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

coordinates z ′ = 2i ln(z/z p ), w ′ = (w − w p ) e −|zp|2 , applying Theorem 1.1and inspecting the function e |zp|2 (e y′ −1) −1, we may check that it is not algebraizableat such points p with z p ≠ 0 (see §7.5). It follows trivially that thehypersurface M HJY is also not locally algebraizable at all the points p withz p = 0, giving the result of [HJY, Theorem 1.1]. Using the same strategy asfor Theorem 1.1, we obtain more generally the following criterion:Theorem 1.4. Let M ϕ : v = ϕ(z¯z) be a Levi nondegenerate real analytichypersurface in C 2 passing through the origin whose Lie algebra of localinfinitesimal CR automorphisms is generated by ∂ w and iz ∂ z . If M ϕ is locallyalgebraizable at the origin, then the first derivative ∂ r ϕ in ϕ (r ∈ R)is algebraic. For instance, the following seven explicit examples are not locallyalgebraizable at the origin : v = e z¯z − 1, v = sin(z¯z), v = tan(z¯z),v = sinh(z¯z), v = tanh(z¯z), v = sin(sin(z¯z)) and v = e ez¯z −1 − 1.Finally, using the same recipe as for Theorems 1.1 and 1.4, we shall providea very simple criterion for the local nonalgebraizability of some hypersurfaceshaving a local Lie CR automorphism group of dimension equal toone exactly. We consider the class R n of Levi nondegenerate real analytichypersurfaces passing through the origin in C n (n ≥ 2) such that the Lie algebraof infinitesimal CR automorphisms of M is generated by exactly oneholomorphic vector field X 1 with holomorphic coefficients not all vanishingat the origin. We call R n the class of strongly rigid hypersurfaces, in orderto distinguish them from the so-called rigid ones whose local CR automorphismgroup may be of dimension larger than 1. By straightening X 1 , wemay assume that X 1 = ∂ w and that M is given by a real analytic equationof the form v = ϕ(z, ¯z) = ϕ(z 1 , . . .,z n−1 , ¯z 1 , . . ., ¯z n−1 ). By making someelementary changes of coordinates (cf. §3.3), we can furthermore assumewithout loss of generality that ϕ(z, ¯z) = ∑ n−1k=1 ε k |z k | 2 + χ(z, ¯z), whereε k = ±1 and χ(0, ¯z) ≡ ∂ zk χ(0, ¯z) ≡ 0.Theorem 1.5. Let M : v = ϕ(z, ¯z) = ∑ n−1k=1 ε k |z k | 2 +χ(z, ¯z) be a stronglyrigid hypersurface in C n with χ(0, ¯z) ≡ ∂ zk χ(0, ¯z) ≡ 0. If M is locallyalgebraizable at the origin, then all the first derivatives ∂ zk ϕ are algebraicfunctions of (z, ¯z).This criterion enables us to exhibit a whole family of non locally algebraizablehypersurfaces in C n :Corollary 1.6. The rigid hypersurfaces M χ1 ,...,χ n−1in C n of equation v =∑ n−1k=1 [ε k |z k | 2 + |z k | 10 + |z k | 14 + |z k | 16 (z k + ¯z k ) + |z k | 18 |z 1 | 2 · · · |z k−1 | 2 +|z k | 2n+16 χ k (z, ¯z)], where the χ k are arbitrary real analytic functions, belongto the class R n of strongly rigid hypersurfaces. Two such tubesM χ1 ,...,χ n−1and M bχ1 ,...,bχ n−1are biholomorphically equivalent if and only if97