Xianghong Gong
email: gong@math.wisc.edu

Laurent Stolovitch

EQUIVALENCE OF NEIGHBORHOODS OF EMBEDDED COMPACT COMPLEX MANIFOLDS AND HIGHER CODIMENSION FOLIATIONS

Keywords: . 2010 Mathematics Subject Classification. 32Q57, 32L30, 32L10, 37F50, 58F36 Neighborhood of a complex manifold, normal bundle, solution of cohomological equations with bounds, holomorphic extension, holomorphic linearization, resonances, small divisors condition, holomorphic foliations

We consider an embedded n-dimensional compact complex manifold in n + d dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert's formal principle program. We will give conditions ensuring that a neighborhood of C n in M n+d is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold's result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in M n+d having C n as a compact leaf, extending Ueda's theory to the high codimension case. Both problems appear as a kind linearization problem involving small divisors condition arising from solutions to their cohomological equations.

We are interested in the classification of the germs of neighborhood of an embedded compact complex manifold C in a complex manifold M. Here two germs (M, C) and ( M , C) are holomorphically equivalent if there is a biholomorphic mapping F fixing C pointwise and sending a neighborhood V of C in M into a neighborhood Ṽ of C in M . These considerations can be useful to extend holomorphic objects such as cohomology classes of holomorphic sections of bundles over C or functions on C to a neighborhood of C in M. Indeed, it might be that such an extension problem is much easy to solve on an equivalent neighborhood. We are also interested in the existence of a non-singular holomorphic foliation of the germ of neighborhood of C in a complex manifold having C as a compact leaf. We refer to it as a "horizontal foliation".

A neighborhood V of an embedded complex manifold C n in M n+d has local holomorphic charts (h j , v j ) = Φ j mapping V j onto Vj in C n+d with n = dim C. Here ∪V j is a neighborhood of C and U j := V j ∩ C is defined by v j = 0. The above-mentioned classification of the germs of neighborhoods of C is then the classification of transition functions Φ kj := Φ k Φ -1 j under holomorphic conjugacy F -1 k Φ kj F j . To such an embedding, one can associate the normal bundle N C (M) of C in M, which has the transition matrices g kj (p), p ∈ U k ∩ U j . To this embedding one can associate another natural embedding, namely the embedding of C as the zero section of N C (M). Under a mild assumption, this last embedding (N C (M), C) naturally serves as a first order approximation of (M, C). Let ϕ j = Φ j | U j and let ϕ kj = ϕ k ϕ -1 j be the transition functions of C. To have a neighborhood of C in M equivalent to a neighborhood of the zero section in N C (M) is equivalent to seeking F j such that Φkj = F -1 k Φ kj F j are of the form N kj (h j , v j ) = (ϕ kj (h j ), t kj (h j )v j ) with t kj (h j ) = g kj , the latter being regarded as the transition functions of a neighborhood of the zero section of N C (M). We call this process a "full linearization" of the neighborhood. The above-mentioned "horizontal foliation" will be obtained as a consequence of a "vertical linearization" of the neighborhood which amounts to seeking F j such that Φkj = (ϕ kj (h j ) + φh kj (h j , v j ), t kj (h j )v j ). Without even considering holomorphic equivalence problem, it is known that there are formal obstructions to linearizing [START_REF] Griffiths | The extension problem in complex analysis. II. Embeddings with positive normal bundle[END_REF][START_REF] Nirenberg | On rigidity of holomorphic imbeddings[END_REF] or to linearizing vertically [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF] a neighborhood; see section 2. Part of the Grauert formal principle [START_REF] Camacho | Fibered neighborhoods of curves in surfaces[END_REF][START_REF] Grauert | Über Modifikationen und exzeptionelle analytische Mengen[END_REF][START_REF] Hironaka | On the equivalence of imbeddings of exceptional complex spaces[END_REF][START_REF] Kosarew | Ein allgemeines Kriterium für das formale Prinzip[END_REF] is to seek geometry conditions that ensure a holomorphic linearization when the formal obstructions are absent. In this paper, we will obtain linearizations of a neighborhood of an embedded compact complex manifold C n at the absence of formal obstructions under small divisor conditions in the form of bounds of solutions of cohomology equations involving all symmetric powers of N * C , the dual of the normal bundle N C of C n in M n+d . Because of the very nonlinear nature of the problem, we need to work with a family of nested domains on which we solve and eventually bound the solutions of 1-cohomological equations. Indeed, we are naturally led to consider shrinking of the domains as we need to get estimates of derivatives of sections (by Cauchy estimates for instance). To be more precise, assume that a 1-cocycle f with value in the sheaf of sections of holomorphic bundle (involving symmetric power S m N * C for some m ≥ 2) on C vanishes in the 1st cohomology group over a covering W. Then there is a 0-cochain w over W such that δw = f . Nevertheless, we need to prove the existence of a (possibly different) solution u satisfying the linear equation δu = f and a "linear" estimate of the form u W ≤ K f W (the norm is either L 2 or the sup-norm).

Because of the nonlinear nature of our problem, we need to solve the linear equation iteratively and estimate solutions of the form δu m = F m (f 2 , . . . , f m , u 2 , . . . , u m-1 ), m ≥ 2.

Here F m (f 2 , . . . , f m , u 2 , . . . , u m-1 ) is a nonlinear function and vanishes in a first cohomology group. Therefore the bound K, depending on m, will compound, which leads to a problem on non-linear estimates. Here come some of the main issues : we need that, at the limit, the sequence of nested domains, over which the solutions are estimated iteratively, remains to cover the manifold. And we need to control the growth of the bound K with respect to m, that gives rise to the so-called small divisors condition. Therefore, the existence of any bound K for linear solutions u without shrinking the covering W is a basic question. The latter was solved affirmatively by Kodaira-Spencer [25, eq. ( 9), p. 499] for the case of line bundles for a general covering. For higher rank vector bundles, we provide a positive solution in the following result : Proposition 1.1. Let C be a compact complex manifold. There exists a family of coverings U r = {U r j }, r * ≤ r < r * , of C such that for any holomorphic vector bundle E over C, and each f ∈ C 1 (U r ′ , E), the space of 1-cochains on U r ′ of holomorphic sections with values in E, satisfying f = δu 0 for some u 0 ∈ C 0 (U r ′ , E), there exist u ∈ C 0 (U r ′ , E) and v ∈ C 0 (U r ′′ , E) such that δu = f and δv = f , and

|u| r ′ ≤ K(E)|f | r ′ , (1.1) |v| r ′′ ≤ D(E) (r ′ -r ′′ ) τ |f | r ′ . (1.2)
Here r ′ , r ′′ are any numbers satisfying r * < r ′′ < r ′ ≤ r < r * and r ′r ′′ ≤ r *r, and τ, K(E), D(E) are independent of r ′ , r ′′ .

Here, we have used the sup-norm (or L 2 -norm) of cochains of holomorphic sections of bundles (see section A.2 for specific notations). We do not know if K(E) and D(E) are comparable when they are applied to the symmetric powers of N * C except when N C is unitary. Hörmander [START_REF] Hörmander | L 2 estimates and existence theorems for the ∂ operator[END_REF][START_REF] Ohsawa | L 2 approaches in several complex variables[END_REF] obtained solutions with bounds for cohomology groups with respect to the ∂ operator acting on the sheaf of (p, q)-forms with L 2 coefficients on C n . The estimate (1.2) was proved by Donin [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF] for a special family of coverings by the L 2 theory. He also raised the question if estimate (1.1) exists, i.e. the basic question mentioned above. Proposition 1.1 gives us a more flexible kind of results and ultimately an estimate that holds without any shrinking for higher rank vector bundles via the above mentioned nested coverings. We also use the L 2 -theory. We first obtain (1.2) by Theorem A.9. Then (1.1) is obtained by Lemma A.2. The constant K(E) is defined for the kind of bundles we need in Definition A. [START_REF] Camacho | Fibered neighborhoods of curves in surfaces[END_REF]. This is summarized in Theorem A.12. The main results of this paper are based on the existence of nested finite coverings proved in subsection A.5.

Proposition 1.1 will be a useful tool in this paper. We now formulate our main results. We say that T C M = T M| C splits if T C M = T C ⊕ N C holomorphically. For instance, T C E splits for any holomorphic vector bundle E over C. Here and in the sequel, we identify C with the zero section of E. We say that N C is flat if the transition matrices of N C are locally constant. We say that N C is unitary if its transition matrices are unitary. Note that the maximum principle implies that a unitary N C is flat. We have the following "vertical linearization" result: Theorem 1.2. Let C n be a compact submanifold of M n+d with splitting T C M and unitary N C . Let η 0 = 1 and

η m := K(N C ⊗ S m (N * C )) max m 1 +•••+mp+s=m η m 1 • • • η mp ,
where the maximum is taken in 1 ≤ m i < m for all i and s ∈ N. Assume that there are positive constants L, L 0 such that η m ≤ L 0 L m , m = 1, 2 . . . .

Assume that H 0 (C, N C ⊗ S ℓ (N * C )) = 0 for all ℓ > 1 . Assume that either H 1 (U, N C ⊗ S ℓ (N * C )) = 0 for all ℓ > 1 or a neighborhood of C is formally vertically linearizable by a formal holomorphic mapping that is tangent to the identity and preserves the splitting of T C M. Then the embedding is actually holomorphically vertically linearizable.

When C is a compact holomorphic curve embedded in a complex surface M with a unitary normal bundle N C , the above vertical linearization is one of main results in Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF] where H 0 (C, N C ⊗ S ℓ (N * C )) = 0 for all ℓ > 1 follows from his small-divisor condition. This has been generalized by T. Koike in higher codimension case under a strong assumption that N C is a direct sum of unitary line bundles [START_REF] Koike | Toward a higher codimensional Ueda theory[END_REF]. The Ueda theory for codimension-one foliations has also been extended by Claudon-Loray-Pereira-Touzet [START_REF] Claudon | Compact leaves of codimension one holomorphic foliations on projective manifolds[END_REF] and Loray-Thom-Touzet [START_REF] Loray | Two dimensional neighborhoods of elliptic curves: formal classification and foliations[END_REF]. We remark that Theorem 1.2 via the flatness of N C ensures the existence of a "horizontal" foliation : Corollary 1.3. Under assumptions of Theorem 1.2, there exists a neighborhood of C n in M n+d that admits an n-dimensional smooth holomorphic foliation having C n as a leaf.

The following results can been understood in the context of the Grauert formal principle for rigidity: If (M, C) is formal equivalent to (N C , C), then they are holomorphically equivalent under suitable assumptions. We first consider the unitary case.

Theorem 1.4. Let C n be a compact submanifold of M n+d N C is unitary. Let η 0 = 1 and

η m := max (K(N C ⊗ S m (N * C )), K(T C ⊗ S m (N * C ))) max m 1 +•••+mp+s=m η m 1 • • • η mp ,
where the maximum is taken in 1 ≤ m i < m for all i and s ∈ N. Assume that there are positive constants L, L 0 such that

(1.3) η m ≤ L 0 L m , m = 1, 2 . . . .
If T C M splits and H1 (U, T C M ⊗ S ℓ (N * C )) = 0 for all ℓ > 1 or more generally if a neighborhood of C in M is linearizable by a formal holomorphic mapping which is tangent to the identity and preserves the splitting of T C M, then there exists a neighborhood of C in M which is holomorphically equivalent to a neighborhood of C (i.e the 0th section) in N C In that case, we say that the embedding C ֒→ M is holomorphically linearizable.

More generally, the following result treats two more general cases.

Theorem 1.5. Let C n be a compact submanifold of M n+d . Suppose that

(1.4) k≥1 log D * (2 k+1 ) 2 k < +∞,
where D * (2 k+1 ) is defined by (5.27). Suppose that either H 0 (C, T C ⊗ S ℓ (N * C )) = 0 for all ℓ > 1, or N C is flat. Assume further that either T C M splits and H 1 (U, T C M ⊗S ℓ (N * C )) = 0 for all ℓ > 1 or (M, C) and (N C , C) are equivalent by a formal holomorphic mapping which is tangent to the identity and preserves the splitting of T C M. Then (M, C) and (N C , C) are actually holomorphically equivalent.

The previous results can be seen as a "full linearization" results. Theorem 1.4 is proved by using a majorant method while Theorem 1.5 is based on a Newton scheme. It is not clear how to compare the two "small divisors conditions" (1.3) and (1.4) althought the counterparts in theory of dynamical systems are equivalent [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF]. The formal principle holds in the following cases: (a) negative N C in the sense of Grauert, by results of Grauert [START_REF] Grauert | Über Modifikationen und exzeptionelle analytische Mengen[END_REF] and Hironaka-Rossi [START_REF] Hironaka | On the equivalence of imbeddings of exceptional complex spaces[END_REF]. In Grauert's case, C n has a system of strictly pseudoconvex neighborhoods and consequently C n is the only compact n-submanifold near C n . In the same spirit, Savelev proved that all neighborhoods of embeddings of P 1 in complex surfaces with a unitary flat normal bundle are holomorphically equivalent [START_REF] Savel'ev | Zero-type imbedding of a sphere into complex surfaces[END_REF]. (b) sufficiently positive N C and dim C > 2, by a result of Griffiths [START_REF] Griffiths | The extension problem in complex analysis. II. Embeddings with positive normal bundle[END_REF]Thm II (i)] showing that a neighborhood is determined by a finite-order neighborhood. In other words, under this condition the holomorphic classification of neighborhoods is "finitely determined". (c) H 1 (C, N C ) = 0 and the case that for each x ∈ C there is x ′ ∈ C such that the fiber of N C at x is generated by global sections of N C vanishing at x ′ , by a result of Hirschowitz (see [START_REF] Hirschowitz | On the convergence of formal equivalence between embeddings[END_REF] for more general results) 1 . (d) 1-positive N C , by a result of Commichau-Grauert [START_REF] Commichau | Das formale Prinzip für kompakte komplexe Untermannigfaltigkeiten mit 1-positivem Normalenbündel[END_REF].

We should remark that the above "full linearization" result was obtained by Arnol'd when C is an elliptic curve and M is a surface, where the vanishing of H 0 (X, T C M ⊗S ℓ M) follows from the non vanishing of "small divisors" [START_REF] Arnol'd | Bifurcations of invariant manifolds of differential equations, and normal forms of neighborhoods of elliptic curves[END_REF][START_REF]Geometrical methods in the theory of ordinary differential equations, Second, Grundlehren der Mathematischen Wissenschaften[END_REF]. Ilyashenko and Pyartli [START_REF] Yu | Neighborhoods of zero type imbeddings of complex tori[END_REF] proved an analogous result for special embeddings of the product flat tori under a strong assumption that N C is a direct sum of flat line bundles. We emphasize that in our linearization Theorem 1.5, for general compact manifolds C n , we impose the vanishing of H 0 (X, T C M ⊗ S ℓ M) for all integers ℓ ≥ 2 whereas there is no restriction on H 0 when C is affine and N C is flat.

As a simple consequence, we have the following Corollary 1.6. Under assumptions of Theorem 1.5 on C and M, any holomorphic section of a holomorphic vector bundle E over C extends to a holomorphic section of a holomorphicvector-bundle extension of E over a neighborhood of C in M.

Corollary 1.7. Let C be a compact complex manifold. Let (M, C) be equivalent to

(C × C d , C
) by a formal holomorphic mapping which is tangent to the identity and preserves the splitting of T C M. Suppose that the small-divisor condition in Theorem 1.5 is satisfied.

Then (M, C) is holomorphically equivalent to (C × C d , C).
We now give an outline of the paper.

In section 2 we study the formal obstructions to the full linearization and vertical linearization problems. The formal obstructions are known from work of Nirenberg-Spencer [START_REF] Nirenberg | On rigidity of holomorphic imbeddings[END_REF], Griffiths [START_REF] Griffiths | The extension problem in complex analysis. II. Embeddings with positive normal bundle[END_REF], Morrow-Rossi [START_REF] Morrow | Submanifolds of P n with splitting normal bundle sequence are linear[END_REF], for the the full linearization problem and by Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF] (see also Neeman [31] and among others) for the vertical linearization problem. The obstructions are described in

H 1 (C, E ⊗ S ℓ N * C ) for a natural vector bundle E that is either T C M or N C . In this paper we emphasize the role of H 0 (C, T C M ⊗ S ℓ N * C ).
In local dynamical systems, the elements in the analogous group appear as finite symmetries in the Ecalle-Voronin theory [START_REF] Ahern | A complete classification for pairs of real analytic curves in the complex plane with tangential intersection[END_REF] and centralizers for the linearizations [START_REF] Gong | Real submanifolds of maximum complex tangent space at a CR singular point, I[END_REF]. The small divisors in local dynamics emerge in the form of the bounds

K(N C ⊗ S ℓ N * C ) and D(T C M ⊗ S ℓ N * C ) in Proposition 1.1.
In work of Arnol'd [START_REF] Arnol'd | Bifurcations of invariant manifolds of differential equations, and normal forms of neighborhoods of elliptic curves[END_REF] and Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF], the vanishing condition of the corresponding zero-th cohomology groups is not explicit; however it follows from their small-divisor conditions.

In section 3, we prove Theorem 1.2 by using Ueda's majorization method [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF]. In our case the majorization relies on an important tool of the (modified) Fischer norm which is invariant under a unitary change of coordinates. The invariance allows us to overcome the main difficulty in our majorization proof to deal with the transition functions of N * C when they are unitary, but not necessarily diagonal. The (modified) Fischer norms have also been useful in other convergence proofs [START_REF] Iooss | Polynomial normal forms with exponentially small remainder for analytic vector fields[END_REF][START_REF] Lombardi | Normal forms of analytic perturbations of quasihomogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation[END_REF][START_REF] Shapiro | An algebraic theorem of E. Fischer, and the holomorphic Goursat problem[END_REF]. In section 4, we also extend the majorant method to the full linearization problem for the special case where N C is unitary. In section 5, we obtain the full linearization in the general case by introducing a Newton scheme, i.e. a rapid convergence scheme as in Brjuno's work [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF]; see also [START_REF] Rüssmann | Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition[END_REF][START_REF] Stolovitch | Singular complete integrability[END_REF]. However, we must cope with the domains of transition functions which are not so regular. These domains, when carefully chosen, have nevertheless a disc structure. This allows us to obtain a proof by using sup-norm estimates.

Finally, the paper contains an appendix which has interests in its own right. It has two results, namely the existence of the two bounds stated in Proposition 1.1 and the existence of nested coverings (see Definition A.1). The existence of bound K(E) was employed by Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF] through the complete system of Kodaira-Spencer [START_REF] Kodaira | A theorem of completeness of characteristic systems of complete continuous systems[END_REF] when dim C = 1 and codim M C = 1. We will prove Proposition 1.1 by using some techniques developed by Donin [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF]. Our proof also relies on a "quantified" version of Grauert-Remmert finiteness theorem [START_REF]Theory of Stein spaces[END_REF]. The existence of bound D(E ′ ⊗ S ℓ E ′′ ) was proved by Donin [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF] for the socalled "normal" coverings. We have used nested coverings in the proof of Proposition 1.1 as well as the convergence proof in Theorem 1.5. We believe that the methods and tools developed in this article will be useful for other kinds of problems.

Full linearizations, horizontal foliations, and vertical linearizations

In this section, we describe the problem of equivalence of a neighborhood of a complex compact submanifold C of M with a neighborhood of the zero section in the normal bundle of C in M as a "full" linearization problem of the transition functions of this neighborhood. We also describe the existence of a holomorphic foliation of a neighborhood of C having C as a leave as a consequence of a vertical linearization problem of the transition functions of this neighborhood.

We will first describe the formal coordinate changes in terms of cohomological groups of holomorphic sections of a suitable sequence of holomorphic vector bundles.

2.1. Transition functions. We recall basic facts on vector bundles, which we refer to [START_REF] Griffiths | Principles of algebraic geometry[END_REF]Chap. 0,Sect. 5].

We first set up notation. If a vector space E has a basis e = {e 1 , . . . , e d }, then a vector v in E can be expressed as

v = ξ µ e µ , ξ = (ξ 1 , . . . , ξ d ) t .
Here, we use the summation notation: ξ µ e µ stands for d µ=1 ξ µ e µ . The ξ µ 's are the coordinates or components of v in the basis e.

We recall that a holomorphic vector bundle E over a complex manifold X is defined by a projection π : E → X and holomorphic trivializations Ψ j : π -1 (D j ) → D j × C r such that each Ψ j : π -1 (D j ) → D j ×C r is a biholomorphism, and Ψ j (E p ) = {p}×C r for E p := π -1 (p). Furthermore {D j } is an open covering of X and the maps

Ψ kj = Ψ k Ψ -1 j : D k ∩ D j × C r → D k ∩ D j × C r satisfy (2.1) Ψ kj (p, ξ j ) = (p, g kj (p)ξ j )
where g kj are transition matrices which are holomorphic and invertible. Thus for ξ µ k e k,µ = ξ µ j e j,µ , we have

ξ µ k = g µ kj,ν ξ ν j , e j,µ = g ν kj,µ e k,ν , (2.2) ξ k = g kj ξ j , e k = (g -1 kj ) t e j . (2.3)
They satisfy the cocycle conditions, (2.4)

g kj g jk = I, on D k ∩ D j ; g ki g ij = g kj , on D k ∩ D j ∩ D i .
We also need to consider the dual bundle E * . Let e * j be the basis dual to e j so that (e * j,µ (e j,ν )) is the identity matrix. Suppose

ζ µ j e * j,µ = ζ µ k e * k,µ ∈ E * . Corresponding to (2.3), we have (2.5) e * k = g kj e * j , ζ k = (g -1 kj ) t ζ j .
Let us also express transition functions for various vector bundles in coordinate charts as above. Let C n be a compact complex manifold embedded in complex manifold M n+d . We cover a neighborhood of C in M by open sets V j so that we can choose coordinate charts (z j , w j ) on V j for M such that

U j := C ∩ V j : w j = 0.
Then U = {U i } is a finite covering of C by open sets on which the coordinate charts

z i = (z 1 i , . . . , z n i ) are defined. Let (2.6) z k = ϕ kj (z j ) = ϕ k ϕ -1 j (z j ) be the transition function of C on U kj := U k ∩ U j . It is a biholomorphic mapping from ϕ j (U kj ) onto ϕ k (U kj ) in C n . Then T C has a basis e j,α := ∂ ∂z α j , 1 ≤ α ≤ n
over U j and its transition matrices s kj have the form

(2.7) s α kj,β (z j ) := ∂z α k ∂z β j U j ∩U k . Thus for η α k ∂ ∂z α k = η α j ∂ ∂z α j on U j ∩ U k , we have η k = s kj (z j )η j . As to the normal bundle N C , its transition matrices t µ kj,ν (z j ) := ∂w µ k ∂w ν j | U j ∩U k on U j ∩ U k are for the basis f j,µ := ∂ ∂w µ j mod T C, 1 ≤ µ ≤ d.
Thus for ξ µ k f j,µ = ξ µ j f k,µ , we have ξ k = t kj (z j )ξ j . With notation (2.1), the transition matrices of T M| C are then of the form

g kj := s kj l kj 0 t kj (z j ) on U j ∩ U k
for some n × d matrices l jk . Note that ∂w j ∂z k | C = 0. Throughout the paper, τ kj (z j ) are the transition matrices of N * C for the base dw j . Note that

τ kj = (t -1 kj ) t . More specifically, if w * j,µ := dw µ j | U j and ζ µ j w * j,µ = ζ µ k w * k,µ , then (2.5) becomes (2.8) ζ * k = (t -1 kj (z j )) t ζ * j , w * k = t kj (z j )w * j .
We remark that the cocycle conditions (2.4) for N C now takes the form (2.9)

t kj (z j )t jk (z k ) = Id on U j ∩ U k , t kj (z j )t jℓ (z ℓ ) = t kℓ (z ℓ ) on U j ∩ U k ∩ U ℓ .
We say that T M splits on C, if there is a (non-canonical) decomposition

(2.10)

T M| C = T C ⊕ ÑC , ÑC ∼ = N C .
Equivalently, there exists a system of coordinate charts such that on C, the transitions matrices of T M| C are of the form

g kj = s kj 0 0 t kj (z j ) on U j ∩ U k .
In other words,

∂z j ∂w k C = 0.
Throughout the paper, we assume that T M splits on C and we fix a splitting (2.10). Then the change of bases of the normal bundle N C has a simple form

z k = ϕ kj (z j ), ∂ ∂w ν k = t µ jk,ν (z k ) ∂ ∂w µ j , on U j ∩ U k .
In summary, for a neighborhood of the embedded manifold C in M with splitting T C M, we can find a covering V = {V i }, with Φ j (V j ) = Ũi × Wi , by open sets on M and coordinates (z i , w i ) defined on V i . We assume that U j := C ∩V i is defined by {w i = 0}. A neighborhood of C will then be described by transition functions on V kj of the form (2.11) Φ kj :

z k = Φ h kj (z j , w j ) := ϕ kj (z j ) + φ h kj (z j , w j ), w k = Φ v kj (z j , w j ) := t kj (z j )w j + φ v kj (z j , w j ).
Here, φ h kj (resp. φ v kj ) are holomorphic functions of vanishing order ≥ 2 along w j = 0: (2.12)

φ h kj (z j , w j ) = O(|w j | 2 ), φ v kj (z j , w j ) = O(|w j | 2
). That φ h kj vanishes at order ≥ 2 follows from the fact that T M| C splits as T C ⊕ N C (see above and [30, proposition 2.9]). Define

N kj (h j , v j ) := (ϕ kj (z j ), t kj (h j )v j ).
Our goals are to apply changes of coordinates to simplify φ h kj , φ v kj , or one of them, according to the problem we study.

