# Topological Moduli Space for Germs of Holomorphic Foliations III: Complete
families

David Marín, Jean-François Mattei and Éliane Salem  Departament de
Matemàtiques
Universitat Autònoma de Barcelona
E-08193 Cerdanyola del Vallès (Barcelona)
Spain

Centre de Recerca Matemàtica, Campus de Bellaterra, E-08193 Cerdanyola del
Vallès, Spain<EMAIL_ADDRESS>Institut de Mathématiques de Toulouse
Université Paul Sabatier
118, Route de Narbonne
F-31062 Toulouse Cedex 9, France<EMAIL_ADDRESS>Sorbonne Université, Université de Paris, CNRS, Institut de Mathématiques de
Jussieu - Paris Rive Gauche, F-75005 Paris, France<EMAIL_ADDRESS>
###### Abstract.

In this work we use our previous results on the topological classification of
generic singular foliation germs on $({\mathbb{C}}^{2},0)$ to construct
complete families: after fixing the semi-local topological invariants we prove
the existence of a minimal family of foliation germs that contain all the
topological classes and such that any equisingular global family with
parameter space an arbitrary complex manifold factorizes through it.

###### Key words and phrases:

Complex dynamical systems, complex ordinary differential equations,
holomorphic vector fields, holomorphic foliations, singularities, holonomy,
moduli space, complete families, universal deformation.

###### 2010 Mathematics Subject Classification:

Primary 37F75; Secondary 32M25, 32S50, 32S65, 34M

D. Marín acknowledges financial support from the Spanish Ministry of Science,
Innovation and Universities, through grant PGC2018-095998-B-I00 and by the
Agency for Management of University and Research Grants of Catalonia through
the grant 2017SGR1725. This work is also supported by the Spanish State
Research Agency, through the Severo Ochoa and María de Maeztu Program for
Centers and Units of Excellence in R&D (CEX2020-001084-M)

August 27, 2024

###### Contents

  1. 1 Introduction
  2. 2 Locally universal family
    1. 2.1 Equisingular global families and deformations
    2. 2.2 Marked foliations and families
    3. 2.3 Local universality
  3. 3 Factorization properties of the locally universal family
    1. 3.1 Local factorization property
    2. 3.2 Global factorization property
  4. 4 Topological equivalences for families and deformations
    1. 4.1 Tame foliations
    2. 4.2 Weak and strong conjugacies of families
    3. 4.3 Conjugacies of families versus conjugacies of deformations

##  1\. Introduction

This paper is the outcome of a series of three works on the topological
classification of germs of singular foliations in the complex plane. In [8],
after fixing the topological invariants already known [7], we have constructed
a moduli space of topological classes. Then in [9], we have studied small
perturbations of a generic foliation by proving the existence of a
“topologically universal” deformation and by representing the “deformation
functor”. In the present paper we rely on these two results and in particular
on the algebraic structure of the moduli space highlighted in [8] to obtain a
“complete family”. Here “complete” means a family that contains all the
topological classes and such that any equisingular global family with
parameter space an arbitrary complex manifold factorizes through it via a
multivalued map.

We shall only consider germs of foliations on $({\mathbb{C}}^{2},0)$ that are
“generalized curves” [1], in the sense that on the exceptional divisor
${\mathcal{E}}_{\mathcal{F}}:=E_{\mathcal{F}}^{-1}(0)$ of the reduction of
singularities map $E_{\mathcal{F}}:M_{\mathcal{F}}\to{\mathbb{C}}^{2}$, there
are no singularities of the reduced foliation
${\mathcal{F}}^{\sharp}:=E_{{\mathcal{F}}}^{-1}({\mathcal{F}})$ of saddle-node
type111i.e. locally defined by a vector field germ whose linear part has
exactly one non-zero eigenvalue., cf. [2, 5]. The foliation
${\mathcal{F}}^{\sharp}$ may have dicritical (i.e. non-invariant) irreducible
components of ${\mathcal{E}}_{\mathcal{F}}$ and also nodal222i.e. locally
defined by a vector field germ such that the ratio of the eigenvalues of its
linear part is strictly positive. singularities.

We will say that two germs of foliations ${\mathcal{F}}$ and $\mathcal{G}$
have same SL-type if there exists a homeomorphism
$\varphi:{\mathcal{E}}_{\mathcal{F}}{\overset{\sim}{\longrightarrow}}{\mathcal{E}}_{\mathcal{G}}$
satisfying:

  1. (SL1)

$\varphi({{\mathrm{Sing}}}({\mathcal{F}}^{\sharp}))={{\mathrm{Sing}}}(\mathcal{G}^{\sharp})$
and $\varphi(D)\cdot\varphi(D^{\prime})=D\cdot D^{\prime}$,

  2. (SL2)

if $D\subset{\mathcal{E}}_{\mathcal{F}}$ is ${\mathcal{F}}^{\sharp}$-invariant
and $p\in D$ is a singular point of ${\mathcal{F}}^{\sharp}$, then the
Camacho-Sad indices of ${\mathcal{F}}^{\sharp}$ at $p$ along $D$ and of
$\mathcal{G}^{\sharp}$ at $\varphi(p)$ along $\varphi(D)$, are equal,

  3. (SL3)

if
${\mathcal{H}}^{{\mathcal{F}}}_{D}:\pi_{1}(D\setminus{{\mathrm{Sing}}}({\mathcal{F}}^{\sharp}),\cdot)\to{{\mathrm{Diff}}}({\mathbb{C}},0)$
denotes the holonomy morphism of ${\mathcal{F}}^{\sharp}$ along an invariant
component $D\subset{\mathcal{E}}_{\mathcal{F}}$ and $\varphi_{\ast}$ denotes
the morphism induced by $\varphi$ at the fundamental groups level, then up to
composition by an inner automorphism of ${{\mathrm{Diff}}}({\mathbb{C}},0)$,
${\mathcal{H}}^{{\mathcal{F}}}_{D}\circ\varphi_{\ast}^{-1}$ is the holonomy
morphism of $\mathcal{G}^{\sharp}$ along $\varphi(D)$.

When all foliations of a family have same SL-type, the family will be called
equisingular. This notion, which presupposes local equireduction of the
considered family, is specifically defined in Section 2.1. For an equisingular
family ${\mathcal{F}}_{P}$ with parameter space a complex manifold $P$, we
will denote by ${\mathcal{F}}_{P}(t_{0}):={\mathcal{F}}_{P}|_{t=t_{0}}$ the
fiber over a particular value $t_{0}\in P$ and by ${\mathcal{F}}_{P,t_{0}}$
the germ of this family along the fiber ${\mathcal{F}}_{P}(t_{0})$.

The notion of tame foliation will be defined in section 4.1. If we exclude
some exceptional configurations on ${\mathcal{E}}_{\mathcal{F}}$, the set of
differential forms defining tame foliations contains a Krull open dense set.
Our main result asserts the existence in this context of an equisingular
global family which is “topologically complete”. We also describe the
(minimal) redundancy of their topological classes.

######  Theorem A.

Let ${\mathcal{F}}$ be a tame foliation. Then there exist
$\tau\in{\mathbb{N}}$, a quotient $\mathbf{D}$ of a finite product of totally
disconnected subgroups of ${\mathbb{U}}(1)$ and an equisingular global family
${\underline{{\mathcal{F}}}}_{\mathbf{U}}$ over
$\mathbf{U}={\mathbb{C}}^{\tau}\times\mathbf{D}$ such that

  1. (1)

for any foliation $\mathcal{G}$ with same SL-type as ${\mathcal{F}}$, there
exists $u_{0}\in\mathbf{U}$ such that $\mathcal{G}$ is
${\mathcal{C}}^{0}$-conjugated to the fiber
${\mathcal{F}}_{\mathbf{U}}(u_{0})$,

  2. (2)

if $P$ is a connected and simply connected manifold, $t_{0}\in P$ and
${\underline{\mathcal{G}}}_{P}$ is an equisingular global family whose fiber
$\mathcal{G}_{P}(t_{0})$ is ${\mathcal{C}}^{0}$-conjugated to a fiber
${\mathcal{F}}_{\mathbf{U}}(u_{0})$, then there exists a holomorphic map
$\lambda:P\to{\mathbf{U}}$ such that $\lambda(t_{0})=u_{0}$, and for any $t\in
P$ the germs of families ${\underline{\mathcal{G}}}_{P,t}$ and
$\lambda^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U},\lambda(t)}$ over the
germ of manifold $(P,t)$, are ${\mathcal{C}}^{0}$-conjugated,

  3. (3)

there exist $p\in\mathbb{N}$, a holomorphic action $*$ of ${\mathbb{Z}}^{p}$
on $\mathbf{U}$ and an action $\star$ of a finite product $\mathbf{I}$ of
braid groups on the quotient $(\mathbf{U}/*{\mathbb{Z}}^{p})$, such that
${\mathcal{F}}_{\mathbf{U}}(u_{1})$ and ${\mathcal{F}}_{\mathbf{U}}(u_{2})$
are ${\mathcal{C}}^{0}$-conjugated if and only if there exists
$g\in\mathbf{I}$ such that
$g\star({\mathbb{Z}}^{p}*u_{1})={\mathbb{Z}}^{p}*u_{2}$.

Tame foliations have the remarkable property that any two
${\mathcal{C}}^{0}$-conjugated tame foliations are also conjugated by an
excellent homeomorphism, i.e. one that lifts through the reduction of
singularities which is holomorphic at the non-nodal singular points, cf.
Theorem 4.1. This result extends to equisingular families of tame foliations:

######  Theorem B.

Let ${\underline{{\mathcal{F}}}}_{Q}$ and ${\underline{\mathcal{G}}}_{Q}$ be
two equisingular global families of foliations over a complex manifold $Q$,
whose fibers are tame. Then the following properties are equivalent:

  1. (1)

for any $u\in Q$ the fibers ${\mathcal{F}}_{Q}(u)$ and $\mathcal{G}_{Q}(u)$
are ${\mathcal{C}}^{0}$-conjugated,

  2. (2)

for any $u\in Q$ the fibers ${\mathcal{F}}_{Q}(u)$ and $\mathcal{G}_{Q}(u)$
are ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated,

  3. (3)

the global families ${\underline{{\mathcal{F}}}}_{Q}$ and
${\underline{\mathcal{G}}}_{Q}$ are locally
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated.

A ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy of families is a
${\mathcal{C}}^{0}$-conjugacy of families that lifts through the local
equireduction maps as a homeomorphism which is holomorphic at the non-nodal
singularities, cf. §2.1.

Theorems A and B will follow from analogous results in the context of marked
foliations that allow to use the moduli space of
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy classes of marked foliations
constructed in [8]. A marking of ${\mathcal{F}}$ by a marked divisor
$({\mathcal{E}},\Sigma,\cdot)$ is a homeomorphism
$f:{\mathcal{E}}\to{\mathcal{E}}_{\mathcal{F}}$ such that
$f(\Sigma)={{\mathrm{Sing}}}({\mathcal{F}}^{\sharp})$ and $f(D)\cdot
f(D^{\prime})=D\cdot D^{\prime}$, cf. §2.2. When ${\mathcal{F}}$ and
$\mathcal{G}$ are endowed with markings
$f:{\mathcal{E}}\to{\mathcal{E}}_{\mathcal{F}}$ and
$g:{\mathcal{E}}\to{\mathcal{E}}_{\mathcal{G}}$ by a common marked divisor
$({\mathcal{E}},\Sigma,\cdot)$, we will say that the marked foliations
$({\mathcal{F}},f)$ and $(\mathcal{G},g)$ have same marked SL-type if
conditions (SL1)-(SL3) are fulfilled and moreover $g^{-1}\circ\varphi\circ f$
is isotopic to the identity of ${\mathcal{E}}$ relatively to $\Sigma$. The
following analogue of Theorem A in the marked setting holds for the larger
class of finite type foliations introduced in [8, §6] and specified in [9,
§5]; in this context we have the uniqueness of the factorization
$\lambda:P\to\mathbf{U}$ of a marked family under a weaker topological
condition on its parameter space $P$.

######  Theorem C.

Let ${\mathcal{F}}^{\diamond}=({\mathcal{F}},f)$ be a marked finite type
foliation which is a generalized curve. Then there exists a marked
equisingular global family of foliations
${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ over
$\mathbf{U}={\mathbb{C}}^{\tau}\times\mathbf{D}$ such that

  1. (0)

$\mathbf{D}$ is a quotient of a finite product of totally disconnected
subgroups of ${\mathbb{U}}(1)$ and $\tau$ is the dimension of the
cohomological space
$H^{1}({\mathsf{A}}_{\mathcal{F}},\mathcal{T}_{\mathcal{F}})$ (that we recall
in (4)) whose finiteness characterizes the finite type of ${\mathcal{F}}$, cf.
[9, Theorem 5.15],

  2. (1)

if $\mathcal{G}^{\diamond}=(\mathcal{G},g)$ is a marked foliation with same
marked SL-type as ${\mathcal{F}}^{\diamond}$, there exists
$u_{0}\in\mathbf{U}$ such that $\mathcal{G}^{\diamond}$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
${\mathcal{F}}_{\mathbf{U}}^{\diamond}(u_{0})$,

  3. (2)

if $P$ is a connected manifold satisfying $H_{1}(P,{\mathbb{Z}})=0$, $t_{0}\in
P$ and ${\underline{\mathcal{G}}}_{P}^{\diamond}$ is a marked equisingular
global family whose fiber $\mathcal{G}^{\diamond}_{P}(t_{0})$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to a fiber
${\mathcal{F}}_{\mathbf{U}}^{\diamond}(u_{0})$ as marked foliations, then
there exists a unique holomorphic map $\lambda:P\to{\mathbf{U}}$ such that
$\lambda(t_{0})=u_{0}$ and for any $t\in P$ the germs of marked families
${\underline{\mathcal{G}}}_{P,t}^{\diamond}$ and
$\lambda^{\ast}{\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U},\lambda(t)}$,
over the germ of manifold $(P,t)$, are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated.

