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Chern-Moser operators and weighted jet determination problems in higher codimension

Counterexamples to the 2-jet determination Chern-Moser Theorem in codimension d>2 have recently been constructed. We extend the Chern-Moser approach for hypersurfaces to real submanifolds of higher codimension in complex space to derive results on jet determination for their automorphism group. Using these techniques, we show that the 2-jet determination Chern-Moser Theorem holds in codimension 2.


Introduction
The local equivalence problem for real submanifolds in complex spaces is a very natural question, which was started on in complex dimension 2 by H. Poincaré and was then explored in the hypersurface case. Two real submanifolds M and M ′ are said to be locally equivalent at p ∈ M and p ′ ∈ M ′ , respectively, if there exist neighborhoods V of p and V ′ of p ′ , and a biholomorphic mapping F : V → V ′ such that F (V ∩ M ) = V ′ ∩ M ′ . Local equivalence of submanifolds is obviously a very restrictive condition. Symmetrically, if such a F exists, it is submitted to strong constraints, as is shown in the hypersurface case by the following classical statement due to Chern and Moser [10]: if (M, p) and (M ′ , p ′ ) are smooth Levi nondegenerate real hypersurfaces, then biholomorphic germs of equivalence are uniquely determined by their jet of order 2 at point p.
Here we consider a generic submanifold M ⊂ C n+d of codimension d, of finite type in the sense of Kohn and Bloom-Graham at a given point p [1,8,18]; this includes in particular the case of Levi nondegenerate hypersurfaces. We will view M as a C r -smooth perturbation of the generic homogeneous submanifold M H (called the model of M , see Section 2 for details) associated to M and given by: where P = (P 1 , . . . , P d ) with P j a (non-zero) weighted homogeneous polynomial of degree m j with no pluriharmonic terms, and the dots corresponding to P j being C r −smooth functions whose derivatives of weighted order less or equal to m j vanish.
Our first goal was to recover the 2-jet determination theorem stated by Beloshapka in any codimension [4] since the original proof did not work, as we explained in [7]. Our strategy was to extend the theory of the generalized Chern-Moser operator [19,20] to smooth generic submanifolds of higher codimension of finite type: indeed, the kernel of the generalized Chern-Moser operator reflects the link between the weighted grading of hol(M H , p), the Lie algebra of the real-analytic infinitesimal CR automorphisms at p of the model M H of M, and the weighted jet determination problem for the stability group Aut(M, p).
Note that codimension 2 is a specific case, since the same techniques actually led the second author to an example of a quadric of codimension 5 in C 9 for which 4-jet determination (and not less) for biholomorphisms holds [22]. Later on, Jan Gregorovic and the second author [16] gave examples of quadrics with jet determination of arbitrarily high order in codimension d > 2. Our main tool is given by the following theorem (see also Theorem 3.10 for a more explicit statement) Theorem 1.2. Let M H be the generic homogeneous submanifold given by (1). Assume that M H is holomorphically nondegenerate. Then (1) There exists k such that hol(M H , 0) admits the weighted grading (2) For any sufficiently finitely smooth perturbation of M H given by (2), there exists a constant k 0 ≥ 1 depending only on M H such that any h ∈ Aut(M, 0) is uniquely determined by its k + k 0 weighted jets at 0.
We should mention that part (1) in Theorem 1.2 is an immediate consequence of [2]. Also, an inspection of the proof of part (2), based on the Taylor expansion of the holomorphic map h, shows that Theorem 1.2 also holds for formal maps sending M to M, where M is a smooth generic submanifold (or formal submanifold) with holomorphically nondegenerate model M H . Theorem 1.1 then comes from the explicit computation of k for which part (1) holds in Theorem 1.2. A crucial point is to use "integrations" of a vector field (Lemma 4.7 and Definition 4.8). This notion was introduced in [19] for the hypersurface case and was the key point to get the counterexample given in [22].
