A note on CR mappings of positive codimension

We prove the following Artin type approximation theorem for smooth CR mappings: given M a connected real-analytic CR submanifold in C^N that is minimal at some point, M' a real-analytic subset of C^N', and H:M->M' a smooth CR mapping, there exists a dense open subset O in M such that for any q in O and any positive integer k there exists a germ at q of a real-analytic CR mapping H^k:(M,q)->M' whose k-jet at q agrees with that of H up to order k.


Introduction
Given germs of real-analytic submanifolds M and M ′ embedded in complex spaces, a fundamental question is to decide whether the formal equivalence of M and M ′ implies their biholomorphic equivalence. While this need not be in general the case in view of a well known example due to Moser-Webster [12] (see also [7,8]), recent results due to Baouendi, Mir, Rothschild and Zaitsev [5,3] provide a partial positive answer when the submanifolds are furthermore assumed to be CR. In [5,3], the positive solution is obtained by approximating in the Krull topology a given formal holomorphic equivalence by a convergent one, following the spirit of Artin's approximation theorem [1]. In this paper, we prove the following Artin type approximation theorem for arbitrary smooth CR mappings of any positive codimension. Theorem 1.1. Let M ⊂ C N be a connected real-analytic CR submanifold that is minimal at some point, M ′ ⊂ C N ′ be a real-analytic subset, and H : M → M ′ be a C ∞ -smooth CR mapping. Then there exists a dense open subset O ⊂ M such that for any q ∈ O and any positive integer k there exists a germ at q of a real-analytic CR mapping H k : (M, q) → M ′ whose k-jet at q agrees with that of H up to order k.
Here minimality is meant in the sense of Tumanov (see Section 2 for the precise definition). To the author's knowledge, Theorem 1.1 is the first result of its kind for mappings of positive codimension between arbitrary real-analytic submanifolds. When the target is a real-algebraic set instead of a real-analytic set, then Theorem 1.1 follows from the work of Meylan, Mir and Zaitsev [10]. Observe that Theorem 1.1 is also new even in the case N = N ′ since there is no rank assumption on the mapping under consideration (compare with [5,3,13]). On the other hand, we do not know whether one may choose in Theorem 1.1 the dense open subset O ⊂ M to be a Zariski open subset independent of the mapping H. Note that when M ′ is real-algebraic, such a choice is possible and follows from the main result of [10]. For more details related to Artin type approximation in CR geometry, we refer the reader to the survey paper [11].
In this paper we shall give a rather elementary and self-contained proof of Theorem 1.1. For this, we will use several main steps of [6] for which we will provide simplified proofs of the results needed for this paper.
We will organize the paper as follows. Section 2 contains some basic definitions and technical lemmas used in Section 3. In Section 3, we give some elementary properties of a complexanalytic set invariantly attached to a graph of a smooth CR-mapping. The last section is devoted to the proof of Theorem 1.1.

Preliminaries
In this section we first recall some basic definitions and prove a lemma used in Section 3. For basic background on CR analysis, we refer the reader to [2]. Let M ⊂ C N be a realanalytic generic submanifold of codimension d. Let us recall that M is said to be minimal at p ∈ M if there is no germ of a real submanifold S ⊂ M through p such that the complex tangent space of M at q is tangent to S at every q ∈ S and dim R S < dim R M (see [2]).
Following [6], for a C ∞ -smooth CR mapping f : M → C N ′ and for p ∈ M , we denote T p (f ) as the germ of the smallest complex analytic set in C N +N ′ containing the germ of the graph of f at (p, f (p)). The integer dim T p (f ) − N will be called the degree of partial analyticity of f at p and denoted by deg p f . We may observe that, if M is minimal at p, the degree of partial analyticity of f at p is non-negative (see Remark 3.2).
We will need the following well known reflection principle (see e.g. [4,9]). In what follows, we say that a C ∞ -smooth mapping h : Ω → C l , with Ω being a real manifold, is not identically zero near a point p 0 if the germ of h at p 0 is non-zero, i.e. if we may find points p as close as we want to p 0 such that h(p) = 0.
We will also use the notion of wedge. For p ∈ M , we consider an open neighborhood U of p in C N and a local defining real-analytic function ρ : U → R d of M near p. If Γ is an open convex cone in R d with vertex at the origin, an open set W of the form {z ∈ U, ρ(z,z) ∈ Γ} is called a wedge of edge M in the direction Γ centered at p.
