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

The first relation gives nothing, since we already know that G ′ 1,1 is independentof t ′ 2. By differentiating the second relation with respect to ǫ 1 at ǫ 1 = 0,we find that G ′ 2,2 is independent of t ′ 1.In summary, after the change of coordinates Φ ′ 2 ◦ Φ′ 1 (t′ ) = t which istangent to the identity map at t ′ = 0, we obtained that107(3.16){G′1 (t ′ ; ǫ 1 ) = (G ′ 1,1(t ′ 1, t ′ 3, . . .,t ′ n), t ′ 2, t ′ 3, . . .,t ′ n),G ′ 2 (t′ ; ǫ 2 ) = (t ′ 1 , G′ 2,2 (t′ 2 , t′ 3 , . . .,t′ n ), t′ 3 , . . .,t′ n ).Using these arguments, the proof of Proposition 3.1 clearly follows by induction.Now, we come back to our CR manifold M ′ having the one-parameterfamilies of algebraic biholomorphisms G ′ i(t ′ ; e i ) given by (3.2) and pairwisecommuting. Applying Proposition 3.1, after a change of complex algebraiccoordinates of the form t ′ = Ψ ′′ (t ′′ ), we may assume that the G ′′i (t′′ ; ǫ i ) arealgebraic and can be written in the specific form(3.17) G ′′i (t′′ ; ǫ i ) ≡ (t ′′1 , . . .,t′′ i−1 , G′′ i,i (t′′ i ; ǫ i), t ′′i+1 , . . .,t′′ n ),with ∂ ǫi G ′′i,i(0; ǫ i )| ǫi =0 = 1. Let t ′′ = Ψ ′ (t ′ ) denote the inverse of t ′ =Ψ ′′ (t ′′ ). We thus have t ′′ = Ψ ′ (t ′ ) = Ψ ′ (Φ(t)), where we remind that t ′ =Φ(t) provides the equivalence between the strong tube M and the algebraicCR generic M ′ .Since Ψ ′ is algebraic, the image M ′′ := Ψ ′ (M ′ ) is also algebraic. Letr j ′(t′ , ¯t ′ ) = 0, j = 1, . . .,d, be defining equations for M ′ . Then r j ′′(t′′, ¯t ′′ ) :=r j(Ψ ′ ′′ (t ′′ ), Ψ ′′ (t ′′ )) = 0 are defining equations for M ′′ . By assumption, forǫ i := e i ∈ R real, the family of algebraic biholomorphisms G ′ i (t′ ; ǫ i ) mapsa small piece of M ′ through the origin into M ′ . It follows trivially thatG ′′i (t ′′ ; ǫ i ) ≡ Ψ ′ (G ′ i(Ψ ′′ (t ′′ ); ǫ i )) maps a small piece of M ′′ through the origininto M ′′ . Furthermore, since dΨ ′′ (0) = Id, it follows that if we denoteX i ′′ := Ψ ′ ∗ (X′ i ), then X′′ i | 0 = ∂ t ′′i| 0 .Next, thanks to the specific form (3.17), by differentiating∂ ǫi G ′′i (t ′′ ; ǫ i )| ǫi =0, we get n vector fields of the form Z i ′′ = c ′′i (t ′′i ) ∂ t ′′By construction, the functions c ′′i (t ′′i ) are algebraic and satisfy c ′′i (0) = 1.Differentiating with respect to e i the identity r j ′′ (G ′′i (t ′′ ; e i ), G ′′i (t′′ ; e i )) = 0for r j ′′(t′′, ¯t ′′ ) = 0, i.e. for t ′′ ∈ M ′′ , we see that Z i ′′ is tangent to M ′′ ,i.e. we see that Z i ′′ is an infinitesimal CR automorphism of M ′′ . Consequently,there exist real constants λ i,l such that Z i ′′ = ∑ nl=1 λ i,l X l ′′.SinceZ i ′′ | 0 = X i ′′ | 0 = ∂ t ′′i| 0 , we have in fact λ i,l = 1 for i = l and λ i,l = 0 fori ≠ l. So Z i ′′ = X i ′′ and we have shown that(3.18) (Ψ ′ ◦ Φ) ∗ (X i ) = X ′′i = Z ′′i = c ′′i (t′′ i ) ∂ t ′′i .i.