The paper is organized as follows. In Section 2, we recall the notion of Bloom-Graham finite type and its basic properties. We also define the notions of model generic submanifold M H associated to M, and of weighted coordinates associated to M H . In Section 3, we show how to reduce the study of the weighted jet determination problem for Aut(M, p), the stability group of M, to the study of hol(M H , p), the set of real-analytic infinitesimal CR automorphisms of M H at p (see Theorem 3.10). In Section 4, we introduce the notion of rigid vector fields and prove results regarding the jet determination problem for hol(M H , p) (see Proposition 4.4). In Section 5, we discuss the quadric model case Q and use Theorem 3.10 to prove Theorem 1.1 by describing hol(M H , p) when d = 2.

Preliminaries
Usually, the study of a local CR equivalence problem begins with the choice of appropriate coordinates, merged into bunches according to their geometric contributions, each bunch being assigned a (numerical) weight. For instance, in the case of a Levi nondegenerate hypersurface, the complex normal direction is assigned the weight 2 while the complex tangential directions are assigned the weight 1 (see [10]). In the case of finite multitype in the sense of Catlin [9] at a given point, the complex normal direction is assigned the weight 1 while the complex tangential directions are assigned (possibly different) rational weights µ j , in order to study the generalized Chern-Moser operator, as done recently [10,19,20].
Let M ⊆ C n+d be a smooth generic submanifold of real codimension d > 1 and p ∈ M be a point of finite type m = (m 1 , . . . , m k ), in the sense of Kohn and Bloom-Graham [1,8,18], where m 1 < · · · < m k are the Hörmander numbers. We consider local holomorphic coordinates (z, w) vanishing at p, where z = (z 1 , z 2 , ..., z n ) and z j = x j +iy j , w ∈ C d and w = u+iv. Assuming that the tangent space to M at 0 is given by {v = 0}, M is described near 0 as the graph of a uniquely determined real vector valued function . . , z n ,z 1 , . . . ,z n , u), dψ(0) = 0.
Writing w = (w m 1 , . . . , w m k ), where w m j are vectors of length l j (such that d = k j=1 l j m j ), we may assume that this d-dimensional equation is actually (4) M : where P m j (z,z, u m 1 , . . . , u m j−1 ) are real vector valued polynomials of length l j satisfying the following conditions of normalization • each one of the l j components of P m j is a homogeneous polynomial of degree m j , that is, [8]), and the dots terms are sums of monomials of order strictly bigger than m j in the formal Taylor expansion of ψ m j .
Coordinates which provide such a description will be called standard coordinates, and M given by (4) is said to be written in standard form.
We assign natural weights to the variables: the tangential variables z 1 , . . . , z n are given weight 1 m 1 while the component variables of w m j are given weight We obtain then the notion of weighted homogeneous polynomial is the model submanifold of M associated to the standard form (4).
Note that standard coordinates are not unique. For instance, in the case of m = (m 1 ), all models are equivalent by a linear action as it is shown in the next section.

The basic identities
In this section, M is assumed to be given by (4), with M H the associated model submanifold. We follow the same approach as in [19] and [20], where the hypersurface case is analyzed.
Definition 3.1. We denote by Aut(M, 0) the set of germs at 0 of biholomorphisms mapping M into itself and fixing 0.
) is a sum of terms of weighted degree bigger or equal to m j m 1 (resp. bigger or equal to 1 m 1 ).
Proof. The statement is obvious for f (z, w) and g m 1 (z, w). Indeed, using Hence, g m 1 (z, w) contains no term of weight less than one.