2.2.

The equivalence of transition functions. The germ of neighborhood of an embedded manifold is well-defined. For the formal normalization, we need to introduce (semi) formal charts and formal neighborhoods of an embedded manifold in a (semi) formal manifold.

Definition 2.1. We call M an (admissible and splitting) formal neighborhood of C if there are holomorphic coordinate charts ϕ j on U j where {U j } is a covering of C and there are formal power series

(z j , w j ) = Φj (p, w) := (ϕ j (p), t j (p)w) + |Q|≥2 Φ j,Q (p)w Q ,
where Φ j,Q are holomorphic functions in U j and each t j is an invertible holomorphic d × d matrix on U j . Note that the formal transition functions Φkj = Φk Φ-1 j have the form Φkj (z j , w j ) = (ϕ kj (z j ), t kj (z j )w j )

+ |Q|>1 Φkj,Q (z j )w Q j , z j ∈ ϕ j (U j ∩ U k ).
(a) When all Φ j are holomorphic, the formal neighborhood M is called the germ of a (holomorphic) neighborhood of C. (b) M is called a linear neighborhood of C if additionally (2.13) Φkj (z j , v j ) = (ϕ kj (z j ), t kj (z j )v j ) and each t kj is an invertible holomorphic matrix in U k ∩ U j . The terminology is meaningful since the Φkj can be realized as the transition functions of a holomorphic vector bundle over C, namely the normal bundle of C in M.

We are mainly interested in the classification of a neighborhood of C for a given C. Therefore, it is reasonable to assume that the local trivialization of C are fixed. In other words, ϕ kj are fixed and we will only consider mappings sending a neighborhood of C into another neighborhood of C that fix C pointwise. Definition 2.2. We shall say that N C is a flat (resp. unitary flat), if we can find constant (resp. with values in group of unitary matrices U d ) transition functions in a possibly refined covering. If T C M := (T M)| C is holomorphically flat, or flat, i.e. in some coordinates both transition functions N C and T C are constant matrices, then by (2.7)

ϕ kj (z j ) = s kj z j + c kj
where s kj are constant matrices and c kj are constant vectors. Then, the transition functions of a neighborhood of the zero section of the normal bundle, Φkj as defined in (2.13) read A kj (z j , w j ) := (s kj z j + c kj , t kj w j ).

We will use the following notation: When N C is flat, we write its transition matrices t kj (z j ) as t kj , indicating that they are independent of z j . Definition 2.3. We shall say that a change of coordinates {F j } preserves the germ of a neighborhood of the zero section of N C with transition maps {N kj } if each F j is biholomorphic and fixes v j = 0 pointwise and F k N kj = N kj F j , in which case we say that {F j } preserves {N kj } for simplicity.

We further observe the following.

Lemma 2.4. Let M, M be two (admissible) neighborhoods of C, of which coordinate charts are {Φ j }, { Φj }, respectively. Let Φ kj = Φ k Φ -1 j and Φkj = Φk Φ-1 j . (a) There is a biholomorphic mapping F : M → M , defined near C and fixing C, if and only if there are biholomorphic mappings F j satisfying

F k Φkj (z j , w j ) = Φ kj F j (z j , w j ), F j (z j , 0) = (z j , 0). (2.14) (b) If F j satisfies (2.14), then F j (z j , w j ) = LF j (z j , w j ) + O(|w j | 2 ), LF j = (z j + s j (z j )w j , u j (z j )w j ), s k (ϕ kj (z j ))t kj (z j ) = Dϕ kj (z j )s j (z j ), u k (ϕ kj (z j ))t kj (z j ) = t kj (z j )u j (z j ).
Assume further that F preserves the splitting. Then s j = 0. (c) Let T C and N C be flat and let F j be (semi) formal biholomorphism fixing C pointwise. Suppose that

F -1 k Φ kj F j = N kj + O(|v| 2 j ). Then {LF j } preserves {N kj }, i.e. LF k N kj (LF j ) -1 = N kj , where F j (h j , v j ) = LF j (h j , v j ) + O(|v j | 2 ), LF j (h j , v j ) = (h j + s j (h j )v j , u j (h j )v j ).
Proof. The points (a), (b) can be verified easily. For (c), let us expand

F k Φ kj (h j , v j ) = N kj • F j (h j , v j ) + O(|v j | 2
) and compare the constant and linear terms in v j . We obtain

ϕ kj (h j ) + s j (ϕ kj (h j ))t kj v j = ϕ kj (h j + s j (h j )v j ) + O(|v j | 2 ), u k (ϕ kj (h j ))t kj v j = t kj u j (h j )v j + O(|v j | 2 ).
Here we have used the assumption that t kj are constant. Since ϕ kj are affine, the two identities still hold if we drop O(|v j | 2 ) from them. This shows that LF k N kj = N kj LF j , again using the fact that t kj are constant and ϕ kj are affine.

Finally, we mention that we will choose the atlas of C so that each ϕ j is a biholomorphism from U j onto the unit polydisc ∆ n in C n and from a neighborhood Ũj of U j onto another larger polydisc. When C is embedded in a complex manifold M, we can extend ϕ j to V j to get a coordinate chart Φ j on V j such that Φ j maps V j onto U j × ∆ d δ . This can be achieved since any holomorphic vector bundle over Ũj is holomorphically trivial. Thus N C | U j splits. Consequently, we can use a flow box of holomorphic normal vector fields to construct the required Φ j . Therefore, if C is embedded into another complex manifold M , we will choose the atlas of a neighborhood of C in M such that the restriction of the chart on U j agrees with ϕ j .

Therefore, we introduce the following.

Definition 2.5. We say that a formal neighborhood {Φ kj } of C is equivalent to a neighborhood { Φkj } of C in M by a formal mapping F that is tangent to the identity and preserves the splitting of T C M, if there are formal maps F j (z j ) = (z j , w j ) + |Q|>1 F j,Q (z j )w Q j such that F j,Q (z j ) are holomorphic functions in U j and as power series in w j F k Φkj (z j , w j ) = Φ kj F j (z j , w j ).

We take F = Φ-1 j F j Φ j , which is well-defined, when Φ kj = Φ k Φ -1 j and Φkj = Φk Φ-1 j . 2.3. The full Linearization of a neighborhood. In this case, our goal is to seek new coordinates (h k , v k ) so that all φ h kj , φ v kj are 0. Let us consider a change of coordinates in a neighborhood of C by modifying the old coordinate charts (z k , w k ) via F k . We write it as

F k : z k = F h k (h k , v k ) := h k + f h k (h k , v k ), w k = F v k (h k , v k ) := v k + f v k (h k , v k ). Here, f h k (h k , v k ) and f v k (h k , v k
) are holomorphic functions vanishing at order ≥ 2 at v k = 0. In particular, C is pointwise fixed by the change as z k = h k on C (i.e. for v k = 0). We require that the inverse of F k is defined in a possibly smaller open sets Vk ⊂ ϕ k (U k ) such that the union of Φ -1 k ( Vk ) remains a neighborhood of C in M. We recall that the cocyle condition (2.9) on the transition matrices t kj has the form t kj (z j )t jk (ϕ kj (z j )) = Id, t kj (ϕ jℓ (z ℓ ))t jℓ (z ℓ ) = t kℓ (z ℓ ). (2.15) Let us assume that the (a priori formal) change of coordinates (2.15), maps a neighborhood C to a neighborhood of the zero section in the normal bundle. This means that, in these new coordinates, we have

N kj := F -1 k Φ kj F j : h k = ϕ kj (h j ), v k = t kj (z j )v j .
Let us write down the above "conjugacy equations". We first consider the horizontal equation of

F k N kj = Φ kj F j .
On the left side of the equation, we have

z k = h k + f h k (h k , v k ) = ϕ kj (h j ) + f h k (ϕ kj (h j ), t kj (h j )v j
). On the other side, we have

z k = ϕ kj (h j + f h j (h j , v j )) + φ h kj (h j + f h j , v j + f v j ).
Let us define the horizontal cohomological operator to be (2. [START_REF] Griffiths | Principles of algebraic geometry[END_REF])

L h kj (f h j ) := f h k (ϕ kj (h j ), t kj (h j )v j ) -s kj (h j )f h j (h j , v j ).
Recall that s kj (h j ) = Dϕ kj (h j ) is the Jacobian matrix of ϕ kj . Hence, we can write the previous horizontal equation as

L h kj (f h j ) = φ h kj (h j + f h j , v j + f v j ) (2.17) + ϕ kj (h j + f h j (h j , v j )) -ϕ kj (h j ) -Dϕ kj (h j )f h j (h j , v j ).
Let us consider the vertical equation. We have, on one side of the equation,

w k = v k + f v k (h k , v k ) = t kj (h j )v j + f v k (ϕ kj (h j ), t kj (h j )v j
). On the other side, we have

w k = t kj (h j + f h j )(v j + f v j ) + φ v kj (h j + f h j , v j + f v j ).
Let us define the vertical cohomological operator to be

(2.18) L v kj (f v j ) := f v k (ϕ kj (h j ), t kj (h j )v j ) -t kj (h j )f v j .
Hence, we can write the previous vertical equation as

L v kj (f v j ) = φ v kj (h j + f h j , v j + f v j ) (2.19) + t kj (h j + f h j (h j , v j )) -t kj (h j ) f v j + t kj (h j + f h j (h j , v j )) -t kj (h j ) v j .
2.4. Horizontal foliations and vertical trivializations. Let us assume that there exists a non singular holomorphic foliation having C as a leaf. We seek holomorphic functions f j = (f j,1 , . . . , f j,d ) defined in a neighborhood V j of U j such that f j = 0 on U j and df j,1 ∧• • •∧ df j,d = 0. Then, we may use (h j , v j ) = (z j , f j (z j , w j )) as a coordinate mapping on V j , which changes variables in vertical components. We then prove that in these new coordinates, the transition functions of a neighborhood of C are of the form Φkj = ( Φh kj , Φv kj ) such that Φ v kj are independent of h j . We remark that N C must be flat if a horizontal foliation exists.

Proposition 2.6. Assume that there is smooth holomorphic horizontal foliation defined in a neighborhood V of C in M. By a refinement of U j , then there exists a change of variables of the form

z k = h k w k = s(h j )v j + O(|v j | 2 )
so that in the new variables, we have

h k = ϕ kj (h j ) + φ h kj (h j , v j ), v k = tkj v j + |Q|>1 c kj,Q v Q j ,
where tkj , c kj,Q are constants.

Proof. By a refinement, we may assume that the foliation on V j is given W j (h j , v j ) = cst by holomorphic functions W j = (W j,1 , . . . , W j,d ) such that W j = 0 on U j and dW j,1 ∧ • • • ∧ dW j,d = 0. We have W k = Φv kj W j , where Φv kj is a biholomorphism of (C d , 0) with Φv kj (0) = 0. Then Wj = (z j , W j ) is a biholomorphism defined on V j and fixing C ∩ V j pointwise, by shrinking V j if necessary in the vertical direction. Since Wj is invertible, we can define Φh

kj = z k W -1 j Then we have Φh kj Wj = z k . Therefore, Wk W -1 j (h j , v j ) = ( Φh kj (h j , v j ), Φv kj (v j )). Set F j = Φ j W -1 j . We have F h j (h j , v j ) = h j . We now get F -1 k Φ k Φ -1 j F j = Wk W -1 j = Φkj .
In this paper, we will approach the horizontal foliation problem via the following vertical linearization when N C is unitary.

2.5. The vertical linearization. Here we seek new coordinates (h j , v j ) from (z j , w j ) such that the vertical component of the new Φ kj agrees with the vertical component of N kj . In Lemma 2.17 we will show that if such formal coordinates exist, then the vertical linearization can be achieved by changing vertical coordinates only, i.e. a coordinate change of the form

w k = F v k (h k , v k ) := v k + f v k (h k , v k ), z k = h k .
For the vertical linearization, we only need to consider the vertical part of transition functions so that in the new variables, we have

h k = Φh kj (h j , v j ) := ϕ kj (h j ) + φh kj (h j , v j ) v k = t kj (h j )v j .
Here, φh kj (h j , v j ) vanishes up to order 2 at v j = 0. The vertical equation reads

t kj (h j )(v j + f v j ) + φ v kj (h j , v j + f v j ) = w k = t kj (h j )v j + f v k ( Φh kj (h j , v j ), t kj (h j )v j ).
Using the previous notation, we finally obtain the following "conjugacy equations"

(2.20) L v kj (f v j ) = φ v kj (h j , v j + f v j ) -f v k ( Φh kj (h j , v j ), t kj (h j )v j ) -f v k (ϕ kj (h j ), t kj (h j )v j ) .
Having determined the coordinate change, let us find the horizontal component φh kj from the horizontal equation

ϕ kj (h j ) + φ h kj (h j , v j + f v j ) = z k = Φh kj (h j , v j ) = ϕ kj (h j ) + φh kj (h j , v j ).
We get

(2.21) φh kj (h j , v j ) = φ h kj (h j , v j + f v j ).
2.6. Coboundary operators in symmetric powers and coordinates. In this subsection, we establish the connections between coordinate changes and formal obstructions to the full linearization and vertical linearization via cohomological groups. In local dynamics, the resonant terms play an important role in the construction of normal forms at least at the formal level, while non-resonant terms play another important role in coordinate changes. In all problems, obstructions are described via the first cohomological groups, while the coordinate changes are described via solutions to the cohomological equations of first order approximation. Let E ′ be a vector bundle of rank τ over C. Let U = {U i } be a covering of C as above. Let e j := {e j,1 , . . . , e j,τ } be a basis over U j and let ξ j := (ξ 1 j , . . . , ξ τ j ) t be coordinates in e j . Let s kj (z j ) be the transition matrices of E ′ over U k ∩ U j . Using notation in (2.3), we have

ξ α k = s α kj,β (z j )ξ β j , e k;α = s β jk,α (z k )e j,β , (2.22) z k = ϕ kj (z j ), ξ k = s kj (z j )ξ j , e k = (s -1 kj (z j )) t e j , (2.23)
where ϕ kj are the transition functions of C. For N * C , by (2.8) we have

ζ k = (t -1 kj ) t (z j )ζ j , w * k = t kj (z j )w * j , z k = ϕ kj (z j ).
The following fact is well-known. We provide a proof for the reader's convenience. Let us first introduce

(2.24) f λ i 0 •••iq (z iq , ζ iq ) := |Q|=L f λ i 0 •••iq;Q (z iq )ζ Q iq , for a cochain {f I } ∈ C q ({U j }, O(E ⊗ S L (N * C )))
given by

f i 0 •••iq (p) = τ λ=1 |Q|=L f λ i 0 •••iq;Q (z iq (p))e i 0 ,λ (p) ⊗ (w * iq (p)) Q , (2.25)
where each f λ i 0 ...iq;Q is a holomorphic function on ϕ iq (U i 0 •••iq ), and

U i 0 •••iq denotes as usual U i 0 ∩ • • • ∩ U iq .
Here we have chosen a representation of cochains in bases that arise from the linearized equations for the problems described above.

Let f i 0 ••• îℓ •••i q+1 denote f i 0 •••i ℓ-1 i ℓ+1 •••i q+1 . Then (δf ) i 0 •••i q+1 = (-1) ℓ f i 0 ••• îℓ •••i q+1 becomes (δf ) i 0 •••i q+1 = q ℓ=1 (-1) ℓ τ λ=1 |Q|=L f λ i 0 ••• îℓ •••i q+1 ;Q (z i q+1 (p))e i 0 ,λ (p) ⊗ (w * i q+1 (p)) Q + τ λ=1 |Q|=L f λ i 1 •••i q+1 ;Q (z i q+1 (p))e i 1 ,λ (p) ⊗ (w * i q+1 (p)) Q -(-1) q τ λ=1 |Q|=L f λ i 0 •••iq;Q (z iq (p))e i 0 ,λ (p) ⊗ (w * iq (p)) Q =: τ λ=1 |Q|=L g λ i 0 •••i q+1 (z q+1 )e i 0 ,λ (p) ⊗ (w * i q+1 (p)) Q .
By (2.22), we have e i 1 ,λ = s µ i 0 i 1 ,λ e i 0 ,µ . In notation (2.24), we can express

gλ i 0 •••i q+1 (z i q+1 , ζ i q+1 ) = q ℓ=1 (-1) ℓ f λ i 0 ••• îℓ •••i q+1 (z i q+1 , ζ i q+1 ) + s λ i 0 i 1 ,µ (ϕ i 1 i q+1 (z q+1 )) f µ i 1 •••i q+1 (z i q+1 , ζ i q+1 ) -(-1) q f λ i 0 •••iq (ϕ iqi q+1 (z i q+1 ), t iqi q+1 (z i q+1 )ζ i q+1 ))
. The above computation especially gives us the following formulae for 0 and 1-cochains.

Lemma 2.7. Let {U j } be an open covering of C. Let t kj be the transition matrices for N C with respect to basis w j and let s kj be the transitions functions of E with respect to base e j . Let

f ij (p) = d λ=1 |Q|=L f λ ij;Q (z j (p))e i,λ (p) ⊗ (w * j (p)) Q , f λ ij (z j , ζ j ) := |Q|=L f λ ij;Q (z j )ζ Q j , u j (p) = d λ=1 |Q|=L u λ j,Q (z j (p))e j,λ (p) ⊗ (w * j (p)) Q , ũλ j (z j , ζ j ) := |Q|=L u λ j;Q (z j )ζ Q j .
The following hold :

(a) f := {f ij } ∈ Z 1 (U, O(E ⊗ S L (N * C ))) if and only if f λ ij (ϕ jk (z k ), t jk (z k )ζ k ) -f λ ik (z k , ζ k ) + s λ ij,ℓ (z j ) f λ jk (z k , ζ k ) = 0. (b) u := {u j } solves the first order cohomological equation δu = f if and only if s λ ij,ℓ (z j )ũ ℓ j (z j , ζ j ) -ũλ i (ϕ ij (z j ), t ij (z j )ζ j ) = f λ ij (z j , ζ j ).
We notice that according to (2.16) and (2.18), we have

-L(f ) = -(L h (f h ), L v (f v )) = δ(f ) := (δ h (f h ), δ v (f v )).

Formal obstructions in cohomology groups. Recall that (2.26)

N kj (h j , v j ) := (ϕ kj (z j ), t kj (h j )v j ).

Let us denote the properties depending on an order m ≥ 1 :

(L m (U)) : the neighborhood of C matches the neighborhood of zero section of the normal bundle up to order m. (V m (U)) : the vertical components of the transition functions of neighborhoods of C in M and in N C match up to order m.

That embedding of C has property (L m ) (resp. (V m )) means that the order along v j = 0 of (φ h kj (h j , v j ), φ v kj (h j , v j )) (resp. φ v kj (h j , v j )) as defined in (2.11) is ≥ m + 1. Definition 2.8.
We shall say that N C is a flat (resp. unitary flat), if we can find constant (resp. with values in group of unitary matrices U d ) transition functions in a possibly refined covering.

We will use the following notation: When N C is flat, we write its transition matrices t kj (z j ) as t kj , indicating that they are independent of z j . Definition 2.9. We shall say that a change of coordinates {F j } preserves the germ of a neighborhood of the zero section of N C with transition maps {N kj } if F k N kj = N kj F j , in which case we says that {F j } preserves {N kj } for simplicity. Lemma 2.10. Let the transition functions Φ kj of a neighborhood of C be given by (2.11)-(2.12).

(a) Assume that C satisfies L m . Then the horizontal and vertical components satisfy Griffiths [15], Morrow-Rossi [START_REF] Morrow | Submanifolds of P n with splitting normal bundle sequence are linear[END_REF] and (b) is proved in Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF] for flat line bundle N * C over a compact curve C. (a) The general case can be verified by using Lemma 2.7 to compare coefficients of w α j on both sides of Φ ij (z j , w j ) = Φ ik • Φ kj (z j , w j ) for |α| ≤ 2m. Indeed, we have

[φ h kj ] ℓ ∈ Z 1 (U, T C ⊗ S ℓ (N * C )), if m < ℓ ≤ 2m; [φ v kj ] ℓ ∈ Z 1 (U, N C ⊗ S ℓ (N * C )), if ℓ = m + 1. If N C is flat, then the vertical component of Φ kj further satisfies [φ v kj ] ℓ ∈ Z 1 (U, N C ⊗ S ℓ (N * C )), m + 1 < ℓ ≤ 2m. (b) Let C satisfy V m . Assume that N C is flat. Then (2.27) [φ v kj ] ℓ ∈ Z 1 (U, N C ⊗ S ℓ (N * C )), ℓ = m + 1. Proof. When ℓ = m + 1, (a) is in
Φ ik = N ik + (φ h ik , φ v ik ) and (φ h ik , φ v ik )(z k , w k ) = O(|w k | m+1 ) with m ≥ 1. Thus N ik • Φ kj (z j , w j ) = N ik • N kj + DN ik • N kj • (φ h kj , φ v kj ) (z j , w j ) + O(|w j | 2m+1 ) = N ik • N kj (z j , w j ) + (s ik (ϕ kj (z j ))φ h kj , t ik (ϕ kj (z j ))φ v kj ) + (0, Dt ik (ϕ kj (z j ))φ h kj (z j )t kj (z j )w j ) + O(|w j | 2m+1 ).
Here s kj are the transition matrices of T C given by (2.7). Therefore,

Φ ik • Φ kj (z j , w j ) = N ik • Φ kj (z j , w j ) + (φ h ik , φ v ik ) • Φ kj (z j , w j ) = N ik • N kj + (φ h ik , φ v ik ) • N kj (z j , w j ) + s ik (ϕ kj (z j ))φ h kj (z j , w j ), t ik (ϕ kj (z j ))φ v kj (z j , w j ) + (0, Dt ik (ϕ kj (z j ))φ h kj (z j )t kj (z j )w j ) + O(|w j | 2m+1 ). Comparing both sides of Φ ij (z j , w j ) = Φ ik • Φ kj (z j , w j )
for the coefficients in w j of order ℓ = m + 1, we obtain the desired conclusion by Lemma 2.7.

(b

) We have Φ kj (z j , w j ) = (ϕ kj (z j ) + φ h kj (z j , w j ), t kj w j + φ v kj (z j , w j )) with φ v kj (z j , w j ) = O(|w j | m+1
). Here t kj are constant. We get from the vertical components of Lemma 2.7 (a). This gives us (2.27).

Φ kj = Φ ki Φ ij that φ v kj (z j , w j ) = t ki φ v ij (z j , w j ) + φ v ki (Φ ij (z j , w j )) = t ki φ v ij (z j , w j ) + ϕ ki (N ij (z j , w j )) + O(|w j | m+2 ), since (Φ ij -N ij )(z j , w j ) = O(|w j | 2 ). This shows that {[φ v kj ] ℓ } ∈ Z 1 (U, N C ⊗ N * ℓ C ) for ℓ = m + 1 by

Automorphisms of neighborhood of the zero section of flat vector bundles.

Let φ kj defined on U k ∩ U j be the transition functions of C. Let Φ kj , defined on V k ∩ V j , be the transition functions of M, and let N kj , defined on Ṽk ∩ Ṽj be the transition functions of N C , with Ṽk = π -1 U k . We identify (C, U j ) as subsets of Ṽj via the zero-section. Recall Φ kj , N kj , and φ kj are the same on U k ∩ U j . By Cartan-Serre theorem, for any integer m, the space of global sections, H 0 (C, T C M ⊗ S m N * C ), is finite dimensional. We say that a vector bundle is flat if its transition matrices are locally constant. Definition 2.11.

