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

[∂ i,ei H ei (t ′ )] t ′ =He −1i(t) , is defined for t ∈ ∆ n(ρ 5 ) and |e i | < ε, has algebraiccoefficients depending on the “time” parameter e i , and hasan algebraic flow, since this coincides with the algebraic mapping(t, e i ) ↦→ H i,ei (t).(6) Let ρ 5 be as in statement (2). Then the dimension c 0 of the real Lie algebraHol(M, ∆ n (ρ 5 )) is finite, bounded by the fixed integer N n,κ0 :=n (n+κ 0)!n! κ 0. Furthermore, each vector field X ∈ Hol(M, ∆! n (ρ 5 )) hascomplex algebraic coefficients.If M is real analytic, the same theorem holds with the word “algebraic”replaced everywhere by the word “analytic”.We shall explain below how the integer κ 0 is related to the minimalityand to the finite nondegeneracy of M at the origin. The next §5 and §6are devoted to the proof Theorem 4.1, namely the existence of the mappingH(t, J κ 0), the existence of the real algebraic totally real submanifold E andthe completion of the proof of properties (1-6).119§5. MINIMALITY AND FINITE NONDEGENERACY5.1. Local CR geometry of complexified real analytic generic submanifolds.Let ζ ∈ C m and ξ ∈ C d denote some independent coordinates correspongingto the complexification of the variables ¯z and ¯w, which we denotesymbolically by ζ := (¯z) c and ξ := ( ¯w) c , where the letter “c” stands for theword “complexified”. We also write τ := (¯t) c , so τ = (ζ, ξ) ∈ C n . Theextrinsic complexification M := (M) c of M is the complex submanifold ofcodimension d defined by(5.1) M := {(z, w, ζ, ξ) ∈ ∆ n (ρ 1 ) × ∆ n (ρ 1 ) : ξ = Θ(ζ, z, w)}.If M is (real, Nash) algebraic, so is M . As remarked, we can choose theequivalent defining equation w = Θ(z, ζ, ξ) for M . In the remainder of §5,we shall essentially deal with M instead of M. In fact, M clearly imbedsin M as the intersection of M with the antiholomorphic diagonal Λ :={(t, τ) ∈ C n × C n : τ = ¯t}.Following [Me1998], [Me2001], we shall complexify a conjugatepair of generating families of CR vector fields tangent to M, namelyL 1 , . . ., L m of type (1, 0) and their conjugates L 1 , . . .,L m whichare of type (0, 1). Here, we can explicitely choose the generatorsL k = ∂/∂z k + ∑ dj=1 [∂Θ j/∂z k (z, ¯z, ¯w)] ∂/∂w j for k = 1, . . .,m. Thentheir complexification yields a pair of collections of m vector fields defined