Xianghong Gong
email: gong@math.wisc.edu

Laurent Stolovitch
email: laurent.stolovitch@univ-cotedazur.fr

ON NEIGHBORHOODS OF EMBEDDED COMPLEX TORI

Keywords: 2020 Mathematics Subject Classification. 32Q57, 32Q28, 32L10, 37F50 Complex torus, neighborhood of a complex manifold, Stein manifold, Grauert's Formal principle, normal bundle, small divisors, KAM scheme

The goal of the article is to show that an n-dimensional complex torus embedded in a complex manifold of dimensional n+d, with a split tangent bundle, has neighborhood biholomorphic a neighborhood of the zero section in its normal bundle, provided the latter has (locally constant) Hermitian transition functions and satisfies a non-resonant Diophantine condition.

In memory of Jean-Pierre Demailly

Introduction

In this paper we show the following Theorem 1.1. Let C be an n-dimensional complex torus embedded in a complex manifold M of dimensional n + d. Assume that T C M , the restriction of T M on C, splits as T C ⊕ N C . Suppose that the normal bundle of C in M admits transition functions that are Hermitian matrices and satisfy a non-resonant Diophantine condition (see Definition 4.5). Then a neighborhood of C in M is biholomorphic to a neighborhood of the zero section in the normal bundle.

We first describe the organization of the proof of our main theorem. A complex torus C can be identified with the quotient of C n by a lattice Λ spanned by the standard unit vectors e 1 , . . . , e n in C n and n additional vectors e ′ 1 , . . . , e ′ n in C n , where Im e ′ 1 , . . . , Im e ′ n are linearly independent vectors in R n . Let Λ ′ be the lattice in the cylinder C := R n /Z n + iR n spanned by e ′ 1 , . . . , e ′ n mod Z n . There are two coverings for the torus C = C n /Λ = C/Λ ′ : the universal covering π : C n → C and the covering by cylinder, π C : C → C that extends to a covering M over M . In section two we recall some facts about factors of automorphy for vector bundles on C via the covering by C n . In section three, we study the flat vector bundles on C. The pull back of the flat vector bundle N C to the cylinder C is the normal bundle N C of C in M. We show that N C is always the holomorphically trivial vector bundle C × C d . By "vertical coordinates", we mean "coordinates on C d ", the normal component of the normal bundle N C , while "horizontal coordinates" mean the tangential components of N C .

Since C is a Stein manifold, a theorem of Siu [START_REF] Tong | Every Stein subvariety admits a Stein neighborhood[END_REF] says that a neighborhood of C in M is biholomorphic to a neighborhood of the zero section in its normal bundle, which is trivial as mentioned above. We show that the holomorphic classification of neighborhoods M of C with flat N C is equivalent to the holomorphic classification of the family of the deck transformations of coverings M of M in a neighborhood of C. These deck transformations are "higher-order" (in the vertical coordinates) perturbations τ 1 , . . . , τ n of τ1 , . . . , τn , where the latter are the deck transformations of the covering of N C over N C . In order to find a biholomorphism between a neighborhood of C in M and a neighborhood of its zero section in N C , it is sufficient to find a biholomorphism that conjugates {τ 1 , . . . , τ n } to {τ 1 , . . . , τn }.

There are two useful features. First, since the fundamental group of C is abelian, the deck transformations τ 1 , . . . , τ n commute pairwise. Second, we can also introduce suitable coordinates on C so that the "horizontal" components of deck transformations have diagonal linear parts. In such a way the classification of neighborhoods of C is reduced to a more attainable classification of deck transformations. While the full theory for this classification is out the scope of this paper, we study the case when N C admits Hermitian transition functions. Since a Hermitian transition matrix must be locally constant, we call such an N C Hermitian flat. The convergence proof for Theorem 1.1 is given in section four. It relies on a Newton rapid convergence scheme adapted to our situation based on an appropriate Diophantine condition among the lattice and the normal bundle. At step k of the iteration scheme, let δ k be the error of the deck transformations {τ

(k) 1 , . . . , τ (k)
n } defined on domain D (k) to τ1 , . . . , τn in suitable norms. By an appropriate transformation Φ (k) , we conjugate to a new set of deck transformations {τ (k+1) 1 , . . . , τ (k+1) n } of which the error to the linear ones is now δ k+1 on a slightly smaller domain D (k+1) . Using our Diophantine conditions, related to the lattice Λ and the normal bundle, we show that the sequence Φ (k) • • • • • Φ (1) converges to a holomorphic transformation Φ on an open domain D (∞) where we linearize {τ 1 , . . . , τ n }.

We now describe closely related previous results. Our work is motivated by work of Arnol'd and Ilyashnko-Pyartli. Our main theorem was proved in [START_REF] Arnol ′ D | Bifurcations of invariant manifolds of differential equations, and normal forms of neighborhoods of elliptic curves[END_REF] when C is an elliptic curve (n = 1) and N C has rank one (d = 1). Il'yashenko-Pyartli [START_REF] Ju | Neighborhoods of zero type imbeddings of complex tori[END_REF] extended Arnol'd's result to the case when the torus is the product of elliptic curves together with a normal bundle which is a direct sum of line bundles, while Theorem 1.1 deals with general complex tori. We also assume that N C is non-resonant, a condition that is weaker than the non-resonant condition used by Il'yashenko-Pyartli. Our small-divisor condition is also weaker. Of course, the study of neighborhood of embedded compact complex manifolds has a long history. Also, see some recent work [START_REF] Hwang | An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principle[END_REF][START_REF] Koike | Linearization of transition functions of a semi-positive line bundle along a certain submanifold[END_REF][START_REF] Loray | Two-dimensional neighborhoods of elliptic curves: formal classification and foliations[END_REF]. We refer to [START_REF] Gong | Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations[END_REF] for some references and a different approach to this range of questions.

Acknowledgments. This work benefits from helpful discussions with Jean-Pierre Demailly. Part of work was finished when X. G. was supported by CNRS and UCA for a visiting position at UCA.

Vector bundles on Tori and factors of automorphy

In this section, we identify vector bundles on a complex torus with factors of automorphy. The latter gives us a useful alternative definition of vector bundles on a higher dimensional complex torus C and the isomorphisms of two vector bundles.

General references for line bundle on complex tori are [START_REF] Birkenhake | Complex abelian varieties, volume 302 of Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Debarre | Complex tori and abelian varieties, volume 11 of SMF/AMS Texts and Monographs[END_REF][START_REF] Iena | Vector bundles on elliptic curves and factors of automorphy[END_REF] and [START_REF] Griffiths | Principles of algebraic geometry[END_REF]p. 307].

Let Λ be a 2n-dimensional lattice in C n . We may assume that Λ is defined by 2n vectors e 1 , . . . , e n , e ′ 1 , . . . , e ′ n of C n , where e i = (0, . . . , 0, 1, 0, . . . , 0) with 1 being at the i-th place, e ′ i = (e ′ i,1 , . . . , e ′ i,n ) and the matrix Im τ := (Im e ′ i,j ) 1≤i,j≤n =: (e ′′ i,j ) 1≤i,j≤n is invertible [BL04, exerc. 2, p. 21]. The compact complex manifold C := C n /Λ is called an (n-dimensional) complex torus. Unless the lattice is equivalent to another one defined by a diagonal matrix e ′ , C is not biholomorphic to a product of onedimensional tori. Let π : C n → C be the universal cover of C. Its group Γ of deck transformations consists of translations

T λ : z → z + λ, λ ∈ Λ.
Note that Γ is abelian and is isomorphic to

Z 2n . Γ is also isomorphic to π 1 (C, 0) since C n is a universal covering of C.
Next, we consider equivalence relations for holomorphic vectors bundles on C and C n , following the realization proof of Theorem 3.2 in [START_REF] Iena | Vector bundles on elliptic curves and factors of automorphy[END_REF]. Let E be a vector bundle of rank d over C. The pull-back bundle π * E on C n is trivial and has global coordinates ξ. Let {U j } be an open covering of C so that coordinates ξ j = (ξ j,1 , . . . , ξ j,d ) t of E are well-defined (injective) on U j . Then we have

(2.1) ξ = h j ξ j (π),
where h j is a non-singular holomorphic matrix on π -1 (U j ). The transition functions g kj satisfy (2.2)

g kj (π) = h -1 k h j , on π -1 (U k ) ∩ π -1 (U j ).
For any z, λ, we know that both π(z + λ) and π(z) are in the same U j for some j. Then we have

ξ(z + λ) = h j (z + λ)ξ j (π(z)) = h j (z + λ)h j (z) -1 ξ(z).
We can define

(2.3) ρ(λ, z) = h j (z + λ)h j (z) -1 , z ∈ π -1 (U j )
as the latter is independent of the choice of j by (2.2). Therefore,

ξ(λ + z) = ρ(λ, z) ξ(z), ρ(λ, z) ∈ GL(d, C), (2.4) ρ : Λ × C n → GL(d, C). (2.5)
Here ρ is called a factor of automorphy. We can verify that

(2.6) ρ(λ + µ, z) = ρ(λ, µ + z)ρ(µ, z).
In particular, if all ρ(λ, z) = ρ(λ) are independent of z, then ρ(Λ) is an abelian group.