An analogue of Theorem B in the marked setting will be given in Theorem 4.4.

We will also compare the conjugation notions of local families and
deformations. A deformation of ${\mathcal{F}}$ is the data of a family
${\underline{{\mathcal{F}}}}_{P,t_{0}}$ over a germ of holomorphic manifold
$P$ at a point $t_{0}$ and a biholomorphism that identifies ${\mathcal{F}}$
with the fiber ${\mathcal{F}}_{P}(t_{0})$. A conjugacy of deformations of
${\mathcal{F}}$ is a conjugacy of the associated families compatible with the
corresponding biholomorphisms, cf. §2.1. In Theorem 4.5 we show that this
compatibility condition is automatically fulfilled in the context of marked
germs of families.

The central point of the paper is Theorem 2.9. It states the
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universality of the germ at any point of the
parameter space of the global family
${\underline{{\mathcal{F}}}}_{\mathbf{U}}$ constructed in [8] that contains
all the topological types in a fixed SL-class. This property will be proven by
explicitly computing the Kodaira-Spencer map of this family at each point that
provides an infinitesimal characterization of
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universality.

In Chapter 3 we look at the problem of existence of factorizations of global
families through ${\underline{{\mathcal{F}}}}_{\mathbf{U}}$. Since Theorem 2.9
provide local factorizations, obtaining a global factorization is reduced to a
gluing problem; the group structure of the moduli space obtained in [8] allows
to translate it into a cohomological problem that can be solved under weak
topological assumptions on the parameter space of the global family.

All the study in Chapters 2 and 3, leading to Theorem C, is made for marked
families modulo ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy and only under the
finite type assumption. But Theorems A and B concern non-marked global
families and ${\mathcal{C}}^{0}$-conjugacies. To work with
${\mathcal{C}}^{0}$-conjugacies we require additional (Krull generic)
hypothesis defining tame foliations in §4.1, which allow to prove Theorem A.
The proof of Theorem B in Section 4.2 is based again in the group structure on
the moduli space using the fact that the mapping class group of the
exceptional divisor is countable.

Through all the paper a ${\mathcal{C}}^{0}$-conjugacy between two foliations
is a homeomorphism sending leaves into leaves and preserving the orientation
of the ambient spaces as well as the orientation of the leaves.

##  2\. Locally universal family

###  2.1. Equisingular global families and deformations

We call global family of (germs of) foliations over a complex manifold $Q$
(not necessarily connected), the data

${\underline{{\mathcal{F}}}}_{Q}:=(M,\pi,\theta,{\mathcal{F}}_{Q})$

of a complex manifold $M$ with $\dim(M)=\dim(Q)+2$, a holomorphic surjective
submersion $\pi:M\to Q$, a holomorphic section $\theta:Q\to M$ of $\pi$, and a
germ along $\theta(Q)$ of a one dimensional holomorphic foliation
${\mathcal{F}}_{Q}$ on $M$ whose leaves are contained in the fibers of $\pi$.
We say that $(M,\pi)$ is a manifold over $Q$. For each $u\in Q$ we consider in
the fiber of $\pi$ over $u$ the germ of foliation at $\theta(u)$ obtained by
restricting ${\mathcal{F}}_{Q}$:

$M(u):=\pi^{-1}(u)\,,\quad{\mathcal{F}}_{Q}(u):={\mathcal{F}}_{Q}|_{(M(u),\theta(u))}\,.$

The family is equireducible if $\theta(Q)$ is the singular locus of
${\mathcal{F}}_{Q}$ and for any point $u_{0}\in Q$ there is an open
trivializing neighborhood $W\ni u_{0}$ and a map called (minimal)
equireduction map over $W$

$E_{{\mathcal{F}}_{W}}:M_{{\mathcal{F}}_{W}}\to M_{W}:=\pi^{-1}(W)$

that is defined by a sequence of blow-ups with etale centers over $W$, and
whose restriction to each fiber

$M_{{\mathcal{F}}_{W}}(u):=\pi^{\sharp\,-1}(u)\,,\quad\pi^{\sharp}:=\pi\circ
E_{{\mathcal{F}}_{W}}\,,\quad u\in W\,,$

is exactly the minimal reduction map of ${\mathcal{F}}_{Q}(u)$, and moreover
the singular locus of the reduced foliation ${\mathcal{F}}_{W}^{\sharp}$ in
$M_{{\mathcal{F}}_{W}}$ is also etale over $W$. A more detailed definition of
this notion is given in [9, §2.2] or in [8, Chapter 10, step (vi)]. Up to
shrinking the neighborhood $W$ of $u_{0}$, the exceptional divisor
${\mathcal{E}}_{{\mathcal{F}}_{W}}=E_{{\mathcal{F}}_{W}}^{-1}(\theta(W))$ and
the singular locus of the reduced foliation ${\mathcal{F}}_{W}^{\sharp}$ in
$M_{{\mathcal{F}}_{W}}$ are topologically trivial: there exists a trivializing
homeomorphism over $W$

$\Psi_{W}:{\mathcal{E}}_{{\mathcal{F}}_{W}}{\overset{\sim}{\longrightarrow}}{\mathcal{E}}_{{\mathcal{F}}_{W}}(u_{0})\times
W\,,\quad{{\mathrm{pr}}}_{W}\circ\Psi_{W}=\pi^{\sharp}_{|{\mathcal{E}}_{{\mathcal{F}}_{W}}}\,,\quad{\mathcal{E}}_{{\mathcal{F}}_{W}}(u):={\mathcal{E}}_{{\mathcal{F}}_{W}}\cap\pi^{\sharp\,-1}(u)\,,$
(1)

that sends the singular locus of ${\mathcal{F}}_{W}^{\sharp}$ on the product
${{\mathrm{Sing}}}({\mathcal{F}}^{\sharp}_{W}(u_{0}))\times W$, with

${\mathcal{F}}_{W}^{\sharp}(u):={\mathcal{F}}_{W}^{\sharp}|_{\pi^{\sharp\,{-1}}(u)}\,.$

Restricted to the fiber of $u\in W$, $\Psi_{W}$ provides a homeomorphism that
identifies the exceptional divisor of the reduction of ${\mathcal{F}}_{W}(u)$,
with that of ${\mathcal{F}}_{W}(u_{0})$,

$\Psi_{u}:{\mathcal{E}}_{{\mathcal{F}}_{W}}(u){\overset{\sim}{\longrightarrow}}{\mathcal{E}}_{{\mathcal{F}}_{W}}(u_{0})\,.$

Thus the holonomy of the foliation ${\mathcal{F}}_{W}^{\sharp}(u)$ along an
invariant component $D_{u}=\Psi^{-1}_{u}(D_{u_{0}})$ may be considered as a
morphism ${\mathcal{H}}_{D_{u}}$ from the fundamental group of
$D_{u_{0}}\setminus{{\mathrm{Sing}}}({\mathcal{F}}_{W}(u_{0}))$ into the group
${{\mathrm{Diff}}}({\mathbb{C}},0)$ of germs of biholomorphisms of
$({\mathbb{C}},0)$.

######  Definition 2.1.

We say that an equireducible family ${\underline{{\mathcal{F}}}}_{Q}$ is
equisingular at $u_{0}\in Q$ if there is a trivializing neighborhood $W$ of
$u_{0}$ such that for any invariant irreducible component
$D_{u_{0}}\subset{\mathcal{E}}_{{\mathcal{F}}_{W}}(u_{0})$ and for any point
$m_{0}\in{{\mathrm{Sing}}}({\mathcal{F}}_{W}(u_{0}))\cap D_{u_{0}}$, we have:

  1. (a)

there exist biholomorphisms $g_{u}\in{{\mathrm{Diff}}}({\mathbb{C}},0)$
depending holomorphically of $u\in W$ such that
$g_{u}\circ{\mathcal{H}}_{D_{u}}\circ g_{u}^{-1}={\mathcal{H}}_{D_{u_{0}}}$,

  2. (b)

the Camacho-Sad function  from $W$ to ${\mathbb{C}}$:

$u\mapsto{{\mathrm{CS}}}({\mathcal{F}}_{W}(u),D_{u},m_{u}),\quad
D_{u}:=\Psi_{W}^{-1}(D_{u_{0}}\times\\{u\\})\,,\quad
m_{u}:=\Psi^{-1}_{W}(m_{0},u)\,,$

is constant.

A ${\mathcal{C}}^{0}$-conjugacy between two global equireducible families
${\underline{{\mathcal{F}}}}_{Q}=(M,\pi,\theta,{\mathcal{F}}_{Q})$ and
${\underline{{\mathcal{F}}}}^{\prime}_{Q}=(M^{\prime},\pi^{\prime},\theta^{\prime},{\mathcal{F}}^{\prime}_{Q})$
over the same parameter space $Q$, is a germ of homeomorphism
$\Phi:(M,\theta(Q)){\overset{\sim}{\longrightarrow}}(M^{\prime},\theta^{\prime}(Q))$
satisfying $\Phi({\mathcal{F}}_{Q})={\mathcal{F}}^{\prime}_{Q}$ and
$\pi^{\prime}\circ\Phi=\pi$. We also assume that $\Phi$ preserves the
orientation of the ambient spaces and the orientation of the leaves. We will
say that $\Phi$ is excellent or of class ${\mathcal{C}}^{{{\mathrm{ex}}}}$, if
its lifting $\Phi^{\sharp}$ through any local equireduction maps
$E_{{\mathcal{F}}_{W}}$ and $E_{{\mathcal{F}}^{\prime}_{W}}$,
$E_{{\mathcal{F}}^{\prime}_{W}}\circ\Phi^{\sharp}=\phi\circ
E_{{\mathcal{F}}_{W}}$, extends to the exceptional divisors, providing a
homeomorphism

$\Phi^{\sharp}:(M_{{\mathcal{F}}_{W}},{\mathcal{E}}_{{\mathcal{F}}_{W}}){\overset{\sim}{\longrightarrow}}(M_{{\mathcal{F}}^{\prime}_{W}},{\mathcal{E}}_{{\mathcal{F}}^{\prime}_{W}})$

which is holomorphic at any singular point of the exceptional divisor that is
a non nodal singular point of the foliation. We will also say that
$\Phi^{\sharp}$ is excellent.

Let $\mu:P\to Q$ be a holomorphic map and let
${\underline{{\mathcal{F}}}}_{Q}=(M,\pi,\theta,{\mathcal{F}}_{Q})$ be a global
family of foliations over $Q$. We consider the fibered product
$\mu^{*}M=M\times_{Q}P\subset M\times P$ with the projection
$\mu^{*}\pi:M\times_{Q}P\to P$,

$\textstyle{\mu^{*}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{*}\pi}$$\scriptstyle{\rho_{\mu}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{Q}$
$\textstyle{(\mu^{*}M)(t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\rho_{\mu}}$$\textstyle{M(\mu(t))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{t\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu(t)}$

and the section $\mu^{*}\theta=(\theta\circ\mu)\times{{\mathrm{id}}}_{P}:P\to
M\times_{Q}P$. Since the restrictions to each fiber of the canonical
submersion $\rho_{\mu}$ are biholomorphisms, there is a unique one-dimensional
foliation germ $\mu^{*}{\mathcal{F}}_{Q}$ on $\mu^{*}M$ along
$(\mu^{*}\theta)(P)$, tangent to the fibers of $\mu^{*}\pi$, such that
$\rho_{\mu}$ sends the leaves of $\mu^{*}{\mathcal{F}}_{Q}$ into the leaves of
${\mathcal{F}}_{Q}$. We will call
$\mu^{*}{\underline{{\mathcal{F}}}}_{Q}=(\mu^{*}M,\mu^{*}\pi,\mu^{*}\theta,\mu^{*}{\mathcal{F}}_{Q})$
the pull-back of the global family ${\underline{{\mathcal{F}}}}_{Q}$ by the
map $\mu:P\to Q$. Since the equisingularity is a local property in the
parameters, by [9, Proposition 3.7] it is preserved by pull-back. Moreover, if
two global equisingular families ${\underline{{\mathcal{F}}}}_{Q}$ and
${\underline{\mathcal{G}}}_{Q}$ are ${\mathcal{C}}^{0}$ (resp.
${\mathcal{C}}^{{{\mathrm{ex}}}}$) conjugated by a homeomorphism $\Phi$ then
so are $\mu^{*}{\underline{{\mathcal{F}}}}_{Q}$ and
$\mu^{*}{\underline{\mathcal{G}}}_{Q}$ by
$\mu^{*}\Phi=\Phi\times{{\mathrm{id}}}_{P}$.

Let $u_{0}$ be a point of $Q$ and let ${\mathcal{F}}$ be a germ of foliation
at a point $m_{0}$ of a two dimensional complex manifold $M_{0}$. An
equisingular deformation of ${\mathcal{F}}$ over the germ of manifold
$(Q,u_{0})$ is the data $({\underline{{\mathcal{F}}}}_{Q,u_{0}},\iota)$ of the
germ at $\theta(u_{0})$ of an equisingular family
${\underline{{\mathcal{F}}}}_{Q}=(M,\pi,\theta,{\mathcal{F}}_{Q})$ together
with the germ of an embedding
$\iota:(M_{0},m_{0})\hookrightarrow(M,\theta(u_{0}))$ that sends
${\mathcal{F}}$ to the restricted foliation germ ${\mathcal{F}}_{Q}(u_{0})$ on
the special fiber $M(u_{0})$. A conjugacy between two equisingular
deformations $({\underline{{\mathcal{F}}}}_{Q,u_{0}},\iota)$ and
$({\underline{{\mathcal{F}}}}^{\prime}_{Q,u_{0}},\iota^{\prime})$ is a
conjugacy $\Phi$ between their associated families,
$\Phi({\underline{{\mathcal{F}}}}_{Q,u_{0}})={\underline{{\mathcal{F}}}}^{\prime}_{Q,u_{0}}$,
such that $\Phi\circ\iota=\iota^{\prime}$.

