real analytic tube generic submanifolds in ℂn to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in ℂn. During the proof, we show that the local CR automorphism group of a minimal, finitely nondegenerate real algebraic generic submanifold is a real algebraic local Lie group. We may state one of the main results as follows. Let M be a real analytic hypersurface tube in ℂn passing through the origin, having a defining equation of the form v= φ(y), where (z,w)=(x+iy,u+iv)∈Open image in new windown−1×Open image in new window. Assume that M is Levi nondegenerate at the origin and that the real Lie algebra of local infinitesimal CR automorphisms of M is of minimal possible dimension n, i.e. generated by the real parts of the holomorphic vector fields ∂z1,...,∂z n−1,∂w. Then M is locally algebraizable only if every second derivative ∂2yky lφ; is an algebraic function of the collection of first derivatives ∂y1φ,⋯,∂ymφ.