FORMAL CLASSIFICATION OF TWO-DIMENSIONAL NEIGHBORHOODS OF GENUS g > 2 CURVES WITH

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ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du

Olivier Thom

Formal classification of two-dimensional neighborhoods of genusg>2curves with trivial normal bundle

Tome 70, n^o5 (2020), p. 1825-1846.

<http://aif.centre-mersenne.org/item/AIF_2020__70_5_1825_0>

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

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FORMAL CLASSIFICATION OF TWO-DIMENSIONAL NEIGHBORHOODS OF GENUS g > 2 CURVES WITH

TRIVIAL NORMAL BUNDLE

by Olivier THOM (*)

Abstract. —In this paper we study the formal classification of two-dimensional neighborhoods of genus g > 2 curves with trivial normal bundle. We first construct formal foliations on such neighborhoods with holonomy vanishing along many loops, then give the formal / analytic classification of neighborhoods equipped with two foliations, and finally put this together to obtain a description of the space of neighborhoods up to formal equivalence.

Résumé. —On étudie dans ce papier la classification formelle des voisinages de dimension deux de courbes de genreg>2 dont le fibré normal est trivial. On construit tout d’abord sur de tels voisinages des feuilletages formels dont l’holono- mie s’annule le long de nombreux lacets, puis on donne la classification formelle / analytique des voisinages équipés de deux feuilletages, et finalement on rassemble tout cela pour obtenir une description de l’espace des voisinages modulo équiva- lence formelle.

1. Introduction 1.1. General setting

LetC be a complex curve of genusg. We are interested in the different 2-dimensional neighborhoods S of C. More precisely, two surfaces S, S⁰ equipped with embeddingsC ,→S,C ,→S⁰ define formally / analytically equivalent neighborhoods if there exists neighborhoodsU, U⁰ofCinS and S⁰and a formal / analytic diffeomorphismϕ:U →U⁰inducing the identity onC. The equivalence of two neighborhoods is thus given by diagrams

Keywords:formal neighborhoods, foliations.

2020Mathematics Subject Classification:32Q57, 32G12, 37F75.

(*) The author has been supported by CNRS, ANR-16-CE40-0008 Foliage.

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C U ⊂S

C U⁰⊂S⁰

id ϕ

We want to understand the classification of such neighborhoods up to equivalence.

The first invariants in this problem are the normal bundle N_C of C in S and the self-intersection C·C = deg(NC) of the curve C. If C · C < 0, Grauert’s theorem (cf. [5] or [2]) tells that if the self-intersection is sufficiently negative (more precisely, if C·C < 2(2−2g)), then S is analytically equivalent toNC(i.e. a neighborhood ofCinS is analytically equivalent to a neighborhood of the zero section in the total space ofNC).

In the caseC·C >0, we can cite the works of Ilyashenko [6] on strictly positive neighborhood of elliptic curves and of Mishustin [9] for neighborhoods of genusg>2 curves with large self-intersection (C·C >2g−2). In both cases, the authors show that there is a huge family of non-equivalent neighborhoods (there are some functional invariants).

In the case C·C= 0, the neighborhoods of elliptic curves have already been studied. Arnol’d showed in [1] that ifSis a neighborhood of an elliptic curve whose normal bundle NC is not torsion, S is formally equivalent to N_C; if moreover N_C satisfies some diophantine condition, then S is analytically equivalent toNC. The case whenCis an elliptic curve andNC

is torsion was studied in [8]; in particular, it is shown that the formal moduli space (i.e. with respect to formal classification) of such neighborhoods is a countable union of finite dimensional spaces.

The goal of this paper is to study the neighborhoods of genus g > 2 curves with trivial normal bundle under formal equivalence.

1.2. Notations

Throughout this paper, we will use the term Diff(C,0) to denote the group of germs of analytic diffeomorphisms ofCat 0; we will writeDiff(y C,0) the group of formal diffeomorphisms ofCat 0.

A formal neighborhood Sp of C is a scheme X = (X,O) withp C as a subscheme such that there is an open covering X = ∪Ui of X with O|p_U_i = (O_C|_U_i)[[y_i]], some coordinatesx_i onC∩U_i and some holomorphic functionsu^(k)_ji withyj =P

k>1u^(k)_ji (xi)y_i^kandu⁽¹⁾_ji not vanishing onUi∩Uj.

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IfSis an analytic neighborhood ofC, then the completionOpofOS along C is the structure sheaf of a formal neighborhood Sp of C. The natural inclusion O_S ,→ Op gives an injection S ,p → S and allows us to see S as a formal neighborhood. We say that two analytic neighborhoodsS, S⁰ are formally equivalent ifSpandSp⁰ are equivalent.

Let S =S

Ui be a covering of an analytic neighborhoodS and (ui, vi) some analytic coordinates onUiwithC∩Ui={vi= 0}. Aregular analytic foliationonS havingC as a leaf can be seen as a collection of submersive analytic functions y_i = P

k>1y^(k)_i (u_i)v^k_i on each U_i such that there exist some diffeomorphismsϕji ∈ Diff(C,0) with yj = ϕji◦yi on Ui∩Uj. In analogy, a regular formal foliationon S aroundC (or on a formal neigh- borhoodSpofC) is a collection of formal power seriesyi=P

k>1y_i^(k)(ui)v^k_i withyj =ϕji◦yi for someϕji∈Diff(y C,0) where the coefficientsy_i^(k)(ui) are still analytic functions onC∩Ui andy⁽¹⁾_i does not vanish on C∩Ui

(otherwise stated, the divisor{yi= 0}is equal to{vi = 0}=C∩Ui).