If $\mu:(P,t_{0})\to(Q,u_{0})$ is a holomorphic map germ and
$({\underline{{\mathcal{F}}}}_{Q,u_{0}},\iota)$ is an equisingular deformation
of ${\mathcal{F}}$ over $(Q,u_{0})$, then
$(\mu^{*}{\underline{{\mathcal{F}}}}_{Q,u_{0}},\mu^{*}\iota)$ is an
equisingular deformation of ${\mathcal{F}}$ over $(P,t_{0})$ where
$\mu^{*}\iota$ is defined by $\rho_{\mu}\circ\mu^{*}\iota=\iota$ (recall that
the restriction of $\rho_{\mu}$ to the fiber over $t_{0}$ is a biholomorphism
onto the fiber over $u_{0}$).

######  Definition 2.2.

Let $({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ be an
equisingular deformation over a germ of manifold
$Q^{\mathbf{\cdot}}:=(Q,u_{0})$, of a foliation ${\mathcal{F}}$. We say that
$({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ is a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal deformation of ${\mathcal{F}}$ if
for any germ of manifold $P^{\mathbf{\cdot}}=(P,t_{0})$ and any equisingular
deformation $({\underline{\mathcal{G}}}_{P^{\mathbf{\cdot}}},\delta)$ of
${\mathcal{F}}$ over $P^{\mathbf{\cdot}}$, there exists a unique germ of
holomorphic map $\lambda:P^{\mathbf{\cdot}}\to Q^{\mathbf{\cdot}}$ such that
the deformations $({\underline{\mathcal{G}}}_{P^{\mathbf{\cdot}}},\delta)$ and
$\lambda^{\ast}({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ of
${\mathcal{F}}$ are ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated.

For each germ of manifold $Q^{\mathbf{\cdot}}$ and each generalized curve
foliation ${\mathcal{F}}$ let us denote by
${{\mathrm{Def}}}_{\mathcal{F}}^{Q^{\mathbf{\cdot}}}$ set of
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy classes of equisingular
deformations of ${\mathcal{F}}$ over $Q^{\mathbf{\cdot}}$.

######  Remark 2.3.

Notice that if $\mu:Q{{}^{\prime}}^{\mathbf{\cdot}}\to Q^{\mathbf{\cdot}}$ is
a germ of biholomorphism, the ${\mathcal{C}}^{{\mathrm{ex}}}$-universality of
$({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ and of
$\mu^{\ast}({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ are
clearly equivalent. On the other hand, it directly results from the definition
that the ${\mathcal{C}}^{{{\mathrm{ex}}}}$-universality of
$({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ only depends on its
${\mathcal{C}}^{{{\mathrm{ex}}}}$-class of conjugacy
$[{\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota]\in{{\mathrm{Def}}}_{\mathcal{F}}^{Q^{\mathbf{\cdot}}}$.
We will then say that
$[{\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota]$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal. ∎

######  Theorem 2.4 ([9, Theorem 3.11 and Corollary 6.8]).

Let ${\mathcal{F}}$ and $\mathcal{G}$ be foliations of finite type which are
generalized curves and let $\phi:\mathcal{G}\to{\mathcal{F}}$ be an excellent
conjugacy of foliations. If
$({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ is an equisingular
deformation of ${\mathcal{F}}$ over $Q^{\mathbf{\cdot}}$, there is an
equisingular deformation
$({\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}},\delta)$ of $\mathcal{G}$
over $Q^{\mathbf{\cdot}}$ and an excellent conjugacy of families
$\Phi:{\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}}\to{\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$
such that $\Phi\circ\delta=\iota\circ\phi$. Moreover, the map

$\phi^{*}:{{\mathrm{Def}}}_{\mathcal{F}}^{Q^{\mathbf{\cdot}}}\to{{\mathrm{Def}}}_{\mathcal{G}}^{Q^{\mathbf{\cdot}}}\,,\quad[{\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota]\mapsto[{\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}},\delta]$

is well defined, bijective and sends the class of a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal deformation of ${\mathcal{F}}$ to
the class of a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal deformation of
$\mathcal{G}$.

###  2.2. Marked foliations and families

Now, we fix for all the sequel a marked divisor
${\mathcal{E}}^{\diamond}=({\mathcal{E}},\Sigma,\cdot)$ in the sense of [8,
§2.1], i.e. a connected compact complex curve with normal crossings
${\mathcal{E}}$, endowed with a finite subset $\Sigma$ of ${\mathcal{E}}$ and
a symmetric map ${{\mathrm{Comp}}}({\mathcal{E}})^{2}\to{\mathbb{Z}}$,
$(D,D^{\prime})\mapsto D\cdot D^{\prime}$, where
${{\mathrm{Comp}}}({\mathcal{E}})$ denotes the set of irreducible components
of ${\mathcal{E}}$. The components of ${\mathcal{E}}$ without any point of
$\Sigma$ are called dicritical components, the others being called invariant
components.

A marked by ${\mathcal{E}}^{\diamond}$ foliation is a pair
${\mathcal{F}}^{\diamond}=({\mathcal{F}},f)$ where

  * •

${\mathcal{F}}$ is a germ (at $m_{0}$) of a holomorphic foliation on a
2-dimensional manifold $(M_{0},m_{0})$,

  * •

$f$ is a homeomorphism, called marking of ${\mathcal{F}}$, from
${\mathcal{E}}$ to the exceptional divisor ${\mathcal{E}}_{\mathcal{F}}$ of
the reduction of ${\mathcal{F}}$ such that: $f(\Sigma)$ is the singular set
${{\mathrm{Sing}}}({\mathcal{F}}^{\sharp})$ of the reduced foliation
${\mathcal{F}}^{\sharp}$, and $D\cdot D^{\prime}$ is equal to the intersection
number of $f(D)$ with $f(D^{\prime})$ in $M_{\mathcal{F}}$ for any components
$D$, $D^{\prime}$ of ${\mathcal{E}}$. Moreover we will also suppose that $f$
is holomorphic at each point of $\Sigma\cup{{\mathrm{Sing}}}({\mathcal{E}})$.

We assume that there exists a foliation that can be marked by
${\mathcal{E}}^{\diamond}$, consequently the dual graph
${\mathsf{A}}_{{\mathcal{E}}}$ of ${\mathcal{E}}$ is necessarily a tree.

Two markings $f$ and $g$ of ${\mathcal{F}}$ by ${\mathcal{E}}^{\diamond}$ will
be called equivalent if the homeomorphism $g^{-1}\circ f$ is isotopic to the
identity map of ${\mathcal{E}}$ by a good isotopy, i.e. an isotopy leaving
fixed $\Sigma$. A ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy between two
marked by ${\mathcal{E}}^{\diamond}$ foliations
${\mathcal{F}}^{\diamond}=({\mathcal{F}},f)$ and
$\mathcal{G}^{\diamond}=(\mathcal{G},g)$ is a germ $\varphi$ of
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy between these foliation germs,
$\varphi({\mathcal{F}})=\mathcal{G}$, such that $g$ and $\varphi^{\sharp}\circ
f$ are equivalent markings by ${\mathcal{E}}^{\diamond}$ of
$\mathcal{G}^{\diamond}$, $\varphi^{\sharp}$ being the lifting of $\varphi$
through the reduction maps.

A pre-marking by ${\mathcal{E}}^{\diamond}$ of an equireducible global family
${\underline{{\mathcal{F}}}}_{Q}$ is a collection

$(f_{u})_{u\in Q},\qquad
f_{u}:{\mathcal{E}}{\overset{\sim}{\longrightarrow}}{\mathcal{E}}_{{\mathcal{F}}_{Q}(u)}\,,$

of markings $f_{u}$ for each foliation ${\mathcal{F}}_{Q}(u)$. Two pre-
markings $(f_{u})_{u\in Q}$ and $(g_{u})_{u\in Q}$ of
${\underline{{\mathcal{F}}}}_{Q}$ of the same global family will be called
equivalent if for each $u\in Q$ the markings $f_{u}$ and $g_{u}$ of
${\mathcal{F}}_{Q}(u)$ are equivalent. A marking of an equireducible global
family ${\underline{{\mathcal{F}}}}_{Q}$ is a pre-marking that satisfies the
following local coherence property: at any point $u_{0}\in Q$ there is an
equireduction neighborhood $W$ of $u_{0}$ and a trivializing homeomorphism
$\Psi_{W}$ as in (1) such that the pre-marking $(f_{u})_{u\in W}$ of
${\underline{{\mathcal{F}}}}_{Q}$ over $W$ is equivalent to the pre-marking
$(\Psi_{u}^{-1}\circ f_{u_{0}})_{u\in W}$, where
$\Psi_{u}:{\mathcal{E}}_{{\mathcal{F}}_{Q}(u)}{\overset{\sim}{\longrightarrow}}{\mathcal{E}}_{{\mathcal{F}}_{Q}(u_{0})}$
is the restriction of $\Psi_{W}$ to the fiber over $u$. A marked by
${\mathcal{E}}^{\diamond}$ global family over a manifold $Q$ is the data

${\underline{{\mathcal{F}}}}_{Q}^{\diamond}=({\underline{{\mathcal{F}}}}_{Q},(f_{u})_{u\in
Q})$

of an equireducible global family over $Q$ and a marking by
${\mathcal{E}}^{\diamond}$ of this family. The fiber at $u\in Q$ of
${\underline{{\mathcal{F}}}}_{Q}^{\diamond}$ is the marked by
${\mathcal{E}}^{\diamond}$ foliation
${\mathcal{F}}_{Q}^{\diamond}(u):=({\mathcal{F}}_{Q}(u),f_{u})$.

######  Remark 2.5.

One can check that the set over $Q$ of the equivalence classes of markings by
${\mathcal{E}}^{\diamond}$ of the foliations ${\mathcal{F}}_{Q}(u)$, $u\in Q$,
can be endowed with a topology such that it becomes a covering over $Q$ (the
local coherence property being equivalent to the existence of continuous local
sections) and the markings of ${\underline{{\mathcal{F}}}}_{Q}$ are continuous
global sections. In particular:

  1. (a)

when $Q$ is connected, two markings of ${\underline{{\mathcal{F}}}}_{Q}$ are
equivalent as soon as, up to a good isotopy, they coincide at some point
$u_{0}\in Q$,

  2. (b)

when $Q$ is connected and simply connected, any marking $f_{u_{0}}$ of the
foliation ${{\mathcal{F}}}_{Q}(u_{0})$ for some $u_{0}\in Q$, extends to a
marking $(f_{u})_{u\in Q}$ of ${\underline{{\mathcal{F}}}}_{Q}$, that is
unique up to equivalence.

∎

A ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy between two marked by
${\mathcal{E}}^{\diamond}$ global families
$({\underline{{\mathcal{F}}}}_{Q},(f_{u})_{u\in Q})$ and
$({\underline{\mathcal{G}}}_{Q},(g_{u})_{u\in Q})$ is a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy of global families
$\Phi({\underline{{\mathcal{F}}}}_{Q})={\underline{\mathcal{G}}}_{Q}$ such
that the restriction $\Phi_{u}$ of $\Phi$ to each fiber is a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy between the corresponding marked
by ${\mathcal{E}}^{\diamond}$ foliations, i.e.
$g_{u}^{-1}\circ\Phi_{u}^{\sharp}\circ f_{u}$ is isotopic to the identity map
of ${\mathcal{E}}$ relatively to $\Sigma$.

For a marked by ${\mathcal{E}}^{\diamond}$ foliation
${\mathcal{F}}^{\diamond}=({\mathcal{F}},f)$ and $D$ an invariant component of
${\mathcal{E}}$, we will denote by
$[{\mathcal{H}}_{D}^{{\mathcal{F}}^{\diamond}}]$ the class, up to composition
by inner automorphisms of ${{\mathrm{Diff}}}({\mathbb{C}},0)$, of the group
morphism

${\mathcal{H}}_{D}^{{\mathcal{F}}^{\diamond}}:\pi_{1}(D^{\ast},o_{D})\to{{\mathrm{Diff}}}({\mathbb{C}},0)\,,\quad
o_{D}\in D^{\ast}:=D\setminus{{\mathrm{Sing}}}({\mathcal{F}}^{\sharp})\,,$

where ${\mathcal{H}}_{D}^{{\mathcal{F}}^{\diamond}}(\dot{\gamma})$ is the
holonomy of the foliation ${\mathcal{F}}^{\sharp}$ along the loop
$f\circ\gamma$ in $f(D)$. We also call Camacho-Sad index of
${\mathcal{F}}^{\diamond}$ at a point $m\in D$ and we write
${{\mathrm{CS}}}({\mathcal{F}}^{\diamond},D,m)$ the Camacho-Sad index of
${\mathcal{F}}^{\sharp}$ along $f(D)$ at the point $f(m)$.

######  Definition 2.6.

We denote by

  * •

${{\mathrm{SL}}}({\mathcal{F}}^{\diamond})$ the collection of all marked by
${\mathcal{E}}^{\diamond}$ foliations $\mathcal{G}^{\diamond}=(\mathcal{G},g)$
having the same marked SL-type as ${\mathcal{F}}^{\diamond}$, i.e. satisfying

$[{\mathcal{H}}_{D}^{{\mathcal{F}}^{\diamond}}]=[{\mathcal{H}}_{D}^{\mathcal{G}^{\diamond}}]\quad\text{and}\quad{{\mathrm{CS}}}({\mathcal{F}}^{\diamond},D,m)={{\mathrm{CS}}}(\mathcal{G}^{\diamond},D,m)$

for any invariant component $D$ of ${\mathcal{E}}$ and any point $m\in D$,

  * •

${{\mathrm{SL}}}_{Q}({\mathcal{F}}^{\diamond})$ the collection of all marked
by ${\mathcal{E}}^{\diamond}$ equisingular global families
${\underline{{\mathcal{F}}}}^{\diamond}_{Q}$ over $Q$ such that any fiber
${\underline{{\mathcal{F}}}}_{Q}^{\diamond}(u)$, $u\in Q$, belongs to
${{\mathrm{SL}}}({\mathcal{F}}^{\diamond})$.

