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

104codimension d passing through the origin in C n given by the equations v j =ϕ j (y), j = 1, . . .,d. Assume that M is biholomorphically equivalent to areal algebraic generic submanifold M ′ .First step. We show that an arbitrary real algebraic element M ′ ∈ Tn d can bestraightened in some local complex algebraic coordinates t ′ = (t ′ 1 , ..., t′ n ) ∈C n in order that its infinitesimal CR automorphisms are the n holomorphicvector fields of the specific form X i ′ = c ′ i(t ′ i) ∂ t ′i, i = 1, . . ., n, where thefunctions c ′ i (t′ i ) are algebraic.Second step. Assuming that there exists a biholomorphic equivalence Φ :M → M ′ satisfying Φ ∗ (∂ ti ) = c ′ i (t′ ) ∂ t ′i, we prove by direct computationthat all the first order derivatives of the mapping ψ ′ (y ′ ) must be algebraic.3.1. Proof of the first step. Let t ′ = Φ(t) be such an equivalence, withΦ(0) = 0 and M ′ := Φ(M) real algebraic. Let X i := ∂ ti , i = 1, . . .,n,be the n infinitesimal CR automorphisms of M and set X ′ i := Φ ∗ (X i ). Ofcourse, we have [X ′ i 1, X ′ i 2] = Φ ∗ ([X i1 , X i2 ]) = 0, so the CR automorphismgroup of M ′ is also n-dimensional and commutative. Let us choose complexalgebraic coordinates t ′ in a neighborhood of 0 ∈ M ′ such that X ′ i | 0 =∂ t ′i| 0 . Let us apply Theorem 2.1 to the real algebraic submanifold M ′ , notingall the datas with dashes. There exists an algebraic mapping H ′ (t ′ ; e) =H ′ (t ′ ; e 1 , . . .,e n ) such that every local biholomorphic self-map of M ′ writesuniquely t ′ ↦→ H ′ (t ′ ; e), for some e ∈ R n . In particular, for every i =1, . . ., n and every small s ∈ R, there exists e s ∈ R n depending on s suchthat exp(sX ′ i)(t ′ ) ≡ H ′ (t ′ ; e s ). From the commutativity of the flows of theX ′ i , i.e. from exp(s 1X ′ i 1(exp(s 2 X ′ i 2(t ′ )))) ≡ exp(s 2 X ′ i 2(exp(s 1 X ′ i 1(t ′ )))),we get(3.1) H ′ (H ′ (t ′ ; e 2 ); e 1 ) ≡ H ′ (H ′ (t ′ ; e 1 ); e 2 ).This shows that the biholomorphisms t ′ ↦→ H e ′(t′ ) := H ′ (t ′ ; e) commutepairwise. In particular, if we define(3.2) G ′ i (t′ ; e i ) := H ′ (t ′ ; 0, . . .,0, e i , 0, . . .,0),we have G ′ i 1(G ′ i 2(t ′ ; e 2 ); e 1 ) ≡ G ′ i 2(G ′ i 1(t ′ ; e 1 ); e 2 ).Next, after making a linear change of coordinates in the e-space, we caninsure that ∂ ei G ′ i (0; e i)| ei =0 = ∂ t ′i| 0 = X ′ i | 0 for i = 1, . . .,n. Finally, complexifyingthe real variable e i in a complex variable ǫ i , we get mappingsG ′ i (t′ i ; ǫ i) which are complex algebraic with respect to both variables t ′ ∈ C nand ǫ i ∈ C and which commute pairwise. We can now state and prove thefollowing crucial proposition (where we have dropped the dashes) accordingto which we can straighten commonly the n one-parameter families ofbiholomorphisms t ′ ↦→ G ′ i (t′ ; ǫ i ).