1.3. Results

We will use the same strategy as in [8]: first construct two “canonical”

regular formal foliationsF,GonS havingCas a leaf, then study the classification of formal / convergent bifoliated neighborhoods (S,F,G), and finally put these together to obtain the formal classification of neighborhoods.

The first step, the construction of “canonical” foliations, is explained in Section 2. Although we will only use it for curves of genus g >2, the construction will also be valid for curves of genus 1. It has already been proved in [4] that there exist formal regular foliations inS havingC as a leaf. Since we need to have some kind of unicity to be able to use these for the classification of neighborhoods, we will need to adapt the construction of [4]. The idea is to construct foliations whose holonomy is trivial along as many loops as possible. For this, we fix a family (α₁, . . . , α_g, β₁, . . . , β_g) of loops inC which is a symplectic basis in homology and denoteA-loops the loopsα_iand B-loops theβ_i. We prove the following:

Theorem 1.1. — LetCbe a curve of genusg>1andSa neighborhood ofC with trivial normal bundle. Then there exists a unique regular formal foliationF on S having C as a leaf, such that the holonomy of F along A-loops is trivial.

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C

Figure 1.1. A bifoliated neighborhood ofC

The second step, the classification of bifoliated neighborhoods, can be found in [13]. We will explain in Section 3 how this classification works in the generic case and show that a bifoliated neighborhood (S,F,G) is characterised by the order of tangencyk between F andG alongC, a 1- formωwhich controls howF andGdiffer at orderk+ 1 and an additionnal invariant

Inv(S,F,G)∈(Diff(C,0))^6g−3/∼

(resp.Inv(S,x F,G)∈(yDiff(C,0))^6g−3/∼for formal neighborhoods), where the relation∼ is given by the action of Diff(C,0)) on (Diff(C,0))^6g−3 by conjugacy on each factor (resp. the action ofDiff(y C,0) on (Diff(y C,0))^6g−3 by conjugacy on each factor). This invariant is given by holonomies of the foliationsFandG computed along a tangency curveT₁, i.e. an irreducible component different from C of the set of points at which F and G are tangent.

Theorem 3.1. — Let C be a curve of genusg >2. Let (S,F,G) and (S⁰,F⁰,G⁰)be two bifoliated neighborhoods ofCwith same tangency order kand1-formω. Supposek>1and thatωhas simple zeroesp1, . . . , p2g−2. Denote T1, T₁⁰ the tangency curves passing through p1 and compute the invariantsInv(S,F,G)andInv(S⁰,F⁰,G⁰)on the tangency curvesT₁, T₁⁰.

Then (S,F,G) and (S⁰,F⁰,G⁰) are analytically (resp. formally) diffeomorphic if and only if

Inv(S⁰,F⁰,G⁰) = Inv(S,F,G) (resp.Inv(Sx ⁰,F⁰,G⁰) =Inv(S,x F,G)).

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Moreover, we know which invariants come from a bifoliated neighborhood: if ((ϕ¹_i)^2g_i=1,(ϕ²_i)^2g_i=1,(ϕ³_j)^2g−2_j=2 ) is a representant of Inv(S,F,G), then theϕ^s_rmust be tangent to identity at orderk. Moreover, if we writeϕ^s_r(t) = t+a^s_rt^k+1 (mod t^k+2), then the periods of ω must be (a²_i −a¹_i)i=1,...,2g

(Equation (3.1) in the text);a³_j must be equal toRpj

p1 ωforj= 2, . . . ,2g−2 (Equation (3.2)); and the (ϕ^s_i)^2g_i=1must be representations of the fundamental group ofCfors= 1,2, i.e. [ϕ^s₁, ϕ^s_1+g]. . .[ϕ^s_g, ϕ^s_2g] = id (Equation (3.3)).

Theorem 3.5. — Let ((ϕ¹_i)^2g_i=1,(ϕ²_i)^2g_i=1,(ϕ³_j)^2g−2_j=2 ) be some analytic / formal diffeomorphisms; letkbe an integer and ωa 1-form. They define a bifoliated analytic / formal neighborhood(S,F,G)with F andG tangent at orderkand with1-formω if and only if everyϕ^s_r is tangent to identity at order (at least)kand if they satisfy the relations (3.1),(3.2)and (3.3).

If the diffeomorphisms ϕ^s_r are only formal, then the neighborhood is a priori only a formal neighborhood ofC. Note here that the relations (3.1) and (3.2) are in fact relations between jets of orderk+ 1 of theϕ^s_r, so the set of bifoliated neighborhoods modulo equivalence has huge dimension.

Indeed, the space of pairs of diffeomorphisms modulo common conjugacy is already infinite dimensional, even formally: if we fix one diffeomorphism ϕ16= id tangent to the identity, then the centralizer ofϕ1has dimension 1 so that the set of pairs (ϕ₁, ϕ₂) modulo common conjugacy has roughly speaking the same cardinality as the set of diffeomorphisms.

