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Formal biholomorphic maps of real analytic hypersurfaces

Let f : (M,p)\to (M',p') be a formal biholomorphic mapping between two germs of real analytic hypersurfaces in \C^n, p'=f(p). Assuming the source manifold to be minimal at p, we prove the convergence of the so-called reflection function associated to f. As a consequence, we derive the convergence of formal biholomorphisms between real analytic minimal holomorphically nondegenerate hypersurfaces. Related results on partial convergence of formal biholomorphisms are also obtained.


Introduction
A formal (holomorphic) mapping f : (C n , p) → (C n , p ′ ), p, p ′ ∈ C n , n ≥ 1, is a vector (f 1 (z), . . . , f n (z)) where each f j (z) ∈ C[[z −p]], the ring of formal holomorphic power series in z − p, and f (p) = p ′ . The mapping f is called a formal biholomorphism if its Jacobian does not vanish at p. If M, M ′ are two smooth real real-analytic hypersurfaces in C n through p and p ′ respectively, we say that a formal mapping f as above sends M into M ′ if ρ ′ (f (z), f (z)) = a(z,z)ρ(z,z), where ρ, ρ ′ are local real-analytic defining functions for (M, p) and (M ′ , p ′ ) respectively and a ∈ C[[z − p,z −p]]. In this paper we study the convergence (and partial convergence) of formal biholomorphic mappings between germs of real analytic hypersurfaces in C n in terms of optimal and natural geometric conditions on the source and target manifolds.
A natural geometric condition which appears in this regularity problem is the concept of holomorphic nondegeneracy. Following Stanton, a real analytic hypersurface M ⊂ C n is called holomorphically nondegenerate if, near any point in M , there is no non-trivial holomorphic vector field, with holomorphic coefficients, tangent to M near that point [16,2]. Baouendi and Rothschild [7] recognized the importance of such a condition and used it to characterize those real algebraic hypersurfaces for which any biholomorphic self-map must be algebraic (see also [6]). In this paper we establish a similar statement for formal biholomorphic mappings of real analytic hypersurfaces (Theorem 2.2 below), namely that any formal biholomorphism between germs of real analytic minimal and holomorphically nondegenerate hypersurfaces is convergent. This statement will in fact be a consequence of our main result, Theorem 2.1, where a description of the analyticity properties of formal biholomorphic maps of minimal real analytic hypersurfaces is given. An application of this theorem to partial convergence of such maps is also given in §7.
The study of the convergence of formal mappings between real analytic CR submanifolds goes back to Chern and Moser [9], who established the convergence of formal biholomorphisms between real analytic Levi-nondegenerate hypersurfaces. More recently, Baouendi, Ebenfelt and Rothschild [3,4,5] addressed this problem in more general situations. In particular, the following two facts follow from their work: i) Given a generic minimal real analytic connected holomorphically nondegenerate submanifold M in C N , N ≥ 2, there exists a proper real analytic subvariety S ⊂ M , such that for any point p ∈ M \ S, any formal biholomorphism sending (M, p) onto another germ of a generic real analytic submanifold in C N , must be convergent.
ii) If M is a connected generic holomorphically degenerate submanifold in C N , then, for any point p ∈ M , there exists a formal biholomorphism sending (M, p) into itself which does not converge.
In view of these facts and other related results in the mapping problems ( [2]), it appears likely that the subvariety S in i) could be taken to be empty. In this paper, we actually prove such a result in the case of hypersurfaces. Our proof is based on an analysis of a so-called reflection function associated to the mapping and the hypersurfaces, which was used in many other situations (cf. [11,13]). We should also mention that, recently in [14], we showed that the real analytic subvariety S in i) can also be taken to be empty when the generic submanifold M is assumed to be real algebraic (and with no such assumption on the target manifold).
Theorem 2.1. Let f : (M, 0) → (M ′ , 0) be a formal biholomorphism between two germs at 0 of real-analytic hypersurfaces in C n . Assume, furthermore, that M is minimal at 0. Then, the formal holomorphic map Such a result has several applications. One of its main applications lies in the following theorem mentioned in the introduction.
