This is the accepted version of the journal article: Marín Pérez, David; Mattei, Jean-François; Salem, Eliane. «Topological moduli space for germs of foliations II : universal deformations». Annali della Scuola normale superiore di Pisa - Classe di scienze, In press 2023. DOI 10.2422/2036- 2145.2021110 06 This version is available at https://ddd.uab.cat/record/272405 under the terms of the license TOPOLOGICAL MODULI SPACE FOR GERMS OF HOLOMORPHIC FOLIATIONS II: UNIVERSAL DEFORMATIONS DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM Abstract. This work deals with the topological classification of singular foliation germs on (C2 , 0). Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representa- tions and we prove the existence of a topological universal deformation through which every equisingular deformation uniquely factorizes up to topological conjugacy. This is done by representing the functor of topological classes of equisingular deformations of a fixed foliation. We also describe the functorial dependence of this representation with respect to the foliation. Contents 1. Introduction 2 2. Group-graphs 4 2.1. Notion of group-graph 5 2.2. Group-graph associated to a sheaf 7 2.3. Cohomology of a group-graph 8 2.4. Pruning 11 2.5. Direct image of a C-graph 12 2.6. Regular group-graph 13 2.7. Tensor product 14 3. Equisingular deformations of foliations 15 3.1. Deformations of foliations 15 3.2. Equisingular deformations 17 3.3. Good trivializing system 18 3.4. Deformation functor 22 4. Group-graphs of automorphisms and transversal symmetries 26 4.1. Group-graph of C ex -automorphisms 26 4.2. Sheaf of transversal symmetries 29 4.3. Group-graph of transversal symmetries 32 5. Finite type foliations and infinitesimal transversal symmetries 34 5.1. Finite type foliations 34 5.2. Sheaf of infinitesimal transversal symmetries 35 5.3. Group-graph of infinitesimal transversal symmetries 38 5.4. Exponential group-graph morphism 40 5.5. Characterization of finite type foliations 42 5.6. Some examples 44 Date: February 27, 2023. 2010 Mathematics Subject Classification. Primary 37F75; Secondary 32M25, 32S50, 32S65, 34M. Key words and phrases. Complex dynamical systems, complex ordinary differential equations, holo- morphic vector fields, singular foliations, holonomy, universal deformations, representable functors, non abelian cohomology. D. Marín acknowledges financial support from the Spanish Ministry of Science, Innovation and Univer- sities, through grants PGC2018-095998-B-I00 and PID2021-125625NB-I00 and by the Agency for Manage- ment of University and Research Grants of Catalonia through the grants 2017SGR1725 and 2021SGR01015. 1 2 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM 6. C ex -universal deformations 46 6.1. C ex -universality 46 6.2. Kodaira-Spencer map 48 6.3. Criteria for universality 50 References 51 1. Introduction This work inserts in a series of three papers whose goal is to obtain a topological clas- sification of singular foliation germs on (C2 , 0) through the construction of a topological moduli space, the description of its algebraic and topological properties and the construc- tion of a family containing all topological types with minimal redundancy. In the article [7], completed by [14] and [8, Appendix], the authors give for a generic germ of foliation F on (C2 , 0) a list of topological invariants: a) the combinatorial reduction of singularities of F, b) the Camacho-Sad indices of the singularities of the reduced foliation F ] , c) the holonomies of F ] along the invariant components of the exceptional divisor EF of the reduction. We call this collection the semi-local invariants of F. In the present paper we are only interested in germs at F of families of foliations with same semi-local invariants as F, that we call equisingular deformations of F. These notions will be specified later. For any generic foliation we prove the existence of a “topological universal deformation” through which any equisingular deformation of F uniquely factorizes up to topological conjugacy. We also provide an infinitesimal criterion of universality. In [8] we have constructed a global family containing all topological types with same semi-local invariants as F. The results that we obtain in this paper will allow us to study in a forecoming paper [9] prop- erties of factorization of this global family. Classically a deformation of a foliation F over a germ of manifold P · = (P, t0 ) is a germ of foliation FP · on (C2 × P, (0, t0 )) defined by a germ of holomorphic vector field X(x, y, t) that coincides on C2 ×{t0 } with a vector field defining F and moreover is tangent to the fibers of the canonical projection prP : C2 × P → P . If λ : (Q, u0 ) → P · is a germ of holomorphic map, the pull-back of FP · by λ is the deformation λ∗ FP · of F over (Q, u0 ) defined by the vector field X(x, y, λ(t)). Two deformations FP · and FP0 · are topologically conjugated if there exists a C 0 -automorphism Φ of (C2 × P, (0, t0 )) that sends the leaves of FP · on that of FP0 · , and satisfies prP ◦ Φ = prP , Φ(x, y, t0 ) = (x, y, t0 ) . As in [8] we say that the deformation FP · is equisingular if the foliations given by the vector fields Xt (x, y) := X(x, y, t) on the fibers C2 × {t} can be “simultaneous reduced” and moreover each of them share the same semi-local invariants as F, see Definition 3.6. We will prove: TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 3 Main Theorem. Every finite type generalized curve1 foliation possesses a topological uni- versal deformation. Topological universality of a deformation FQ· of F means that for any germ of manifold P · and any equisingular deformation FP · of F over P · , there exists a unique holomorphic map germ λ : P · → Q· such that FP · is topologically conjugated to λ∗ FQ· . In fact we will prove the stronger result that the topological conjugacy between FP · and λ∗ FQ· is realized by an excellent (or C ex ) homeomorphism, i.e. it lifts through the equireduction maps of FP · and λ∗ FQ· and its lifting fulfills a regularity property, see Definition 3.3. We obtain a universal deformation of F by representing the functor Def F that associates · to any germ of regular manifold P · , the set Def PF of C ex -conjugacy classes of deformations of F over P · . To describe the dependence of this representation with respect to F we define, up to excellent conjugacy, the pull-back of an equisingular deformation of F by a C ex -conjugacy φ : G → F. We thus get a contravariant deformation functor · Def : Man· × Fol → Set· , (P · , F) 7→ Def PF , · which associates to a foliation F and a germ of manifold P · , the set Def PF . Here Man· is the category of germs of complex manifolds, the morphism sets O(P · , Q· ) consisting of holomorphic map germs compatible with the pointing, and Fol is the category whose objects are the germs of foliations which are generalized curves of finite type, the morphisms being C ex -conjugacies. In fact, we will construct a suitable (pointed by 0) cohomogical C- vector space H 1 (A, TF )· associated to F and an isomorphism of functors ∼ Def −→ (P · , F) 7→ O(P · , H 1 (A, TF )· .  (1) The paper is organized in the following way: - In Chapter 2 we further develop the key notion of group-graph already introduced in [8]. This notion is well adapted to our problem and it may also be useful in other sit- uations, which simultaneously deal with local and semi-local objects. In absence of nodal singularities and dicritical components the group-graphs considered in the sequel are asso- ciated to sheaves but otherwise we need to consider general group-graphs as we did in [8]. We also define the notion of regular group-graph and we describe its cohomology (see Theorem 2.15). - The notion of equisingular deformation is introduced in Chapter 3. Its characteristic property, stated in Theorem 3.8, is the triviality along each irreducible component of the exceptional divisor of the equireduction. This allows (Theorem 3.11) to define for a C ex - · · conjugacy φ : G → F, the pull-back map φ∗ : Def PF → Def PG , and the functor Def. · - In Chapter 4 we consider the group-graph AutPF , over the dual graph AF of EF , of excellent automorphisms of the constant deformation of F over P · . For an equisingular deformation, the trivializing maps given by Theorem 3.8 provide a cocycle with values in this group-graph. In this way we obtain a natural transformation from the functor Def 1 i.e. a germ of foliation F such that the foliation F ] obtained after reduction is without saddle-node (i.e. singularity given by a vector field germ whose linear part has exactly one non-zero eigenvalue); however F ] may have nodal singularities (i.e. defined by a vector field germ such that the ratio of the eigenvalues of its linear part is strictly positive) and the exceptional divisor of the reduction may have irreducible components non invariant by F ] . For more details we refer to [2]. 4 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM · to the functor that associates to F and P · the cohomology space H 1 (AF , AutPF ). This transformation is an isomorphism of functors (Theorem 4.4)   ∼ · Def −→ P · , F) 7→ H 1 (AF , AutPF ) . (2) · By taking the quotient of AutPF by the normal subgroup-graph of automorphisms fixing · · each leaf, we obtain a simpler group-graph SymPF with same cohomology as AutPF (Propo- sition 4.11). - The notion of finite type foliation is defined and cohomologically characterized (The- · orem 5.15) in Chapter 5. For such a foliation the cohomology of the group-graph SymPF over AF is completely given by restricting it to an appropriate subgraph RF ⊂ AF (The- · orem 5.3). The advantage of this restriction is that over RF the group-graph SymPF is · isomorphic (via the “exponential morphism”) to the abelian group-graph TFP of C-vector spaces of infinitesimal transverse symmetries of the constant deformation, see Defi- nition 5.8. This study gives the natural isomorphisms · ∼ · ∼ · ∼ · H 1 (AF , AutPF )−→H 1 (AF , SymPF )−→H 1 (RF , SymPF )−→H 1 (RF , TFP ) . (3) · The structure of TFP over RF is the tensor product TF ⊗C MP · of the group-graph of infinitesimal symetries of F with the maximal ideal of OP · (Lemma 5.11). Finally, using the results of Section 2.7 we get: · ∼ ∼ ∼ H 1 (RF , TFP )−→H 1 (RF , TF ⊗C MP · )−→H 1 (RF , TF ) ⊗C MP · −→O(P · , H 1 (RF , TF )· ) , that achieves, using (2) and (3), the construction of the natural isomorphism (1). Finally in Section 5.6 we discuss some examples of foliation germs whose separatrix set is the double cusp (y 2 + x3 )(y 3 + x2 ) = 0, to illustrate the notions of equireducibility, equisingularity, finite type with explicit group-graph cohomology computations, Kodaira-Spencer map and C ex -universal deformation. - In Chapter 6, using that the restriction of the group-graph TF to RF is regular (Propo- sition 5.12) and Theorem 2.15, we specify in Theorem 6.4 the structure of the finite di- mensional universal parameter space H 1 (RF , TF )· . We also construct a Kodaira-Spencer map ∂[FP · ] : Tt0 P −→ H 1 (RF , TF ) ∂t t=t0 associated to an equisingular deformation FP · , that will provide in Theorem 6.7 an infini- tesimal criterion of universality. 2. Group-graphs We recall that a graph is the data of a pair A = (VeA , EdA ) where VeA is a set and EdA ⊂ P(VeA ) is a collection of subsets of two distinct elements v, v 0 of VeA , denoted by hv, v 0 i. The elements of VeA are called vertices of A and those of EdA are called edges of A. We denote by IA := {(v, e) ∈ VeA × EdA | v ∈ e} the set of oriented edges of A. A morphism of graphs ϕ : A0 → A is a map ϕ : VeA0 → VeA such that if e = hv, v 0 i ∈ EdA0 either ϕ(v) 6= ϕ(v 0 ) and ϕ(e) := hϕ(v), ϕ(v 0 )i ∈ EdA , or ϕ(v) = ϕ(v 0 ) and ϕ(e) := ϕ(v) ∈ VeA . TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 5 2.1. Notion of group-graph. Definition 2.1. Let C be a category. A C-graph over A is a collection G of objects of C, denoted2 by Gv and Ge , for each vertex v ∈ VeA and each edge e ∈ EdA , and of C- morphisms ρev : Gv → Ge for each (v, e) ∈ IA , which are called restriction morphisms. When C is the category Gr of groups we say that G is a group-graph; if all groups G? , ? ∈ VeA ∪ EdA , are abelian, we say that G is abelian and when all groups G? , ? ∈ VeA ∪ EdA are trivial we say that G is the trivial group-graph and we denote it by 0 or 1. The category of C-graphs over A is the category denoted by CA , whose objects are the C-graphs over A and whose morphisms α : F → G are the data of C-morphisms αv : Fv → Gv and αe : Fe → Ge , v ∈ VeA , e ∈ EdA , such that the following diagram v α Fv −→ Gv ξve ↓ ↓ ρev αe Fe −→ Ge commutes for each (v, e) ∈ IA , ξVe and ρev being the restriction maps of F and G. In all the sequel we suppose that C is a subcategory of the category of groups. A C-graph H is a sub-C-graph of a C-graph G if H? is a subgroup of G∗ for any ∗ ∈ VeA ∪ EdA , the inclusion map H? ,→ G? being C-morphisms, and the restriction maps Hv → He being given by the restriction map ρev of G, a fortiori ρev (Hv ) ⊂ He . When each group H? is a normal subgroup of G∗ we say that H is a normal sub-C-graph of G; then the map ρev factorizes as a map ρev : Gv /Hv → Ge /He , defining the quotient C-graph G/H, with (G/H)? = G? /H? , the maps ρev being the restriction maps. If G (resp. G0 ) is a C-graph over a graph A (resp. A0 ), a morphism of C-graphs φ : G → G0 over a morphism of graphs ϕ : A0 → A is a collection of C-morphisms φ? : Gϕ(?) → G0? , ? ∈ VeA0 ∪ EdA0 such that, if e = hv, v 0 i then the following diagram commutes φv Gϕ(v) / G0 v ϕ(e) ρϕ(v) ρ0 ev  φe  Gϕ(e) / G0 e ϕ(e) If ϕ(e) = ϕ(v) then ρϕ(v) is the identity. A consequence of the commutativity of this diagram is that ρev sends the kernel of φv into the kernel of φe and ρ0 ev sends the image of φv into the image of φe . This allows to define the C-graph kernel ker φ over A by (ker φ)? = ker(φ? ), which is a sub-C-graph of G and the C-graph image φ(G) over A0 by φ(G)? = φ? (Gϕ(?) ), which is a sub-C-graph of G0 . We can thus consider exact sequences of C-graphs over a common graph. If ϕ0 : A00 → A0 is another graph morphism and φ0 : G0 → G00 is a C-graph morphism over ϕ0 , then the composition defined by φ0 ◦ φ := {φ0? ◦ φϕ0 (?) : Gϕ(ϕ0 (?)) 7→ G00? | ? ∈ VeA00 ∪ EdA00 } is a C-graph morphism G → G00 over ϕ ◦ ϕ0 . Hence the collection of all the pairs (A, G) where A is a graph and G is a C-graph over A together with the C-graphs morphisms consisting of the pairs (ϕ, φ) : (A, G) → (A0 , G0 ) with ϕ : A0 → A and φ : G → G0 over ϕ, forms a category that we will denote by CG. A C-graph morphism (idA , φ) over the 2The notation G(v) and G(e) is also used in the text. 6 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM identity of A is just a morphism of group-graphs over A as defined previously. Thus, CA is a subcategory of CG. Definition 2.2. The pull-back by a graph morphism ϕ : A0 → A of a C-graph G over A is the C-graph over A0 defined by (ϕ∗ G)? = Gϕ(?) , ? ∈ VeA0 ∪ EdA0 , the restriction morphism (ϕ∗ G)v → (ϕ∗ G)e for e = hv, v 0 i ∈ EdA0 being the restriction morphism Gϕ(v) → Gϕ(e) when ϕ(e) ∈ EdA , and the identity map of Gϕ(v) otherwise. We call canonical morphism the C-graph morphism ıϕ : G → ϕ∗ G over ϕ defined by the identity maps ıϕ ? := idGϕ(?) : Gϕ(?) −→ (ϕ∗ G)? , ? ∈ VeA0 ∪ EdA0 . In this way, the data of a morphism of C-graphs φ : G → G0 over a morphism of graphs ϕ : A0 → A is just the data of a morphism of C-graphs φ̆ : ϕ∗ G → G0 over A0 . Remark 2.3. Let F : G → G0 be a morphism of C-graphs over f : R0 → A. Let r : R → A be a morphism of graphs. If f factorizes as f = r ◦ f¯ for some morphism of graphs f¯ : R0 → R then F factorizes as F = F̄ ◦ ır where F̄ : r∗ G → G0 is a morphism of C- graphs over f¯. Indeed, if we define F̄? := F? : (r∗ G)f¯(?) = Gr(f¯(?)) = Gf (?) → G0? for each ? ∈ VeR0 ∪ EdR0 then F = F̄ ◦ ır .  Remark 2.4. If j = 1, 2, let Gj be a group-graph over Aj and Kj a normal sub-group-graph of Gj , then any group-graph morphism g : G1 → G2 over a graph-morphism ϕ : A2 → A1 sending K1 to K2 factorizes as a morphism ḡ between the quotient group-graphs: 1 / K1 / G1 / G1 /K1 /1 g ḡ    1 / K2 / G2 / G2 /K2 /1 We easily check this property when A1 = A2 and ϕ = id. Since, by definition ϕ∗ (G1 /K1 ) = ϕ∗ G1 /ϕ∗ K1 , the general case follows taking the pull-back by ϕ in the first row.  Remark 2.5. Every graph A can be seen as a category whose objects are the vertices and the edges of A, and whose morphisms (other than the identities) are the inclusion maps iba : {a} ,→ b of a vertex in an edge. a b a b iba A C-graph over A is just a covariant3 functor G : A → C and morphisms of C-graphs are just morphisms (i.e. natural transformations) of functors. This explains the adopted notation CA = {F : A → C covariant functor}. Under this identification, a morphism of graphs ϕ : A0 → A is a covariant functor between the corresponding categories. If G ∈ CA 0 then the pull-back ϕ∗ G ∈ CA is the composition of functors G ◦ ϕ and 0 ϕ∗ : CA → CA , G 7→ ϕ∗ G = G ◦ ϕ 3The contravariant version leads to the dual notion of graph of C, for instance graph of groups in the sense of Serre [13]. TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 7 becomes a contravariant functor defining the pull-back by ϕ of a morphism of C-graphs α : G1 → G2 over A as the morphism ϕ∗ α : ϕ∗ G1 → ϕ∗ G2 of C-graphs over A0 given by (ϕ∗ α)? = αϕ(?) for ? ∈ VeA0 ∪ EdA0 .  