Finally, the last step (the formal classification of neighborhoods) is done in Section 4. For the pair of canonical foliations constructed, the tangency orderk will be the Ueda index of the neighborhood (introduced by Ueda in [14] and named by Neeman in [11]), i.e. the highest order such that there is a tangential fibration on Spec(OS/I^k+1) whereIis the ideal sheaf ofCin S. Similarly,ω can be interpreted in terms of the Ueda class ofS. We will define the spaceV(C, k, ω) of neighborhoods with trivial normal bundle, fixed Ueda index equal tok and fixed Ueda class given by ω in order to state the final theorem:

Theorem 4.1. — LetCbe a curve of genusg>2,16k <∞andωa 1-form onC with simple zeroes. Then there is an injective map

Φ :V(C, k, ω),→Diff(y C,0)^g×Diff(y C,0)^g×Diff(y C,0)^2g−3/∼ where the equivalence relation ∼ is given by the action of Diff(y C,0) on Diff(y C,0)^N by conjugacy on each factor.

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A tuple of diffeomorphisms ((ϕ^(j)_i )i)³_j=1 is in the image ofΦif and only if the ϕ^(j)_i are tangent to the identity at order k and if they satisfy the compatibility conditions(3.1)and(3.2).

Note that we do not need to check condition (3.3) because the foliations that we use, being canonical, automatically satisfy it.

2. Construction of foliations

On the curve C we can choose loopsαi, βi,i= 1, . . . , g forming a symplectic basis ofH₁(C,C), i.e.α_i·α_j =β_i·β_j = 0 andα_i·β_j= 1 ifi=j and 0 otherwise. We callA-loops the loops αi andB-loops the βi. Similarly, if ωis a 1-form onC, we will callA-period (resp.B-period) ofω any integral R

α_iω (resp.R

β_iω).

Figure 2.1. A- andB-loops

Definition 2.1. — A foliation will be calledA-canonical if its holonomy representationρ satisfies ρ(α_i) = id for all i = 1, . . . , g and if the linear part ofρ is trivial. We define the notion of B-canonicity similarly; unless otherwise stated, the term “canonical” will meanA-canonical.

Choose an open covering (U_i) of some neighborhood ofC; letV_i=U_i∩C andV = (Vi) the associated open covering of C. Denote byC the trivial rank one local system onC and byO_C the trivial line bundle onC.

First, let us give the following definitions:

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Definition 2.2. — Let (a_ij) be a cocycle in Z¹(V,C) and letγ be a loop onC. We define the period of(aij)alongγ to be the sum

Z

γ

(aij) =

n

X

p=1

ai_pi_p+1

where the open sets(V_i_p)ⁿ_p=1form a simple covering ofγ andV_i_p∩V_i_p+1∩ γ6=∅.

This definition is more convenient in our setting, but it is of course equivalent to other more intrinsic ones (for example using the pairing between homology and cohomology); for instance, this application only depends on the class [γ] ofγin the fundamental group ofC. Taking periods along the αi andβi gives applications

PA, PB:Z¹(V,C)→C^g.

Putting these together gives an application P : Z¹(V,C) → C^2g which induces an injectionP:H¹(V,C)→C^2g.

On the other hand, the exact sequence

0→C→ OC→Ω¹→0 gives the exact sequence in cohomology

(2.1) 0→H⁰(C,Ω¹)→^δ H¹(C,C)→H¹(C,O_C)→0.

We have dim(H⁰(C,Ω¹)) = dim(H¹(C,O_C)) =gand dim(H¹(C,C)) = 2g so thatP :H¹(C,C)→C^2g, being injective, is bijective. It is well-known that a 1-form whoseA-periods vanish is zero, so that the applicationPA◦δ: H⁰(C,Ω¹)→C^g is a bijection.

Constructing a foliation on S is equivalent to constructing functionsyi

onU_iwhich are reduced equations of C∩U_i such that y_j=ϕ_ji(y_i),

where theϕ_ji are diffeomorphisms of (C,0). As before, ifγ is a loop, we can define the product

Hγ((ϕji)) =ϕi₁i_n◦ · · · ◦ϕi₃i₂◦ϕi₂i₁

which will be the holonomy of the foliation given by theyi along the loop γ. Once again this definition is fitted for our purposes but is equivalent to more classical ones. For example in Section 3.1 it will be more convenient to see holonomy as the monodromy of first integrals. The equivalence can be easily checked from the fact that the analytic continuation of a first

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integral y_i on U_i to a nearby open set U_j with y_j = ϕ_ji(y_i) is given by ϕ⁻¹_ji (yj).

To construct these functions y_i, we are going to proceed by steps, but first, we need another definition.

Definition 2.3. — A set of functions(yi)on the open setsUi is called A-normalized at orderµif they_i are regular functions onU_i vanishing at order1onC and

(2.2) yj=ϕ^(µ)_ji (yi) +a^(µ+1)_ji y_i^µ+1,

onUi∩Uj, wherea^(µ+1)_ji is a function onUi∩Uj, theϕ^(µ)_ji are polynomials of degree µ which are also diffeomorphisms tangent to identity and the holonomiesHα_k((ϕ^(µ)_ji ))are the identity moduloy_i^µ+1 for allk= 1, . . . , g.

The idea of the proof of Theorem 1.1 is first to construct some functions (y_i) which are A-normalized at order 1, and then to show that every A- normalized at order µ set of functions (yi) can be transformed into an A-normalized at order (µ+ 1) set of functions by changes of coordinates yi 7→ yi−biy^µ+1_i for some functions bi on Ui. At the limit, we will thus obtain a formal foliation onS with trivial holonomy alongA-loops.