The above construction from a vector bundle E on C to a factor of automorphy can be reversed. Namely, given (2.4)-(2.6), define the vector bundle E on C as the quotient vector space of C n × C d via the equivalence relation

(2.7) (z, ξ) ∼ (z + λ, ρ(λ, z)ξ), z ∈ C n , ξ ∈ C d , λ ∈ Λ.
We denote the projection from the cylinder

C := R n /Z n + iR n = C n /Z n onto C by π C . Therefore, we can define π * C E on the cylinder C by the equivalence relation (2.8) (z, ξ) ∼ (z + λ, ρ(λ, z)ξ), z ∈ C n , ξ ∈ C d , λ ∈ Z n .
Of course, global coordinates ξ, ρ, and g jk are not uniquely determined by E. However, their equivalence classes are determined. Two vector bundles E, Ẽ are isomorphic if their corresponding transitions g jk , gjk satisfy gjk = h -1 j g jk h k where h j are non-singular holomorphic matrices. Replacing global coordinates ξ by ν ξ where ν :

C n → GL(n, d) is holomorphic, we can verify that (2.9) ν(λ + z)ρ(λ, z)ν(z) -1 =: ρ(λ, z)
is also a factor of automorphy. Define two factors of automorphy ρ, ρ to be equivalent if (2.9) holds. Therefore, the classification of holomorphic vector bundles is identified with the classification of factors of automorphy.

Flat vector bundles

In this section, we will show that the pull-back of a flat vector bundle E on C to the cylinder C = C n /Z n is always trivial.

When E is flat, we can choose global coordinates as follows. We know that π * E is also flat and we can choose its global flat basis, or global flat coordinates ξ by using analytic continuation on C n and pulling back flat local coordinates of E. In other words, in (2.1) h j are locally constants while ξ j are locally flat coordinates. Then ρ(λ, z) depend only on λ, in which case we write ρ(λ) for ρ(λ, z). As remarked above, ρ(Λ) is abelian. When E is unitary (flat), by the same reasoning π * E is unitary and we can choose h j and ρ(λ) to be unitary.

A d × d Jordan block J d (λ) is a matrix of the form λI d + N d , where I d is the d × d identity matrix and N d is the d × d matrix with all entries being 0, except all the (i, i + 1)-th entries being 1. A matrix T commutes with J if and only if

T = T d (a) := a 0 I d + i>0 a i N i d .
Note that N i d = 0 for i ≥ d. Following [Gan98, p. 218], we call the above T as well as the following two types of matrices, regular upper triangular matrices w.r.t. J:

A = (0, T d (a)), or B = T d ′ (a ′ ) 0 ,
where 0 denotes in A (resp. B) a 0 matrix of d rows (resp. d ′ columns). Given a Jordan matrix J = diag(J d1 (λ 1 ), . . . , J d k (λ k )).

the matrices that commute with J are precisely the block matrices

X = (X αβ ) m×m
where X αβ = 0 if λ α = λ β , while X αβ is a regular upper triangular (d α ×d β ) matrix if λ α = λ β . Such a matrix X is said to be a regular upper triangular matrix w.r.t. J From the structure of matrices commuting with a Jordan matrix, we can verify the following two results. Proposition 3.1. Let A 1 , . . . , A m be 2 × 2 matrices commuting pairwise. Then there is a non-singular matrix S such that all S -1 A j S are Jordan matrices.

Example 3.2. The 3 × 3 matrices λI 3 + N 3 , µI 3 + N 2 3 commute, but they cannot be transformed into the Jordan normal forms simultaneously.

The following results on logarithms are likely classical. However, we cannot find a reference. Therefore, we give proofs emphasizing commutativity of logarithms of matrices.

We start with the following.

Lemma 3.3. Let A 1 , . . . , A m be pairwise commuting matrices. Then there is a non-singular matrix S such that

S -1 A j S = ( Âαβ ) 1≤α,β≤s =: Âj , 1 ≤ j ≤ m
where Â1 is a Jordan matrix and all Âj are upper triangular matrices.

Proof. We may assume that A 1 is a Jordan matrix J = diag(J d1 (λ 1 ), . . . , J ds (λ s )). Then A j = (X j αβ ) 1≤α,β≤s are regular w.r.t J. Note that pairwise commuting non-singular matrices have nontrivial common eigenspaces. The eigenspace of J are spanned e d ′ 1 , . . . , e d ′ s with

d ′ 1 = 1 and d ′ j = d 1 + • • • d j-1 + 1.
To simplify the indices, we may assume that e 1 is an eigenvector of all A j . Then the first column of A j is a j e 1 with a j = 0. The new matrices Ãj , obtained by removing all first rows and first columns, still commute pairwise. In particular all Ãj are regular to the new Jordan matrix J = Ã1 . By induction on d, we can find a non-singular matrix S which is regular to J so that all S-1 Ãj S are upper-triangular. Let S = diag(1, S). Now S -1 = diag(1, S-1 ).

We can check that all Âj := S -1 A j S are upper triangular. Then Â1 , J have the same entries, with only one possible exception Â1;12 = s 11 J 12 .

If Â1;12 = 0, dilating the first coordinate can transform Â1 into the original J, while Âj remain upper-triangular. If s 12 J 12 = 0, then J 12 must be 0, i.e. Â1 = J, because one cannot transform a Jordan matrix, A 1 = J, into a new Jordan matrix, Â1 , by reducing an entry 1 to 0 and keeping other entries unchanged.

The above simultaneous normalization of upper-triangular matrices allows us to define the logarithms. The construction of logarithms of non-singular matrices can be found in [START_REF] Gantmacher | The theory of matrices[END_REF]p. 239]. Here we need to find a definition that is suitable to determine the commutativity of the logarithm of pairwise commuting non-singular matrices.

Recall that for a d×d matrix A, the generalized eigenspace E λ (A) with eigenvalue λ is the kernel of (A -λI) d , while C d is the direct sum of all E λ (A). A matrix B that commutes with A leaves each E λ (A) invariant, i.e. B(E λ (A)) ⊂ E λ (A). Thus if A 1 , . . . , A m commute pairwise, we can decompose C d as a direct sum of liner subspaces V j such that each V j is invariant by A i and admits exactly one eigenvalue of A i . Thus to define ln A j , we will assume that each A j has a single eigenvalue on C d if we wish.

Given a non-singular matrix A, a logarithm of A is a matrix ln A satisfying e ln A = A where the exponential matrix e B = B n n! is always well-defined. However, ln A is not unique.

For a non-singular upper triangular matrix (3.1)

A = λI d + a, a := (a ij ) 1≤i,j≤d , a ij=0 ∀i ≥ j,
we have a k = 0 for k ≥ d. Using the identity e B+C = e B e C for two commuting matrices B, C, we see that e ln A = A for

(3.2) ln A := (ln λ)I d - k>0 (-λ -1 a) k k
with 0 ≤ Im ln(.) < 2π. For a non-singular Jordan matrix, we can define ln diag(J d1 (λ 1 ), . . . , J dm (λ m )) = diag(ln J d1 (λ 1 ), . . . , ln J dm (λ m )).

Note that ln λ α = ln λ β if and only if λ α = λ β . Since matrices that commute with a fixed matrix is closed under multiplication by a scalar, addition and multiplication.

It is thus clear that if A is an upper triangular matrix that is regular to a nonsingular Jordan matrix J = diag(J d1 (λ 1 ), . . . , J ds (λ s )), then ln A remains regular to J. Equivalently and more importantly, ln A is regular to the Jordan normal form of ln J, which is diag(J d1 (ln λ 1 ), . . . , J ds (ln λ s )).

Proposition 3.4. Let A 1 , . . . , A m be pairwise commuting d × d matrices. Then there is a non-singular matrix S such that Âj := S -1 A j S are block diagonal matrices of the form

(3.3) Âj = diag( Âj,d1 , . . . , Âj,d k )
where all Âj,di are upper triangular d i × d i matrices, and each Âj,di has only one eigenvalue λ j,i . Assume further that all A j are non-singular. Then

(3.4) ln A j := S diag(ln Âj,d1 , . . . , ln Âj,d k )S -1 , 1 ≤ j ≤ m
commute pairwise and e ln Aj = A j , where ln Âj,di are defined by (3.1)-(3.2).

Proof. Note that for pairwise commuting matrices A 1 , . . . , A m , we have a decomposition C d = t i=1 V i where each A j preserves V i and has only one eigenvalue

λ j,i . Their restrictions of A 1 , . . . , A m on V i remain commutative pairwise. Let dim V i = d i , d ′ 0 = 0, d ′ i+1 -d ′ i = d i . By Lemma 3.3, we can find a basis e * d ′ i-1 +1 , . . . , e * d ′ i-1 +di for V i such that A 1 | Vi , . . . , A m | Vi are upper triangular matrices.
Using the new basis e * 1 , . . . , e * d we can find the matrix S for the decomposition (3.3). Therefore, Â1,di , . . . , Âm,di commute pairwise.

Assume now that all λ j,i are non-zero. It remains to show that ln Â1,di , . . . , ln Âm,di commute pairwise. Write Âj,i = λ j,i

I di + W j,i
where W j,i are upper triangular matrices and W di j,i = 0. By a straightforward computation, we have

[W j,i , W j ′ ,i ] = [ Âj,i , Âj ′ ,i ] = 0.
Therefore, W k j,i commutes with W ℓ j ′ ,i for any k, ℓ. Consequently, the finite sum ln Âj,i = ln λj,i

I di - k>0 (-λ -1 j,i W j,i ) k k
commutes with ln Âj ′ ,i . Therefore, ln A 1 , . . . , ln A m in (3.4) commute pairwise.

