Stability of the embeddability under perturbations of the CR structure for compact CR manifolds

We study the stability of the embeddability of compact 2-concave CR manifolds in complex manifolds under small horizontal perturbations of the CR structure.

The study of local and global embeddability of CR manifolds in complex manifolds has occupied a large number of mathematicians in the last forty years. Most of the results concern the case of strictly pseudoconvex CR manifolds of hypersurface type, very few is known in the other cases.
The stability of the embeddability property of a CR manifold M was first studied by N. Tanaka, [14], for strictly pseudoconvex CR manifolds of hypersurface type and real dimension greater or equal to 5 embedded in some C N . Few years later R. S. Hamilton, [5] et [6], was interested in the stability of the embeddability property for hypersurfaces provided the perturbation of the original CR structure is the restriction of a perturbation of a complex structure on some complex manifold X of which M is the boundary. It was proved in [8] and [7] that this last condition is satisfied for 2-concave hypersurfaces with a perturbation preserving the contact structure.
Here we consider CR manifolds of higher codimension type with mixed Levi signature and we are interested in the stability of the embeddability of such CR manifolds in complex manifolds under small perturbations of the CR structure preserving the complex tangent bundle.
Our main result is the following theorem: Looking to the proof (see section 4), one can easily see that CP n can be replaced by C N × CP n 1 × · · · × CP n k in Theorem 1 In the case of a general complex manifold X, Theorem 1 was proved by P. L. Polyakov in [12] under the stronger hypothesis that M is 3-concave, and E 0 is a generic embedding, but with a larger loss of regularity. We have to notice that, compare to our situation, Polyakov has no restriction on the kind of the perturbation of the CR structure. For example let us consider the CR manifold M = {(z, ζ) ∈ CP n × CP n | n+1 j=1 z j ζ j = 0} which is 2-concave if n ≥ 3, then by Theorem 1 any sufficiently small horizontal CR perturbation of M is still embeddable in CP n × CP n , but, if n ≥ 4, M satisfies the vanishing cohomological condition (see [12]), and is 3-concave, so by Polyakov's result any sufficiently small CR perturbation (without any restriction) of M is still embeddable in CP n × CP n .
Note that our hypothesis of horizontality of the perturbation allows us to work with anisotropic Hölder spaces and to avoid a Nash-Moser process in the proof of the theorem.
Finally note that in the case of a Levi non degenerate hypersurface, by a theorem of Gray [4] saying that all contact structures on a compact manifold near a fixed contact structure are equivalent, any perturbation of the CR structure can be reduced to an horizontal one (in that case the horizontal perturbations are the perturbations which preserve the natural contact structure). So we can immediately derive from Theorem 1 the following sharp corollary Corollary 2. Let M be a 2-concave non degenerate real quadric of CP n defined by with p ≥ 2, q ≥ 3 and p + q = n. If we equip M with a sufficiently small perturbation of class C l+3 of the CR structure induced by the complex structure of CP n , then the new CR manifold is still embeddable in CP n as a CR submanifold of class C l .
The case of the 1-concave non degenerate quadrics was studied by Biquard in [2]. He proved that the space of obstruction to the stability of the embeddability is infinite dimensional.
The paper is organized as follows: Sections 1 and 2 consist in the description of the general setting and in the definitions of the the main objects used in this paper. Section 3 is devoted to the proof of Theorem 1 in the general case of a complex manifold. We first remark that the problem can be reduced to the solvability of some tangential Cauchy-Riemann equation for the perturbed structure. Using global homotopy formulas with good estimates we are lead to a fixed point theorem, which gives the solution.
In section 4, we consider the case when X = CP n .
In particular, if (M, H 0,1 M) is a CR manifold and f a complex valued function, then f is a CR function if and only if for any L ∈ H 0,1 M we have Lf = 0.
If the almost CR structure is a CR structure, i.e. if it is integrable, and if s ≥ 1, then we can define an operator called the tangential Cauchy-Riemann operator by setting  Therefore we can associate to each ω ∈ H 0 p M an hermitian form on H p M. This is called the Levi form of M at ω ∈ H 0 p M. In the study of the ∂ b -complex two important geometric conditions were introduced for CR manifolds of real dimension 2n − k and CR-dimension n − k. The first one by Kohn in the hypersurface case, k = 1, the condition Y(q), the second one by Henkin in codimension k, k ≥ 1, the q-concavity.
A CR manifold M satisfies Kohn's condition Y (q) at a point p ∈ M for some 0 ≤ q ≤ n − 1, if the Levi form of M at p has at least max(n − q, q + 1) eigenvalues of the same sign or at least min(n − q, q + 1) eigenvalues of opposite signs.
A CR manifold M is said to be q-concave at p ∈ M for some 0 ≤ q ≤ n − k, if the Levi form L ω at ω ∈ H 0 p M has at least q negative eigenvalues on H p M for every nonzero ω ∈ H 0 p M. In [13] the condition Y(q) is extended to arbitrary codimension. Definition 1.3. An abstract CR manifold is said to satisfy condition Y(q) for some 1 ≤ q ≤ n − k at p ∈ M if the Levi form L ω at ω ∈ H 0 p M has at least n − k − q + 1 positive eigenvalues or at least q + 1 negative eigenvalues on H p M for every nonzero ω ∈ H 0 p M.
Note that in the hypersurface type case, i.e. k = 1, this condition is equivalent to the classical condition Y(q) of Kohn for hypersurfaces and in particular if the CR structure is strictly peudoconvex, i.e. the Levi form is positive definite or negative definite, condition Y(q) holds for all 1 ≤ q < n − 1. Moreover, if M is q-concave at p ∈ M, then q ≤ (n − k)/2 and condition Y(r) is satisfied at p ∈ M for any 0 ≤ r ≤ q − 1 and n − k − q + 1 ≤ r ≤ n − k. Definition 1.4. Let (M, H 0,1 M) be an abstract CR manifold, X be a complex manifold and F : M → X be an embedding of class C l , then F is called a CR embedding if dF (H 0,1 M) is a subbundle of the bundle T 0,1 X of the holomorphic vertor fields of X and Let F be a CR embedding of an abstract CR manifold into a complex manifold X and