Lemma 2.4. — There exists an A-normalized at order 1 set of functions and the foliations associated to two such sets of functions coincide at order1.

Proof. — Take any reduced equations (yi) ofC and compute yj in the coordinatey_i:

yj=a⁽¹⁾_ji yi.

The cocycle (a⁽¹⁾_ji |C) defines the normal bundleNC=OC ofC so is cohomologous to the trivial cocycle: there exist functionsb_i onU_i such that

a⁽¹⁾_ji |C =bj|C

b_i|_C. Putzi=yi/bi to obtain

zj=zi+a⁽²⁾_ji z_i². for some functionsa⁽²⁾_ji .

Regarding uniqueness, consider two sets of functions (y_i) and (z_i) A- normalized at order 1. Then (yi) and (zi) define two sections y¹ and z¹ on the normal bundle N_C. Necessarily, y¹ and z¹ are colinear, hence the

result.

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Lemma 2.5. — Let (y_i) be a set of functions A-normalized at order µ. Then there exist functions (bi) on Ui such that the coordinates zi = y_i−b_iy_i^µ+1areA-normalized at orderµ+ 1. Moreover, two sets of functions A-normalized at order (µ+ 1)which coincide at order µ define the same foliation at orderµ+ 1.

Proof. — Since (yi) isA-normalized at orderµ, it satisfies yj=ϕ^(µ)_ji (yi) +a^(µ+1)_ji y_i^µ+1.

In the following, denote by Diff¹_µ(C,0) = Diff¹(C,0)/Diff^µ+1(C,0) the group ofµ-jets of diffeomorphisms tangent to the identity. The tuple (ϕ^(µ)_ji)ji

is a cocycle inH¹(C,Diff¹_µ(C,0)); it is entirely determined by its holonomy representationH((ϕ^(µ)_ji )) : π1(C)→ Diff¹_µ(C,0). We would like to extend this cocycle to some cocycle inH¹(C,Diff¹_µ+1(C,0)). SinceHα_k((ϕ^(µ)_ji )) is trivial fork= 1, . . . , g, extendH_α_k((ϕ^(µ)_ji )) to ρ_α_k = id. Next, extend the diffeomorphismsHβ_k((ϕ^(µ)_ji )) to diffeomorphismsρβ_k ∈Diff¹_µ+1in any way.

ThenQg

k=1[ρ_α_k, ρ_β_k] = id so the (ρ_γ)_γdefine a representation ofπ₁(C) into Diff¹_µ+1 which corresponds to a cocycle (ψji) such thatHα_k((ψji)) = ρα_k

andH_β_k((ψ_ji)) =ρ_β_k. We can then write

y_j=ψ_ji(y_i) +a⁰_ji^(µ+1)y^µ+1_i for somea⁰_ji^(µ+1). Next,

y_k =ψ_kj(y_j) +a⁰_kj^(µ+1)y_j^µ+1

=ψkj

ψji(yi) +a⁰_ji^(µ+1)y_i^µ+1

+a⁰_kj^(µ+1)

ψji(yi) +a⁰_ji^(µ+1)y^µ+1_i ^µ+1

=ψkj(ψji(yi)) + (a⁰_ji^(µ+1)+a⁰_jk^(µ+1))y^µ+1_i +· · ·

Since ψ_ki=ψ_kjψ_ji, we obtain a⁰_ki^(µ+1)|_C=a⁰_kj^(µ+1)|_C+a⁰_ji^(µ+1)|_C and thus (a⁰_ji^(µ+1)|C) is a cocycle inH¹(C,OC). By the exact sequence (2.1), it is cohomologous to a constant cocycle (cji) ∈ H¹(C,C): there exists functionsb_i onU_i such thata⁰_ji^(µ+1)|_C−c_ji=b_j|_C−b_i|_C. Still using the exact sequence (2.1), we see that two cocycles (cji),(c⁰_ji) cohomologous to (a⁰_ji^(µ+1)|C) differ only by the periods of a 1-form. As noted before, PA◦δ : H⁰(C,Ω¹) → C^g is bijective so we can choose (cji) with trivial A-periods, and such a (c_ji) is unique. Put ϕ^(µ+1)_ji (y) = ψ_ji(y) +c_jiy^µ+1

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andz_i=y_i−b_iy^µ+1_i to obtain

zj =ψji(zi) + (a⁰_ji^(µ+1)−bj+bi)z_i^µ+1+o(z^µ+1_i )

=ϕ^(µ+1)_ji (zi) +o(z_i^µ+1).

Since the choice of (cji) ∈ H¹(C,C) is unique, if two sets of functions (zi),(z_i⁰) are both A-normalized at order µ+ 1 and coincide at order µ, then they differ at order µ+ 1 by a coboundary (d_i) ∈ H⁰(C,C): z_i⁰ = zi+diz_i^µ+1+· · · Hence, they define the same foliation at orderµ+ 1.

Putting all this together, we obtain Theorem 1.1.