Lemma 3.5. Let A 1 , . . . , A n be non-singular upper triangular d × d matrices.

Suppose that A 1 , . . . , A n commute pairwise. There exists a linear mapping w → ṽz (w

) := v(z)w in C d , entire in z ∈ C n such that v(z) ∈ GL d (C), v(0) = Id and v(e j ) = A j for j = 1, . . . , n. Furthermore, v(z + z ′ ) = v(z)v(z ′ ) for all z, z ′ ∈ C n .
Proof. By Proposition 3.4, we define pairwise commuting matrices ln A 1 , . . . , ln A n such that e ln Aj = A j . Then the Lie brackets of the vector fields for ẇ = ln A j w, j = 1, . . . , n vanish and their flows ϕ t j (w) commute pairwise. Note that

ϕ 0 j (w) = w, ϕ 1 j (w) = e ln Aj w = A j w. Define ṽz (w) = ϕ z1 1 • • • ϕ zn n (w). We conclude ṽz ṽz ′ (w) = ṽz+z ′ (w), that is v(z + z ′ ) = v(z)v(z ′ ).
By Proposition 3.4, we define ln ρ(e 1 ), . . . , ln ρ(e 2n ) and they commute pairwise. We now define ln ρ(λ) for all λ = 2n j=1 m j e j ∈ Λ as follows

ln ρ   2n j=1 m j e j   := 2n j=1 m j ln ρ(e j ).
Thus the matrices ln ρ(λ) for λ ∈ Λ commute pairwise.

Proposition 3.6. Let E be a flat vector bundle on C. Then π * C E admits a factor of automorphy ρ satisfying ρ(e j ) = Id for j = 1, . . . , n; in particularly, π * C E is holomorphically trivial.

Proof. Let d be the rank of E. Let A j = ρ(e j ) -1 for j = 1, . . . , n. With v(e j ) = A j and v(0

) = Id d , we first see that ρ(λ, z) := v(z + λ)ρ(λ)v(z) -1
satisfies ρ(e j , 0) = Id d . We want to show that ρ(λ, z) depends only on λ. Fix λ = 2n j=1 m j e j ∈ Λ. By definition, the matrix ln ρ(λ) commutes with each ln ρ(e j ), j = 1, . . . , 2n. Thus the flow ϕ t λ of ẇ = ln ρ(λ)w commutes with the flows of ẇ = ln ρ(e j )w, j = 1, . . . , 2n. As in the proof of the previous lemma, we know that ϕ t λ (w) is linear in w and entire in t ∈ C. For z ∈ C n and w ∈ C d , let ṽz (w) be as defined in the previous lemma. Thus we have ϕ t λ ṽz (w) = ṽz ϕ t λ (w). Taking derivatives in w and plugging in t = 1, we get

exp(ln ρ(λ))v(z) = v(z) exp(ln ρ(λ)). Since exp(ln ρ(λ)) = ρ(λ), we have ρ(λ)v(z) = v(z)ρ(λ) for all λ ∈ Λ, z ∈ C n . Considering z = z 1 e 1 + • • • + z n e n ∈ Λ, we have v(z + λ) = v(z)v(λ). Hence, ρ(λ, z) is independent of z.
We have achieved ρ(λ) = ν(z + λ)ρ(z)ν(z) -1 and ρ(e j ) = Id d for j = 1, . . . , n. Therefore, π| * C E is trivial, by the equivalence relation (2.8). It is known that there are Stein manifolds with non-trivial vector bundle [START_REF] Forster | Über die Anzahl der Erzeugenden von projektiven Steinschen Moduln[END_REF]. Furthermore, we conclude the section by emphasizing that the triviality π * | C E relies on the extra assumption that it is a pull-back bundle. The following result is likely known, but we include a short proof for completeness.

Proposition 3.7. The set of holomorphic equivalence classes of flat holomorphic line bundles on C can be identified with H 1 ( C, C * ). The latter is non-trivial.

Proof. Each element {c jk } in H 1 ( C, C * ) is clearly an element in H 1 ( C, O * ). We want to show that if {c jk }, {c jk } ∈ H 1 ( C, C * ) represent the same element in H 1 ( C, O * ), then they are also the same in H 1 ( C, C * ). Indeed, we can cover C by convex open sets U 1 , U 2 , U 3 , U 4 such that U 1 ∩ U 2 ∩ U 3 ∩ U 3 is non empty. Thus {U i } is a Leray covering. If cjk = h j c jk h -1
k , where each h j is a non-vanishing holomorphic function on U j . Take p in all U j . We get cjk = c j c jk c -1

k for c j = h j (p). Note that H 1 ( C, C * ) is non-trivial. Otherwise, the exact sequences 0 → Z → C → C * → 0 and 0 → H 0 ( C, Z) → H 0 ( C, C) → H 0 ( C, C * ) → 0 would imply that H 1 ( C, Z) ∼ = H 1 ( C, C), a contradiction.

Equivalence of neighborhoods and commuting deck transformations

In this section, we will discuss how the classification of neighhorhoods U of a compact complex manifold C is related to the classification of deck transformations of a holomorphic covering Ũ → U , where U, Ũ are chosen carefully and Ũ contain C * that covers C. When C * is additionally Stein, we can choose Ũ to be a neighborhood of C * in its normal bundle N C * ( Ũ ) by applying a result of Siu. After preliminary results in Lemma 4.1 and Lemma 4.2, we will return to our previous study where C is a complex torus, and its covering of C is the Stein manifold C * = C, and N C (M ) is Hermitian flat. We then prove the main result of this paper by using a KAM rapid iteration scheme.

Let us start with ι : C ֒→ M , a holomorphic embedding of a compact complex manifold C. We shall still denote ι(C) by C. Let U be a neighborhood of C in M such that U admits a smooth, possibly non-holomorphic, deformation or strong retract C [Mun00, p. 361]; namely there is a smooth mapping R : [START_REF] Munkres | Topology[END_REF]p. 361]). When M is N C , we can find a holomorphic deformation retraction from a suitable neighborhood of its the zero section onto C, by using a Hermitian metric on N C .

U × [0, 1] → U such that R(•, 0) = Id on U , R(•, t) = Id on C, and R(•, 1)(U ) = C. Thus, π 1 (U, x 0 ) = π 1 (C, x 0 ) for x 0 ∈ C (see
Let X be a complex manifold and X be a universal covering of X. Then the group of deck transformations of the covering is identified with π 1 (X, x 0 ). The set of equivalence classes of coverings of X is identified with the conjugate classes of subgroups of π 1 (X, x 0 ); see [Mun00, Thm. 79.4, p. 492]. Furthermore, π 1 (X, x 0 ) acts transitively and freely on each fiber of the covering and X is the quotient of X by π 1 (X, x 0 ). Lemma 4.1. Let C be a compact complex manifold. Let π : C * → C be a holomorphic covering and π(

x * 0 ) = x 0 . Suppose that (M, C) (resp. (M ′ , C)) is a holo- morphic neighborhood of C. There is a neighborhood U in M (resp. U ′ in M ′ ) of C and a holomorphic neighborhood Ũ (resp. U ′ ) of C * such that p : Ũ → U (resp p : U ′ → U ′ ) is an extended covering of the covering π : C * → C and C (resp. C * ) is a smooth strong retract of U, U ′ (resp. Ũ , U ′ ). Consequently, π 1 ( Ũ , x * 0 )) = π 1 (C * , x * 0 ), π 1 (U, x 0 ) = π 1 (C, x 0 ).
Suppose that (M, C) is biholomorphic to (M ′ , C). Then U, U ′ , Ũ , U ′ can be so chosen that there is covering transformation sending Ũ onto U ′ and fixing C * pointwise. Conversely, if there is a convering transformation sending Ũ onto U ′ fixing C * pointwise, then (U, C), (U ′ , C) are holomorphically equivalent.

Proof. Since π : C * → C is a covering map, according to [Vic94, Thm. 4.9], it extends to a covering map p : Ũ → U such that Ũ contains C * and p|

C * = π. Suppose that R is a strong retraction of U onto C. We can lift R(z, •) : [0, 1] → U to a continuous mapping R(z, •) : [0, 1] → Ũ such that R(z, 0) = z and p R(z, .) = R(p(z), .
) for all z ∈ Ũ . One can verify that R is a strong retraction of Ũ onto C * . Suppose that a biholomorphic map f sends (M, C) onto (M ′ , C) fixing C pointwise. We may assume that f is a biholomorphic mapping from U onto U ′ . Then we can lift the mapping f π : Ũ → U ′ to obtain a desired covering biholomorphism

F , since (f π) * π 1 ( Ũ , x * 0 ) = π * π 1 (C * , x * 0 ). Conversely, a covering biholomorphism from Ũ onto Ũ ′ fixing C * pointwise clearly induces a biholomorphism from U onto U ′ fixing C pointwise.
With the covering, we can identity (M, C) with ( M , C * )/ ∼ where p ∼ p if and only if p, p are in the same stack of the covering π.