(1) A formal tangent vector field Y j on Ṽj vanishing at U j is identified with

Y j = ℓ≥1 Y ℓ j with Y ℓ j ∈ Γ(U j , T C M ⊗ S ℓ N * C ) via |Q|=ℓ a α Q (h j )v Q j ∂ ∂h α j + b λ Q (h j )v Q j ∂ ∂v λ j → |Q|=ℓ a α Q (z j )(w * j ) Q ∂ ∂z α j + b λ Q (z j )(w * j ) Q ∂ ∂w λ j .
Here (h j , v j ) is the coordinate map for v λ j ∂ ∂w λ j ∈ (N C ) p and we identity h j with

z j | U j and ∂ ∂v j with ∂ ∂w j | U j . ( 2
) A formal automorphism of Ṽj at U j that is tangent to the identity is an automorphism of a formal neighborhood of the 0-section of Ṽj , fixing U j pointwise.

Lemma 2.12. Let {F j } j be a collection of formal automorphisms of Ṽj fixing U j pointwise. Let {Y j } j be a collection of formal tangent vector fields of Ṽj vanishing at U j . We have

(1) {F j } j defines an automorphism F of a formal neighborhood of the 0-section in N C if and only

F k • N kj = N kj • F j for all k, j (2) Suppose that N C is flat. {Y j } j defines a vector field Y on a formal neighborhood of the 0-section in N C if and only if {Y ℓ j } ∈ H 0 (C, T C M ⊗ S ℓ N * C ) for all ℓ.
(3) Suppose that N C is not flat. {Y j } j defines a vector field on a formal neighborhood of the 0-section in N C if and only if

{Y j } ∈ H 0 twisted (C, T C M ⊗ ⊕ ℓ≥2 S ℓ N * C ) with respect to the linear operator δ nf ({(Y h j , Y v j )}) = {( Ỹ h kj , Ỹ v kj )} with Ỹ h kj = Y h k (N kj (h j , v j )) -Dφ kj (h j )Y h j (h j , v j ), Ỹ v kj = Y v k (N kj (h j , v j )) -t kj (h j )Y v j (h j , v j ) -Dt kj (h j )v j .Y h j (h j , v j ). Proof. Let (h j , v j ) be the coordinates in N C over U j . Note that {Y j } defines a global tangent vector filed of N C if and only if DN kj (Y j ) = Y k . A homogeneous vector field of degree ℓ on Ṽj is an element Y ℓ j ∈ C 0 (U j , T C M ⊗ S ℓ N * C ) defined by Y ℓ j (h j , v j ) = n m=1 Y ℓ,h j,m (h j , v j ) ∂ ∂h j,m + d r=1 Y ℓ,v j,r (h j , v j ) ∂ ∂v j,r =: Y ℓ,h j + Y ℓ,v j . Recall that N kj (h j , v j ) = (φ kj (h j ), t kj (h j )v j ). Thus DN kj Y ℓ,h j + Y ℓ,v j = Dφ kj (h j )Y ℓ,h j (h j , v j ) + t kj (h j )Y ℓ,v j (h j , v j ) + n j=1 d r,s=1 ∂t kj,rs (h j ) ∂h j,m Y ℓ,h j,m (h j , v j )v j,s ∂ ∂v k,r
,

where the last term is in C 0 (U k ∩U j , N C ⊗S ℓ+1 N * C ). When N C is flat, we see that DN kj Y j = Y k if and only if DN kj Y ℓ j = Y ℓ k
for each ℓ and that the latter holds if and only if

(2.28) Y ℓ,h k (φ kj (h j ), t kj v j ) = Dφ kj (h j )Y ℓ,h j (h j , v j ), Y ℓ,v k (φ kj (h j ), t kj v j ) = t kj Y ℓ,v j (h j , v j ). In other words, {Y ℓ j } j defines a global section of T C M ⊗ S ℓ N * C .
Lemma 2.13. Let F j be a formal automorphism of Ṽj in N C , which is tangent to identity and preserves the splitting of T C (N C ) along U j . Then, F j is the time-1 map of a unique formal vector field Y j in Ṽj , vanishing on U j up to order ≥ 2.

Proof. Let F j be given by

hj = h j + |α|≥2 A j,α (h j )v α j , ṽj = v j + |β|≥2 B j,β (h j )v β j .
Drop the index j. We want to express it as the time-1 map of a tangent vector field

Y = ℓ≥2 n m=1 Y ℓ,h m (h, v) ∂ ∂h m + d r=1 Y ℓ,v r (h, v) ∂ ∂v r ,
where Y ℓ,h m (h, v), Y ℓ,v r (h, v) are homogeneous polynomials in v of degree ℓ. The flow of Y with time θ is given by

h θ m = h m + |α|≥2 A θ m,α (h)v α , v θ r = v r + |α|≥2 B θ r,α (h)v α ,
where A θ , B θ satisfy A 0 = B 0 = 0 and

|α|≥2 v α j dA θ m,α (h j ) dθ = ℓ≥2 Y ℓ,h m (h θ , v θ ), |α|≥2 v α j dB θ r,α (h) dθ = Y ℓ,v r (h θ , v θ ).
Inductively, we can verify that

A 1 m,α -Y h m,α , B 1 m,α -Y v r,α are uniquely determined by Y ℓ,h m ′ ,β , Y ℓ,v r ′ ,β
with ℓ < |α|.

Note that the formal time-1 mapping of DN kj (Y j ) on Ṽk ∩ Ṽj can also be defined and it equals N kj F j N -1 kj where F j is the time-1 map of Y j . Thus the uniqueness assertion in the lemma implies the following.

Proposition 2.14. Any automorphism F of a formal neighborhood of C in N C , which is tangent to identity and preserves the splitting of T C (N C ), is the time-1 map of a unique vector field defined on a formal neighborhood of C in N C and vanishing on C. Assume further that N C is flat. Then any tangent vector field Y of N C that vanishes on C to order two admits a decomposition

Y = ℓ≥2 Y ℓ , Y ℓ ∈ H 0 (C, T C M ⊗ S ℓ N * C ).
We write

δ m = (δ h m , δ v m ) corresponding to the splitting T C M = T C ⊕ N C . Let us set G m := Range(δ m ). We have a decomposition (2.29) Z 1 (U, T C M ⊗ S m N * C ) = G m ⊕ N m where N m ≃ H 1 (U, T M C ⊗ S m N * C ). Let C 0 (U, T M C ⊗ S m N * C ) = R m ⊕ ker δ m with δ m (R m ) = G m .
We emphasize that the decomposition (2.29) is not unique. For our convergence result, a natural decomposition will be given via a possibly non-unique minimizing solution. Consequently, ⊕ is interpreted as merely a decomposition suitable for convergence proof.

Lemma 2.15. Suppose that N C is flat. Any formal transformation F j of Ṽj which is tangent to identity and preserves the splitting of T C (N C ) can be uniquely factorized as

F j = G -1 j • H j where H j -I ∈ m≥2 R m , G j is
an automorphism of Ṽj , and terms of order m in G j , H j are uniquely determined by the terms of order at most m in F j . Furthermore,

G i N ik = N ik G k for all i, k. Proof. We know that F j = exp m C m j is the time-1 map of m≥2 C m j . We want to decompose exp m C m j = (exp m A m j )(I + m H m j ).
By Campbell-Hausdorff formula, we are led to the equation

H m j = C m j -A m j + E m j
where E m j depends only on C ℓ j , A ℓ j for ℓ < m. We determine A m j , B m j by decomposing C m j and E m j as follow : Let π be the (non-canonical) projection from

C 0 (U, T M C ⊗ S m N * C ) onto ker δ m . Let {A m j } j := π({C m j + E m j }). Then {H m j } ∈ R m .
Next, we study the dependence of cohomology classes of [φ h kj ] ℓ , [φ v kj ] ℓ in coordinates. We first consider the full set of linear cohomological equations.

2.9. Formal coordinates in the absence of formal obstructions. For a power series u(z j , w j ), let u ≤m (z j , w j ) be the Taylor polynomial of u about w j = 0 with degree m. Thus we can define

u = u ≤m + u >m , u >m (z j , w j ) = O(|w j | m+1 ), [u] m = u ≤m -u <m , [u] m ℓ = u ≤m -u <ℓ .
In order to describe the coboundary operator in next lemma, we define the linear operator D by

(( Du)f )(h j , v j ) := ∂u ∂h j (h j , 0)f h (h j , v j ) + ∂u ∂v j (h j , 0)f v (h j , v j ),
for a function u(h j , v j ). The standard differential D is given by

((Du)f )(h j , v j ) = ∂u ∂h j (h j , v j )f h (h j , v j ) + ∂u ∂v j (h j , v j )f v (h j , v j ). Thus (2.30) (Du -Du)f (h j , v j ) = (Du(h j , v j ) -Du(h j , 0))f (h j , v j ).
For a multiindex α = (α h , α v ), define

( Dα u)(h j ) = ∂ |α| u ∂h α h j ∂v αv j (h j , 0). Lemma 2.16. Let Φ kj = N kj + φ kj satisfy condition L m with m ≥ 1. Suppose that F j (h j , v j ) = (h j , v j ) + f j (h j , v j ) with f j (h j , v j ) = O(|v j | 2 ) are formal mappings such that {F -1 k Φ kj F j } ∈ L m . Then, on U j ∩ U k , l = 2, . . . , m, (δ{[f j ] ≤l }) kj (h j , v j ) = -N kj ((I + [f j ] ≤l-2 )(h j , v j )) -N kj (h j , v j ) (2.31) -DN kj (h j , v j )[f j ] ≤l-2 (h j , v j ) ≤l -0, (Dt kj (h j )[f h j ] ≤l-1 (h j , v j ))v j . (a) If f j (h j , v j ) = O(|v j | m+1 ) for all j, then N kj + φkj = F -1 k Φ kj F j + O(|v j | 2m+1 ) hold if and only if on U j ∩ U k (2.32) (δ{[f i ] ≤2m }) kj = [ φkj -φ kj ] ≤2m -0, (Dt kj (h j )[f h j ] ≤2m-1 )v j . (b) If {F j }
defines a germ of biholomorphism of order m at the zero section of the normal bundle, i.e.

F -1 k N kj F j (h j , v j ) = N kj (h j , v j ) + O(|v j | m+1 ) and if f h j (h j , v j ) = O(|v j | m ), then V ≤m j (h j , v j ) := (h j , v j + [f v j ] ≤m ) preserves {N kj }. (c) Suppose F -1 k Φ kj F j ∈ L 2m . Assume further that either N C is flat or (2.33) H 0 (C, T C ⊗ S p N * C ) = 0, 2 ≤ p ≤ 2m.
Then there exist

Fj = I + O(|v j | m+1 ) where [ F h j ] 2m m+1 are uniquely determined by [Φ kj ] 2m m+1 such that F -1 k Φ kj Fj ∈ L 2m . There exists a unique decomposition { Fj = H j • V j • Fj } in the form H j (h j , v j ) = (h j + H j (h j , v j ), v j ), (2.34) V j (h j , v j ) = (h j , v j + V j (h j , v j )), (2.35) [ Fj ] i = 0, ∀2 ≤ i ≤ 2m, [H j ] ℓ = [V j ] ℓ = 0, ∀ℓ > 2m. (2.36) Furthermore, [H j ] ℓ = [V j ] ℓ = 0 for ℓ ≤ m,
and H j are uniquely determined by

(2.37) (δ h {H i }) kj = -[φ h kj ] ≤2m . Moreover, φkj = H -1 k Φ kj H j -N kj satisfy φh kj (h j , v j ) = O(|v j | 2m+1 ) and φv kj (h j , v j ) = O(|v j | m+1 ), and V i satisfy (2.38) (δ v {V i }) kj = -[ φv kj ] ≤2m .
Proof. Let Φ kj = N kj + φ kj and Φkj = N kj + φkj . Suppose that both φ kj and φkj are of order ≥ m + 1 (i.e. O(|v j | m+1 )) and

F k Φ kj = Φkj F j . Recall that F k = I + f k .
To use the coboundary operator, we write

f k (N kj ) -DN kj f j + φ kj -φkj = f k (N kj -f k (N kj + φ kj )) A (2.39) + φkj (I + f j ) -φkj B + N kj (I + f j ) -N kj -DN kj f j C .
Since f j has order ≥ 2 at v j = 0, by the Taylor expansion at N kj and at I respectively, both A and B are of order ≥ m + 2 (w.r.t v j ) at the origin. For the same reason, the C is of order ≥ 4. We recall that, for each ℓ ∈ N * , the coboundary operator δ sends

C 0 (U, T C M ⊗ S ℓ (N * C )) into C 1 (U, T C M ⊗ S ℓ (N * C )) as sections. It is defined in coordinates by (δf ) kj = DN kj f j (h j , v j ) -f k (N kj (h j , v j )) on U j ∩ U k when f = {f j } ∈ C 0 (U, T C M ⊗ S ℓ (N * C )).
As δ preserves the degree ℓ of f j in v j , we shall omit its dependence in ℓ. Truncating the Taylor expansion of (2.39) at v j = 0 up to degree m will lead to the first point.

Since f j (h j , v j ) = O(|v j | 2 ), then A, B are of order ≥ m + 1. Using (2.30), we obtain

C = N kj (I + f j (h j , v j )) -N kj (h j , v j ) -DN kj (h j , v j )f j (h j , v j ) + (DN kj (h j , v j ) -DN kj (h j , 0))f j (h j , v j ).
We have

(DN kj (h j , v j ) -DN kj (h j , 0))f j (h j , v j ) = (0, Dt kj (h j )f h j (h j , v j )v j ). Thus, C = (0, (Dt kj (h j )f h j (h j , v j )v j ) + a(1) -a(0) -a ′ (0) with a(λ) = N kj (h j + λf h j , v j + λf v j ). Note that a(1)-a(0) -a ′ (0) = 1 0 (1 -λ)a ′′ (λ) dλ = |α|=2 |α|! α! 1 0 (1 -λ)D α N kj (I + λf j )f α j dλ = |α|=2 |α|! α! 1 0 (1 -λ)D α N kj (I + λ[f j ] ≤m-2 )([f j ] ≤m-2 ) α dλ + O(|v j | m+1 ) = b(1) -b(0) -b ′ (0) + O(|v j | m+1 ) for b(λ) = N kj (I + λ[f j ] ≤m-2
). This proves (2.31).

For point (a), we use (2.39) again. This time, we have

A(h j , v j ) = O(|v j | 2m+1 ) and B(h j , v j ) = O(|v j | 2m+1 ), while C = (0, DN kj (h j )[f h j ] ≤2m-1 v j ) + O(|v j | 2m+1
). We have derived (2.32).

For point (b), note that

F -1 k N kj F j = N kj + O(|v j | m+1 ) is equivalent to F k N kj = N kj F j + O(|v j | m+1
). From the vertical components, we obtain

t kj (h j )v j + f v k (ϕ kj (h j ), t kj (h j )v j ) = t kj (h j + f h j )(v j + f v j (h j , v j )) + O(|v j | m+1 ). Since f h j = O(|v j | m ) and f v j = O(|v j | 2 ), the m-jet (w.r.t. v j ) above reads t kj (h j )v j + [f v k ] ≤m (ϕ kj (h j ), t kj (h j )v j ) = t kj (h j )(v j + [f v j ] ≤m (h j , v j )). That is that V ≤m k N kj = N kj V ≤m j , as V ≤m j (h j , v j ) = (h j , v j + [f j ] ≤m (h j , v j )
). The point (c) follows from Proposition 2.14 when N C is flat. For the remaining case, it follows from points (a) and (b) as follows.

By (2.32) and H 0 (C, T C ⊗S ℓ N * C ) = 0, we obtain [f h j ] m 2 = 0. By (b), we know that [F j ] ≤m preserve N kj . Then Fj = F j ([F j ] ≤m ) -1 meet the requirement. The uniqueness of [ F h j ] ℓ for m < ℓ ≤ 2m follows from the assumption on H 0 too.

We are seeking a unique decomposition

F j = H j • V j • Fj . Let us write F -1 k Φ kj F j = N kj + φkj with φkj = O(|v j | 2m+1 ). From the horizontal component of (2.32) in which [ φh kj ] ≤2m = 0 and condition (2.33), we uniquely determine {[f h j ] ≤2m }. Take H j (h j , v j ) = (h j + [f j ] ≤2m (h j , v j ), v j ). Then (2.40) H -1 k Φ kj H j (h j , v j ) = (ϕ kj (h j ), t kj (h j )v j + φv kj (h j , v j )) + O(|v j | 2m+1 ). We still have (H -1 k F k ) -1 (H -1 k Φ kj H j )(H -1 j F j ) ∈ L 2m . We have H -1 j F j (h j , v j ) = V j (h j , v j ) + O(|v j | 2m+1 ), V j (h j , v j ) = (h j , v j + V j (h j , v j )), (2.41)
where φv kj , V j contain only terms of orders ℓ in

v j for m + 1 ≤ ℓ ≤ 2m. Since F j = H j V j + O(|v j | 2m+1 ), we have V -1 k (H -1 k Φ kj H j )V j ∈ L 2m .
From the vertical components of (2.40)-(2.41), and (2.32) in which we take Dt kj [f h j ] ≤2m-1 = 0, we see that (2.32) becomes (2.38)

, i.e. (δ v [V ] ℓ ) kj = -[ φv kj ] ℓ for ℓ = m + 1, . . . , 2m. To
show the uniqueness of [F j ] ≤2m , we may assume that Φ kj = N kj + O(|v j | 2m+1 ). Then the uniqueness follows from the above arguments.

The following is in Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF], when both the dimension and codimension of C are one.

Lemma 2.17. Let Φ kj satisfy condition V m with m ≥ 1. Suppose that N C is flat and

H 0 (C, N C ⊗ S ℓ (N * C )) = 0 for 1 < ℓ ≤ m. Then [φ v kj ] m+1 ∈ H 1 (U, N C ⊗ S m+1 (N * C )
) is independent of coordinates of the neighborhoods of C. Furthermore, there are formal biholomorphic mappings

F j = I + (f h j , f v j ) with f j (h j , v j ) = O(|v j | 2 ) satisfy (2.42) {F -1 k Φ kj F j } ∈ V m+1 if and only if [φ v kj ] m+1 = 0 in H 1 (U, N C ⊗ S m+1 (N * C )). When (2.42) holds, { F -1 k Φ kj Fj } is still in V m+1 , for Fj (h j , v j ) = (h j , v j + [f v j ] m+1 (h j , v j )). Proof. Let Φkj := F -1 k Φ kj F j . We want to show that [ φv kj ] m+1 = [φ v kj ] m+1 in H 1 (U, N C ⊗ S m+1 (N * C )), provided that Φkj (h j , v j ) = N kj (h j , v j ) + ( φh kj , φv kj ), Φ kj (h j , v j ) = N kj (h j , v j ) + (φ h kj , φ v kj )
, and

(2.43) φv kj (h j , v j ) = O(|v j | m+1 ), φ v kj (h j , v j ) = O(|v j | m+1 ). First, we have F j (h j , v j ) = (h j , v j ) + O(|v j | 2 ). Suppose that [f v j ] ≤m * -1 = 0 for 2 ≤ m * ≤ m. Comparing vertical components of Φ kj • F j = F k • Φkj , we obtain t kj • v j + f v j (h j , v j ) ≤m * = (Φ v kj • F j ) ≤m * (h j , v j ) = (F v k • Φkj ) ≤m * (h j , v j ) = (F v k ) ≤m * • N kj (h j , v j ). Here the last identity is obtained from Φkj (h j , v j )-N kj (h j , v j ) = O(|v j | 2 ), [F v j ] ≤m * (h j , v j ) = v j + [f v j ] m *
, and (2.43). Looking at terms of order m * in w j , we see that

{[f v j ] ℓ } is a global section of N C ⊗ S ℓ (N * C ) for ℓ = m * . This shows that [f v j ] ≤m * = 0 and we can take m * = m, i.e. [f v j ] ≤m = 0. We also have [Φ v kj F j ] m+1 = t kj [f v j ] m+1 + [φ v kj ] m+1 and [F v k Φkj ] m+1 = [f v k ] m+1 • N kj + [ φv kj ] m+1 . This shows that (2.44) [ φv kj ] m+1 -[φ v kj ] m+1 = t kj [f v j ] m+1 -[f v k ] m+1 • N kj . The latter is equivalent to [ φv kj ] m+1 = [φ v kj ] m+1 in H 1 (U, N C ⊗ S m+1 (N * C ))
, which follows from Lemma 2.7 (b). The last assertion is equivalent to (2.44) with [ φv kj ] m+1 = 0.

A majorant method for the vertical linearization

Let C be an n-dimensional complex compact manifold embedded in an (n+d)-dimensional complex manifold. We assume that the normal bundle N C is flat and unitary. Let {t kj } be its transition (constant) matrices in a suitable covering U = {U j } of C, we have

t kj t * kj = Id. Let K(N C ⊗ S m (N * C
)) be the "norm" of the cohomological operator acting on C 0 (U, N C ⊗ S m (N * C )) as defined in Theorem A.12. Let us consider the sequence of numbers {η m } m≥1 with η 1 = 1 and

(3.1) η m = K(N C ⊗ S m (N * C )) max m 1 +•••+mp+s=m η m 1 • • • η mp , m > 1,
where 1 ≤ m i < m for all i and s ∈ N.

In this section, we shall prove the following Theorem 3.1. Let C be a compact complex submanifold in M with

T C M = T C ⊕ N C .
Assume that the embedding is vertically linearizable by a formal holomorphic mapping which is tangent to the identity and preserves the splitting of

T C M or that H 1 (C, N C ⊗S ℓ (N * C )) = 0 for all ℓ ≥ 2.
We also assume that N C is unitary flat and that H 0 (C, N C ⊗ S ℓ (N * C )) = 0 for all ℓ ≥ 2. Assume that for the η m defined above, there are positive constants L 0 , L such that η m ≤ L 0 L m for all m. Then the embedding is actually holomorphically vertically linearizable.

Remark 3.2. In the previous Theorem 3.1, if a neighborhood of C is formally vertically linearizable by a minimizing vertical mapping which is tangent to the identity and preserves the splitting of T C M, then the assumption "H 0 (C,

N C ⊗ S ℓ (N * C )) = 0, ℓ > 1" is not necessary.
Here by a formal minimizing vertical mapping it means a map of the form Proof. According to Theorem 3.1, there is a neighborhood of the C in M with suitable holomorphic coordinates patches (V j , (h j , v j )) with (h j , v j ) ∈ C n ×C d and C∩V j = {v j = 0}, such that, on V j ∩ V k , we have

(h j , v j + f v j (h j , v j )) with {f v j } ∈ C 0 (C, ℓ≥2 N C ⊗ S ℓ (N * C )) such that each {[f v j ] ℓ } j is
v k = t kj v j , h k = φkj (h j , v j ).
We then define the foliation in chart V j by dv j = 0.

The rest of the section is devoted to the proof of Theorem 3.1. We follow the method of majorant developed by T. Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF] for 1-dimensional unitary normal bundle over compact complex curve.

(3.2) L v kj (f v j ) = φ v kj (h j , v j + f v j ) -f v k ( Φh kj (h j , v j ), t kj v j ) -f v k (ϕ kj (h j ), t kj v j )
where

Φh kj (h j , v j ) = ϕ kj (h j ) + φ h kj (h j , v j + f v j ), L v kj (f v j ) = f v k (ϕ kj (h j ), t kj v j ) -t kj f v j .
Let us expand φ h kj (h j , v j + f v j ) in power of v j by using φ h kj (h j , w j ) =:

Q∈N d 2 φ h kj,Q (h j )w Q j φ h kj (h j , v j + f v j (h j , v j )) =: Q∈N d 2 h ′ kj,Q (h j )v Q j =: h ′ kj (h j , v j ).
We have

(3.3) Q∈N d 2 h ′ kj,Q (h j )v Q j = Q∈N d 2 φ h kj,Q (h j )(v j + f v j (h j , v j )) Q .
Let us also set

Q∈N d 2 h ′′ kj,Q (h j )v Q j := f v k ( Φh kj (h j , v j ), t kj v j ) -f v k (ϕ kj (h j ), t kj v j ).
As we shall see below, the functions [h ′ ] m and [h ′′ ] m are defined by induction on m ≥ 2 as they depend on [f ] l , l = 2, . . . , m -1. Therefore, the homogeneous polynomial of degree m ≥ 2 of the Taylor expansion of solution of the conjugacy equation satisfies

(3.4) L v kj ([f v j ] m ) = [h ′ kj ] m + [h ′′ kj ] m .
According to Lemma 2.17, there is a solution to the above equation either by the formal assumption or by the assumption that the cohomology class of [h ′ kj ] m + [h ′′ kj ] m is 0, i.e. it is a coboundary. Indeed, since the normal bundle is flat, this class is independent of the coordinates system and the neighborhood is formally vertically linearizable.

