On the local algebraizability of real analytic generic submanifolds of   C^n
  We prove that the local (pseudo)group of biholomorphisms stabilizing a minimal, finitely nondegenerate real algebraic submanifold in C^n is a real algebraic local Lie group (the works of S.M. Baouendi, P. Ebenfelt, L.-P. Rothschild and D. Zaitsev published in this direction restrict, without understandable reason, to the isotropy group of a fixed central point). We deduce necessary conditions for the local algebraizability of real analytic rigid tubes of arbitrary codimension in C^n. Without using the Elie Cartan equivalence algorithm, we explain the up to now only known example Im w = e^(|z^2|), due to X. Huang, S. Ji and S.S. Yau in 2001, of a nonalgebraizable Levi nondegenerate real analytic hypersurface of C^2. These elementary criteria provide a first answer to an open problem raised by S.M. Baouendi, P. Ebenfelt and L.-P. Rothschild, in the survey article : ``Local geometric properties of real submanifolds in complex space'', Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 309--336. A second answer (necessary and sufficient condition) for local algebraizability in the homogeneous case will appear subsequently. 
Herve Gaussier and Joel Merker