######  Remark 2.7.

When $Q$ is connected, a marked equisingular global family
${\underline{{\mathcal{F}}}}_{Q}^{\diamond}$ belongs to
${{\mathrm{SL_{Q}({\mathcal{F}}^{\diamond})}}}$ as soon as one of its fibers
${\mathcal{F}}_{Q}^{\diamond}(u_{0})$ belongs to
${{\mathrm{SL}}}({\mathcal{F}}^{\diamond})$. Indeed the Camacho-Sad indices of
${\mathcal{F}}_{Q}^{\diamond}(u)$ depend continuously on $u$ and they are
determined up to $2\pi i{\mathbb{Z}}$ by the holonomy maps around the singular
points. The constancy of
$u\mapsto\left[{\mathcal{H}}_{D}^{{\mathcal{F}}^{\diamond}_{Q}(u)}\right]$
follows from the equisingularity of ${\underline{{\mathcal{F}}}}_{Q}$ and the
coherence property of the marking. ∎

###  2.3. Local universality

Let us suppose now that ${\mathcal{F}}^{\diamond}=({\mathcal{F}},f)$ is a
marked by ${\mathcal{E}}^{\diamond}$ foliation with ${\mathcal{F}}$ a finite
type generalized curve, on an ambient space $(M_{0},m_{0})$. Theorem D in [8,
§2.6] gives a description of the collection

${{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}]):=\\{[\mathcal{G}^{\diamond}]\;;\;\mathcal{G}^{\diamond}\in{{\mathrm{SL({\mathcal{F}}^{\diamond})}}}\\}$

of all ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy classes
$[\mathcal{G}^{\diamond}]$ of marked by ${\mathcal{E}}^{\diamond}$ foliations
$\mathcal{G}^{\diamond}\in{{\mathrm{SL({\mathcal{F}}^{\diamond})}}}$: it is a
set endowed with an abelian group structure given by an exact sequence

$0\to{\mathbb{Z}}^{p}\stackrel{{\scriptstyle\alpha}}{{\to}}{\mathbb{C}}^{\tau}\stackrel{{\scriptstyle\Lambda}}{{\to}}{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\stackrel{{\scriptstyle\beta}}{{\to}}\mathbf{D}\to
1\,,$ (2)

where

  1. (i)

$\tau=\dim_{{\mathbb{C}}}H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})$
is the dimension of a cohomological space (that we recall in (4)) whose
finiteness characterizes the finite type of ${\mathcal{F}}$, cf. [9, §5.5].

  2. (ii)

$\mathbf{D}$ is a quotient of a finite product of totally disconnected
subgroups of ${\mathbb{U}}(1):=\\{|z|=1\\}\subset{\mathbb{C}}$, that according
to [12] can be uncountable when ${\mathcal{F}}^{\sharp}$ possesses a
singularity which is non-linearizable and non-resonant, cf. [8, Example 4,
§8].

######  Definition 2.8.

We call moduli map of a marked global family
${\underline{{\mathcal{F}}}}^{\diamond}_{Q}\in{{\mathrm{SL}}}_{Q}({\mathcal{F}}^{\diamond})$
the map

${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}^{\diamond}_{Q}}:Q\to{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\,,\quad
u\mapsto[{{\mathcal{F}}}^{\diamond}_{Q}(u)]\,.$

We proved in [8] that for any map
$\zeta:\mathbf{D}\to{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])$ such that
$\beta\circ\zeta={{\mathrm{id}}}_{\mathbf{D}}$, there exists a marked
equisingular global family

${\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}=({\underline{{\mathcal{F}}}}_{\mathbf{U}},(f_{z,d})_{z,d})\in{{\mathrm{SL}}}_{\mathbf{U}}({\mathcal{F}}^{\diamond})\,,\quad{\underline{{\mathcal{F}}}}_{\mathbf{U}}=\left(M_{0}\times{\mathbf{U}},\pi,\theta,{\mathcal{F}}_{\mathbf{U}}\right)\,,$
$\mathbf{U}:={\mathbb{C}}^{\tau}\times\mathbf{D}\,,\quad\pi:M_{0}\times{\mathbf{U}}\to{\mathbf{U}}\,,\quad\pi(m,z,d):=(z,d)\,,\quad\theta(z,d):=(m_{0},z,d)\,,$

where $\mathbf{D}$ is endowed with the discrete topology, such that if we
denote by the dot $\cdot$ the group operation in
${{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])$, we have:

${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}}(z,d)=\Lambda(z)\cdot\zeta(d)\,.$
(3)

The goal of this section is to prove that this global family satisfies a local
universal property:

######  Theorem 2.9.

Let ${\mathcal{F}}^{\diamond}=({\mathcal{F}},f)$ be a marked by
${\mathcal{E}}^{\diamond}$ foliation of finite type which is a generalized
curve. Let $\zeta$ be a section of the map
$\beta:{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\to\mathbf{D}$ in the exact
sequence (2). Then there exists a marked global family
${\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}=({\underline{{\mathcal{F}}}}_{\mathbf{U}},(f_{z,d})_{z,d})\in{{\mathrm{SL}}}_{\mathbf{U}}({\mathcal{F}}^{\diamond})$,
$\mathbf{U}:={\mathbb{C}}^{\tau}\times\mathbf{D}$, such that

  1. (1)

the moduli map
${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}$ is
surjective and relation (3) is satisfied,

  2. (2)

for any point ${\tilde{u}}\in{\mathbf{U}}$, the deformation
$({\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}},\iota_{{\tilde{u}}})$
of the foliation ${\underline{{\mathcal{F}}}}_{\mathbf{U}}({\tilde{u}})$ over
the germ of manifold $({\mathbf{U}},{\tilde{u}})$, given by the germ of
${\underline{{\mathcal{F}}}}_{\mathbf{U}}$ at ${\tilde{u}}$ and the embedding
$\iota_{\tilde{u}}(m):=(m,{\tilde{u}})$, is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal.

To prove this result we will see that the marked global family
${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ constructed in [8, §10,
Step (vii)], which fulfills property (1), also satisfies the assertion (2).
For this we will use the following cohomological criterion of universality
proven in [9].

######  Theorem 2.10 ([9, Theorem 6.7]).

Let $({\underline{{\mathcal{F}}}}_{Q,u_{0}},\iota)$ be an equisingular
deformation of a finite type generalized curve ${\mathcal{F}}$. The following
properties are equivalent:

  1. (1)

$({\underline{{\mathcal{F}}}}_{Q,u_{0}},\iota)$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal,

  2. (2)

the Kodaira-Spencer map $\frac{\partial{\mathcal{F}}_{Q}}{\partial
u}\Big{|}_{u=u_{0}}:T_{u_{0}}Q\to
H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})$ is an
isomorphism.

Let us explain first what is the cohomological space
$H^{1}({\mathsf{A}}_{\mathcal{F}},\mathcal{T}_{\mathcal{F}})$ and the Kodaira-
Spencer map of an equisingular deformation.

The dual graph ${\mathsf{A}}_{{\mathcal{E}}}$ of the divisor ${\mathcal{E}}$
is the tree with vertex set ${\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{E}}}}$
formed by the irreducible components of ${\mathcal{E}}$ and edge set
${\mathsf{Ed}}_{{\mathsf{A}}_{{\mathcal{E}}}}$ consisting in unordered pairs
$\langle D,D^{\prime}\rangle$ of distinct irreducible components of
${\mathcal{E}}$ with $D\cap D^{\prime}\neq\emptyset$. We also consider the set
of oriented edges ${\mathcal{I}}_{{\mathsf{A}}_{{\mathcal{E}}}}$ consisting in
pairs
$(D,\mathsf{e})\in{\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{E}}}}\times{\mathsf{Ed}}_{{\mathsf{A}}_{{\mathcal{E}}}}$
with $D\in\mathsf{e}$.

Let us denote by
${\underline{{\mathcal{X}}}}_{{\mathcal{F}}}\subset{\underline{{\mathcal{B}}}}_{{\mathcal{F}}}$
the sheaves of tangent and basic holomorphic vector fields of
${\mathcal{F}}^{\sharp}$ on $M_{{\mathcal{F}}}$. We will use the cohomology
$H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})$ of the
group-graph (cf. [8, §3]) $\mathcal{T}_{\mathcal{F}}$ of infinitesimal
transverse symmetries over the dual graph ${\mathsf{A}}_{{\mathcal{F}}}$ of
the exceptional divisor ${\mathcal{E}}_{\mathcal{F}}$, introduced in [9,
Definition 5.8]. The group-graph $\mathcal{T}_{\mathcal{F}}$ is given by

$\mathcal{T}_{{\mathcal{F}}}(D)=\lim\limits_{\stackrel{{\scriptstyle\longleftarrow}}{{\tiny
D\subset
U}}}H^{0}(U,{\underline{{\mathcal{B}}}}_{{\mathcal{F}}}/{\underline{{\mathcal{X}}}}_{{\mathcal{F}}})$

if $D\subset{\mathcal{E}}_{{\mathcal{F}}}$ is not a dicritical component of
${\mathcal{F}}$ and zero otherwise,

$\mathcal{T}_{{\mathcal{F}}}(\langle
D,D^{\prime}\rangle)=\lim\limits_{\stackrel{{\scriptstyle\longleftarrow}}{{\tiny
D\cap D^{\prime}\subset
U}}}H^{0}(U,{\underline{{\mathcal{B}}}}_{{\mathcal{F}}}/{\underline{{\mathcal{X}}}}_{{\mathcal{F}}})$

if $D\cap D^{\prime}$ is not reduced to a nodal singularity of
${\mathcal{F}}^{\sharp}$ and zero otherwise, where $U$ runs over all the open
sets of $M_{{\mathcal{F}}}$ containing $D$ or $D\cap D^{\prime}$. By
definition, $H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})$
is the $1$-cohomology vector space of the following complex

$\bigoplus_{D\in{\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{F}}}}}\mathcal{T}_{{\mathcal{F}}}(D)\stackrel{{\scriptstyle\partial^{0}}}{{\longrightarrow}}\bigoplus_{(D,\langle
D,D^{\prime}\rangle)\in{\mathcal{I}}_{{\mathsf{A}}_{{\mathcal{F}}}}}\mathcal{T}_{{\mathcal{F}}}(\langle
D,D^{\prime}\rangle)\stackrel{{\scriptstyle\partial^{1}}}{{\longrightarrow}}\bigoplus_{\langle
D,D^{\prime}\rangle\in{\mathsf{Ed}}_{{\mathsf{A}}_{{\mathcal{F}}}}}\mathcal{T}_{{\mathcal{F}}}(\langle
D,D^{\prime}\rangle)),$ (4)

where $\partial^{0}((X_{D})_{D})=(X_{D^{\prime}}-X_{D})_{(D,\langle
D,D^{\prime}\rangle)}$,

$\partial^{1}((X_{D,\langle D,D^{\prime}\rangle})_{(D,\langle
D,D^{\prime}\rangle)})=(X_{D,\langle
D,D^{\prime}\rangle}+X_{D^{\prime},\langle D,D^{\prime}\rangle})_{\langle
D,D^{\prime}\rangle}.$

If ${\mathcal{F}}$ is of finite type then
$H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})$ is of finite
dimension by [9, Theorem 5.15] and we can give an explicit basis by applying
Remark 5.10, Proposition 5.12 and Theorem 2.15 as in the proof of the last
assertion of Theorem 6.4 of [9]. For this we fix a marking
$f:{\mathcal{E}}\to{\mathcal{E}}_{\mathcal{F}}$ and we consider the set

$S_{{\mathcal{F}}^{\diamond}}=\\{\star\in{\mathsf{Ve}}_{{\mathsf{A}}_{\mathcal{E}}}\cup{\mathsf{Ed}}_{{\mathsf{A}}_{{\mathcal{E}}}}:\dim_{\mathbb{C}}\mathcal{T}_{{\mathcal{F}}}(f(\star))=1\\}$

and we choose a subset

${\mathcal{A}}^{\prime\prime}\subset{\mathcal{A}}^{\prime}=\\{\langle
D,D^{\prime}\rangle\in S_{{\mathcal{F}}^{\diamond}}\,|\,D\text{ or
}D^{\prime}\notin S_{{\mathcal{F}}^{\diamond}}\\}$

obtained by removing from ${\mathcal{A}}^{\prime}$ an element in each
connected component of $S_{{\mathcal{F}}^{\diamond}}$ not reduced to a single
edge, cf [9, §2.6]. Finally we choose for each
$\mathsf{e}\in{\mathcal{A}}^{\prime\prime}$ one vertex $D\in\mathsf{e}$. This
gives us an orientation for each edge
$\mathsf{e}\in{\mathcal{A}}^{\prime\prime}$. Let us denote by

${\mathcal{A}}=\\{(D,\mathsf{e}):\mathsf{e}\in{\mathcal{A}}^{\prime\prime}\\}\subset{\mathcal{I}}_{{\mathsf{A}}_{{\mathcal{E}}}}\,.$

Such a set will be called set of active oriented edges for
${\mathcal{F}}^{\diamond}$.