Perturbation of CR structures
In this section we shall define the notion of perturbation of a given CR structure on a manifold and introduce some new complex associated to the perturbed structure.
The horizontal perturbation H 0,1 M of the CR structure H 0,1 M will be integrable if and only if given which we simply write in the form But going back to the definition of H 0,1 M, this means where ∂ b is the tangential Cauchy-Riemann operator associated to the original CR struc- Let us consider the two operators then a fonction f is CR for the new structure We are lead to consider two pseudo-complex : In degree q ≥ 1, since the two structures H 0,1 M and H 0,1 M are integrable, the operators can be defined in the following way : and if β ∈ C ∞ 0,q (M) and L 1 , . . . , L q+1 ∈ Γ(M, H 0,1 M), we set ) is a differential complex, which is nothing else than the tangential Cauchy-Riemann complex on M associated to the new CR structure H 0,1 M.
which proves the lemma.

Embedding of small horizontal perturbations in complex manifolds
Let (M, H 0,1 M) be an abstract compact CR manifold of class C ∞ and E 0 : Let H 0,1 M be an horizontal perturbation of H 0,1 M, we are looking for an embedding the associated tangential Cauchy-Riemann operator. We will consider only small (the sense will be precised later) perturbations of the original structure, thus it is reasonable to assume that the diffeomorphism We equip the manifold X with some Riemannian metric (for example, if X = CP n , take the Fubini-Study metric). The idea is to look for some F in the subset of the restrictions to M 0 of C l -diffeomorphisms of X parametrized by sections of the vector bundle T X by mean of the exponential map. Let us consider the following diagram where U is a neighborhood of the zero section and σ a section of T X over M 0 In fact E will be a CR embedding if and only if F is CR as a map from M 0 = (M 0 , H 0,1 M 0 ) into X, which means that we have to find a section σ of T X over M 0 such that the image of the new CR structure H 0,1 M 0 by the tangent map to exp • σ is contained in T 0,1 (X × X).
More precisely, using that for all x ∈ M 0 we can write F (x) = π(x, F (x)) = π(exp x σ(x)), with π the second projection from X × X onto X, the map F will be CR if and only if for all vector fields As the differential of the map exp at a point is given by the map (u, ξ) → (u, u + ξ), this is equivalent to (d(Id) + d(σ))(L Φ ) = 0, i.e.
Since d(Id)(L Φ ) = −Φ(L), all that means that we have to solve the equation The remaining of the section will be devoted to the proof of the following theorem:

Reduction to a fixed point theorem
Let E be a CR bundle over M which satisfies H 0,1 (M, E) = 0 and g be a ∂ Φ b -closed (0, 1)-form in C l+2 0,1 ( M, E) and let us consider the following equation: where g is the (0, 1)-form on M relatively to the initial structure H 0,1 M defined by g(L) = g(L − Φ(L)) for L ∈ Γ(M, H 0,1 M).
A natural tool to solve such an equation is a global homotopy formula for the ∂ boperator with good estimates.
Assume M is 2-concave, then by [11], since M is embeddable and 1-concave, M is locally generically embeddable and we may apply the results in [1] and [10] on local estimates and global homotopy formulas for the tangential Cauchy-Riemann operator.
In [1] the following result is proved and T r is continuous as an operator between C l n,r (M ) and C l+1/2 n,r−1 (U ). (ii) If f ∈ C 1 n,r (M ), 0 ≤ r ≤ 1, has compact support in U , then, on U , and in [10] we have derived from the previous proposition a global homotopy formula by mean of a functional analytic construction. such that (i) For all l ∈ N and 1 ≤ r ≤ 2, and A r is continuous as an operator between C l 0,r (M, E) and C l+1/2 In fact we need better estimates than the previous ones to reduce the solvability of our equation 3.2 to a fixed point theorem.
Assume there exist Banach spaces B l (M), l ∈ N, with the following properties : Assume Φ is of class C l+2 , then the map is continuous, and the fixed points of Θ are good candidates to be solutions of (3.1).

A fixed point theorem
In this section we assume that all the assumptions of the previous section are satisfied.
Let δ 0 such that, if Φ l+2 < δ 0 , then the norm of the bounded endomorphism A 1 •Φ ∂ b of B 2l+1 (M, E) is equal to ǫ 0 < 1. We shall prove that, if Φ l+2 < δ 0 , the map Θ admits a unique fixed point, which is a solution of the equation Consider first the uniqueness of the fixed point. Assume v 1 and v 2 are two fixed points of Θ, then and, by the hypothesis on Φ, For the existence we proceed by iteration. We set v 0 = Θ(0) = A 1 ( g) and, for n ≥ 0, v n+1 = Θ(v n ). Then for n ≥ 0, we get Therefore, if Φ l+2 < δ 0 , the sequence (v n ) n∈N is a Cauchy sequence in the Banach space B 2l+1 (M, E) and hence converges to a form v, moreover by continuity of the map Θ, v satisfies Θ(v) = v.
It remains to prove that v is a solution of (3.1). Since H 0,1 (M, E) = 0, it follows from (3.4) and from the definition of the sequence (v n ) n∈N that For any L 1 , L 2 ∈ Γ(M, H 0,1 M), Let us consider the (0, 1)-form u for the new structure defined by u = g − ∂ Φ b v n , the associated (0, 1)-form u for the initial structure satisfies and W Φ (L 1 , L 2 ) depends on Φ at the order 1. Finally and going back to E), moreover, by definition of an horizontal perturbation, for any section L of H 0,1 M the vector field Φ(L) is complex tangent.
Thus it follows by induction that d ′′ Φ v n ∈ B 2l 0,1 (M, E) for all n ∈ N and, if Φ is of class C l+3 , by (3.5) we have the estimate Let δ such that if Φ l+3 < δ, then the maximum A 1 • Φ ∂ b and Φ l+3 A 2 is equal to ǫ < 1. Assume Φ l+3 < δ, then by induction we get (3.8) Since ǫ < 1, the righthand side of (3.8) tends to zero, when n tends to infinity and by continuity of the operator d ′′ Φ from B 2l+1 (M, E) into B 2l 0,1 (M, E), we get that v is a solution of (3.1).

