Nonalgebraizable real analytic tubes in C-n

摘要

We give necessary conditions for certain real analytic tube generic submanifolds in C-n to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in C-n. 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 C-n passing through the origin, having a defining equation of the form v = phi(y), where (z,w) = (x + iy, u + iv) epsilon Cn-1 x C. 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 partial derivative(z1),...,partial derivative(zn-1),partial derivative(w). Then M is locally algebraizable only if every second derivative partial derivative(ykyl)(2) phi is an algebraic function of the collection of first derivatives partial derivative(y1)phi,...,partial derivative(ym)phi.