######  Theorem 2.11 ([9]).

If ${\mathcal{F}}^{\diamond}=({\mathcal{F}},f)$ is a marked by
${\mathcal{E}}^{\diamond}$ foliation of finite type which is a generalized
curve and ${\mathcal{A}}\subset{\mathcal{I}}_{{\mathsf{A}}_{{\mathcal{E}}}}$
is a set of active oriented edges for ${\mathcal{F}}^{\diamond}$ then the map

$\delta_{{\mathcal{A}}}^{{\mathcal{F}}^{\diamond}}:\bigoplus_{(D,\langle
D,D^{\prime}\rangle)\in{\mathcal{A}}}\mathcal{T}_{{\mathcal{F}}}(\langle
f(D),f(D^{\prime})\rangle)\to
H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})$

defined by $\delta_{{\mathcal{A}}}^{{\mathcal{F}}^{\diamond}}((X_{D,\langle
D,D^{\prime}\rangle})_{{\mathcal{A}}})=[(Y_{D,\langle
D,D^{\prime}\rangle})_{{\mathcal{I}}_{{\mathsf{A}}_{{\mathcal{F}}}}}]$, where

$Y_{D,\langle
D,D^{\prime}\rangle}=\left\\{\begin{array}[]{cl}X_{f^{-1}(D,\langle
D,D^{\prime}\rangle)}&\text{if }f^{-1}(D,\langle
D,D^{\prime}\rangle)\in{\mathcal{A}},\\\ -X_{f^{-1}(D^{\prime},\langle
D,D^{\prime}\rangle)}&\text{if }f^{-1}(D^{\prime},\langle
D,D^{\prime}\rangle)\in{\mathcal{A}},\\\
0&\text{otherwise,}\end{array}\right.$

is an isomorphism, $f^{-1}((D,\langle D,D^{\prime}\rangle))$ denoting
$(f^{-1}(D),\langle f^{-1}(D),f^{-1}(D^{\prime})\rangle)$.

######  Remark 2.12.

Let
${\underline{{\mathcal{F}}}}_{Q}^{\diamond}=(M,\pi,\theta,{\mathcal{F}}_{Q},(f_{u})_{u\in
Q})$ be a global equisingular family of foliations over a connected manifold
$Q$ endowed with a marking
$(f_{u}:{\mathcal{E}}\to{\mathcal{E}}_{{\mathcal{F}}_{Q}(u)})_{u\in Q}$ by
${\mathcal{E}}^{\diamond}$. Assume that some fiber ${\mathcal{F}}_{Q}(u_{0})$
of ${\mathcal{F}}_{Q}$ is of finite type then all the fibers
${\mathcal{F}}_{Q}(u)$ are of finite type because the definition (cf. [8,
Definition 6.7]) only depends on their holonomies. For the same reason the set
$S_{{\mathcal{F}}_{Q}^{\diamond}(u)}$ does not depend on $u\in Q$.
Consequently the map
$\delta_{{\mathcal{A}}}^{{\mathcal{F}}_{Q}^{\diamond}(u)}$ is an isomorphism
for any $u\in Q$. ∎

The simplest example of equisingular family over $Q$ is the constant family
${\mathcal{F}}_{Q}^{{{\mathrm{ct}}}}$ that is obtained in the ambient space
$M_{0}\times Q$ by lifting a given foliation ${\mathcal{F}}$ on $M_{0}$ to
each submanifold $M_{0}\times\\{u\\}$ through the projection map $M_{0}\times
Q\to M_{0}$. We now make more precise the fact that every equisingular family
after reduction is equivalent to a constant family along each irreducible
component of the exceptional divisor.

######  Theorem 2.13 ([9, Theorem 3.8]).

Let ${\underline{{\mathcal{F}}}}_{Q}$ be an equisingular family of foliations
over $Q$ and $u_{0}\in Q$. If the fiber
${\mathcal{F}}_{u_{0}}={\mathcal{F}}_{Q}(u_{0})$ is a generalized curve, then
we can associate to each
$D\in{\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{F}}_{u_{0}}}}$, a homeomorphism
germ

$\Psi_{D}:(M_{{\underline{{\mathcal{F}}}}_{Q,u_{0}}},D){\overset{\sim}{\longrightarrow}}(M_{{\mathcal{F}}_{u_{0}}}\times
Q,D\times\\{u_{0}\\})\,,$

so that:

  1. (i)

$\Psi_{D}$ is a map over $Q$, i.e.
${{\mathrm{pr}}}^{\sharp}_{Q}\circ\Psi_{D}=\pi^{\sharp}$, the identity map
over $u_{0}$ and it is excellent: $\Psi_{D}$ is holomorphic at each point of
${{\mathrm{Sing}}}({\mathcal{E}}_{{\mathcal{F}}_{u_{0}}})\cup{{\mathrm{Sing}}}({\mathcal{F}}^{\sharp}_{u_{0}})$
except perhaps at the singular points of
${\mathcal{E}}_{{\mathcal{F}}_{u_{0}}}$ that are nodal singularities of
${\mathcal{F}}^{\sharp}_{u_{0}}$,

  2. (ii)

$\Psi_{D}$ conjugates the foliation ${\mathcal{F}}_{Q,u_{0}}^{\sharp}$ to the
foliation ${\mathcal{F}}_{Q,u_{0}}^{{{\mathrm{ct}}}\,\sharp}$ obtained after
equireduction of the constant deformation
${\mathcal{F}}_{Q,u_{0}}^{{{\mathrm{ct}}}}$,

  3. (iii)

the germ of $\Psi_{D}\circ\Psi_{D^{\prime}}^{-1}$ at the intersection point
$\\{s_{DD^{\prime}}\\}=(D\cap D^{\prime})\times\\{u_{0}\\}$ of two irreducible
components $D$ and $D^{\prime}$, is the identity when either $s_{DD^{\prime}}$
is a nodal singular point of ${\mathcal{F}}^{\sharp}_{u_{0}}$ or
$s_{DD^{\prime}}$ is a regular point of ${\mathcal{F}}^{\sharp}_{u_{0}}$.

The collection $(\Psi_{D})_{D}$ of these homeomorphisms indexed by the
components of ${\mathcal{E}}$ is called good trivializing system for
${\underline{{\mathcal{F}}}}_{Q,u_{0}}$.

For each $(D,\langle
D,D^{\prime}\rangle)\in{\mathcal{I}}_{{\mathsf{A}}_{{\mathcal{E}}}}$ the
composition $\phi_{D,\langle
D,D^{\prime}\rangle}:=\Psi_{D^{\prime}}\circ\Psi_{D}^{-1}$ is a local
biholomorphism at $(D\cap D^{\prime})\times\\{u_{0}\\}$ preserving the
constant family
$({\mathcal{F}}_{u_{0}}^{\sharp})^{{{\mathrm{ct}}}}_{Q,u_{0}}$. For every
$v\in T_{u_{0}}Q$ the Lie derivative
$L_{v}({{\mathrm{pr}}}_{M_{{\mathcal{F}}_{u_{0}}}}\circ\phi_{D,\langle
D,D^{\prime}\rangle})$ defines a germ of basic vector field of
${\mathcal{F}}_{u_{0}}^{\sharp}$ at $D\cap D^{\prime}$. It is shown in [9,
§6.2] that the Kodaira-Spencer map

$\frac{\partial{\mathcal{F}}_{Q}}{\partial u}\Big{|}_{u=u_{0}}:T_{u_{0}}Q\to
H^{1}({\mathsf{A}}_{{\mathcal{F}}_{u_{0}}},\mathcal{T}_{{\mathcal{F}}_{u_{0}}})$

given by
$v\mapsto[(L_{v}({{\mathrm{pr}}}_{M_{{\mathcal{F}}_{u_{0}}}}\circ\phi_{D,\langle
D,D^{\prime}\rangle}))_{D,\langle D,D^{\prime}\rangle}]$ does not depend on
the choice we have made of a good trivializing system for
${\mathcal{F}}_{Q,u_{0}}$.

###### Proof of Theorem 2.9.

In a first step we recall the construction of
${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$. In the second step we
fix ${\tilde{u}}\in\mathbf{U}$ and we will determine a good trivializing
system $(\Upsilon_{D})_{D}$ for
${\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}}$. Finally, in the last
step, we compute the Kodaira-Spencer map of
${\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}}$ and we apply Theorem
2.10 to end the proof.

-Step 1.  Let us fix $d\in\mathbf{D}$ and a marked foliation $\mathcal{G}^{\diamond}=(\mathcal{G},g)$ belonging to $\zeta(d)$. First, let us recall that there are germs of ${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphisms compatible with the markings

$\psi_{D}:(M_{\mathcal{G}},g(D)){\overset{\sim}{\longrightarrow}}(M_{\mathcal{F}},f(D))\,,\quad
D\in{\mathsf{Ed}}_{{\mathsf{A}}_{{\mathcal{E}}}}\,,$ (5)

that conjugate $\mathcal{G}^{\sharp}$ to ${\mathcal{F}}^{\sharp}$. The
biholomorphism germs

$\varphi_{D,\langle
D,D^{\prime}\rangle}:=\psi_{D^{\prime}}\circ\psi_{D}^{-1}:(M_{\mathcal{F}},s_{\langle
D,D^{\prime}\rangle}){\overset{\sim}{\longrightarrow}}(M_{\mathcal{F}},s_{\langle
D,D^{\prime}\rangle})\,,$ (6)

with

$\\{s_{\langle D,D^{\prime}\rangle}\\}:=f(D\cap D^{\prime})\,,\quad\langle
D,D^{\prime}\rangle\in{\mathsf{Ed}}_{{\mathsf{A}}_{{\mathcal{E}}}}\,,$

leave ${\mathcal{F}}^{\sharp}$ invariant. Thanks to [8, §11.5] we may require
that $\varphi_{D,\langle D,D^{\prime}\rangle}$ is the identity map when
$s_{\langle D,D^{\prime}\rangle}$ is a nodal singular point of
${\mathcal{F}}^{\sharp}$ or a singular point of the divisor belonging to a
dicritical component.

We fix a set of active oriented edges
${\mathcal{A}}\subset{\mathcal{I}}_{{\mathsf{A}}_{{\mathcal{E}}}}$ for
${\mathcal{F}}^{\diamond}$, we choose for each
$(D,\mathsf{e})\in{\mathcal{A}}$ a germ at $f(s_{\mathsf{e}})$ of basic and
not tangent holomorphic vector field $X_{\mathsf{e}}$ to
${\mathcal{F}}^{\sharp}$ on $M_{\mathcal{F}}$ and we fix a bijection
$\kappa:{\mathcal{A}}\to\\{1,\ldots,\tau\\}$. Now, $d\in\mathbf{D}$ being
fixed, in all the sequel we will simply denote by
${\underline{{\mathcal{F}}}}^{\diamond}_{{\mathbb{C}}^{\tau}}$ the marked
global family ${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ restricted
to ${\mathbb{C}}^{\tau}\times\\{d\\}$.

The ambient space of ${\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau}}$ is
constructed by gluing neighborhoods $U_{D}\times{\mathbb{C}}^{\tau}$ of
$f(D)\times{\mathbb{C}}^{\tau}$ in $M_{\mathcal{F}}\times{\mathbb{C}}^{\tau}$,
using an appropriate family of biholomorphisms

${\mathfrak{u}}:=(\Phi_{D,\mathsf{e}})_{D\in\mathsf{e}\in{\mathsf{Ed}}_{{\mathsf{A}}}}\,,$
$\Phi_{D,\mathsf{e}}:\left(U_{D}\times{\mathbb{C}}^{\tau},\\{s_{\mathsf{e}}\\}\times{\mathbb{C}}^{\tau}\right){\overset{\sim}{\longrightarrow}}\left(U_{D^{\prime}}\times{\mathbb{C}}^{\tau},\\{s_{\mathsf{e}}\\}\times{\mathbb{C}}^{\tau}\right)\,,\quad\text{with
}\mathsf{e}=\langle D,D^{\prime}\rangle\,.$

Writing $(m,t)\mapsto\exp(Z)[t](m)$ the flow at the time $t$ of a vector field
$Z$, we set:

  * •

$\Phi_{D,\mathsf{e}}(m,z)=(\varphi_{D,\mathsf{e}}(m),z)$,
$z=(z_{1},\ldots,z_{\tau})$, if
$(D,\mathsf{e}),(D^{\prime},\mathsf{e})\notin{\mathcal{A}}$,

  * •

$\Phi_{D,\mathsf{e}}(m,z)=(\varphi_{D,\mathsf{e}}\,\circ\,{{\mathrm{exp}}}(X_{\mathsf{e}})[z_{\kappa(\mathsf{e})}](m),z)$,
if $(D,\mathsf{e})\in{\mathcal{A}}$,

  * •

$\Phi_{D,\mathsf{e}}=\Phi^{-1}_{D^{\prime},\mathsf{e}}$ if
$(D^{\prime},\mathsf{e})\in{\mathcal{A}}$,

where $\varphi_{D,\mathsf{e}}$ are the biholomorphism germs in (6). We
consider the following germ of manifold

$\left(M_{{\mathfrak{u}}},{\mathcal{E}}_{{\mathfrak{u}}}\right):=\Big{(}\bigcup_{D\in{\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{E}}}}}U_{D}\times{\mathbb{C}}^{\tau}\times\\{D\\}\Big{/}\sim_{{\mathfrak{u}}}\;,\;\bigcup_{D\in{\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{E}}}}}f(D)\times{\mathbb{C}}^{\tau}\times\\{D\\}\Big{/}\sim_{{\mathfrak{u}}}\Big{)}\,,$

the equivalence relation $\sim_{{\mathfrak{u}}}$ being defined by:

$U_{D}\times{\mathbb{C}}^{\tau}\times\\{D\\}\;\ni\;(m,z,D)\sim_{{\mathfrak{u}}}\;(\Phi_{D,\langle
D,D^{\prime}\rangle}(m,z),D^{\prime})\,\in\;U_{D^{\prime}}\times{\mathbb{C}}^{\tau}\times\\{D^{\prime}\\}$

when $(m,z)$ belongs to the domain of $\Phi_{D,\langle D,D^{\prime}\rangle}$
and with the single relation $(m,z,D)\sim_{{\mathfrak{u}}}(m,z,D)$ otherwise.
As the biholomorphisms $\Phi_{D,\mathsf{e}}$ leave invariant the projections
$U_{D}\times{\mathbb{C}}^{\tau}\to{\mathbb{C}}^{\tau}$ and the constant family
${\mathcal{F}}^{\sharp\,{{\mathrm{ct}}}}_{{\mathbb{C}}^{\tau}}$, the gluing
process provides a holomorphic submersion
$\pi_{{\mathfrak{u}}}:M_{{\mathfrak{u}}}\to{\mathbb{C}}^{\tau}$ and a
foliation tangent to the fibers of $\pi_{{\mathfrak{u}}}$, which we denote by
${\mathcal{F}}_{{\mathfrak{u}}}$.

