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

infinite families of pairwise non biholomorphically equivalent and non locallyalgebraizable hypersurfaces. Our computations for the construction offamilies of manifolds with a control on the structure of their automorphismgroup are all based on the Lie theory of symmetries of differential equations.For the convenience of the reader, we recall briefly the procedure (see[Su2001a,b], [GM2001a,b,c] for more details).7.2. Hypersurfaces and differential equations. Let M be a real analytichypersurface in C n . Assume that M is Levi nondegenerate at one of itspoints p. Then there exist some local holomorphic coordinates (z, w) =(z, u + iv) ∈ C n−1 × C vanishing at p such that M is given by the realanalytic equation(7.1) v = ϕ(z, ¯z, u) = ε 1 |z 1 | 2 + · · · + ε n−1 |z n−1 | 2 + ψ(z, ¯z, u),where ε k = ±1, k = 1, . . ., n − 1 and where ψ = O(3). Passing to theextrinsic complexification M of M, we may consider the variables ¯z and¯w as independent complex parameters ζ ∈ C n−1 and ξ ∈ C. Then theassociated complex defining equation is of the form(7.2) w = Θ(z, ζ, ξ) = ξ + 2i(ε 1 z 1 ζ 1 + · · · + ε n−1 z n−1 ζ n−1 + Ξ(z, ζ, ξ)),where Ξ = O(3). By [Me1998] (cf. §5.1 above), for τ p = (ζ p , ξ p ) fixed, thefamily of complexified Segre varieties S τp := {(t, τ p ) : w = Θ(z, τ p )} isinvariantly and biholomorphically attached to M.Following [Se1931] and [Su2001a,b], we may consider this family as afamily of graphs of the solutions of a second order completely integrablesystem of partial differential equations as follows. By differentiating the leftand the right hand sides of (7.2) with respect to z k , we get(7.3) ∂ zk w = ∂ zk Θ(z, τ) = 2i(ε k ζ k + ∂ zk Ξ(z, τ)),for k = 1, . . .,n − 1. Here, we consider w as a function of z. Usingthe analytic implicit function theorem to solve τ in the 1 + (n − 1) = nequations (7.2) and (7.3), we may express τ in terms of w, of z and of thefirst order derivative w zl , which yields(7.4) τ = Π(z, w, (∂ zl w) 1≤l≤n−1 ),where Π is holomorphic in its variables. If we take the second derivativew zk1 z k2of w and replace the value of τ, we get the desired system of partialdifferential equations:(7.5)∂ 2 z k1 z k2w = ∂ 2 z k1 z k2Θ(z, τ) = ∂ 2 z k1 z k2Θ(z, Π(z, w, (∂ zl w) 1≤l≤n−1 )) =:=: F k1 ,k 2(z, w, (∂ zl w) 1≤l≤n−1 ).133