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

which yields after differentiating with respect to z ′ :(3.42) {Θ′z ′(z′ , ¯z ′ , ¯w ′ )/[b ′ (Θ ′ (z ′ , ¯z ′ , ¯w ′ ))] ≡ 2i ∂ z ′f(z ′ ) ¯f ′ (¯z ′ ) ∂ r ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ ))115≡ − 2 f ′ (z ′ ) ¯f ′ (¯z ′ ) ∂ r ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ ))/[a ′ (z ′ )].Here, we consider the function ϕ as a function ϕ(r) of the real variabler ∈ R. Since the left hand side is an algebraic function and a ′ (z ′ ) is alsoalgebraic, there exists an algebraic function A ′ (z ′ , ¯z ′ ) such that we can write(3.43) A ′ (z ′ , ¯z ′ ) ≡ f ′ (z ′ ) ¯f ′ (¯z ′ ) ∂ r ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ )).Next, using the property ϕ(r) = r + O(r 2 ), differentiating (3.43) withrespect to ¯z ′ at ¯z ′ = 0, we obtain that f ′ (z ′ ) is algebraic. Coming backto (3.43), this yields that ∂ r ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ )) is algebraic. Since f ′ (z ′ ) is alsoalgebraic, we finally obtain that ∂ r ϕ(r) is algebraic. Excepting the exampleswhich will be treated in §7.5, the proof of Theorem 1.4 is complete.The next three sections are devoted to the statement of Theorem 4.1,which implies directly Theorem 2.1 (§4), and to its proof (§§5-6).§4. LOCAL LIE GROUP STRUCTURE FOR THE CR AUTOMORPHISMGROUP4.1. Local representation of a real algebraic generic submanifold. Weconsider a connected real algebraic (or more generally, real analytic) genericsubmanifold M in C n of codimension d ≥ 1 and CR dimension m =n − d ≥ 1. Pick a point p ∈ M and consider some holomorphic coordinatest = (t 1 , . . .,t n ) = (z 1 , . . .,z m , w 1 , . . .,w d ) ∈ C m × C d vanishingat p in which T 0 M = {Im w = 0}. If we denote w = u + iv, it followsthat there exists (Nash) real algebraic power series ϕ j (z, ¯z, u) withϕ j (0) = 0 and dϕ j (0) = 0 such that the defining equations of M are ofthe form v j = ϕ j (z, ¯z, u), j = 1, . . .,d in a neighborhood of the origin.By means of the algebraic implicit function theorem, we can solve with respectto ¯w the equations w j − ¯w j = 2i ϕ j (z, ¯z, (w + ¯w)/2), j = 1, . . ., d,which yields ¯w j = Θ j (¯z, z, w) for some power series Θ j which are complexalgebraic with respect to their 2m + d variables. Here, we haveΘ j = w j + O(2), since T 0 M = {Im w = 0}. Without loss of generality,we shall assume that the coordinates are normal, namely the functionsΘ j (¯z, z, w) satisfy Θ j (0, z, w) ≡ w j and Θ j (¯z, 0, w) ≡ w j . It may beshown that the power series Θ j = w j + O(2) satisfy the vectorial functionalequation Θ(¯z, z, Θ(z, ¯z, ¯w)) ≡ ¯w in C{z, ¯z, ¯w} d and conversely that to everysuch power series mapping satisfying this vectorial functional equation,there corresponds a unique real algebraic generic manifold M (cf. for instancethe manuscript [GM2001c] for the details). So we can equivalently