The ambient space of ${\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau}}$ is
the manifold over ${\mathbb{C}}^{\tau}$, obtained by contracting
${\mathcal{E}}_{{\mathfrak{u}}}$ to a $\tau$-dimensional manifold
$S_{{\mathfrak{u}}}$:

$E_{{\mathfrak{u}}}:(M_{{\mathfrak{u}}},{\mathcal{E}}_{{\mathfrak{u}}})\longrightarrow(M^{\flat}_{{\mathfrak{u}}},S_{{\mathfrak{u}}})\,,\quad\pi_{{\mathfrak{u}}}^{\flat}:(M^{\flat}_{{\mathfrak{u}}},S_{{\mathfrak{u}}})\longrightarrow{\mathbb{C}}^{\tau}\,,\quad\pi_{{\mathfrak{u}}}^{\flat}\circ
E_{{\mathfrak{u}}}:=\pi_{{\mathfrak{u}}}\,.$

Restricted to $S_{{\mathfrak{u}}}$ the submersion
$\pi_{{\mathfrak{u}}}^{\flat}$ is a biholomorphism, we will denote by
$\theta_{{\mathfrak{u}}}$ its inverse. Finally the foliation
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}$ constructed in [8, §10, Step (vii)] is
the direct image $E_{{\mathfrak{u}}}({\mathcal{F}}_{{\mathfrak{u}}})$ and we
have an equisingular global family over ${\mathbb{C}}^{\tau}$

${\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau}}:=\left(M^{\flat}_{{\mathfrak{u}}},\pi_{{\mathfrak{u}}}^{\flat},\theta_{{\mathfrak{u}}},{\mathcal{F}}_{{\mathbb{C}}^{\tau}}\right)\,.$

By a classical property of blow-ups, there is a germ $F$ of biholomorphism
that conjugates the reduced foliation
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}$ to
${\mathcal{F}}_{{\mathfrak{u}}}$:

$F:(M_{{\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau}}},{\mathcal{E}}_{{\mathcal{F}}_{{\mathbb{C}}^{\tau}}}){\overset{\sim}{\longrightarrow}}(M_{{\mathfrak{u}}},{\mathcal{E}}_{{\mathfrak{u}}})\,,\quad
F({\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp})={\mathcal{F}}_{{\mathfrak{u}}}\,,$
(7)

In order to endow this family with a marking by ${\mathcal{E}}^{\diamond}$ we
highlight that we have:

$\Phi_{D,\mathsf{e}}(m,0)=(\varphi_{D,\mathsf{e}}(m),0)\,,\quad
D\in\mathsf{e}\in{\mathsf{Ed}}_{{\mathsf{A}}_{{\mathcal{E}}}}\,.$

Therefore, according to relations (6) the maps $\psi_{D}$ introduced in (5)
glue as a germ of ${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism
$\Psi_{\mathcal{G}}$ which conjugates $\mathcal{G}^{\sharp}$ to the foliation
${\mathcal{F}}^{\sharp}_{{\mathbb{C}}^{\tau}}(0)$ obtained by restricting
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}$ to the fiber
$M_{{\mathfrak{u}}}(0):={\pi_{{\mathfrak{u}}}^{-1}(0)}$:

$\Psi_{\mathcal{G}}:(M_{\mathcal{G}},{\mathcal{E}}_{\mathcal{G}})\;{\overset{\sim}{\longrightarrow}}\;(M_{{\mathfrak{u}}}(0),{\mathcal{E}}_{{\mathfrak{u}}}(0))\,,\quad\Psi_{\mathcal{G}}(\mathcal{G}^{\sharp})={\mathcal{F}}^{\sharp}_{{\mathbb{C}}^{\tau}}(0)\,,$

with
${\mathcal{E}}_{{\mathfrak{u}}}(0):={\mathcal{E}}_{{\mathfrak{u}}}\cap\pi_{{\mathfrak{u}}}^{-1}(0)$.
The homeomorphism $\Psi_{\mathcal{G}}\circ
g:{\mathcal{E}}\to{\mathcal{E}}_{{\mathfrak{u}}}(0)$ defines a marking of
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}(0)$, that extends to a marking
$(f_{z})_{z}$ of the global family
${\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau}}$ thanks to property (b) of
Remark 2.5. Since $S_{{\mathfrak{u}}}$ is Stein, up to a biholomorphism over
${\mathbb{C}}^{\tau}$, we can assume by classical arguments that
$M_{{\mathfrak{u}}}^{\flat}$ is a neighborhood of
$\\{m_{0}\\}\times{\mathbb{C}}^{\tau}\subset M_{0}\times{\mathbb{C}}^{\tau}$
and $\pi_{{\mathfrak{u}}}^{\flat}$ is the projection map onto
${\mathbb{C}}^{\tau}$.

-Step 2.  Now we also fix $\tilde{z}\in{\mathbb{C}}^{\tau}$ and we identify ${\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}}$, ${\tilde{u}}:=({\tilde{z}},d)$, to the deformation $({\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau},{\tilde{z}}},\iota_{\tilde{z}})$ over the manifold germ $({\mathbb{C}}^{\tau},{\tilde{z}})$ of the fiber ${\mathcal{F}}_{{\mathbb{C}}^{\tau}}(\tilde{z})={\mathcal{F}}_{\mathbf{U}}({\tilde{u}})$, defined by the germ ${\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau},{\tilde{z}}}$ at ${\tilde{z}}$ of the family ${\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau}}$ and the inclusion map $\iota_{\tilde{z}}:\pi_{{\mathfrak{u}}}^{\flat\,-1}({\tilde{z}})\hookrightarrow M_{{\mathfrak{u}}}^{\flat}$.

If we restrict the map germ (7) to the fiber over ${\tilde{z}}$, we obtain a
biholomorphism germ $F_{\tilde{z}}$ between the ambient space
$M_{{\mathcal{F}}_{{\mathbb{C}}^{\tau}}(\tilde{z})}$ of
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}(\tilde{z})$ and the manifold
germ
$(\pi_{{\mathfrak{u}}}^{-1}({\tilde{z}}),{\mathcal{E}}_{{\mathfrak{u}}}\cap\pi_{{\mathfrak{u}}}^{-1}({\tilde{z}}))$.
This manifold is also the manifold germ

$\big{(}\;M_{{\mathfrak{u}}(\tilde{z})},{\mathcal{E}}_{{\mathfrak{u}}(\tilde{z})}\,\big{)}:=\Big{(}\bigcup_{D\in{\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{E}}}}}U_{D}\times\\{D\\}\big{/}\sim_{{\mathfrak{u}}(\tilde{z})},\;\bigcup_{D\in{\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{E}}}}}f(D)\times\\{D\\}\big{/}\sim_{{\mathfrak{u}}(\tilde{z})}\;\Big{)}\,,$

defined by the gluing process given by the equivalence relation
$\sim_{{\mathfrak{u}}(\tilde{z})}$ defined by the family:

${\mathfrak{u}}(\tilde{z}):=(\Phi^{\tilde{z}}_{D,\mathsf{e}})_{D\in\mathsf{e}\in{\mathsf{Ed}}_{{\mathsf{A}}_{{\mathcal{E}}}}}\,,$
$\Phi^{\tilde{z}}_{D,\mathsf{e}}:(U_{D},s_{\mathsf{e}}){\overset{\sim}{\longrightarrow}}(U_{D^{\prime}},s_{\mathsf{e}})\,,\quad
m\mapsto\Phi_{D,\mathsf{e}}(m,\tilde{z})\,,\quad\mathsf{e}=\langle
D,D^{\prime}\rangle\,.$

Clearly $F_{\tilde{z}}$ conjugates
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}(\tilde{z})$ to the foliation
${\mathcal{F}}_{{\mathfrak{u}}({\tilde{z}})}$ obtained by gluing
${\mathcal{F}}^{\sharp}$ restricted to each $U_{D}$, i.e.

$F_{\tilde{z}}:\big{(}M_{{\mathcal{F}}_{{\mathbb{C}}^{\tau}}(\tilde{z})},{\mathcal{E}}_{{\mathcal{F}}_{{\mathbb{C}}^{\tau}}(\tilde{z})}\big{)}{\overset{\sim}{\longrightarrow}}\big{(}\;M_{{\mathfrak{u}}(\tilde{z})},{\mathcal{E}}_{{\mathfrak{u}}(\tilde{z})}\,\big{)}\,,\quad
F_{\tilde{z}}({\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}(\tilde{z}))={\mathcal{F}}_{{\mathfrak{u}}({\tilde{z}})}\,.$

Let us denote by $\\{U_{D}\times{\mathbb{C}}^{\tau}\\}\subset
M_{{\mathfrak{u}}}$ the image of the canonical embedding
$U_{D}\times{\mathbb{C}}^{\tau}\hookrightarrow M_{{\mathfrak{u}}}$, and by

$g_{D}:\\{U_{D}\times{\mathbb{C}}^{\tau}\\}\subset
M_{{\mathfrak{u}}}\longrightarrow U_{D}\times{\mathbb{C}}^{\tau}$

the inverse of this embedding. We have the following relations of “change of
charts”

$g_{D^{\prime}}=\Phi_{D,\langle D,D^{\prime}\rangle}\circ g_{D}\,.$ (8)

Similarly, $\\{U_{D}\\}$ denoting the image of the canonical embedding
$U_{D}\hookrightarrow M_{{\mathfrak{u}}(\tilde{z})}$ and

$g_{D}^{\tilde{z}}:\\{U_{D}\\}\subset
M_{{\mathfrak{u}}(\tilde{z})}\longrightarrow U_{D}$

denoting its inverse, we also have:

$g_{D^{\prime}}^{\tilde{z}}=\Phi^{\tilde{z}}_{D,\langle
D,D^{\prime}\rangle}\circ g^{\tilde{z}}_{D}\,.$ (9)

Notice that $g_{D}$ conjugates the foliation ${\mathcal{F}}_{{\mathfrak{u}}}$
restricted to $\\{U_{D}\times{\mathbb{C}}^{\tau}\\}$ to the constant
deformation $({\mathcal{F}}^{\sharp})_{{\mathbb{C}}^{\tau}}^{{{\mathrm{ct}}}}$
of ${\mathcal{F}}^{\sharp}$, restricted to $U_{D}\times{\mathbb{C}}^{\tau}$.
Similary $g_{D}^{\tilde{z}}$ conjugates
${\mathcal{F}}_{{\mathfrak{u}}({\tilde{z}})}$ restricted to $\\{U_{D}\\}$ to
${\mathcal{F}}^{\sharp}$ restricted to $U_{D}$. Hence
$g_{D}^{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}}$ conjugates the
constant deformation of ${\mathcal{F}}_{{\mathfrak{u}}({\tilde{z}})}$ on
$\\{U_{D}\\}\times{\mathbb{C}}^{\tau}$ over ${\mathbb{C}}^{\tau}$, denoted by
$({\mathcal{F}}_{{\mathfrak{u}}({\tilde{z}})})_{{\mathbb{C}}^{\tau}}^{{{\mathrm{ct}}}}$,
to the constant deformation
$({\mathcal{F}}^{\sharp})_{{\mathbb{C}}^{\tau}}^{{{\mathrm{ct}}}}$ of
${\mathcal{F}}^{\sharp}$ restricted to $U_{D}\times{\mathbb{C}}^{\tau}$. If we
write
$[U_{D}\times{\mathbb{C}}^{\tau}]:=F^{-1}(\\{U_{D}\times{\mathbb{C}}^{\tau}\\})$
and $[U_{D}]:=F_{\tilde{z}}^{-1}(\\{U_{D}\\})$, then
$F_{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}}$ conjugates the
constant deformation of
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}({\tilde{z}})$ over
${\mathbb{C}}^{\tau}$, denoted by
$({\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}({\tilde{z}}))^{{{\mathrm{ct}}}}_{{\mathbb{C}}^{\tau}}$,
to
$({\mathcal{F}}_{{\mathfrak{u}}({\tilde{z}})})_{{\mathbb{C}}^{\tau}}^{{{\mathrm{ct}}}}$:

$[U_{D}\times{\mathbb{C}}^{\tau}]\stackrel{{\scriptstyle
F}}{{{\overset{\sim}{\longrightarrow}}}}\\{U_{D}\times{\mathbb{C}}^{\tau}\\}\stackrel{{\scriptstyle
g_{D}}}{{{\overset{\sim}{\longrightarrow}}}}U_{D}\times{\mathbb{C}}^{\tau}\stackrel{{\scriptstyle(g_{D}^{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}})^{-1}}}{{\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}}}\\{U_{D}\\}\times{\mathbb{C}}^{\tau}\stackrel{{\scriptstyle(F_{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}})^{-1}}}{{\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}}}[U_{D}]\times{\mathbb{C}}^{\tau}\,.$
$\phantom{aa}{\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}\phantom{aa}\mapsto\phantom{aaaa}{\mathcal{F}}_{{\mathfrak{u}}}\phantom{aaaa}\mapsto\phantom{a}({\mathcal{F}}^{\sharp})_{{\mathbb{C}}^{\tau}}^{{{\mathrm{ct}}}}\phantom{aaaa}\mapsto\phantom{aaaa}({\mathcal{F}}_{{\mathfrak{u}}({\tilde{z}})})_{{\mathbb{C}}^{\tau}}^{{{\mathrm{ct}}}}\phantom{aaaa}\mapsto\phantom{ai}({\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}({\tilde{z}}))_{{\mathbb{C}}^{\tau}}^{{{\mathrm{ct}}}}$