Applying the above to (N C , C) and a covering π| C * : C * → C, we have a covering π :

N C → N C such that C * ⊂ N C , π 1 ( N C , x * 0 ) = π 1 (C * , x * 0 ), π 1 (N C , x 0 ) = π 1 (C, x 0 ).
To simplify the notation, we denote U, Ũ by M, M respectively. Thus we have commuting diagrams for the coverings:

N C ←֓ C * ֒→ M π ↓ π C * ↓ p ↓ N C ←֓ C ֒→ M .
The set of deck transformations of p (resp. π) will be denoted by {τ 1 , . . . , τ n } (resp. {τ 1 , . . . , τn }). If π : Ũ → U is a covering map, Deck( Ũ ) denotes the set of deck transformations. Here, z = (z 1 , . . . , z n ) belongs to the fundamental domain

N C into N C * ( N C ). Furthermore, L, L ′ fix C * pointwise. We have p * π 1 (N C * ( M ), x * 0 ) = p * L -1 * π 1 ( M , x * 0 ) = π 1 (C, x 0 ) = π 1 (L -1 M , x * 0 ) = π * (L ′ ) -1 * (π 1 ( N C , x * 0 )). Both π • L ′-1 : N C * ( N C )) → M and p • L -1 : N C * ( M ) → M
ω 0 =    2n j=1 t j e j ∈ C n : t ∈ [0, 1) 2n    , e n+i := τ i , i = 1, . . . , n.
Thus h belongs to the fundamental domain Ω 0 defined

Ω 0 := {(e 2πiζ1 , . . . e 2πiζn ) : ζ ∈ ω 0 }, Ω + 0 = {(|z 1 |, . . . , |z n |) : z ∈ Ω 0 }. Thus Ω 0 is a Reinhardt domain, being {(ν 1 R 1 , . . . , ν n R n ) : |ν j | = 1 : R ∈ Ω + 0 }. We have Ω + 0 = (e -2πR1 , . . . , e -2πRn ) : R = n i=1 t i Im τ i , t ∈ [0, 1) n .
For ǫ > 0, define a (Reinhardt) neighborhood Ω ǫ of Ω 0 by

ω ǫ :=    2n j=1 t j e j : t ∈ [0, 1) n × (-ǫ, 1 + ǫ) n    , Ω ǫ := {(e 2πiζ1 , . . . e 2πiζn ) : ζ ∈ ω ǫ }.
For any ℓ-tuple of indices in {n + 1, . . . , 2n}, we set

ω j1...j ℓ ǫ :=    2n j=1 t j e j : t k ∈ (-ǫ, ǫ), k -n ∈ {j 1 , . . . , j ℓ }; t k ∈ (-ǫ, 1 + ǫ), k -n ∈ {j 1 , . . . , j ℓ }    , (4.1) ωj1...j ℓ ǫ :=    2n j=1 t j e j : t k -1 ∈ (-ǫ, ǫ), k -n ∈ {j 1 , . . . , j ℓ }; t k ∈ (-ǫ, ǫ), k -n ∈ {j 1 , . . . , j ℓ }    . (4.2)
Note that ωj1...j ℓ ǫ and ω j1...j ℓ ǫ are subset of ω ǫ , and

ω 1...n ǫ = { 2n j=1 t j e j ∈ ω ǫ : t ∈ [0, 1) n × (-ǫ, ǫ) n }. Then Ω j1...j ℓ ǫ := Ω ǫ ∩ ℓ k=1 T -1 j k Ω ǫ = {(
:= ∩ ǫ>0 Ω 1•••n 0 and Ω1•••n 0 = ∩ ǫ>0 Ω 1•••n ǫ are diffeomorphic to the real torus (S 1 ) n . We remark that T i T j maps Ω ij ǫ into Ω ǫ for i = j, while T i • T i does not map Ω ii ǫ,r into Ω ǫ .
With ∆ r = {z ∈ C : |z| < r}, we also define

(4.5) ω ǫ,r := ω ǫ × ∆ d r , Ω ǫ,r := Ω ǫ × ∆ d r .
Throughout the paper, a mapping (z ′ , v ′ ) = ψ 0 (z, v) from ω ǫ,r into C n+d that commutes with z j → z j + 1 for j = 1, . . . , n will be identified with a well-defined mapping (h ′ , v ′ ) = ψ(h, v) from Ω ǫ,r into C n+d , where z, h and z ′ , h ′ are related as in (4.8). A function on ω ǫ,r that has period 1 in all z j is identified with a function on Ω ǫ,r . We shall use these identifications as we wish.

Proposition 4.3. Let C be the complex torus and π C : C = C n /Z n → C be the covering. Let (M, C) be a neighborhood of C. Assume that N C is flat.

(i) Then one can take ω ǫ0,r0 = ω ǫ0 × ∆ d r0 such that (M, C) is biholomorphic to the quotient of ω ǫ0,r0 by τ 0 1 , . . . , τ 0 n . Let τ j be the mapping defined on Ω ǫ0,r0 corresponding to τ 0 j . Then τ 1 , . . . , τ n commute pairwise wherever they are defined, i.e.

τ i τ j (h, v) = τ j τ i (h, v) ∀i = j for (h, v) ∈ Ω ǫ0,r0 ∩ τ -1 i Ω ǫ0,r0 ∩ τ -1 j Ω ǫ0,r0 . (ii) Let ( M , C
) be another such neighborhood having the corresponding generators τ1 , . . . , τn of deck transformations defined on Ω ǫ0,r0 . Then (M, C) and ( M , C) are holomorphically equivalent if and only if there is a biholomorphic mapping F from Ω ǫ,r into Ω ǫ,r for some positive ǫ, r, ẽ, r such that

F τj (h, v) = τ j F (h, v), j = 1, . . . , n,
wherever both sides are defined, i.e. (h, v) ∈ Ω ǫ,r ∩ τ -1 j Ω ǫ,r ∩Ω ǫ,r ∩F -1 Ω ǫ,r .

Proof. We now apply Lemma 4.2, in which C * is replaced by C = R n /Z n + iR n is a Stein manifold. Assume that N C is flat. Then according to Proposition 3.6,

N C ( N C ) = N C ( M ) = π * C (N C ) is the trivial vector bundle C × C d with coordinates (h, v), while C × {0} is defined by v = 0. Set P + ǫ := n i=1 t i Im τ i : t ∈ (-ǫ, 1 + ǫ) n ,
Ω + ǫ := (e -2πR1 , . . . , e -2πRn ) : R ∈ P + ǫ .

Note that P + ǫ , P + 0 are n-dimensional parallelotopes, and Ω + ǫ contains (1, . . . , 1), the image of 0 ∈ P + ǫ , corresponding to the real torus (S 1 ) n . Since Ω ǫ is Reinhardt, we have

(4.6) Ω ǫ ⊃ Ω + ǫ , (∂Ω ǫ ) + = ∂(Ω + e ), (∂Ω ǫ ) + := {(|h 1 |, . . . , |h n |) : h ∈ ∂Ω ǫ }.
We now apply the above general results to the case where C is a complex torus, C * = C and N C (M ) is Hermitian flat.

As in [START_REF] Ju | Neighborhoods of zero type imbeddings of complex tori[END_REF], the deck transformations of ( ÑC , C) are generated by n biholomorphisms τ1 , . . . , τn that preserve C. τj (h, v) = (T j h, M j v), M j := diag(µ j,1 , . . . , µ j,d ) (4.7) with h, T j being defined by : (4.8) h = (e 2πiz1 , • • • , e 2πizn ), T j := diag(λ j,1 , . . . , λ j,n ), λ j,k := e 2πiτ jk .

Here the invertible (d× d)-matrix M j is the factor of automorphy ρ(e n+j ) of π * C N C . Recall that each deck transformation τ 0 j (z, v), j = 1, . . . , n, is a holomorphic map defined on ω ǫ,r . In coordinates (h, v), τ 0 j becomes τ j defined on Ω ǫ,r . Since T C M splits, it is a higher-order perturbation of τj with τ j (h, 0) = (T j h, 0)

The above computation is based on the assumption that N C is flat. We now assume that N C is Hermitian and flat, i.e. the N C admits locally constant Hermitian transition matrices. Then all ρ(e n+j ) are Hermitian and constant matrices. Since they pairwise commute, they are simultaneously diagonalizable. Hence, we can assume that M j = diag(µ j,1 , . . . , µ j,d ). We recall from (4.8) that T j = diag(λ j,1 , . . . , λ j,n ) are already diagonal.

Definition 4.4. The normal bundle N C is said to be non-resonant if, for each (Q, P ) ∈ N d × Z n with |Q| > 1, each i = 1, . . . , n, and each j = 1, . . . , d, there exist i h := i h (Q, P, i) and i v := i v (Q, P, j) that are in {1, . . . , n} such that

λ P i h µ Q i h -λ i h ,i
= 0 and λ P iv µ Q iv -µ iv ,j = 0. Definition 4.5. The pullback normal bundle N C , or N C , is said to be Diophantine if for all (Q, P ) ∈ N d × Z n , |Q| > 1 and all i = 1, . . . , n, and j = 1, . . . , d, max ℓ∈{1,...,n}

λ P ℓ µ Q ℓ -λ ℓ,i > D (|P | + |Q|) τ , (4.9) max ℓ∈{1,...,n} λ P ℓ µ Q ℓ -µ ℓ,j > D (|P | + |Q|) τ . (4.10)
We shall choose i h (resp. i v ) to be the index that realizes the maximum of (4.9) (resp. (4.10)).

Remark 4.6. If the right-hand sides are replaced by 0, then N C is non-resonant.

Proposition 4.7. The properties of being non-resonant and Diophantine is a property of the (abelian) group and not the choice of the generators.

Proof. Recall that λ ℓ = (λ ℓ,1 , . . . , λ ℓ,n ) = (e 2πiτ ℓ,1 , . . . , e 2πiτ ℓ,n ).