In fact, the natural context to consider these notions is that of abstract simplicial com- plexes: Remark 2.6. Recall that an abstract simplicial complex ∆ is a nonempty subset of P(S) 6 F0 ⊂ F whose elements are called faces, such that for each F ∈ ∆, 0 < |F | < ∞ and if ∅ = 0 then F ∈ ∆. The dimension of F ∈ ∆ is dim F = |F | − 1, the dimension of ∆ is dim ∆ = sup{dim F : F ∈ ∆}. A simplicial complex of dimension ≤ 1 is just a graph. The k-skeleton ∆k of a simplicial complex ∆ is the subcomplex of ∆ consisting S of all faces of dimension at most k. We will identify ∆0 with the set of vertices F ⊂ S F ∈∆ of ∆. Each simplicial complex ∆ can be thought of as a small category whose objects are the elements of ∆ and whose morphisms are the inclusions, i.e. if F ⊂ F 0 ∈ ∆ then Hom∆ (F, F 0 ) = {iF F 0 : F ,→ F 0 }. A simplicial map between (abstract) simplicial complexes f : ∆ → Γ is defined by a map f0 : ∆0 → Γ0 such that f (F ) := f0 (F ) ∈ Γ for all F ∈ ∆. Any simplicial map f : ∆ → Γ can be thought of as a functor. The category SC of simplicial complexes and simplicial maps contains the full subcate- gory SCk of simplicial complexes of dimension ≤ k. In particular G := SC1 is the category of graphs. If ∆ is a graph then ∆ = ∆1 and ∆1 \ ∆0 is the set of edges. Passing to the k-skeleton defines a functor SC → SCk . For every category C we consider the collection CSC of C-simplicial complexes which are pairs (∆, G) with ∆ a simplicial complex and G ∈ C∆ := {∆ → C covariant functor}, i.e. G is an assignment ∆ 3 F 7→ G(F ) jointly with a C-morphism ρG 0 0 F F 0 : G(F ) → G(F ), that we call restriction, if F ⊂ F ∈ ∆. We will say that G is a C-simplicial complex over ∆. There is a natural definition of morphism of C-simplicial complexes over a map of simplicial complexes completely analogous to the one considered for C-graphs which makes CSC a category.  2.2. Group-graph associated to a sheaf. Let S be a C-sheaf on a topological space D and C a collection of sets of D. Consider the following graph A (not necessarily finite): its vertices are the elements of C and its edges are all the sets hD, D0 i formed by two distinct elements of C, such that D ∩ D0 6= ∅. For any W ⊂ D (not necessarily open) we recall that the group of continuous sections of S over W is S(W ) := lim −→ S(U ), where UW is U ∈UW the set of open neighborhoods of W . In the case that W = {p}, S({p}) is just the stalk S(p) of S at p ∈ D. If W 0 ⊂ W then UW ⊂ UW 0 and the inductive limit of the restriction morphisms of S define a restriction morphism S(W ) → S(W 0 ). We define the C-graph S over A associated to S in the following way: • SD := S(D) for D ∈ VeA , • ShD,D0 i := S(D ∩ D0 ) for hD, D0 i ∈ EdA , hD,D0 i • the restriction maps ρD are the restriction morphisms considered before. Any morphism of C-sheaves over D induces a morphism of C-graphs over A, defining a covariant functor: CShD −→ CA , S 7→ S , from the category of C-sheaves over D to the category of C-graphs over A. We highlight that this functor is not exact in general. For instance, assume that 0 → S 0 → S → S 00 → 0 is an exact sequence of sheaves of abelian groups on a topological space D and that there is an open set D0 ∈ C such that H 1 (D0 , S 0 |D0 ) 6= 0 and H 1 (D0 , S|D0 ) = 0. Then the long exact sequence in sheaf cohomology gives S(D0 ) → S 00 (D0 ) → H 1 (D0 , S 0 |D0 ) → 0 = H 1 (D0 , S|D0 ). 8 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM 00 → 0 is not exact, the sequence of group- Since the sequence of abelian groups SD0 → SD 0 graphs 0 → S 0 → S → S 00 → 0 can not be exact. Let D0 be another topological space with a collection C 0 of subsets of D0 and let S 0 be a C-sheaf over D0 . Let φ : D0 → D be a homeomorphism such that φ(C) = C 0 . If D ∈ VeA0 and hD, D0 i ∈ EdA0 then φ(D) ∈ VeA , φ(D ∩ D0 ) = φ(D) ∩ φ(D0 ) and φ induces a graph morphism Aφ : A0 → A , ? 7→ φ(?) ; ? ∈ VeA0 ∪ EdA0 . Given a morphism of C-sheaves S → S 0 over φ : D → D0 , i.e. a morphism g : φ−1 S → S 0 of C-sheaves over D0 , we have C-morphisms g D : (φ−1 S)(D) = S(φ(D)) → S 0 (D) , g D∩D0 : (φ−1 S)(D ∩ D0 ) = S(φ(D ∩ D0 )) = S(φ(D) ∩ φ(D0 )) → S 0 (D ∩ D0 ) , for D ∈ VeA0 and hD, D0 i ∈ EdA0 . Since (A∗φ S)(hD, D0 i) = S(hφ(D), φ(D0 )i) = S(φ(D) ∩ φ(D0 )) we obtain a C-graph morphism associated to the sheaf morphism g g : A∗φ S → S 0 . Notice that A∗φ S coincides with the C-graph associated to the sheaf φ−1 S over D0 , and g can be seen as the C-graph morphism associated to the morphism of sheaves g : φ−1 S → S 0 . The situation we will deal with in the sequel is the following: D is an analytic set (and more specifically a hypersurface in a complex manifold), C is the collection of irreducible components of D. The graph A is called the dual graph of D. In this way we have a functor CShan → CG, S 7→ S , where CShan is the subcategory of the category of C-sheaves over analytic sets whose morphisms are over homeomorphisms. 2.3. Cohomology of a group-graph. This notion was introduced in [8]. For group- graphs associated to sheaves considered in subsection 2.2, with C a locally finite open covering U of D and S abelian, this notion will coincide with the Čech cohomology groups Ȟ i (U, S), i = 0, 1. Let G be a group-graph over a graph A. The 0-cohomology set is the subgroup H 0 (A, G) of C 0 (A, G) := v∈VeA Gv whose elements are the families (gv ) satisfying the relations Q ρev (gv ) = ρev0 (gv0 ) whenever e = hv, v 0 i. In order to define the 1-cohomology set H 1 (A, G) of a group-graph (G, (ρev )(e,v)∈IA ) we first define the set of cocycles Z 1 (A, G) as the set of families Y (gv,e ) ∈ Gv,e , with Gv,e := Ge , (v,e)∈IA such that gv,e gv0 ,e = 1 whenever e = hv, v 0 i. Then H 1 (A, G) is the quotient set of Z 1 (A, G) by the following action of C 0 (A, G): (gv ) ?G (gv,e ) := ρev (gv )−1 gv,e ρev0 (gv0 ) .  The set H 1 (A, G) contains the privileged element 1 defined by gv,e = 1. In this way, from now on H 1 (A, G) will be consider as a pointed set. TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 9 Remark 2.7. When G is an abelian group-graph, then H 1 (A, G) is an abelian group. Specifically, we have in this case an exact sequence of groups (with additive notations) ∂0 C 0 (A, G) −→ Z 1 (A, G) → H 1 (A, G) → 0 , ∂ 0 ((gv )) = (gv,e ) , gv,e := gv0 − gv , e = hv, v 0 i . More formally, H i (A, G) is the i-th cohomology group of the cochain complex of abelian groups ∂0 ∂1 Y C ∗ (A, G) : C 0 (A, G) → C 1 (A, G) → C 2 (A, G) := Ge , e∈EdA with: ∂ 1 ((gv,e )) = (gv,e + gv0 ,e ), if e = hv, v 0 i.  Every morphism φ : G → G0 of C-graphs over a graph morphism ϕ : A0 → A induces maps φ0 : C 0 (A, G) → C 0 (A0 , G0 ), φ0 ((gv )v ) = (φv0 (gϕ(v0 ) ))v0 , φ1 : C 1 (A, G) → C 1 (A0 , G0 ), φ1 ((gv,e )) = (gv0 0 ,e0 ), where φe0 (gϕ(v0 ),ϕ(e0 ) ) if ϕ(e0 ) is an edge of A,  gv0 0 ,e0 = 1 otherwise. The image of the restriction H 0 (φ) of the group morphism φ0 to the subgroup H 0 (A, G) is contained in H 0 (A0 , G0 ). Moreover, φ1 sends Z 1 (A, G) into Z 1 (A0 , G0 ), the following diagram is commutative ?G C 0 (A, G) × Z 1 (A, G) / Z 1 (A, G) φ0 φ1 φ1   ?G0  C 0 (A0 , G0 ) × Z 1 (A0 , G0 ) / Z 1 (A0 , G0 ) inducing a map H 1 (φ) : H 1 (A, G) → H 1 (A0 , G0 ) . (4) i i In this way one can check that the correspondences (A, G) 7→ H (A, G) and (ϕ, φ) 7→ H (φ) define covariant functors H i : CG → Set· , i = 0, 1 , (5) from the category of C-graphs to the category of pointed sets. Moreover when C is one of the following sub-categories of Gr: • the category Ab of abelian groups, • the category Vec of C-vector spaces, and linear maps, we obtain covariant functors with values in the same category pointed by 0: H i : CG → C· , i = 0, 1 . In particular, H i (φ), i = 0, 1, are C-morphisms. Remark 2.8. The canonical morphism ıϕ : G → ϕ∗ G induces maps H i (ıϕ ) : H i (A, G) → H i (A0 , ϕ∗ G) and we have H i (φ) = H i (φ̆) ◦ H i (ıϕ ) , i = 0, 1, where φ̆ : ϕ∗ G → G0 is the C-graph morphism over A0 associated to φ and ıϕ : G → ϕ∗ G is the canonical morphism.  i p Proposition 2.9. Let 1 → G0 ,→ G → G00 → 1 be a short exact sequence of group-graphs over a tree A and suppose that all restriction maps ρ0ev : G0v  G0e are surjective. Then the induced morphism H 1 (p) : H 1 (A, G) → H 1 (A, G00 ) is an isomorphism. 10 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM Proof. First we define an orientation ≺ of each edge of A in the following way: we choose a vertex v0 ∈ VeA ; as A is a tree, for each vertex v ∈ VeA there is a unique geodesic in VeA joining v to v0 , i.e. a unique minimal sequence of vertices v0 , . . . , v` , such that v` = v and hvi−1 , vi i, i = 1, . . . `, are edges of A; then we set vi−1 ≺ vi . Notice that for any vertex v 6= v0 there is only one edge hv 0 , vi such that v 0 ≺ v. The surjectivity of p∗ := H 1 (p) follows from that of p. Indeed for any (hv,e ) ∈ Z 1 (A, G00 ) and each edge e = hv 0 , v 00 i with v 0 ≺ v 00 , we can choose an element gv0 ,e ∈ Gv0 ,e = Ge such that pe (gv0 ,e ) = hv0 ,e . Setting gv00 ,e := gv−1 1 0 ,e we obtain an element (gv,e ) of Z (A, G) satisfying p∗ ([(gv,e )]) = [(hv,e )]). To prove the injectivity of p∗ let us consider two cohomological classes [(gv,e )] and [(hv,e )] ∈ H 1 (A, G) such that p∗ ([(gv,e )]) = p∗ ([(hv,e )]). TheQcocycles (pe (gv,e )) and (pe (hv,e )) being cohomologous, there exists (gv00 ) ∈ C 0 (A, G00 ) = v∈VeA G00v satisfying the following equalities in G00e , for any e = hv, wi ∈ EdA , v ≺ w: ρ00e 00 −1 v (gv ) pe (gv,e ) ρ00e 00 w (gw ) = pe (hv,e ) . By surjectivity of pv : Gv → G00v , v ∈ VeA , there are gv ∈ Gv such that gv00 = pv (gv ) and, thanks to the commutative diagrams  iv G0v  pv / Gv / G00 (6) v ρ0e ρev ρ00e v    v  ie / G0e  pe Ge / G00 e for any e = hv, wi, we obtain the equalities in Ge pe ρev (gv )−1 gv,e ρew (gw ) = pe (hv,e ) .  Therefore there exists ge0 ∈ G0e such that (?e ) ρev (gv )−1 gv,e ρew (gw )ie (ge0 ) = hv,e . Q We will construct a cocycle (kv ) ∈ v∈VeA Gv that satisfies the equality (??e ) ρev (kv )−1 gv,e ρew (kw ) = hv,e . for each edge e = hv, wi, v ≺ w, of A, using an induction process indexed by the lengths ` of the geodesics v0 , . . . , v` = v joining in A any vertex v ∈ VeA to the previously chosen vertex v0 . One call ` the distance of v to v0 and we denote ` = dA (v, v0 ). Consider the following assertion: Q (Hn ) there exists (kv ) ∈ v∈VeA ,dA (v,v0 )≤n Gv such that: (αn ) the relations (??e ) are fulfilled for every edge e = hv, wi, v ≺ w, with dA (v, v0 ) and dA (w, v0 ) ≤ n, (βn ) for every v ∈ VeA , 1 ≤ dA (v, v0 ) ≤ n, there exists fv0 ∈ G0v such that kv = gv iv (fv0 ). We will prove in a) that assertion H1 is true, and in b) that assertion Hn+1 is true as soon as assertion Hn is satisfied. - a) Let us consider the relation (?e ) for each edge e = hv, wi, with v = v0 . The restriction maps ρ0 ew : G0w → G0e being surjective, we choose gw 0 ∈ G0 such that g 0 = ρ0e (g 0 ). Using w e w w again the commutativity of all diagrams (6) we deduce the equality ρev0 (gv0 )−1 gv0 ,e ρew (gw iw (gw 0 )) = hv0 ,e . 0 ) and f 0 = g 0 , we obtain the assertion H . Setting kv0 = gv0 , kw = gw iw (gw w w 1 - b) Now let us suppose Hn satisfied, we will prove Hn+1 . Let us fix families Y Y (gv ) ∈ Gv and (ge0 ) ∈ G0e v∈VeA e∈EdA TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 11 fulfilling the relation (?e ) for every e ∈ EdA . Let us fix also a collection Y (fv0 ) ∈ G0v v∈VeA , dA (v,v0 )≤n such that the elements kv := gv iv (fv0 ) ∈ Gv , v ∈ VeA , dA (v, v0 ) ≤ n , (7) satisfy the relation (??e ) for every edge e of A whose vertices are at distances to v0 at most n. Let w be a vertex of A such that dA (w, v0 ) = n + 1. As noticed above, there is a unique edge ew = hvw , wi of A with vw ≺ w. Therefore vw is the unique vertex of A such that dA (vw , w) = n and hvw , wi is an edge of A. The relations (?ew ) and (7) give the equality: ρevww (ivw (fv0 w )) ρevww (kvw )−1 gvw ,ew ρeww (gw ) iew (ge0 w ) = hvw ,ew . 0 ∈ G0 such that ρ0ew (g 0 ) = g 0 . We have: As in step a), let gw w w w ew iew (ge0 w ) = iew (ρ0eww (gw 0 0 )) = ρeww (iw (gw )) , thus 0 ρevww (ivw (fv0 w )) ρevww (kvw )−1 gvw ,ew ρeww (gw iw (gw )) = hvw ,ew . On the other hand the element ρevww (ivw (fv0 w )) = iew (ρ0evww (fv0 w )) ∈ Gew belongs to the normal subgroup of Gew ker(pew ) = iew (G0ew ) = iew (ρ0eww (G0w )) = ρeww (iw (G0w )) . The following element of Gew : geew := g −1 ρevww (ivw (fv0 w )) g , g := ρevww (kvw )−1 gvw ,ew ρeww (gw iw (gw 0 )) , 0 ∈ G0 such that is also an element of ker(pew ). There exists gew w 0 geew = ρeww (iw (e gw )) . We finally obtain: 0 ρevww (kvw )−1 gvw ,ew ρeww (gw iw (gw )) ρeww (iw (e0 gw )) = hew ,vw , and ρevww (kvw )−1 gvw ,ew ρeww (gw iw (gw 0 0 gew )) = hew ,vw . We set 0 0 kw := gw iw (gw gew ) ∈ Gw , fw0 := gw 0 0 gew and we repeat this construction for each vertex whose distance to v0 is n + 1. The family (kv ), v ∈ VeA , dA (v, v0 ) ≤ n + 1, that we obtain satisfies assertion Hn+1 .  2.4. Pruning. A path in a tree A with origin c0 and extremity c` is a sequence L = (c0 , . . . , c` ), cj ∈ VeA ∪ EdA such that: • if cj , j < `, is a vertex, then cj+1 is an edge and cj ∈ cj+1 , • if cj , j < `, is an edge, then cj+1 is a vertex and cj 3 cj+1 . If R is a sub-tree of a A we can define for any vertex v of A \ R the notion of geodesic in A from v to R, as the unique minimal path Lv = (c0 , . . . , c` ) in VeA ∪ EdA such that c0 = v, c` ∈ VeR and c`−1 , . . . , c0 ∈/ VeR ∪ EdR . When v is a vertex of R, the geodesic Lv is reduced to the single element v. We define a partial order relation on VeA by setting v ≺R w if and only if the geodesic Lv is contained in the geodesic Lw . We will say that R is repulsive for a group-graph G over A, if for every edge e = hv, v 0 i ∈ EdA with v ≺R v 0 , the restriction map ρev0 : Gv0 → Ge is surjective. From [8, Theorem 3.11 and Remark 3.12] we have: 12 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM Theorem 2.10. Let R be a subtree of a tree A that is repulsive for a C-graph G over A. Then the map H 1 (ır ) : H 1 (A, G) → H 1 (R, r∗ G) , (gv,e )v∈e∈EdA 7→ (gv,e )v∈e∈EdR induced by the canonical C-graph morphism ır : G → r∗ G over the inclusion graph mor- phism r : R ,→ A, is a bijection of pointed sets. Moreover, if C = Ab or C = Vec then H 1 (ır ) is a C-isomorphism. 2.5. Direct image of a C-graph. Let ϕ : A → A0 be a morphism of graphs and let G be a C-graph over A. We define the direct image of G by ϕ as the C-graph ϕ∗ G over A0 given for v 0 ∈ VeA0 and e0 ∈ EdA0 by Y Y (ϕ∗ G)v0 := H 0 (ϕ−1 (v 0 ), G) ⊂ Gv , (ϕ∗ G)e0 := Ge ϕ(v)=v 0 ϕ(e)=e0 e0 and (ϕ∗ ρ) ((gv )v ) := (ρev (gv ))e , where v ∈ VeA and e ∈ EdA . It is implicitely understood v0 that the product over the empty set is the trivial group. There is a canonical morphism jϕ Q : ϕ∗ G → G of C-graphs over ϕ defined by the natural projections (jϕ )? : (ϕ∗ G)ϕ(?) ⊂ G• → G? for every ? ∈ VeA ∪ EdA . It can be ϕ(•)=ϕ(?) checked that if G0 is a C-graph over A0 then the maps a b HomA (G, ϕ∗ G0 ) −→ Homϕ (G, G0 ) ←− HomA0 (ϕ∗ G, G0 ) given by a(φ) = jϕ ◦ φ and b(φ̆) = φ̆ ◦ ιϕ are bijective. The preimage ϕ−1 (v 0 ) of a vertex v 0 ∈ VeA0 by a graph morphism ϕ : A → A0 is always a subgraph of A. If G is a group-graph over A we will denote by H 1 (ϕ−1 (v 0 ), G) the 1-cohomology set of the pull-back of G by the inclusion map ϕ−1 (v 0 ) ,→ A. Lemma 2.11. Let ϕ : A → A0 be a morphism of graphs, let G be a group-graph over A and consider the map H 1 (jϕ ) : H 1 (A0 , ϕ∗ G) → H 1 (A, G) defined in (4). (a) The image of H 1 (jϕ ) is the set of cohomology classes of 1-cocycles (he )e ∈ Z 1 (A, G) with he = 1 if ϕ(e) ∈ VeA0 . (b) H 1 (jϕ ) : H 1 (A0 , ϕ∗ G) → H 1 (A, G) is always injective. (c) If H 1 (ϕ−1 (v 0 ), G) = 1 for all v 0 ∈ VeA0 , then H 1 (jϕ ) : H 1 (A0 , ϕ∗ G) → H 1 (A, G) is surjective. Proof. By fixing an orientation for each edge of A and A0 we have bijections Y Y Z 1 (A0 , ϕ∗ G) ' (ϕ∗ G)e0 and Z 1 (A, G) ' Ge . e0 ∈EdA0 e∈EdA Under these identifications the map H 1 (jϕ ) is induced by Y Y Y Y jϕ1 : Z 1 (A0 , ϕ∗ G) ' (ϕ∗ G)e0 = Ge → Ge ' Z 1 (A, G) e0 ∈EdA0 e0 ∈EdA0 ϕ(e)=e0 e∈EdA which puts 1 in the factor Ge when ϕ(e) ∈ / EdA0 , this proves assertion (a). To prove assertion (b) let us fix (ge0 )e0 , (he0 )e0 ∈ Z 1 (A0 , ϕ∗ G) and (kv )v ∈ C 0 (A, G) safisfying (kv ) ?G jϕ1 (ge0 ) = jϕ1 (he0 ) in Z 1 (A, G). For any v 0 ∈ VeA0 we check that kv0 := (kv )v∈ϕ−1 (v0 )∩VeA ∈ H 0 (ϕ−1 (v 0 ), G). Then kv0 ∈ (ϕ∗ G)v0 and (kv0 )v0 ∈ C 0 (A0 , ϕ∗ G) satisfies (kv0 )?ϕ∗ G (ge0 ) = (he0 ) in Z 1 (A0 , ϕ∗ G). To prove assertion (c), let us fix a 1-cocycle (ge )e ∈ Z 1 (A, G). Since H 1 (ϕ−1 (v 0 ), G) = 1 for each v 0 ∈ VeA0 there is (kv )v∈ϕ−1 (v0 )∩VeA ∈ C 0 (ϕ−1 (v 0 ), G) such that (kv ) ? (ge ) = 1 in Z 1 (ϕ−1 (v 0 ), G). Then (kv )v∈VeA ∈ C 0 (A, G) satisfies (kv ) ? (ge ) = (he ) with he = 1 if ϕ(e) ∈ VeA0 . We conclude using assertion (a).  TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 13 2.6. Regular group-graph. The support of a group-graph G over a graph A is the set of vertices and edges where the corresponding group is non-trivial: supp(G) = {? ∈ VeA ∪ EdA | G? 6= {1}} , (8) with 1 denoting the identity element. Remark 2.12. Let G be a group-graph over A and let A0 be a subgraph of A obtained by removing some edges of A which are not in the support of G. Then the morphism ∼ H 1 (ıj ) : H 1 (A, G)−→H 1 (A0 , j ∗ G) induced by the canonical morphism ıj : G → j ∗ G over the inclusion j : A0 ,→ A is an isomorphism.  Definition 2.13. We will say that a group-graph G over A is regular if the restriction morphisms ρev : Gv → Ge are isomorphisms as soon as v, e ∈ supp(G). Let A0 be a subtree of a tree A. An edge e = hv, v 0 i ∈ EdA is adjacent to A0 if v ∈ VeA \ VeA0 and v 0 ∈ VeA0 . We define the contraction A/A0 as the tree whose vertices are VeA/A0 = (VeA \ VeA0 ) t {vA0 } and whose edges are the edges of A which do not belong to A0 and are not adjacent to A0 , jointly with an additional edge ẽ = hv, vA0 i for each adjacent edge e = hv, v 0 i to A0 with v∈/ VeA0 , ∼ EdA \ EdA0 −→ EdA/A0 , e 7→ e or ẽ. (9) There is a natural surjective graph morphism cA0 : A → A/A0 given by cA0 (v) = vA0 if v ∈ VeA0 and cA0 (v) = v otherwise. If A00 ⊂ A0 ⊂ A are subtrees of a tree A then we have a natural isomorphism ∼ j : A/A0 −→(A/A00 )/(A0 /A00 ) such that cA0 /A00 ◦ cA00 = j ◦ cA0 . (10) If G is a C-graph over A the direct image G̃ := (cA0 )∗ G over A/A0 satisfies G̃vA0 = H 0 (A0 , G), G̃ẽ = Ge if e ∈ EdA is adjacent to A0 and G̃? = G? otherwise. Lemma 2.14. Let G be a regular C-graph over a tree A and let A0 be a subtree of A such that all its edges are contained in the support of G. Then (cA0 )∗ G is a regular C-graph over A/A0 . Proof. It is easy to check when A0 has only one edge. In the general case, we proceed by induction on the number of edges of A0 using isomorphisms (10).  