Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

94not holomorphic) equivalence, the question whether M is biholomorphicallyequivalent to a real algebraic submanifold is subtle. In this article,we study the question whether every real analytic CR submanifold is locallyalgebraizable. One of the interests of algebraizability lies in the reflectionprinciple, which is better understood in the algebraic category. Indeed,in the fundamental works of Pinchuk [Pi1975], [Pi1978] and of Webster[We1977], [We1978] and in the recent works of Sharipov-Sukhov [SS1996],Huang-Ji [HJ1998], Verma [Ve1999], Coupet-Pinchuk-Sukhov [CPS2000],and Shafikov [Sha2000], [Sha2002], the extendability of germs of CR mappingswith target in a real algebraic hypersurface is achieved. On the contrary,even if some results previously shown under an algebraization hypothesiswere proved recently under general assumptions (see the strong resultobtained by Diederich-Pinchuk [DP2003]), most of the results cited aboveare still open in the case of a real analytic target hypersurface.1.1. Brief history of the question. By the work of Moser and Webster[MW1983, Thm. 1], it is known that every real analytic two-dimensionalsurface S ⊂ C 2 at an isolated elliptic (in the sense of Bishop) complex tangencyp ∈ S is biholomorphic to one of the surfaces S γ,δ,s := {(z 1 , z 2 ) ∈C 2 : y 2 = 0, x 2 = z 1¯z 1 + (γ + δ(x 2 ) s )(z1 2 + ¯z2 1 )}, where p corresponds tothe origin, where 0 < γ < 1/2 is Bishop’s invariant and where δ = ±1 ands ∈ N or δ = 0. The quantities γ, δ, s form a complete system of biholomorphicinvariants for the surface S near p. In particular, every elliptic surfaceS ⊂ C 2 is locally algebraizable. To the authors’ knowledge, it is unknownwhether there exist nonalgebraizable hyperbolic surfaces in C 2 . In fact, veryfew examples of nonalgebraizable submanifolds are known. In [Eb1996],the author constructed a nonminimal (and non Levi-flat) real analytic hypersurfaceM through the origin in C 2 which is not locally algebraizable (cf.[BER2000, p. 330]). In a recent article [HJY] the authors prove that thestrongly pseudoconvex real analytic hypersurface Im w = e |z|2 − 1 passingthrough the origin in C 2 is not locally algebraizable at any of its points. Usingan associated projective structure bundle Y introduced by Chern, theyshow that for every rigid algebraic hypersurface in C n , there exists an algebraicdependence relation between seven explicit Cartan-type holomorphicinvariant functions on Y . However a computational approach shows thatwhen M is of the specific form Im w = e |z|2 − 1, no algebraic relation canbe satisfied by these seven invariants.1.2. Presentation of the main results. Our aim is to present a geometricalapproach of the problem, valid in arbitrary dimension and in arbitrarycodimension, and to exhibit a large class of nonalgebraizable real analyticgeneric submanifolds. We consider the class Tn d of generic real analyticsubmanifolds in C n passing through the origin, of codimension d ≥ 1