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

Then the sum L + L is tangent to M and its flow stabilizes the two invariantfoliations, obtained by intersecting M by {(z, w) = cst.} or by{(ζ, ξ) = cst.}. In [Me2005a, Me2005b], these two foliations, denotedF , F , are called (conjugate) Segre foliations, since its leaves are the complexificationsof the (conjugate) classical Segre varieties ([Se1931, Pi1975,Pi1978, We1977, DW1980, BJT1985, DF1988, BER1999, Su2001, Su2002,Su2003, GM2003a]) associated to M, viewed in its ambient space C n+m .The next definition is also classical ([Be1979, Lo1981, EKV1985, Kr1987,KV1987, Be1988, Vi1990, St1996, Be1997, BER1999, Lo2002, FK2005a,FK2005b]):Definition 6.60. By hol(M) is meant the Lie algebra of local holomorphicvector fields L = ∑ ni=1 X i (z, w) ∂ + ∑ m∂z i j=1 Y j (z, w) ∂ whose real∂w jflow exp ( tL ) (z, w) induces one-parameter families of local biholomorphictransformations of C n+m stabilizing M. Equivalently,(6.61) 2 ReL = L + Lis tangent to M. Again equivalently, L + L is tangent to M = M c .Then obviously hol(M) is a real Lie algebra.Theorem 6.62. ([Ca1932a, BER1999, GM2004]) The complexificationhol(M) ⊗ C identifies with SYM ( E (M c ) ) . Furthermore, if M is finitelynondegenerate and minimal at the origin, both are finite-dimensional andhol(M) is totally real in SYM ( E (M c ) ) .The minimality assumption is sometimes presented by saying that the Liealgebra generated by T c M generates TM at the origin ([BER1999]). However,it is more natural to proceed with the fundamental pair of foliationsassociated to M ([Me2001, GM2004, Me2005a, Me2005b]). AnticipatingSections 10 and 11 to which the reader is referred, we set.Definition 6.63. A real analytic generic submanifold M ⊂ C n+m is minimalat one of its points p if the fundamental pair of foliations of its complexificationM is covering at p (Definition 10.17).Further informations may be found in Section 10. We conclude by formulatingapplications of Theorems 5.13 and 5.24.Corollary 6.64. The bound dim hol(M) 8 for a Levi nondegenerate hypersurfaceM ⊂ C 2 is attained if and only if it is locally biholomorphic tothe sphere S 3 ⊂ C 2 .Corollary 6.65. The bound dim hol(M) n 2 + 4n + 3 for a Levi nondegeneratehypersurface M ⊂ C n+1 is attained if and only if it is locallybiholomorphic to the sphere S 2n+1 ⊂ C n+1 .263