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

120over ∆ n (ρ 1 ) × ∆ n (ρ 1 ) by⎧L k := ∂ +⎪⎨ ∂z k(5.2)⎪⎩ L k := ∂ +∂ζ kd∑j=1d∑j=1∂Θ j∂z k(z, ζ, ξ)∂∂w j,∂Θ j∂ζ k(ζ, z, w) ∂∂ξ j,k = 1, . . .,m,k = 1, . . .,m.The reader may check directly that L k (w j − Θ j (z, ζ, ξ)) ≡ 0, whichshows that the vector fields L k are tangent to M . Similarly, L k (ξ j −Θ j (ζ, z, w)) ≡ 0, so the vector fields L k are also tangent to M . Furthemore,we may check the commutation relations [L k , L k ′] = 0 and[L k , L k ′] = 0 for all k, k ′ = 1, . . ., m. It follows from the Frobeniustheorem that the two m-dimensional distributions spanned by each of thesetwo collections of m vector fields has the integral manifold property. Thisis not surprising since the vector fields L k are the vector fields tangent tothe intersection of M with the sets {τ = τ p = ct.}, which are clearly m-dimensional complex integral manifolds. Following [Me1998], [Me2001],we denote these manifolds by S τp := {(t, τ p ) : w = Θ(z, ζ p , ξ p )}, whereτ p is a constant, and we call them complexified Segre varieties. Similarly,the integral manifolds of the vector fields L k are the conjugate complexifiedSegre varieties S tp:= {(t p , τ) : ξ = Θ(ζ, z p , w p )}, where t p is fixed.The union of the manifolds S τp induces a local complex algebraic foliationF of M by m-dimensional leaves. Similarly, there is a second foliation Fwhose leaves are the S tp.The following symbolic picture summarizes our constructions. However,we warn the reader that the codimension d ≥ 1 of the union of the twofoliations F and F in M is not visible in this two-dimensional figure.Mτ pL0F{t = t p}S tpS τpFLt pΛ{τ = τ p}tThe complexification of a real analyticCR-generic manifold M carries twocomplex foliations F L and F L directedby the complefixied CR-vector fields L and Lwhose leaves coincide with the complexifiedSegre varieties S τp and S tp .These leaves also coincide with the intersectionof M with the horizontal slices {τ = τ p}and with the vertical slices {t = t p}.FIGURE 4: GEOMETRY OF THE COMPLEXIFICATION MNow, we introduce the “multiple” flows of the two collections of conjugatevector fields (L k ) 1≤k≤m and (L k ) 1≤k≤m . For an arbitrary point