Theorem 2.2. Any formal biholomorphic mapping between germs of minimal, holomorphically nondegenerate, real analytic hypersurfaces in C n is convergent.
As explained in the introduction, the interest in such a result lies in the fact that, in view of [3], the condition of holomorphic nondegeneracy is optimal for the class of formal biholomorphisms. However, it remains an open problem to decide whether or not the condition of minimality is necessary in Theorem 2.2. Another application of Theorem 2.1 deals with partial convergence of formal biholomorphisms. For this, we refer the reader to §7. Remark 1. After this work was completed, I received a preprint by J. Merker, "Convergence of formal biholomorphic mappings between minimal holomorphically nondegenerate real analytic hypersurfaces", in which a similar statement to Theorem 2.2 is given. In that preprint, Theorem 2.1 above is also stated for the special case where M ′ is a rigid and polynomial hypersurface.

Definitions and notations
3.1. Real analytic hypersurfaces. Let M be a smooth real real-analytic hypersurface in C n , n ≥ 2. Since the situation is purely local, we shall always work near a point p ∈ M , which will be assumed, without loss of generality, to be the origin. Let ρ be a real analytic defining function for M near 0 i.e. We shall assume, without loss of generality, that the coordinates z = (z ′ , z n ) ∈ C n are chosen so that ∂ρ ∂z n (0) = 0. In this case, we define the following holomorphic vector fields tangent to M which are the complexifications of the usual (0,1) vector fields tangent to M . We recall that for a point w near 0, its associated Segre surface is the complex hypersurface defined by Q w = {z ∈ (C n , 0) : ρ(z,w) = 0}. Observe that by the complex analytic implicit function theorem, each Segre variety can be described as a graph of the form Φ denoting a convergent power series in some neighborhood of the origin in C 2n−1 satisfying the relations Φ(0) = 0 and The coordinates z are said to be normal with respect to M if the additional condition holds. It is well-known that given a real-analytic hypersurface through the origin, one can always construct such coordinates [9]. Thus, from now on and for simplicity, we will always assume that the z-coordinates are chosen to be normal for the manifold M .
The real analytic hypersurface M is called minimal at 0 (in the sense of Trépreau and Tumanov), or, equivalently of finite type (in the sense of Kohn and Bloom-Graham) if it does not contain any complex-analytic hypersurface through 0. To use such a nondegeneracy condition, we will need the Segre set mappings associated to M (see [2]) up to order 3, which, in normal coordinates, are the following three maps : These maps are of fundamental importance since they parametrize the socalled Segre sets (up to order three) associated to M . Moreover, the interest in such Segre sets or Segre set mappings lies in the fact that the minimality assumption is equivalent to the fact that the generic rank of v 2 (and also v 3 ) equals n (see [2]). This will be useful for the proof of Theorem 2.1.
All the notations introduced in this section will be used for the source hypersurface M . For the target real analytic hypersurface M ′ , we shall use the notations introduced in §2, before the statement of Theorem 2.1. In particular, we denote by z the coordinates in the source space and by ζ the coordinates at the target space.
3.2. Some commutative algebra. We recall here, for the reader's convenience, some basic definitions, needed in §7, about regular local rings and their ideals. All these definitions can be found for instance in [12,17].
Let A be a Noetherian ring. If I and J are two ideals of A, we use the notation I < J to mean that I ⊂ J and I = J. Given a prime ideal I ⊆ A, the height of I is defined by the formula where I 1 , . . . , I k are prime ideals of A. If J is any ideal of A, we define the height of J by the formula If A is furthermore assumed to be a local ring, one defines the Krull dimension of A to be the height of its maximal ideal. Observe that if A is a Noetherian local ring and if I is a proper ideal of A, the quotient ring A/I is also a Noetherian local ring. This allows one to consider the Krull dimension of such a ring. Finally, a Noetherian local ring is said to be regular if its maximal ideal has δ generators, where δ is the Krull dimension of the ring A. The rings of formal holomorphic power series or convergent power series in p indeterminates, p ∈ N * , are regular rings of Krull dimension p [17].