We call active edge of a regular C-graph G over a tree A any edge a = hv, v0 i ∈ EdA such that Ga 6= {0} and Gv0 = {0}. If Gv 6= 0, the vertex v will be called active vertex associated to a and denoted by va . If Gv = Gv0 = {0} and Ga 6= {0}, we select one of the two vertices v or v0 as active vertex associated to a. Let (Sα )α∈I be the collection of path connected components of supp(G), i.e. the maximal subsets of supp(G) such that any two elements can be joined by a path in supp(G). We say that Sα is an active component if it contains an active edge, or equivalently an active vertex. We denote by I 0 the set of indices α ∈ I such that Sα is active and not reduced to a single edge. Let A be the collection of all active edges. Now, let us choose one edge aα in each active component Sα , α ∈ I 0 , and let us write A 0 := A \ {aα ; α ∈ I 0 } . Theorem 2.15. Let G be a regular C-graph over a tree A. If A0 = ∅ then H 1 (A, G) = 1, otherwise we consider the map Y [δG ] : Ga → H 1 (A, G) a∈A 0 14 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM / A0 Ga → Z 1 (A, G) defined by δG ((ga )a ) = (gv,e ) with gv,e = 1 if e ∈ Q induced by δG : a∈A 0 and gva ,a = ga−1 , gv0 ,a = ga for a = hva , v0 i ∈ A 0 . Then [δG ] is bijective and if C = Ab or C = Vec then [δG ] is a C-isomorphism. Moreover, if C = Vec and all the vector spaces G? , ? ∈ supp(G), have the same dimension d then dim H 1 (A, G) = (a − p) · d where a is the number of active edges, p is the number of active connected components of supp(G) not reduced to a single edge. Proof. We reason by induction on the number n(A, G) of path connected components of supp(G) not reduced to a single vertex or a single edge. If n(A, G) = 0 the statement is clear. If n(A, G) > 0 we consider a path connected component Sα of supp(G) not reduced to a single edge nor a single vertex. It contains a nonempty maximal subgraph Cα0 . If Sα is an active component we consider the graph Cα given by EdCα = EdCα0 ∪ {aα } ⊂ supp(G) and VeCα = VeCα0 ∪ {vaα , v0 } where aα = hvaα , v0 i is the active edge previously chosen to define A 0 . If Sα is not an active component then we set Cα := Cα0 . Let c : A → à = A/Cα be the contraction of the subtree Cα ⊂ A. By Lemma 2.14 the C-graph G̃ = c∗ G over à is regular and A 0 ' A˜0 under the bijection (9). Moreover we have the following commutative diagram [δG ] δG ' / Z 1 (A, G) / H 1 (A, G) Q Ga a∈A 0 O O jc1 H 1 (jc ) Q δ G̃ã G̃ / Z 1 (Ã, G̃) / H 1 (Ã, G̃) 8 ã∈A˜0 [δ ] G̃ where the left vertical arrow, induced by the bijection (9) using that G̃ã = Ga , is the identity. It is clear that every vertex of Cα ∩ supp(G) is repulsive for the restriction of G to Cα . By applying Theorem 2.10 we deduce that H 1 (Cα , G) = 1 so that hypothesis (c) in Lemma 2.11 is fulfilled for the contraction map c : A → Ã. Consequently H 1 (jc ) is bijective (or a C-isomorphism when C = Ab or C = Vec). It is easy to see that if Sα is an active component then ṽ := c(Cα ) ∈ Veà does not belong to the support of G̃, i.e. G̃ṽ = H 0 (Cα , G) = 1. If Sα is not active then {ṽ} is a path connected component of supp(G̃). In both cases n(Ã, G̃) = n(A, G) − 1. By the inductive hypothesis [δG̃ ] is bijective (or a C-isomorphism). Therefore [δG ] is bijective (or a C-isomorphism). The last assertion is trivial.  2.7. Tensor product. If T is a Vec-graph over a graph A and W is a C-vector space we can define the Vec-graph T ⊗C W in an obvious way and we obtain a functor ⊗C : VecG × Vec → VecG , VecG being the category of C-vector space-graphs. The commutative property between tensor product and direct sum gives an isomorphism between the functors (T, W ) 7→ C ∗ (A, T ⊗C W ) and (T, W ) 7→ C ∗ (A, T ) ⊗C W, from VecG × Vec to the category of vector space complexes. It induces an isomorphism  ∼ (T, W ) 7→ H 1 (A, T ⊗C W ) −→ (T, W ) 7→ H 1 (A, T ) ⊗C W  (11) between functors from the category VecG × Vec to Vec. TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 15 3. Equisingular deformations of foliations 3.1. Deformations of foliations. Consider a germ F of singular foliation at the origin of C2 , given by a germ Z = a(x, y)∂x + b(x, y)∂y of holomorphic vector field with {a(x, y) = b(x, y) = 0} = {0}. Let Q· = (Q, u0 ) be a germ of manifold. A deformation of F over Q· is a germ of foliation FQ· on (C2 × Q, (0, u0 )) defined by a germ of vertical (tangent to the fibers of the canonical projection prQ : C2 × Q → C2 ) vector field X = A(x, y, u)∂x + B(x, y, u)∂y , whose restriction to C2 × {u0 } is equal to F, D prQ · X = 0 , ι ∗ F Q· = F , ι : C2 ,→ C2 × Q , ι(x, y) := (x, y, u0 ) . The germ Q· is called parameter space of FQ· . If λ is a germ of holomorphic map from a germ of manifold P · = (P, t0 ) to Q satisfying λ(t0 ) = u0 , the pull-back of FQ· by λ is the deformation λ∗ FQ· of F over P · , defined by the vector field λ∗ X := A(x, y, λ(t))∂x + B(x, y, λ(t))∂y . When Q = {u0 }, λ is the constant map and λ∗ FQ· is called constant deformation over P · and is denoted by FPct· . Two deformations FQ· and FQ 0 of F with same parameter space Q· are topologically · 0 conjugated, or C -conjugated, if there is a germ of homeomorphism Φ that is a defor- mation of idC2 , that sends the leaves of FQ· on that of FQ 0 · ∼ 0 Φ : (C2 × Q, (0, u0 ))−→(C2 × Q, (0, u0 )) , prQ ◦ Φ = prQ , Φ ◦ ι = ι, Φ(FQ· ) = FQ ·; we will say that Φ is a conjugacy of deformation from FQ· to FQ 0 and we will denote · 0 Φ : FQ· → FQ· . We will say that a deformation is trivial if it is conjugated to the constant deformation. 0 , the pull-back λ∗ Φ of Φ by a map germ λ : P · → Q· , Remark 3.1. (a) If Φ : FQ· → FQ · defined by ∼ λ∗ Φ : (C2 × P, (0, t0 ))−→(C2 × P, (0, t0 )) , λ∗ Φ(x, y, t) := Φ(x, y, λ(t)) , is a conjugacy from the deformation λ∗ FQ· to λ∗ FQ 0 . (b) If µ : N · → P · is a germ of · holomorphic map, we have the relation (λ ◦ µ) FQ· = µ∗ λ∗ FQ· . ∗  Let us recall that a deformation FQ· is called equireducible if there exists a map germ called equireduction map EFQ· : (MFQ· , Eu0 ) → (C2 × Q, (0, u0 )) (12) obtained by composition of proper holomorphic map germs EFQ· = E1 ◦ · · · ◦ Ek , Ej : (Mj , Kj ) → (Mj−1 , Kj−1 ) , (M0 , K0 ) = (C2 × Q, (0, u0 )) , (Mk , Kk ) = (MFQ· , Eu0 ), fulfilling the following properties (i)-(iii) below: for 1 ≤ j ≤ k let us write E j := E1 ◦ · · · ◦ Ej : (Mj , Kj ) → (C2 × Q, (0, u0 )) , πj := prQ ◦ E j : M j → Q, j j −1 (F · ) on M , then for j = 1, . . . , k, we must and let us denote by FQ · the foliation (E ) Q j have: j (i) on an open neighborhood of Kj in Mj the singular locus of FQ is regular and the restriction of πj to it is a covering map over an open neighborhood of u0 in Q; (ii) Ej is a blow-up map germ with center a union Cj of components of the singular locus j−1 of FQ · and Kj = Ej−1 (Kj−1 ); moreover C1 is the singular locus Sing(FQ· ) of FQ· ; 16 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM (iii) there is an open neighborhood U ⊂ Q of u0 such that for any u ∈ U the restriction of FQk to π −1 (u) is a reduced foliation at each of its singular points; moreover the · k restriction of E k to πk−1 (u) is the minimal reduction map of the germ at pr−1Q (u) ∩ −1 Sing(FQ· ) of the restriction of FQ to prQ (u). We will write: ] EFQ· := EF−1Q· (C1 ) , π ] := πk : (MFQ· , Eu0 ) → Q· , FQ k · := FQ· ; (13) By induction on j = 1, . . . , k, we check that π ] is a submersion. The exceptional di- visor EFQ· is an hypersurface with normal crossing and the restriction of π ] to each of its irreducible components is a holomorphically trivial fibration with fiber P1 . Its special fiber Eu0 = EF−1Q· (0, u0 ) = EFQ· ∩ π ] −1 (u0 ) . (14) is a curve with normal crossings and irreducible components biholomorphic to P1 ; the restriction of EFQ· to the special fiber Mu0 := π ] −1 (u0 ) of MFQ· is identified to the reduction map EF : (MF , EF ) → C2 of F, EF ' EFQ· |Mu0 : (Mu0 , Eu0 ) −→ C2 × {u0 } ' C2 , (Mu0 , Eu0 ) ' (MF , EF ) , (15) ] and the special fiber of FQ · , ] Fu] 0 := FQ · |M u , (16) 0 is identified to the reduced foliation F ] := EF−1 (F) on MF . Notice that any constant deformation FQ ct is equireducible and its reduction map is the product map of the reduction · map of F with the identity map of Q: EF ct· = EF × idQ : (MF × Q, EF × {u0 }) → (C2 × Q, (0, u0 )) , (m, u) 7→ (EF (m), u) ; Q Using the fact that pull-back process induces biholomorphisms at the fibers level one checks the following property: Proposition 3.2. The pull-back µ∗ FQ· of an equireducible deformation FQ· over Q· of a foliation F by a holomorphic map germ µ : P · → Q· , is an equireducible deformation of F over P · and its equireduction map is the pull-back µ∗ EFQ· of the equireduction map of F Q· . For equireducible deformations we may consider a special class of C 0 -conjugacies: Definition 3.3. Let FQ· and FQ 0 be two deformations over Q· = (Q, u ) of a foliation F · 0 2 ∼ 2 and let F : (C × Q, (0, u0 ))−→(C × Q, (0, u0 )), prQ ◦ F = prQ , be a homeomorphism that 0 . We will say that F is excellent or of class sends the leaves of FQ· to the leaves of FQ · ex C , if (1) F lifts through the reduction maps of these foliations EFQ· : (MFQ· , Eu0 ) → C2 × Q , EF 0 · : (MF 0 · , Eu0 0 ) → C2 × Q , Q Q i.e. there is a (unique) germ of homeomorphism F ] : (MFQ· , Eu0 ) → (MF 0 · , Eu0 0 ) Q satisfying EF 0 · ◦ F ] = F ◦ EFQ· , Q (2) F] is holomorphic in a neighborhood of each point of Sing(Eu0 ) ∪ Sing(Fu] 0 ) ⊂ Eu0 , except perhaps at the singular points of Eu0 that are nodal singularities of the special fiber Fu] 0 of FQ ] · , cf. (16). TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 17 Remark 3.4. According to Camacho-Sad index Theorem, there is a non-nodal singular ] point of FQ · in each invariant component of the special fiber Eu0 of the exceptional divisor of the reduction of FQ· ; consequently the holomorphy property (2) in Definition 3.3 induces ] the transversal holomorphy of F ] at any regular point of the foliation FQ ·.  Remark 3.5. If µ : P · → Q· is a holomorphic map germ and F is a C ex -conjugacy between two equireducible deformations FQ· and GQ· of the same foliation F, then µ∗ F is a C ex -conjugacy between the deformations µ∗ FQ· and µ∗ GQ· .  3.2. Equisingular deformations. Let us consider an equireducible foliation FQ· , over a germ of manifold Q· = (Q, u0 ), of a foliation F on (C2 , 0). We keep all previous nota- tions (13)-(16). We will denote by Diff(C × Q, (0, u0 )) the group of germs of holomorphic automorphisms of (C × Q, (0, u0 )) fixing the point (0, u0 ) and by Diff Q (C × Q, (0, u0 )) := {h ∈ Diff(C × Q, (0, u0 )) | prQ ◦ h = prQ } , (17) the subgroup of automorphisms over Q. Now let us fix a point oD in each Fu0 -invariant component D of Eu0 that is a non- singular point of this foliation and let us choose a germ of holomorphic submersion gD : (MFQ· , oD ) → (C × Q, (0, u0 )) , gD (oD ) = (0, u0 ) , ] that is a map over Q· , i.e. prQ ◦ gD = π ] , constant on the leaves of FQ · . We will say ] that gD is a transversal factor to FQ · at the point oD . Classically the holonomy of ] FQ · along D realized on gD is the group representation of the fundamental group of the ] punctured component D∗ := D \ Sing(FQ ) F] · HDQ : π1 (D∗ , oD ) → Diff Q (C × Q, (0, u0 )) (18) that associates to the class of a loop γ in D∗ , γ(0) = oD , the automorphism hγ over Q· such that gD ◦ h−1γ is the analytic extension (equivalently the extension as first integral of ] FQ· ) of gD along γ. Up to composition by inner automorphisms of Diff Q (C × Q, (0, u0 )), this representation does not depend on the choice of the point oD in D∗ or that of the transversal factor gD . For a germ of holomorphic map µ : P · → Q· we will identify to MF the special fibers of the reductions of FQ· and of µ∗ FQ· , see (15). The pull-back by µ of a submersion over Q· , ] resp. a first integral over Q· of FQ · · , being a submersion over P , resp. a first integral over ] P · of µ∗ FQ · , we have: ] • the pull-back µ∗ gD of a transversal factor gD to FQ · · , considered as a map over Q , ] is a transversal factor to µ∗ FQ · at the same point of the same invariant component ] D of EF , and the holonomy of µ∗ FQ · represented on it is ] µ∗ FQ · F] · HD = µ∗ ◦ HDQ , (19) where µ∗ : Diff Q (C × Q, (0, u0 )) → Diff P (C × P, (0, t0 )) , h 7→ µ∗ h : (z, t) 7→ h(z, µ(t) ;  ] • if HD denotes the holonomy group of FQ · along D, i.e. the image of the morphism F] · ] HDQ , then µ∗ (HD ) is the holonomy group of µ∗ FQ · along D. Let us denote by Diff(C, 0) × {idQ } ⊂ Diff Q (C × Q, (0, u0 )) the subgroup of automor- phisms that do not depend on u ∈ Q. 18 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM Definition 3.6. We say that a deformation FQ· of F over Q· is equisingular, if it ] is equireducible and the holonomy representation of the reduced foliation FQ · along any invariant component D of the special fiber Eu0 of the exceptional divisor EFQ· is conjugated to a morphism with values in Diff(C, 0) × {idQ }: there exists ψD ∈ Diff Q (C × Q, (0, u0 )) such that F] · τψD ◦ HDQ : π1 (D∗ , oD ) → Diff(C, 0) × {idQ } ⊂ Diff Q (C × Q, (0, u0 )) −1 where τψD is the inner automorphism φ 7→ ψD ◦ φ ◦ ψD of Diff Q (C × Q, (0, u0 )). In other words, an equireducible foliation FQ· is equisingular if and only if for any invariant F] · component D of Eu0 , the holonomy representation HDQ is conjugated to the holonomy ct ] representation along D of the constant foliation FQ · , i.e. F] · F ct·] τψD ◦ HDQ = HDQ . (20) for an appropriate ψD ∈ Diff Q (C × Q, (0, u0 )). Proposition 3.7. The pull-back by a holomorphic map germ µ : P · → Q· of an equisingular deformation FQ· over Q· is an equisingular deformation over P · . Proof. Let us suppose equality (20) satisfied, and let us denote by κP · : P · → P · the constant map t 7→ t0 . Since κ∗P · µ∗ FQ· is the constant deformation of F over P · , it suffices to prove the equality ] ] µ∗ FQ · κ∗ · µ∗ FQ · τµ∗ ψD ◦ HD = HDP , (21) κP · : P·→ P·being the constant map t 7→ t0 . Trivially we have: τµ∗ ψD ◦ µ∗ = µ∗ ◦ τψD . Hence, it follows from (19) and (20): ] µ ∗ FQ · F] · F] · ] κ∗ · FQ · ] µ∗ κ∗Q· FQ · τµ∗ ψD ◦ HD = τµ∗ ψD ◦ µ∗ ◦ HDQ = µ∗ ◦ τψD ◦ HDQ = µ∗ ◦ HDQ = HD , the last equality follows from the fact that the constant deformation κ∗Q· FQ· is equisingular and thus fulfills the corresponding relation (19). Equality (21) results from the trivial relation κQ· ◦ µ = µ ◦ κP · that gives µ∗ κ∗Q· FQ· = κ∗P · µ∗ FQ· .  3.3. Good trivializing system. In all the sequel we will make the hypothesis that the considered foliations F are generalized curves, i.e. the reduced foliations F ] have no saddle-node singularities. Consequently at each singular point s of F ] in an invariant component D of EF , the holonomy around s and the Camacho-Sad index CS(F ] , D, s) determine the analytical type of the germ of F ] at s. We will see that this property will imply the “C ex -rigidity” of F ] along each component D of EF , in the meaning that the germ along D of the reduced foliation associated to any equisingular deformation of F, is C ex -conjugated to that of the constant deformation. Let us consider an equisingular deformation FQ· of F. Let us keep the previous notations (13)-(16) and let us denote by ι] : (MF , EF ) ,→ (MFQ· , Eu0 ) , EFQ· ◦ ι] = ι ◦ EF , (22) the lifting throught the reduction and equireduction maps of the canonical immersion ι : (C2 , 0) ,→ (C2 × Q, (0, u0 )) , (x, y) 7→ (x, y, u0 ) . (23) We will also denote by j ] : MF ,→ MF × Q the canonical immersion m 7→ (m, u0 ), by prQ : C2 × Q → Q and pr]Q : MF × Q → Q the canonical projections, and we again write π ] := prQ ◦ EFQ· : (MFQ· , Eu0 ) → Q. TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 19 Theorem 3.8. If F is a generalized curve, then we can associate to each irreducible com- ponent D of EF , a homeomorphism germ ∼ ΨD : (MFQ· , ι] (D))−→(MF × Q, D × {u0 }) , so that: (i) ΨD is a map over Q· , i.e. pr]Q ◦ ΨD = π ] , and corresponds to the identity map over u0 , i.e. ΨD ◦ ι] = j ] ; (ii) ΨD is holomorphic at each point of Sing(Eu0 ) ∪ Sing(Fu] 0 ) except perhaps at the sin- gular points of Eu0 that are nodal singularities of Fu] 0 ; ] ct ] (iii) ΨD conjugates the foliation FQ · to the foliation FQ· obtained after equireduction of the constant deformation FQ ct ; · (iv) the germ of ΨD ◦ Ψ−1 D0 at the intersection point {sDD0 } = (D ∩ D0 ) × {u0 } of two irreducible components D and D0 , is the identity when either sDD0 is a nodal singular point of Fu] 0 or sDD0 is a regular point of Fu] 0 . The collection (ΨD )D of these homeomorphisms indexed by the components of EF is called good trivializing system for FQ· . Proof. We will proceed in five steps. -Step 1: construction of ΨD on a neighborhood Ω of ι] (D \ Sing(F ] )) with D invariant. ] Let us fix a point oD ∈ D \ Sing(F ] ) and a transversal factor to FQ · g : (MFQ· , ι] (oD )) → (C × Q, (0, u0 )) . Let us also fix a C ∞ submersion ρ : W → ι] (D) defined on a neighborhood W of ι] (D) in MFQ· , such that: (i) the restriction of ρ to ι] (D) is the identity map, (ii) the restriction ρ0 of ρ to the special fiber Mu0 := π ] −1 (u0 ) is a submersion, (iii) ρ is holomorphic at ι] (oD ) and also at each point s ∈ Sing(Eu0 ) ∪ Sing(Fu] 0 ), (iv) the fibers ρ−1 (s), s ∈ Sing(Eu0 ) ∪ Sing(Fu] 0 ), are invariant by FQ] ·. There is a unique section σ : (C × Q, (0, u0 )) → (MFQ· , ι] (oD )) of g, whose image coincides with the fiber ρ−1 (ι] (oD )). We do a similar construction for the constant deformation. First, at the point ŏD := j ] (oD ) we have the following transversal factor ğ = ğ0 × idQ : (MF × Q, (ŏD , u0 )) → (C × Q, (0, u0 )) , ğ0 := prC ◦ g ◦ ι] , with prC : C × Q → C the first projection. Next, we define the following submersion ρ̆ onto D × {u0 } ρ̆ : ι] −1 (W ) × Q → D × {u0 } , (m, u) 7→ (ι] −1 ◦ ρ0 ◦ ι] (m), u0 ) . Finally we consider the section σ̆ of ğ whose image coincides with ρ̆−1 (ŏD ). Now let us fix an element