3. Classification of bifoliated neighborhoods

A bifoliated neighborhood ofC is a tuple (S,F,G) whereS is a neighborhood ofC andF,Gare distinct foliations on ShavingC as a common leaf. Two bifoliated neighborhoods (S,F,G) and (S⁰,F⁰,G⁰) are said to be equivalent if there are two neighborhoods U ⊂ S, U⁰ ⊂ S⁰ of C and a diffeomorphismφ:U →U⁰ fixingC such that

φ∗F =F⁰ and φ∗G=G⁰.

In this section, we want to study the classification of bifoliated neighborhoods under this equivalence relation. We will consider here analytic equivalence, but the formal classification can be obtained by replacing the word

“analytic” by “formal” everywhere.

A neighborhood will have a lot a formal foliations, and the canonical ones may diverge even though others might converge (cf. [8]). We will thus consider here a general bifoliated neighborhood (S,F,G), with the additional assumptions thatF andG coincide at order 1 and their holomy representations are tangent to the identity. The study can be done without these assumptions (cf. [13]), but the pair of canonical foliations satisfies them and it simplifies the results (for example, in general an affine structure is involved which under our assumptions is only a translation structure, i.e.

a 1-form).

3.1. First invariants

If (S,F,G) is a bifoliated neighborhood, each foliation comes with the holonomy representation of the leafC:

ρF, ρG :π1(C)→Diff(C,0),

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Fix a base point p₀ ∈C, a transversalT₀ passing throughC at p₀ and a coordinatet onT0 (i.e. a functiont∈(C,0)7→q(t)∈T0. Let γ be a loop onC based atp₀; choose the minimal first integralF ofF aroundT₀such thatF(q(t)) =t. The analytic continuationF^γ ofF alongγis again a first integral ofF, hence is of the form

F^γ=ϕ⁻¹_γ ◦F.

We defineρ_F(γ) =ϕ_γ.

A second invariant is the order of tangency between F and G along C:

take two 1-formsαandβonS defining locally the foliationsFandG. The 2-formα∧βvanishes onCso the order of vanishing ofα∧β alongCgives a global invariantk+ 1 which does not depend on the choice ofαandβ. The order of tangency betweenF and G is defined to be this integer k.

Our assumption that the foliations coincide at order 1 exactly means that k>1.

The next invariant is a 1-form onC associated to this pair of foliations.

Choose as before a pointp₀ ∈C, a transversalT₀ at p₀ and a coordinate t 7→ q(t) on T0. Take local minimal first integrals F and G of F and G such thatF(q(t)) =G(q(t)) =t. By definition ofk, G=F +aF^k+1+. . . in a neighborhood of p for a local function a on C. Take the analytic continuations F^γ, G^γ and a^γ of F, G and a along a loop γ. Then we can use the fact that the holonomy representations ofF and G are tangent to the identity to get

G^γ =F^γ+ (a^γ)(F^γ)^k+1+· · ·

ρ_G(γ)⁻¹◦G=ρ_F(γ)⁻¹◦F+ (a^γ)(ρ_F(γ)⁻¹◦F)^k+1+· · · G=ρ_G(γ) ρ_F(γ)⁻¹◦F+a^γF^k+1+· · ·

=ρ_G(γ)◦ρ_F(γ)⁻¹◦F+a^γF^k+1+. . .

=F+aF^k+1+· · ·

Since ρ_G(γ)◦ρ_F(γ)⁻¹ has constant coefficients, there exists constants c^γ such that a^γ =a+c^γ. Then the 1-form ω = da is a well-defined 1-form onC.

Note that we saw in the process that (3.1) ρG(γ)◦ρF(γ)⁻¹(y) =y+

Z

γ

ω

y^k+1+· · · , thus the formω is entirely determined byρ_F andρ_G.

Note also that the holonomy representation and the form ω depend on the choice of the transversalT0 and of a coordinate t on it. A change of

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coordinateet=ϕ(t) induces conjugacies on ρ_F andρ_G and changesω into some multiple of it:ρe_F(γ) =ϕ◦ρ_F(γ)◦ϕ⁻¹,ρe_G(γ) =ϕ◦ρ_G(γ)◦ϕ⁻¹ and eω=ϕ⁰(0)^−kω.

3.2. Tangency set

IfF andGare local minimal first integrals ofFandG, then the tangency set betweenF andG is defined to be

Tang(F,G) ={dF ∧dG= 0} \C.

This definition does not depend on the choice ofF andGand gives a well- defined analytic subset ofS. As noted before, dF∧dGvanishes onC, but here we are only interested in the transverse part of the tangency set.

Note that if we write G=F+aF^k+1+. . ., then we obtain dF∧dG= F^k+1dF∧(da+· · ·). Sinceω= da,

Tang(F,G)∩C={ω= 0}.

In particular, the set Tang(F,G) intersectsCat 2g−2 points counted with multiplicities. In the sequel, we will suppose that we are in the generic case:

ωhas 2g−2 distinct zeroes. This also means that Tang(F,G) is the union of 2g−2 curves which are transverse toC.

Denote p1, . . . , p2g−2the zeroes of ω andTi the tangency curve passing throughpi. If we fix some simple pathsγij betweenpi andpj, we can look at the holonomy transports

ϕ^F_ij, ϕ^G_ij:T_i→T_j

following the leaves ofFandGalongγij. To simplify, suppose that theγ1j

only intersect each other atp1 and thatγij =γ_1i⁻¹·γ1j.