3.2.

A modified Fischer norm for symmetric powers. We define a scaler product on the space of polynomials C[x 1 , . . . , x d ] as follows. First, we set

(3.5) x R , x Q mf := (r 1 !)•••(r d !) |R|! if R = Q 0 otherwise , Q C Q x Q 2 mf := Q |C Q | 2 Q! |Q|! ,
where R = (r 1 , . . . , r d ) and andC Q are constants. The subscript mf stands for "modified Fischer". The associated norm will be denoted by |.| k . The Fischer (resp. modified Fischer) scalar product has been used in [START_REF] Fischer | Über die Differentiationsprozesse der Algebra[END_REF][START_REF] Iooss | Polynomial normal forms with exponentially small remainder for analytic vector fields[END_REF][START_REF] Shapiro | An algebraic theorem of E. Fischer, and the holomorphic Goursat problem[END_REF] (resp. [START_REF] Lombardi | Normal forms of analytic perturbations of quasihomogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation[END_REF]). Let ω be an open set on C n . For a vector of polynomials g = (g 1 , . . . ,

|R| = r 1 + • • • + r d ,
g k ) ∈ O k (ω) ⊗ C[x 1 , . . . , x d ],
we set

(3.6) |g| 2 mf,ω := sup z∈ω |g(z, •)| 2 mf := sup z∈ω k j=1 Q∈N d Q! |Q|! |g j,Q (z)| 2 .
We now apply the Fischer norm (resp. modified Fischer norm) to f ∈ C q (U, E ⊗ S L N * C ). Returning to notation in (2.25), we write

f i 0 ...iq (p) = rank E λ=1 |Q|=L f λ i 0 ...iq;Q (z iq (p))e i 0 ,λ (p) ⊗ (w * iq (p)) Q ,
where e i 0 is the base of E over U i 0 and w * iq is the base of

N * C on U iq . Define |f | 2 mf,U := max (i 0 ,...,iq)∈I q+1 sup z iq ∈ϕ iq (U i 0 ...iq ) rank E λ=1 Q Q! |Q|! f λ i 0 •••iq;Q (z iq ) 2 . (3.7)
When there is no confusion, we shall in the sequel write "f " instead of "mf ". The following two propositions are a "version with parameters" of [28, propositions 3.6-3.7] (see also [START_REF] Iooss | Polynomial normal forms with exponentially small remainder for analytic vector fields[END_REF]). We only give the proof of the last two points of next proposition.

Proposition 3.4. Let O n (ω) ⊗ C[x 1 , . . . , x d ] be the set of polynomials f (x, z) in x with coefficients holomorphic in z ∈ ω ⊂ C n . (a) Let f, g ∈ O n (ω) ⊗ C[x 1 , . . . , x d ] be homogeneous polynomials of degree k, k ′ respec- tively. Then |f g| f,ω ≤ |f | f,ω |g| f,ω . (b) Let f ∈ O n (ω) ⊗ C[x 1 , . . . , x d ] and let fP (z, x) = 1 P ! ∂ P z f (z, x). Then | fP | f,ω ′ ≤ |f | f,ω (dist * (ω ′ , ∂ω)) |P | , ∀ω ′ ⊂ ω, dist * (ω ′ , ∂ω) := dist(ω ′ , ∂ω)/ √ n. (c) Let T be a d × d unitary matrix. Let f ∈ O d n (ω) ⊗ C[x 1 , . . . , x d ]. Then, |T f | f,ω = |f | f,ω . (d) Let T be a d × d unitary matrix. Let f ∈ O n (ω) ⊗ C[x 1 , . . . , x d ] and f T (z, x) := f (z, T x). Then, |f T | f,ω = |f | f,ω .
Proof. We only prove the last two points. Fix z ∈ ω ′ . The polydisc center at z with radius δ := dist(ω ′ , ∂ω)/ √ n is contained in ω. By the Cauchy formula, we have

fP (z, x) = 1 δ |P | [0,2π] n f (z + δ(e iθ 1 , . . . , e iθn ), x)(e iθ 1 , . . . , e iθn ) -P dθ 1 2π • • • dθ n 2π = 1 δ |P | Q∈N d x Q [0,2π] n f Q (z + δ(e iθ 1 , . . . , e iθn ))(e iθ 1 , . . . , e iθn ) -P dθ 1 2π • • • dθ n 2π .
We emphasize that the sum is finite. By the Cauchy-Schwarz inequality applied to the integral, we have

| fP (z, •)| 2 mf = 1 δ 2|P | Q∈N d |x Q | 2 mf [0,2π] n f Q (z + δ(e iθ 1 , . . . , e iθn ))(e iθ 1 , . . . , e iθn ) -P dθ 1 2π • • • dθ n 2π 2 ≤ 1 δ 2|P | Q∈N d |x Q | 2 mf [0,2π] n |f Q (z + δ(e iθ 1 , . . . , e iθn ))| 2 dθ 1 2π • • • dθ n 2π = 1 δ 2|P | [0,2π] n Q∈N d |x Q | 2 mf |f Q (z + δ(e iθ 1 , . . . , e iθn ))| 2 dθ 1 2π • • • dθ n 2π ≤ 1 δ 2|P | [0,2π] n |f | 2 ω dθ 1 2π • • • dθ n 2π = 1 δ 2|P | |f | 2 ω .
For the last point, we have, for a homogeneous polynomial f in x of degree m with holomorphic coefficients in ω the identity:

|f m | 2 ω = 1 π d m! sup z∈ω C d |f (z, x)| 2 e -|x| 2 dV (x).
In particular, the integral is invariant under the transformation x → T x when T is unitary (and constant).

Proposition 3.5. For a formal power series

f (h, v) = k f k (z, v) with f k (z, v) being a
homogeneous polynomial in v of degree k of which the coefficients are functions holomorphic in z ∈ U, the following properties are equivalent: (a) f is uniformly convergent for v in a neighborhood of the origin, uniformly in U.

(b) There exist M, R > 0 such that for every k,

|f k | mf,U ≤ M R k .
For convenience, we will use the following orthonormal Fischer base of S L N * C :

e * j,Q = |Q|! Q! (w * j ) Q , |Q| = L, Q ∈ N d .
The transition matrices t L kj of S L N * C is then determined in the following way : Let |t L kj | = 1, L = 1, 2, . . . . We will apply results in the appendix to the transition matrices t L kj . 3.3. A majorization in the modified Fischer norm for the vertical linearization. Let {f v j } be the formal solution of (3.2). We use notation (3.7). Let ϕ j (U j ) = ∆ n and

F k = |P |=L F k,P e *
U kj := U k ∩ U j . Define Ûkj = ϕ j (U kj ). Then, ϕ kj ( Ûkj ) = Ûjk . Let us first assume that H 0 (C, N C ⊗ S ℓ (N * C )) = 0 for all ℓ ≥ 2.
We shall see later on how to get rid of this assumption to prove the general result.

Let us assume that there exists a vertical formal transformation F := {F j } fixing C, being tangent to identity on it, and preserving the splitting of T C M that linearizes vertically a neighborhood of C in M. Let us write

F j (h j , v j ) := (h j , v j + f j ), f j = k≥2 [f j ] k , {[f j ] k } ∈ C 0 (C, N C ⊗ S k (N * C )).
Assume that there is a sequence {A k } k≥2 of positive numbers such that

(3.9) ∀k < m |[f j ] k | Ûj ≤ η k A k . Let us set A(t) = k≥2 A k t k with t ∈ C. Let us first estimate both |[h ′ kj ] m | Ûkj and |[h ′′ kj ] m | Ûkj in term of J m-1 A(t) := A 2 t 2 + • • • + A m-1 t m-1 .
Since φ h kj is holomorphic in h j ∈ Ûkj and v j in a neighborhood of the origin. We can assume that there is a positive R such that sup

h j ∈ Ûkj |φ h kj,Q (h j )| ≤ R |Q| for all Q ∈ N d 2 , where φ h kj,Q is defined by (3.3) and N d k := {Q ∈ N d : |Q| ≥ k}. For Q ∈ N d 2 , we have (v j + f v j (h j , v j )) Q m = (m 1,1 ,...,m 1,q 1 ,...,m d,1 ,...,m d,q d ) d i=1 m i,1 +•••+m i,q i =m d i=1 [f j,i ] m i,1 • • • [f j,i ] m i,q i
where we have set

f v j = (f j,1 , . . . , f j,d ), [f j,i ] 1 = v j,i and [f j,i ] 0 = 0.
In the following, all m i,j are positive integers. Hence, by the first point of Proposition 3.4, we have (3.10)

(v j + f v j (h j , v j )) Q m Ûkj ≤ (m 1,1 ,...,m 1,q 1 ,...,m d,1 ,...,m d,q d ) d i=1 m i,1 +•••+m i,q i =m d i=1 |[f j,i ] m i,1 | Ûj • • • |[f j,i ] m i,q i | Ûj . Let m ≥ 2, for Q ∈ N d 2 , |Q| ≤ m, let us set E Q,m = (m 1,1 , . . . , m 1,q 1 , . . . , m d,1 , . . . , m d,q d ) ∈ N |Q| 1 : d i=1 m i,1 + • • • + m i,q i = m . Let M i = (m (i) 1,1 , . . . , m (i) 1,q (i) 1
, . . . , m

(i) d,1 , . . . , m (i) d,q (i) d ) ∈ N |Q (i) | 1 with |Q (i) | ≤ m i and m i = d j=1 m (i) j,1 + • • • + m (i) j,q (i) j , i = 1, 2. Define the concatenation M 1 ⊔ M 2 to be (M 1 , M 2 ).
We also have 2

j=1 d i=1 m (j) i,1 + • • • + m (j) i,q (j) i = m 1 + m 2 .
Hence, we emphasize that the concatenation (3.11)

  2≤|Q 1 |≤m 1 E Q 1 ,m 1   ⊔   2≤|Q 2 |≤m 2 E Q 2 ,m 2   ⊂ 2≤|Q|≤m 1 +m 2 E Q,m 1 +m 2 .
As a consequence, according to (3.3) and (4.4), we have

  Q∈N d ,|Q|=m h ′ kj,Q (h j )v Q j   m Ûkj ≤ m |Q|=2 R |Q| M ∈E Q,m d i=1 |[f j,i ] m i,1 | Ûj • • • |[f j,i ] m i,q i | Ûj ≤ m |Q|=2 R |Q| M ∈E Q,m d i=1 η m i,1 A m i,1 • • • η m i,q i A m i,q i ≤   m |Q|=2 η Q,m R |Q| (t + J m-1 (A(t)) |Q|   m ≤ E m [g m (t)] m , (3.12)
where we have set

η Q,m := max M ∈E Q,m d i=1 η m i,1 • • • η m i,q i , E m := max Q∈N d 2≤|Q|≤m η Q,m , g m (t) := m |Q|=2 R |Q| (t + J m-1 (A(t)) |Q| , g(t) := |Q|≥2 R |Q| (t + A(t)) |Q| .
Hence, as formal power series, we have

(3.13) g(t) = 1 1 -R(t + A(t)) d -dR(t + A(t)) -1. Let U * = {U * i }
:= ϕ j (U j ∩ U * k ). We have   Q∈N d ,|Q|=m h ′′ kj,Q (h j )v Q j   m = P ∈N n 1 m 1 +m 2 =m 1 P ! ∂ P h f k (ϕ kj (h j ), t kj v j ) m 1 φ h kj (h j , v j + f v j ) P m 2 = P ∈N n 1 m 1 +m 2 =m 1 P ! ∂ P h f k (ϕ kj (h j ), t kj v j ) m 1 h ′ kj (h j , v j ) P m 2 .
Here, both indices m 1 and m 2 are ≥ 2. Since the Fischer norm is submultiplicative, we have

h ′ kj (h j , v j ) P m 2 Û * kj ≤ E m 2      m 2 |Q|=2 R |Q| (t + J m-1 (A(t)) |Q|   |P |    m 2 . Indeed, h ′ kj (h j , v j ) P m 2 = n i=1 (h ′ kj,i ) p i m 2 = i (m i,1 +•••+m i,p i )=m 2 n i=1 [h ′ kj,i ] m i,1 • • • [h ′ kj,i ] m i,p i .
According to (3.11) and by (3.12), we have

n i=1 [h ′ kj,i ] m i,1 • • • [h ′ kj,i ] m i,p i Û * kj ≤ n i=1 E m i,1 g m i,1 (t) m i,1 • • • E m i,p i g m i,p i (t) m i,p i ≤ max 2≤|Q|≤m 2 η Q,m 2 n i=1 g m i,1 (t) m i,1 • • • g m i,p i (t) m i,p i .
Hence, we have

i (m i,1 +•••+m i,p i )=m 2 n i=1 [h ′ kj,i ] m i,1 • • • [h ′ kj,i ] m i,p i Û * kj ≤ E m 2 [g(t) |P | ] m 2 .
We have, by definition

∂ P h f k (ϕ kj (h j ), t kj v j ) m 1 = ∂ P h [f k ] m 1 (ϕ kj (h j ), t kj v j ).
Recall that the Fischer norm is unitary invariant and by Proposition 3.4, we have

∂ P h [f k ] m 1 (ϕ kj (h j ), t kj v j ) 2 Û * kj = ∂ P h [f k ] m 1 (ϕ kj (h j ), v j ) 2 Û * kj ≤ P ! dist * ( Û * k , ∂ Ûk ) |P | 2 |[f k ] m 1 | 2 Ûk .
Let us set M := inf k dist( Û * k , ∂ Ûk ). As a consequence, we have

  Q∈N d ,|Q|=m h ′′ kj,Q (h j )v Q j   m Û * kj ≤ m 1 +m 2 =m P ∈N n |P |≥1 1 M |P | |[f k ] m 1 | Ûk E m 2 [g(t) |P | ] m 2 ≤ m 1 +m 2 =m |[f k ] m 1 | Ûk     E m 2 P ∈N n |P |≥1 g(t) M |P |     m 2 ≤ max m 1 +m 2 =m η m 1 E m 2 A(t) M M -g(t) n -1 m . (3.14)
Collecting estimates (3.12) and (3.14), we obtain

L v kj ([f v j ] m ) Û * kj ≤ E m g(t) + max m 1 +m 2 =m η m 1 E m 2 A(t) M M -g(t) n -1 m .
Let us extend this to an estimate on Ûkj = ϕ j (U j ∩ U k ). Following again Ueda's argument [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF]

let us express the fact that [h]

m := [h ′ ] m + [h ′′ ] m is a 1-cocycle with values in N C ⊗ S m (N * C ). Let p ∈ U k ∩ U j . Then p ∈ U k ∩ U j ∩ U * i for some i. According to (3.4) and Lemma 2.7, at p ∈ U k ∩ U j ∩ U * i we have (3.15) t ki |Q|=m h ik,Q (z k (p))(t kj v j ) Q -t ki |Q|=m h ij,Q (z j (p))(v j ) Q + |Q|=m h kj,Q (z j (p))(v j ) Q = 0.
Here by (3.7) the Fischer norms of h kj on all subdomains must be computed in the base e v k of N C on U k and the base w * j of N * C on U j . We can apply the previous estimates (3.12) and (3.14) to the first two sums respectively on Û * ik and Û * ij . To estimate the first sum, we need to change coordinates. From section 2, t kj (resp. s kj ) are transition matrices of N

C (resp. T C). Recall that {[h kj ] m } ∈ Z 1 (U r * , N C ⊗ S m N * C ) and h ik (p) = d λ=1 |Q|=m h λ ik;Q (z k (p))e v i,λ (p) ⊗ (w * k (p)) Q = d λ ′ =1 d λ=1 |Q|=m h λ ik;Q (z k (p))t λ ′ ki,λ (z k (p))e v k,λ ′ (p) ⊗ (t kj w * j (p)) Q =: hkj (z k (p), w * j ). Thus, |Q|=m h ik,Q (z k (p))(t kj v j ) Q = hkj (z k (p), v j )
. By the unitary invariance by multiplication and composition of the Fischer norm and by definition (3.7), we have for fixed

z k (p) ∈ Û * ik , | hkj (z k (p), v j )| 2 mf = d λ ′ =1 |Q|=m d λ=1 t λ ′ ki,λ (z k )h λ ik;Q (z k ) (t kj v j ) Q 2 mf = d λ ′ =1 |Q|=m d λ=1 t λ ′ ki,λ (z k )h λ ik;Q (z k ) v Q j 2 mf = d λ ′ =1 |Q|=m Q! |Q|! d λ=1 t λ ′ ki,λ (z k )h λ ik;Q (z k ) 2 ≤ λ ′ |Q|=m Q! |Q|! d λ=1 h λ ik;Q (z k ) 2 ≤ d|h ik | 2 Û * ik ,
where the second last inequality is obtained by the Cauchy-Schwarz inequality. In a similar way, we have a similar estimate for the second sum in (3.15) on ϕ j (U k ∩ U j ∩ U * i ). For the third sum in (3.15), we note that the entries of the unitary matrix t ki have modulus at most one. Thus, there exist constants M ′ , M such that the third sum in (3.15) satisfies

|h kj | Ûkj ≤ M ′ max i (|h ik | Û * ik + |h ij | Û * ij ) ≤ M max   E m , max m 1 +m 2 =m m 1 ,m 2 ≥2 η m 1 E m 2   g(t) + A(t) M M -g(t) n -1 m .
We now adapt the estimate in Lemma A.2 (see also Theorem A.12). Recall that [h kj ] ≤m depends only on [f ] ≤m-1 and the hypothesis (3.9). By the formal assumption, we have a solution to (3.4):

L kj ([f v j ] m ) = [h kj ] m . By assumptions, H 0 (C, N C ⊗S ℓ (N * C )) = 0, for all ℓ ≥ 2.
Hence, the solution of the previous equation is unique. By Lemma A.2, (A.5) and (3.8), the solution satisfies the estimate:

|{[f v j ] m }| U ≤ C(1 + K * (N C ⊗ S m N * C )|){[h kj ] m }| U . Here, C depends neither on N C nor on S m N * C . Therefore, we have |[f v j ] m | Ûj ≤ K(N C ⊗ S m (N * C )) max k L v kj ([f v j ] m ) Û * kj .
By definition (3.1), we have

K(N C ⊗ S m (N * C )) max   E m , max m 1 +m 2 =m m 1 ,m 2 ≥2 η m 1 E m 2   ≤ η m .
Hence, we have

(3.16) |[{f v }] m | Û ≤ M η m g(t) + A(t) M M -g(t) n -1 m .

Let us consider the functional equation

A(t) = F (t, A(t)) := M g(t) + A(t) M M -g(t) n -1 ,
where g(t) is a function of A by (3.13). This equation has a unique analytic solution vanishing at the origin at order 2.

We now can prove the theorem. Indeed by assumption, there are positive constants M, L such that η m ≤ ML m for all m ≥ 2. Since A(t) converges at the origin, then A m ≤ D m for some positive D. According to (3.16), we have also proved

|[{f v }] m | Û ≤ η m A m , so that, finally, |[{f v }] m | Û ≤ M(DL) m for all m ≥ 2. Hence, f v = m≥2 [{f v }] m
converges at the origin and this proves the theorem.

Let us see how we can prove Remark 3.2. The issue is that, when considering a solution

[f v j ] m of the cohomological equation L kj ([f v j ] m ) = R m
, the estimate given by Lemma A.2 and Proposition A.4 might be obtained by another solution. Hence, the formal solution might not be the good one for the estimate. Furthermore, we cannot replace a solution at degree m as we wish to ensure that higher order terms in the vertical component can be eliminated formally. We now explain the general result as formulated in the theorem. We will assume that there are formal mappings

Fj (h j , v j ) = (h j , v j ) + 0, ℓ>2 f v j,ℓ (h j , v j ) satisfying the following (1) { F -1 k Φ kj Fj -N kj } v =
0 for all k, j. In other words, { Fj } formally linearizes Φ kj vertically. In particular,

{( F m k ) -1 Φ kj F m j -N kj } v = [φ v kj ] m + R m kj ({[φ kj ] ℓ , [ f v k ] ℓ } 2≤ℓ<m ) + O(|v j | m+1 ) for F m j = (h j , v j ) + 0, 2≤ℓ≤m f v j,ℓ (h j , v j ) .
(The last assertion can be check easily since ( F m j ) -1 Fj (h j , v j ) = (h j , v j )+O(|v j | m+1 )).

(2) Each { f v j,m } j is a "minimizer" in the sense that it satisfies the equation

{δ v f v m } kj = [φ v kj ] m + [R m ({[φ kj ] ℓ , [ f v k ] ℓ } 2≤ℓ<m )] m and the estimate | f v m | ≤ K(N C ⊗ S m (N * C ))|[φ v ] m + [R m kj ({[φ kj ] ℓ , [ f v k ] ℓ } 2≤ℓ<m )] m |.
As a consequence, the scheme of convergence applies to that formal solution { Fj } and we are done.

A majorant method for the full linearization with a unitary normal bundle

In this section, we shall devise a proof of Theorem 1.4, that is of the linearization of the neighborhood problem in the case N C is unitary (and flat) following a majorant method scheme.