Let G be the group generated by the τℓ 's. Then, {τ ℓ } ℓ defines another set of generators of

G if τℓ = τ a ℓ,1 1 • • • τ a ℓ,n n , ℓ = 1, . . . , n where A = (a i,j ) 1≤i,j≤n ∈ GL n (Z) with det A = ±1. Then, the eigenvalues of τℓ are λℓ,i = n k=1 λ a ℓ,k k,i , μℓ,j = d k=1 µ a ℓ,k k,j .
Hence, we have λP

ℓ μQ ℓ λ-1 ℓ,i = (λ P 1 µ Q 1 λ -1 1,i ) a ℓ,1 • • • (λ P n µ Q n λ -1 n,i
) a ℓ,n . Fix P, Q and i. Taking the logarithm, we have as n-vectors

(4.11) ln λP ℓ μQ ℓ λ-1 ℓ,i ℓ=1,...,n = A ln λ P ℓ µ Q ℓ λ -1 ℓ,i ℓ=1,...,n
, mod 2πi.

Since A, A -1 ∈ GL n (Z), given P, Q and i,

ln λP ℓ μQ ℓ λ-1 ℓ,i ℓ=1,...,n = 0 mod 2πi iff ln λ P ℓ µ Q ℓ λ -1
ℓ,i ℓ=1,...,n = 0 mod 2πi. Similarly, by considering ln λP ℓ μQ ℓ μ-1 ℓ,i , we obtain that the non-resonant condition does not depend on the choice of generators. Given P, Q, i, if one of the λ P ℓ µ Q ℓ λ -1 ℓ,i 's is not close to 1, then (ln λP

ℓ μQ ℓ λ-1 ℓ,i ) ℓ is bounded way from 0. On the other hand, if all λ P ℓ µ Q ℓ λ -1 ℓ,i 's are close to 1, then | ln λ P ℓ µ Q ℓ λ -1 ℓ,i | (with Im ln in (-π, π]) is comparable to |λ P ℓ µ Q ℓ λ -1 ℓ,i -1|.
Furthermore, taking the module of (4.11), we obtain

A -1 -1 (ln λ P ℓ µ Q ℓ λ -1 ℓ,i ) ℓ ≤ (ln λP ℓ μQ ℓ λ-1 ℓ,i ) ℓ ≤ A (ln λ P ℓ µ Q ℓ λ -1 ℓ,i ) ℓ ,
where (a ℓ ) ℓ = max ℓ |a ℓ |. If the latter is bounded below by

C (|P |+|Q|) τ , so is (ln λP ℓ μQ ℓ λ-1 ℓ,i ) ℓ .
Theorem 4.8. Let C n be an n-dimensional complex torus, holomorphically embedded into a complex manifold M n+d . Assume that T C M splits. Assume the normal bundle N C has (locally constant) Hermitian transition functions. Assume that N C is Diophantine. Then some neighborhood of C is biholomorphic to a neighborhood of the zero section in the normal bundle.

Remark 4.9. When C is a product of 1-dimensional tori with normal bundle which is a direct sum of line bundles, the above result is due to Il'yashenko-Pyartli [START_REF] Ju | Neighborhoods of zero type imbeddings of complex tori[END_REF].

We have

τ j (h, v) = τj (h, v) + (τ h j (h, v), τ v j (h, v)).
Here, functions

τ • j (h, v) = Q∈N d ,|Q|≥2 τ • j,Q (h)v Q are holomorphic in (h, v) in a neighborhood of Ω ǫ,r with values in C n+d .
Definition 4.10. Set Ω ǫ,r := Ω ǫ × ∆ d r , Ωǫ,r := Ω ǫ,r ∪ n i=1 τi (Ω ǫ,r ). Denote by A ǫ,r (resp. Ãǫ,r ) the set of holomorphic functions on Ω ǫ,r (resp. Ωǫ,r ). If f ∈ A ǫ,r (resp. f ∈ Ãǫ,r ), we set

f ǫ,r := sup (h,v)∈Ωǫ,r |f (h, v)|, ||| f ||| ǫ,r := sup (h,v)∈ Ωǫ,r | f (h, v)|.
As such, each f ∈ A ǫ,r can be expressed as a convergent Taylor-Laurent series

f (h, v) = P ∈Z n f Q,P h P v Q for (h, v) ∈ Ω ǫ,r = Ω ǫ × ∆ d r .
Recall that each holomorphic function on E j in Lemma 4.11 admits a unique Taylor-Laurent series expansion on Ω1

•••n ǫ ′ × ∆ d ǫ ′ when ǫ ′ > 0 is sufficiently small.
We can state the following, for later use :

Lemma 4.11. For i = j, the set τj Ω ǫ,r ∩ τi (Ω ǫ,r ) is a connected Reinhardt domain containing Ω1...n ǫ × ∆ d r ′ in C n+d when r ′ > 0 is sufficiently small. 4.1. Holomorphic functions on Ω ǫ,r . In this section, we study elementary properties and estimate holomorphic functions f on Ω ǫ,r . Lemma 4.12. An element f (h) of A ǫ , that is a holomorphic function in a neighborhood of Ω ǫ , admits a Laurent series expansion in h (4.12)

f (h) = P ∈Z n c P h P .
The series converges normally on Ω ǫ . Moreover, the Laurent coefficients

f (h) = 1 (2πi) n ∂A(|h1|-ǫ,|h1|+ǫ) • • • ∂A(|hn|-ǫ,|hn|+ǫ) f (ζ) dζ 1 . . . dζ n (ζ 1 -h 1 ) • • • (ζ n -h n )
.

By Laurent expansion in one-variable, we get the expansion (4.12) in which c P are given by (4.13) if we take r j = |h j | ± ǫ according to the sign of p j ∈ Z ∓ . We want to show that c P is independent of s ∈ Ω + ǫ . Note that Ω + ǫ is a connected open set. For any two points s, s can be connected by a union of line segments in Ω Finally the series converges uniformly on each compact subset of Ω ǫ . Indeed, for a small perturbation of h, we can choose ǫ(h) to be independent of h. Then we can see easily that the series converges locally uniform in h.

Recall that

P + ǫ = n i=1 t i Im τ i ∈ R n : t ∈ (-ǫ, 1 + ǫ) n , Ω + ǫ = (e -2πR1 , . . . , e -2πRn ) : R ∈ P + ǫ .
Lemma 4.13. There is a constant κ 0 > 0 that depends only on Im τ 1 , . . . , Im τ n such that if P ∈ R n and ǫ > ǫ ′ , there exists R ∈ P + ǫ such that for all R ′ ∈ P + ǫ ′ we have

(4.15) (R ′ -R) • P ≤ -κ 0 (ǫ -ǫ ′ )|P |.
Proof. Without loss of generality, we may assume that P is a unit vector. Let π(x)P be the orthogonal projection from x ∈ R n onto the line spanned by P . Choose R ∈ P + ǫ so that π(R) has the largest value for R ∈ P + ǫ . Note that R must be on the boundary of P + ǫ and latter is contained in the half-space H defined by π(y) ≤ π(R) for y ∈ R n . Hence, ∂H is orthogonal to P . Then for any

R ′ ∈ P + ǫ ′ , π(R) -π(R ′ ) = dist(R ′ , ∂H) ≥ dist(∂P + ǫ , P + ǫ ′ ) ≥ (ǫ -ǫ ′ )/C. Therefore, we obtain (R ′ -R) • P = -(π(R) -π(R ′ )) ≤ -(ǫ -ǫ ′ )/C.
Remark 4.14. Since P + ǫ is a parallelotope, we can choose R to be a vertex of

P + ǫ .
In what follows, we denote the fixed constant :

(4.16) κ := 2πκ 0 .

Lemma 4.15 (Cauchy estimates). If f ∈ A ǫ,r , then for all

(P, Q) ∈ Z n × N d , (4.17) |f Q,P | ≤ M r |Q| max s∈Ω + ǫ s P . Furthermore, if f = Q∈N d ,P ∈Z n f QP h P v Q ∈ A ǫ,r and 0 < δ < κǫ, then f ǫ-δ/κ,re -δ ≤ C sup Ωǫ,r |f | δ ν ,
where C and ν depends only on n and d.

Proof. According to Lemma 4.12 and Cauchy estimates on polydiscs, we have for

any s ∈ Ω + ǫ , |f Q,P | ≤ sup Ωǫ |f Q (h)| s P ≤ sup Ωǫ,r |f | s P r |Q| .
According to Lemma 4.12 and Cauchy estimates for polydiscs, we have if

(h, v) ∈ Ω ǫe -δ ′ ,re -δ , then for all s ∈ Ω + ǫ , |f Q (h)| ≤ sup v∈∆ d r |f (h, v)| r |Q| , |f Q,P h P | ≤ 1 (2πi) n |ζ1|=s1,...,|ζn|=sn f Q (ζ) h P ζ P dζ 1 ∧ • • • ∧ dζ n ζ 1 • • • ζ n . Set s j = e -2πRj , |h j | = e -2πR ′ j , R = (R 1 , . . . , R n ) and R ′ = (R ′ 1 , . . . , R ′ n ). By Lemma 4.13, (4.18) inf (|ζ1|,...,|ζn|)=s∈Ω + ǫ sup h∈Ω ǫ-δ ′ h P ζ P = inf R∈P + ǫ sup R ′ ∈P + ǫ-δ ′ e -2π<R-R ′ ,P > ≤ e -κδ ′ |P | ,
where the positive constant κ, defined by Lemma 4.13 and (4.16), is independent of

P, ζ 1 , ζ 2 . Thus (4.19) |f Q,P h P v Q | ≤ sup Ωǫ,r |f |e -κδ ′ |P | r |Q| e -δ|Q| r |Q| ≤ sup Ωǫ,r |f |e -δ ′ κ|P | e -δ|Q| .
Hence, setting δ ′ := δ/κ, we have

f ǫ-δ/κ,re -δ ≤ Q∈N d ,P ∈Z n |f Q,P h P v Q | ≤ C sup Ωǫ,r |f | δ ν ,
where C and ν depends only on n and d.