The following result is a lemma from [6] for which we provide a more elementary proof.
Lemma 2.2. Let M ⊂ C N be a real-analytic generic submanifold, minimal at p ∈ M , let F : (M, p) → C s and u : (M, p) → C t be two germs of C ∞ -smooth CR mappings and let ψ : function. Assume that ψ z,z, u(z), F (z) ≡ 0 for z ∈ M near p and that the function Then there exists q ∈ M as close as we want to p such that deg q F < s.
Proof. This result will be proved by induction on the integer s. First, we consider the case where s = 1.
Let A be the set of points p 1 near p in M such that the holomorphic function C ∋ w → ψ p 1 , p 1 , u (p 1 ), w is not identically zero near F (p 1 ) in C. We may find p 1 ∈ A as close as we want to p. Indeed, since ψ z,z, u (z), w is not identically zero near (p, F (p)) in M ×C, there exists (p 1 , p ′ 1 ) as close as we want to (p, F (p)) such that ψ p 1 , p 1 , u (p 1 ), p ′ 1 = 0. So, the holomorphy of C ∋ w → ψ p 1 , p 1 , u (p 1 ), w implies it cannot be identically zero near F (p 1 ) in C. Moreover, for z ∈ A fixed, since the holomorphic function C ∋ w → ψ z, z, u (z), w doesn't vanish identically near F (z) in C, there exists a unique positive integer k z such that ∂ kz ψ ∂w kz z, z, u (z), F (z) = 0 and, for any integer k < k z , ∂ k ψ ∂w k z, z, u (z), F (z) = 0. Now we fix a sufficiently small open neighborhood V of p in M and consider the integer We may pick p 1 ∈ A ∩ V such that k p 1 = K, and we have Since ψ is holomorphic near p,p, u (p), F (p) in C 2N +t+1 by the implicit function theorem, there exists a germ at p 1 , p 1 , u (p 1 ) of a holomorphic function are given by the equation w = Θ (z, ζ, ν). On the other hand, we may observe that the function is identically zero near p 1 in M . Suppose, in order to reach a contradiction, that it is false. In this case, we may find p 2 as close as we want to p 1 in M such that and, from the remark on the zeros is close enough to p, we may assume that M is minimal at p 1 (since M is real-analytic and minimal at p), and consequently we may apply Proposition 2.1 to obtain the existence of a holomorphic extension F of F near p 1 in C N . Thus the graph of F is contained, near p 1 , in the graph of F , which is a complex analytic set of dimension N . Consequently, the dimension of T p 1 (F ) is less than or equal to N . So we proved that, for an arbitrary small neighborhood V of p in M , there exists p 1 ∈ V such that deg p 1 F ≤ 0. This finishes the proof of the lemma for s = 1. Now, we assume that the lemma holds for s − 1, and for any t ∈ N, any germs of CR mappings F : (M, p) → C s−1 , u : (M, p) → C t and any germ at p,p, u (p), F (p) of a holomorphic function ψ : C 2N +t+s−1 , p,p, u (p), F (p) → C. Our aim is to prove the same result for s.
First, we consider the case where ψ z,z, u (z), F ′ (z) , w s ≡ 0 for (z, w s ) ∈ M × C near (p, F s (p)). Taking the Taylor series of ψ in w s at F s (p), we obtain that, for any k ∈ N, ψ k z,z, u (z), F ′ (z) ≡ 0 for z ∈ M near p and that there exists k 0 such that ψ k 0 z,z, u (z), w ′ doesn't vanish identically near (p, F ′ (p)) in M × C s−1 . So, by the induction hypothesis, there exists q ∈ M as close as we want to p such that deg q F ′ < s − 1, which implies deg q F < s. This completes the proof for this case.
To finish the proof, we have to consider the case where ψ z,z, u (z), F ′ (z) , w s doesn't vanish identically near (p, F s (p)) in M × C. By the same method as in the case s = 1, we show that, for point p 1 as close as we want to p where M is minimal, there exists a germ at If every Θ α z,z, u (z) is CR near p 1 in M , then by Proposition 2.1 (recall that M is minimal at p 1 ) the function M × C s−1 ∋ (z, w ′ ) → Θ z,z, u (z), w ′ ∈ C can be holomorphically extended near (p 1 , F ′ (p 1 )) in C N +s−1 . We denote the extension by Θ. The graph of F is contained in the graph of Θ, which is a complex submanifold of C N +s of dimension N + s − 1. This is equivalent to saying that the degree of partial analyticity of F at p 1 is smaller than s − 1.