To define this, fix some coordinates ti, tj on Ti and Tj; there exists a simply connected neighborhoodU of the pathγ_ij. Let Lbe a leaf ofF in U. It intersects Ti at exactly one point (let ti be its coordinate). In the same way, lett_j be the coordinate ofL∩T_j. We setϕ^F_ij(t_i) =t_j. This gives a germ of diffeomorphismϕ^F_ij ∈Diff(C,0) which depends on the choices of coordinates onTi andTj. Their composition

ϕ^↔_ij = (ϕ^G_ij)⁻¹◦ϕ^F_ij

is a diffeomorphism ofT_i so only depends on the choice of a coordinate on Ti; a change of coordinatet⁰_i=ϕ(ti) acts by conjugacyϕ^0↔_ij =ϕ◦ϕ^↔_ij◦ϕ⁻¹.

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Figure 3.1. The pathsγ1j

Each diffeomorphism ϕ^↔_ij will give an invariant, but considering all of them will be redundant. It will be sufficient to consider only theϕ^↔_1j. Indeed, ifi, j are two indices different from 1,

ϕ^↔_ij(ti) =

(ϕ^G_i1)⁻¹ϕ^↔_1j(ϕ^G_i1)

ϕ^↔_i1(ti).

Said differently, if we equipTi with the coordinate ti obtained by trans- porting t1 along the leafs of G following γ1i, we have ϕ^G_i1 = id and thus ϕ^↔_ij =ϕ^↔_1j◦(ϕ^↔_1i)⁻¹.

We can show as in the previous subsection that the holonomy transports ϕ^↔_ij are related to the 1-formωby the relation:

(3.2) ϕ^↔_ij(ti) =ti− Z

γij

ω

!

t^k+1_i +· · ·

3.3. Classification of bifoliated neighborhoods

We say that a bifoliated neighborhood (S,F,G) is generic if there are 2g−2 distinct tangency curvesT1, . . . , T_2g−2betweenF andGand if they intersectC transversely at some distinct pointsp₁, . . . , p_2g−2.

On each neighborhood, we can fix one of these points, for example p1, fix a coordinate t on T₁, fix paths γ_1j between p₁ and p_j and compute every invariant on the transversalT1 with coordinatet. We thus have the

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t₁• •ϕ^F₁₂(t1)

•ϕ^↔₁₂(t₁)

T₁ T₂ C

Figure 3.2. Holonomy transports

holonomy representationsρ_F, ρ_Gand the holonomy transportsϕ^↔_1j between T1 and another tangency curveTj.

The holonomy representationsρ_F andρ_G are entirely determined by the images of the basis α1, . . . , αg, β1, . . . , βg: these are any diffeomorphisms such that

(3.3) [ρ_F(α1), ρ_F(β1)]. . .[ρ_F(αg), ρ_F(βg)] = id [ρ_G(α₁), ρ_G(β₁)]. . .[ρ_G(αg), ρ_G(βg)] = id. Every invariant diffeomorphism found

ϕ=ρ_F(αi), ρ_F(βi), ρ_G(αi), ρ_G(βi), ϕ^↔_1j

depend on the choice of the coordinatet. A change of coordinatet⁰ =ψ(t) induces a conjugacy onϕ:ϕ⁰ =ψ◦ϕ◦ψ⁻¹. So we define the invariant of a neighborhood (S,F,G) to be

Inv(S,F,G)

=h

((ρ_F(α_i))^g_i=1,(ρ_F(β_i))^g_i=1,(ρ_G(α_i))^g_i=1,(ρ_G(β_i))^g_i=1,(ϕ^↔_1j)^2g−2_j=2 )i

∈Diff(C,0)^2g×Diff(C,0)^2g×Diff(C,0)^2g−3/∼ where∼is the action of Diff(C,0) by conjugacy on each factor.

Theorem 3.1. — Let C be a curve of genusg >2. Let (S,F,G) and (S⁰,F⁰,G⁰) be two bifoliated neighborhoods of C with tangency order k and1-formω. Suppose k>1 and that ω has simple zeroesp1, . . . , p_2g−2. Denote T₁, T₁⁰ the tangency curves passing through p₁ and compute the invariantsInv(S,F,G)andInv(S⁰,F⁰,G⁰)on the tangency curvesT1, T₁⁰.

Then (S,F,G)and(S⁰,F⁰,G⁰)are diffeomorphic if and only if Inv(S⁰,F⁰,G⁰) = Inv(S,F,G).

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Before starting the proof, let us write some lemmas.

Lemma 3.2. — Letpbe a point inC, letF,Gbe two reduced equations ofCaroundpand(x, y)some local coordinates withC={y= 0}. Suppose that F and G are tangent at order k and that the zero divisor of dF ∧ dGis (k+ 1)C (i.e. there are no other tangencies). There exists a unique diffeomorphismφfixingCpointwise such that

(F, G)◦ϕ= (y, y+a(x)y^k+1).

The functionais unique and satisfiesda|C=ω.

This lemma can already be found in [8] (Lemma 4.9).

Lemma 3.3. — Letpbe a point inC, letF,Gbe two reduced equations ofCaroundpand(x, y)some local coordinates withC={y= 0}. Suppose that there is a transversal T to C such that the zero divisor of dF ∧dG is (k+ 1)C +T. Then there exists a unique diffeomorphism φ fixing C pointwise such that

(F, G)◦ϕ= (y, b(y) +a(x)y^k+1).