Solution of the embedding problem
In the setting of the beginning of section 2, let us consider the following anisotropic Hölder spaces of functions: -A α (M 0 ), 0 < α < 1, is the set of continuous functions on M 0 which are in C α/2 (M 0 ).
Fix some 0 < α < 1 and set B l (M 0 ) = A l+α (M 0 ). This sequence (B l (M 0 ), l ∈ N) is a sequence of Banach spaces which satisfies properties (i) to (iv) listed in section 3.1 Moreover it is proved in [9] that the operators A r , r = 1, 2, from Theorem 3.3 are linear continuous operators between the anisotropic Hölder spaces B l 0,r (M 0 , E) and B l+1 0,r−1 (M 0 , E). One can also consider the anisotropic Hölder spaces introduced by Folland and Stein when they studied the tangential Cauchy-Riemann complex on the Heisenberg group and more generally on strictly pseudoconvex CR manifolds.
Let M be a generic CR manifold of class C ∞ of real dimension 2n and CR dimension 2n − k and D be a relatively compact domain in M . Let X 1 , . . . , X 2n−2k be a real basis of HM . A C 1 curve γ : where |c j (t)| 2 ≤ 1. The Folland-Stein anisotropic Hölder spaces Γ p+α (D ∩ M ) are defined in the following way: for any admissible complex tangent curve γ through x 0 .
-Γ p+α (D ∩ M ), p ≥ 1, 0 < α < 1, is the set of continuous fonctions in M such that X C f ∈ Γ p−1+α (D ∩ M ), for all complex tangent vector fields X C to M .
Fix some 0 < α < 1, the sequence (Γ p+α , p ∈ N) is also a sequence of Banach spaces which satisfies properties (i) to (iv) listed in section 3.1 Continuity properties for the operators A r and B r , r = 1, 2, defined in Theorem 3.3 are proved in [9]. More precisely, for all p ∈ N and 0 < α < 1, the operators A r , r = 1, 2, from Theorem 3.3 are continuous from Γ p+α 0,r (M ) into Γ p+1+α 0,r−1 (M ). We can apply now the method developed in section 3.
Id and a unique solution is given by a fixed point σ of the map Θ if Φ l+3 is sufficiently small, moreover this fixed point is contained in the neighborhood U of the zero section in T X on which the exponential map is defined, once again if Φ l+3 is small enough, since σ l < Φ l+2 . We deduce that E = F • E 0 , with F defined by F (x) = exp x σ(x), is the embedding we are looking for, which ends the proof of Theorem 3.1.

Stability of embeddability in CP N
In this section we will consider the case when the compact CR manifold (M, H 0,1 M) is embeddable in some CP N . Let E 0 : M → M 0 ⊂ CP N be a C ∞ -smooth CR embedding, it is defined by some homogeneous coordinates (f 1 , . . . , f N ), each f j , j = 1, . . . , N , being a CR function such that on the set U j = {x ∈ M | f j = 0}, ( f 1 f j , . . . , Let us consider an horizontal perturbation H 0,1 M of H 0,1 M, we are looking for a CR embedding E of (M, H 0,1 M) in CP N . As we will consider only small perturbations of the original structure, we are lead to look for some E given by some small perturbation (f 1 − g 1 , . . . , f N − g N ) of the original homogeneous coordinates (f 1 , . . . , f N ), which have to satisfy ∂ Note that the second member of these equations is controlled by the form Φ which defines the perturbation, so that if we can solve the equation ∂ Φ b g = f with C l -estimates uniformly with respect to Φ, then for Φ sufficiently small the homogeneous coordinates (f 1 − g 1 , . . . , f N − g N ) will define the CR embedding we are looking for.

