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

that for every local holomorphic self-mapping h of M belonging to H ρ 2,ρ 1M,κ 0 ,ε ,we have the representation formula(6.20) h(t) = H(t, J κ 0h(0)),for all t ∈ C n with |t| < ρ 4 . Furthermore the mapping H depends neither onthe choice of smaller radii ˜ρ 1 ≤ ρ 1 , ˜ρ 2 ≤ ρ 2 , ˜ρ 3 ≤ ρ 3 and ˜ρ 4 ≤ ρ 4 satisfying0 < ˜ρ 4 < ˜ρ 3 < ˜ρ 2 < ˜ρ 1 nor on the choice of a smaller constant ˜ε < ε,so that the first sentence of property (3) in Theorem 4.1 holds true. Finally,if M is real analytic, the same statement holds with the word “algebraic”everywhere replaced by the word “analytic”.It remains now to construct the submanifold E whose existence is statedin Theorem 4.1 and to establish that H ρ 2,ρ 1M,κ 0 ,εmay be endowed with the structureof a local real algebraic Lie group.6.5. Local real algebraic Lie group structure. In order to construct thissubmanifold E, we introduce the κ 0 -th jet mapping J κ 0: H ρ 2,ρ 1M,κ 0 ,ε →C Nn,κ 0 defined by Jκ 0(h) := (∂t α h(0)) |α|≤κ0 = J κ 0h(0). The followinglemma is crucial.Lemma 6.5. Shrinking ε if necessary, the set(6.21) E := J κ 0(H ρ 2,ρ 1M,κ 0 ,ε ) = {Jκ 0h(0) : h ∈ H ρ 2,ρ 1M,κ 0 ,ε }is a real algebraic totally real submanifold of the polydisc {J κ 0∈ C Nn,κ 0 :|J κ 0− J κ 0Id | < ε}.Proof. Let h ∈ H ρ 2,ρ 1M,κ 0 ,ε. Substituting the representation formula h(t) =H(t, J κ 0h(0)) given by Theorem 6.4 in the defining equations of M, we get(6.22) r j (H(t, J κ 0h(0)), H(τ, J κ 0¯h(0))) = 0,for j = 1, . . .,d and (t, τ) ∈ M with |t|, |τ| < ρ 4 . As (t, τ) ∈ M , wereplace ξ by Θ(ζ, t) and we use the 2m + d coordinates (t, ζ) on M . So, byexpanding the functions (6.22) in power series with respect to (t, ζ), we canwrite(6.23)∑r j (H(t, J κ 0), H(ζ, Θ(ζ, t), J κ 0 )) = t α ζ β C j,α,β (J κ 0, J κ 0 ).α∈N n , β∈N mHere, we obtain an infinite collection of complex-valued real algebraic functionsC j,α,β defined in {|J κ 0− J κ 0Id| < ε} with the property that a mappingH(t, J κ 0) sends M ∩ ∆ n (ρ 4 ) into M if and only if(6.24) C j,α,β (J κ 0, J κ 0 ) = 0, ∀ j, α, β.Consequently, the set E defined by the vanishing of all the equations (6.24)is a real algebraic subset.129