Two convergence results
In this section, we first state and prove a convenient lemma which will be used twice in the paper. This lemma may already be known.
Lemma 4.1. Let (u i (t)) i∈I be a family of convergent power series in C{t}, t = (t 1 , . . . , t q ), q ∈ N * . Let also (K i (ς)) i∈I be a family of convergent power series in C{ς}, ς = (ς 1 , . . . , ς r ), r ∈ N * . Assume that: (i) There exists R > 0 such that the radius of convergence of any Proof. By Artin's approximation theorem [1], one can find a convergent power series mapping ϑ(t) = (ϑ 1 (t), . . . , ϑ r (t)) ∈ (C{t}) r such that ϑ(0) = 0 and for all i ∈ I, Since the radius of convergence of the K i , i ∈ I, is at least R, the radius of convergence of any K i • ϑ, i ∈ I, is at least R ′ . Thus, the family (u i (t)) i∈I has a radius of convergence at least equal to R ′ and for all t ∈ C q with |t| < R ′ , one has |u To derive Theorem 2.2 from Theorem 2.1, we will need the following consequence of Artin's approximation theorem, which is contained for instance in [15]. (See also §7 for another formulation of such a result.) Proof. We reproduce here the arguments of [15]. Write By assumption, we know that we have det Q(x, g(x), g(x)) ≡ 0. This implies that one can find an integer k g such that if H(x) is any formal power series which agrees up to order k g with g then det Q(x, g(x), H(x)) ≡ 0. For this integer k g , according to Artin's approximation theorem, one can find a convergent power series H 0 (x) satisfying R(x, H 0 (x)) = 0 and agreeing with g(x) up to order k g .
and thus g is convergent.

The reflection principle
Let f : (M, 0) → (M ′ , 0) be a formal biholomorphic mapping between germs at 0 of real analytic hypersurfaces in C n . We shall use the notations introduced in §2 and §3. In particular, we denote f = f (z) = (f 1 (z), . . . , f n (z)) = (f ′ (z), f n (z)) in the ζ-coordinates. In this section, we shall make no further assumptions on M and M ′ .
As in [13,14], we define the following formal holomorphic power series The goal of this section is to prove the following proposition. 0) be a formal biholomorphism between germs at 0 of real analytic hypersurfaces in C n , and R defined by (5.1). Then for any multi-index γ ∈ N n , the formal holomorphic map Here, v 1 is the first Segre set mapping for M as defined in (3.3).
Before proceeding to the proof of Proposition 5.1, we need a preliminary lemma (Lemma 5.2 below). Since f maps formally M into M ′ , there exists Equivalently, we havē . By applying the vector fields L j , j = 1, . . . , n − 1, as defined by (3.1), to (5.2) and using the fact that f is invertible, one obtains the following known statement (see [4] for instance).
Lemma 5.2. Under the assumptions of Proposition 5.1, one has, for any multindex α ∈ N n−1 , the formal power series identitȳ where each χ α is a convergent power series of its arguments.
Proof of Proposition 5.1. We write the expansion For the sake of clarity, we shall first give the proof of the Proposition in the case γ = 0.
The case γ = 0. We restrict all the identities given by Lemma 5.2 to the (n − 1)-dimensional subspace This gives, for any multiindex α ∈ N n−1 , where φ ′ α is given by (5.3). Observe that for each multiindex α, u α (z ′ ) is convergent. To is convergent, we claim that it suffices to show that there exists a > 0 and R 0 > 0 such that the radius of convergence of each u α is at least a, and such that the following Cauchy estimates hold: Indeed, if (5.5) holds, then the formal holomorphic power series This proves that under the assumption (5.5), R(v 1 (z ′ ), λ) is a convergent power series in (z ′ , λ). It remains to prove (5.5). SinceΦ ′ is holomorphic in a neighborhood of 0 ∈ C 2n−1 , in view of (5.3), one can find δ > 0 and a constant R > 0 such that for any multiindices α ∈ N n−1 , ν ∈ N n , In view of (5.4) and (5.7) (in the case ν = 0), we can apply Lemma 4.1 to conclude that there exists a > 0 such that the family (u α (z ′ )) α∈N n−1 is convergent in B n−1 (0, a) and such that (5.5) holds with R 0 = R. This finishes the proof of Proposition 5.1 in the case γ = 0.
