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

(1) The complex Lie algebra Hol(M, ∆ 1 ) is of finite dimension c ∈ Nwhich depends only on the local geometry of M in a neighborhood ofp.(2) There exists a nonempty open polydisc ∆ 2 ⊂ ∆ 1 also centered at pand a C n -valued mapping H(t; e) = H(t; e 1 , . . .,e c ) with H(t; 0) ≡ twhich is defined in a neighborhood of the origin in C n × R c and whichis algebraic with respect to both its variables t ∈ C n and e ∈ R c suchthat for every holomorphic map h : ∆ 2 → ∆ 1 with h(∆ 2 ∩ M) ⊂∆ 1 ∩ M which is sufficiently close to the identity map, there exists aunique e ∈ R c such that h(t) = H(t; e).(3) The mapping (t, e) ↦→ H(t; e) constitutes a K-algebraic local Lietransformation group action. More precisely, there exist a local multiplicationmapping (e, e ′ ) ↦→ µ(e, e ′ ) and a local inversion mappinge ↦→ ι(e) such that H, µ and ι satisfy the axioms of local algebraic Liegroup action as defined in §2.3.(4) The c “time dependent” holomorphic vector fields(2.1) X i (t; e i ) := [∂ ei H i ](H −1i,e i(t); e i ),103where H i,ei (t) := H i (t; e i ) := H(t; 0, . . ., 0, e i , 0, . . ., 0), have algebraiccoefficients and have an algebraic flow, given by (t, e i ) ↦→H i (t; e i ).In the case where M is real analytic, the same theorem holds true with theword “algebraic” everywhere replaced by the word “analytic”.A special case of Theorem 2.1 was proved in [BER1999b] where, apparently,the authors do not deal with the notion of local Lie groups andconsider the isotropy group of the point p, namely the group of holomorphicself-maps of M fixing p. The consideration of the complete local Lie groupof biholomorphic self-maps of a piece of M in a neighborhood of p (notonly the isotropy group of p) is crucial for our purpose, since we shall haveto deal with strong tubes M ∈ Tn d for which the isotropy group of p ∈ Mis trivial. Sections §4, §5 and §6 are devoted to the proof of Theorem 4.1, aprecise statement of Theorem 2.1. We mention that our method of proof ofTheorem 2.1 gives a non optimal bound for the dimension of Hol(M, ∆ 1 ).To our knowledge, the upper bound c ≤ (n + 1) 2 − 1 is optimal only incodimension d = 1 and in the Levi nondegenerate case.§3. PROOF OF THEOREM 1.1We take in this section Theorem 2.1 for granted. As explained in §1.3above, we shall conduct the proof of Theorem 1.1 in two essential steps(§§3.1 and 3.2). The strategy for the proof of Theorems 1.4 and 1.5 is similarand we prove them in §§3.3 and 3.4. Let M ∈ Tn d be a strong tube of