Suppose now that the statement is true for g m k (z, w), k < j, and suppose by contradiction that there is in g m j (z, w) a term of minimal order where c is a real constant. But this can not cancel with any other term, using the conditions of normalization of P ms , s = 1 . . . k. We decompose the formal Taylor expansion of ψ m j , denoted by Ψ m j , into weighted homogeneous polynomials Ψ m j ,ν of weighted degree ν, Let h = (z j ′ , w ′ ) ∈ Aut(M, 0) given by (6). Putting f = (f 1 , . . . , f n ), and g = (g m 1 , . . . , g m k ), we consider the mapping given by (7) T = (f, g), and, again, decompose each power series f j and g m j into weighted homogeneous polynomials f j,µ and g m j ,µ of weighted degree µ, Since h ∈ Aut(M, 0), substituting (6) into v ′ = ψ(z ′ ,z ′ , u ′ ) we obtain the transformation formula (8) ψ(z + f (z, u + iψ(z,z, u)), z + f (z, u + iψ(z,z, u)),u + + Re g(z, u + iψ(z,z, u)) = ψ(z,z, u) + Im g(z, u+iψ(z,z, u)).
If µ 0 > 1, the (non trivial) vector field where M H is given by (5).
Here the notation ∂ ∂w m j stands for the l j -dimensional vector of the corresponding partial derivatives, and g m j · ∂ ∂wm j for the usual dot product.
Definition 3.6. We say that the vector field if F j is a weighted homogeneous polynomial of weighted degree µ + 1 m 1 , and G j is a homogeneous polynomial of weighted degree µ + m j m 1 .
Remark 3.7. We write where G µ consists of weighted homogeneous vector fields of weight µ. Note that each weighted homogeneous component X µ of X is in hol(M H , 0) if X ∈ hol(M H , 0).
Example 3.8. The vector fields W m k ,j , j = 1, . . . , l k , given by Example 3.9. The vector field defined by lies in G 0 .
Theorem 3.10. Let M ⊂ C n+d be a smooth generic submanifold of codimension d that is of finite type at 0 given by (4). Let M H be the model hypersurface given by (5).

Proof.
Using Examples 3.8 and 3.9, we see that − m k m 1 ≤ µ < µ 0 − m k m 1 , with µ 0 > m k m 1 . Then we apply Proposition 3.5. Proof of Theorem (1.2) An inspection of the proof of Proposition 3.5 shows that the conclusion of Theorem 3.10 holds if M is assumed to be of class C m k +1 . Remark 3.11. If m = (2), the conclusion of Theorem 3.10 holds if M is assumed to be of class C 3 .

The components G µ .
Definition 4.1. We denote by G µ R the set of vector fields in G µ that are rigid, that is, whose coefficients depend only on z.
Remark 4.2. Note that W m k ,j are rigid, while E is not.
We recall the following definition We prove that X itself is complex tangent to M H . First, using (4) and in particular the fact that no P mr contains pluriharmonic terms, we obtain that g m l = 0. Then we get by assumption that for any r = 1, . . . , k, By the reality of P mr , we may rewrite (19) as On the other hand, since P mr is of weighted degree m r m 1 , we have First we claim that B ααs, r are zero for all α,α, s, r. By contradiction, assume there is α,α, s, r, with B ααs, r = 0. By assumption, |α| ≥ m r m 1 whereas |α| < m r m 1 , using (23). On the other hand, by (22), we obtain that there exists a nonzero term with weight in z less than m r m 1 , and inz greater than or equal m r m 1 . That gives a contradiction, hence all B ααs, r are zero.
Therefore we obtain that X is complex tangent to M H , and since M H is holomorphically nondegenerate, X = 0.
Let h = (z + f, w + g) ∈ Aut(M, 0) be given by (6). If M H is holomorphically non degenerate, then there exist N 1 , N 2 , N 3 such that g is uniquely determined by the following set of derivatives [3] (24) { ∂ |α| f ∂z α , We have the following lemma whose easy proof is left to the reader. Let Y ∈ G µ \ G µ R and let {W m 1 ,j , j = 1, . . . , d} be given by (15). For every 1 ≤ j ≤ d, there exist an integer k j ≥ 0 and a vector field denoted by where the string of brackets is of length k j .