The case |γ| > 0. We proceed now to the proof of Proposition 5.1 for general γ ∈ N n . For this, we need the following lemma.
is convergent.
Remark 2. It is clear that Proposition 5.1 still holds in higher codimension with the same proof. More precisely, the following holds. Let M, M ′ be two germs through the origin in C n , n ≥ 2, of smooth real real-analytic generic submanifolds of CR dimension N and of real codimension d. Let f : (M, 0) → (M ′ , 0) be a formal biholomorphic map. Assume that the coordinates at the target space ζ = (ζ ′ , ζ * ) ∈ C N × C d are chosen so that, near the origin, M ′ = {(ζ ′ , ζ * ) ∈ (C n , 0) :ζ * =Φ ′ (ζ,ζ ′ )}, for some C d -valued holomorphic mapΦ ′ = (Φ ′ 1 , . . . ,Φ ′ d ) near 0 ∈ C 2N +d . Assume also that the coordinates at the source space are chosen to be normal coordinates for M . Then, if we define C n × C N ∋ (z, λ) → R(z, λ) :=Φ ′ (f (z), λ) ∈ C d , the following holds. For any multiindex γ ∈ N n , the formal holomorphic power series mapping

Proofs of Theorem 2.1 and Theorem 2.2
For the proof of Theorem 2.1, we first need to prove a lemma (also used in [14]) which will allow us to bypass the second Segre set and to work directly on the third Segre set. Proof. First observe that T (x, u) is a formal power series solution of the analytic system in the unknown W , Thus, an application of Artin's approximation theorem [1] gives, for any positive integer e, an r-tuple of convergent power series T e (x, u) ∈ (C{x, u}) r solution in W of (6.1), and which agrees up to order e with T (x, u). The Lemma follows by noticing that ϕ(T e (x, u); x, u, y) ≡ 0 in C [[x, u, y]] if and only if T e (x, u) is solution of (6.1). The proof of Lemma 6.1 is complete.
The following proposition will also be useful in the proof of Theorem 2.1 (see [8] for a proof for instance).

Transcendence degree and partial convergence of formal maps
In this last section, we want to indicate how Theorem 2.1 can be viewed as a result of partial convergence for formal biholomorphic mappings of real analytic hypersurfaces. Before explaining what we mean by this, we need to recall the following. If M is a real analytic hypersurface in C n and p ∈ M , let K(p) be the quotient field of C{z − p}, and H(M, p) be the vector space over K(p) consisting of the germs at p of (1,0) vector fields, with meromorphic coefficients, tangent to M (near p). We then define the degeneracy of M at p, denoted d(M, p), to be the dimension of H(M, p) over K(p). It is shown in [7]  Theorem 2.1 gives the following result of partial convergence for formal biholomorphic mappings of real analytic hypersurfaces. By this, we mean that we have the following. Proof. We again use the notations of §2. As in the proof of Theorem 2.2, we have, using the expansion (5.3), Thus, we know, by Theorem 2.1, that for any multi-index β ∈ N n−1 , , of generic maximal rank equal to n − d(M ′ ) in a neighborhood U ′ of 0 in C n (see [7]). Then, if we define g j (ω) = φ ′ β j (ω), j = 1, . . . , n − d(M ′ ), we obtain the desired statement of the Theorem.
