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

118(1) Every local biholomorphic self-mapping h ∈ H ρ 2,ρ 1M,κ 0 ,εof M (which isdefined on the large polydisc ∆ n (ρ 1 )) is represented by(4.6) h(t) = H(t, J κ 0h(0)),on the smallest polydisc ∆ n (ρ 4 ). In particular, each h ∈ H ρ 2,ρ 1M,κ 0 ,ε isa complex algebraic biholomorphic mapping. Furthermore, the κ 0 -jetof h at the origin belongs to the real algebraic submanifold E, namelywe have C l (J κ 0h(0), J κ 0¯h(0)) = 0, l = 1, . . ., υ.(2) Conversely, shrinking ε if necessary, given an arbitrary jet J κ 0in Ethere exists a smaller positive radius ρ 5 < ρ 4 such that the mappingdefined by h(t) := H(t, J κ 0) for |t| < ρ 5 sends M ∩ ∆ n (ρ 5 ) CRdiffeomorphicallyonto its image which is contained in M ∩ ∆ n (ρ 4 ).We may therefore say that the set H ρ 2,ρ 1M,κ 0 ,εof local biholomorphic selfmappingsof M is parametrized by the real algebraic submanifold E.(3) For every choice of two smaller positive radii ˜ρ 1 ≤ ρ 1 and ˜ρ 2 ≤ ρ 2 with˜ρ 2 < ˜ρ 1 , there exists a positive radius ˜ρ 4 ≤ ρ 4 with ˜ρ 4 < ˜ρ 2 , and a positive˜ε ≤ ε such that the same complex algebraic mapping H(t, J κ 0)as in statement (1) above represents all local biholomorphic selfmappings˜h ∈ H eρ 2,eρ 1M,κ 0 ,eε of M, namely we have ˜h(t) = H(t, J κ 0˜h(0))for all |t| < ˜ρ 4 as in (4.6). Furthermore, the corresponding real algebraictotally real submanifold Ẽ coincides with E in the polydisc{|J κ 0− J κ 0Id| < ˜ε} and it is defined by the same real algebraic equationsC l (J κ 0, J κ 0 ) = 0, l = 1, . . .,υ, as in equation (4.5). In fact,the algebraic mapping H(t, J κ 0) and the real algebraic totally realsubmanifold E depend only on the local geometry of M in a neighborhoodof the origin, namely on the germ of M at 0.(4) The set H ρ 2,ρ 1M,κ 0 ,ε, equipped with the law of composition of holomorphicmappings, is a real algebraic local Lie group. More precisely, let thepositive integer c 0 denote the real dimension of E, which is independentof ρ 1 , ρ 2 and consider a parametrization(4.7) R c 0∋ e = (e 1 , . . .,e c0 ) ↦→ j κ0 (e) ∈ E ⊂ C Nn,κ 0of the real algebraic totally real submanifold E. Then there exist a realalgebraic associative local multiplication mapping (e, e ′ ) ↦→ µ(e, e ′ )and a real algebraic local inversion mapping e ↦→ ι(e) such that if wedefine H(t; e) := H(t, j κ0 (e)), then H(H(t; e); e ′ ) ≡ H(t; µ(e, e ′ ))and H(t; e) −1 ≡ H(t; ι(e)), with the local Lie transformation groupaxioms, as defined in §2.3, being satisfied by H, µ and ι.(5) For i = 1, . . .,c 0 , consider the one-parameter families of transformationsdefined by H(t; 0, . . ., 0, e i , 0, . . .,0) =: H i (t; e i ) =:H i,ei (t). Then for each i = 1, . . .,c 0 , the vector field X i | (t;ei ) :=