Study of the formal CR reflection mapping
Résumé
Searching normal forms for real analytic submanifolds of C^n involves convergence problems. In 1983, J.K. Moser and S.M. Webster provided examples of real analytic surfaces in C^2 having an isolated hyperbolic (in the sense of E. Bishop) complex tangency, which are formally but not holomorphically normalizable (because of the presence of small divisors), even if the normal form is itself real analytic or algebraic. On the contrary, it appears that such a nonconvergence phenomenon does not appear for submanifolds of C^n whose CR dimension is locally constant, in view of recent results by S.M. Baouendi, P. Ebenfelt and L.P. Rothschild. These results hold true with hypotheses which are relatively simple, but satisfied at a Zariski-generic. Notably, these authors establish that every invertible formal CR mapping between two submanifolds of C^n which are real analytic, generic, finitely nondegenerate and minimal (in the sense of J.-M. Trépreau and A.E. Tumanov) is convergent. In this paper, we establish a more general convergence theorem, which is valid without any nondegeneracy condition, and which confirms the rigidity of the CR category (see Theorem 1.23). This result may be interpreted as a formal Schwarz reflection principle for CR mappings. We deduce that every formal CR equivalence between two submanifolds of C^n which are real analytic, generic and minimal is convergent if and only if both submanifolds are holomorphically nondegenerate (in the sense of N. Stanton). Finally, we establish that two submanifolds of C^n which are real analytic, generic and minimal are formally CR equivalent if and only if they are biholomorphically equivalent.