This leads to the following definition in the case m = (m 1 ).
By abuse of notation, we will also refer to D −k (X) where k = |κ| = d i=1 κ i , since the resulting integrated vector fields will be treated similarly. 5. The components G µ for the quadric.
We wish to discuss Theorem 3.10 in the case of a smooth generic submanifold of codimension d that is of finite type m = (2). Once written in standard form, we get that the model submanifold is a quadric Q, that is, with A j , j = 1, . . . , d being linearly independent Hermitian matrices.
Remark 5.5. We also get that D −1 (G 0 R ) = {0} can only happen if the codimension d is bigger than 2, since for d = 1 Chern Moser's Theorem [10] shows that the mixed derivatives ∂ 2 f j ∂w∂z k , j, k = 1, . . . , n, are not needed.
And it actually happens, as in the following example.
Example 5.6. Let M be given by Let X and Y be the vector fields It is easy to check that Theorem 5.7. Let M ⊂ C n+2 be a smooth generic submanifold of codimension d = 2 that is of finite type m = (2) at 0, written in standard form. Assume the associated model quadric Q is holomorphically non degenerate, then Before proving Theorem 5.7, let us explain how it leads to theorem 1.1, providing a generalization of the 2-jet determination Chern-Moser Theorem in the case of codimension 2.
According to Lemma 4.7, we need to study the κ-integrations of any rigid vector field. Theorem 5.7 provides the precise κ-integrations needed, and shows that at most 2-integrations are needed, depending on the rigid vector field. Using Theorem 3.10 and Remark 3.11, we then conclude that any h = (z + f, w + g) ∈ Aut(M, 0) is uniquely determined by the following partial derivatives • the first complex tangential derivatives ∂f j ∂z k (0), j, k = 1, . . . , n, corresponding to (i) in Theorem 5.7, • the first and second order normal derivatives ∂f j ∂w l (0), ∂g u ∂w l (0), ∂ 2 g u ∂w s ∂w l (0), j = 1, . . . , n, u, l, s = 1, 2 corresponding to (ii) and (iii) in Theorem 5.7.
Proof. In this special case of codimension d = 2 with m = (2), we refer to (26) by setting Q = (Q 1 , Q 2 ) where Q i = tz A i z (i = 1, 2), instead of P m 1 , P m 2 .
• Proof of (i) -Suppose by contradiction that there exist X, Y ∈ G R 0 such that Without loss of generality, we may assume that [X, Y ] = 0, since otherwise, we have w 2 [X, Y ] ∈ D −1 (G 0 R ), which is not possible unless [X, Y ] = 0. Then, by assumption, we obtain the following equation Using the fact that [X, Y ] = 0, and the fact that the model quadric Q is holomorphically nondegenerate, we obtain that X = Y = 0.
• Proof of (ii) -Let Z 1 ∈ G − 1 2 R be of the form with a i ∈ C, and b i (z) linear. If Z 1 integrates, we obtain, after a possible permutation of the variables w 1 and w 2 , an equation of the form (33) Q 1 Im Z 1 (Q) + Q 2 Im Z 2 (Q) + ReX 1 (Q) = 0, where Z 2 ∈ G − 1 2 R and X 1 is a vector field of weight 1 2 . If D −2 (Z 1 ) exists, then we obtain the following system where X j are vector fields of weight 1 2 . It is not hard to see that since Q 1 and Q 2 are linearly independent Hermitian forms, the only solution to the system (34) is the trivial solution. Hence, using Corollary 5.3 and (33), the following system of equations holds where Z 3 ∈ G − 1 2 R . Using (35), we conclude that Z 1 (Q) = 0, and hence Z 1 = 0, which gives the contradiction.
But (37) is not possible by assumption of finiteness. Indeed, (37) would imply Q 1 = αQ 2 , which contradicts that M is of finite type m = (2).