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

274D L 1 L 1 ′ L 2 L 3D 0 −L 1 −L 1 ′ −2 L 2 −3 L 3L 1 L 1 0 −L 2 0 0L 1 ′ L 1 ′ L 2 0 −2 L 3 0L 2 2 L 2 0 2 L 3 0 0L 3 3 L 3 0 0 0 0Table 3.shows that the subalgebra spanned by L 1 , L ′1 , L 2, L 3 is isomorphic tothe unique irreducible 4-dimensional nilpotent Lie algebra n 1 4 ([OV1994,BES2005]). Then SYM(E 4 ) is a semidirect product of K with n 1 4 . Theauthor ignores whether it is rigid. The following accessible research will bepursued in a subsequent publication.Open problem 8.38. Classify systems (E 4 ) up to point transformations. Deducea complete classification, up to local biholomorphisms, of all real analyticgeneric submanifolds of codimension 2 in C 3 , valid at a Zariski-genericpoint.8.39. Almost everywhere rigid hypersurfaces. When studying and classifyingdifferential objects, it is essentially no restriction to assume their Liesymmetry groups to be of dimension 1, the study of objects having no infinitesimalsymmetries being an independent field of research. In particular,if M ⊂ C n+1 (n 1) is a connected real analytic hypersurface, we maysuppose that dimhol(M) 1, at least. So let L be a nonzero holomorphicvector field with L + L tangent to M.Lemma 8.40. ([Ca1932a, St1996, BER1999]) If in addition M is finitelynondegenerate, then(8.41) Σ := { p ∈ M : L (p) ∈ T c pM }is a proper real analytic subset of M.In other words, at every point p belonging to the Zariski-dense subsetM\Σ, the real nonzero vector L (p) + L (p) ∈ T p M supplements Tp cM.Straightening L in a neighborhood of p, there exist local coordinates t =(z 1 , . . .,z n , w) with T0 cM = {w = 0}, T 0M = {Im w = 0}, whence Mis given by Im w = h(z, ¯z, Re w), and with L = ∂ . The tangency of∂w∂+ ∂ = ∂ to M entails that h is indendepent of u. Then the complex∂w ∂ ¯w ∂uequation of M is of the precise form(8.42) w = ¯w + i Θ(z, ¯z),with Θ = 2h simply. The reality of h reads Θ(z, ¯z) ≡ Θ(¯z, z).