Conjugacy of the deck transformations.

Let us show that there is a biholomorphism Φ = (h, v) + φ(h, v) of some neighborhood Ω ǫ,r , fixing C pointwise (i.e. Φ(h, 0) = (h, 0)) such that

Φ • τi = τ i • Φ, i = 1, . . . , n. This reads τi + φ(τ i ) = τi (Id + φ) + τ • i (Id + φ), that is, for all i = 1, . . . n, (4.20) L i (φ) = τ • i (Id + φ) + (τ i (Id + φ) -τi -Dτ i .φ) = τ • i (Id + φ). Here we define L i (φ) := φ(τ i ) -Dτ i .φ and (L h i (φ h ), L v i (φ v )) := L i (φ). We have (L h i (φ h ), L v i (φ v )) = φ h (T i h, M i v) -T i φ h (h, v), φ v (T i h, M i v) -M i φ v (h, v
) . Since τj are linear, then Dτ j = τj . Expand the latter in Taylor-Laurent expansions as

L h i (φ h ) = Q∈N d 2 P ∈Z n λ P i µ Q i × Id n -T i φ h Q,P h P v Q , φ h Q,P ∈ C n , L v i (φ v ) = Q∈N d 2 P ∈Z n λ P i µ Q i × Id d -M i φ v Q,P h P v Q , φ v Q,P ∈ C d .
Recall the notations λ ℓ = (λ ℓ,1 , . . . , λ ℓ,n ) and µ ℓ = (µ ℓ,1 , . . . , µ ℓ,d ). With P = (p 1 , . . . , p n ) ∈ Z n and Q = (q 1 , . . . , q d ) ∈ N d , we have

λ P ℓ µ Q ℓ := n i=1 λ pi ℓ,i n j=1 µ qj ℓ,j . Lemma 4.16. Let τ j ∈ A n+d ǫ,r with τ j = τj -F j and F j (h, v) = O(|v| q+1 ) with q ≥ 1. Suppose that τ i τ j = τ j τ i in a neighborhood of Ω 1•••n 0 × {0} in C n+d . Then L i F j -L i F i = O(|v| 2q+1 ).
Proof. Recall that τi (h, v) are linear maps in h, v. Also, τi τj sends Ω j ǫ × ∆ d ǫ into C n+d . Since τ j ∈ A n+d ǫ,r and τ j = τj + O(|v| 2 ), the continuity implies that τ i τ j is well-defined on the product domain Ω j ǫ ′ × ∆ d r ′ when ǫ ′ , r ′ are sufficiently small. Fix h ∈ Ω j ǫ ′ . By Taylor expansions in v, we obtain

τ i τ j (h, v) = τi τj (h, v) + τj F j (h, v) + F i • τj (h, v) + O(|v| 2q+1 ).
Since τi τj = τj τi and

τ i τ j = τ j τ i in a neighborhood of Ω 1•••n 0 × {0}, we get L i F j = L j F i = O(|v| 2q+1 ) in a possibly smaller neighborhood of Ω 1•••n 0 × {0}.
We will apply the following result to F j = τj -J 2q (τ j ), where J 2q (τ j ) denotes the 2q-jet at 0 in the variable v. These are holomorphic on Ω ǫ,r . Recall that

Ω ij ǫ ′ ,r ′ := Ω ǫ ′ ,r ′ ∩ τ -1 i (Ω ǫ ′ ,r ′ ) ∩ τ -1 j (Ω ǫ ′ ,r ′ ) Proposition 4.17. Assume N C is Diophantine. Fix ǫ 0 , r 0 , δ 0 in (0, 1). Let 0 < ǫ ′ < ǫ < ǫ 0 , 0 < r ′ < r < r 0 , 0 < δ < δ 0 , and δ κ < ǫ. Suppose that F i ∈ A ǫ,r , i = 1, . . . , n, satisfy (4.21) L i (F j ) -L j (F i ) = 0 on Ω ij ǫ ′ ,r ′ . There exist functions G ∈ Ãǫ-δ/κ,re -δ such that L i (G) = F i on Ω ǫ-δ/κ,re -δ . (4.22) Furthermore, G satisfies G ǫ-δ/κ,re -δ ≤ max i F i ǫ,r C ′ δ τ +ν , (4.23) G • τi ǫ-δ/κ,re -δ ≤ max i F i ǫ,r C ′ δ τ +ν . (4.24)
for some constant κ, C ′ that are independent of F, q, δ, r, ǫ and ν that depends only on n and d. Furthermore, if F j (h, v) = O(|v| q+1 ) for all j, then

(4.25) G(h, v) = O(|v| q+1 ), G(h, v) = J 2q G(h, v).
Proof. Since F i ∈ A ǫ,r , we can write

F i (h, v) = Q∈N d ,|Q|≥2 P ∈Z n F i,Q,P h P v Q ,
which converges normally for (h, v) ∈ Ω ǫ,r . We emphasize that F i,Q,P are vectors, and its kth component is denoted by F i,k,Q,P . For each (Q, P ) ∈ N d × Z n , each i = 1, . . . , n, and each j = 1, . . . , d, let i h := i h (Q, P, i), i v := i v (Q, P, j) be in {1, . . . , n} as in Definition 4.4. Let us set

G h i := Q∈N d ,2≤|Q|≤2q P ∈Z F h i h ,i,Q,P λ P i h µ Q i h -λ i h ,i h P v Q , i = 1, . . . , n (4.26) G v j := Q∈N d ,2≤|Q|≤2q P ∈Z F v iv ,j,Q,P λ P iv λ Q iv -µ iv ,j h P v Q , j = 1, . . . .d. (4.27)
According to (4.21), we have (4.28)

(λ P i h µ Q i h -λ i h ,i )F h m,i,Q,P = (λ P m µ Q m -λ m,i )F h i h ,i,Q,P , 2 ≤ |Q| ≤ 2q. Therefore, using (4.28), the ith-component of L m (G) reads L m (G h i ) = Q∈N d ,2≤|Q|≤2q P ∈Z n (λ P m µ Q m -λ m,i ) F h i h ,i,Q,P (λ P i h µ Q i h -λ i h ,i ) h P v Q = Q∈N d ,2≤|Q|≤2q P ∈Z n F h m,i,Q,P h P v Q .
Proceeding similarly for the vertical component, we have obtained, the formal equality :

(4.29) L m (G) = F m , m = 1, . . . ,
|G h i,Q,P |, |G v j,Q,P | ≤ max i |F i,Q,P | (|P | + |Q|) τ D .
Let (h, v) ∈ Ω ǫ-δ/κ,re -δ . According to (4.19), we have

G h Q,P h P v Q ≤ max i F i ǫ,r e -δ(|P |+|Q|) (|P | + |Q|) τ D ≤ max i F i ǫ,r e -δ/2(|P |+|Q|) (2τ e) τ Dδ τ .
Summing over P and Q, we obtain

G ǫ-δ/κ,re -δ ≤ max i F i ǫ,r C ′ δ τ +ν ,
for some constants C ′ , ν that are independent of F, ǫ, δ. Hence, G ∈ A ǫ-δ/κ,re -δ .

Let us prove (4.24). Let B := 2 max k,i,j (|λ k,i |, |µ k,j |). Then, there is a constant D ′ such that max ℓ∈{1,...,n}

λ P ℓ µ Q ℓ -λ ℓ,i ≥ D ′ max k |λ P k µ Q k | (|P | + [Q|) τ , (4.31) max ℓ∈{1,...,n} λ P ℓ µ Q ℓ -µ ℓ,j ≥ D ′ max k |λ P k µ Q k | (|P | + [Q|) τ . (4.32) Indeed, if max k |λ P k µ Q k | < B,
| := max k |λ P k µ Q k | ≥ B, then |λ k0,i | ≤ B 2 ≤ |λ P k 0 µ Q k 0 | 2
. Hence, we have

λ P k0 µ Q k0 -λ k0,i ≥ |λ P k0 µ Q k0 | -|λ k0,i | ≥ |λ P k0 µ Q k0 | 2 .
We have verified (4.31). Similarly, we can verified (4.32). Finally, combining all cases gives us, for m = 1, . . . , n,

|[G • τm ] QP | = G Q,P λ P m µ Q m ≤ max ℓ |F ℓ,Q,P | |λ P m µ Q m | |λ P i h µ Q i h -λ i h ,i | ≤ max ℓ |F ℓ,Q,P | |λ P m µ Q m |(|P | + |Q|) τ D ′ max k |λ P k µ Q k | ≤ max ℓ |F ℓ,Q,P | (|P | + |Q|) τ D ′ .
Hence, Gm := G • τm ∈ A ǫ-δ/κ,re -δ . We can define G ∈ Ãǫ-δ/κ,re -δ such that G = Gm τ -1 m on τm Ω ǫ,r . We verify that G extends to a single-valued holomorphic function of class Ãǫ,r . Indeed, Gi τ -1 i = Gj τ -1 j on τi Ω ǫ,r ∩ τj Ω ǫ,r , since the latter is connected by Lemma 4.11 and the two functions agree with G on τi Ω ǫ,r ∩τ j Ω ǫ,r ∩Ω ǫ,r that contains a neighborhood of Ωǫ × {0} in C n+d .

