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

220To conclude this extensive introduction which was designed for readerswanting to quickly embrace the contents, we would like to draw theattention on the work [22], whose manual calculations where finalized inmanuscript form already in 2003 24 , and which will soon confirm the abovetheorem by following another route, viz. by calculating explicitly the socalledChern(-Moser) tensor differential forms, which might interest somecontemporary CR geometers better than the (essentially equivalent) originalCartan-Hachtroudi(-Tanaka) approach.§2. SEGRE VARIETIES AND DIFFERENTIAL EQUATIONSReal analytic hypersurfaces in C n+1 . Let us therefore consider an arbitraryreal analytic hypersurface M in C n+1 with n 2, and let us localizeit around one of its points, say p ∈ M. Then there exist complex affinecoordinates:(z, w) = (z 1 , . . .,z n , w) = ( x 1 + √ −1y 1 , . . .,x n + √ −1 y n , u+iv ) = ( x+ √ −1y, u+ √ −1 v )vanishing at p in which T p M = {u = 0}, so that M is represented in aneighborhood of p by a graphed defining equation of the form:u = ϕ(x, y, v) = ϕ ( x 1 , . . .,x n , y 1 , . . .,y n , v ) ,where the real-valued function:∑ϕ = ϕ(x, y, v) =k∈N n , l∈N n , m∈N|k|+|l|+m2ϕ k,l,m x k y l v m ∈ R { x, y, u } ,which possesses entirely arbitrary real coefficients ϕ k,l,m , vanishes at theorigin: ϕ(0) = 0, together with all its first order derivatives: 0 = ∂ x kϕ(0) =∂ y lϕ(0) = ∂ v ϕ(0). By simply rewriting this initial real equation of M as:w+w= ϕ ( z+z, z−z2 2 2 √ , w−w−1 2 −1) √ ,and then by solving the so written equation with respect to w, one obtainsan equation of the shape:w = Θ ( z, z, w ) ∑= Θ k,l,m z k z l w m ∈ C { z, z, w } ,k∈N n , l∈N n , m ∈ N|k|+|l|+m1whose right-hand side converges of course near the origin (0, 0, 0) ∈ C n ×C n × C and whose coefficients Θ k,l,m ∈ C are complex. Since dϕ(0) = 0,one has Θ = −w + order 2 terms.The paradox that any such complex equation provides in fact two realdefining equations for the real hypersurface M which is one-codimensional,24 At the conference Cauchy-Riemann Analysis and Geometry organized by Ingo Lieband Gerd Schmalz at the Max-Planck Institut of Bonn, 22–27 September 2003, the authorgave a talk the title of which was “Explicit Chern-Moser tensors”.