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

122Theorem 5.2. The minimality of M at 0 is a biholomorphically invariantproperty. It depends neither on the choice of a defining equation for Mnor on the choice of a system of generating complexified CR vector fields(L k ) 1≤k≤m and (L k ) 1≤k≤m . Also, minimality is equivalent to the fact thatthe Lie algebra generated by the complexified CR vector fields (L k ) 1≤k≤mand (L k ) 1≤k≤m spans TM in a neighborhood of 0. Furthermore, thereexists an invariant integer ν 0 , called the Segre type of M at 0 satisfyingν 0 ≤ d + 1 which is the smallest integer such that the mappings Γ k and Γ kare of generic rank equal to 2m+d over ∆ mk (δ k ) for all k ≥ ν 0 +1. Finally,with this integer ν 0 , the odd integer µ 0 := 2ν 0 + 1, called the Segre type Mat 0 is the smallest integer such that the mappings Γ k and Γ k are of rankequal to 2m + d at the origin in ∆ mk (δ k ).Let µ 0 := 2ν 0 + 1 be the Segre type of M at 0 (notice that this is alwaysodd). In the remainder of this section, we assume that M is minimal at 0 andwe exploit the rank condition on Γ k . More precisely we choose a positive ηwith 0 < η ≤ δ µ0 such that Γ µ0 has rank 2m+d at every point of the polydisc∆ mµ0 (η). Without loss of generality, we can also assume that Γ µ0 (∆ mµ0 (η))contains M ∩ (∆ n (ρ 4 ) × ∆ n (ρ 4 )). Simple examples in the hypersurfacecase show that ρ 4