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

140Lemma 7.5. The Lie algebra Hol(M χ ) of the real analytic hypersurfacesM χ ⊂ C 2 of equation v = z¯z + z 5¯z 5 (z + ¯z) + z 10¯z 10 χ(z, ¯z) is onedimensionaland generated by ∂ w . Furthermore M χ is biholomorphicallyequivalent to M ′ χ ′ if and only if χ = χ′ .Proof. The derivatives ∂ z w and ∂ 2 zw of w with respect to z are given by :{ 2 ∂z w = 2i¯z + 12iz 5¯z 5 + 10iz 4¯z 6 + O(¯z 10 )),(7.27)∂ 2 zz w = 60iz4¯z 5 + 40iz 3¯z 6 + O(¯z 10 ).Replacing ¯z in the second equation by its expression given by the first equationwe obtain the following second order differential equation, interpretedin the jet space :(7.28)W 2 = [15z 4 /8] (W 1 ) 5 − [5iz 3 /8] (W 1 ) 6 − [225z 9 /64] (W 1 ) 9 + O((W 1 ) 10 ).Solving the partial differential equations involving Q, ∂ z Q, ∂ w Q, ∂ z R and∂ w R given in the coefficients of (W 1 ) 4 , (W 1 ) 5 , (W 1 ) 6 , (W 1 ) 7 and (W 1 ) 9we obtain Q ≡ ∂ z Q ≡ ∂ w Q ≡ ∂ z R ≡ ∂ w R ≡ 0 which is the desired information.Finally, proceeding exactly as in the end of the proof of Lemma 7.4,we see that the M χ are pairwise biholomorphically not equivalent.The dimension of Hol(M) for the five examples of Corollary 1.3, for theseven examples of Theorem 1.4, for the seven examples of Corollary 1.7and for the hypersurface v = e z¯z − 1 at a point p with z p ≠ 0 was computedwith the package diffalg of Maple Release 6. Since at a pointp with z p ≠ 0 the hypersurface v = e z¯z − 1 is biholomorphically equivalentto the hypersurface M a of equation v = ϕ a (y) := e a(ey−1) − 1with a = |z p | 2 , this defines a strong tube. Applying Theorem 1.1 andLemma 3.3, we see that M a is not locally algebraizable at the origin, becauseϕ a yy (y) = aey e a(ey−1) + a 2 e 2y e a(ey−1) and ϕ a y (y) = aey e a(ey−1) arealgebraically independent. Finally all the examples of Corollary 1.3, Theorem1.4 and Corollary 1.6 are not locally algebraic since they satisfy therequired transcendence conditions.§8. ANALYTICITY VERSUS ALGEBRAICITYIntuitively there seems to be much more analytic mappings, manifoldsand varieties than algebraic ones. Our goal is to elaborate a precise statementabout this. By complexification, every local real analytic object yields a localcomplex analytic object, so we shall only work in the holomorphic category.Let ∆ n be the complex polydisc of radius one in C n and ∆ n its closure.Let k ∈ N. We consider the space O k (∆ n ) := O(∆ n ) ∩ C k (∆ n ) of holomorphicfunctions extending up to the boundary as a function of class C k. This is a Banach space for the C k norm ||ϕ|| k := ∑ kl=0 sup z∈∆ n|ϕ z l(z)|.