Let us recall the horizontal cohomological operator

L h kj (f h j ) := f h k (ϕ kj (h j ), t kj v j ) -s kj (h j )f h j (h j , v j )
, where s kj (h j ) = Dϕ kj (h j ). We then have the horizontal equation (2.17)

L h kj (f h j ) = φ h kj (h j + f h j , v j + f v j ) (4.1) + ϕ kj (h j + f h j (h j , v j )) -ϕ kj (h j ) -Dϕ kj (h j )f h j (h j , v j ). Let us recall the vertical cohomological operator L v kj (f v j ) := f v k (ϕ kj (h j ), t kj v j ) -t kj f v j , and vertical equation (2.19) (recall that N C is flat) (4.2) L v kj (f v j ) = ϕ v kj (h j + f h j , v j + f v j ).
By assumption, there exists a formal solution

f j = (f h j , f v j ) = k≥2 [f j ] k with {[f j ] k } ∈ C 0 (C, T C M ⊗ S k (N * C )). In case we assume H 1 (C, T C M ⊗ S k (N * C )) = 0,
for all k ≥ 2, this follows from Lemma 2.10. We now use the "norm" of the cohomological operator acting on C 0 (U, T C M ⊗ S m (N * C )) as defined by Theorem A.12. We have, for m ≥ 2 Km :

= max (K(N C ⊗ S m (N * C )), K(T C ⊗ S m (N * C )))
. As in the foliation problem, we consider the sequence of numbers {η m } m≥1 with η 1 = 1 and, if m ≥ 2 (4.3) η m := Km max

m 1 +•••+mp+s=m η m 1 • • • η mp ,
where, in the maximum, 1 ≤ m i < m for all i and s ∈ N. In what follows, f • j (resp. φ • kj ) stands for either f h j or f v j (resp. φ h kj or φ v kj ). As in the previous section, let us expand φ • kj (h j + f h j , v j + f v j ) appeared in (4.1) and (4.2) in power series of v j and let us define φ • kj (z j , w j ) =:

Q∈N d 2 φ • kj,Q (z j )w Q j φ • kj (h j + f h j (h j , v j ), v j + f v j (h j , v j )) =: Q∈N d 2 h • kj,Q (h j )v Q j =: h • kj (h j , v j ).
Then we obtain

Q∈N d 2 h • kj,Q (h j )v Q j = Q∈N d 2 φ • kj,Q (h j + f h j (h j , v j ))(v j + f v j (h j , v j )) Q .
We further expand the first expression on the right-hand side as

h• kj,Q := φ • kj,Q (h j + f h j (h j , v j )) = P ∈N n 1 P ! ∂ P h φ • kj,Q (h j )(f h j (h j , v j )) P .
Hence, for any m ≥ 2,

[h • kj ] m = m 1 +m 2 =m Q∈N d 2 P ∈N n 1 P ! ∂ P h φ • kj,Q (h j ) (f h j (h j , v j )) P m 1 (v j + f v j (h j , v j )) Q m 2 .
Let {f • j } be the formal solution of (4.1) and (4.2). Let us first assume that H 0 (C, T C M ⊗ S ℓ (N * C )) = 0 for all ℓ ≥ 2. We shall see later on how to get rid of the assumptions. Assume that there is a sequence {A k } k≥2 of positive numbers such that

∀k < m |[f j ] k | Ûj ≤ η k A k . Let us set A(t) = k≥2 A k t k with t ∈ C.
Since φ • kj is holomorphic in h j ∈ Ûkj and v j in a neighborhood of the origin, we can assume that there is a positive R such that (4.4) sup

h j ∈ Ûkj |φ • kj,Q (h j )| ≤ R |Q| .
According to (3.10) and the proof of (3.12), we obtain

(v j + f v j (h j , v j )) Q m 2 Ûkj ≤ (m 1,1 ,...,m 1,q 1 ,...,m d,1 ,...,m d,q d ) d i=1 m i,1 +•••+m i,q i =m 2 d i=1 |[f j,i ] m i,1 | Ûj • • • |[f j,i ] m i,q i | Ûj ≤ M ∈E Q,m 2 d i=1 η m i,1 A m i,1 • • • η m i,q i A m i,q i ≤ η Q,m 2 t + J m 2 -1 A(t) |Q| m 2 .
On the other hand, let

U * = {U * i } be an open covering of C such that U * i is relatively compact in U i . We shall write Û * k := ϕ k (U * k ). Let us set M := min k dist( Û * k , ∂ Ûk ).
Let us consider the index j as fixed and let us estimate the Fischer norm of

[ h• kj ] m 1 on Û * kj := ϕ j (U j ∩ U * k ). We get [ h• kj ] m 1 Û * kj = P ∈N n 1 P ! ∂ P h φ • kj,Q (h j ) (f h j (h j , v j )) P m 1 Û * kj ≤ P ∈N n 1 dist( Û * k , ∂ Ûk ) |P | φ • kj,Q Ûkj (f h j (h j , v j )) P m 1 Û * kj ≤ P ∈N n 1 M |P | R |Q| (f h j (h j , v j )) P m 1 Û * kj .
Since f j is of order ≥ 2 at v j = 0, we have |P | ≤ m 1 2 in the above sum. According to estimate (3.10) and following the proof of (3.12), we obtain

[ h• kj,Q ] m 1 Û * kj ≤ m 1 2 P ∈N n ,|P |=0 1 dist( Û * k , ∂ Ûk ) |P | R |Q| η P,m 1 A(t) |P | m 1 . (4.5)
Combining inequalities (4.5) and (4.5), we obtain

[h • kj ] m Û * kj ≤ m 1 +m 2 =m Q∈N d 2 P ∈N n 1 P ! ∂ P h φ • kj,Q (h j ) (f h j (h j , v j )) P m 1 (v j + f v j (h j , v j )) Q m 2 Û * kj ≤ m 1 +m 2 =m m 2 Q∈N d |Q|=2 m 1 2 P ∈N n |P |=0 1 M |P | R |Q| η P,m 1 A(t) |P | m 1 η Q,m 2 t + J m 2 -1 A(t) |Q| m 2 ≤ m 1 +m 2 =m m 2 Q∈N d |Q|=2 m 1 2 P ∈N n |P |=0 A(t) M |P | m 1 η P,m 1 η Q,m 2 Rt + RJ m 2 -1 A(t) |Q| m 2 ≤ Ẽm 1 1 -A(t) M n 1 1 -(Rt + RA(t)) d -1 -d(Rt + RA(t)) m .
Here, we have set

Ẽm = max m 1 +m 2 =m max P ∈N n ,Q∈N d |P |≤ m 1 2 ,2≤|Q|≤m 2 , η P,m 1 η Q,m 2 .
It remains to estimate the rest of terms in (4.1). We define

B m : = ϕ kj (h j + f h j (h j , v j )) -ϕ kj (h j ) -Dϕ kj (h j )f h j (h j , v j ) m = m 2 l=2 |P |=l 1 P ! ∂ P h ϕ kj (h j ) (f h j ) P m .
Hence, as above, we have

|B m | Û * kj ≤ |ϕ kj | Ûkj m 2 l=2 |P |=l 1 M |P | (A(t)) |P | m ≤ |ϕ kj | Ûkj 1 1 -A(t) M n -1 -n A(t) M m .
By the same reasoning as in the foliation section, the previous estimates on Û * kj extend to estimates on Ûkj , by multiplication by a constant M.

Let us define constant C 0 := max kj |ϕ kj | Ûkj Since we have

|[f • j ] m | Ûj ≤ Km max k L kj ([f • j ] m ) Ûkj , then |[f • j ] m | Ûj ≤ M Km C 0 1 1 -A(t) M n -1 -n A(t) M m + Ẽm 1 1 -A(t) M n 1 1 -(Rt + RA(t)) d -1 -d(Rt + RA(t)) m .
We emphasize that due to the vanishing assumption of the spaces H 0 (U,

T C M ⊗ S m (N * C )), m ≥ 2, the solution of cohomological equation L kj ([f • j ] m ) = R m
is unique and is equal to the minimizing solution obtained in Lemma A.2 and Proposition A.4. Consider the following analytic functional equation :

A(t) = M C 0 1 1 -A(t) M n -1 -n A(t) M + 1 1 -A(t) M n 1 1 -(Rt + RA(t)) d -1 -d(Rt + RA(t))
.

It has a unique analytic solution A of order ≥ 2 at the origin. Since we have Km max

(1, Ẽm ) ≤ η m , |[f • j ] m | Ûj ≤ A m η m , m ≥ 2 then m≥2 [f • j ] m converges in a neighborhood of the origin.
Let us see how the general case reduces to the previous one. The issue is that, when considering a solution [f

• j ] m of the cohomological equation L kj ([f • j ] m ) = R m ,
the estimate given by Lemma A.2 and Proposition A.4 might be obtained by another solution. Hence, the formal solution might not be the good one for the estimates. So we will need to correct it. As we already emphasized, equations (4.1) and (4.2) read

L kj ({[f ] ℓ i }) = R kj,ℓ ([f ] ℓ ′ , ℓ ′ < ℓ; [Φ] l , l ≤ ℓ)
where R kj,ℓ is an analytic function of its arguments. Let us start at ℓ = 2.

(1) R kj,2 is just a function of the [Φ kj ] 2 's and and we have

L kj ([f ] 2 ) = R kj,2 . Let {[ fj,2 ] 2 }
] 2 := [f j ] 2 -[ fj,2 ] 2 . We have {[k j ] 2 } ∈ H 0 (U, T C M ⊗ S 2 (N * C )). (2) According to lemma Lemma 2.14, F j,2 := F j exp(-[k j ] 2 ) linearizes Φ kj since F -1 j,2 Φ kj F j,2 = exp(-[k j ] 2 ) -1 N kj exp(-[k j ] 2 ) = N kj . F j,2
is tangent to identity and its 2nd order term is the minimizer [ fj ] 2 .

(3) Assume that F j,ℓ linearizes Φ kj , is tangent to identity at the origin and has the minimizers solution up to degree ℓ as Taylor expansion at 0. This means that

F j,ℓ = Id + ℓ l=2 [ fj,l ] l + l≥ℓ+1 [f j,ℓ ] l .
Let us write the conjugacy equation. By induction we have, for all 2 ≤ l ≤ ℓ,

L kj ({[ fi,l ] l }) = R kj,l ({[ fi,l ′ ] l ′ } i , l ′ < l; [Φ] m , m ≤ l).
Furthermore, it satisfies at degree ℓ + 1

L kj ({[f i,ℓ+1 ] ℓ+1 }) = R kj,ℓ+1 ({[ fi,ℓ ′ ] ℓ ′ } i , ℓ ′ ≤ ℓ; [Φ] m , m ≤ ℓ + 1).
Let [ fi,ℓ+1 ] ℓ+1 be the minimizer solution of the above cohomological equation. Let [k i,ℓ+1 ] ℓ+1 = [f i,ℓ+1 ] ℓ+1 -[ fi,ℓ+1 ] ℓ+1 . As above, it defines an element of H 0 (U, T C M ⊗ S ℓ+1 (N * C )). Let us set F j,ℓ+1 = F j,ℓ exp([k j,ℓ+1 ] ℓ+1 ) -1 . Then it linearizes Φ kj and has the minimizers solution up to degree ℓ + 1 as Taylor expansion at 0:

F j,ℓ+1 = Id + ℓ+1 l=2 [ fj,l ] l + l≥ℓ+2 [f j,ℓ+1 ] l . (4) Since F j,ℓ+1 F -1 j,ℓ = I + O(ℓ + 1)
, the sequence {F j,ℓ } ℓ converges in the space of formal power series to Fj . Furthermore, { Fj } linearizes {Φ kj } as each {F j,ℓ } j does. The Taylor expansion of Fj at the origin is

Fj = Id + l≥2 [ fj,l ] l .
(5) We can estimate the [ fj,l ] l as we did above in the case of vanishing cohomology since the Taylor coefficient are minimizer solutions of the same equations. Hence, we are done.

In summary, we have proved the following theorem.

Theorem 4.1. Let C be an embedded compact manifold in M. Assume that the embedding is linearizable by a formal holomorphic mapping which is tangent to the identity and preserves the splitting of T C M, and N C is unitary. Suppose that {η m } m≥1 defined by (4.3) satisfy η m ≤ L 0 L m , for some positive numbers L 0 , L and for all m. Then the embedding is actually holomorphically linearizable.

We remark that in general there is a rigid theory on deformations in an analytic family of complex complex manifolds due to Kodaira [START_REF] Kodaira | A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds[END_REF]. Strengthening Corollary 3.3, we finish the section with the following corollary. This may be regarded as a rigidity for a simplify connected manifold. Proof. We already know that M admits a horizontal foliation by Corollary 3.3. To show that each leaf is biholomorphic to C, we may assume that M = N C and we will use the projection π : N C → C. We fix x 0 ∈ C. We take a point p ∈ π -1 (x 0 ) close to C. Let L be the (connected) leaf of the foliation containing p. Then L intersects each fiber of N C at a unique point. To verify this, we connect a point in x ∈ C to x 0 by a continuous path γ in C with γ(0) = x 0 and γ(1) = x. By continuation along leaves, we can find a lifted continuous path γ and the germ L * γ (t) at γ(t) of a leaf L γ (t) such that π(γ(t)) = γ(t). Note that L * γ (t ′ ), L * γ (t) are contained in the same leaf on which π is injective, when t ′ is sufficiently close to t. The lifting γ(1) is independent of γ. Indeed if γ θ (a ≤ θ ≤ b) is a continuous family of paths connecting x 0 to x. Let L γ θ be the leaf associated to γ θ . Then γ θ (t) ∈ L γ θ 0 (t) when θ is sufficiently close to θ 0 , as L γ θ (0) = L γ a (0) as a leaf near p.

Obviously, x → γ(1) gives a biholomorphism from C onto the leaf through p. And (x, v) → γ(1) defines a biholomorphisms from C × B into N C , where B is a small neighborhood of 0 ∈ π -1 (x 0 ).

The full linearization

The main purpose of this section is to solve the linearization problem in the general setting (i.e. N C not necessarily being flat) under a general hypotheses on the existence of bounds to the cohomology equations. At the end of the section we will illustrate the results with Arnold's examples [START_REF] Arnol'd | Bifurcations of invariant manifolds of differential equations, and normal forms of neighborhoods of elliptic curves[END_REF], following computations by Arnol'd [START_REF]Geometrical methods in the theory of ordinary differential equations, Second, Grundlehren der Mathematischen Wissenschaften[END_REF].

We shall devise a Newton scheme to solve the linearization of the neighborhood problem. Let us recall the condition.

(L m ) : The neighborhood of C agrees with the neighborhood of the zero section of the normal bundle up to order m.

That embedding of C has property (L m ) means that the order of (φ h kj (h j , v j ), φ v kj (h j , v j )) along v j = 0 as defined in (5.16) is ≥ m + 1.

Assuming that (L m ) holds. We shall assume either that H 0 (C, T C ⊗S p N * C ) = 0, 2 ≤ p ≤ 2m or that N C is flat. According to Lemma 2.16 (c) and (d), the following linearization step in the Newton method is fulfilled : ) for m = 2, 3, . . . such that for all r ′ , r ′′ with r * < r ′′ < r ′ < r < r * and r ′ -r ′′ ≤ r * -r, and all

(N m ) : If {Φ kj } ∈ L m , then {F -1 k Φ kj F j } ∈ L 2m for some {F j = I + f j } with f j (h j , v j ) = O(|v j | m+1 ).
f ∈ Z 1 (U r ′ , E ′ ⊗S m N * C ) with f = 0 in H 1 (U r ′ , E ′ ⊗ S m N * C ), there is a solution u ∈ C 0 (U r ′′ , E ′ ⊗ S m N * C ) to δu = f such that (5.1) max j sup |u j | L ∞ ( Û r ′′ j ) ≤ D(E ′ ⊗ S m N * C ) (r ′ -r ′′ ) τ max k,j |f kj | L ∞ ( Û r ′ kj ) , where D(E ′ ⊗ S m N * C ) is independent of r ′ , r ′′ and f and τ = τ (N * C ) is independent of m.
In what follow, we shall express sections of bundles in coordinates. It is more convenient to express domains by using the trivialization of the vector bundle N C . Recall that the N C has trivializations N j and transition functions N kj . Let B r d be the ball of radius r in C d centered at the origin. Thus, we define

V r j = N j (V r j ) = Ûr j × B r d , V r i 0 •••iq := V r i 0 ∩ • • • ∩ V r iq , (5.2) V r i 0 •••iq := N iq (V r i 0 •••iq ) ⊂ ϕ iq (U r i 0 •••iq ) × C d , V r jk = N kj ( V r kj ), N kj = N -1 jk on V r kj , (5.3)
N ki N ij = N kj on V r kij . (5.4)
Denote the corresponding domains by Ṽ r j , Ṽ r kj when N j are replaced by Φ j . Then we still have the above relations when N j , N kj are replaced by Φ j , Φ kj . We know that Φ kj are perturbations of the transition functions N kj of the normal bundle of C in M, which are defined on different domains but in the same space. We will however work on domains V r kj for Φ kj , instead of Ṽ r kj . With notation of section 2.6, for L ≥ 1 and for r * ≤ r ≤ r * , we consider a cochain

{f I } ∈ C q+1 (U r , O(T C M ⊗ S L (N * C ))
), given by

f I := f i 0 •••iq (p) = n+d λ=1 |Q|=L f λ i 0 •••iq;Q (z iq (p))e i 0 ,λ (p) ⊗ (w * iq (p)) Q where I = (i 0 , . . . , i q ) ∈ I q+1 . Recall that V r I = N iq (V r i 0 ∩ • • • ∩ V r iq ). Define |f I | r = sup (h iq ,v iq )∈ V r I | Q f I,Q (h iq )v Q iq |.
We also set |{f

I }| r = max I |f I | r . Note that V r j = Ûj × B r d are product domains. Also, Ûr kj × B c * r d ⊂ V r kj ⊂ Ûr kj × B c * r d , c * ≤ 1 ≤ c * . Define B r kj (h j ) to be {v j ∈ B r d : t kj (h j )v j ∈ B r d }.
The skewed domain V r kj can be described as follows:

(h j , v j ) ∈ V r kj if and only if h j ∈ Ûr kj , v j ∈ B r kj (h j ). Next, we note that the d-torus action (h j , v j ) → (h j , (ζ 1 v 1 , . . . , ζ d v d )) with ζ ∈ (S 1 ) d does not preserve V r kj when t kj (h j ) is not diagonal. Nevertheless, the V r kj has a disc structure : (h j , ζv j ) ∈ V r kj , ∀(h j , v j ) ∈ V r kj , ∀ζ ∈ ∆. Indeed, suppose that (h j , v j ) ∈ V r kj . Then h j ∈ Ûr kj and (h j , v j ) = N j (p) with p ∈ V r k ∩ V r j and N k (p) = (h k , v k ) ∈ V r k . By definition, V r j = Ûj × B r d . Take p = N -1 j (h j , ζv j ). We have p ∈ V r j and N k (p) = (h k , t kj (ζv j )) = (h k , ζt kj (v j )) ∈ Ûr kj × B r d .
Throughout this section, we use

|u j | ρ = sup (h j ,v j )∈ V ρ j |u j (h j , v j )|, |u kj | ρ = sup (h j ,v j )∈ V ρ kj |u kj (h j , v j )|
where u j , u kj are functions on V r j and V r kj , respectively. We also define |{u I }| ρ = max I |u I | ρ . We now prove the following. Lemma 5.1. Let u kj be a holomorphic function on V r kj with r * < r < r < r * . Suppose that (5.5) V r * kj = ∅. For 0 < θ < 1 with θr > r * , we have

|u kj | θr ≤ θ m |u kj | r , if u kj (h j , v j ) = O(|v j | m ); |[u kj ] ℓ | r ≤ |u kj | r ; ∞ ℓ=i |[u kj ] ℓ | θr ≤ θ i 1 -θ |u kj | r .
Proof. Let u = u kj . The first inequality follows from the Schwarz lemma applied to the holomorphic function ζ → u(h j , ζv j ) on the unit disk for fixed (h

j , v j ) ∈ V r kj . Note that [u] i (h j , ζv j ) = ζ i [u] i (h j , v j ).
Thus the second inequality follows directly by averaging,

[u] ℓ (h j , v j ) = 1 2πi ζ∈∂∆ u(h j , ζv j ) dζ ζ ℓ+1 , (h j , v j ) ∈ V r kj .
The last inequality follows from the first two inequalities.

For the rest of this section, we rename r in the Donin Condition by r which is fixed now. We will let r vary in (r * , r). Lemma 5.2. Let r * < θr < r < r < r * < 1. Fix k, j ∈ I. Suppose that (1θ 4 )r < r *r and (5.5) holds.

(a) We have dist( V θr j , ∂ V r j ) ≥ r(1 -θ)/C 0 , dist( V θr kj , ∂ V r kj ) ≥ r(1 -θ)/C 0 , (5.6)
for some constant C 0 . (b) Assume further that θ 4 r > r * . There exists a constant C * 0 such that if

F j = I + f j satisfy (5.7) |f j | θ 2 r ≤ (1 -θ)r/C * 0 , then we have F j ( V θ 2 r j ) ⊂ V θr j , F j ( V θ 2 r kj ) ⊂ V θr kj , (5.8)
F -1 j ( V θ 4 r j ) ⊂ V θ 3 r j , F j F -1 j = I on V θ 4 r j . (5.9)
Proof. (a) The V r j is the product domain Ûr j × B r d . Thus the first inequality in (5.6) holds trivially since Ûr j is a polydisc. Note that V r kj are open sets. Then

δ := dist((h, v), ( h, ṽ)) = dist( V θr kj , ∂ V r kj ) is attained by (5.10) (h, v) ∈ ∂ V θr kj , ( h, ṽ) ∈ ∂ V r kj . If h ∈ ∂ Ûr kj , we immediately get δ ≥ dist( Ûθr kj , ∂ Ûr kj ) ≥ (1 -θ)r/C by Lemma A.6. Assume that h ∈ Ûr kj .
Then by the continuity of the function t kj , ṽ must be in ∂B r kj ( h). Otherwise, both h ∈ Ûr kj and ṽ ∈ B r kj ( h) are interior points of the two sets, then any small perturbation of ( h, ṽ) still satisfies the second condition in (5.10). The last assertion implies that ( h, ṽ) cannot be a boundary point and we get a contradiction. Therefore, we have ṽ ∈ ∂B r d or t kj ( h)ṽ ∈ ∂B r d . The first case yields |ṽ -v| ≥ dist(B θr d , ∂B r d ) = (1θ)r. We now consider the second case. By assumption t kj is holomorphic in ω for a neighborhood ω of Ûkj . Thus there is δ * > 0 depending only on Ûkj such that if h ∈ Ûkj and | h -h| < δ * , then the line segment γ connecting h, h is contained in ω. Suppose that | h -h| < (1θ)r/C 1 for C 1 to be determined so that (1θ)r/C 1 < δ * . Applying the mean-value-theorem to t kj (γ) and using

t kj (h)v ∈ B θr d , we get C 4 |ṽ -v| ≥ |t kj ( h)(ṽ -v)| ≥ |t kj ( h)ṽ -t kj (h)v)| -|(t kj ( h) -t kj (h))v| ≥ (1 -θ)r -C 5 | h -h||v| ≥ (1 -θ)r/2, when C 1 is sufficiently large. Thus we get dist( Ûθr kj , ∂ Ûr kj ) ≥ (1 -θ)r/C as in the first case. If | h -h| ≥ (1 -θ)r/C 1 , the required estimate is immediate.
(b) Note that θ > r * . By choosing a larger C * 0 , (5.8) follows from (5.6) immediately. We want to find F -1 . By (5.7) and the Cauchy estimate, we know that (5.11)

|∂ h j f j (h j , v j )| + |∂ v j f j (h j , v j )| ≤ C 6 /C * 0 , ∀(h j , v j ) ∈ V θ 3 r j .
Note that V r j = Ûr j × B r d is convex. By (5.11) and the fundamental theorem of calculus, we have

|f j (p 1 ) -f j (p 0 )| ≤ C 7 |p 1 -p 0 |/C * 0 , ∀p 0 , p 1 ∈ V θ 3 r j . Suppose that C * 0 > 2C 7 . Then F j : V θ 3 r j → V θ 2 r j
is injective, and T (h j , v j ) = ( hj , ṽj )f j (h j , v j ) defines a contraction mapping on V θ 3 r j , if ( hj , ṽj ) ∈ V θ 4 r j and C * 0 is sufficiently large. This gives us (5.9).

In this section, we change notation and let

f • j = (f h j , f v j ), φ • kj = (φ h kj , φ v kj ).
Lemma 5.3. Let r * < θr < r < r < r * < 1. Suppose that Vkj satisfies (5.5). There exists

a constant C * 1 such that if |φ • kj | r ≤ (1 -θ)r/C * 1 (5.12)
then we have

Φ kj ( V θr kj ) ⊂ V r jk .
Proof. Note that θ > r * . Since Φ kj -N kj = φ • kj and N kj ( V θr kj ) = V θr jk , the assertion follows from (5.6) and (5.12) for sufficiently large C * 1 .

Proposition 5.4. Let r * < θ 7 r < r < r < r * < 1. Assume that Vkj satisfies (5.5). Suppose that Φ kj = N kj + φ • kj satisfy (5.12). Let

F j = I + f j satisfy f j (h j , v j ) = O(|v j | 2 ). Suppose Φkj = F -1 k Φ kj F j = N kj + φ• kj . There exists a constant C * 2 such that if (5.13) |{f j }| θ 2 r ≤ (1 -θ)r/C * 2 , and φ• kj (h j , v j ) = O(|v j | m ), then |{ φ• kj }| θ 7 r ≤ C 2 θ m (|{f j }| θ 2 r + |{φ • kj }| r , (5.14) |{ φ• kj }| θ 7 r ≤ C 2 θ m (1 -θ)r. (5.15) Proof. Let us write Φkj = N kj + φ• kj and F -1 k = I + g k . Thus φh kj = g h k • Φ kj • F j + φ h kj • F j + (ϕ kj (I + f h j ) -ϕ kj ), φv kj = g v k • Φ kj • F j + φ v kj • F j + (t kj (h j + f h j ) -t kj (h j )) × (v j + f v j ) + t kj (h j ) × f v j (h j , v j
). According to (5.9), we have

F k (I + g k ) = I on V θ 4 r k . Thus g k = -f k • F -1 k implies that |g k | θ 4 r ≤ |f k | θ 3 r . For (h j , v j ) ∈ V θ 6 r kj , using dist( Ûθ 6 r kj , ∂ Ûθ 5 r kj ) ≥ (1 -θ)θ 5 r/C 0 , we can obtain |t kj (h j + f h j (h j , v j ))-t kj (h j )| ≤ C 3 |f h (h j , v j )| and | ϕ kj (h j +f h j (h j , v j ))-ϕ kj (h j , v j )| ≤ C 3 |f j (h j , v j )|.
Nesting domains and using (5.12), (5.13) and hence (5.7), we obtain by Lemma 5.2 in which r is replaced by θ 5 r :

|{ φ• kj }| θ 6 r ≤ C 4 (|{f j }| θr + |{φ • kj }| r , |{ φ• kj }| θ 6 r ≤ C 4 (1 -θ)r.
Applying Schwarz inequality, we get (5.14)-(5.15).

