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

Proposition 3.1. Let t ↦→ G i (t; ǫ i ), i = 1, . . ., n, be n one complex parameterfamilies of complex algebraic biholomorphic maps from a neighborhoodof 0 in C n onto a neighborhood of 0 in C n satisfying G i (t; 0) ≡ t,∂ ǫi G i (0; ǫ i )| ǫi =0 = ∂ ti | 0 and pairwise commuting: G i1 (G i2 (t; ǫ 2 ); ǫ 1 ) ≡G i2 (G i1 (t; ǫ 1 ); ǫ 2 ). Then there exists a complex algebraic biholomorphismof the form t ′ ↦→ Φ ′ (t ′ ) =: t of C n fixing the origin with dΦ ′ (0) = Id suchthat if we set G ′ i(t ′ ; ǫ i ) := Φ ′ −1 (G i (Φ ′ (t ′ ); ǫ i )), where t ′ = Φ(t) denote theinverse of t = Φ ′ (t ′ ), then we have(3.3) G ′ i (t′ ; ǫ i ) ≡ (t ′ 1 , . . .,t′ i−1 , G′ i,i (t′ i ; ǫ i), t ′ i+1 , . . .,t′ n ),where the functions G ′ i,i are complex algebraic, depend only on t′ i (and onǫ i ) and satisfy G ′ i,i(t ′ i; 0) ≡ t ′ i and ∂ ǫi G ′ i,i(0; ǫ i )| ǫi =0 = 1.Proof. First of all, we define the complex algebraic biholomorphism(3.4) Φ ′ 1 : (t′ 1 , t′ 2 , . . .,t′ n ) ↦−→ G 1(0, t ′ 2 , . . .,t′ n ; t′ 1 ) =: t.We have dΦ ′ 1(0) = Id, because G 1 (t; 0) ≡ t and ∂ ǫ1 G 1 (0; ǫ 1 )| ǫ1 =0 = ∂ t1 | 0 .Furthermore, since ∂ ǫ1 G 1 (0; ǫ 1 )| ǫ1 =0 is transversal to {(0, t 2 , . . .,t n )}, italso follows that a small neighborhood of the origin in C n t is algebraicallyfoliated by the (n − 1)-parameter family of complex curves C ′t :=′ 2 ,...,t′ n{G 1 (0, t ′ 2, . . .,t ′ n; t ′ 1) : |t ′ 1| < δ} where δ > 0 is small and t ′ 2, . . ., t ′ n arefixed. The existence of this foliation shows that the relation(3.5) t ∗ ∼ t iff there exists ǫ 1 such that t ∗ = G 1 (t; ǫ 1 )is a local equivalence relation, whose equivalence classes are the leaves(see FIGURE 1).C ′t ′ 2 ,...,t′ n105t 2, . . . , t nt ∗ ∼ tC ′ t ′ 2 ,...,t′ nG 1(G 1(0, t ′ 2 , . . . , t′ n ; t′ 1 ); ǫ1)G 1(0, t ′ G 2 , . . . , t′ n ; t′ 1 )1(0, t ′ 2 , . . . , t′ n ; 0)0t 1FIGURE 1: LOCAL ALGEBRAIC STRAIGHTENING OF THE ORBITS OF G 1(t; ǫ 1)Consequently, as we clearly have(3.6) (0, t ′ 2 , . . .,t′ n ) ∼ G 1(0, t ′ 2 , . . .,t′ n ; t′ 1 ) ∼ G 1(G 1 (0, t ′ 2 , . . ., t′ n ; t′ 1 ); ǫ 1),using the transitivity of the relation ∼, it follows that there exists a complexnumber ε t ′ ,ǫ 1depending on t ′ and on ǫ 1 such that(3.7) G 1 (G 1 (0, t ′ 2 , . . ., t′ n ; t′ 1 ); ǫ 1) = G 1 (0, t ′ 2 , . . .,t′ n ; ε t ′ ,ǫ 1).