Remark 4. One should observe that the convergent power series mapping g in Theorem 7.1 is obtained in a constructive way from the target manifold M ′ . Indeed, this is a consequence of the statement of Theorem 2.1. Now, we want to make explicit links with the notion of transcendence degree introduced in [10] in the C ∞ mapping problem. For this, we first set the corresponding definitions in the formal case.   Proof. Let V be as in the proposition. We can assume that, near 0 ∈ C N +N ′ , where each b j (x, y) ∈ C{x, y}. To this complex analytic set V , as is customary, we associate the following ideal of C{x, y} defined by By the Noetherian property, we can assume that I(V ) is generated by a family (h i (x, y)) i=1,... ,k ⊆ C{x, y}. Furthermore, by the Nullstellensatz [12,17], for i = 1, . . . , k, there exists an integer µ i such that h µ i i (x, y) is in the ideal generated by the b j (x, y), j = 1, . . . , p. This implies that for The following result, a consequence of the Artin approximation theorem, is contained in [17] (p.63). Assume that the height of the ideal I(V ) (generated by the family (h j (x, y)) 1≤j≤k in C{x, y}) is equal to N ′ . Then, any formal solution Y(x) ∈ (C[[x]]) N ′ , Y(0) = 0, of the system h 1 (x, y) = . . . = h k (x, y) = 0 (in the unknown y) is convergent. Thus, to obtain the convergence of our original formal power series H, it suffices to check that the height of I(V ) is equal to N ′ . Since C{x, y} is a local regular ring of Krull dimension N + N ′ , by Proposition 6.12, p.22 of [17], we have the formula height (I(V )) + dim C{x, y}/I(V ) = N + N ′ , where dim C{x, y}/I(V ) is the Krull dimension of the ring C{x, y}/I(V ). Since the Krull dimension of such a ring coincides with the dimension of the complex analytic set V (cf. [12], p.226, Proposition 1), which is, here, equal to N , we obtain that the height of I(V ) is N ′ . This completes the proof of Proposition 7.3, and hence, the proof of Proposition 7.2.
Let A = ∪ k i=1 Γ i be the decomposition of A into irreducible components. For any positive integer σ, one can find, according to the Artin approximation theorem [1], a convergent power series mapping f σ (z) ∈ (C{z}) n defined in some small neighborhood U σ of 0 in C n , which agrees with f (z) up to order σ (at 0) and such that the graph of f σ , denoted G(f σ ), is contained in A. Since G(f σ ) is contained in A, it must be contained in an irreducible component of A. Thus, by the pigeonhole principle, at least one subsequence of (f σ ) σ∈N * is contained in one of such irreducible components, say Γ 1 . There is no loss of generality in assuming that such a subsequence is (f σ ) σ∈N * itself. We first observe that this implies that the graph of f is formally contained in Γ 1 .
Moreover, since f is a formal biholomorphism, the family (f σ ) σ∈N * is also a family of local biholomorphisms. In particular, this implies that the generic rank of the family of holomorphic functions ((g i • f 1 )(z)) 1≤i≤r , is r. As a consequence, if z 0 is close enough to 0 in C n and is chosen so that the rank of the preceding family at z 0 equals r, the implicit function theorem shows that A is an n + d(M ′ )-dimensional complex submanifold near (z 0 , f 1 (z 0 )) ∈ Γ 1 . Since Γ 1 is irreducible, it is pure-dimensional; thus Γ 1 is an n+d(M ′ ) pure-dimensional complex analytic set formally containing the graph of f . By definition of the transcendence degree, this implies that D f ≤ d(M ′ ).
The following example illustrates the applications of Theorem 7.1 and Corollary 7.4.
We conclude by observing that Theorem 2.2 can be regarded as a direct consequence of Corollary 7.4. Indeed, when M ′ is holomorphically nondegenerate, as mentioned above, d(M ′ ) = 0 and hence D f = 0 by Corollary 7.4. It then follows from Proposition 7.2 that f is convergent in that case. We should also mention that Theorem 2.2, Theorem 7.1 and Corollary 7.4 are all consequences of our main result, namely Theorem 2.1.