When we apply the above to iteration, the new Φ kj in the sequence of iteration is defined by

(F (m) k ) -1 (• • • ((F (1) k ) -1 Φ kj F (1) j ) • • • )F (m) j on V r m+1 kj with F (m) j ( V r m+1 kj ) ⊂ V rm kj .
Let us find [f j ] 2m m+1 (h j , v j ), a polynomial of order ≥ m + 1 and of degree ≤ 2m in v j (holomorphic in h j ), such that {F -1 k Φ kj F j } ∈ L 2m holds for some {F j = I + [f j ] 2m m+1 }. Let us consider the neighborhood written in the new coordinates {F j }. We obtain for (h k , v k ) = Φkj (h j , v j ):

h k = Φh kj (h j , v j ) := ϕ kj (h j ) + φh kj (h j , v j ), v k = Φv kj (h j , v j ) := t kj (h j )v j + φv kj (h j , v j ). (5.16)
We assume that φ• kj := ( φh kj , φv kj ) has order ≥ 2m + 1 at v j = 0. Let us write down the horizontal and vertical equations for the linearization problem: F k Φkj = Φ kj F j . We obtain the horizontal equation

ϕ kj (h j ) + φh kj (h j , v j ) + f h k (ϕ kj + φh kj , t kj (h j )v j + φv kj ) = ϕ kj (h j + f h j (h j , v j )) + φ h kj (h j + f h j , v j + f v j ). The vertical equation reads t kj (h j )v j + φv kj (h j , v j ) + f v k (ϕ kj + φh kj , t kj (h j )v j + φv kj ) = t kj (h j + f h j )(v j + f v j ) + φ v kj (h j + f h j , v j + f v j
). We will interpret the above identity as power series in v j with coefficients being holomorphic in ϕ j (U k ∩ U j ). In what follows, degrees or orders of sections are considered w.r.t. v j at v j = 0.

5.2.

A Newton method for the full linearization. For this problem, the two previous equations can be written as (5.17)

L kj (f j ) = 0, Dt kj (h j )f h j v j + F kj (f j ), where L kj (f j ) stands for (L h kj (f h j ), L v kj (f v j )) as defined by (2.16),(2.18): L h kj (f h j ) := f h k (ϕ kj (h j ), t kj (h j )v j ) -s kj (h j )f h j (h j , v j ), (5.18) L v kj (f v j ) := f v k (ϕ kj (h j ), t kj (h j )v j ) -t kj (h j )f v j (h j , v j ). (5.19)
Recall that s kj (h j ) = Dϕ kj (h j ) is the Jacobian matrix of ϕ kj . Furthermore, we have the horizontal error term

F h kj (f j ) := φ h kj (h j + f h j , v j + f v j ) -φh kj (5.20) + f h k (ϕ kj , t kj (h j )v j ) -f h k (ϕ kj + φh kj , t kj (h j )v j + φv kj ) + ϕ kj (h j + f h j (h j , v j )) -ϕ kj (h j ) -Dϕ kj (h j )f h j (h j , v j ),
as well as the vertical error term

F v kj (f j ) := φ v kj (h j + f h j , v j + f v j ) -φv kj + Dt kj (h j )f h j f v j (5.21) + f v k (ϕ kj , t kj (h j )v j ) -f v k (ϕ kj + φh kj , t kj (h j )v j + φv kj ) + t kj (h j + f h j (h j , v j )) -t kj (h j ) -Dt kj (h j )f h j (v j + f v j
). We collect 2m jets from (5.17), (5.20), (5.21). Since f j = O(m + 1) and φ• kj = O(2m + 1), this gives us

[(δ h f h ) kj ] ≤2m = -[φ h kj ] ≤2m , (5.22) [(δ v f v ) kj ] ≤2m = -Dt kj (h j )[f h j ] ≤2m-1 v j -[φ v kj ] ≤2m . (5.23)
Under formal assumptions, according to Lemma 2.16 (c), equations (5.22)

-(5.23) have a solution ([f h j ] 2m m+1 , [f v j ] 2m m+1 ). We first consider the case that H 0 (C, ⊕ 2m k=2 T C ⊗ S k (N * C )) = 0.
Then, for any r * < r ′′ < r ′ < r < r * with r ′′ = θr ′ = θ 2 r, r ′r ′′ < r *r, the solution to (5.22) is unique and by Theorem A.12 that unique solution satisfies the estimate

(5.24) |{[f h k ] }| r ′ ≤ D(T C ⊗ S l (N * C )) (r -r ′ ) τ |{[φ h kj ] l }| r , l = m + 1, . . . , 2m.
In particular, {[f h k ] 2m m+1 } has been determined. The solvability of (5.23) and Theorem A.12 imply that we can find a solution {[f v k ] 2m m+1 } such that for l = m + 1, . . . , 2m,

(5.25) |{[f v k ] l }| r ′′ ≤ D(N C ⊗ S l (N * C )) (r ′ -r ′′ ) τ c D(T C ⊗ S l-1 (N * C )) (r -r ′ ) τ |{[φ h kj ] l-1 }| r + |{[φ v kj ] l }| r .
Here c depends only on the Dt kj over the initial covering. If H 0 (C, ⊕ 2m k=m+1 T C ⊗ S k (N * C )) = 0, we are in the flat case, that is Dt kj = 0. Thus, we can find a solution {[f v k ] 2m m+1 } such that for l = m + 1, . . . , 2m, (5.26)

|{[f v k ] l }| r ′′ ≤ D(N C ⊗ S l (N * C )) (r ′ -r ′′ ) τ |{[φ v kj ] l }| r .
Let us set

D * (2m) := 1 + max 2≤l≤2m (1 + cK(T C ⊗ S l-1 (N * C )))D(N C ⊗ S l (N * C ) . (5.27)
Hence, in any case, estimates (5.24)-(5.26) lead to

|{[f • k ] l }| θ 2 r ≤ C 1 D * (2m) (r -θ 2 r) 2τ |{[φ • kj ] l }| r
for all θ and r satisfying r * ≤ θ 2 r < r < r < r * and all m + 1 ≤ l ≤ 2m. Assume further that θ 6 r > r * and (1θ 7 )r < r *r. We obtain, by Proposition 5.4 with m = 2m + 1

| φ• kj | θ 7 r ≤ C 1 D * (2m)θ 2m+1 (r -θ 2 r) 2τ |φ • kj | r ≤ θ 2m+1 (1 -θ)r/C 0 , provided |{φ • kj }| r ≤ (1 -θ)r/C 0 , (5.28) D * (2m) (r -θ 2 r) 2τ |{φ • kj }| r ≤ (1 -θ)r/C 0 . (5.29)
Note that condition (5.28) follows from (5.29)

as D * (ℓ) ≥ 1. Rename Φ kj , φ • kj , F j , f • j , Φkj , φ• kj respectively as Φ (0) kj , φ (0) kj , F (0)
j , f

(0) j , Φ (1)
kj , φ

kj . Thus Φ

(1)

kj = (F (0) k ) -1 Φ (0) kj F (0) j .
Repeating this formally, we obtain

Φ (ℓ+1) kj = (F (ℓ) k ) -1 Φ (ℓ) kj F (ℓ) j , F (ℓ) j = I + f (ℓ) j , Φ (ℓ+1) kj = N kj + φ (ℓ+1) kj . Set r ℓ+1 = θ 7
ℓ r ℓ and m ℓ = 2 ℓ . We also have

F (ℓ) j ( V r ℓ+1 j ) ⊂ V r ℓ j , (5.30) |φ (ℓ+1) kj | r ℓ+1 ≤ θ 2m ℓ +1 ℓ (1 -θ ℓ )r ℓ /C 0 (5.31) provided r * ≤ θ 7 ℓ r k < 1, 0 < θ k < 1; (5.32) C 1 D * (2m ℓ ) (r ℓ -θ 2 ℓ r ℓ ) 2τ |{φ (ℓ) kj }| r ℓ ≤ (1 -θ ℓ )r ℓ /C 0 . (5.33)
To set parameters, we follow Russmann [START_REF] Rüssmann | Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition[END_REF]; see [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF][START_REF] Stolovitch | Singular complete integrability[END_REF] for different choices of parameters. As in [START_REF] Rüssmann | Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition[END_REF], we now use an addition assumption that

m ℓ = 2 ℓ 0 +ℓ , r ℓ+1 = θ 7 ℓ r ℓ , r 0 = 1, 1 -θ ℓ = δ ℓ , δ ℓ = C * log D * (m ℓ+2 ) m ℓ+2 .
Note that in [START_REF] Rüssmann | Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition[END_REF]Lemma 6.2] and [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF][START_REF] Stolovitch | Singular complete integrability[END_REF], ω(m ℓ+1 ) is used to define δ ℓ . Shifting the index by 1, we use D * (m ℓ+2 ) to simplify the argument. We can find ℓ 0 = ℓ 0 (C * ) such that 0 < θ ℓ < 1 for all ℓ and furthermore

∞ ℓ=0 θ 7 ℓ = ∞ ℓ=0 (1 -δ ℓ ) 7 ≥ exp - ∞ ℓ=0 7C * 2 log D * (m ℓ+2 ) m ℓ+2 . Since 2 -k log D * (2 k
) < ∞, the latter is larger than r * , provided ℓ 0 > ℓ 0 (C * ). Inductively, we want to show that if (5.33) ℓ holds, then (5.33) ℓ+1 also holds. Indeed, with (5.33) ℓ , we can use (5.31) ℓ+1 to obtain

C 1 D * (m ℓ+2 ) (r ℓ+1 -θ 2 ℓ+1 r ℓ+1 ) 2τ |{φ ℓ+1 kj }| r ℓ+1 × C 0 (1 -θ ℓ+1 )r ℓ+1 ≤ D * (m ℓ+2 )θ 2m ℓ -6 ℓ (r ℓ+1 -θ 2 ℓ+1 r ℓ+1 ) 2τ × 1 -θ ℓ 1 -θ ℓ+1 (by (5.31)) ≤ C 2 D * (m ℓ+2 )θ 2m ℓ -6 ℓ (1 -θ ℓ+1 ) 2τ +1 = C 2 D * (m ℓ+2 )(1 -δ ℓ ) 2m ℓ -6 δ 2τ +1 ℓ+1 .
We need to check that the last expression is less than one by using logarithm. Note that log(1δ) < -δ, ∀δ ∈ (0, 1).

Therefore,

log C 2 D * (m ℓ+2 )(1 -δ ℓ ) 2m ℓ -6 δ 2τ +1 ℓ+1 < log C 2 -(2m ℓ -6)δ ℓ + log D * (m ℓ+2 ) -(2τ + 1) log δ ℓ+3 = log C 2 -(2m ℓ -6)C * log D * (m ℓ+2 ) m ℓ+2 + log D * (m ℓ+2 ) -(2τ + 1) log C * log D * (m ℓ+3 ) m ℓ+3 = log C 2 - (2m ℓ -6)C * 3 log D * (m ℓ+2 ) m ℓ+2 + log D * (m ℓ+2 ) - (2m ℓ -6)C * 3 log D * (m ℓ+2 ) m ℓ+2 + - (2m ℓ -6)C * 3 log D * (m ℓ+2 ) m ℓ+2 -(2τ + 1) log C * log D * (m ℓ+3 m ℓ+3 .
When ℓ 0 is sufficiently large, then m ℓ+2 > 24. This implies that if C * > 12, the sum in each of first two braces is negative. Since log increases, we have by (5.34)

-log D * (m ℓ+3 ) ≤ log 1 m ℓ+3 , -log C * log D * (m ℓ+3 ) m ℓ+3 ≤ -log -C * log 1 m ℓ+3 m ℓ+3 .
With m ℓ > 6, the difference in the last brace is bounded above by

(2m ℓ -6)C * 3 log 1 m ℓ+2 m ℓ+2 -(2τ + 1) log C * log m ℓ+3 m ℓ+3 ≤ - 1 12 C * + 2τ + 1 log m ℓ+2 ,
which is negative when C * > 24τ + 12. We have determined C * . This allows us to determine ℓ 0 (C * ) so that 0 < θ ℓ < 1 and ∞ ℓ=0 θ 7 ℓ > r * . Therefore, (5.33) ℓ holds if it holds for initial value ℓ = 0. Using a dilation v j → ǫv j for ǫ > 0, we may replace Φ kj (h j , v j ) by (ϕ kj (h j ) + φ h kj (h j , ǫv j ), t kj (h j )v j + ǫ -1 φ v kj (h j , ǫv j )). This yields (5.33) 0 when ǫ is sufficiently small, as φ • kj (h j , v j ) = O(|v j | 2 ). To finish the proof, we set Ψ (ℓ)

j := F (0) j • • • • • F (ℓ) j . We have Ψ (ℓ) j ( V r ℓ+1 j ) ⊂ V r ℓ j , Ψ (ℓ+1) j (h j , v j ) -Ψ (ℓ) j (h j , v j ) = O(|v j | ℓ ).
Consequently, the sequence Ψ (ℓ) j is bounded in V r∞ j . Fix 0 < θ < 1. By the Schwarz lemma, we get sup

Û r∞ j ×B θr∞ d |Ψ (ℓ+1) j -Ψ (ℓ) j | ≤ Cθ ℓ .
Therefore, of Ψ (ℓ) j converges uniformly on Ûr∞

j × B θr∞ d to a holomorphic mapping Ψ ∞ j . Then F := N -1 j Ψ ∞ j Φ j is well defined. Indeed, N -1 k Ψ ∞ k Φ k = N -1 j Ψ ∞ j Φ j is equivalent to Ψ ∞ k (Φ k Φ -1 j ) = (N k N -1 j )Ψ ∞ j .
Since Ψ ∞ j are tangent to the identity, they are germs of biholomorphisms. Therefore, F linearizes a small neighborhood of C in M.

Therefore, we have proved the following full linearization result.

k≥1 log D * (2 k+1 ) 2 k < +∞,
there is a neighborhood of the compact manifold C in M that is biholomorphic to a neighborhood of the zero section of normal bundle of C in M.

When C is affine and N C is flat, the formal equivalence assumption can be relaxed by assuming that the neighborhoods are equivalent under a formal biholomorphisms fixing C pointwise. This follows from Lemma 2.4 (c).

We now present two examples to illustrate the results in this paper.

5.

3. An example of Arnol'd. This is originally studied by Arnold [START_REF] Arnol'd | Bifurcations of invariant manifolds of differential equations, and normal forms of neighborhoods of elliptic curves[END_REF], [3, §27] for linearization of a neighborhood. See also Ilyashenko-Pyartli [START_REF] Yu | Neighborhoods of zero type imbeddings of complex tori[END_REF] for linearization for flat tori in higher dimensions.

Example 5.6. [3, §27]. Let C be defined by identifying points in C via

h = 0 mod (2π, 2ω), h ∈ C,
where ω = a + ib with b > 0 and a ≥ 0. Consider domains in C defined by parallelograms

U 1 = P (-rπ -rω, (1 + r)π -rω, (1 + r)π + (1 + r)ω, -rπ + (1 + r)ω) U 4 = U 1 + π, U 3 = U 4 + ω, U 2 = U 3 -π.
Suppose that r > 0 is sufficiently small. Then U i ∩ U j has two connected components U ij,0 and U ij,1 with Assume that f has been normalized so that

U 14,1 = U 14,0 -π, U 34
va(h, v) = v n a n (h) + O(n + 1), vb(h, v) = v n b n (h) + O(n + 1), n = 1, 2, . . . .
For the purpose of illustration, we will only restrict to a special unitary line bundle case where |λ| = 1. Then by the non-resonance condition that λ is not a root of unity, we may assume that as in [3, p. 211]

g(h, v) = (h + v n B n (h), v(1 + v n A n (h)) + O(n + 1).
This leads to decoupled equations of the form In Fourier coefficients a n,ℓ and a non-resonant condition, the Fourier coefficients of A n are given by

λ n A n (h + 2ω) -A n (h) = -a n (h), λ n B n (h + 2ω) -B n (h) = -b n (h).
A n,j = a n,j λ n e 2ωj √ -1 -1 .
Assume that a n are holomorphic and 2π periodic in h for S δ : | Im h| < δ. Suppose that

|λ n e 2jω √ -1 -1| ≥ c|λ n -1|. Then |A n,j | ≤ C |λ n -1| |a n | L 2 (S δ ) e -|j|δ , |A n,j e jh | ≤ C |λ n -1| |a n | L 2 (S δ ) e -|j|(δ-δ ′ ) , -δ ′ < Im h < Im ω + δ ′ .
Furthermore, we can verify that

|A n | L 2 (S δ ′ ) ≤ C (δ -δ ′ )|λ n -1| |a n | L 2 (S δ ) .
Note that t kj are locally constant with values 1, λ, λ -1 . Therefore, we have verified

D((T C ⊕ N C ) ⊗ S n N * C ) ≤ C |λ n -1|
.

By Lemma A.2, we get an estimate with equivalent bounds (up to a scalar) but in the original domain, i.e. without shrinking domains. Strictly speaking, the above covering {U r j } has non smooth boundary. The intersection is non-transversal either. However, this covering can be easily modified to get a generic covering defined early, replacing Ûj by smooth strictly convex domains Ûj and then replacing Ûj by Ûj + c j for suitable small constants.

5.4.

Counter-examples. We now show that a certain small-condition is necessary to ensure the vertical and full linearizations. We will achieve this by establishing a connection between the classical linearization problem for germs of one-dimensional holomorphic mappings and the vertical linearization of foliated neighborhood of an elliptic curve.

We keep the notation in subsecton 5.3. Let us start with a power series

(5.41) a(h, v) = n≥2 a n v n := a(v).
Set b(h, v) = 0. Then we have a neighborhood of C associated to

(5.42) f (h, v) = (h + 2ω, λv + a(v)).
Since the vertical part of the transition functions depends only on v, then M is already admits a horizontal foliation with center C being compact.

Proposition 5.7. Let M λ,ω,a be neighborhood of C defined by transition functions Φ kj given by (5.36)- (5.38) where f is given by (5.41)- (5.42). Suppose that λ, ω satisfy the nonresonant condition

(5.43) λ n e 2jω √ -1 -1 = 0, n = 2, 3, . . . , j ∈ Z.
Then M λ,ω,a is vertically (resp. formally) linearizable by a mapping tangent to the identity if and only if the germ of holomorphic mapping ϕ(v) = λv + a(v) is holomorphically (resp. formally) linearizable.

Proof. Suppose that M is vertically linearizable by a holomorphic mapping that is tangent to the identity. By Proposition 2.6, it is vertically linearization by a mapping G j such that

G j (h j , v j ) = (h j , v j + O(|v j | 2 )).
By the non-resonant condition (5.43), we can verify that (5.39) is equivalent to that the g in (5.39) has the form g(h, v) = (h, ψ(v)) and ϕ is linearized by ψ.

The existence of non-holomorphically linearizable ϕ is well-known. By theorems of Bruno [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF] and Yoccoz [START_REF] Yoccoz | Petits diviseurs en dimension 1[END_REF], Proposition 5.7 shows that M λ,ω,â with â(v) = v 2 is vertically linearizable and hence linearizable if and only if λ is a Bruno number, that is

k≥1 log max 2≤j≤2 k |λ j -1| -1 2 k < +∞.
∂B d r |z Q | 2 dσ d = (d -1)!Q! (|Q| + d -1)! r 2d-1+2|Q| , B d r |z Q | 2 dV d = d!Q! (|Q| + d)! r 2|Q|+d .
Therefore, there is a precise asymptotic behavior of Fischer norm and the Bergman norm:

c d g 2 L 2 (B d r ) ≤ |g| 2 f,r ≤ C d g 2 L 2 (B d r ) , 1/4 < r < 4. (5.44)
We also have Bergman's inequality for L 2 holomorphic functions [14, p. 189

]: |f | ∞, V (1-θ)r j ≤ C d (θr) d sup h j |f (h j , •)| L 2 (B d r ) , (5.45) sup h j |f (h j , •)| L 2 (B d r ) ≤ C d |f | ∞,B d r , 1/4 < r < 4. (5.46)
In general, we get

|φ • kj | L ∞ ( Vkj,(1-θ)r ) ≤ C d (θr) d sup h j |φ • kj (h j , •)| L 2 (B d kj,r (h j )) , (5.47) sup h j |φ • kj | L 2 (B d kj,r (h j )) ≤ C d |φ • kj | L ∞ ( V r kj ) , 1/4 < r < 4. (5.48)
Note that when t kj are unitary, the skewed domain V r kj defined in (5.2) are actually product domains V r kj = Ûr kj × B r d . Therefore, the Fischer norm and Bergman norm bound each other with constants depending only on θ and d. We can fix θ too by applying Lemma A.2 as we did in sections 3 and 4. Therefore, any estimate of cohomology equations in Fischer norms has a counter part in super norm on the unit ball in C d and vice versa.

Note that the small divisors condition (5.49) |λ n -1| ≥ Cn -τ , n = 1, 2, . . . for some constants C, τ is equivalent to Ueda's condition in terms of dist(N n C , 1) for the foliation problem when C is an elliptic curve of type zero. In this case the linearized essential use of the uniqueness theorem. This allows us to introduce the small divisors in (A.2) to the cohomology equation (A.1).

A.2. Bounds of solutions of cohomology equations. We now start to introduce nested coverings of C. This will be an essential ingredient of the small divisors for the cohomology equation. We cover C by finitely many open sets U i , i ∈ I such that there are open sets V i in M with V i ∩ C = U i . We also assume that there are biholomorphic mappings Φ i from V i onto the polydisc ∆ r * n+d of radius r * , where n is the dimension of C and n + d is the dimension of M. Assume further that Φ

i (U r * i ) = ∆ r * n × {0} for ϕ i × {0} = Φ i | U i . Set U r = {U r i : i ∈ I} with U r i = ϕ -1 i (∆ r n ).
We assume that r * < 1 and U r * with r * < r * , remains a covering of C. When U r

I := U r i 0 ∩ • • • U r iq is non-empty,
} is nested, if each connected component of U ρ k ∩ U r * j intersects U r * k ∩ U r * j when r * ≤ ρ ≤ r * . In particular, U r * k ∩ U r * j is non-empty if and only if U ρ k ∩ U r * j is non-empty.
Let N(U r * i ) be the union of all U r * k that intersect U r * i ; as in [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF] we will call the union the star of U r * i . Refining U r * if necessary, we may assume that there is a biholomorphism ϕ i from a neighborhood of the star onto an open set in C n . If E ′ , E ′′ are holomorphic vector bundles over C, we will fix a trivialization of E ′ over U i by fixing a holomorphic basis e

′ k = {e ′ k,1 , . . . , e ′ k,m } in U r * k . We also fix a holomorphic base e ′′ j = {e ′′ j,1 , . . . , e ′′ j,d } of E ′′ in U r * j . On U r * I = U r * i 0 ∩ • • • ∩ U r * iq ,
f 2 D := f 2 L 2 (D) := f 1 2 D + • • • + f d 2 D , |f | 2 D := sup z∈D |f 1 (z)| 2 + • • • + |f d (z)| 2 , |f | ∞,D := sup z∈D max{|f 1 (z)|, . . . , |f d (z)|}.
For a d ×d matrix t of functions on D, denote by |t| D , t D , |t| ∞,D respectively the operator norms defined by

|t| D = sup |f | D =1 |tf | D , t D = sup f D =1 tf D , |t| ∞,D = sup |f | ∞,D =1 |tf | ∞,D . Therefore, t D ≤ |t| D as tf D ≤ (sup z∈D |t(z)|) f D = |t| D f D . Then we define the L 2 norm for f ∈ C q (U r , O(E ′ ⊗ E ′′ ))
||f || ≤ C|f |, sup |f | ≤ f ≤ C rank(E ′ ⊗ E ′′ ) sup |f |, |t| ∞ ≤ |t| ≤ C rank(E ′ ⊗ E ′′ )|t| ∞ ,
where C does not depend on E ′ , E ′′ . The first result of this appendix is to find a way to obtain solutions with bounds to (A.1) on the original covering, if a solution with a bound exists on a shrinking covering. This relies on the nested coverings defined above. We first study the L 2 norms case.