If there is a multi-index α ∈ N s−1 such that the mapping Θ α z,z, u (z) is not CR, then there exists a vector fieldL = N j=1 a j (z,z) ∂ ∂z j near p in C N , where a 1 , . . . , a N are realanalytic functions near p, such thatL| M is a CR vector field and L Θ z,z, u (z), w ′ ≡ / 0 near (p 1 , F ′ (p 1 )) in M × C s−1 . Using the chain rule, we may observe that there exists a holomorphic function Ψ 1 near p 1 , p 1 ,L u(z) z=p 1 , F ′ (p 1 ) in C 2N +t+s−1 such that (2.5)L Θ z,z, u (z), w ′ = Ψ 1 z,z,L u(z) , w ′ near (p 1 , F ′ (p 1 )) in M × C s−1 . Since M is minimal at p, Tumanov's extension theorem (see [2]) implies that there exists a holomorphic extensionũ of u in a wedge W of edge M centered at p. We may assume that the mappingũ is C ∞ -smooth on W up to the edge M . Moreover, for any z ∈ W ∪ (M ∩ V ), where V is a sufficiently small neighborhood of p in C N , Now, for any j ∈ {1, . . . , N }, ∂ũ ∂z j is holomorphic in W and C ∞ up to the edge M . Thus, the mapping U whose components are the restrictions to M near p of the derivatives ofũ is a CR mapping near p in M . Consequently, from the identity (2.5), and sinceL(u(z)) =L(ũ(z)), for z ∈ M close enough to p, we may find a germ at p 1 , p 1 , U (p 1 ), F ′ (p 1 ) of a holomorphic function Ψ :

the induction assumption implies
that the degree of partial analyticity of F ′ is strictly smaller than s − 1 for points in M as close as we want to p 1 and therefore as close to p. This finishes the proof of Lemma 2.2.
As in [6], one gets from Lemma 2.2 the following result. be a germ at p,p, F (p), F (p) of a holomorphic function. Assume that ψ z,z, F (z), F (z) ≡ 0 for z ∈ M near p and that the function (z, v, w) → ψ(z,z, v, w) is not identically zero near p, F (p), F (p) in M × C 2s . Then, there exists q ∈ M as close as we want to p such that deg q F < s.
Proof. First, we assume that ψ z,z, F (z), w does not vanish identically near (p, F (p)) in M × C s . Since ψ z,z, F (z), F (z) ≡ 0 for z ∈ M near p, we may apply Lemma 2.2 and deduce that the degree of partial analyticity of F at q is strictly smaller than s for q arbitrarily close to p.
Now we treat the case where ψ z,z, F (z), w ≡ 0 for (z, w) ∈ M × C s near (p, F (p)). For this we consider the Taylor series of ψ in w at F (p): The assumption implies that, for any α ∈ N s , ψ α z,z, F (z) ≡ 0 near p in M . However, by assumption, there is a multi-index α 0 such that ψ α 0 (z,z, v) ≡ / 0 near p, F (p) in M × C s , and Lemma 2.2 gives the desired result.

Properties of T p (H)
In this section, we fix a real-analytic generic submanifold M ⊂ C N and a C ∞ -smooth CR mapping H : M → M ′ on M with values in a real-analytic set M ′ ⊂ C N ′ . We shall give some properties of the degree of partial analyticity of H and of the complex analytic set T p (H) for p ∈ M . All the results of this section can be found in [6], but since the proofs we shall give are rather elementary compared to [6], we include them in this note for completeness.
The following lemma is a direct consequence of the boundary uniqueness theorem.   H(p)). This implies that the dimension of T p (H) at p is greater than N , i.e. that the degree of partial analyticity of H at p is non-negative.
The two following lemmas describe the regular points of the complex analytic set T p (H). be the set of points q ∈ U 1 p for which T p (H) is not regular at (q, H(q)). In view of the classical definition of regular points of a complex analytic set, Σ 1 p is a closed subset of M ∩ U 1 p . To prove that its interior is empty, assume by contradiction that we may find an open subset V of M contained in Σ 1 p . Thus, for any q ∈ V , the graph of H near (q, H(q)) is contained in the set of the singular points of T p (H), which is a complex analytic set of dimension strictly smaller than N + s. So, the dimension of T q (H) is also strictly smaller than N + s for any q ∈ V . This is impossible, since the degree of partial analyticity of H is constant equal to s on V .