In what follows, we shall set τ0 = Id and for any f ∈ ( Ãǫ,r ) n+d and F ∈ ( Ãǫ,r ) n+d , we set

|||f ||| ǫ,r := f ǫ,r + n i=1 τ -1 i f • τi ǫ,r = n i=0 τ -1 i f • τi ǫ,r , F (i) := τ -1 i F • τi ∈ A ǫ,r , F (0) := F, i = 1, . . . , n.
4.3. Iteration scheme. We shall prove the main result through a Newton scheme.

Let us define sequences of positive real numbers

δ k := δ 0 (k + 1) 2 , r k+1 := r k e -5δ k , ǫ k+1 := ǫ k - 5δ k κ such that (4.33) σ := k≥0 δ k < 2δ 0
We assume the following conditions hold : Let m = 5 be fixed. We define ǫ k+1 < ǫ

δ 0 < κ 20
(ℓ) k < ǫ k and r k+1 < r (ℓ) k < r k , ℓ = 1, . . . m as follows : ǫ (ℓ) k = ǫ k - ℓδ k κ , ǫ k+1 := ǫ (m) k , r (ℓ)
k = r k e -ℓδ k , r k+1 := r (m) k .
We emphasize that condition (4.34) ensures ǫ (ℓ) k > 0. Let us assume that for each i = 1, . . . , n, τ

(k) i = τi + τ •(k) i , is holomorphic and on Ω ǫ k ,r k it satisfies (4.36) ||τ •(k) || ǫ k ,r k < δ µ k , τ •(k) (h, v) = O(|v| q k +1 ).
We further assume that τ

(k) i , τ (k) j commute on Ω ij ǫ ′ ,r ′ ⊂ Ω ǫ k ,
r k for some positive ǫ ′ < ǫ k , r ′ < r k . We take q k+1 = 2q k + 1 ≥ q 0 2 k for q 0 ≥ 1 to be determined. We will define a sequence Φ (k) with Φ (k) (h, 0) = (h, 0). Let us write on appropriate domains

Φ (k) = Id + φ (k) , (Φ (k) ) -1 := Id -ψ (k) , τ (k+1) i = Φ (k) • τ (k) i • (Φ (k) ) -1 , i = 1, . . . , n.
Note that there is a constant C > 0 (depending only on the µ i,j 's) such that if

ǫ ′′ < ǫ ′ , Cr ′′ < r ′ then Ω ij ǫ ′′ ,r ′′ ⊂ Ω ǫ ′ × ∆ d r ′ , Ω ǫ ′′ × ∆ d r ′′ ⊂ Ω ij ǫ ′ ,r ′ . Since the τ (k) i , τ (k) j commute on Ω ǫ ′ × ∆ d ǫ ′ for some ǫ ′ > 0 and Φ (k) (h, v) = (h, v) + O(|v| 2 ), then τ (k+1) i , τ (
k+1) j still commute on the same kind of domains for a possibly smaller ǫ ′ . By Lemma 4.16, we obtain :

L j (τ •(k) i ) -L i (τ •(k) j ) =O(|v| 2q k +1 ). (4.37)
We want to find φ (k) so that

τ (k+1) i := τi + τ •(k+1) i = Φ (k) • τ (k) i • (Φ (k) ) -1 (4.38) = τi (Id -ψ (k) ) + τ •(k) i (Id -ψ (k) ) + φ (k) τi (Id -ψ (k) ) + τ •(k) i (Id -ψ (k) )
is defined and bounded on Ω ǫ k+1 ,r k+1 .

We now define Φ (k) by applying Proposition 4.17 with these F i := -J 2q k (τ

•(k) i
). Let φ (k) stands for G. Therefore, given 0 < δ < κǫ

(1) k , φ (k) is holomorphic and bounded on Ωǫ k -δ κ ,r k e -δ and it satisfies on Ω ǫ (1)

k -δ κ ,r (1)
k e -δ , (4.39)

L i (φ (k) ) := φ (k) (τ i ) -Dτ i .φ (k) = -J 2q k (τ •(k) i ), i = 1, . . . , n.
Writing formally Φ (k) • Ψ (k) = Id and using linearity of τi , we obtain

ψ (k) (i) = φ (k) (i) (Id -ψ (k) (i) ), i = 0, . . . , n, (4.40) recalling the notation φ (k) (i) = τ -1 i φ (k) • τi .
According to Proposition 4.17, we have

(4.41) |||φ (k) ||| ǫ k -δ κ ,r k e -δ ≤ ||τ •(k) || ǫ k ,r k C ′ δ τ +ν ≤ C ′ δ µ k δ -τ -ν .
We recall that the constatn C ′ does not depend on k, nor on τ •(k) .

Lemma 4.18. There is constant D > 0 (independent of k) such that, given a positive number µ, so that δ < min{ D, ln 2, r0 2 } satisfies (4.42) 2 m+2 C ′ δ µ k δ -τ -ν-3 < 1, where m = 5. Then, for all 0 ≤ ℓ ≤ m and i = 0, . . . , n, the maps Φ

(k) (i) := Id + φ (k) (i) are biholomorphisms Φ (k) (i) : Ω ǫ k -(ℓ+1)δ κ ,r k e -(ℓ+1)δ → Ω ǫ k -ℓδ 2κ ,r k e -ℓδ 2 with inverse Ψ (k) (i) := Id -ψ (k) (i) : Ω ǫ k -(ℓ+2)δ κ ,r k e -(ℓ+2)δ → Ω ǫ k -(ℓ+1)δ κ ,r k e -(ℓ+1)δ satis- fying Φ (k) (i) • Ψ (k) (i) = τ -1 i (Φ (k) • Ψ (k) ) • τi = Id on Ω ǫ k -(ℓ+2)δ κ ,r k e -(ℓ+2)δ , provided q 0 > C(δ 0 , µ).
Proof. We have 1 -e -δ > δ/2 and 1 2 < e -δ for 0 < δ < ln 2. Assuming δ < r0 2 , we have δ < r k so that

dist(∆ d r k e -(1+ℓ)δ , ∂∆ d r k e -ℓδ ) = r k e -ℓδ (1 -e -δ ) > δ 2 2 ℓ+1
. On the other hand, by (4.6), we have

dist(Ω ǫ k -(ℓ+1)δ κ , ∂Ω ǫ k -ℓδ κ ) = dist(Ω + ǫ k -(ℓ+1)δ κ , ∂Ω + ǫ k -ℓδ κ ). Let (e -2πR , e -2πR ′ ) ∈ Ω + ǫ k -(ℓ+1)δ κ × ∂Ω + ǫ k -ℓδ κ . Since the matrix (Im τ j i ) 1≤i,j≤n is invertible, there is a constant C (independent of k) such that |e -2πR -e -2πR ′ | ≥ C-1 1 -e -2π(R-R ′ ) > 2π C-1 |R -R ′ | ≥ 2π C-1 δ κ . Let us set D := 2 2π Cκ . Assuming δ < D, we have δ < D < 2 ℓ+2 D ≤ 2 m+2 D. Assume 2 m+2 C ′ δ µ k δ -τ -ν-3 < 1. Then according to (4.41), we have, for (h, v) ∈ Ω ǫ k -ℓδ κ ,r k e -ℓδ , |φ (k)
(i) (h, v)| ≤ C ′ δ µ k δ -τ -ν < δ 3 2 m+2 < δ 2 δ 2 2 ℓ+1 (4.43) < δ 2 dist(Ω ǫ k -(ℓ+1)δ κ ,r k e -(ℓ+1)δ , ∂Ω ǫ k -ℓδ κ ,r k e -ℓδ ). By the Cauchy inequality, we have, for (h, v) ∈ Ω ǫ k -(ℓ+2)δ κ ,r k e -(ℓ+2)δ (4.44) |Dφ (k) (i) (h, v)| ≤ C ′ δ µ k δ -τ -ν dist(Ω ǫ k -(ℓ+2)δ κ ,r k e -(ℓ+2)δ , ∂Ω ǫ k -(ℓ+1)δ κ ,r k e -(ℓ+1)δ ) ≤ δ 2 < 1/2.
We can apply the contraction mapping theorem to (4.40) together with the last inequality of (4.43). We find a holomorphic solution ψ

(k) (i) such that for (h, v) ∈ Ω ǫ k -(ℓ+3)δ κ ,r k e -(ℓ+3)δ |ψ (k) (i) (h, v)| ≤ ||φ (k) (i) || ǫ k -δ(Corrected)(ℓ+2) κ ,r k e -(ℓ+2)δ ≤ C ′ δ µ k δ -τ -ν ≤ δ 3 2 m+2 (4.45) < δ 2 m-ℓ-1 dist(Ω ǫ k -(ℓ+3)δ κ ,r k e -(ℓ+3)δ , ∂Ω ǫ k -(ℓ+2)δ κ ,r k e -(ℓ+2)δ ).
Hence, we have found a mapping Ψ We recall that δ 0 satisfies (4.34) and (4.35). Setting δ := δ k and if q 0 large enough, we have max ǫ k -4δ/κ,r k e -4δ ≤ C. On the other hand, on Ω ǫ k -4δ/κ,r k e -4δ/κ , we have τ