Lemma A.2. Let U r = {U r i : i ∈ I} with r * ≤ r ≤ r * be a family of nested finite coverings of C. Suppose that f ∈ C 1 (U r * , E ′ ⊗ E ′′ ) and f = 0 in H 1 (U r * , E ′ ⊗ E ′′ ). Assume that there is a solution v ∈ C 0 (U r * ) such that (A.4) δv = f, v U r * ≤ K f U r * .
Then there exists a solution u ∈ C 0 (U r * ) such that δu = f on U r * and

(A.5) u U r * ≤ C(|{t ′ kj }| U r * + K|{t ′ kj }| U r * |{t ′′ kj }| U r * ) f U r * ,
f jk = (δv) jk , U r * j ∩ U r * k , (A.6) v U r * ≤ K f U r * . (A.7) Take any v * ∈ C 0 (U r * , E ′ ⊗ E ′′ ) such that δv * = f . Then (δv * -δv) jk = 0 in U r * j ∩ U r * k , because (δv * ) jk = f jk on the larger set U r * j ∩ U r * k . Since {U r * j } is a covering of C then w := v j -v * j is a global section of E ′ ⊗ E ′′ .
This shows that v j , via v * j , extends to a holomorphic section in U r * j . In fact, v j is the restriction of u j = v * j + w defined on U r * j . We now derive the bound for u j . Suppose that U r * j ∩ U r * k is non-empty. By the assumptions, each component of U r

* j ∩ U r * k intersects U r * j ∩ U r * k . We have u j = u k + f jk on U r * j ∩ U r * k
and hence the uniqueness theorem implies that it holds on U r * j ∩U r * k too. And on U r * j ∩U r * k , we have u k = v k and u j = v kf kj . We express the identity in coordinates u j = ũj e j , v k = ṽk e k = vkj e j , f kj = fkj e kj = fkj e jj .

Let t ′ kj , t ′′ kj respectively be the transition matrices of e ′ j , e ′′ j for E ′ , E ′′ . Then tkj = t ′ kj ⊗ t ′′ kj are the transition matrices of e kj for E ′ ⊗ E ′′ . Then we have

vkj = t ′ jk ⊗ t ′′ jk ṽk , fkj = t ′ jk ⊗ I d fkj . Thus, ũj = vkj -fkj = t ′ jk ⊗ t ′′ jk ṽk -t ′ jk ⊗ I d fkj . We have ũj L 2 (U r * j ∩U r * k ) = ũj • ϕ -1 j L 2 (ϕ j (U r * j ∩U r * k )) ≤ (t ′ jk ⊗ t ′′ jk ṽk ) • ϕ -1 j L 2 (ϕ j (U r * j ∩U r * k )) + (t ′ jk ⊗ I d fkj ) • ϕ -1 j L 2 (ϕ j (U r * j ∩U r * k )) .
Here

t jk • ϕ -1 j = t jk • ϕ -1 k • ϕ kj . By the properties of operator norm and t ′ kj ⊗ t ′′ kj D ≤ |t ′ kj ⊗ t ′′ kj | D ≤ |t ′ kj | D |t ′′ kj | D for D = ϕ j (U r * j ∩ U r * k ), we have (t ′ jk ⊗ t ′′ jk ṽk ) • ϕ -1 j 2 D ≤ C * |t ′ jk | 2 D × |t ′′ jk | 2 D × ṽk 2 ϕ k (U r * j ∩U r * k )
, where the constant C * comes from the Jacobian of z k = ϕ kj (z j ). By (A.7), we have ṽk

• ϕ -1 k 2 L 2 ≤ K 2 f 2 L 2 . We also have (t ′ jk ⊗ I d fkj ) • ϕ -1 j ϕ j (U r * j ∩U r * k ) ≤ |t ′ jk • ϕ -1 j | ϕ j (U r * j ∩U r * k ) × f ϕ j (U r * j ∩U r * k ) . Since U r * j is covered by {U r * j ∩ U r * k }, we get the desired bound from ũj L 2 (U r * j ) ≤ k ũj L 2 (U r * j ∩U r * k ) .
The argument for the norm | • | is verbatim and we can take the above constant C * to be one.

Corollary A.3. With notations and assumptions in Lemma A.2, the solution u also satisfies

|u| ∞,U r * ≤ C(|{t ′ kj }| U r * + K|{t ′ kj }| U r * |{t ′′ kj }| U r * ) rank(E ′ ⊗ E ′′ )|f | ∞,U r * , where C does not depend on E ′ , E ′′ .
The above lemma leads us to the following proposition and definition.

Proposition A.4. Let U r = {U r i : i ∈ I} with r * ≤ r ≤ r * be a family of nested coverings of a compact complex manifold C. Let E ′ (resp. E ′′ ) be a holomorphic vector bundle over C with bases {e ′ j } (resp. {e ′′ j }) and transition matrices t ′ kj (resp. {t ′′ kj }). Suppose that there is a finite number K such that for any

f ∈ C 1 (U r * , E ′ ⊗ E ′′ ) with f = 0 in H 1 (U r * , E ′ ⊗ E ′′ ), there is a solution v ∈ C 0 (U r * , E ′ ⊗ E ′′ ) satisfying (A.4). Then there is a possible different solution v ∈ C 0 (U r * , E ′ ⊗ E ′′ ) satisfying (A.4) in which K is replaced by K * (E ′ ⊗ E ′′ ) = sup u 1 inf u 0 u 0 U r * : δu 0 = δu 1 on U r * , (A.8) δu 1 U r * = 1, u i ∈ C 0 (U r i , E ′ ⊗ E ′′ ) . Proof. By the assumption, K * = K * (E ′ ⊗ E ′′ ) is well-defined and K * ≤ K. Fix u 1 ∈ C 0 (U r i , E ′ ⊗ E ′′ ). Suppose that δu 1 = f and f U r * = 1.
By the definition (A.8), there exists u j 0 such that δu m 0 = f on U r * and u m 0 U r * ≤ K * + 1/m. By the Cauchy formula on polydiscs, (u m 0 ) j • ϕ -1 j is locally bounded in ϕ j (U j ) in sup-norm. We may assume that as m → ∞, (u m 0 ) j converges uniformly to u ∞ 0 on each compact subset of U j for all j. This shows that (u ∞ 0 ) j • ϕ -1 j L 2 (E) ≤ K * for any compact subset E of ϕ j (U j ). Since E is arbitrary, we obtain u ∞ 0 U r * ≤ K * . By the uniform convergence, we also have δu ∞ 0 = f on U r * . support such that φr;r * k = 1 in ϕ k (U r k ), and (A. [START_REF] Hirschowitz | On the convergence of formal equivalence between embeddings[END_REF] φr;r

* k (x) < 3/4, if dist(ϕ k (x), ϕ k (U r k )) > min i c i,k /C * .
Note that the latter can be achieved with

|∇ φr;r * k | < C1 /min i c i,k ≤ C 2 κ -1 /(r ′ -r ′′ ).
In U r * k , define a non-negative smooth function

φ r ′′ ;r ′ i;k = φr ′′ ;r ′ i;k 1 -φr;r * k + φr ′′ ;r ′ k • ϕ k ,
where the smoothness follows from the denominator being bigger than 1/4 by (A.17 Recall that φ r ′′ ;r ′ ℓ;k has compact support in U r ′ ℓ ∩U r * k . Thus it is smooth on ω := U r ′ j ∩U r * k ∩U r ′ ℓ and vanishes on an open set D containing U r ′ j ∩ U r * k \ ω. On the other hand, f r ′ ;r * ℓj;k is holomorphic in ω. Hence the product φ r ′′ ;r ′ ℓ;k f r ′ ;r * ℓj;k is smooth in U r ′ j ∩ U r * k . Then v j;k = ∂w j;k is a smooth (0, 1) form in U r ′ j ∩ U r * k . Let A denote the sheaf of smooth functions on C. We now pull back the forms from the polydisc ∆ n via ϕ k . For each fixed k, we have {w j;k } j ∈ C 0 (U r ′ ;r * k , A m ). Let us denote t ′ kj ⊗ I by t ′ kj . By f ij = f ikf jk and (A.14), we have

t ′ ki ⊗ t ′′ kj f r ′ ij = t ′ ki f r ′ ik -t ′ kj f r ′ jk .
Since i φ r ′′ ;r ′

ℓ;k (t ′ kℓ ⊗ t ′′ ki f r ′ ℓi -t ′ kℓ ⊗ t ′′ kj f r ′ ℓj ) = ℓ φ r ′′ ;r ′ ℓ;k (t ′ kj f r ′ jk -t ′ ki f r ′ ik ) = t ′ kj f r ′ jk -t ′ ki f r ′ ik .
The latter is holomorphic. Thus (δv) ij;k = ∂(δw ṽℓ k e kk,ℓ , we have a bounded linear operator S : v k → u k such that ∂((ϕ -1 k ) * u k ) = (ϕ -1 k ) * v k . Returning to the complex manifold via ϕ k , we have

) ij;k = 0 on U r ′ i ∩ U r * k ∩ U
u k U r * k = u k • ϕ -1 k L 2 (∆ n r * ) ≤ C v k • ϕ -1 k L 2 (∆ n r * ) ≤ Cκ -1 |{t ′ kj }||{t ′′ kj }| r ′ -r ′′ f L 2 (U r * ) .
Here we have used (A.20), estimate (A. [START_REF] Hörmander | L 2 estimates and existence theorems for the ∂ operator[END_REF]) and the definition of norm (A.4). Note that the C is independent of the rank since we applied the L 2 componentwise. Set ĝr ′ ;r j;k = w j;ku k on U r ′ j ∩ U r k . We obtain ĝr ′ ,r i;kĝr ′ ,r j;k = f r ′ ;r * ij;k , U r ′ i ∩ U r k ∩ U r ′ j , (A.21) max j ĝr ′ ;r

j;k U r ′ j ∩U r k ≤ Cκ -1 |{t ′ kj }||{t ′′ kj }| r ′ -r ′′ f U r ′ . (A.22)
We have obtained (A.13).

To verify (A.12), we will use the same base e k and take the product of (A.21) with e k in order to obtain on U r ′′ i ∩ U r ′′ j ∩ U r k ∩ U r ℓ g r ′ ;r i;kg r ′ ;r j;k = f r ′ ;r * ij;k e k = f r ′ ij = f r ′ ;r * ij;ℓ e ℓ = g r ′ ;r i;ℓg r ′ ;r j;ℓ and thus (A.23) g r ′ ;r j;ℓg r ′ ;r j;k = g r ′ ;r i;ℓg r ′ ;r i;k , on U r ′′ i ∩ U r ′′ j ∩ U r k ∩ U r ℓ . Then we have a (well-defined) holomorphic section g r kℓ := g r ′ ;r i;ℓg r ′ ;r i;k , U r k ∩ U r ℓ .

We verify that {g r kℓ } ∈ Z 1 (U r , O m ). Set u r ′′ i := g r ′′ ;r i;i . Since r ′ ≤ r we actually have {u r ′′ i } ∈ C 0 (U r ′ , E ′ ⊗ E ′′ ). However, only on U r ′′ i ∩ U r ′′ j , we can verify via (A.23) that g r ijf r ′ ij = (g r ′′ ;r i;jg r ′′ ;r j;j ) -(g r ′′ ;r i;jg r ′′ ;r i;i ) = u r ′′ iu r ′′ j .

The above result is a type of Grauert's smoothing decomposition, which can also be obtained by open mapping theorem. See for instance [14, p. 200]. However, this yields an unknown bound in the estimates.

A.4. Finiteness theorem with bounds. The above smoothing decomposition does not provide a solution to the cohomology equations, i.e. if f = 0 in H 1 (U r ′ , O(E ′ ⊗ E ′′ )), then there exists u ∈ C 0 (U r ′′ , O(E ′ ⊗ E ′′ )) such that δu = f on U r ′′ , for some r ′′ ≤ r ′ . We will follow [START_REF]Theory of Stein spaces[END_REF] to derive the finiteness theorem with explicit bounds. In particular, this provides solutions of first cohomology equations with bounds on shrinking domains.

We first recall the resolution atlases from [14, p. 194], specializing them for the vector bundles. Assume that we have coordinate charts

ϕ k : U r * k → P k := ϕ k (U r * k ) = ∆ r * n .
Define U r * I = U r * i 0 ∩ • • • ∩ U r * iq for I ∈ I q+1 . Then ϕ I = (ϕ i 0 , . . . , ϕ iq ) is defined on U r * I with range Ûr * I . Unless otherwise stated, we omit the superscript r * in U r * I . We can define a proper embedding ϕ I : U I → ÛI ֒→ P I := ∆ r * nq , n q = n(q + 1). Then the push-forward of the vector bundle E ′ ⊗ E ′′ | U I defines a coherent analytic sheaf (ϕ I ) * (E ′ ⊗ E ′′ ) over P I by trivial zero extension; see [14, p. 5, p. 195] and [13, p. 239]. A section f ∈ Γ(U I , E ′ ⊗ E ′′ ) yields a section fI of (ϕ I ) * (E ′ ⊗ E ′′ ) over P I by fI • ϕ I (x) = (f I (x), . . . , f I (x)), fI | P I \ ÛI = 0.

Note that U r * has a Stein neighborhood. Then following notation in [14, p. 196] we have an epimorphism by Cartan's Theorem A:

ǫ I : O ℓ | ∆ r * nq → (ϕ I ) * (E ′ ⊗ E ′′ )| U I , ℓ ≥ rank(E ′ ⊗ E ′′ )
, where ǫ I is defined by finitely many global sections defined in a neighborhood of P I . When E ′ ⊗ E ′′ is a vector bundle, we take ℓ to be the minimal value, the rank of E ′ ⊗ E ′′ , and specify the above ǫ I by taking Here we want to obtain a more general description without restricting to a vector bundle. Define C q (U) :=

I∈I q+1
O ℓ (P I ).

(Set O ℓ (P I ) = 0 when U r * I is empty.) We recall that P I = ∆ r * nq is independent of the order of multi-indices. Thus C q (U) ∼ = (O(∆ r * nq )) L := O L (∆ r * nq ). Here L ≤ |I q+1 |ℓ. Let O h (∆ r nq ) be the space of holomorphic functions on ∆ r nq with finite L 2 norm on ∆ r nq . Set P r I = ∆ r nq for I ∈ I q+1 . We define a Hilbert space

C q h (U r ) := I∈I q+1 O ℓ h (P r I ) := O L h (∆ r nq ),
which is a subspace of C q (U r ).

Using the collection ǫ = {ǫ I : I ∈ I q+1 }, we define C q h (U r , E ′ ⊗ E ′′ ) := ǫ(C q h (U r )) ∼ = C q h (U r )/(ker ǫ ∩ C q h (U r )), which is the vector space of q-cochains, equipped with the standard coboundary operator δ.

Remark A.8. Our cochains are not necessary alternating. As in [14, p. 35], we let C q a (U, E ′ ⊗ E ′′ ) denotes alternating cochains. For the isomorphism of the two kinds of Cečh cohomology groups; see [14, p. 35] and Serre [START_REF] Serre | Prolongement de faisceaux analytiques cohérents[END_REF]. Since we are interested in the cohomological solutions with bounds, we fix our nation without requiring that the cochains be alternating.

Let • ∆ r nq be the Hilbert space norm on C q h (U r ) and set ζ • U r = inf{ v ∆ r nq : v ∈ C q h (U r ), ǫ(v) = ζ}, ζ ∈ C q h (U r , E ′ ⊗ E ′′ ). The inclusion C q h (U r , E ′ ⊗ E ′′ ) ֒→ C q (U r , E ′ ⊗ E ′′ ) is continuous and compact [START_REF]Theory of Stein spaces[END_REF]Thm. 3,p. 197]). We also define Z q h (U r ) := ǫ -1 (Z q h (U r , E ′ ⊗ E ′′ )), ζ U r := inf{ v ∆ nq r : v ∈ Z q h (U r ), ǫ(v) = ζ}, ∀ζ ∈ Z q h (U r , E ′ ⊗ E ′′ ), v := ǫ(v).

Then Z q h (U r , E ′ ⊗ E ′′ ) is an isometric subspace of C q h (U r , E ′ ⊗ E ′′ ) via inclusion. Let {g 0 , g 1 , . . . } be a monotone orthogonal base of Z 1 h (U r ) ( [14, p. 141, p. 201]). An important feature of the monotone base is that the vanishing orders of g j at the origin satisfy ord 0 g 0 ≤ ord 0 g 1 ≤ • • • , lim i→∞ ord 0 g i = ∞.

By [START_REF]Theory of Stein spaces[END_REF]Thm. 1,p. 192 and p. 201], for a given ν there is an µ such that g i (Z) = O(|Z| ν ), i > µ, Z ∈ ∆ nq r . (A. [START_REF] Kodaira | A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds[END_REF] In fact, let the index set be I = {1, . . . , L}. Set ω((f 1 , . . . , f L )) = min{(α, Q) : f α,Q = 0} by using order < on I × N m defined by (α, P ) < (β, Q) if |P | < |Q|, or if |P | = |Q| and there is an ℓ such that p ℓ < q ℓ and p ℓ ′ = q ℓ ′ for all ℓ ′ > ℓ, or if P = Q and α < β. Then the basis {g j } satisfies ω(g j ) < ω(g j+1 ).

We now return to the case q = 1 with n q = 2n. In the sequel, {|t ′ kj |} = {|t ′ kj |} U r * and {|t ′′ kj }| = {|t ′′ kj |} U r * . Theorem A.9 (Donin-Grauert-Remmert). Let C be a compact complex manifold and let U r (r * < r < r * < 1) be a family of open coverings of C as in Lemma A.6 such that (A.11) holds for all k, j. Let E = E ′ ⊗ E ′′ be a holomorphic vector bundle of positive rank m over C and fix a holomorphic base e ′ j (resp. e ′′ j ) for E ′ (resp. E ′′ ) over U r * j . Suppose that r * < r ′′ < r ′ < r < r * and r ′r ′′ ≤ r *r. Let θ = r ′ /r. Let {g 0 , g 1 , . . . } be a monotone orthogonal base of Z 1 h (U r ) as above. Assume that µ, ν satisfy (A.24) and

t := C n κ -1 (r ′ -r ′′ )(r -r ′ ) 2n θ ν < 1/2.

(A.25)

There exist g m 0 , . . . , g m µ * such that their equivalence classes in H 1 (U r , E) form a C-linear basis of subspace spanned by g 0 , • • • , g µ in H 1 (U r , E). For any f ∈ Z 1 h (U r ′ , E) there exists v ∈ C 0 h (U r ′′ , E) satisfying f = δv + µ * 0 c i g m i with Furthermore, all c j = 0 when f = 0 in H 1 (C, E).

|c i | ≤ C n κ -1 A r (E) r -r ′ f U r ′ , (A.26) v U r ′′ ≤ C n κ -1 B r -(E) r -r ′ f U r ′ , ∀r -∈ [r ′ ,
Remark A.10. The solution operator f → v may not be linear. See a proof by Donin [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF] to get a linear solution operator for which the constant C * results from a lemma of Schwartz.

Remark A.11. The previous theorem gives a solution v, defined on a smaller domain, to the equation f = δv (i.e cohomological equations) whenever f is 0 in the first cohomology group. It also provides a bound of the solution in terms of the data. We emphasize that this bound depends on the bundle E ′ ⊗ E ′′ . In the applications we have in mind, we will have to consider a sequence of bundles {S m E ′′ } m , and we will need to control the growth of these bounds as m goes to infinite, similarly to the small divisors appearing in local dynamical systems.

Proof. Recall that q = 1 and n 1 = 2n. We may assume that g j ∆ r 2n = 1. By the definition of µ, ν and the monotone basis, we have for any v ∈ Z 1 h (U r ), where C n (rr ′ ) -2n is the constant M in [START_REF]Theory of Stein spaces[END_REF]Thm. 6,p. 191].

Replacing the smoothing lemma in [14, p. 200] by Theorem A.7, we derive some estimates following the proof of the finiteness lemma in [14, p. 201]. By assumption, we have

t = C n κ -1 (r ′ -r ′′ )(r -r ′ ) 2n θ ν < 1/2, θ = r ′ r < 1.
Let ζ 0 := f ∈ Z 1 h (U r ′ , E ′ ⊗ E ′′ ). By Theorem A.7, we have for some

ξ 0 ∈ Z 1 h (U r , E ′ ⊗ E ′′ ) ζ 0 = ξ 0 + δη 0 , ξ 0 U r ≤ t ′ ζ 0 U r ′ , η 0 U r ′′ ≤ t ′ ζ 0 U r ′ , with t ′ := Cn|{t ′ kj }||{t ′′ kj }| κ(r ′ -r ′′ )
. Let v denote ǫ(v). Then ξ 0 = v 0 for some v 0 satisfying v 0 ∆ r 2n = ξ 0 U r ; see [14, p. 198]. Consider

w 1 = v 0 - µ j=0 (v 0 , g j ) ∆ r 2n g j , ζ 1 = w 1 .
According to (A.30), we have

ζ 1 U r ′ ≤ w 1 U r ′ ≤ C n (r -r ′ ) 2n (r ′ /r) ν v 0 ∆ r 2n ≤ t ζ 0 U r ′ .

Corollary 3 . 3 .

 33 a possibly non-unique Donin (minimizing) solution of a suitable cohomology equation. Under assumptions of Theorem 3.1, there exists, in a neighborhood of C in M, a smooth holomorphic d-dimensional foliation having C as a leaf.

3. 1 .

 1 Conjugacy equations and cohomological equations. Let us first recall (2.21) and (2.20) :

Corollary 4 . 2 .

 42 Keep the assumptions in Theorem 4.1. Assume further that C is simply connected. Then a neighborhood of C in M is biholomorphic to C × B d where B d is the unit ball in C d .

5. 1 .

 1 Domains for iteration and the Donin condition. Following Lemma A.6 and Proposition A.19, we shall consider a family of nested coverings U r = {U r i } i∈I of C with r * ≤ r ≤ r * . Let us fix a trivialization of N * C (resp. T C) over U r * i by fixing a holomorphic basis e i = (e i,1 , . . . , e i,n+d ) of T C M on U r * i . We first define various domains. Let Ûr j := ϕ j (U r j ) = ∆ r n and U r kj := U r k ∩ U r j . We have U r kj = U r jk . Define Ûr kj = ϕ j (U r kj ). Then ϕ kj (U r kj ) = Ûr jk . Donin Condition. Let U r be a family of nested covering of C for r * < r < r * . Let E ′ = T C or N C . Suppose that there are constants D(E ′ ⊗S m N * C

( 5

 5 .34) D * ℓ ≥ ℓ, ℓ ≥ 1. Indeed, when D * (k) = max(D * (k), k) replaces with D * (k), the sequence D * (k) still increases and 2 -k log D * (2 k ) converges. For a constant C * ≥ 1 to be determined later, define

Theorem 5 . 5 .

 55 Let a neighborhood of the compact manifold C in M be equivalent to a neighborhood of the zero section of normal bundle N C of C in M by a formal holomorphic mapping which is tangent to the identity and preserves the splitting of T C M. Assume that 0 (C, T C ⊗ S ℓ (N * C )) = 0 for all ℓ > 1 or that the normal bundle N C is flat. If {D * (2 k )} defined by (5.1) and (5.27) satisfies(5.35)

, 1 =

 1 U 34,0ω, U 23,1 = U 23,0π, U 12,1 = U 12,0ω.Let Ûj = U j and Vj = Ûj × ∆ δ . Define M = ∪ Vj / ∼, V j = {[x] : x ∈ Vj }, Φ j : V j → Vj and the transition functions Φ kj on V kj = V k ∩ V j of M as follows. Let f (h, v) = (h + 2ω + vb(h, v), λv(1 + va(h, v))), | Im h| < δwhere a, b are 2π periodic holomorphic function in h. Define Φ 12,0 = I, Φ 43,0 = I, Φ 12,1 = f | V12 , Φ 43,1 = f | V43 , (5.36) Φ 14 = I, Φ 23 = I, (5.37)Φ 13,0 = I, Φ 13,1 = f | V13,1 , Φ 42,0 = I, Φ 42,1 = f | V42,1 . (5.38)The linearization of a neighborhood of C in M is equivalent to G -1 k Φ kj G j = Φkj where Φkj are constructed as above by replacing f with f defined by f (h, v) = (h + 2ω, λv).Thus T M has transition functions:Φ14 = I, Φ23 = I, Φ12,0 = I, Φ43,0 = I, Φ12,1 = f | V12 , Φ43,1 = f | V43 . Then we have g := G 1 = G 4 on V1 ∩ V4 , g := G 2 = G 3 on V2 ∩ V3 , g := G 1 = G 2on V12,0 and g := G 3 = G 4 on V34,0 . In other words, g is 2π periodic and defined on-δ Im ω < Im h < 2(1 + δ) Im ω. The cohomology equation is reduced to G -1 1 Φ 12 G 2 = Φ12 and G -14 Φ 43 G 3 = Φ43 . Equivalently, we need to solve (5.39) g -1 f g = f .