Lemma 3.4. In the above setting, assume that M is minimal at p ∈ M , that the degree of partial analyticity of H is constant equal to s near p in M , and write t = N ′ − s. Then there are an open neighborhood U 2 p ⊂ U 1 p of p in M (with U 1 p given by Lemma 3.3) and a closed set with empty interior Σ 2 p ⊂ M ∩ U 2 p such that, for any q ∈ U 2 p \ Σ 2 p , there are holomorphic coordinates (u ′ , v ′ ) ∈ C s × C t near H(q) for which H = (F, G) ∈ C s × C t , and a germ at (q, F (q)) of a holomorphic mapping T q : C N +s , (q, F (q)) → C t such that T p (H) is given near (q, H(q)) by the equation v ′ = T q (z, u ′ ). In particular T p (H) is regular at (q, H(q)).
Proof. Since M is minimal at p and H is CR on M , Tumanov's extension theorem implies that there exists a holomorphic extension H of H in a wedge W of edge M centered at p which is C ∞ -smooth up to the edge. By Lemma 3.1 the graph of H is contained in T p (H) near (p, H(p)). It means that we may choose an open neighborhood ∆ of p in C N such that (z, H(z)) ∈ T p (H) for any z ∈ ∆ ∩ W.
On the other hand, if U 1 p is the open neighborhood of p given by Lemma 3.3, we define U 2 p = U 1 p ∩ ∆ and Σ 2 p = Σ 1 p ∩ U 2 p . Now, for a fixed point q ∈ U 2 p \ Σ 2 p , T p (H) is regular at We fix a point q ∈ U 3 p \ Σ 3 p and a positive integer k. Since the graph of H is contained in T p (H) near (q, H (q)), we have the following mapping identity: near q in M . Let F k be the k-th order Taylor polynomial of F at q (that is holomorphic since H is CR; see [2]). We define the holomorphic mapping G k near q in C N by setting Thus H k = F k , G k is a holomorphic mapping near q in C N with values in C N ′ . Moreover, by definition of H k and from (4.1), the k first derivatives of H k at q coincide with that of H. So to complete the proof of Proposition 4.1, we have to show that H k sends M into M ′ near q. From the definition of H k and the local defining equation of T p (H) near (q, H (q)) in C N +N ′ , we obtain that there exists an open neighborhood Ω q of (q, H (q)) in C N +N ′ such that where G H k is the graph of the mapping H k . Since π ′ (T p (H) | M ∩ Ω q ) ⊂ M ′ by Lemma 3.5, the conclusion of the proposition follows.
Now, we are able to prove our main result, Theorem 1.1.
Proof of Theorem 1.1. We first note that Theorem 1.1 holds in the case where N = 1 or N ′ = 1. We may therefore assume that N, N ′ ≥ 2.
We first treat the case where M is generic. Since M is real-analytic and connected, there exists a real-analytic subvariety Σ 1 of M such that M is minimal at each point p ∈ M \ Σ 1 . Moreover, from Lemma 3.6, there exists a closed set with empty interior Σ 2 ⊂ M such that the degree of partial analyticity of H is constant on each connected components of M \ Σ 2 . Thus, Σ = Σ 1 ∪ Σ 2 is a closed subset of M with empty interior. Fix a point p ∈ M \ Σ; by Proposition 4.1, we may find an open subset U 3 p of p in M and a closed set with empty interior Σ 3 p ⊂ M ∩ U 3 p such that, for every q ∈ U 3 p \ Σ 3 p , the conclusion of Theorem 1.1 holds at q. Thus, the set O = p∈M \Σ U 3 p \ Σ 3 p does the job, and this finishes the proof of Theorem 1.1 for the generic case.
If M is not generic, for any p ∈ M \Σ 1 (where Σ 1 again denotes the set of nonminimal points of M ) we may assume, thanks to a local holomorphic change near p, that M = M p × {0} ⊂ C N −r 1 z 1 × C r 1 z 2 , where r 1 is a non-negative integer and M p is a connected real-analytic generic submanifold which is minimal (see [2]). From the generic case treated above, there exists a dense open subset O p ⊂ M p such that, for any non-negative integer k and anyq ∈ O p , there exists a germ atq of a real-analytic CR mapping H k 1 : ( M p , q) → M ′ whose k-jet atq agrees with that of M p ∋ z 1 → H(z 1 , 0). Since p∈M \Σ 1 O p × {0} is a dense open subset of M , the proof of Theorem 1.1 is complete.