Solving the
We are interested in solving the equation with estimates, uniformly with respect to Φ, in bidegree (0, 1) for an horizontal perturbation, given by a form Φ, of the CR structure of an abstract compact CR manifold (M, H 0,1 M) of class C ∞ , when Φ is sufficiently small. We will follow the method used by [13] leading to a strong Hodge decomposition theorem and homotopy formulas. We equip M with a hermitian metric such that H 0,1 M and H 1,0 M are orthogonal and we consider the Kohn-Laplacian For M equipped with the perturbed structure H 0,1 M, we can also consider the Kohn-Laplacian replacing the ∂ b -operator by the new operator ∂ Φ b . We first establish some a priori estimates for Φ b .
Theorem 4.1. Let (M, H 0,1 M) be an abstract compact CR manifold of class C ∞ and H 0,1 M be an horizontal perturbation of H 0,1 M defined by a (0, 1)-form Φ ∈ C l 0,1 (M, H 1,0 M), l ≥ 1, such that Φ C 0 < 1. Assume that the Levi form of M satisfies condition Y (1) at x 0 ∈ M. Then there exists δ > 0 and a sufficiently small neighborhood U of x 0 such that, if Φ C 1 < δ, for all α ∈ D 0,1 (U ), where . and . s are respectively the L 2 and the Sobolev norms.
. . , n − k, defines a basis of HM. We denote by ω Φ 1 , . . . , ω Φ n−k the dual basis of L Φ 1 , . . . , L Φ n−k , then, for all j = 1, . . . , n − k, we have Let α be a (0, r)-form for the new structure H 0,1 M with smooth coefficients, then . . , i r ) and we have where . . . denote terms which do not involve the derivatives of α.
If r = 0, then and since the perturbation is horizontal, i.e. Φ ∈ C l 0,1 (M, H 1,0 M), we get If r = 1, then where R(Φ, α) is controlled by (1 + Φ C 1 ) α , and if (L Φ j ) * denotes the Hilbert adjoint . The a priori estimates proved in Theorem 3.1 of [13] gives the existence of a constant C 0 such that for any α ∈ C ∞ 0,r (M), r = 0, 1, the following estimate is satisfied Using the previous calculations we get the existence of two constants C and C ′ such that Choose δ < 1 sufficiently small such that C ′ δ < 1, then if Φ C 1 < δ there exists a constant C ′′ such that This implies the theorem since under the condition Y(1) the real and the imaginary part of the vector fields L j satisfy Hörmander's finite type condition of type 2.
In the spirit of the pionnier works by Kohn-Nirenberg and Folland-Kohn, following the methods used in section 3 of [13], the a priori estimates obtained in Theorem 4.1 gives the existence and the regularity of solutions for the Φ b and the ∂  (i) For all s ∈ N and r = 0, 1, G Φ b is continuous from W s 0,r (M) into W s+1 0,r (M), more precisely there exists δ > 0 such that, if Φ C 2+s < δ, there exists a constant C s independent of Φ such that G Φ b (α) s+1 ≤ C s α s , for α ∈ W s 0,r (M). (ii) For any f ∈ L 2 (M)

Stability theorem
Let us go back to the setting of the beginning of Section 4. Note that if the CR manifold (M, H 0,1 M) is 2-concave its Levi form satisfies condition Y(1) and so we can apply the previous results on the solvability of the ∂ Assume that Φ C l+2 is sufficiently small to apply Corollary 4.3, then there exists a constant C independent of Φ such that if g j satisfies (4.3) then and taking Φ even smaller (f 1 − g 1 , . . . , f N − g N ) will define homogeneous CR coordinates on (M, H 0,1 M). So we have proved Then there exists a positive real number δ such that if Φ C l+2 < δ, then the CR manifold (M, H 0,1 M) is embeddable in CP n as a CR submanifold of class C l .