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

Redeveloping the determinant, the vanishing of the coefficients of x 1 x 1 , ofx 1 a 1 , of a 1 a 1 yields the system⎧0 ≡ Λ ⎪⎨1,1 Λ 2,0x 2 a− 2 Λ 2,02 aΛ 1,12 x, 2(8.126) 0 ≡ Λ 1,1 Λ 1,1x⎪⎩2 a− Λ 1,12 aΛ 1,12 x− 4 Λ 2,02 aΛ 0,22 x, 20 ≡ Λ 1,1 Λ 0,2x 2 a− 2 Λ 0,22 xΛ 1,12 a. 2Since Λ 1,1 (0) = 2 ≠ 0, we may divide by Λ 1,1 , obtaining a PDE system withthe three functions Λ 2,0x 2 a, Λ 1,12 x 2 a, Λ 0,22 x 2 ain the left hand side. We observe that2the normalizations of Lemma 8.55 entail(8.127)Λ 2,0 = a 2 +O(x 2 a 2 ), Λ 1,1 = 2+O(x 2 a 2 ), Λ 0,2 = x 2 +O(x 2 a 2 ).By cross differentiations in the mentioned PDE system, it follows that all theTaylor coefficients of Λ 2,0 , Λ 1,1 , Λ 0,2 are uniquely determined. As alreadydiscovered in [GM2003b], the unique solution(8.128) Λ 2,0 a 22x 2=1 − x 2 a 2, Λ1,1 =1 − x 2 a 2, Λ0,2 =1 − x 2 a 2,guarantees, when the remainder O 3 (x 1 , a 1 ) vanishes, that the determinant(8.45) indeed vanishes identically.Conversely, suppose that dim SYM(E 5 ) = 10 is maximal.With ε ≠ 0 small, replacing (x 1 , x 2 , y, a 1 , a 2 , b) by(εx 1 , x 2 , ε 2 y, εa 1 , a 2 , εεb) in (8.122) and dividing by εε, the remainderterms become small:(8.129) y = b + 2 x1 a 1 + x 1 x 1 a 2 + a 1 a 1 x 21 − x 2 a 2 + O(ε).Then all the remainders in the equations ∆ E5 of the skeleton are O(ε). Weget ten generators similar to (8.120), plus an O(ε) perturbation. Thanks tothe rigidity of so(5, C), Theorem 5.15 provides a change of generators, closeto the 10×10 identity matrix, leading to the same structure constants as thoseof the ten vector fields (8.120). As in the end of the proof of Theorem 5.13,we may then straighten some relevant vector fields (exercise) and finallycheck that their tangency to the skeleton implies that it is the model one.Theorem 8.117 is proved.Corollary 8.130. Let M ⊂ C 3 be a connected real analytic hypersurfacewhose Levi form has uniform rank 1 that is 2-nondegenerate at every point.Then(8.131) dimhol(M) 10,and the bound is attained if and only if M is locally, in a neighborhood ofZariski-generic points, biholomorphic to the model M 0 .289