The functionbis unique andais the primitive ofω which is zero atp.

The function b is of course entirely determined by the equationG|T = b(F|T).

Proof. — Put ye = F, b the function determined by G|T = b(F|T), H=G−b(y) and supposee xis a reduced equation ofT. Then dF∧dG= (∂_xH)dF ∧dx so by the hypotheses on the tangency divisor, ∂_xH = 2xey^k+1u for some invertible function u. Then H = x²ye^k+1v with v invertible so forφ(x,ey) =x√

v, we haveG=b(ey) +φ(x,y)e²ey^k+1.

Ifψ=φ|C, then the coordinatexe=ψ⁻¹◦φ(x,y) is equal toe xonC and (F, G) = (y, b(e ey) +ψ(ex)²ey^k+1). Thus the diffeomorphism ϕ(x, y) = (ex,y)e

is as sought.

Lemma 3.4. — Let(S,F,G)be a bifoliated neighborhood whose1-form ωhas simple zeroes, letT₁, T_jbe two tangency curves andγ_1ja simple path betweenp1=T1∩Candpj =Tj∩C. SupposeFandGare some submersive first integrals ofFandG aroundp₁such thatF|_T₁=G|_T₁. By Lemma 3.3, the analytic continuations ofF andGalongγ1j can be writtenF =y and G=b(y) +a(x)y^k+1 for some coordinates(x, y)aroundpj.

Then b(y) =ϕ^↔_1j(y)ifϕ^↔_1j is computed in the coordinatet=y onT1. Proof. — Indeed,b is characterised by G|_T_j =b◦F|_T_j, and ϕ^↔_1j by the fact that the leaf ofFpassing throughT1at the point of coordinateF =y0

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intersects (tangentially) onT_jthe leaf ofGpassing throughT₁at the point of coordinateF =ϕ^↔_1j(y0). This means that the first integralϕ^↔_1j◦F ofF coincides withGonT_j, ieϕ^↔_1j◦F|_T_j =G|_T_j, hence the result.

Proof of Theorem 3.1. — Take two bifoliated neighborhoods (S,F,G) and (S⁰,F⁰,G⁰) with the same tangency index k, 1-form ω and the same invariants computed in some coordinatest, t⁰ onT1 andT₁⁰.

Begin by fixing simply connected neighborhoodsY, Y⁰ of∪^2g_j=2γ_1j in S andS⁰. We begin by showing that (Y,F,G) and (Y⁰,F⁰,G⁰) are diffeomorphic, and we will then show that this diffeomorphism can be extended to S andS⁰.

Figure 3.3. The neighborhood Y

Since Y and Y⁰ are simply connected, the foliations on these sets have first integralsF, G, F⁰, G⁰and we can suppose thatF(t) =G(t) andF⁰(t⁰) = G⁰(t⁰) on T₁ and T₁⁰. Since ω = ω⁰, Lemma 3.3 tells us that there is a (unique) diffeomorphismψbetween a neighborhood ofp1inSand a neighborhood ofp1inS⁰ such thatF⁰◦ψ=F andG⁰◦ψ=G. We can take the analytic continuation ofF, G, F⁰andG⁰along one of the pathsγ_1j. For any pointpin this path, Lemma 3.2 tells us that the pairs (F, G) and (F⁰, G⁰) are equivalent (by a unique diffeomorphism) if and only if the numbera(p) is the same for both couples. Buta(p) =Rp

p₁ω where the integral is taken along the pathγ_1j so it is the case. By uniqueness, the diffeomorphismψ can be extended along the pathγ1j arbitrarily near the pointpj.

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At the point p_j, Lemmas 3.3 and 3.4 show that the pairs (F, G) and (F⁰, G⁰) are also conjugated by a unique diffeomorphism in a neighborhood ofp_j. Hence, we can extendψto a diffeomorphismψ:Y →Y⁰ conjugating the pairs (F, G) and (F⁰, G⁰).

By Lemma 3.2, we can also extend ψ along any simple path. Then we only need to show thatψ can be extended along a non-trivial loop. Letγ be a non-trivial loop onC based atp1, ϕ_F =ρ_F(γ) and ϕ_G =ρ_G(γ). The extensions of F and G along γ are ϕ⁻¹_F ◦F and ϕ⁻¹_G ◦G; we know that F⁰◦ψ=F andG⁰◦ψ=G, soϕ⁻¹_F ◦F⁰◦ψ=ϕ⁻¹_F ◦F andϕ⁻¹_G ◦G⁰◦ψ= ϕ⁻¹_G ◦G. Henceψis the diffeomorphism conjugating (ϕ⁻¹_F ◦F, ϕ⁻¹_G ◦G) with (ϕ⁻¹_F ◦F⁰, ϕ⁻¹_G ◦G⁰) and by unicity this means thatψcan be extended along any loop. Thusψ can be extended to a diffeomorphism between (S,F,G)

and (S⁰,F⁰,G⁰).

3.4. Construction of bifoliated neighborhoods

We saw three restrictions for a set of diffeomorphisms to be an invariant of some bifoliated neighborhood: these are the compatibility relations (3.1), (3.2) and (3.3). These are the only restrictions; to obtain a simpler result, we will consider the 1-formω as an invariant here.