ψD ∈ Diff Q (C × Q, (0, u0 )) that conjugates the holonomy ] ct ] representation along ι] (D) of FQ · realized on g, to that of FQ· realized on ğ: F] · F ct·] −1 τψD ◦ HDQ = HDQ , τψD (φ) := ψD ◦ φ ◦ ψD , as in Definition 3.6 and equation (20). By classical theory of path lifting in leaves of regular 1-dimensional foliations, there is a homeomorphism Ψ : Ω → Ω̆ where Ω is an open neighborhood of ι] (D \ Sing(F ] )) in W ⊂ MFQ· and Ω̆ is an open neighborhood of (D \ Sing(F ] )) × {u0 } in MF × Q, satisfying the following properties: 20 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM • when restricted to ι] (D \ Sing(F ] )), Ψ coincides with the map ∼ Ψ[ : ι] (D)−→D × {u0 } , p 7→ (ι] −1 (p), u0 ) , • Ψ sends the fiber ρ−1 (ι] (oD )) to the fiber ρ̆−1 (ŏD ) and its restriction to ρ−1 (ι] (oD )) is equal to σ̆ ◦ ψD ◦ g, ] ct ] • Ψ conjugates the restriction of FQ · to Ω to that of FQ· to Ω̆, • Ψ is a lift of Ψ[ , that is ρ̆ ◦ Ψ = Ψ[ ◦ ρ. By construction, Ψ is a map over Q· , i.e. pr]Q ◦Ψ = π ] and its germ along ι] (D\Sing(F ] )) is unique. Moreover, ρ being holomorphic at the singular points, Ψ is also holomorphic on the intersection of Ω with neigborhoods of these points. -Step 2: extension at a non-nodal singular point. The proof of Mattei-Moussu’s theo- rem [11] given in [6, Theorem 5.2.1] shows that the closures of Ω and Ω̆ at the non-nodal singular points of Fu] 0 are neighborhoods of these points; in fact, the estimates made in [6] are uniform in the parameters, see also [3]. Since Ψ constructed in Step 1 is holomorphic near these singularities we conclude that Ψ extends holomorphically at these points by classical Riemann’s theorem. -Step 3: construction of ΨD when D is dicritical. Classically, the holomorphic type ] ] of FQ · along a dicritical divisor ι (D) only depends on the self-intersection number of ι] (D) in the special fiber π ] −1 (u0 ). Thus there exists a germ of biholomorphism Ψ : ∼ ] ct ] (MFQ· , ι] (D))−→(MF × Q, D × {u0 }) over Q· that conjugates FQ · to FQ· . Up to conju- ct ] gating by a biholomorphism of (MF × Q, D × {u0 }) leaving FQ · invariant we may also suppose that Ψ ◦ ι = j . It remains to modify Ψ at each point where ι] (D) meets an- ] ] other component ι] (D0 ) so that at this point the germ of Ψ coincides with that of the homeomorphism constructed in Step 1 for D0 . This follows from the following lemma. Lemma 3.9. Let us consider two germs of biholomorphisms over Cq  ∼ g j : C2 × Cq , D1 × {0} −→ C2 × Cq , g j (D1 × {0}) ,  j = 1, 2, of the following form: g j (x, y, u) = (g1j (x, u), g2j (x, y, u), u) , u = (u1 , . . . , uq ) , with g1j : C × Cq , D1 × {0} → C, satisfying  g1j (0, u) = g2j (x, 0, u) = 0 , g1j (x, 0) = x , g2j (x, y, 0) = y . (24) Here D1 denotes the closed unit disk on C. Then for suitable real numbers 0 < r1 < R1 < 1, there exists a homeomorphism germ ∼ g : (C2 × Cq , D1 × {0})−→(C2 × Cq , g(D1 × {0})), g(x, y, u) = (g1 (x, u), g2 (x, y, u), u) , of the same form, satisfying also Properties (24), such that g(x, y, u) = g 1 (x, y, u) if |x| ≤ r1 , g(x, y, u) = g 2 (x, y, u) if R1 ≤ |x| ≤ 1 . Proof of Lemma 3.9. Left to the reader.  -Step 4: Extension at a nodal singular point s ∈ / Sing(Eu0 ). The extension of Ψ will be done “by linearity” as follows. Let ζ = (ζ1 , · · · , ζq ) : (Q, u0 ) → (Cq , 0) be a chart on Q· . ] Since the holonomy around s is a trivial family, Camacho-Sad index of FQ · restricted to TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 21 the fibers of π ] is constant along the singular locus. By linearization (with parameters) there is a local chart χ = (w1 , w2 , z1 , . . . , zq ) : (MF Q· , s) → (C2 × Cq , 0) , zj = ζj ◦ π ] , ] −1 (L), where L is the one dimensional foliation on Cq+2 such that FQ · = χ x,y,u1 ,...,uq , with singular set {(0, 0)} × Cq , given by the linear differential equations system xdy − αydx = du1 = · · · = duq = 0 , α ∈ R>0 . (25) We may suppose that the x-axis corresponds to ι] (D) and that ρ corresponds to the linear projection on the first coordinate w1 in C . At the point s̆ := (ι] −1 (s), u0 ) ∈ MF × Q, 2 with the local chart χ̆ = (w1 ◦ ι] , w2 ◦ ι] , ζ1 , . . . , ζq ) : (MF × Q, s̆) → (C2 × Cq , 0) , the component D × {u0 } corresponds again to the x-axis, ρ̆ is the linear projection and ct ] −1 (L). Notice that χ̆ ◦ Ψ ◦ χ−1 is a holomorphic automorphism leaving we have: FQ · = χ̆ invariant the foliation L, defined on a neigbourhood in Cq+2 of a punctured disk D∗ = {0 < |x| ≤ ε, y = 0, u = 0}. It has the following expression:   χ̆ ◦ Ψ ◦ χ−1 (x, y, u) = x, Ψ(x, e y, u), u , u = (u1 , . . . , uq ) , Ψ(x, e 0, u) = 0 , Ψ(x, e y, 0) = (x, y, 0) . ∗ On {x} × C , x ∈ D , the holonomy of L along the loop γx (t) = (e2πit x, 0, . . . , 0), q+1 t ∈ [0, 1], is the linear automorphism h(x, y, u) = (x, e2παi y, u). The commutativity of χ̆ ◦ Ψ ◦ χ−1 with these holonomy maps, Ψ(x, e e2παi y, u) = e2παi Ψ(x, e y, u) , gives Ψ(x, e y, u) = A(x, u)y , A(x, u) 6= 0 , where A is a holomorphic map defined on an open set of Cq+1 x,u1 ,...,uq that contains the compact set defined by ε/2 ≤ |x| ≤ ε, |uj | ≤ η for j = 1, . . . , q. By the invariance of L under χ̆ ◦ Ψ ◦ χ−1 , we have the equality: dx dΨ e dx dy (−α + ) ∧ (−α + ) ∧ du1 ∧ · · · ∧ duq = 0 . x Ψ e x y Hence: dA dx dy ∧ (−α + ) ∧ du1 ∧ · · · ∧ duq = 0 . A x y dy Since the differential form −α dx 2 x + y in C posseses only constant holomorphic first in- tegrals, A does not depend on the variable x. It extends trivially to a holomorphic map defined on {|x| ≤ ε, |uj | ≤ η, j = 1, . . . , q}. Thus the automorphism χ̆ ◦ Ψ ◦ χ−1 extends to a neighborhood of the origin in Cq+2 , as a holomorphic automorphism Ψ leaving L invariant. We conclude that the desired extension of Ψ is given by χ̆−1 ◦ Ψ ◦ χ. -Step 5: Extension at a nodal singular point s ∈ Sing(Eu0 ). If by Step 3 we extend at a such a point s the homeomorphisms along the components D and D0 meeting at s constructed in Step 1, we obtain two germs at s of biholomorphisms Ψ and Ψ0 that do not fulfill the requested property (iv). Thanks to the following lemma, whose proof is left to the reader, we modify them so that they coincide as germs at s. 2 ∼ Lemma 3.10. Let g j : D1 × Dqη −→Wj , j = 1, 2, be two biholomorphisms leaving invariant the linear foliation L defined by (25), such that (1) g j (x, y, 0) = (x, y, 0), 22 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM (2) g 1 (x, y, u) = (x, g21 (x, y, u), u), with g21 (x, 0, u) = 0, (3) g 2 (x, y, u) = (g12 (x, y, u), y, u) with g12 (0, y, u) = 0, where Dη = {|z| < η} ⊂ C. Then for η > 0 small enough, there are suitable real numbers 0 < C1 < C2 < 1 < C20 < C10 such that there exists a homeomorphism germ 2 ∼ 2 g : D1 × Dqη −→D1 × Dqη , (x, y, u) 7→ (g1 (x, y, u), g2 (x, y, u), u) satisfying also Properties (1)-(3) above, that is equal to g1 when |y| ≤ C1 |x|α , to g2 when |y| ≥ C10 |x|α and to the identity map when C20 |x|α < |y| ≤ C2 |x|α . This achieves the proof of Theorem 3.8.  3.4. Deformation functor. Let us consider the pointed set · Def Q F := {[FQ ] : FQ equisingular deformation of F}/ ≈C · · ex of all C ex -conjugacy classes [FQ· ] of germs of equisingular deformations FQ· over Q· of a fixed foliation F. This set is pointed by the class of the constant deformation. · The assignment Q· 7→ Def Q F is a contravariant functor, because according to Remark 3.1, to a germ µ : P · → Q· corresponds the well defined pull-back map · · µ∗ : Def Q P F → Def F , [FQ· ] 7→ [µ∗ FQ· ] . ∼ Theorem 3.11. Let φ : (C2 , 0)−→(C2 , 0) be a homeomorphism germ that is a C ex -conjugacy between two germs of foliations G and F = φ(G) which are generalized curves. Let Q· = (Q, u0 ) be a germ of manifold. Then there exists a bijective map · ∼ · φ∗ : Def Q Q F −→ Def G defined by the following property: (?) φ∗ ([FQ· ]) = [GQ· ] if and only if there exists a germ of homeomorphism over Q ∼ Φ : (C2 × Q, (0, u0 ))−→(C2 × Q, (0, u0 )) , prQ ◦ Φ = prQ , that sends the leaves of GQ· on that of FQ· , is excellent, and satisfies Φ(x, y, u0 ) = (φ(x, y), u0 ) . ∼ Moreover, if ψ : (C2 , 0)−→(C2 , 0), ψ(K) = G, is a C ex -conjugacy between a germ of foliation K and G, then · ∼ Q· (φ ◦ ψ)∗ = ψ ∗ ◦ φ∗ : Def Q F −→Def K . (26) · Proof. Under the hypothesis of the theorem, let us consider a class c ∈ Def Q F and an equisingular deformation FQ· of F in c. In a first step we will construct an equisingular deformation GQ· of G and a C ex -homeomorphism Φ satisfying Φ(GQ· ) = FQ· , such that Φ ◦ ι = ι ◦ φ, with ι : C2 ,→ C2 × Q, ι(x, y) := (x, y, u0 ). Then in a second step we · will verify that the class [GQ· ] ∈ Def Q G does not depend on the choice of the deformation FQ· in c. Finally in a third step we check that the map φ∗ that associate to each class · c = [FQ· ] ∈ Def QF the class of the deformation GQ defined in the first step, fulfills the · property (?) and the functorial relation. -Step 1. We again denote by ι] the lifting (22) of ι through the reduction and equireduction maps EF and EFQ· , by j ] : MF ,→ MF × Q the lifting of ι through through EF and EF ct· , Q that is j ] (m) := (m, u0 ), and finally by φ] : (MG , EG ) → (MF , EF ) , EF ◦ φ] = φ ◦ EG , TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 23 the lifting of φ through EF and the reduction map EG : (MG , EG ) → (C2 , 0) of G. The following homeomorphism φ]Q· : (MG × Q, EG × {u0 }) −→ (MF × Q, EF × {u0 }) , (m, u) 7→ (φ] (m), u) , ct ] · ] is excellent and sends the reduced constant foliation GQ · over Q with special fiber G , to ct ] the constant foliation FQ · . According to Theorem 3.8, let us fix a good trivializing system for FQ· ∼ ] ct ] ΨD : (MFQ· , ι] (D))−→(MF × Q, D × {u0 }) , ΨD (FQ · ) = F Q· , ΨD ◦ ι] = j ] , indexed by the irreducible components D of EF . At the intersection points {sDD0 } := (D ∩ D0 ) × {u0 }, D ∩ D0 6= ∅, the cocycles ∼ ΦDD0 := (φ]Q· )−1 ◦ ΨD ◦ Ψ−1 ] D0 ◦ φQ· : (MG × Q, sDD )−→(MG × Q, sDD ) 0 0 (27) are germs of biholomorphisms over Q· fulfilling the properties ct ] ct ] ΦDD0 (GQ · ) = GQ· , ΦDD0 ◦ j ] = j ] . Indeed according to (ii) and (iv) in Theorem 3.8, if the intersection point D ∩ D0 is not a nodal singular point of F ] , the germs of φ]Q· at the point sDD0 and of ΨD0 at Ψ−1 D (sDD0 ) −1 are holomorphic; otherwise, at ΨD (sDD0 ) the germs ΨD and ΨD0 coincide and ΦDD0 is the identity map. Let us consider the manifold germ (N, EG0 ) := tD (MG × Q, D × {u0 })/ (ΦDD0 ) , θ : (N, EG0 ) → Q· , obtained by gluing neighborhoods in MG × Q of the irreducible components j ] (D) using these cocycles, and endowed with the germ of submersion θ obtained by gluing the germs of the canonical projection prQ : (MG × Q, D × {u0 }) → Q. Since ΦDD0 are the identity on the special fiber MG × {u0 }, j ] induces an embedding ∆ : (MG , EG ) ,→ (N, EG0 ) that is a biholomorphism germ onto (θ−1 (u0 ), EG0 ). The gluing maps leaving invariant ct ] 0 0 the constant foliation GQ · , they define in the ambient space (N, EG ) a foliation germ GQ· tangent to the fibers of θ, that coincides with ∆(G ] ) on θ−1 (u0 ). Thanks to the relations ] ] Ψ−1 −1 −1 D0 ◦ φQ· ◦ ΦDD0 = ΨD ◦ φQ· given by (27), the collection of homeomorphisms ] ct ] ] ΦD := Ψ−1 ] ] D ◦ φQ· : (MG × Q, j (D)) → (MFQ· , ι (D)), ΦD (GQ · ) = F Q· , glue as a homeomorphism over Q· ∼ Φ0 : (N, EG0 )−→(MFQ· , ι] (EF )) , prQ ◦ Φ0 = θ , 0 to that of F ] . As the maps φ] and Ψ , this map is excellent that send the leaves of GQ· Q· D in the meaning that it is also holomorphic at the non-nodal points of the corresponding foliation. It satisfies: Φ 0 ◦ ∆ = ι] ◦ φ ] ; (28) On the other hand, the preimage by Φ0 of the exceptional divisor EFQ· := EF−1Q· ({0} × Q) is an hypersurface EQ which is also exceptional in N (see [10, p. 306]): there is a holomorphic map germ C : (N, EG0 ) → (C2 × Q, (0, u0 )) such that prQ ◦ C = θ , C(EQ ) = {0} × Q , that is a biholomorphism from complementary of EQ to the complementary of {0} × Q. This last property allows to define a germ of holomorphic foliation GQ· on (C2 × Q, (0, u0 )), 24 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM 0 by C. Up to perform an additional biholomorphism we also that is the direct image of GQ · require that ∆ contracts to the embedding ι, i.e. C ◦ ∆ = ι ◦ EG , so that 0 ] ] GQ· |C2 ×{0} = C(GQ · |θ −1 (u ) ) = C(∆(G )) = ι(EG (G )) = ι(G). 0 In other words, GQ· is a deformation of G. By construction this deformation is equisingular and more precisely there is a biholomorphism germ ∼ F : (N, EG0 )−→(MGQ· , Eu0 0 ) , such that 0 ] EGQ· ◦ F = C , F (GQ ·) = G · , Q F ◦ ∆ = k] , (29) k ] being the lifting of ι through the reduction map EG and the equireduction map EGQ· : (MGQ· , Eu0 0 ) → (C2 × Q, (0, u0 )) of the deformation GQ· , (N, EG0 ) 8 ∆ C ( (MG , EG ) F (C2 × Q, (0, u0 )) . 6 k] &  EGQ· (MGQ· , Eu0 0 ) Now let us notice that since C(EQ ) = {0} × Q, the homeomorphism germ Φ0 contracts through C and EFQ· to a germ of map Φ : (C2 × Q, (0, u0 )) → (C2 × Q, (0, u0 )) , EFQ· ◦ Φ0 = Φ ◦ C , that by construction is a germ of homeomorphism satisfying: prQ ◦ Φ = prQ , Φ(GQ ) = FQ , Φ ◦ ι = ι ◦ φ. To achieve Step 1, it remains to check that Φ is excellent. Indeed, Φ0 ◦ F −1 is a lifting of Φ, Φ0 ◦ F −1 : (MGQ· , Eu0 0 ) → (MFQ· , Eu0 ) , EFQ· ◦ (Φ0 ◦ F −1 ) = Φ ◦ C ◦ F −1 = Φ ◦ EGQ· . Since Φ0 is excellent we deduce that Φ is also excellent. -Step 2. Notice first that up to C ex -conjugacy the deformation GQ· obtained by this construction does not depend on the choice of the good trivializing system (ΨD )D . If 0 and Ğ · are similarly obtained from another good trivializing system (Ψ̆ ) (N̆ , ĔG0 ), ĞQ· Q D D −1 then the homeomorphisms ΨD ◦ Ψ̆D : (MF × Q, D × {u0 }) → (MF × Q, D × {u0 }) glue to an excellent homeomorphism that conjugates ĞQ 0 and G 0 and contracts to an excellent · Q· conjugacy between the deformations ĞQ· and GQ· of G. Now let us show that [GQ· ] does not depend on the choice of the representative FQ· · of c ∈ Def Q F . Let F̆Q be another representative of c, ĞQ a deformation of G and · · Φ̆ : (C2 × Q, (0, u0 )) → (C2 × Q, (0, u0 )) a germ of excellent homeomorphism such that Φ̆(ĞQ· ) = F̆Q· , prQ ◦ Φ̆ = prQ and Φ̆ ◦ ι = ι ◦ φ. Then ĞQ· is C ex -conjugated to GQ· . Indeed, if ξ is an C ex -homeomorphism such that ξ(F̆Q· ) = FQ· and ξ ◦ ι = ι, then the C ex -homeomorphism Υ := Φ−1 ◦ ξ ◦ Φ̆ trivially satisfies Υ(ĞQ· ) = GQ· and Υ ◦ ι = ι. This implies that the map φ∗ is well-defined. -Step 3. The direct implication of (?) is clear. To see the converse, we apply the previous argument to the case F̆Q· = FQ· . The functorial relation follows directly from (?) and TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 25 · if φ = idC2 then φ∗ is the identity map on Def Q ∗ F . This implies that φ is bijective and ∗ −1 −1 (φ ) = (φ ) . ∗  We check that for any holomorphic map germ µ : P · → Q· and any deformation GQ· ∈ φ∗ ([F Q· ]) we have: φ∗ ([µ∗ FQ· ]) = [µ∗ GQ· ] , i.e. the following diagram is commutative: · φ∗ · Def Q / Def Q (30) F G µ∗ µ∗   φ∗ Def PF · / Def P · . G Lemma 3.12. Under the assumptions of Theorem 3.11, if µ : P · → Q· and λ : R· → P · are holomorphic maps between germs of manifolds, φ : G → F and ψ : K → G are C ex - conjugacies and if we write · (µ, φ)∗ := φ∗ ◦ µ∗ : Def Q P· F → Def G , then we have (λ, ψ)∗ ◦ (µ, φ)∗ = (µ ◦ λ, φ ◦ ψ)∗ . Proof. It suffices to check that the following diagram is commutative using (26), dia- gram (30) and Remark 3.5, · Def Q F (µ◦λ, φ◦ψ)∗ (31) (µ, φ)∗ µ∗  # φ∗ (µ◦λ)∗ Def PF · / Def P · G (λ, ψ)∗ λ∗ λ∗ "   #  φ∗ ψ∗ Def R · / Def R· / Def R· F G : K (φ◦ψ)∗  Let us denote now by • Fol the category whose objects are the germs of foliations on (C2 , 0) which are generalized curves and whose morphisms φ : G → F are the germs of C ex - conjugacies, φ(G) = F; • Set· the category of pointed sets whose objects are the pairs (A, a) formed by a set and a point of this set, the morphisms F : (A, a) → (B, b) being maps from A to B such that F (a) = b; • Man· the subcategory of Set· , consisting of pairs (A, a) with A endowed with a complex manifold structure, the morphisms being holomorphic pointed sets mor- phisms µ : P · → Q· . Definition 3.13. The deformation functor is the contravariant functor · Def : Man· × Fol → Set· , (Q· , F) 7→ Def Q F defined by associating to any morphism (µ, φ) : (P · , G) → (Q· , F), the pull-back map · · (µ, φ)∗ : Def Q P F → Def G , [FQ· ] 7→ φ∗ (µ∗ ([FQ· ])) = φ∗ ([µ∗ FQ· ]) . The fact that Def is a functor follows from Lemma 3.12. 26 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM As a direct consequence of Theorem 3.11, if [GP · ] = (µ, φ)∗ ([FQ· ]) with µ : P · := (P, t0 ) → Q· , then for t ∈ P sufficiently close to t0 the foliations GP · |C2 ×{t} and FQ· |C2 ×{µ(t)} are C ex -conjugated. 