(k) (i) := Id -ψ (k) (i) such that Ωǫ k -(ℓ+3)δ
•(k+1) i = τi (φ (k) -ψ (k) ) + τ •(k) i (Id -ψ (k) ) -τ •(k) i (4.47) + φ (k) τi -τi ψ (k) + τ •(k) i (Id -ψ (k) ) -φ (k) (τ i ) + (φ (k) (τ i ) -τi φ (k) + τ •(k) i ).
The last term is equal to τ

•(k) i -J 2q k (τ •(k) i ) = O(|v| 2q k +1
). We also have φ (k)ψ (k) = O(|v| 2q k +1 ), φ (k) = O(|v| q k +1 ). Thus Improving the estimate by the Schwarz inequality, we obtain τ

•(k+1) i ǫ k -5δ
κ ,r k e -5δ ≤ Ce -q k+1 δ . We have e -x < 1/x for x > 0. For δ := δ k < min{ D, ln 2, r0 2 }, let µ satisfy 2 m+2 C ′ δ µ-τ -ν-3 k < 1, in which case assumptions of Lemma 4.18 are satisfied. We then obtain Cδ -µ k+1 e -q k+1 δ ≤ C (k + 2) 2 δ 0 µ+1 1 q 0 2 k < 1 provided q 0 > C(δ 0 , µ).

Finally, quite classically, since (4.44) and (4.33), the sequence of diffeomorphisms

{Φ k • Φ k-1 • • • • • Φ 1 } k converges uniformly
on the open set Ω ǫ 0 2 ,r0e -σ to a diffeomorphism Φ which satisfies Φ • τi = τ i • Φ, i = 1, . . . , n provided q 0 ≥ C(δ 0 , µ) and ||τ •(0) i || ǫ0,r0 ≤ δ µ 0 for some δ 0 , µ are fixed. The condition q 0 > C(δ 0 , µ, τ, ν) can be achieved by using finitely many Φ 0 , . . . , Φ m . Then initial condition ||τ •(0) i || ǫ0,r0 ≤ δ µ 0 can be achieved easily by a dilation in the v variable. Indeed, we apply the dilation in v to the original τ 1 , . . . , τ n with q 0 = 1. This allows us to construct Φ 0 in Proposition 4.19 with k = 0 and define Φ 0 τ j Φ -1 0 to achieve q 1 ≥ 2. Then Φ 0 τ j Φ -1 0 , j = 1, . . . , n still commute pairwise on Ω ǫ ′ × ∆ d ǫ ′ for some ǫ ′ > 0. Applying the procedure again, this allows us to find Φ 1 , . . . , Φ k-1 to achieve q k ≥ 2 k > C(δ 0 , µ). Finally using dilation we can apply the full version of Proposition 4.19 for all k to construction a new sequence of desired mapping Φ 0 , Φ 1 , . . . . Hence, the torus C has a neighborhood in M biholomorphic to a neighborhood of the zero section in its normal bundle N C since there is a biholomorphism fixing C that conjugate the deck transformations of the covering of the latter to those of the former.

Remark 4.20. The assumption "T C M splits" can be replaced by the condition that for all P ∈ Z n , for all Q ∈ N d , |Q| = 1 and for all i = 1, . . . , n max ℓ∈{1,...,n}

λ P ℓ µ Q ℓ -λ ℓ,i > D (|P | + 1) τ .

Lemma 4. 2 .

 2 Let C, C * , M be as in Lemma 4.1. Suppose that C * is a Stein manifold. Let ω * 0 be an open set of N C such that π(ω * ) contains C. Then there is an open subset ω * of ω * 0 such that πω * contains C and (M, C) is holomorphically equivalent to the quotient space of ω * by Deck( N C ). Proof. By a result of Siu [Siu77, Cor. 1], we find a biholomorphism L from a holomorphic strong retraction neighborhood of C * in M , still denoted by M into N C * ( M ) and a biholomorphism L ′ from a strong retraction neighborhood of C * in N C , still denoted by

  are coverings and the above identifications show that the lifts of the two coverings yield a biholomorphism between neighborhoods of C * in M and N C * ( N C ) fixing C * pointwise.

T

  e 2πiζ1 , . . . e 2πiζn ) : ζ ∈ ω j1...j ℓ j k Ω ǫ = {(e 2πiζ1 , . . . e 2πiζn ) : ζ ∈ ωj1...j ℓ } (4.4) are connected non-empty Reinhardt domains. Moreover, Ω 1•••n 0

  n |ζ1|=s1,...,|ζn|=sn f (ζ)ζ -P -(1,...,1) dζ 1 ∧ • • • ∧ dζ n are independent of s ∈ Ω + ǫ and (4.14) |c P | ≤ sup Ωǫ |f | inf s∈Ω + ǫ s -P . Proof. Obviously, estimate (4.14) follows from (4.13). Define A(a, b) = {w ∈ C : a < |w| < b}. Fix h ∈ Ω ǫ . Then |h| := (|h 1 |, . . . , |h n |) ∈ Ω + ǫ . The latter is an open set and we have for a small positive number ǫ = ǫ(h),

  + ǫ which are parallel to coordinate axes in R n . Using such line segments, say a line segment [a, b] × (s 2 , . . . , s n ) in Ω + ǫ , we know that f (ζ)ζ -P -(1,...,1) is holomorphic in ζ 1 in the closure of A(a, b) when |ζ 2 | = s 2 , . . . , |ζ n | = s n . By Cauchy theorem, the integrals are independent of s 1 ∈ [a, b] for these s 2 , . . . , s n . This shows that (4.13) is independent of s.

κ)

  ,r k e -(ℓ+3)δ := Ψ(k) (i) (Ω ǫ k -(ℓ+3)δ κ ,r k e -(ℓ+3)δ ) ⊂ Ω ǫ k -(ℓ+2)δ κ ,r k e -(ℓ+2)δ . = Id on Ω ǫ k -(ℓ+3)δ κ ,r k e -(ℓ+3)δ . Therefore, Φ (k) (i) : Ωǫ k -(ℓ+3)δ κ ,r k e -(ℓ+3)δ → Ω ǫ k -(ℓ+1)δ κ ,r k e -(ℓ+1)δ is an (onto) biholomorphism such that (Φ (k) (i) ) -1 = Ψ (k) (i) .Proposition 4.19. Keep conditions on δ, µ in Lemma 4.18 as well as conditions (4.34),(4.35). If δ 0 small enough there is possibly larger µ > 0 such that if for all i = 1, . . . , n, τ(0) i ∈ (A ǫ0,r0 ) n+d with |||τ •(0) i ||| ǫ0,r0 ≤ δ µ 0, then for all k ≥ 0 we have the following :τ •(k+1) i ∈ (A ǫ k+1 ,r k+1 ) n+d , τ •(k+1) i = O(|v| 2q k +1 ), ||τ •(k+1) i || ǫ k+1 ,r k+1 ≤ δ µ k+1 . (4.46) Proof. Let us first show that τ (k+1) i := Φ (k) • τ (k) i • (Φ (k) ) -1 is well defined on Ω ǫ k -ℓδ κ ,r k e -ℓδfor m ≥ ℓ ≥ 5 and all i = 1, . . . , n. Here m is fixed from Lemma 4.18. Indeed, we haveτ • (Φ (k) ) -1 . Since τ •(k) i is of order ≥ 2q k-1 + 1, Schwarz inequality gives τ -1 i τ •(k) i ǫ k -(ℓ-1)δ κ ,r k e -(ℓ-1)δ ≤ max i τ -1 i ǫ0,r0 Ce -(2q k-1 +1)δ τ •(k) i ǫ k -(ℓ-2)δ κ,r k e -(ℓ-2)δ .

  Ce -(2q k-1 +1)δ ≤ max i 2 q0δ0 < 1.According to Lemma 4.18, (4.42) and the distance estimate in (4.43), we have(I + τ -1 i τ •(k) i ) • (Φ (k) ) -1 (Ω ǫ k -ℓδ κ ,r k e -ℓδ ) ⊂ Ω ǫ k -(ℓ-3)δ κ ,r k e -(ℓ-3)δ so that τ (k+1) i is defined on Ω ǫ k -ℓδ κ ,r k e -ℓδ for ℓ = 4 ≤ m since φ (k) (i) is defined on Ω ǫ k -δ/κ,r k e -δ/κ . For the rest of the proof, we fix ℓ = 4. From the argument above, we haveτ (k+1) i (Ω ǫ k -4δ/κ,r k e -4δ/κ ) ⊂ τi (Φ k (i) (Ω ǫ k -δ/κ,r k e -δ/κ )) ⊂ τi (Ω ǫ k ,r k ) ⊂ τi (Ω ǫ0,r0 ).Hence, τ •(k+1) i is uniformly bounded on Ω ǫ k -4δ/κ,r k e -4δ/κ w.r.t k : τ •(k+1) i

=

  O(|v| 2q k +1 ).

  n.

	Let us estimate these solutions. According to Definition 4.5 and formulas (4.26)-
	(4.27), we have
	(4.30)	max i,j

† Partially supported by a grant from the Simons Foundation (award number: 505027) and NSF grant DMS-2054989. † † Research of L. Stolovitch has been supported by the French government through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01.