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

Example 1.21. (Continued) With n = m = 1, i.e. M ⊂ C 2 and κ = 1, i.e.M is Levi nondegenerate of equation(1.22) w = ¯w + i z¯z + O 3 ,where z, ¯z are assigned weight 1 and w, ¯w weight 2, B. Segre [Se1931]obtained w zz = F(z, w, w z ). J. Faran [Fa1980] found some examples ofsuch equations that cannot come from a M ⊂ C 2 . But the following wasleft unsolved.Open problem 1.23. Characterize equations y xx = F(x, y, y x ) associated toa real analytic, Levi nondegenerate (i.e. κ = 1) hypersurface M ⊂ C 2 . Canon read the reality condition (1.12) on F ? In case of success, generalize toarbitrary M ⊂ C n+m .Example 1.24. (Continued) Similarly, the system (E 2 ) comes from a Levinondegenerate hypersurface M ⊂ C n+1 ([Ha1937, CM1974, Ch1975,Su2001]. Exercise: why (E 3 ) cannot come from any M ⊂ C ν ?Example 1.25. (Continued) With n = 1, m = 2 and κ = 1, the system (E 4 )comes from a M ⊂ C 3 which is Levi nondegenerate and satisfies(1.26) T c M + [T c M, T c M] + [ T c M, [T c M, T c M] ] = TMat the origin, namely which has equations of the following form, after someelementary transformations ([Be1997, BES2005]):(1.27)w 1 = ¯w 1 + i z¯z + O 4 ,w 2 = ¯w 2 + i z¯z(z + ¯z) + O 4 ,where z, ¯z are assigned weight 1 and w 1 , w 2 , ¯w 1 , ¯w 2 weight 2.Example 1.28. (Continued) With n = 2, m = 1 and κ = 2, the system (E 5 )comes from a hypersurface M ⊂ C 3 of equation ([Eb1998, GM2003b,FK2005a, FK2005b, Eb2006, GM2006]):(1.29) w = ¯w + i 2 z1¯z 1 + z 1 z 1¯z 2 + ¯z 1¯z 1 z 21 − z 2¯z 2 + O 4 ,where z 1 , ¯z 1 , z 2 , ¯z 2 are assigned weight 1 and w, ¯w weight 2, with theassumption that the Levi form has rank exactly one at every point, and withthe assumption that M is 2-nondegenerate at 0.1.30. Jet spaces, contact forms and Frobenius integrability. Throughoutthe present Part I, we assume that the system (E ) is completely integrable,namely that the Pfaffian system naturally associated to (E ) in the appropriatejet space is involutive in the sense of Frobenius. This holds automatically incase (E ) comes from a generic submanifold M ⊂ C n+m . In general, we willconstruct a submanifold of solutions associated to (E ). So, we must explaincomplete integrability.237