4. Group-graphs of automorphisms and transversal symmetries 4.1. Group-graph of C ex -automorphisms. Given a foliation F and a germ of manifold · Q· = (Q, u0 ), let us consider the following sheaf AutQ F over the exceptional divisor EF of Q· the reduction of F: if U is an open subset of EF , then AutF (U ) is the group of germs along U × {u0 } of C ex -homeomorphisms over Q· Φ : (MF × Q, U × {u0 }) −→ (MF × Q, U × {u0 }) ] ct ] leaving invariant the constant family FQ · with fiber the reduced foliation F and moreover being the identity map on the special fiber MF × {u0 }. The same definition works when · U is not open in EF and in that case AutQ F (U ) coincides with the inductive limit of Q· AutF (V ) for V open subset of EF containing U , cf. Section 2.2. The property “excellent” means here that at each point m in an invariant component of EF the germ Φm of Φ is a holomorphic germ if m ∈ Sing(EF ) ∪ Sing(F ] ), except perhaps if m is a nodal singularity of F ] belonging to Sing(EF ), and that Φm is transversely holomorphic if m is a regular point of F ] . According to [1] if D is an invariant component of EF and if one saturates by F ] a neighborhood of Sing(F ] ) ∩ D, one obtains a set that contains all the regular points of F ] in D. Therefore when U contains D, the above transversal holomorphy property is automatically induced by the holomorphy at the singular points; for this reason we did not need to require it in Definition 3.3 of C ex -conjugacy. Definition 4.1. We call group-graph of automorphisms over Q· of F and we denote · by AutQ F the following group-graph over the dual graph AF of EF : · · (i) AutQ Q F (D) = AutF (D), if D ∈ VeAF is invariant; · (ii) AutQ F (D) = {ID }, if D ∈ VeAF is dicritical ; · · · (iii) AutQ Q Q 0 F (e) is the stalk AutF (s) of the sheaf AutF at the point s defined by e = hD, D i, D ∩ D0 = {s}, if s is neither a regular point nor a nodal singular point of F ] ; · (iv) AutQ 0 0 F (e) = {Ie }, if e = hD, D i, D ∩ D = {s} and s is either a regular point or a ] nodal singular point of F ; Q · · (v) the restriction map ρeD : AutF (D) → AutQ F (e) is the restriction· map of the sheaf · AutF when D is invariant and e fulfills condition (iii); ρeD : AutQ Q F (D) → {Ie } is the trivial map otherwise; where ID , resp. Ie , denotes the germ along D × {u0 }, resp. at the point (s, u0 ), of the identity map idMF ×Q . · Remark 4.2. Notice that restricted to its support, see (8), AutQ F coincides with the group- Q· graph associated to the sheaf AutF defined in Section 2.2. The elements of VeAF ∪ EdAF not belonging to this support are exactly the elements given by (ii) and (iv): the vertices that are dicritical components of EF , the edges hD, D0 i with D or D0 dicritical and the · edges hD, D0 i for witch F ] has a nodal singularity at the point D∩D0 . Clearly supp(AutQ F ) Q· F α is a sub-graph of EF called cut-graph of F. We denote by supp(AutF ) = α∈A AF its decomposition into connected components which we call cut-components of AF . We have: · · H 1 (AF , AutQ H 1 (AαF , AutQ Y F ) = F ). (32) α∈A TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 27 This decomposition, produced by the points (ii) and (iv) and Remark 2.12 in the above definition, may seem artificial. However the cocycles (ΨD ◦ Ψ−1 D0 ) that we will consider are constructed using good trivializing systems (ΨD )D provided by Theorem 3.8. Consequently the property (iv) of that theorem guarantees that ΨD ◦ Ψ−1 D0 is trivial when D or D is 0 dicritical or when F ] has a nodal singularity at D ∩ D0 .  ∼ Now let us consider a germ of C ex -homeomorphism φ : (C2 , 0)−→(C2 , 0) which conju- gates two foliations G and F, φ(G) = F, and the corresponding C ex -conjugacy ∼ φ]Q· : (MG × Q, EG × {u0 })−→(MF × Q, EF × {u0 }) , (p, u) 7→ (φ] (p), u) , ct ] ct ] between the contant families GQ · and FQ· . Let us denote by φE : EG → EF the restriction · of φ] to the exceptional divisors. If U ⊂ EG is an open set and Φ belongs to φ−1 E AutQ F (U ) = Q · ] −1 ] Q · AutF (φE (U )), then φQ· ◦ Φ ◦ φQ· belongs to AutG (U ). As described in Section 2.2, the homeomorphism φE induces an isomorphism between the dual graphs of EG and EF Aφ : AG → AF , D 7→ φE (D) , hD, D0 i 7→ hφE (D), φE (D0 )i . (33) We thus obtain the following isomorphism of group-graphs over Aφ : · · φ∗ : AutQ Q F → AutG , · ] −1 · AutQ ] Q F (Aφ (?)) 3 Φ 7→ φQ· ◦ Φ ◦ φQ· ∈ AutG (?) , ? ∈ VeAF ∪ EdAF . On the other hand let µ : →P· Q· be a holomorphic map between germs of manifolds. ct ] The pull-back being a functor and, by definition, FQ · being the pull-back by a constant map, it follows: ct ] ct ] µ∗ F Q · = FP · and µ∗ φ]Q· = φ]P · . Thus we have the equality µ∗ (φ]Q−1 · ◦ Φ ◦ φ]Q· ) = φ]P−1 · ◦ µ∗ Φ ◦ φ]P · . We finally obtain the following commutative diagram of group-graph morphisms · φ∗ · AutQ / AutQ (34) F G µ∗ µ∗   φ∗ AutPF · / AutP · . G Using the relations φ∗ ◦ ψ ∗ = (ψ ◦ φ)∗ and (µ ◦ λ)∗ = λ∗ ◦ µ∗ we deduce as in (31) that the following assignments · (Q· , F) 7→ (AF , AutQ F ), · P· (µ, φ) : (P · , G) → (Q· , F) 7→ (µ, φ)∗ := µ∗ ◦ φ∗ : AutQ   F → AutG , (35) define a contravariant functor with values in the category GrG of group-graphs. When restricted to generalized curves this functor is denoted by Aut : Man· × Fol → GrG . (36) From now on F will be a generalized curve. For any deformation FQ· of F over Q· , let us choose a good trivializing system (ΨD )D∈VeAF meaning that the properties (i)-(iv) of Theorem 3.8 are satisfied. The family (ΦD,e )D∈e , defined by ΦD,e = ΨD ◦ Ψ−1 D0 , e = hD, D0 i , (37) · is an element of Z 1 (AF , AutQ F ). 28 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM · Lemma 4.3. The cohomology class C(FQ· ) ∈ H 1 (AF , AutQ F ) of the above cocycle (ΦD,e )D∈e does not depend on the choice of a good trivializing system; moreover it only depends on · the C ex -class [FQ· ] ∈ Def Q F . Proof. We check that if (ΨD )D∈VeAF and (Ψ0D )D∈VeAF are two good trivializing systems Q · for FQ· , then the homeomorphisms ΨD ◦ Ψ0−1 D belong to AutF (D) and define a 0-cocycle whose action on the cocycle (ΨD ◦ Ψ−1 0 0 −1 D0 ) gives the cocycle (ΨD ◦ Ψ D0 ). Hence C(FQ ) is · ex well defined. On the other hand if Φ is an C -conjugacy between another deformation GQ· of F over Q· and FQ· , Φ(GQ· ) = FQ· , we easily verify that (ΨD ◦ Φ] )D∈VeAF is a good trivializing system for GQ· with the same associated cocycle.  Theorem 4.4. For any germ of manifold Q· and any foliation F which is a generalized curve, the map · · · CFQ : Def Q 1 Q · F → H (AF , AutF ) , [FQ· ] 7→ C(FQ· ) , Q· is bijective. Moreover the collection of the maps CF define a natural isomorphism ∼ C : Def −→H 1 ◦ Aut , between the contravariant functor Def : Man· × Fol → Set· introduced in Definition 3.13 · and the contravariant functor (Q· , F) 7→ H 1 (AF , AutQ F ) obtained by composing the con- travariant functor Aut : Man· × Fol → GrG with the covariant cohomological functor H 1 : GrG → Set· defined in (5) (pointed by the class of the identity). · Proof. The maps CFQ are well defined thanks to Lemma 4.3. We proceed in three steps: -Step 1: functoriality of C. We must prove that, given a germ of holomorphic map µ : P · → Q· and an C ex -conjugacy φ : G → F between generalized curves, the following diagram is commutative: · φ∗ · Def Q / Def Q F · G · Q CF CGQ µ∗ v v · H 1 (φ∗ ) · H 1 (AF , AutQ / H 1 (AG , AutQ ) µ∗ F ) G H 1 (µ∗ )   φ∗ H 1 (µ∗ ) Def PF · / Def P · G · · P CF CGP  u  u H 1 (φ∗ ) · H 1 (AF , AutPF ) / H 1 (AG , AutP · ) G Let us check first the commutativity of the lateral faces: If (ΨD )D∈VeAF is a good triv- ializing system for FQ· then (µ∗ ΨD )D∈VeAF is also a good trivializing system for µ∗ FQ· . Consequently we have: · Q · CFP ([µ∗ FQ· ]) = [µ∗ ΨD ◦ µ∗ Ψ−1 1 ∗ −1 1 ∗ D0 ] = H (µ )([ΨD ◦ ΨD0 ]) = H (µ ) ◦ CF ([FQ ]). · To check the commutativity of the top face, we notice that by definition c := H 1 (φ∗ ) ◦ · · ] −1 CFQ ([FQ· ]) is the cohomology class in H 1 (AG , AutQ −1 ] G ) of the cocycle (φQ· ◦ΨD ◦ΨD0 ◦φQ· ). It coincides with the cocycle (27) used in the proof of Theorem 3.11 to construct the de- · formation GQ· ∈ φ∗ ([FQ· ]). Therefore c = CGQ ([GQ· ]). The same arguments give the commutativity of the lower face. That of the back and front faces of the cube results from TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 29 the relations (30) and (34) respectively. · -Step 2: injectivity of CFQ . Let (ΨD )D∈VeAF resp. (Ψ0D )D∈VeAF be good trivializing 0 , inducing the same cohomology systems for two equisingular deformations FQ· , resp. FQ · · · class in H 1 (AF , AutQ Q F ). There exist ΦD ∈ AutF (D), D ∈ VeAF , such that the following relation: 0 −1 ΦD ◦ ΨD ◦ Ψ−1 −1 0 D 0 ◦ ΦD 0 = Ψ D ◦ Ψ D 0 is satisfied for any pair (D, D0 ) of irreducible components of EF such that {sDD0 } = D ∩ D0 is neither a nodal singularity or a regular point of F ] . This relation also means that the homeomorphisms KD := Ψ0 −1 D ◦ ΦD ◦ ΨD defined on neighborhoods of D × {u0 } coincide on neighborhoods of (sDD0 , u0 ) and induce a C ex -conjugacy between FQ· and FQ 0 . · · · -Step 3: surjectivity of CFQ . Given a cocycle (ΦD,e ) ∈ Z 1 (AF , AutQ F ), the construction of an equisingular deformation FQ equipped with a good trivializing system satisfying (37), · may be done by a gluing process as in the proof of Theorem 3.11.  4.2. Sheaf of transversal symmetries. Let us fix again a foliation F and a germ of manifold Q· = (Q, u0 ). For an open set U ⊂ EF we will say that an automorphism Q· Φ ∈ AutF (U ) fixes the leaves, if it leaves invariant the codimension one foliation F ] × · Q· Q. We denote by FixQ F ⊂ AutF , the subsheaf of normal subgroups consisting of these automorphisms. We will describe in an explicit way the quotient sheaf · · · SymQ F = AutQ Q F /FixF . To do that let us consider the normal subgroup Diff 0Q (C × Q, (0, u0 )) = {φ ∈ Diff Q (C × Q, (0, u0 )) | φ(z, u0 ) ≡ (z, u0 )}, of the group Diff Q (C × Q, (0, u0 )) defined in (17), and for any subgroup G ⊂ Diff Q (C × Q, (0, u0 )) let us adopt the following notations: • CQ· (G) is the centralizer of G, i.e. the subgroup of Diff Q (C × Q, (0, u0 )) whose elements commute with any element of G; • CQ0 (G) = C · (G) ∩ Diff 0 (C × Q, (0, u )); · Q Q 0 • in the monogenous case G = hhi, we write CQ· (h) and CQ 0 (h) instead of C · (hhi) · Q 0 and CQ· (hhi). Now let us fix an invariant component D of EF . For m ∈ D \ Sing(F ] ), let us choose a germ of holomorphic submersion g : (MF , m) −→ (C, 0) · constant on the leaves of Any φ ∈ AutQ F ]. F (m) factorizes through g × idQ , defining an 0 element g∗ (φ) ∈ Diff Q (C × Q, (0, u0 )) such that g∗ (φ) ◦ (g × idQ ) = (g × idQ ) ◦ φ . The holomorphy of g∗ (φ) results from the fact that φ is transversely holomorphic by defi- nition. Clearly · g∗ : AutQ 0 F (m) → Diff Q (C × Q, (0, u0 )) (38) is a surjective group morphism. Lemma 4.5. The following sequence · · g∗ 1 → FixQ Q 0 F (m) → AutF (m) → Diff Q (C × Q, (0, u0 )) → 1 (39) is exact. 30 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM Proof. For the exactness at the central term, let us first notice that the germ at (m, u0 ) · of an element φ ∈ AutQ ] F (m) preserves the codimension one foliation F × Q if and only if there is a factorization g∗ (φ)[ : g×idQ prC (MF × Q, (m, u0 )) / (C × Q, (0, u0 )) / (C, 0) φ g∗ (φ) g∗ (φ)[  g×idQ  prC  (MF × Q, (m, u0 )) / (C × Q, (0, u0 )) / (C, 0) e u), u), g∗ (φ)[ exists if and only if φ(p, where prC (z, u) = z. Since g∗ (φ)(p, u) = (φ(p, e u) does not depend on u. But φ(z, [ e u0 ) = z, therefore g∗ (φ) exists if and only if g∗ (φ) = idC×Q .  Lemma 4.6. If U ⊂ D is open4 and connected and p ∈ U , then we have the exact sequence: · · · 1 → FixQ Q Q F (U ) → AutF (U ) → SymF (p) . (40) Proof. The statement is trivial if U = W ∩ D and W ⊂ MF is an open subset trivializing the foliation F ] . If U ∩ Sing(F ] ) = ∅ we cover U by open subsets in MF trivializing F ] and we conclude by connectedness of U . For the last case p ∈ Sing(F ] ) we take a point · q ∈ U \ Sing(F ] ) close to p and we note that if the germ of an element φ ∈ AutQ F (U ) at p Q· Q· is in FixF (p) then the germ of φ at q also belongs to FixF (q). By applying the exactness of sequence (40) substituting U and p by U \ Sing(F ] ) and q respectively, we deduce that · φ ∈ FixQF (U \ Sing(F ] )). It remains to see that the germ of φ at p0 ∈ U ∩ Sing(F ] ) belongs · to FixQF (p 0 ). For this we use the holomorphy of φ at p0 and the following characterization: · φ ∈ FixQ 0 ∗ 0 F (p ) ⇔ (φ ω) ∧ ω ≡ 0, where ω is the germ at p of a holomorphic 1-differential form defining the codimension one foliation F × Q. ]  Let us fix an invariant component D of EF and let us denote by iD : D ,→ EF the inclusion map. Let us also fix a transverse fibration ρ : (MF × Q· , D) → D satisfying properties (i)-(iv) described in the step 1 of the proof of Theorem 3.8 and let us consider the subsheaf over D · Q· AutQ −1 F ,ρ ⊂ iD AutF of automorphisms preserving the fibration ρ. Lemma 4.7. If F is a generalized curve, for any connected open set U of D and any point m ∈ U \ Sing(F ] ), the following assertions hold: · (i) The sheaf AutQ ] F ,ρ is locally constant over D \ Sing(F ) and the morphism g∗ defined in (38) induces an isomorphism · AutQ 0 F ,ρ (m) ' Diff Q (C × Q, (0, u0 )) ; · · (ii) The restriction map AutQ Q ] F ,ρ (U ) → AutF ,ρ (U \ Sing(F )) is an isomorphism and g∗ · induces an isomorphism AutQ 0 F ,ρ (U ) ' CQ· (HU ), where HU is the holonomy group F ct·] HU := HDQ (π1 (U \ Sing(F ] ), m)) ⊂ Diff 0Q (C × Q, (0, u0 )) ; · · (iii) For any p ∈ U , the natural map AutQ Q F ,ρ (U ) → AutF ,ρ (p) is injective. Proof. Assertion (i) follows from the fact that the restriction of g × idQ to each fiber of ρ is a local diffeomorphism onto (C × Q, (0, u0 )). Assertion (ii) is a consequence of Mattei- Moussu’s Theorem as in step 2 of the proof of Theorem 3.8. To prove assertion (iii), let · us assume that the germ of φ ∈ AutQ / Sing(F ] ) then F ,ρ (U ) at p is the identity. If p ∈ 4U may not be open in E . F TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 31 φ|U \Sing(F ] ) = id by assertion (i) and φ = id using also assertion (ii). If p ∈ Sing(F ] ) then there is q ∈ / Sing(F ] ) close to p such that the germ of φ at q is the identity; we apply the previous case and we conclude by the holomorphy of the germ of φ at p.  Proposition 4.8. If F is a generalized curve, then the composition of the group sheaves morphisms · · · AutQ −1 Q −1 Q F ,ρ ,→ iD AutF → iD SymF (41) is an isomorphism. · · Proof. We have to see that AutQ Q F ,ρ (p) → SymF (p) is an isomorphism for each p ∈ D. The case p ∈ D \ Sing(F ] ) follows from assertion (i) in Lemma 4.7 and the exact sequence · (39) in Lemma 4.5. Next, we fix p ∈ Sing(F ] ) and we take [φp ] ∈ SymQ F (p). There is a · neighborhood U of p in D and φU ∈ AutQ F (U ) such that [φU ] 7→ [φp ]. In the commutative diagram below · φ̃p ∈ AutQ / SymQ· (p) 3 [φp ] F ,ρ (p) F O O Q· · φ̃U ∈ AutQ / AutF· (U ) 3 [φU ] F ,ρ (U ) Q FixF (U )  _ ∼ b c   · AutQ ] ∼ / SymQ· (U \ Sing(F ] )) F ,ρ (U \ Sing(F )) a F the arrow a is an isomorphism by the regular case already considered and the arrow b is · also an isomorphism by assertion (ii) of Lemma 4.7. Hence there is φ̃U ∈ AutQ F ,ρ (U ) such · that [φ̃U ] and [φU ] are sent to the same element in SymQ F (U \ Sing(F ] )). Using the exact sequence (40) we deduce that the arrow c is injective and consequently φ̃U is sent to [φU ]. By the commutativity of the top square the germ φ̃p of φ̃U at p projects onto [φp ]. This shows that the composition (41) is surjective at p. The injectivity of the composition (41) at p follows, as in the proof of assertion (iii) in Lemma 4.7, using the holomorphy of φ̃U and the injectivity at the regular points, which has already been shown.  Corollary 4.9. If F is a generalized curve, for any connected open set U of D, the following assertions hold: · (i) The sheaf SymQF is locally constant on D \ Sing(F ] ); · (ii) The morphism g∗ induces an isomorphism SymQ F 0 (H ); (U ) ' CQ· U (iii) We have the exact sequence: · · · 1 → FixQ Q Q F (U ) → AutF (U ) → SymF (U ) → 1 . · · Proof. Assertions (i) and (ii) are obvious from the isomorphism AutQ Q F ,ρ ' SymF and assertions (i) and (ii) in Lemma 4.7. To check the exactness of the sequence in assertion · Q· (iii) it only remains to show the surjectivity of AutQ F (U ) → SymF (U ). This is so because the composition · · · AutQ Q Q F ,ρ (U ) ,→ AutF (U ) → SymF (U ) is an isomorphism thanks to Proposition 4.8.  32 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM 4.3. Group-graph of transversal symmetries. Let us again fix a foliation F that is a · Q· generalized curve. We consider the normal subgroup-graph FixQ F ⊂ AutF defined by · · · FixQ Q Q F (?) = AutF (?) ∩ FixF (?) , ? ∈ VeAF ∪ EdAF , · · where FixQ Q 0 0 F (e) denotes FixF (D ∩ D ) if e = hD, D i ∈ EdAF . Definition 4.10. The group-graph of transversal symmetries is the quotient group- · graph SymQ F defined by the group-graph exact sequence · · · πQ · · · 1 → FixQ Q F Q Q Q F → AutF −→ SymF = AutF /FixF → 1 . (42) For each invariant component D ∈ VeAF , using the exact sequence (iii) in Corollary 4.9 with U = D, we have a natural5 isomorphism: · · · ∼ · SymQ Q Q Q F (D) = AutF (D)/FixF (D)−→SymF (D) , (43) when F is a generalized curve. We check that if (µ, φ) : (P · , G) → (Q· , F) is a morphism in the category Man· × Fol, · P· then the morphism (µ, φ)∗ defined in (35) sends the group-graph FixQ F into FixG and it factorizes (see Remark 2.4) as a morphism of group-graphs over the graph morphism Aφ : AG → AF defined in (33), that we also denote by · · (µ, φ)∗ : SymQ P F → SymG . This allows to define a contravariant functor from Man· × Fol to GrG · Sym : (Q· , F) 7→ (AF , SymQ F ), (µ, φ) 7→ (µ, φ)∗ . Q · The collection {πF } of quotient maps (42) defines a natural transformation Aut → Sym . Q · By applying the functor H 1 : GrG → Set· to the morphisms πF we obtain maps · · H 1 (AF , AutQ · 1 Q · F ) → H (AF , SymF ) (44) defining a natural transformation H 1 ◦ Aut → H 1 ◦ Sym . It follows immediately from Lemma 4.12 below and Proposition 2.9 applied to the exact sequence (42) that: Proposition 4.11. For any germ of manifold Q· and any generalized curve F, the map (44) is bijective and consequently the natural transformation H 1 ◦ Aut → H 1 ◦ Sym is an isomorphism of contravariant functors from Man· × Fol to Set· . Lemma 4.12. Assume that F is a generalized curve. For any edge e = hD, D0 i of AF · Q· with D invariant, the restriction map FixQ F (D) → FixF (e) is surjective. 5 If A → A0 is a morphism of sheaves of groups over X sending a normal subgroup F into F 0 then for any open subset U ⊂ X the following diagram is commutative: A(U )/F (U ) / (A/F )(U )   A0 (U )/F 0 (U ) / (A0 /F 0 )(U ) TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 33 Proof. At the point {s} = D ∩ D0 we take local coordinates (x, y) : (MF , s) → (C2 , 0) such that the foliation F ] is defined by a vector field x∂x + yB(x, y)∂y with B(0, 0) 6= 0. Let us · Q· consider Φ ∈ FixQ F (e) = FixF (s). Let u = (u1 , . . . , uq ) be a centered coordinate system on Q· . In the chart χ = (x, y, u) ct ] ] the foliation FQ · is given by the vector field Z = x∂x +yB(x, y)∂y and the foliation F ×Q is defined by yB(x, y)dx − xdy = 0. Let us denote by ϕ = χ ◦ Φ ◦ χ−1 the expression of Φ in this chart. Since ϕ(x, y, 0) = (x, y, 0) and the points (x, y, u) and ϕ(x, y, u) belong to the R ϕ(x,y,t) R ϕ(x,y,t) dy same leaf Lx,y,u of F ] ×Q the function τ (x, y, t) = (x,y,t) dx x Lx,y,u = (x,y,t) yB(x,y) Lx,y,u is well defined and holomorphic in an open neighborhood Ω of C = {(x, y, u) : ε ≤ |x| ≤ 2ε, |y| ≤ ε, |u| ≤ δ} for 0 < δ  ε small enough, moreover τ (x, y, 0) = 0. By definition, the flow ΦZ Z ∞ function t of Z satisfies Φτ (p) (p) = ϕ(p) for p ∈ C. Let α : C → R be a C with compact support on x(Ω), that is equal to 1 in a neighborhood of {ε ≤ |x| ≤ 2ε}. The map p 7→ ξ(p) := ΦZ ∞ diffeomorphism, because its restriction to α(x(p))τ (p) (p) is a C u = 0 is the identity and moreover it is a local diffeomorphism as it can be easily checked by computing its Jacobian matrix. Clearly the map φ = ξ −1 ◦ ϕ coincides with ϕ on a neighborhood of s, it preserves the codimension 1 foliation F ] × P and φ(x, y, u) = (x, y, u) for ε ≤ |x| ≤ 2ε. Thus, Φ extends to a neighborhood of D as the identity and defines an · element of FixQ F (D).  · Now we will give an explicit expression of the group-graph SymQ F which will depend on the choice of the following additional data: Definition 4.13. A geometric system for an invariant component D of EF consists in: • a point oD ∈ D \ (Sing(EF ) ∪ Sing(F ] )) and a germ of holomorphic submersion g : (MF , oD ) → (C, 0) which is constant along the leaves of F ] ; • a collection {Up }p∈Sing(F ] )∩D of connected and simply connected open subsets of D such that Up ∩ Sing(F ] ) = {p} and oD ∈ p∈Sing(F ] )∩D Up . T For e = hD, D0 i with D ∩ D0 = {p} we denote by hD,e ∈ HD ⊂ Diff Q (C × Q, (0, u0 )) (45) ct ] the holonomy of FQ · along of a path in Up \ {p} of index 1 with respect to p, which belongs F ct·] to the holonomy group HD image of the morphism HDQ in (18). Proposition 4.14. Assume that F is a generalized curve. If D ∈ VeAF and e = hD, D0 i ∈ EdAF , after choosing a geometric system for D, the morphism (38) with m = oD induces isomorphisms · ∼ · ∼ GD,e : SymQ 0 F (e)−→CQ· (hD,e ) and GD : SymQ 0 F (D)−→CQ· (HD ) . · · Under these isomorphisms the restriction map SymQ Q F (D) → SymF (e) is just the inclusion 0 (H ) ,→ C 0 (h CQ · D Q· D,e ). · · · Proof. We have: SymQ Q Q F (e) = SymF (e) ' SymF (Up ), thanks to assertion (i) of Corol- lary 4.9, where {p} = D ∩ D0 . By assertion (ii) in Corollary 4.9 with U = Up , g∗ induces an · isomorphism SymQ F 0 (h (Up ) ' CQ · D,e ). The second isomorphism follows immediately from (43) and assertion (ii) of Corollary 4.9 with U = D.  34 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM 5. Finite type foliations and infinitesimal transversal symmetries 5.1. Finite type foliations. Given a foliation F which is a generalized curve, we will say that a vertex D, resp. an edge hD, D0 i, belonging to a cut-component AαF , α ∈ A, of AF (see Remark 4.2) is red for F if, using the notations in (45) with Q = {u0 }, the holonomy group HD of F ] is not finite, resp. the holonomy diffeomorphism hD,e (or equivalently hD0 ,e ) is not periodic. Classically a vertex D, resp. an edge hD, D0 i, is red if every holo- morphic first integral of F ] defined in a neighborhood of D, resp. D ∩ D0 , is constant. Notice that the red part RαF of AαF is a sub-graph. When it is connected and non- empty, we consider the partial order relation ≺RαF on VeAαF defined in Subsection 2.4. When RαF = ∅ we will consider the partial order relation ≺{v} on VeAαF defined by the subgraph {v} reduced to some single vertex v. Definition 5.1. We say that F is of finite type if for each α ∈ A one of the following conditions holds: (i) RαF 6= ∅ is connected and for any edge e = hD, D0 i ∈ (EdAαF \ EdRαF ) with D0 ≺RαF D, the holonomy group HD is generated by the holonomy map hD,e ; (ii) RαF = ∅ and AαF contains a vertex v such that we have: HD = hhD,e i for any edge e = hD, D0 i ∈ EdAαF with D0 ≺{v} D. We will denote by Folft ⊂ Fol the full subcategory of finite type foliations. · When F is of finite type, for every germ of manifold Q· the subgraph RαF is SymQ F - repulsive in AαF in the meaning of Section 2.4. Indeed for D ∈ VeAαF and e = hD, D0 i ∈ · EdAαF , thanks to Proposition 4.14, we have isomorphisms SymQ 0 F (D) ' CQ· (HD ) and · · SymQ 0 Q F (e) ' CQ· (hD,e ). As we will see later the cohomology of SymF is given by its restriction to the subgraph G RF := RαF ⊂ AF . α∈A Definition 5.2. We call restricted group-graph of transversal symmetries the group- · Q· graph RSymQ ∗ F = rF SymF over RF defined as the pull-back by the inclusion rF : RF ,→ AF : · · RSymQ Q F (?) = SymF (?) , ? ∈ VeRF ∪ EdRF , Notice that for any morphism φ : G → F in the category Fol, the graph isomorphism ∼ Aφ : AG → AF restricts to a graph isomorphism Rφ : RG −→RF . If µ : P · → Q· is a morphism in Man· , we consider the left diagram of group-graphs morphisms over the right diagram of graph morphisms: · (µ,φ)∗ Aφ SymQ / SymP · AF o AG F G O _ O F f ırF ırG over rF rG  %  ? ? · Rφ RSymQ F̄ / RSymP · RF o RG F G where ιrF and ιrG denote the canonical morphisms, see Definition 2.2. Since Aφ (RG ) ⊂ RF , the morphism F = ırG ◦ (µ, φ)∗ over f = Aφ ◦ rG factorizes through ırF , according to · Q· Remark 2.3, and defines a morphism of group-graphs F̄ : RSymQ F → RSymG over Rφ . By abuse of notation we will denote F̄ as (µ, φ)∗ . This allows to consider the contravariant functor · RSym : Man· × Fol → GrG, (Q· , F) 7→ (RF , RSymQ F ), (µ, φ) 7→ (µ, φ)∗ . TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 35 · · The collection of canonical morphisms ırF : SymQ Q F → RSymF of group-graphs over the graph morphisms rF : RF ,→ AF defines a natural transformation R : Sym → RSym between contravariant functors from Man· × Fol to GrG. It induces a natural transfor- mation R := H 1 (R) : H 1 ◦ Sym → H 1 ◦ RSym (46) between contravariant functors from Man· × Fol to Set· . By applying (32) and Theo- rem 2.10 to each subtree RαF ⊂ AαF , α ∈ A, we directly obtain: Theorem 5.3. For any germ of manifold Q· and any finite type foliation which is a generalized curve, the map · · ∼ · RQ 1 Q · 1 Q · F : H (AF , SymF ) −→H (RF , RSymF ) is bijective and the natural transformation R considered in (46) is an isomorphism of contravariant functors when restricted to the subcategory Man· × Folft . · We will see in the next section that the group-graph RSymQ F is abelian, so that the two functors in (46) restricted to Man· × Folft are isomorphic and take values in the category Ab of abelian groups, which can be seen as a subcategory of Set· by pointing by zero, see Section 2.3. 5.2. Sheaf of infinitesimal transversal symmetries. Given a foliation F let us con- sider now the following sheaves X F ⊂ B F over EF of tangent and basic holomorphic vector fields of F ] : the stalk B F (m) of B F at m ∈ EF is the C-vector space of germs at m of holomorphic vector fields in MF leaving invariant the foliation F ] and the divisor EF ; X F (m) is the subspace of B F (m) consisting of vector fields tangent to F ] . The quotient sheaf T F := B F /X F is called sheaf of infinitesimal transversal symmetries of F ] . · Similarly, given Q· = (Q, u0 ) a germ of manifold, we define B Q F the sheaf over EF Q· of OQ,u0 -modules whose stalks are the spaces B F (m) of germs at (m, u0 ) of holomorphic ct ] vector fields in MF ×Q leaving invariant the constant foliation FQ · and the divisor EF ×Q, that are vertical (i.e. tangent to the fibers of the projection MF × Q → Q) and zero on · Q· the special fiber MF × {u0 }; X QF ⊂ B F is the subsheaf consisting of vector fields which ct ] are tangent to FQ · and the quotient sheaf · · · TQ Q Q F := B F /X F ct ] is called the sheaf of infinitesimal transversal symmetries of FQ · . Notice that, if as usual we denote by MQ,u0 the maximal ideal of OQ,u0 , we have: · BQ F ⊗OQ,u0 (OQ,u0 /MQ ) = {0} = · 6 BF . · We will give local expressions for the stalks T F (m) and T Q F (m) at a point m in an ∼ invariant component D of EF . Let us fix in MF a local chart z = (z1 , z2 ) : Ω−→D2r satisfying r > 1, z(m) = (0, 0) , D = {z2 = 0} , EF = {z1 z2 = 0} ,  ∈ {0, 1} . We suppose that Ω ∩ Sing(F ] ) is either empty or reduced to {m}. We also fix a chart ∼ u : Ω0 −→Dqη , η > 0, on Q· with u(u0 ) = 0. · Let us denote by Vm , resp. by VmQ , the space of germs of vector fields Z in the submani- fold {z1 = 1} of Ω, resp. of Ω×Ω0 , at the point of coordinates (1, 0), resp. (1, 0, . . . , 0), that satisfy: (a) Z = 0 when z2 = 0, and (b) hm∗ (Z) = Z where hm is the classical holonomy 36 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM ct ] map of F ] , resp. of FQ · , along the loop z(t) = (e 2πti , 0), resp. z(t) = (e2πti , 0), u(t) = 0, t ∈ [0, 1], realized on the transverse manifold {z1 = 1}. If Y is a vector field on an open set U ⊂ MF we will consider the constant vertical extension YQct· on U × Q, i.e. the unique vertical vector field on U × Q related to Y by the projection U × Q → U . Lemma 5.4. Assume that F is a generalized curve. With the previous notations we have: (1) if F ] at m is singular and it is either (a) non-resonant, non-linearizable but formally linearizable or (b) resonant non-formally linearizable nor normalizable, then: · · T F (m) = {0} , TQ F (m) = {0} , Vm = {0} , VmQ = {0} ; (2) if F ] at m is not as in case (1) and any germ of holomorphic first integral of F ] at m is constant, then we may choose the coordinates z1 , z2 so that · T F (m) = C [Z] , T Q  ct  F (m) = MQ,u0 ZQ· , · Vm = C · Z|{z1 =1} , VmQ = MQ,u0 · ZQ ct · |{z1 =1} , ct | ct where ZQ · {z1 =1} denotes the restriction of ZQ· to {z1 = 1} and Z is the following vector field on Ω: ∂ (a) Z = z2 ∂z 2 when F ] is linearizable at m, (z1a z2b )k (b) Z = z ∂ 1+ζ(z1a z2b )k 2 ∂z2 when F ] is singular resonant normalizable at m, and z1 , z2 is chosen so that F ] is given by ω = 0 where ω := bz1 (1 + ζ(z1a z2b )k )dz2 + az2 (1 + (ζ − 1)(z1a z2b )k )dz1 , with a, b, k ∈ N∗ , (a, b) = 1, ζ ∈ C; (3) ifF] at m has a non-constant first integral F , then by choosing F minimal and z1 , z2 such that F (z1 , z2 ) = z1a z2b , a, b ∈ N, b 6= 0, (a, b) = 1, we have:     ∂ Q· ∂ T F (m) = C{F } z2 , T F (m) = MQ,u0 C{F, u} z2 , ∂z2 ∂z2 ∂ · ∂ and: Vm = C{z2b } · z2 , VmQ = MQ,u0 C{z2b , u} · z2 . ∂z2 ∂z2 {z1 =1} Proof. Classically T F (m) and Vm are zero except when F ] is either regular, or linearizable or resonant normalizable. In these last cases T F (m) is a free module of rank one over the ring OF ] ,m ⊂ OMF ,m of germs of holomorphic first integrals (perhaps constant) of F ] . We deduce the expression of T F (m) after checking that the vector fields Z in (2) and z2 ∂∂z2 in (3) are basic and OF ] ,m = C, resp. OF ] ,m = C{F }, in case (2), resp. (3), cf. [12, §5.1.2]. · The expressions of T Q F (m) are versions with parameters of these results. · In the cases (2a) and (3) the holonomy map hm is linear and Vm and VmQ is obtained · by a direct computation. In order to obtain VmQ in case (2b) one first notices that the flow ct | Φt (z2 , u) = (φ(z2 , t), u) of ZQ · {z1 =1} satisfies φ(z2 , t) ∈ C[t]{z2 }; therefore any biholomor- phism germ that commutes with a single element of this flow also commutes with all the Q· other elements. Since h◦q m = Φ2iπq , the flow of any element X ∈ Vm commutes with that ct | ct of ZQ · {z1 =1} . It follows that X ∈ MQ,u0 ZQ· |{z1 =1} .  · In order to describe T Q F (U ) for any open set U ⊂ D, we fix a geometric system as in Definition 4.13. For any X ∈ BF (oD ) there is a holomorphic vector field g∗ (X) on (C, 0) such that X(0) = 0 and g∗ (X) ◦ g = Dg(X). Moreover, X 7→ g∗ (X) is C-linear. Let us adopt the following notations: TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 37 • V(H) is the vector space of holomorphic germs of vector fields on (C, 0) vanishing at 0 and invariant under the action of the subgroup H ⊂ Diff(C, 0); • VQ0 is the vector space of holomorphic germs of vector fields on (C × Q, (0, u )) · 0 which are vertical with respect to C×Q → C and vanish along ({0}×Q)∪(C×{u0 }); 0 (G) denotes the subspace of • If G ⊂ Diff Q (C × Q, (0, u0 )) is a subgroup, then VQ · VQ0 consisting of vector fields invariant by G. · · Similarly if X ∈ B QF (oD ) there is a (unique) germ of vector field, again denoted by g∗ (X), such that g∗ (X) ◦ (g × idQ· ) = D(g × idQ· )(X). According to the model (3) with a = 0 and b = 1 in Lemma 5.4 we have the following exact sequence: · · g∗ 0 → XQ Q 0 F (oD ) → B F (oD ) → VQ· → 0 . (47) This proves that the sheaf T F is locally constant on D \ Sing(F ] ). Remark 5.5. Let X be a section of T F over a connected open subset V of EF . If the germ of X at some point p of V is zero, then X = 0. Indeed if p is a regular point, by local triviality, the section is zero along the whole regular part of D. The vanishing at the remaining singularities follows by analytic continuation. If p is a singular point, then the germ of X at a regular point close to p is zero and we conclude as before. The same · property holds for TFQ .  · Remark 5.6. The monodromy of T Q ] F restricted to D \ Sing(F ) corresponds to the ho- ct ] 0 Q· lonomy of the foliation FQ· in the following sense: if Z is the extension of Z ∈ T F (oD ) (as germ of a locally constant sheaf) along a loop γ in D∗ with origin oD , then g∗ (Z 0 ) = F ct·] hγ ∗ (g∗ (Z)), where hγ = HDQ (γ̇), see (18). Indeed we have: g 0 = g ◦ h−1 γ and on the other hand, since the expression g∗ (Z) remains constant when we perform along γ the analytic extension of g and the extension of Z as section of a locally constant sheaf, we also have g∗0 (Z 0 ) = g∗ (Z), where g 0 is the analytic extension of g along γ.  Proposition 5.7. Assume that oD ∈ U ⊂ D. The following sequence is exact: · · g 0 → XQ Q U∗ 0 F (U ) → B F (U ) → VQ· (HU ) → 0 , (48) Q· where gU ∗ is the composition of the morphism g∗ in (47) with the natural map B F (U ) → · F ct·] BQ Q ] F (oD ) and HU := HD (π1 (U \ Sing(F ), oD )). If e = hD, D0 i and {p} = D ∩ D0 , by applying this proposition to U = Up , and to U = D we obtain isomorphisms · ∼ · ∼ GTD,e : TFQ (e)−→VQ 0 · (hD,e ) and GTD : TFQ (D)−→VQ 0 · (HD ) . (49) Q· Q· Under these isomorphisms the restriction map TF (D) → TF (e) corresponds to the in- 0 (H ) ,→ V 0 (h clusion VQ· D Q· D,e ). 0 (H ) results from Remark 5.6 which also gives Proof. The fact that gU ∗ takes values in VQ · U the exactness of the sequence when U does not meet Sing(F ] ). It remains to see that the restriction map · · TQ Q ] F (U ) → T F (U \ Sing(F )) , Z 7→ Z|U \Sing(F ] ) is an isomorphism. We may suppose that U is a disk such that U ∩ Sing(F ] ) = {m}. Thus, the map gU ∗ in (48) induces an isomorphism · ∼ TQ ] 0 F (U \ Sing(F ))−→VQ· (HU ) . We may also suppose that U is the domain Ω of a chart (z1 , z2 ) as in Lemma 5.4. The restriction of g × idQ· to {z1 = 1} ⊂ MF × Q· induces a linear isomorphism from VmQ to 38 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM 0 (H ). We conclude by noting that, according to Lemma 5.4, any element of V Q extends VQ U m · to a vector field in B Q F (U ).  In the same way we prove the exactness of the following sequence: U∗ g 0 → X F (U ) → B F (U ) → V(HU ) → 0 . (50) 5.3. Group-graph of infinitesimal transversal symmetries. A C ex -conjugacy does not induce a map between the sheaves of basic holomorphic vector fields, but it will do for the sheaves of transverse infinitesimal symmetries. For this reason we do not consider the · Q· Q· quotient of the group-graphs associated to B Q F and X F but a group-graph TF associated · to the sheaf T Q F . As in the case of the group-graph of automorphisms (see Definition 4.1) we set: Definition 5.8. The vector space-graph over AF of infinitesimal transversal sym- · ct , denoted by T , resp. T Q , is defined, for ? ∈ Ve metries of F, resp. of FQ · F F AF ∪ EdAF , by: · (1) TF (?) = {0} and TFQ (?) = {0} if ? ∈ VeAF is a dicritical component of EF or ? = hD, D0 i ∈ EdAF and the foliation F ] has a nodal singularity at the point D ∩ D0 ; · · (2) TF (D) = T F (D) and TFQ (D) = T Q F (D) if D ∈ VeAF is invariant; · · (3) TF (hD, D0 i) = T F (D ∩ D0 ) and TFQ (hD, D0 i) = T Q 0 0 F (D ∩ D ) if hD, D i ∈ EdAF and D ∩ D0 is not a nodal singularitiy of F ] ; · · · (4) the restriction map TFQ (D) → TFQ (e) is the trivial map TFQ (D) → {0} in case (1) and it is the restriction map of sheaves in cases (2) and (3). · · The support of TFQ is contained in the cut-graph of F which is the support of AutQ F , see Remark 4.2. The pull-back by a holomorphic map germ µ : P · → Q· of a vertical vector field X is also a vertical vector field and its flow is the pull-back of the flow of X. Thus, the pull-back · Q· Q· operation defines sheaf morphisms from the sheaves B Q F , X F and T F respectively to the · · · sheaves B PF , X PF and T PF , inducing a morphism of vector space-graphs · · µ∗ : TFQ → TFP . On the other hand, let φ be an C ex -conjugacy between G and a foliation F, φ(G) = F. ∼ Since the germs of homeomorphisms φ] : (MG , EG )−→(MF , EF ) and ∼ φ]Q := φ] × idQ : (MG × Q, EG × {u0 })−→(MF , EF × {u0 }) are holomorphic at the singular points and transversely holomorphic elsewhere, we can define the inverse image morphisms of sheaves over EG · · φ∗ : φ−1 E TF →TG and φ∗Q : φ−1 E TQ Q F →TG , where φE : EG → EF is the restriction of φ] to the exceptional divisors, as in Section 4.1. Indeed, let us fix m ∈ EG and [Z] ∈ T F (φE (m)), which is the class of Z ∈ B F (φE (m)). If m ∈ Sing(G ] ) ∪ Sing(EG ) then φ] is holomorphic at m and we define φ∗ ([Z]) as the class of the usual inverse image (φ] )∗ (Z) ∈ B G (m). Otherwise, there is a homeomorphism germ ξ at φ] (m) fixing the leaves of F ] such that ξ ◦ φ] is holomorphic and we define φ∗ ([Z]) as the class of (ξ ◦ φ] )∗ (Z), which does not depend on the choice of ξ. We can similarly define the sheaf morphism φ∗Q . We will denote in the same way by ∼ · ∼ · φ∗ : TF −→TG and φ∗ : TFQ −→TGQ (51) TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 39 the vector space-graph morphisms over Aφ : AG → AF defined in (33), which are associated to the sheaf morphisms φ∗ and φ∗Q , see Section 2.2. We can check that the second morphism φ∗ satisfies the relations µ∗ ◦ φ∗ = φ∗ ◦ µ∗ , allowing us to define the following contravariant functors (denoted by the same letter) T : Fol → VecG , F 7→ TF , φ 7→ φ∗ , · T : Man· × Fol → VecG , (Q· , F) 7→ TFQ , (µ, φ) 7→ (µ, φ)∗ := φ∗ ◦ µ∗ , where VecG denotes the category of of C-vector space-graphs and linear maps. As we did for the group-graph of transversal symmetries we consider the restriction of infinitesimal transversal symmetries vector space-graphs to the red subgraph RF ⊂ AF : Definition 5.9. We call restricted group-graph of infinitesimal transversal sym- ct , the group-graph RT ∗ Q· ∗ Q · metries of F, resp. FQ · F = rF TF , resp. RT F = rF TF , over RF defined as the pull-back by the inclusion rF : RF ,→ AF : · · RT F (?) = TF (?), resp. RT Q Q F (?) = TF (?) , ? ∈ VeRF ∪ EdRF . We denote by RT : Fol → VecG, resp. RT : Man· × Fol → VecG, the functors · F 7→ RTF , resp. (Q· , F) 7→ RT Q F . Remark 5.10. As for transversal symmetries, the collections of canonical morphisms · · ırF : TF → RTF and ırF : TFQ → RTFQ of vector space-graphs over the graph morphisms rF : RF ,→ AF define natural transformations, again denoted by R : T → RT , and also R := H 1 (R) : H 1 ◦ T → H 1 ◦ RT . If F is of finite type, thanks to the exact sequence (50), in each cut-component AαF of AF the red part RαF is repulsive for the group-graph TF restricted to AαF , see Section 2.4. By applying again (32) and Theorem 2.10 we directly obtain that the natural maps ∼ RF : H 1 (AF , TF )· −→H 1 (RF , RTF )· . (52) are bijective, thus R is an isomorphism of contravariant functors. In the same way, using the exact sequence (48) we obtain a natural isomorphism · · ∼ · RQ 1 Q · 1 Q · F : H (AF , TF ) −→H (RF , RTF ) .  Lemma 5.11. Assume that F is a generalized curve. Let us again denote by ZQ ct the · constant vertical extension of a vector field Z on an open set of MF , defined just before Lemma 5.4. The extension of scalars sheaf morphism6 · · ExtQ Q F : T F ⊗C MQ → T F , · ct [Z] ⊗ a 7→ [aZQ ·] , define an isomorphism of vector space-graphs · ∼ · ExtQ Q F : RTF ⊗C MQ −→RTF · which induces a natural isomorphism Ext between the contravariant functors (Q· , F) 7→ · RTF ⊗C MQ· and (Q· , F) 7→ RTFQ , from Man· × Fol to VecG. In this way we obtain a natural isomorphism · ∼ H 1 (Ext−1 ) : H 1 (RF , RTFQ )· −→H 1 (RF , RTF ⊗C MQ· )· (53) between functors from Man· × Fol to Vec, as subcategory of pointed sets. 6We highlight that ExtQ· is not an isomorphism of sheaves. F 40 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM Proof. Consider an invariant component D of EF , an edge e = hD, D0 i and the point {p} := D∩D0 . Assume that D and e are red for F. We can then use the isomorphisms (49), the exact sequence (50) with U = Up as in Definition 4.13, and cases (1) and (2) in Lemma 5.4. With the notations used in this lemma and these sequences, we have the following commutative diagrams whose vertical arrows are isomorphisms: · · ExtQ F (D) · ExtQ F (p) · TF (D) ⊗C MQ,u0 / T Q (D) TF (e) ⊗C MQ,u0 / T Q (e) F F o ġD∗ ⊗C idMQ,u0 GT D o o ġU ∗ ⊗C idMQ,u0 GT D,e o  Ext(D)   Ext(e)  V(HD ) ⊗C MQ,u0 / V 0 · (HD ) V(hD,e ) ⊗C MQ,u0 / V 0 · (hD,e ) Q Q where Ext(D) and Ext(e) are the maps Z ⊗C a 7→ aZQ ct . To prove that the top horizontal · arrows are isomorphisms it suffices to prove this property for the bottom arrows. Since the holonomy of the constant deformation “does not depend on the parameter” this fact directly results from the definitions of VQ 0 (H ) and V 0 (h · D Q· D,e ) and dimC V(hD,e ), dimC V(HD ) ≤ 1. Finally, this collection of isomorphisms induces the isomorphism of functors Ext since µ∗ φ∗ ([aZQ ct ]) = (µ∗ a)(φ∗ ([Z ct ])) for any morphism (µ, φ) in the category Man· × Fol.  · Q· Proposition 5.12. Assume that F is a generalized curve. The vector space-graphs RTF · and RTFQ over RF are regular (see Definition 2.13). Moreover, in each red subgraph RαF ⊂ AαF the complementary of its support is a subgraph. · Proof. By Lemma 5.4 for each ? ∈ VeRF ∪ EdRF , either both RTF (?) and RTFQ (?) are zero, · or RTF (?) is isomorphic to C and RTFQ (?) is isomorphic to the maximal ideal MQ,u0 of OQ,u0 . Assume that D ∈ VeRF is invariant and e = hD, D0 i ∈ EdRF does not correspond to a nodal singular point at D ∩ D0 . By Remark 5.5 either the restriction map RTF (D) → RTF (e) is an isomorphism or RTF (D) = 0 and RTF (e) ' C, the situation RTF (D) 6= 0 · and RTF (e) = 0 being impossible. According to Lemma 5.11, we deduce that RTFQ ' RTF ⊗C MQ,u0 is also regular.  ct ] 5.4. Exponential group-graph morphism. The flows of basic vector fields of FQ · ct ] leave invariant the foliation FQ · . As in [8, Lemma 9.1] we see that the exponential maps · · · · BQ Q F (m) → AutF (m), Z 7→ exp(Z)[1], m ∈ EF , send X Q Q F (m) in FixF (m) and factorize · · into maps expm : T Q Q F (m) → SymF (m), thus defining a morphism of sheaves of sets · · · ExpQ F :TQ Q F → SymF . Using the isomorphism (43) it induces maps · · · · ExpQ Q Q Q F (?) : TF (?) → SymF (?) ' SymF (?) , ? ∈ VeAF ∪ EdAF . In general these maps are not group morphisms but this will be the case when the OQ,u0 - · module TFQ (?) is free of rank one or null, cf. [8, §9]. Therefore to define an exponential group-graph morphism we must restrict the group-graph of infinitesimal symmetries of F ct ] Q· or FQ · to the group-graph RTF over the sub-graph RF of AF . Using the definitions of the isomorphisms GT? in (49) and the definitions of the isomor- phisms G? in Proposition 4.14 with the same geometric system, cf. Definition 4.13, we TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 41 have the following commutative diagrams · GT · GT TFQ (e) / 0 D,e TFQ (D) D / V 0 · (HD ) ∼ VQ· (hD,e ) ∼ Q (54) · · ExpQ F (e) exp ExpQ F (D) exp     · GD,e · GD SymQ / C 0 · (hD,e ) SymQ / C 0 · (HD ) F (e) ∼ Q F (D) ∼ Q where e = hD, D0 i and hD,e is the holonomy map defined in (45). Indeed, when the direct image g∗ (Z) of a basic vector field Z is defined, its flow is also the direct image of the flow of Z by g. Theorem 5.13. Given a foliation which is a generalized curve and a germ of manifold Q· , · the morphisms ExpQ F (?) induce a group-graph isomorphism over RF Q · · ∼ · ExpF : RTFQ −→RSymQ F . · The collection {ExpQ F } defines an isomorphism of contravariant functors ∼ Exp : RT −→RSym , from Man· × Fol to the category of abelian group-graphs, the functor RT taking values in the subcategory of C-vector space group-graphs. In order to prove this theorem we will need an auxiliary result. Lemma 5.14. If h ∈ Diff(C, 0) is non-periodic then the exponential map induces a group isomorphism 0 ∼ 0 exp : VQ · (h) → CQ· (h). Proof. If h is formally linearizable then there is a formal coordinate w such that w ◦h = λw with λ ∈ C∗ . If φ ∈ CQ 0 (h) then w ◦ φ = ν(t)w with ν ∈ O ∗ · t Q,u0 and ν(0) = 1. Indeed, i P φ̃(w, t) := w ◦ φt = i≥1 φi (t)w belongs to C{w, t} and φ̃(λw, t) = λφ̃(w, t) implies that φi (t) ≡ 0 for i > 1 and ν(t) = φ1 (t) 6= 0 is holomorphic. There is ξ ∈ MQ,u0 such that ν(t) = exp(ξ(t)). If w is convergent then φt = exp(ξ(t)w∂w ). If w(z) is divergent then |λ| = 1 and CQ 0 (h) is the set of φ(z, t) = (φ (z), t) such that φ (z) = w −1 ◦ ν(t)w(z) is · t t convergent. If w is divergent then VQ 0 (h) = 0 and ν(t) takes values in a discrete subset of · the unit circle S1 ⊂ C. We conclude that ν ≡ 1 by holomorphy. If h is resonant there is a formal coordinate w such that w ◦ h = `r ◦ exp sX with wp+1 0 rt ◦ exp τ (t)X with X = 1+λw p ∂w for some integer p ≥ 1. If φ ∈ CQ· (h) then w ◦ φt = ` rt ∈ Z. Since φ̃(w, t) = w ◦ φt = i≥1 φi (t)wi ∈ C{w, t} and exp τ X(w) = w + τ wp+1 + · · · P 2iπrt we conclude that the holomorphic function φ1 (t) = e p is identically equal to 1 and the function t 7→ τ (t) is holomorphic and vanishes at t = 0. If w is convergent then φt = exp(τ (t)X) with τ ∈ MQ,u0 . If w is divergent then VQ 0 (h) = 0 and C 0 (h) is the · Q· −1 set of φ(z, t) = (φt (z), t) such that w ◦ exp(τ (t)X) ◦ w is convergent. This implies that τ (t) ∈ Q by the Écalle-Liverpool’s Theorem [5] and consequently τ ≡ 0.  Proof of Theorem 5.13. It suffices to see that for D ∈ VeRF and e ∈ EdRF the right vertical arrows in the diagrams (54) are isomorphisms. For the diagram in the left this follows from Lemma 5.14 by taking h = hD,e . It only remains to examine the diagram in the right of (54) with D red. Since HD is infi- nite there is a non-periodic element h0 ∈ HD . Indeed, when HD is non-abelian it contains a non-trivial commutator, which is tangent to the identity, hence non-periodic. When HD is abelian, if all its elements were periodic then HD would be finite. We must prove that 42 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM 0 (H ) → C 0 (H ) is an isomorphism. By Lemma 5.14 the the exponential map exp : VQ · D Q· D bottom horizontal map in the following diagram is an isomorphism: exp 0 (H ) VQ / C 0 · (HD ) · D _ Q  _   0 (h ) VQ· 0 ∼ / C 0 · (h0 ) . exp Q This shows that the top horizontal map is injective. To prove the surjectivity we distinguish two cases: (a) TF (D) = 0. In this case VQ 0 (H ) = 0. By contradiction, we must see that if · D 0 0 CQ· (HD ) 6= {idC×Q } then VQ· (HD ) 6= {0}. If (f (z, u), u) ∈ CQ 0 (H ) \ {id · D C×Q } there ∗ is a holomorphic germ λ : (C, 0) → (Q, u0 ) and n ∈ N such that z 6≡ g(z, t) := f (z, λ(t)) = z + tn a(z) mod tn+1 with a(z) 6≡ 0. For every h ∈ HD we have: g(h(z), t) = h(g(z, t)). Working modulo tn+1 we deduce that h(z) +tn a(h(z)) = h(z) + h0 (z)tn a(z), i.e. a(h(z)) = h0 (z)a(z). This means that 0 6≡ a(z)∂z ∈ ∩h∈HD V(h) = V(HD ) 6= 0 and consequently VQ 0 (H ) ' V(H ) ⊗ M · D D C Q,u0 6= 0. (b) TF (D) 6= 0. In this case, 0 6= TF (D) ' V(HD ) ⊂ V(h0 ) and since h0 is non-periodic classically, we have: dimC V(h0 ) ≤ 1. Consequently V(HD ) = V(h0 ) has dimension 1 and VQ 0 (H ) = V(H ) ⊗ M 0 · D D C Q,u0 = V(h0 ) ⊗C MQ,u0 = VQ· (h0 ). Using Lemma 5.14 we have: 0 0 0 0 0 CQ · (h0 ) = exp(VQ· (h0 )) = exp(VQ· (HD )) ⊂ CQ· (HD ) ⊂ CQ· (h0 ) . 0 (H )) = C 0 (H ). Hence exp(VQ · D Q· D We let the reader check that if (µ, φ) : (P · , G) → (Q· , F) is a morphism in the category Man· × Fol, then the following diagram of group-graph morphisms is commutative: · (µ,φ)∗ RTFQ / RT P · G · · ExpQ F ExpP G   · (µ,φ)∗ RSymQ / RSymPG . · F  5.5. Characterization of finite type foliations. In this section we prove that, under a technical hypothesis, a foliation F is of finite type if and only if the cohomology vector space H 1 (AF , TF ) is of finite dimension, which justifies the name that we have adopted. Theorem 5.15. Let F be a foliation which is a generalized curve. If there is no cut- component AαF of AF entirely green, then F is of finite type if and only if dimC H 1 (AF , TF ) < ∞. Before proving the theorem we need to state some auxiliary results. Remark 5.16. If K, K0 are subgraphs of AF , then we have: dimC H 1 (K0 , TF ) ≤ dimC H 1 (K, TF ) as soon as K0 ⊂ K .  Lemma 5.17. If an edge e ∈ EdAF is green and D ∈ e then the following properties are equivalent: (1) the holonomy group HD is generated by hD,e ; (2) the restriction morphism ρeD : TF (D) → TF (e) is surjective; (3) the image of ρeD has finite codimension in TF (e); TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 43 where hD,e are defined using a given geometric system, cf. Definition 4.13. An immediate consequence of this lemma is the following: Corollary 5.18. If there is no cut-component of AF entirely green, then F is of finite type if and only if in each cut-component AαF of AF , α ∈ A, the red part RαF is connected and repulsive for the group-graph TF . To lighten the text, in this proof we will denote by T the vector space-graph TF and by T? the vector space TF (?). Proof of Lemma 5.17. If D is not green then dimC TD ∈ {0, 1}, dimC Te = ∞, hD,e is periodic and HD is infinite. Thus, none of the three assertions hold. If D is green then there is a transverse factor z : (MF , oD ) → (C, 0) at a regular point oD ∈ D such that HD = hz 7→ e2iπ/nD zi and hD,e (z) = e2iπ/nD,e z where nD , nD,e ∈ Z. The proof of [8, Proposition 6.4] shows that Te /ρeD (TD ) ' C{z nD,e }/C{z nD } is either zero (when nD = nD,e ) or it has infinite codimension (when nD 6= nD,e ).  Let us highlight that by Remark 5.5 the restriction maps ρeD : TD → Te , with e = hD, D0 i ∈ EdAF , of the group graph TF are always injective. We now provide "orientations" to the edges e of AF in the following way: D e D0 (i) ◦ → ◦ means that ρeD is not bijective and ρeD0 is bijective, D e D0 (ii) ◦ ← ◦ means that ρeD bijective and ρeD0 is not bijective, D D0 (iii) ◦ ↔ ◦ means that both ρeD and ρeD0 are bijective, D D0 (iv) ◦ ←7→ ◦ means that both ρeD and ρeD0 are not bijective. Lemma 5.19. In a cut-component AαF of AF , let K be a geodesic of one of following types: D0 e0 D1 e1 en−1 Dn e Dn+1 n (1) • ? ··· ? ←− ? , with n ≥ 1 ; D0 0 e D1 (2) • ←7→ ? , the edge e0 being necessarily green; D0 e0 D1 e1 en−1 Dn en Dn+1 (3) • ? ··· ? • , with n ≥ 1 ; D0 e0 D1 (4) • • , the edge e0 being green; D e D0 where the green vertices are denoted by ?, the red vertices by • and ◦ ◦ denotes any "orientation" (i)-(iv). Then the dimension of H 1 (K, T ) is infinite. Proof. First consider case (1). Thanks to Remark 5.16, even if we restrict to a smaller geodesic, we can suppose that all arrows e0 , . . . , en−1 are either simple arrows directed to ej−1 ej−1 Dn , i.e. ?Dj−1 −→ ?Dj or double arrows ?Dj−1 ←→ ?Dj ; therefore all the restriction e e maps ρDj−1j : TDj → Tej−1 , j = 0, . . . , n, are isomorphisms. Every map ρDjj being injective we can identify all the spaces Tej , j = 0, . . . n, and TDj , j = 0, . . . , n + 1 with subspaces of Ten . With these identifications we have: TD0 ⊆ Te0 = TD1 ⊆ · · · ⊆ TDn = Ten ) TDn+1 . (55) Since Dn+1 ∈ en are green, from Lemma 5.17 it follows that dimC (Ten /TDn+1 ) = ∞ . (56) With the identifications (55) the coboundary morphism for the subgraph K can be written as n+1 n ∼ Y Y 0 0 1 ∂K : C (K, T ) = TDj −→ Z (K, T )−→ Tej , j=0 j=0 0 ∂K ((Xj )j=0,...,n+1 ) = (Xj − Xj−1 )j=1,...,n+1 . 44 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM The surjective linear map n Y n X β: Tej → Ten , (Xj )j=0,...,n 7→ Xj j=0 j=0 induces the following diagram whose rows and columns are all exact: ∂K0 / / H 1 (K, T ) /0 Qn+1 Qn j=0 TDj j=0 Tej α β βe    TD0 × TDn+1 σ / Te / Te /(TD + TD ) /0 n n 0 n+1    0 0 0 with α((Xj )j=0,...,n+1 ) := (X0 , Xn+1 ) and σ(X0 , Xn+1 ) := Xn+1 −X0 . Since the dimension of TD0 is finite and the codimension of TDn+1 in Ten is infinite according to (56), we deduce that the dimension of Ten /(TD0 +TDn+1 ) is infinite and consequently dimC H 1 (K, T ) = +∞. Case (2) can be treated as case (1). In case (3), if K does not contain a subgraph of type (1) nor (2), even by renumbering, then the configuration must be D0 0 1e D1 n e en−1 Dn e Dn+1 • −→ ? ←→ · · · ←→ ? ←− • and we can make again the identifications (55). The spaces TD0 and TDn+1 having both finite dimension, we obtain the conclusion. Case (4) is trivial because TD0 and TD1 have finite dimension and dimC Te0 = ∞.  Proof of Theorem 5.15. We will use the characterization of finite type foliations given in Corollary 5.18. Notice that the red part RαF of a cut-component AαF is not repulsive with respect to TF if and only if it contains a geodesic of type (1) or (2) because the configuration • ←− ? cannot occur. On the other hand, RαF is not connected if and only if it contains a geodesic of type (3) or (4). It follows from Lemma 5.19 that if F is not of finite type then a cut-component AαF contains a geodesic K with dim H 1 (K, TF ) = ∞ and consequently dimC H 1 (AF , TF ) ≥ dimC H 1 (AαF , TF ) = ∞, cf. Remark 5.16. Conversely, if F has finite type, from Remark 5.10, Proposition 5.12 and Theorem 2.15 we deduce that H 1 (AF , TF ) has finite dimension.  5.6. Some examples. Consider the logarithmic foliation Lα defined by the multivalued first integral (y 2 + x3 )α (y 3 + x2 ), α ∈ C. The dual graph associated to the exceptional divisor of the desingularization π of the separatrices (y 2 + x3 )(y 3 + x2 ) = 0 is given by M− a− D− b− D0 b+ D+ a+ M+ • • • • • . (57) Let s± be the intersection points of the strict transforms of S− = {y 3 + x2 = 0} and S+ = {y 2 + x3 = 0} with D− and D+ respectively. We can compute the Camacho-Sad indices 1 1 1+α CS(π ∗ Lα , D− , a− ) = − , CS(π ∗ Lα , D− , s− ) = − , CS(π ∗ Lα , D− , b− ) = − 2 6 + 4α 3 + 2α and 1 α α+1 CS(π ∗ Lα , D+ , a+ ) = − , CS(π ∗ Lα , D+ , s+ ) = − , CS(π ∗ Lα , D+ , b+ ) = − . 2 6α + 4 3α + 2 If α0 ∈ R− one of the Camacho-Sad indices is a real positive number and in any neigh- borhood of α0 there is a positive rational number α1 and a positive irrational number α2 such that π ∗ Lα1 is not reduced but π ∗ Lα2 is reduced. Consequently the local deformation TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 45 (Lα )α∈(C,α0 ) is not equireducible. If α0 ∈ C \ R− then the local deformation (Lα )α∈(C,α0 ) is equireducible with equire- duction map π but not equisingular (and a fortiori not an unfolding) because there are singular points in the exceptional divisor of π with Camacho-Sad index varying with α. Let us illustrate now the computation of the cohomology group H 1 (AF , TF ) in the case of a foliation germ F such that π ∗ F is reduced. In the diagram below the arrows correspond to restriction maps in the group-graph TF , the first line corresponds to the groups associated to the vertices and the second line corresponds to the groups associated to the edges of the graph (57): TF (M− ) TF (D− ) TF (D0 ) TF (D+ ) TF (M+ ) &&  & y % x  xx TF (a− ) TF (b− ) TF (b+ ) TF (a+ ) Using the surjectivity at the extremities of the group-graph TF we can apply pruning Theorem 2.10 to obtain a new group-graph with the same cohomology: TF (D− ) TF (D0 ) TF (D+ ) % y % y TF (b− ) TF (b+ ) By applying Remark 2.7 we obtain the exact sequence: ∂ TF (D− ) ⊕ TF (D0 ) ⊕ TF (D+ ) → TF (b− ) ⊕ TF (b+ ) → H 1 (AF , TF ) → 0 (58) where ∂(X− , X0 , X+ ) = (X0 − X− , X0 − X+ ). (59) We consider four cases and for each of them we use the local models given in Lemma 5.4. The first three cases are examples of finite type foliations. (1) If F = Lα with α ∈ C \ R then (58) becomes ∂ C ⊕ C ⊕ C → C ⊕ C → H 1 (ALα , TLα ) = 0. (2) If α ∈ (−3/2, −1) \ Q then b− is the only nodal singularity of π ∗ Lα . In this case TLα does not coincide with the group-graph associated to the sheaf T Lα of germs of infinitesimal transversal symmetries of π ∗ Lα , see Definition 5.8, because TLα (b− ) = 0 6= C = T Lα (b− ). The exact sequence (58) is ∂ C ⊕ C ⊕ C → 0 ⊕ C → H 1 (ALα , TLα ) → 0 and we also obtain that H 1 (ALα , TLα ) = 0. If α ∈ (−1, −2/3) \ Q then b+ is the only nodal singularity of π ∗ Lα and the same conclusion holds. (3) Let F be a perturbation of Lα , α ∈ C\R, with same Camacho-Sad indices such that the holonomy groups of D± are non-abelian. Such a foliation can be constructed using Lins-Neto’s theorem [4]. In that case TF (D± ) = 0, (58) becomes ∂ 0 ⊕ C ⊕ 0 → C ⊕ C → H 1 (AF , TF ) → 0 and the expression (59) gives H 1 (AF , TF ) = C. 46 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM (4) Let F be a perturbation of Lα , α ∈ Q+ , with same Camacho-Sad indices such that the holonomy groups of D± are non-abelian and the holonomy group of D0 is finite. In that case TF (D± ) = 0, TF (D0 ) ' C{z}, (58) becomes 0 ⊕ C{z} ⊕ 0 → C{z} ⊕ C{z} → H 1 (AF , TF ) → 0 and we deduce that dim H 1 (AF , TF ) = ∞. Hence F is not of finite type. This example, whose dual graph AF contains a geodesic with colored vertices red-green- red, also illustrates Corollary 5.18. In cases (1) and (2) we obtain that H 1 (AF , TF ) = 0. We will see in next chapter that if F is a finite type foliation then dim H 1 (AF , TF ) is the dimension of the universal parameter space of equisingular deformations and when it is zero every equisingular deformation is topologically trivial. We conclude this section with an example of equisingular deformation which is not an unfolding (for the precise definition of this notion see [10]). Let F be a perturbation of Lα , α ∈ C \ R, with same Camacho-Sad indices such that the holonomy groups of D± are non-abelian as in case (2) above. There is a local non-zero transverse symmetry X of F at b+ . We consider two open subsets U+ and U− whose union is a neighborhood of the exceptional divisor of π and whose intersection is a small neighborhood of b+ . We glue U+ and U− by the time t flow of X obtaining a complex surface Ut with a foliation F̃t . This gluing does not change the self-intersections of the irreducible components of the exceptional divisor and consequently we can contract it to obtain a foliation germ Ft in (C2 , 0). The results in next chapter allow to prove that the Kodaira-Spencer map (68) of the deformation (Ft )t∈(C,0) ∂[Ft ] : T0 C = C → H 1 (AF , TF ) ' TF (b+ ) = C · [X], c 7→ [cX], ∂t t=0 is an isomorphism. Hence (Ft )t∈(C,0) is a C ex -universal deformation, consequently not topologically trivial and a fortiori not an unfolding. 6. C ex -universal deformations 6.1. C ex -universality. We will show the existence of a C ex -universal deformation for finite type foliations through the representability of the corresponding deformation functor. Definition 6.1. Let FQ· be an equisingular deformation over a germ of manifold Q· := (Q, u0 ), of a foliation F. We say that FQ· is a C ex -universal deformation of F if for any germ of manifold P · = (P, t0 ) and any equisingular deformation GP · of F over P · , there exists a unique germ of holomorphic map λ : P · → Q· such that the deformations GP · and λ∗ FQ· of F are C ex -conjugated. Remark 6.2. Notice that if µ : Q0 · → Q· is a germ of biholomorphism, the C ex -universality of FQ· and of µ∗ FQ· are clearly equivalent. On the other hand, it directly results from the · definition that the C ex -universality of FQ· only depends on its class fQ· := [FQ· ] ∈ Def Q F . We will then say that fQ· is C ex -universal.  Let us consider the maps · · ΛPfQ· : O(P · , Q· ) → Def PF , λ 7→ [λ∗ FQ· ] , where O(P · , Q· ) always denotes the set of holomorphic map germs P · → Q sending t0 to u0 . By definition we have: · fQ· is C ex -universal ⇐⇒ for any P · the map ΛPfQ· is bijective. TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 47 · One easily checks that (ΛPfQ· )P · defines a natural transformation ∼ ΛfQ· : FQ· : −→Def F where FQ· , Def F : Man· → Set· are the following contravariant functors: · FQ· (P · ) := O(P · , Q· ), FQ· (λ) = · ◦ λ , DefF (P · ) := Def PF , Def F (λ) := λ∗ , where the first set is pointed by the constant map κu0 : P · → Q· and the second one is pointed by the class of the constant deformation FQ ct , see Section 3.4. Thus f · is C ex - · Q universal if and only if ΛfQ· is an isomorphism of functors. Classically Q· being fixed, any isomorphism of functors ∼ · ∼ · Λ : FQ· −→DefF , Λ = (ΛP : O(P · , Q· )−→Def PF )P · is of this type: · Λ = ΛfQ· with fQ· := ΛQ (idQ· ) . It is Yoneda’s Lemma which may be summarized in the diagrams below whose commuta- tivity results from the functoriality of Λ: · idQ·  ΛQ / Def F (Q· ) / fQ· O(Q· , Q· ) _ _ · ◦λ λ∗  ·    λ ΛP / DefF (P · ) O(P · , Q· ) / ΛP · (λ) = λ∗ fQ· Finally, to find a germ of manifold Q· and a C ex -universal deformation FQ· is equivalent to represent the functor Def F , i.e. to find a germ of manifold Q· and an isomorphism ∼ of functors DefF −→FQ· : ·     Q· ∼ Q· fQ· ∈ Def Q F is C ex -universal ⇐⇒ ∃ ξ : DefF −→ FQ· , ξ (fQ · ) = idQ· . (60) · As we will also need later the naturality of ξ Q relative to the foliation F ∈ Fol, we will prove a slightly stronger result. If φ : G → F is a C ex -conjugacy between two foliations G and F, we will denote by ∼ [φ∗ ] := H 1 (φ∗ ) : H 1 (AF , TF )−→H 1 (AG , TG ), (61) ∼ the morphism induced by the vector space-graph isomorphism φ∗ : TF −→TG defined in (51). We define the contravariant factorizing functor Fac : Man· × Fol → Set· as Fac(Q· , F) := O(Q· , H 1 (AF , TF )· ) , this set being pointed by the zero map, and if (µ, φ) : (P · , G) → (Q· , F), then Fac(µ, φ) := Facµφ is the following linear map: Facµφ : O(Q· , H 1 (AF , TF )· ) → O(P · , H 1 (AG , TG )· ) , λ 7→ [φ∗ ] ◦ λ ◦ µ , (62) where H 1 (AF , TF )· is the vector space H 1 (AF , TF ) pointed by the origin. Theorem 6.3. For any finite type foliation F which is a generalized curve and for any germ of manifold Q· there is a bijection Q · · ∼ ξF : Def Q · 1 · F −→O(Q , H (AF , TF ) ) Q · such that the collection {ξF }(Q· , F ) defines an isomorphism of contravariant functors ∼ ξ : Def −→Fac , (63) when both functors are restricted to the subcategory Man· × Folft of the category Man· × Fol, see Definition 5.1. 48 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM Proof. We successively apply Theorem 4.4, Proposition 4.11, Theorem 5.3, Theorem 5.13, Lemma 5.11, natural isomorphisms (53), (11) and (52). We obtain for any (Q· , F) ∈ Man· × Folft , the following isomorphisms: Th. 4.4 Prop. 4.11 Th. 5.3 · ∼ · ∼ · ∼ · Def Q F −→ H 1 (AF , AutQ F ) · −→ H 1 (AF , SymQ · 1 Q · F ) −→ H (RF , RSymF ) Th. 5.13 (53) (11) ∼ · ∼ ∼ −→ H 1 (RF , RTFQ )· −→ H 1 (RF , RTF ⊗C MQ· )· −→ H 1 (RF , RTF )· ⊗C MQ· (52) ∼ ∼ −→H 1 (AF , TF )· ⊗C MQ· −→O(Q· , H 1 (AF , TF )· ) , (64) the last natural isomorphism being as usual (c ⊗ a) 7→ (t 7→ a(t)c). Each of them defines in fact a natural transformation between contravariant functors from Man· × Folft to Set· . The functor isomorphism ξ is defined as the composition of all the isomorphisms in (64).  Theorem 6.4. For any foliation of finite type (which is a generalized curve) there exists a C ex -universal deformation FQ· , with base Q· = H 1 (AF , TF )· , P ([F · ]) satisfies · such that for any equisingular deformation FP · of F, we have that λ := ξF P [λ∗ FQ· ] = [FP · ]. Moreover, H 1 (AF , TF ) is a C-vector space of dimension the rank of H1 (RF /(RF \ supp(RTF ))). Here RF /(RF \ supp(RTF )) denotes the graph obtained by contracting to a single ver- tex the complementary of the support of RTF , which is a subgraph of RF according to Proposition 5.12. Q −1 · Proof. By (60) with Q· = H 1 (AF , TF )· we can choose for FQ· any element in (ξF ) (idQ· ). To obtain the description of H 1 (AF , TF ) we use the isomorphism H 1 (AF , TF ) ' H 1 (RF , RTF ) given by (52) and Proposition 5.12. We then apply Theorem 2.15 to each connected com- ponent of RF , taking d = 1 and noting that a − p = rk H1 (RF /(RF \ supp(RTF ))).  6.2. Kodaira-Spencer map. This map assigns to each equisingular deformation its as- sociated “infinitesimal deformation”. We will define for any germ of manifold Q· = (Q, u0 ) and any foliation F ∈ Fol, a group-graph morphism · · ΘQ Q 2 F : AutF → TF ⊗C (MQ,u0 /MQ,u0 ) so that this collection is a natural transformation Θ between the functor Aut considered · in (36) and the functor (Q· , F) 7→ TF ⊗C (MQ,u0 /M2Q,u0 ). The definition of ΘQ F ,e for e := hD, D0 i ∈ EdAF is based on the following fact: let (u1 , . . . , uq ) be a centered coordinate system on Q· and let us denote by prMF the canonical projection MF × Q → MF ; if a germ of biholomorphism Φ at the point (s, u0 ) ∈ MF × Q, with {s} := D ∩ D0 , leaves invariant ct ] ∂ prM ◦ Φ the constant deformation FQ · , then F ∂uk u=u0 , k = 1, . . . , q, are germs of vector fields h ∂ pr ◦ Φ i MF in MF at s, basic for the foliation F ] . We denote by ∂uk u=u0 its class in TF (e) and, when s is not a nodal singularity of F ] , we set: · · ΘQ Q 2 F , e : AutF (e) → TF (e) ⊗C (MQ,u0 /MQ,u0 ) , q " # · ∂ prMF ◦ Φ ΘQ X F , e (Φ) := ⊗ u̇k . (65) ∂uk u=u0 k=1 · The definition of ΘQ F , D for D ∈ EdAF invariant is less direct because the homeomorphisms · Φ ∈ AutQ F (D) are not holomorphic a priori. We will fix the germ of a submersion g : TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 49 (MF , oD ) → (C, 0) at a regular point oD ∈ D constant along the leaves of F ] and we will use the composition of group morphisms · · G AutQ Q D 0 F (D) → SymF (D) → CQ· (HD ), Φ 7→ g∗ Φ, cf. Proposition 4.14, and the isomorphism ∼ ġD∗ : TF (D)−→V(HD ) given by the exact sequence (50) with U = D. One easily checks that if h(z) is a germ of biholomorphism of (C, 0) and (φ(z, u), u) is a germ of biholomorphism of (C × Q, (0, u0 )) ∂φ over Q satisfying φ(z, u0 ) = z and φ(h(z), u) = h(φ(z, u)), then ∂u k u=u0 , k = 1, . . . , q, are vector field germs on (C, 0) invariant by h. We set: · · ΘQ Q 2 F , D : AutF (D) → TF (D) ⊗C (MQ,u0 /MQ,u0 ) , q ! Q· X −1 ∂ prC ◦ g∗ Φ ΘF , D (Φ) := ġD∗ ⊗ u̇k , (66) ∂uk u=u0 k=1 where prC again denotes the canonical projection C × Q → C. One can check that defini- tions (65) and (66) do not depend on the choice of the germ of first integral submersion g at some regular point oD ∈ D nor on that of the coordinate system on Q· . To see that these Q· group morphisms define a group-graph  morphism  we need to show that for Φ ∈ AutF (D), −1 ∂ prC ◦g∗ Φ the germ at {s} = D ∩ D0 of ġD∗ ∂uk is equal to the class in T F (s) of the germ u=u0 ∂ prM ◦ Φ F at s of ∂uk u=u0 , k = 1, . . . , q. Thanks to Remark 5.5 it suffices to check this equality at a regular point s ∈ D close to s. We may suppose that oD = s0 . Using the map g∗ in 0 the exact sequence (47), the commutativity of the operations of partial derivatives at s0 and direct image by the first integral g: ! ∂ prMF ◦ Φ ∂ prC ◦ g∗ Φ g∗ = , ∂uk u=u0 ∂uk u=u0 gives the required equality. · It is easy to check that the collection {ΘQ F } defines a natural transformation of functors Θ : Aut → T ⊗C M/M2 . Now we apply the cohomological functor to Θ and we use the natural identification between MQ,u0 /M2Q,u0 and the cotangent vector space Tu∗0 Q of Q at u0 . We obtain natural maps · ∼ · H 1 (Θ) ∼ Def Q 1 Q 1 2 F −→H (AF , AutF ) −→ H (AF , TF ⊗C (MQ,u0 /MQ,u0 ))−→ ∼ H 1 (AF , TF ) ⊗C (MQ,u0 /M2Q,u0 ) −→H 1 (AF , TF ) ⊗C Tu∗0 Q = L(Tu0 Q, H 1 (AF , TF )) , (67) where L(E, E 0 ) denotes the space of C-linear maps from the C-vector space E to the C- vector space E 0 . We call Kodaira-Spencer map for (Q· , F) the composition (67) of these maps: · Q· KSQ 1 F : Def F → L(Tu0 Q, H (AF , TF )) . We will also write · ∂[FQ· ] KSQ F ([FQ ]) =: · . (68) ∂u u=u0 Consider now the contravariant functor DFac : Man· × Fol → Set· defined by DFac(Q· , F) := L(Tu0 Q, H 1 (AF , TF )) , DFac(µ, φ) := DFacµφ , with DFacµφ : L(Tu0 Q, H 1 (AF , TF )) → L(Tt0 P, H 1 (AG , TG )) defined by DFacµφ (`) := [φ∗ ] ◦ ` ◦ Dt0 µ 50 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM if (µ, φ) : (P · , G) → (Q· , F) is a morphism in the category Man· × Fol. Since Dt0 ([φ∗ ] ◦ λ ◦ µ) = [φ∗ ] ◦ Du0 λ ◦ Dt0 µ the derivation maps Q · DF : O(Q· , H 1 (AF , TF )) → L(Tu0 Q, H 1 (AF , TF )) , λ 7→ Du0 λ constitute a natural transformation D : Fac → DFac (69) according to (62). One can check the following: Proposition 6.5. For any morphism (µ, φ) : (P · , G) → (Q· , F) in Man· × Fol and any · deformation [FQ· ] ∈ Def Q F , we have the following commutative diagram: ∂((µ,φ)∗ [FQ· ]) ∂t / H 1 (AG , TG ) t=t0 Tt0 P O Dt0 µ ∂([FQ· ]) [φ∗ ]  ∂u / H 1 (AF , TF ) u=u0 Tu0 Q · in other words, the collection {KSQ F }(Q· ,F ) defines a natural transformation KS : Def → DFac between contravariant functors from Man· × Fol to Set· . 6.3. Criteria for universality. Let us suppose now that the foliation F has finite type. Using the representation of the deformation functor, the Kodaira-Spencer transformation becomes the usual derivation: Proposition 6.6. Restricted to the subcategory Man· × Folft the natural transformation KS is equal to the composition of the natural transformation derivative (69) with the natural ∼ isomorphism ξ : Def −→Fac defined in (63) KS = D ◦ ξ Proof. Let us fix (Q· , F) ∈ Man· × Folft . Since F is assumed to be of finite type, ξ is an isomorphism of functors and it suffices to see that, after the identifications O(Q· , H 1 (AF , TF )) ' H 1 (AF , TF ) ⊗C MQ,u0 and L(Tu0 Q, H 1 (AF , TF )) ' H 1 (AF , TF ) ⊗C MQ,u0 /M2Q,u0 , the following map · · KSQ Q −1 F ◦ (ξF ) : H 1 (AF , TF ) ⊗C MQ,u0 → H 1 (AF , TF ) ⊗C MQ,u0 /M2Q,u0 coincides with the tensor product of the identity map of H 1 (AF , TF ) and the quotient map MQ,u0 → MQ,u0 /M2Q,u0 , a 7→ ȧ. By following the functor morphisms in (64) and (67) and formula (66) we obtain that · Q· −1 X ∂  (KSQF ◦ (ξ F ) )([XD,e ] ⊗ a(u)) = exp(a(u)X D,e )[1] ⊗ u̇k ∂uk u=u0 k X ∂  = exp(XD,e )[a(u)] ⊗ u̇k ∂uk u=u0 k X  ∂a  = (u0 )XD,e ⊗ u̇k ∂uk k X ∂a = [XD,e ] ⊗ (u0 )u̇k = [XD,e ] ⊗ ȧ . ∂uk k TOPOLOGICAL UNIVERSAL FAMILIES OF HOLOMORPHIC FOLIATIONS 51  This interpretation of KS provides an infinitesimal criterium of universality. Theorem 6.7. Let F be a finite type foliation which is a generalized curve. For any equisingular deformation FP · of F over a germ of manifold P · , the following properties are equivalent: (1) FP · is C ex -universal, ∼ (2) there is a biholomorphism germ µ : R· −→P · such that µ∗ FP · is C ex -universal, ∼ (3) for any biholomorphism germ µ : R· −→P · the deformation µ∗ FP · is C ex -universal, · P ([F · ]) : P · → H 1 (A , T )· is a biholomorphism germ, (4) the map ξF P F F ∂[FP · ] (5) the Kodaira-Spencer map ∂t is an isomorphism. t=t0 Proof. The equivalence of the first three assertions follows directly from the definition of C ex -universality. The proof of (1) =⇒ (4) is classical7: after setting Q· := H 1 (AF , TF ) one considers · Q· the class fQ· ∈ Def QF such that ξF (fQ· ) = idQ· , which is C ex -universal, according to (60). · P ([F · ]) : P · → Q· satisfies f · := [F · ] = λ∗ f · . On the other Therefore, the map λ := ξF P P P Q hand, since fP · is assumed to be C ex -universal, there is µ : Q· → P · such that fQ· = µ∗ fP · . The uniqueness of factorizations and the relations µ∗ λ∗ fQ· = fQ· , λ∗ µ∗ fP · = fP · , give λ ◦ µ = idQ· and µ ◦ λ = idP · . The implication (4) ⇒ (1) is a consequence of Theorem 6.4 and Remark 6.2. According to Proposition 6.6, ∂[F∂tP · ] is the derivative of the map ξF P · ([F · ]), thus the P t=t0 equivalence (4) ⇐⇒ (5) is trivial.  Corollary 6.8. Let φ be an C ex -conjugacy between two foliations F, G ∈ Fol of finite type, · Q· φ(G) = F. Then fQ· ∈ Def Q ex ∗ F is C -universal if and only if gQ := φ (fQ ) ∈ Def G is · · universal. Proof. Let us suppose fQ· C ex -universal. According to Theorem 6.7, gQ· is C ex -universal as soon as λ∗ gQ· = φ∗ (λ∗ fQ· ) is C ex -universal for some biholomorphism germ λ : P · → Q· . Q· −1 Therefore we may suppose that Q· := H 1 (AF , TF )· and fQ· = (ξF ) (idQ· ). Then we set: P · := H 1 (AG , TG )· , λ := [φ∗ ]−1 : P · → Q· . Since ξ is a natural transformation we have the following commutative diagram: Q · · ξF Def Q / O (Q· , Q· ) F (λ,φ)∗ Facλ φ  P· ξG  Def PG · / O (P · , P · ) · · We check that Facλφ (idQ· ) = idP · , hence ξGP (λ∗ gQ· ) = ξGP ((λ, φ)∗ (fQ· )) = idP · . Thanks to criterion (4) in Theorem 6.7, λ∗ gQ· is C ex -universal.  References [1] C. Camacho and R. Rosas, Invariant sets near singularities of holomorphic foliations, Ergodic Theory Dynam. Systems 36 (2016), 2408–2418. [2] F. Cano, D. Cerveau and J. Deserti, “Théorie élémentaire des feuilletages holomorphes singuliers”, Éditions Belin, Echelles, 2013. 7In fact, in the category whose objects are the classes of equisingular deformations of F and the morphisms are pull-backs, a class of an equisingular deformation is C ex -universal if and only if it is a final object. It is well-known that the final objects are canonically isomorphic, i.e. by a unique isomorphism. 52 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ELIANE SALEM [3] A. A. 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Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Cerdany- ola del Vallès (Barcelona), Spain, Centre de Recerca Matemàtica, Campus de Bellaterra, E-08193 Cerdanyola del Vallès, Spain E-mail address: David.Marin@uab.cat Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, Route de Nar- bonne, F-31062 Toulouse Cedex 9, France E-mail address: jean-francois.mattei@math.univ-toulouse.fr Sorbonne Université, Université de Paris, CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche, F-75005 Paris, France E-mail address: eliane.salem@imj-prg.fr