Theorem 3.5. — Let ((ϕ¹_i)^2g_i=1,(ϕ²_i)^2g_i=1,(ϕ³_j)^2g−3_j=2 ) be some diffeomorphisms; letkbe an integer andω a1-form. They define a bifoliated neighborhood (S,F,G)with F and G tangent at order k and with1-form ω if and only if everyϕ^s_ris tangent to identity at order (at least)k and if they satisfy the relations(3.1),(3.2)and(3.3).

Let us recall that the results stated in this section are also true in the formal setting: replacing diffeomorphisms by formal diffeomorphisms everywhere gives the formal results.

Proof. — Denote by ρ₁ and ρ₂ the representations given by the diffeomorphisms (ϕ¹_i) and (ϕ²_i). ConsiderCe=Dxthe universal cover ofC,X a small neighborhood of a fundamental domain,Ui a small neighborhood of pi inX andCq=X\(U2∪. . .∪U2g−2).

Consider next the trivial bundle Sq =Cq×Cy along with two functions F =y and G =y+a(x)y^k+1 (with a(x) =Rx

p1ω). We now want to glue the borders ofSqtogether: for this, we need to show that there exists for each loopγ a diffeomorphism ψ_γ defined when it makes sense such that ψγ|Cˇ =γ (where we identify the loop γ ∈ π1(C) with the corresponding

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Figure 3.4. The neighborhoodX

deck transformation ofC), ande

(ρ₁(γ)◦F, ρ₂(γ)◦G) = (F◦ψ_γ, G◦ψ_γ).

Thanks to the compatibility condition (3.1) and Lemma 3.2, the couples (ρ₁(γ)◦F, ρ₂(γ)◦G) and (F◦γ, G◦γ) are diffeomorphic so we can indeed find such aψγ. We can then glue the borders ofSqtogether to obtain a surface which is a neighborhood ofC with holesH_i aroundp_i (i= 2, . . . ,2g−2) and two foliations F andG transverse outside the holes. The holonomies of these foliations areρ_F=ρ₁ andρ_G =ρ₂by construction.

To fill these holes, takeCia neighborhood ofpi inX slightly larger than Ui and consider the patchPi =Ci×C^y. Consider on Pi the couple

(F ,e G) = (y, ϕe ³_i(y) + (x−p_i)²y^k+1).

By Lemma 3.2 and compatibility Condition (3.2), for every pointpnear the boundary of the holeHi, there exists a unique diffeomorphism ψbetween a neighborhood of p in Sq and a neighborhood of p in Pi sending (F, G) to (F ,e G). By uniqueness, these diffeomorphisms glue to a diffeomorphisme between neighborhoods of the boundaries of Hi and Pi and we can then glue the patch P_i onto H_i using this diffeomorphism. By Lemma 3.4, we then haveϕ^↔_1i =ϕ³_i which concludes the proof.

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4. Formal classification of neighborhoods

We know how to construct two canonical foliations on any neighborhood, and we know the classification of bifoliated neighborhoods, so we only need to put this together.

Denote F andG theA- andB-canonical foliations. Note that ifF =G, then they define a fibration tangent toC, so this case can be treated by Kodaira’s deformation theory. Suppose this is not the case and F 6= G;

denote byktheir order of tangency. SinceF=Gup to orderkthere exists a fibration tangent toC up to orderkonS, and conversely, a fibration up to orderk⁰ is equal toFandG at orderk⁰ by unicity. Hencekis the Ueda index of the curveC as defined in [14, p. 589] or [11, Definition 1.6].

Moreover, let (u_ij)∈H¹(C,O_C) be the Ueda class of the neighborhood (it is not explicitely named in [14], but it is Definition 1.5 of [11]). Let also (aij),(bij)∈H¹(C,C) be the cocycles defining the (k+ 1)-th order holonomy ofFandG. By definition, the images of (aij) and (b_ij) under the map H¹(C,C)→H¹(C,OC) are both (uij). Thus by the exact sequence (2.1), the cocycle (b_ij−a_ij) is given by a 1-form: this 1-form is exactlyω. To sum up, we have constructed an applicationH¹(C,OC)→H⁰(C,Ω¹); this is a bijection because we can find (a_ij) (and thus (u_ij)) from ω as the cocycle with nullA-periods and withB-periods equal to those ofω.

By extension, we will call this form the Ueda form of the neighborhood.

The Ueda form is well-defined only up to a multiplicative constant, but the set of its zeroes is well-defined. The situation will be quite different depending on the tangency set betweenFandG, so suppose thatωhas only simple zeroes (so that the tangency set consists of 2g−2 simple transversal tangency curves). Denote byV(C, k, ω) the space of 2-dimensional formal neighborhoods of C with trivial normal bundle, Ueda index k < ∞ and Ueda form (a multiple of)ωmodulo formal equivalence.

Theorem 4.1. — LetCbe a curve of genusg>2,16k <∞andωa 1-form onC with simple zeroes. Then there is an injective map

Φ :V(C, k, ω),→Diff(y C,0)^g×Diff(y C,0)^g×Diff(y C,0)^2g−3/∼ where the equivalence relation ∼ is given by the action of Diff(C,0) on Diff(C,0)^N by conjugacy on each factor.

A tuple of diffeomorphisms ((ϕ^(j)_i )i)³_j=1 is in the image ofΦif and only if the ϕ^(j)_i are tangent to the identity at order k and if they satisfy the compatibility conditions(3.1)and(3.2).