The homeomorphism

$\Upsilon_{D}:=(F_{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}})^{-1}\circ(g_{D}^{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}})^{-1}\circ
g_{D}\circ
F:[U_{D}\times{\mathbb{C}}^{\tau}]{\overset{\sim}{\longrightarrow}}[U_{D}]\times{\mathbb{C}}^{\tau}\,,$

is a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-trivialization of
${\mathcal{F}}^{\sharp}_{{\mathbb{C}}^{\tau}}$ as deformation of
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}({\tilde{z}})$, i.e. a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy from
${\mathcal{F}}_{{\mathbb{C}}^{\tau}}^{\sharp}$ to the constant deformation of
${\mathcal{F}}^{\sharp}_{{\mathbb{C}}^{\tau}}({\tilde{z}})$. The collection
$(\Upsilon_{D})_{D\in{\mathsf{Ve}}_{{\mathsf{A}}_{{\mathcal{E}}}}}$ is a good
trivializing system for
${\underline{{\mathcal{F}}}}_{{\mathbb{C}}^{\tau},{\tilde{z}}}$. Consider the
germ of biholomorphism

$\Upsilon_{D,\mathsf{e}}:\big{(}([U_{D}]\cap[U_{D^{\prime}}])\times{\mathbb{C}}^{\tau},s_{\mathsf{e}}\big{)}{\overset{\sim}{\longrightarrow}}\big{(}([U_{D}]\cap[U_{D^{\prime}}])\times{\mathbb{C}}^{\tau},s_{\mathsf{e}}\big{)}\,,$
$\Upsilon_{D,\mathsf{e}}:=\Upsilon_{D^{\prime}}\circ\Upsilon_{D}^{-1}\,,\quad\mathsf{e}={\langle
D,D^{\prime}\rangle}\,.$

If we compute the expression of $\Upsilon_{D,\mathsf{e}}$ in the following
“chart”

$\chi:=(g_{D}^{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}})\circ(F_{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}}):[U_{D}]\times{\mathbb{C}}^{\tau}{\overset{\sim}{\longrightarrow}}U_{D}\times{\mathbb{C}}^{\tau}\,$

we get:

${\widetilde{\Upsilon}}_{D,\mathsf{e}}:=\chi\circ\Upsilon_{D,\mathsf{e}}\circ\chi^{-1}=(g_{D}^{\tilde{z}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}})\circ((g_{D^{\prime}}^{{\tilde{z}}})^{-1}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}})\circ
g_{D^{\prime}}\circ g_{D}^{-1}\,$

and, thanks to (8) and (9) we obtain:

${\widetilde{\Upsilon}}_{D,\mathsf{e}}=(\Phi_{D,\mathsf{e}}^{{\tilde{z}}}\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}})^{-1}\circ\Phi_{D,\mathsf{e}}\,.$

Using the explicit expressions of $\Phi_{D,\mathsf{e}}$ and
$\Phi_{D,\mathsf{e}}^{\tilde{z}}$, we finally have, writing
$z=(z_{1},\ldots,z_{\tau})\in{\mathbb{C}}^{\tau}$ and
$\tilde{z}=(\tilde{z}_{1},\ldots,\tilde{z}_{\tau})$,

  1. (i)

${\widetilde{\Upsilon}}_{D,\mathsf{e}}(m,z)=(\exp(X_{\mathsf{e}})[z_{\kappa(\mathsf{e})}-{\tilde{z}}_{\kappa(\mathsf{e})}](m),z)$,
if $(D,\mathsf{e})\in{\mathcal{A}}$,

  2. (ii)

${\widetilde{\Upsilon}}_{D,\mathsf{e}}={\widetilde{\Upsilon}}_{D^{\prime},\mathsf{e}}^{-1}$,
if $(D^{\prime},\mathsf{e})\in{\mathcal{A}}$,

  3. (iii)

${\widetilde{\Upsilon}}_{D,\mathsf{e}}(m,z)=(m,z)$, if
$(D,\mathsf{e}),(D^{\prime},\mathsf{e})\notin{\mathcal{A}}$.

-Step 3. From the previous expression of ${\widetilde{\Upsilon}}_{D,\mathsf{e}}$ we deduce the following partial derivatives:

$\left.\frac{\partial{{\mathrm{pr}}}_{U_{D}}\circ{\widetilde{\Upsilon}}_{D,\mathsf{e}}}{\partial
z_{k}}\right|_{z={\tilde{z}}}=\left\\{\begin{array}[]{rl}X_{\mathsf{e}}&\hbox{if
}\;(D,\mathsf{e})\in{\mathcal{A}}\hbox{ and }k=\kappa((D,\mathsf{e})),\\\
-X_{\mathsf{e}}&\hbox{if }\;(D^{\prime},\mathsf{e})\in{\mathcal{A}}\hbox{ and
}k=\kappa((D^{\prime},\mathsf{e})),\\\ 0&\text{otherwise.}\end{array}\right.$

Since $\Upsilon_{D,\mathsf{e}}=[(g_{D}^{{\tilde{z}}}\circ
F_{{\tilde{z}}})\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}}]^{-1}\circ{\widetilde{\Upsilon}}_{D,\mathsf{e}}\circ[(g_{D}^{{\tilde{z}}}\circ
F_{{\tilde{z}}})\times{{\mathrm{id}}}_{{\mathbb{C}}^{\tau}}]$, we deduce that

$\left.\frac{\partial{{\mathrm{pr}}}_{M_{{\mathcal{F}}_{{\tilde{z}}}}}\circ\Upsilon_{D,\mathsf{e}}}{\partial
z_{k}}\right|_{z={\tilde{z}}}=\left\\{\begin{array}[]{rl}X_{\mathsf{e}}^{{\tilde{z}}}&\hbox{if
}\;(D,\mathsf{e})\in{\mathcal{A}}\hbox{ and }k=\kappa((D,\mathsf{e})),\\\
-X_{\mathsf{e}}^{{\tilde{z}}}&\hbox{if
}\;(D^{\prime},\mathsf{e})\in{\mathcal{A}}\hbox{ and
}k=\kappa((D^{\prime},\mathsf{e})),\\\ 0&\text{otherwise,}\end{array}\right.$

where $X_{\mathsf{e}}^{{\tilde{z}}}=(g_{D}^{{\tilde{z}}}\circ
F_{{\tilde{z}}})^{*}X_{\mathsf{e}}$ is a basis of the one-dimensional vector
space
$\mathcal{T}_{{\mathcal{F}}_{{\tilde{z}}}}(f_{{\tilde{z}}}(s_{\mathsf{e}}))$,
${\mathcal{F}}_{{\tilde{z}}}^{\diamond}$ denoting the marked foliation
$({\mathcal{F}}_{{\mathbb{C}}^{\tau}}({\tilde{z}}),f_{{\tilde{z}}})$, because
$g_{D}^{{\tilde{z}}}\circ F_{{\tilde{z}}}:[U_{D}]\to U_{D}$ is a
biholomorphism conjugating ${\mathcal{F}}_{{\tilde{z}}}^{\sharp}$ and
${\mathcal{F}}^{\sharp}$. Hence the linear map
$\gamma_{\tilde{z}}:T_{{\tilde{z}}}{\mathbb{C}}^{\tau}\to\bigoplus_{(D,\mathsf{e})\in{\mathcal{A}}}\mathcal{T}_{{\mathcal{F}}_{{\tilde{z}}}}(f_{{\tilde{z}}}(s_{\mathsf{e}}))$
defined by
$\gamma_{\tilde{z}}\left(\frac{\partial}{\partial{z_{\kappa(D,\mathsf{e})}}}\right)=X_{\mathsf{e}}^{\tilde{z}}$
is an isomorphism. By construction, and according to Theorem 2.11 and Remark
2.12, the Kodaira-Spencer map
$\frac{\partial{\mathcal{F}}_{{\mathbb{C}}^{\tau}}}{\partial
z}\Big{|}_{z={\tilde{z}}}=\delta_{{\mathcal{A}}}^{{\mathcal{F}}_{{\tilde{z}}}^{\diamond}}\circ\gamma_{\tilde{z}}$
is also an isomorphism. By Theorem 2.10 we conclude that the deformation
$({\underline{{\mathcal{F}}}}_{\mathbf{U},\tilde{u}},\iota_{\tilde{u}})$ of
${\mathcal{F}}_{\tilde{u}}$ is ${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal.
This achieves the proof of Theorem 2.9. ∎

##  3\. Factorization properties of the locally universal family

###  3.1. Local factorization property

We will now prove that the global family
${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ of Theorem 2.9 is
complete in a similar meaning to that given by Kodaira-Spencer in [4] in the
context of complex manifolds:

######  Theorem 3.1.

Let ${\mathcal{F}}^{\diamond}=({\mathcal{F}},f)$ be a marked by
${\mathcal{E}}^{\diamond}$ foliation of finite type which is a generalized
curve and let ${\underline{\mathcal{G}}}_{P}^{\diamond}$ be a marked global
family in ${{\mathrm{SL}}}_{P}({\mathcal{F}}^{\diamond})$. Let us consider
$t_{0}\in P$ and ${\tilde{u}}\in{\mathbf{U}}$ such that the marked foliation
$\mathcal{G}^{\diamond}_{P}(t_{0})$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to the fiber
${\mathcal{F}}^{\diamond}_{\mathbf{U}}({\tilde{u}})$ of the marked global
family ${\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}$ given by Theorem
2.9. Then there exists a unique germ of holomorphic map
$\lambda:(P,t_{0})\to({\mathbf{U}},{\tilde{u}})$ such that the germ of
${\underline{\mathcal{G}}}^{\diamond}_{P}$ at $t_{0}$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated, as marked family, to the germ at
$t_{0}$ of
$\lambda^{\ast}{\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}$.

###### Proof.

Let $\phi$ be an ${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism such that

$\phi(\mathcal{G}^{\diamond}_{P}(t_{0}))={\mathcal{F}}^{\diamond}_{\mathbf{U}}({\tilde{u}})\,.$

We will denote by $({\underline{\mathcal{G}}}_{P,t_{0}},{\delta})$ the
deformation of ${\mathcal{G}}_{P}(t_{0})$ over the germ of manifold
$(P,t_{0})$ defined by the germ of ${\underline{\mathcal{G}}}_{P}$ at $t_{0}$
and by the embedding $\delta$ given by the inclusion map of the ambient space
of $\mathcal{G}_{P}(t_{0})$ in that of $\mathcal{G}_{P}$. According to Theorem
2.9 and Theorem 2.4 the deformation
$({\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}},{\iota_{\tilde{u}}})$
is ${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal and any deformation

$({\underline{\mathcal{G}}}_{\mathbf{U},{\tilde{u}}},\kappa)\in\phi^{*}([{\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}},{\iota_{\tilde{u}}}])\;\in\;\;{{\mathrm{Def}}}_{\mathcal{G}_{P}(t_{0})}^{(\mathbf{U},{\tilde{u}})}$

is a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal deformation of
$\mathcal{G}_{P}(t_{0})$. There exist a holomorphic map germ
$\lambda:(P,t_{0})\to(\mathbf{U},{\tilde{u}})$ and a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy $\Phi_{P,t_{0}}$ from the
deformation $({\underline{\mathcal{G}}}_{P,t_{0}},\delta)$ to
$\lambda^{\ast}({\underline{\mathcal{G}}}_{\mathbf{U},{\tilde{u}}},\kappa)$.
By definition of $\phi^{*}$, the associated families
${\underline{\mathcal{G}}}_{\mathbf{U},{\tilde{u}}}$ and
${\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}}$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated as germs of families over
$(\mathbf{U},{\tilde{u}})$, by a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism
$\Phi_{\mathbf{U},{\tilde{u}}}$ which is equal to $\phi$ over ${\tilde{u}}$.
Since $\phi$ is compatible with the markings of
$\mathcal{G}^{\diamond}_{P}(t_{0})$ and
${\mathcal{F}}_{\mathbf{U}}^{\diamond}({\tilde{u}})$, it follows from Remark
2.5 that the homeomorphism
$\lambda^{\ast}\Phi_{\mathbf{U},{\tilde{u}}}\circ\Phi_{P,t_{0}}$ that
conjugates the associated family ${\underline{\mathcal{G}}}_{P,t_{0}}$ to
$\lambda^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}}$, is
compatible with the markings of the families
${\underline{\mathcal{G}}}_{P,t_{0}}^{\diamond}$ and
$\lambda^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U},{\tilde{u}}}^{\diamond}$.
The uniqueness of $\lambda$ results from the following lemma. ∎

######  Lemma 3.2.

Let ${\underline{\mathcal{G}}}^{\diamond}_{P}$ be a marked global family in
${{\mathrm{SL}}}_{P}({\mathcal{F}}^{\diamond})$ over a connected complex
manifold $P$. Two holomorphic liftings $\lambda,\mu:P\to\mathbf{U}$ of the
moduli map of ${\underline{\mathcal{G}}}_{P}^{\diamond}$ through the moduli
map of ${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$,

${{{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}}\circ\lambda={{{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}}\circ\mu={{\mathrm{mod}}}_{{\underline{\mathcal{G}}}_{P}^{\diamond}}\,,$
(10)
$\textstyle{\mathbf{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}}$$\textstyle{{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{{\mathrm{mod}}}_{{\underline{\mathcal{G}}}_{P}^{\diamond}}}$$\scriptstyle{\lambda}$$\scriptstyle{\mu}$

coincide as soon as they take the same value at some point $t_{0}\in P$.

