Nonalgebraizable real analytic tubes in C^n

Abstract

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) \in C^{n-1} \times 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_{z_1}, ..., \partial_{z_{n-1}}, \partial_w. Then M is locally algebraizable only if every second derivative \partial^2_{y_ky_l}\phi is an algebraic function of the collection of first derivatives \partial_{y_1} \phi,..., \partial_{y_m} \phi.