(5. 40 )

 40 Note that a n , b n are holomorphic in | Im h| < δ and we are seeking a solution on a large strip -δ ′ < Im h < Im ω + δ ′ .

  it will be convenient to use the base e i 0 ...iq := e ′ i 0 ⊗ e ′′ iq := {e ′ i 0 ,k ⊗ e ′′ iq,j : 1 ≤ k ≤ m, 1 ≤ j ≤ d}. Throughout the paper • D and | • | D denote respectively the L 2 and sup norms of a function in D, when D is a domain in C n . If f = (f 1 , . . . , f d ) is a vector of functions, we define the L 2 norm, metric, and sup norms as follows:

  by a I e I := md µ=1 a µ I e I,µ , f U r := max I=(i 0 ,...,iq)∈I q+1 a I • ϕ -1 iq ϕ iq (U I ) : f i = a I e I in U I . Sometimes we denote f U r * by f for abbreviation. We define similarly the metric norm |f | U r * , or |f |, and the sup-norm |f | ∞,U r * or sup |f |. It is obvious that

  ) and (A.18). Thus, φ r ′′ ;r′ i;k has compact support in U r ′ i ∩ U r * k and i φ r ′′ ;r ′ i;k = 1 in U r k = i (U r ′′ i ∩ U r k ), as φr;r * k = 1 on U r k . We can verify that (A.19) |∇(φ r ′′ ;r ′ i;k • ϕ -1 k )| < C ′ κ -1 /(r ′r ′′ ). Consider the expression (A.20) w j;k = ℓ φ r ′′ ;r ′ ℓ;k f r ′ ;r * ℓj;k .

= 1 =

 1 φr;r * k • ϕ k on U r k , then by δf = 0 and (A.14), we get on U r ′ i ∩ U r k ∩ U r ′ j w i;kw j;k = ℓ φ r ′′ ;r ′ ℓ;k ( f r ′ ;r * ℓi;k -f r ′ ;r * ℓj;k ) = ℓ φ r ′′ ;r ′

  r ′ j . This shows thatv k := v j;k is actually a ∂-closed (0, 1) form in U r * k . Thus (ϕ -1 k ) * v k is a ∂-closed (0,1)-form on the polydisk ∆ n r * . By the L 2 theory [20, Thm. 4.4.3] applied to each component of v k = m ℓ=1

ǫ

  I : g I → gI := (ϕ I ) * {g I • ϕ I e I }.

  r), (A.27)g j = µ * i=0 c ji g m i + δη * j , η * j ∈ C 0 (U r , E), (A.28) A r (E) = |{t ′ kj }||{t ′′ kj }| max 0≤i≤µ * µ j=0 |c ji |, B r -(E) = |{t ′ kj }||{t ′′ kj }| µ j=0 {η * j } U r -. (A.29)

  g j )g j ∆ r ′ 2n ≤ C n (rr ′ ) 2n (r ′ /r) ν v ∆ r 2n (A.[START_REF] Morrow | Submanifolds of P n with splitting normal bundle sequence are linear[END_REF]

  This can be computed from the transition matrices of N * C by expressing the basis w * k,1 , . . . , w *

	k,d
	in terms of w * j,1 , . . . , w * j,d . Since t L kj maps orthonormal basis into orthonormal basis, by
	Proposition 3.4 we know that t L kj are unitary matrices, i.e. in operator norm defined in
	(A.4),
	(3.8)

k,P . We have (F k,P ) |P |=L = t L kj (F j,P ) |P |=L .

  be the minimizer solution of this equation obtained by Lemma A.2 and Proposition A.4 and let [k j

  5.5. A foliation example.Here we specialize Ueda's theory for elliptic curves. Let us first discuss the Fischer norms and Bergman norm when the N C is unitary. Let us recall two formulae from Zhu[42, p. 22]:

  it is still Stein [14, p. 127]. Definition A.1. Let {U r j } be an open covering of C for each r ∈ [r * , r * ]. We say that the family of coverings {U r j

  where t ′ kj , t ′′ kj are the transition matrices of E ′ , E ′′ , respectively, and C depends only on the number |I| of open sets in U r * and transition functions of C. In particular, C does not depend on E ′ , E ′′ .

	Proof. By assumptions, we have

Recently, Jun-Muk Hwang proved instances of Hirschowitz's conjecture on the Formal Principle[START_REF] Hwang | An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principle[END_REF]. The authors thank Takeo Ohsawa for acknowledging this work.

Partially supported by a grant from the Simons Foundation (award number: 505027). † † Research of L. Stolovitch was supported by ANR grant "ANR-14-CE34-0002-01" for the project "Dynamics and CR geometry".

1

equation corresponds to equation (5.40) we can take the small divisor 1/K * (N C ⊗ S n N * C ) to be |λ n -1|.

Finally, we should mention that the assumption η m ≤ L 0 L m is satisfied under Siegel's small divisor condition |λ n -1| ≥ Cn -τ by a method of Siegel; see Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF] for the vertical linearization problem. It is also satisfied under the Bruno condition [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF] which is a condition weaker than (5.49). For the details, we refer to [START_REF] Brjuno | Analytic form of differential equations. I, II[END_REF][START_REF] Pöschel | On invariant manifolds of complex analytic mappings near fixed points[END_REF].

Appendix A. L 2 bounds of cohomology solutions and small divisors A.1. A question of Donin. Let E be a holomorphic vector bundle on a compact complex manifold C. The main purpose of this section is to obtain L 2 and sup-norm bounds for the cohomology equation

where f ∈ Z 1 (U, O(E)) and U is a suitable covering of C. Our goal is to show that if f = 0 in H 1 (C, O(E)), then there is a solution u such that

Here • U is the L 2 -norm for cochains of the covering U. The main assertion is that the solution u admits estimate on the original covering U without any refinement, which is important to the application in this paper. For this purpose, we will choose the covering U which consists of biholomorphic images of the unit polydisc, which are in the general position. The question on the existence of such an estimate and solutions was raised by Donin who asked the general question if O(E) is replaced by a coherent analytic sheaf F on C and f is any p-cocycle, with p > 0, of a covering U [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF]. The result in this appendix provides an affirmative answer to Donin's question for p = 1 and the sheaf of holomorphic sections of a holomorphic vector bundle. Furthermore, we will introduce the small divisor for (A.1) in (A.2). Although some of results in this appendix can be further developed for a general setting, we limit to the case of H 1 (C, O(E ′ ⊗ E ′′ )); this suffices applications in this paper. One may take E ′′ to be the trivial bundle to deal with a general vector bundle E. In the applications we have in mind, C is embedded into a complex manifold M and we will take E ′′ to be symmetric powers Sym ℓ N * C of N * C , the dual of the normal bundle of C in M. In this paper, S ℓ E denotes the symmetric power Sym ℓ E of a vector bundle E over C. We are mainly concerned with how various bounds depend on ℓ as ℓ → ∞ when we employ the important Fisher metric on S ℓ N * C for unitary the normal bundle N C . This will be crucial in our applications.

To prove (A.2), we will first use the original estimate of Donin [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF], without solving the cohomology equation. This serves as a smoothing decomposition in the sense of Grauert [START_REF]Theory of Stein spaces[END_REF] by expressing

where g is defined on a larger covering while u is defined on a shrinking covering. We will then combine with the proof of finiteness theorem of cohomology groups from Grauert-Remmert [START_REF]Theory of Stein spaces[END_REF] to refine the decomposition (A.3) by expressing g in a base of cocycles. Finally, we will obtain (A.2) by avoiding shrinking of covering. This last step is motivated by a method of Kodaira-Spencer and Ueda [START_REF] Ueda | On the neighborhood of a compact complex curve with topologically trivial normal bundle[END_REF]. We take a different approach by an Definition A. [START_REF] Camacho | Fibered neighborhoods of curves in surfaces[END_REF]. Let E ′ , E ′′ , e ′ j , e ′′ j , t ′ kj , t ′′ kj be as in Proposition A.4. Let t ′′ kj (S m E ′′ ) be the transition matrices of the symmetric power S m E ′′ induced by t ′′ kj . For m = 2, 3, . . . , we shall call

the generalized small divisors of E ′ ⊗ E ′′ with respect to e ′′ j , t ′′ kj . A.3. Donin's smoothing decomposition. Grauert's smoothing decomposition for cochains of analytic sheaves is an important tool. Here we will follow an approach of Donin [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF], by specializing for vector bundles.

We first need to introduce coverings by analytic polydiscs.

Lemma A.6. Let C be a compact complex manifold. Let {U r * i : i ∈ I} be a finite open covering of C, and let ϕ j map U r j biholomorphically onto ∆ n r for r * < r < r * < 1. Assume further that ϕ i is a biholomorphism defined in a neighborhood of the star N(U r * i ) onto a domain in C n . Suppose that r * < r ′ i < r i < r * , and

Then the distance is attained for some z ′ ∈ ∂∆ n r ′ and z ∈ ∂∆ n r . By the definition of κ, we get the desired estimate.

We will recall the following smoothing decomposition of Donin [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF]. Here we restrict to the case of H 1 and the holomorphic vector bundle to indicate the specific bounds in the estimates.

Theorem A.7 (Donin [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF]). Let C be a compact complex manifold and let U r (r * < r < r * < 1) be a family of open coverings of C as in Lemma A. [START_REF] Claudon | Compact leaves of codimension one holomorphic foliations on projective manifolds[END_REF]. Let E ′ ⊗ E ′′ be a holomorphic vector bundle of rank m over C and fix a holomorphic base e ′ j (resp. e ′′ j ) for E ′ (resp. E ′′ ) over U j . Let r * < r ′′ < r ′ < r < r * , and

where κ is defined (A.10). The constant

As in [START_REF] Donin | Cohomology with estimates for coherent analytic sheaves over complex spaces[END_REF], we will apply L 2 -theory for (0, 1)-forms on a bounded pseudoconvex domain in C n . In our case the domain is actually a polydisc. Fix a holomorphic base e ′ k = (e ′ k,1 , . . . , e ′ k,m ) for the vector bundle E ′ in U r * k with transition functions t ′ kj (z j ). Analogously, let t ′′ kj (z j ) be the transition matrices for basis e ′′ k of E ′′ for U r * . For brevity, we write t kj for t kj (z j ).

We can write

) is a covering of the polydisc ∆ n r * . By Lemma A.6, we have

Let χ be a non-negative smooth function in R so that χ(t) = 1 for t < 3/4 and χ(t) = 0 for t > 7/8. By smoothing the Lipschitz function χ(

Then by the mean-value theorem and the first inequality of (A.16), we get (A.17)

In general, we have

Then we have

So far we have followed the proof of the finiteness lemma in [14, p. 201]. We now finish the proof of the theorem. Let us first find the linearly independent elements g i 0 , . . . , g iµ * . Assume first that all g i = 0 in H 1 := H 1 (U r , E ′ ⊗ E ′′ ). Then δη j = g j with η j ∈ C 0 (U r , E). Assume now that g m 0 = 0 in H 1 for some m 0 . Then we have two cases again: either g i = c i0 g m 0 + δη i on U r for all i ∈ {0, . . . , µ} \ m 0 , or it fails for some m 1 . We repeat this to exhaust all elements so that (A.31)

while g m 0 , . . . , g mµ * are linearly independent in H 1 . (Note that the above expression means the trivial identity g j = g j when j is not in {m 0 , . . . , m µ * }.) We have obtained (A.28) with the decomposition

The solution η * j in (A.31) can be bounded in U r -for any r -< r. Of course we need to estimate η * j on U r ′ . Thus, r -≥ r ′ . We have

We have obtained the required estimates.

Finally, let us assume that f = 0 in H 1 (C, E) in order to show that all c j = 0 and thus f = δv. Since each U r ′′ is Stein, we also have f = 0 in H 1 (U r , E). Thus f = δṽ with ṽ ∈ C 0 (U r ′′ , E). We get δ(ṽv) = µ * j=0 c j g m j . By the linear independence, we conclude that c j = 0. We are done.

Theorem A.12. Let C be a compact complex manifold and let U r (r * ≤ r ≤ r * < 1) be nested coverings of C as in Proposition A. [START_REF] Hörmander | L 2 estimates and existence theorems for the ∂ operator[END_REF]. Let µ, ν, r, r ′ , r ′′ , r * , r * be given in Theorem A.9, which satisfy

where K * (E ′ ⊗ E ′′ ), defined by (A.8), satisfies

where κ and B r -are defined by (A.10) and (A.29). The same conclusion holds if both sides are in sup norms

Remark A. [START_REF] Grauert | Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften[END_REF]. The main conclusion is that (A.32) holds without shrinking the covering {U r ′ i } on which f is defined. The solution operator f → u may not be linear. The small divisor conditions are carried by B r -which is determined by (A.25) and (A.29), while the bounds in Theorem A.7 as smoothing lemma does not involve small divisors.

Proof. By the Leray theorem, we know that [f ] = 0 in H 1 (U r ′ , E). By Theorem A.9, we have a solution u ∈ C 0 (U r ′′ , E) so that

Then the conclusion follows from Lemma A.2.

When the super norm is used, we first obtain a solution u = {u k } for U r * for r * = (r ′′ + r ′ )/2, while (A.34) takes the form

Then the conclusion follows from Lemma A.2 again.

A.5. Existence of nested coverings. In this subsection, our main goal is to construct nested coverings by using transversality theorems and analytic polyhedrons. We recall that C n is a n-dimensional compact complex manifold. We shall omit to mention its dimension in what follows.

We first deal with the transversality for a piecewise smooth boundary of an analytic polyhedron and we then define the general position property of several analytic polyhedrons.

Definition A.14.

(a) Let M j be a C 1 real hypersurface defined by r j = 0, where r j is a C 1 function in an open set ω j of a complex manifold C and dr j = 0 on M j . We say that M 1 , . . . , M N are in the general position, if

Let ω be a proper open set of a complex manifold C and let f ∈ O N (ω). We say that (A. [START_REF] Rüssmann | Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition[END_REF])

is an analytic N-polyhedron in ω if Q is non-empty and relatively compact in ω, and Q does not contain any compact connected component. We say that Q is generic, if

We will apply transversality theorems. This requires us to use open submanifolds in C n which may not be closed in

we may assume that the non-empty intersection of any number of Q 1 N , . . . Q N N intersects Q N . By (A.36), the closed set Q N does not intersect the closed subset of ω defined by

Removing the above sets from ω, we find a neighborhood ω

it is a codimension k smooth submanifold in ω * . For brevity we will call ω * a neat neighborhood of Q. We will take ω = ω * without specifying ω * . Definition A.15. Let ω i be open sets in C. For i = 0, . . . , p, assume that φ i ∈ O N i (ω i ) and Q N i (φ i , ω i ) is an analytic polyhedron in ω i . We say that they are in the general position, if all faces Q j N i for 1 ≤ j ≤ N i and 0 ≤ i ≤ p are in general position. More precisely, ω * N i ∩ Q j N i are in the general position, where each ω * i is a neat neighborhood of

Let us describe some elementary properties of generic analytic polyhedrons. If Q N (f, ω) is defined in ω by (A.35), we denote for ρ = (ρ 1 , . . . , ρ N )

By the assumption of the general position, we see that the faces of Q are in the general position.

Let X, Y be smooth real manifolds without boundary and W a smooth submanifold of Y . Following [10, p. 50], we say that a smooth mapping h :

For an analytic

be the set of all edges, with the first N edges being the faces.

are in the general position. Equivalently, each edge Q s N ℓ intersects transversally with each edge of the intersection of any number of Q N 0 , . . . , Q N ℓ-1 , for ℓ = 1, . . . , p.

Proof. Since each edge of a polyhedron is the intersection of its faces, it is clear that if Q N 0 , . . . , Q Np are in the general position, then the edges

Conversely, let φ i = (φ i,1 , . . . , φ i,N i ) and let ψ 1 , . . . , ψ m be a subset of φ 0,1 , . . . , φ 0,N 0 , . . . , φ p,1 , . . . , φ p,Np . We emphasize that we do not assume that the latter are distinct functions, although φ i,1 , . . . , φ i,N i are distinct by the general position property of the faces of Q N i . Suppose that ψ ℓ is in {φ i ℓ ,1 , . . . , φ i ℓ ,N i ℓ }. We need to show that (A. [START_REF] Stolovitch | Singular complete integrability[END_REF])

. Without loss of generality, we may assume that

with ψα β being a non-empty subset of components of φ α β . Without loss of generality, we may assume that ψα

The equivalence of (A.37) and (A.38) implies that (A.39) follows from the assumption that

Lemma A.18 (Golubitsky-Guillemin [10, p. 53]). Let X, B, and Y be smooth manifolds with W a submanifold of Y . Let ψ : B → C ∞ (X, Y ) be a mapping (not necessarily continuous) and define Ψ :

Proposition A.19. Let C be a compact complex manifold of dimension n. Let {U i : i = 1, . . . , m} be a finite open covering of C. Assume that ϕ j is a biholomorphism from a neighborhood ω j of the star N(U j ) of U j onto ωj ⊂ C n such that U j = ϕ -1 j (∆ n ) = Q n (ϕ j , ω j ). There exists δ > 0 satisfying the following:

(a) For each j, there are a relatively compact open set ωj (resp. Ũj ) in ω j (resp. ωj ) and a dense open set A j of ∆ δ n such that if c j ∈ A j , then φj := ϕ jc j is a biholomorphic mapping from Ũj onto ∆ n , and Ũ1 := Q n ( φ1 , ω1 ), . . . , Ũm := Q n ( φm , ωm ) are generic n-polyhedrons in the general position, where { Ũ1 , . . . Ũm } remains an open covering of C and ωj is a neighborhood of N( Ũj ). In particular each φj , a translation of ϕ j , is injective on ωj . (b) There is 0 < r * < 1 such that if r * ≤ ρ i ≤ 1, then Ũρ 0 i 0 , . . . , Ũρq iq are generic n-polyhedrons in the general position, where Ũρ

We will apply the transversality theorem for real submanifolds in C n . Therefore, we will use old coordinate charts ϕ j to map edges of polyhedrons [START_REF] Hirschowitz | On the convergence of formal equivalence between embeddings[END_REF], mainly the density assertion in the lemma, finitely many times in

where ω ′ 2 is a relatively compact open subset of ω 2 which is independent of δ, and U 2 ⊂ ω ′ 2 . We also remark that (A.18) can be applied for finitely many times since

Therefore, (A.41) implies that every edge of Ũ2 intersects each edge of Ũ1 . We have determined Ũ2 = φ-1 2 (∆ n ). We have verified (a) when m = 2. Let us assume that it also holds for m ≥ j. By Lemma A.16, each edge of a non-empty intersection of any number of Ũ1 , . . . , Ũj is a smooth submanifold. We remark the above transversality argument mainly uses the fact that ϕ 2 is a biholomorphism, while each edge of Ũ1 is a smooth submanifold.

To repeat the above argument for m = 2 in details, we list all edges of all possible intersections of Ũ1 , . . . , Ũj as W ′ 1 , . . . , W ′ L so that each W j is an edge of some analytic polyhedron U ′ j , where U ′ j is the intersection of some of Ũ1 , . . . , Ũj ′ which are in general position by the induction hypothesis as mentioned above. Therefore, by Lemma A.16, each U ′ ℓ is generic. Now we are in the situation of m = 2 by considering the sets of two analytic polyhedrons {U ′ ℓ , U j+1 } one by one for ℓ = 1, . . . , j ′ . Here U j+1 = ϕ -1 j+1 (∆ n ) with ϕ j+1 being biholomorphic in a neighborhood of N(U j+1 ). Therefore, we can find φj+1 = ϕ j+1c j+1 such that each edge of Ũj+1 intersects each W ′ ℓ transversally on Ũj+1 ∩ U ′ ℓ . The above argument shows the existence of c 1 , . . . , c N in ∆ δ n when δ is sufficiently small. The openness property on A j is clear, since by shrinking ωj slightly the general position and generic properties are preserved under small perturbation of c j . Then density of A j when δ is sufficiently small can also be achieved; indeed when c j is sufficiently small, we may shrink ω j slightly and apply the above argument by replacing ϕ jc j with ϕ j . Finally, { Ũ1 , . . . , ŨN } still covers C when δ is sufficiently small. We have verified (a).

The assertion (b) follows from (a) and Proposition A.17. Indeed, we first note that when r * is less than 1, but it is sufficiently close to 1, the ∂Q ρ ( φj ) is in a given neighborhood of ∂Q( φj , ωj ), as Q ρ ( φj , ωj ) does not have any compact connected component. By the relative compactness of Q n ( φi , ωi ), the condition (A.36) with f j being replaced by f j /ρ j and the general position condition remain true when ρ j are in [r * , 1] when r * < 1 is sufficiently close to 1. The proof is complete.

The following is a basic property of a generic analytic polyhedron.

Proposition A.20. Let C be a compact complex manifold of dimension n. Let Q N (f, ω) be a generic analytic N-polyhedron C defined by (A.35) and (A.36). There exists r * ∈ (0, 1) satisfying the following.

) and the latter is non-empty. (b) There are finitely many open sets ω ′′ j in C and smooth diffeomorphisms φ j sending ω ′′ j onto ω′′ j in R 2n such that {ω ′′ j } covers ∂Q N (f, ω), and for any p 0 , p

Note that {µ 1 , . . . , µ m } is uniquely determined by x. By the transversality condition (A.36), we have m ≤ 2n. Choose an open set ω ′ such that x ∈ ω ′ ⊂ ω and

In particular, we have

By (A.36), we can take (|f µ 1 |, . . . , |f µm |) to be the first m components of a smooth diffeomorphism ϕ : ω ′ → ω, shrinking ω ′ if necessary. Taking a smaller open subset ω ′′ of ω ′ with x ∈ ω ′′ , we may assume that tζ ∈ ω, ∀ζ ∈ ω′′ := ϕ(ω ′′ ), 1δ ≤ t ≤ 1, for some δ ∈ (0, 1].

Since ∂Q is compact, there exists {x j , ω ′′ j , ω ′ j : j = 1, . . . , k} satisfying the following: (a) The k is finite. For each j, we have that x j ∈ ω ′′ j ⊂ ω ′ j ⊂ ω, x j ∈ ∂Q, and ω ′ j is an open subset of ω. For each j, we have m j and µ j,1 < . . . < µ j,m j , which are the numbers associated to x j , so that (A.42) holds for

Here we set M j = 0 if m j = N. (c) The (|f µ j,1 |, . . . , |f µ j,m j |) are the first m j components of a smooth diffeomorphism φ j from ω j onto a subset ωj of C n . There exists δ * > 0 such that ω′′ j := φ j (ω ′′ j ) satisfies Indeed, let φ j (x j ) = (1, . . . , 1, xj ) with xj ∈ R 2n-m j . We can take (A.44) ω′′ j = (1δ * , 1 + δ * ) m j × B δ ′′ 2n-m j (x j ) where B δ ′′ 2n-m j (x j ) is the ball in R 2n-m j centered at xj with a sufficiently small radius δ ′′ . Note that Suppose that r * ≤ ρ ′ i ≤ ρ i ≤ 1 for i = 1, . . . , N. Let Ω be a connected component of Q ρ N . Since Ω does not have a compact connected component, there exists z * ∈ ∂Ω satisfying |f i (z * )| = ρ i for some i. Since ρ i > M * , then z * ∈ ω ′′ j for some j. Let us assume that z * ∈ ω ′′ 1 , and (µ 1,1 , . . . , µ 1,m 1 ) = (1, . . . , m 1 ). Thus φ 1 = (|f 1 |, . . . , |f m 1 |, fm 1 +1 , . . . , f2n ). We now replace z * by some z * ∈ Ω ∩ ω ′′ 1 . We consider a path defined by t → γ(t) := φ -1 1 (tφ 1 (z * )), 1δ * ≤ t ≤ 1. Note that by (A.43), γ is well defined and is contained in ω 1 . We now have i (∆ n ) (i = 1, . . . , m), which are in the general position. By Lemma A.16, each U i ∩ U j , if non-empty, is a generic analytic polyhedron. Applying Proposition A.20 (a) to all nonempty U i ∩ U j , we know that {U r i = ϕ -1 i (∆ r n ) : i = 1, . . . , m} for r * ≤ r ≤ 1 is a family of nested coverings. Therefore, we can apply Theorem A.9 and Theorem A.12.