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

102Indeed, from the group property Φ(Φ(z, w; σ); σ ′ ) ≡ Φ(z, w; σ + σ ′ ), it isclassical and immediate to deduce that if we define the parameter independentvector field X 0 (z, w) := ∂ σ Φ(z, w; σ)| σ=0 = z ∂ z , then it holds that∂ σ Φ(z, w; σ) = e σ z ∂ z = X 0 (Φ(z, w; σ)). So the infinitesimal generatorof the action is independent of the parameter g. However, the main troublehere is that the algebraicity of the action is necessarily lost since the flow ofX 0 is not algebraic (the reader may check that each right (or left) invariantvector field on an algebraic local Lie group defines in general a nonalgebraicone-parameter subgroup, e.g. for SO(2, R), SL(2, C)).Consequently we may allow the infinitesimal generators of an algebraiclocal Lie group action x ′ = Φ(x; g), defined by X i (x; g i ) :=[∂ gi Φ i ](Φ −1i,g i(x); g i ) to depend on the group parameter g i , even if the families(Φ i,gi (x)) gi ∈K do not constitute one-dimensional subgroups of transformations.2.4. Algebraicity of complex flow foliations. Suppose now that M is a realalgebraic generic submanifold in C n , for instance a hypersurface which isLevi nondegenerate at a “center” point p ∈ M corresponding to the originin the coordinates t = (t 1 , . . .,t n ). Let X ∈ Hol(M) be an infinitesimalCR automorphism. Even if, for fixed real s, the biholomorphicmapping t ↦→ exp(sX)(t) is complex algebraic, i.e. the n components ofthis biholomorphism are complex algebraic functions by Webster’s theorem[We1977], we know by considering the infinitesimal CR automorphismX 1 := i(z + 1)∂ z of the strong tube Im w = |z + 1| 2 + |z + 1| 6 − 2 in C 2passing through the origin, that the flow of X is not necessarily algebraicwith respect to all variables (s, t).Nevertheless, we shall show that the local CR automorphism group ofM is a local algebraic Lie group whose general transformations are of theform t ′ = H(t; e 1 , . . .,e c ), where t ∈ C n and (e 1 , . . ., e c ) ∈ R c and whereH is algebraic with respect to all its variables. Thus the “time” dependentvector fields defined by X i (t; e i ) := [∂ ei H i ](H −1i,e i(t); e i ), where H i,ei (t) :=H i (t; e i ) := H(t; 0, . . ., 0, e i , 0, . . ., 0), have an algebraic flow, simply givenby (t, e i ) ↦→ H i (t; e i ). It follows that each foliation defined by the complexintegral curves of the time dependent complex vector fields X i , i = 1, . . .,c,is a complex algebraic foliation, see §3 below. Now, we can state the maintechnical theorem of this paper, whose proof is postponed to §4, §5 and §6.Theorem 2.1. Let M ⊂ C n be a real algebraic connected geometricallysmooth generic submanifold of codimension d ≥ 1 and CR dimension m =n − d ≥ 1. Let p ∈ M and assume that M is finitely nondegenerate andminimal at p. Then for every sufficiently small nonempty open polydisc ∆ 1centered at p, the following three properties hold: