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

and of CR dimension m = n − d ≥ 1, whose local CR automorphismgroup is n-dimensional, generated by the real parts of n holomorphic vectorfields having holomorphic coefficients X 1 , . . .,X n which are linearlyindependent at the origin and which commute: [X i1 , X i2 ] = 0. We shallcall Tn d the class of strong tubes of codimension d. Indeed, since thereexists a straightened system of coordinates t = (t 1 , . . .,t n ) over C n inwhich X i = ∂ ti , we observe that every submanifold M ∈ Tn d is tubifiableat the origin. By this, we mean that there exist holomorphic coordinatest = (z, w) = (x + iy, u + iv) ∈ C m × C d vanishing at the origin in whichM is represented by d equations of the form v j = ϕ j (y). Hence M is a tube,i.e. a product of the submanifold {v j = ϕ j (y), j = 1, . . .,d} ⊂ R n y,vby the n-dimensional real space R n x,u . Since M ∈ T n d , the only infinitesimalCR automorphisms of M are the real parts of the vector fields∂ z1 , . . .,∂ zm , ∂ w1 , . . .,∂ wd , explaining the terminology. Notice that not everytube belongs to the class Tn d . For instance in codimension d = 1, theHeisenberg sphere v = ∑ n−1k=1 y2 k and more generally the Levi nondegeneratequadrics v = ∑ n−1k=1 ε k yk 2, where ε k = ±1, have a CR automorphismgroup of dimension (n+1) 2 −1 > n and so do not belong to Tn 1 . We assumethat M ∈ Tn d is minimal at the origin, namely the local CR orbit of 0 in Mcontains a neighborhood of 0 in M. Furthermore, we assume that M ∈ Tndis finitely nondegenerate at 0, namely that there exists an integer l ≥ 1 suchthat Span {L β ∇ t (r j )(0, 0) : β ∈ N m , |β| ≤ l, j = 1, . . ., d} = C n , wherer j (t, ¯t) = 0, j = 1, . . .,d are arbitrary real analytic defining functions for Mnear 0 satisfying ∂r 1 ∧· · ·∧∂r d ≠ 0 on M, where ∇ t (r j )(t, ¯t) is the holomorphicgradient with respect to t of r j and where L β denotes (L 1 ) β1 · · ·(L m ) βmfor an arbitrary basis L 1 , . . ., L m of (0, 1)-vector fields tangent to M ina neighborhood of 0. In particular Levi nondegenerate hypersurfaces arefinitely nondegenerate. Finally, assuming only that ϕ j (0) = 0, j = 1, . . ., d,we shall observe in Lemma 3.2 below that a tube v j = ϕ j (y) of codimensiond is finitely nondegenerate at the origin if and only if there exist multi-indicesβ∗ 1, . . .,βm ∗ ∈ Nm with |β∗ k| ≥ 1 and integers 1 ≤ j1 ∗ , . . .,jm ∗ ≤ d such thatthe real mapping(1.1) ψ(y) :=(∂|β 1 ∗ | ϕ j 1 ∗(y)∂y β1 ∗), . . ., ∂|βm ∗ | ϕ j m ∗(y)=: y ′ ∈ R m∂y βm ∗is of rank m at the origin in R m . Our main theorem provides a necessarycondition for the local algebraizability of strong tubes :Theorem 1.1. Let M be a real analytic generic tube of codimension d inC n given in coordinates (z, w) = (x + iy, u + iv) ∈ C m × C d by theequations v j = ϕ j (y), where ϕ j (0) = 0, j = 1, . . ., d. Assume that Mis minimal and finitely nondegenerate at the origin, so the real mapping95