###### Proof.

Let $d\in\mathbf{D}$ be the image of $\lambda(t_{0})=\mu(t_{0})$ by the
morphism $\beta$ in (2). It follows from relation (3) and (10) that for any
$t\in P$ we have:
$\Lambda(\lambda(t))\cdot\zeta(d)=\Lambda(\mu(t))\cdot\zeta(d)$. According to
the exact sequence (2), there is $N_{t}\in{\mathbb{Z}}^{p}$ such that
$\lambda(t)-\mu(t)=\alpha(N_{t})$. The following sets

$K_{N}:=\\{t\in P\;;\;\alpha(N_{t})=N\\}\,,\quad N\in{\mathbb{Z}}^{p}\,,$

are closed analytic subsets of $P$ given by the global equations
$\lambda(t)-\mu(t)=N$. They cannot all be proper subsets of $P$, because
$P=\cup_{N\in{\mathbb{Z}}^{p}}K_{N}$. Therefore there exists
$N_{0}\in{\mathbb{Z}}^{p}$ such that $\lambda(t)-\mu(t)=\alpha(N_{0})$ for any
$t\in P$. As $\lambda(t_{0})=\mu(t_{0})$, we have: $N_{0}=0$, which ends the
proof. ∎

######  Corollary 3.3.

Let ${\mathcal{F}}^{\diamond}$ be a marked by ${\mathcal{E}}^{\diamond}$
foliation of finite type which is a generalized curve. For any marked global
family
${\underline{\mathcal{G}}}_{P}^{\diamond}\in{{\mathrm{SL}}}_{P}({\mathcal{F}}^{\diamond})$
over a connected manifold $P$ the map
$\beta\circ{{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{P}^{\diamond}}:P\to\mathbf{D}$
is constant, where
$\beta:{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\to\mathbf{D}$ still
denotes the last group morphism in the exact sequence (2).

###### Proof.

It suffices to prove that the map
$\beta\circ{{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}$
is locally constant. Let $t_{0}$ be a point in $P$. There is
${\tilde{u}}\in\mathbf{U}$ such that
$[{\mathcal{F}}^{\,\diamond}_{\mathbf{U}}({\tilde{u}})]=[\mathcal{G}^{\diamond}_{P}(t_{0})]\in{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])$.
Theorem 3.1 provides a holomorphic map germ
$\lambda:(P,t_{0})\to(\mathbf{U},{\tilde{u}})$ such that, according to (3),
for $t\in P$ close to $t_{0}$, we have:

${{\mathrm{mod}}}_{{\underline{\mathcal{G}}}_{P}^{\diamond}}(t)={{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}(\lambda(t),d)=\Lambda(\lambda(t))\cdot\zeta(d)\;\in\;\Lambda({\mathbb{C}}^{\tau})\cdot\zeta(d)=\beta^{-1}(d)\,.$

∎

###  3.2. Global factorization property

We are interested now in factorizing up to
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy marked global families through the
marked global family ${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$
provided by Theorem 2.9. However there could already exist obstructions to the
factorization of the ambient space of the family in the ambient space of
${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$, which is trivial over
$\mathbf{U}$. To obtain such a factorization we need to weaken the relation of
conjugacy.

######  Definition 3.4.

Two equisingular (resp. marked equisingular) global families over a manifold
$Q$ are locally ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated if their germs at
any point of $Q$ are ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated as families
(resp. as marked families, see Section 2.1).

The object of this section is to prove the following theorem of factorization
up to local ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy.

######  Theorem 3.5.

Let ${\mathcal{F}}^{\diamond}$ be a marked by ${\mathcal{E}}^{\diamond}$
foliation of finite type which is a generalized curve and let
${\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}$ be the marked
equisingular global family given by Theorem 2.9. Let $P$ be a connected
manifold satisfying $H_{1}(P,{\mathbb{Z}})=0$ and let
${\underline{\mathcal{G}}}_{P}^{\diamond}$ be a global family in
${{\mathrm{SL}}}_{P}({\mathcal{F}}^{\diamond})$. Then for any $t_{0}\in P$ and
$({\tilde{z}},d)\in{\mathbf{U}}$ such that the marked foliations
$\mathcal{G}^{\diamond}_{P}(t_{0})$ and
${\mathcal{F}}_{\mathbf{U}}^{\diamond}({\tilde{z}},d)$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated, there exists a unique
holomorphic map $\lambda:P\to{\mathbf{U}}$ satisfying
$\lambda(t_{0})=({\tilde{z}},d)$, such that the marked global families
${\underline{\mathcal{G}}}_{P}^{\diamond}$ and
$\lambda^{\ast}{\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}$ are
locally ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated.

###### Proof.

\- Step 1: Construction of $\lambda$.  According to Corollary 3.3,
${{\mathrm{mod}}}_{{\underline{\mathcal{G}}}^{\diamond}_{P}}$ takes values in
$\beta^{-1}(d)$. Thus for any $t\in P$, there exist
$z_{t}\in{\mathbb{C}}^{\tau}$ and a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism $\phi_{t}$ such that
$\phi_{t}(\mathcal{G}_{P}(t))={\mathcal{F}}_{\mathbf{U}}^{\diamond}(z_{t},d)$.
Let us denote by ${\underline{\mathcal{G}}}^{\diamond}_{P,\,t}$ the germ of
${\underline{\mathcal{G}}}^{\diamond}_{P}$ at $t$. According to Theorem 3.1
there exists a holomorphic map germ
$\lambda_{t}:(P,t)\to(\mathbf{U},(z_{t},d))$ such that
${\underline{\mathcal{G}}}^{\diamond}_{P,t}$ and
$\lambda_{t}^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated, as germs of families. Therefore
there exist an open covering $(V_{i})_{i\in I}$, $I\subset\mathbb{N}$, of $P$
and holomorphic maps $\lambda_{i}:V_{i}\to{\mathbb{C}}^{\tau}$ such that the
restriction of ${\underline{\mathcal{G}}}^{\diamond}_{P}$ to $V_{i}$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
$(\lambda_{i},d)^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$.
Thus we have:

${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}\circ(\lambda_{i},d)={{\mathrm{mod}}}_{{\underline{\mathcal{G}}}^{\diamond}_{P}}|_{V_{i}}\,.$

We can also require that this covering is locally finite and that the open
sets $V_{i}$ and $V_{i}\cap V_{j}$, $i,j\in I$, are connected. When $V_{i}\cap
V_{j}$ is non empty, the restrictions of $(\lambda_{i},d)$ and
$(\lambda_{j},d)$ to this open set are two factorizations of the moduli map of
${\underline{\mathcal{G}}}_{P}^{\diamond}$ through
${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}$.
Fixing a point $t_{ij}$ in $V_{i}\cap V_{j}$, we have:

$[{\mathcal{F}}_{\mathbf{U}}^{\diamond}(\lambda_{i}(t_{ij}),d)]=[{\mathcal{F}}_{\mathbf{U}}^{\diamond}(\lambda_{j}(t_{ij}),d)]=[\mathcal{G}_{P}^{\diamond}(t_{ij})]\,.$

The relation (3) gives
$\Lambda(\lambda_{i}(t_{ij}))\cdot\zeta(d)=\Lambda(\lambda_{j}(t_{ij}))\cdot\zeta(d)$;
thus $(\lambda_{j}(t_{ij})-\lambda_{i}(t_{ij}))$ belongs to the kernel of
$\Lambda$ and there exist $N_{ij}\in{\mathbb{Z}}^{p}$ such that
$\lambda_{j}(t_{ij})-\lambda_{i}(t_{ij})=\alpha(N_{ij})$. As by assumption we
have: $H_{1}(P,{\mathbb{Z}})=0$, the Čech cohomology group
$H^{1}(P,{\mathbb{Z}}^{p})$ is trivial and there exist
$N_{i}\in{\mathbb{Z}}^{p}$, ${i\in I}$, such that $N_{j}-N_{i}=N_{ij}$ as soon
as $V_{i}\cap V_{j}\neq\emptyset$. Notice that the maps
$(\alpha(N_{i})+\lambda_{i},d):V_{i}\to\mathbf{U}$ are again liftings of
${{\mathrm{mod}}}_{{\underline{\mathcal{G}}}_{P}^{\diamond}}|_{V_{i}}$ through
${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}$.
Indeed we have:

$\displaystyle{{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}\circ(\alpha(N_{i})+\lambda_{i},d)=$
$\displaystyle\Lambda(\alpha(N_{i})+\lambda_{i})\cdot\zeta(d)$
$\displaystyle=$
$\displaystyle\Lambda(\lambda_{i})\cdot\zeta(d)={{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}\circ(\lambda_{i},d)={{\mathrm{mod}}}_{{\underline{\mathcal{G}}}_{P}^{\diamond}}|_{V_{i}}\,.$

Since $\alpha(N_{i})+\lambda_{i}(t_{ij})=\alpha(N_{j})+\lambda_{j}(t_{ij})$,
thanks to Lemma 3.2 and the connectedness of $V_{i}\cap V_{j}$, the maps
$\alpha(N_{i})+\lambda_{i}$, $i\in I$, glue to a global holomorphic map

$\lambda:P\longrightarrow{\mathbb{C}}^{\tau}\times\\{d\\}\subset\mathbf{U}\,,\quad{{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}\circ(\lambda,d)={{\mathrm{mod}}}_{{\underline{\mathcal{G}}}^{\diamond}_{P}}\,.$

Step 2: Properties of $\lambda$.  First we notice that for any
$N\in{\mathbb{Z}}^{p}$ we also have the equality
${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}\circ(\alpha(N)+\lambda,d)={{\mathrm{mod}}}_{{\underline{\mathcal{G}}}^{\diamond}_{P}}$.
Consequently we can assume that $\lambda(t_{0})=({\tilde{z}},d)$. On the other
hand, the global families ${\underline{\mathcal{G}}}^{\diamond}_{P}|_{V_{i}}$
and
$(\lambda_{i},d)^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$
being ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated, the local
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy between
${\underline{\mathcal{G}}}_{P}^{\diamond}$ and
$\lambda^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ results
from the lemma below. ∎

######  Lemma 3.6.

If $\mu:(Q,u_{0})\to({\mathbb{C}}^{\tau},z_{0})$ is a holomorphic map germ and
$N\in{\mathbb{Z}}^{p}$, then the germs at $u_{0}$ of the marked families
$(\mu,d)^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ and
$(\alpha(N)+\mu,d)^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$
are ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated.

###### Proof.

Let us denote by ${\underline{{\mathcal{F}}}}_{\mathbf{U},(z,d)}^{\diamond}$
the germ of ${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ at $(z,d)$
considered as a deformation of the foliation
${\mathcal{F}}^{\diamond}_{\mathbf{U}}(z,d)$, the embedding map being the
inclusion $M_{0}\times\\{(z,d)\\}\hookrightarrow M_{0}\times\mathbf{U}$.

As
$(\alpha(N)+\mu,d)^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}=(\mu,d)^{\ast}(\Delta^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond})$,
with

$\Delta:(\mathbf{U},(z_{0},d))\longrightarrow(\mathbf{U},(\alpha(N)+z_{0},d))\,,\quad\Delta(z,d):=(\alpha(N)+z,d)\,,$

it suffices to see that
${\underline{{\mathcal{F}}}}_{\mathbf{U},(z_{0},d)}^{\diamond}$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
$\Delta^{\ast}({\underline{{\mathcal{F}}}}_{\mathbf{U},(\alpha(N)+z_{0},d)}^{\diamond})$
as a family. To lighten the text let us write

${\mathcal{F}}^{\diamond}_{0}:={\mathcal{F}}_{\mathbf{U}}^{\diamond}(z_{0},d)\quad\hbox{and}\quad{\mathcal{F}}^{\diamond}_{N}:={\mathcal{F}}_{\mathbf{U}}^{\diamond}(\alpha(N)+z_{0},d)=(\Delta^{*}{\mathcal{F}}_{\mathbf{U}}^{\diamond})(z_{0},d)\,.$

There exists a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism $\phi$ such
that $\phi({\mathcal{F}}^{\diamond}_{N})={\mathcal{F}}^{\diamond}_{0}$. Let
${\underline{{\mathcal{K}}}}^{\diamond}_{\mathbf{U},(z_{0},d)}$ be a
deformation of ${\mathcal{F}}^{\diamond}_{N}$ over the germ of manifold
$(\mathbf{U},(z_{0},d))$ that belongs to the class
$\phi^{*}([{\underline{{\mathcal{F}}}}_{\mathbf{U},(z_{0},d)}^{\diamond}])$.
According to Theorem 2.9 the deformations
${\underline{{\mathcal{F}}}}_{\mathbf{U},(z_{0},d)}^{\diamond}$ and
${\underline{{\mathcal{F}}}}_{\mathbf{U},(\alpha(N)+z_{0},d)}^{\diamond}$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal; it follows from Theorem 2.4 and
Remark 2.3 that the deformation
${\underline{{\mathcal{K}}}}_{\mathbf{U},(z_{0},d)}^{\diamond}$ of
${\mathcal{F}}_{N}^{\diamond}$ is ${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal.
On the other hand, since $\Delta$ is a biholomorphism, the deformation
$\Delta^{\ast}({\underline{{\mathcal{F}}}}_{\mathbf{U},(\alpha(N)+z_{0},d)}^{\diamond})$
of ${\mathcal{F}}_{N}^{\diamond}$ is also
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal over the same parameter space
$(\mathbf{U},(z_{0},d))$, again by Remark 2.3. By uniqueness of
${\mathcal{C}}^{{{\mathrm{ex}}}}$-universal deformations,
${\underline{{\mathcal{K}}}}^{\diamond}_{\mathbf{U},(z_{0},d)}$ and
$\Delta^{\ast}({\underline{{\mathcal{F}}}}_{\mathbf{U},(\alpha(N)+z_{0},d)}^{\diamond})$
are ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated deformations of
${\mathcal{F}}_{N}^{\diamond}$. We end the proof by noting that by definition
of
$\phi^{*}([{\underline{{\mathcal{F}}}}_{\mathbf{U},(z_{0},d)}^{\diamond}])$,
the families ${\underline{{\mathcal{K}}}}^{\diamond}_{\mathbf{U},(z_{0},d)}$
and ${\underline{{\mathcal{F}}}}_{\mathbf{U},(z_{0},d)}^{\diamond}$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated. ∎

Now, we consider a weaker notion of conjugacy requiring the equality of moduli
maps, in other words, the ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy fiber by
fiber for each value of the parameter.

######  Theorem 3.7.

Let ${\mathcal{F}}^{\diamond}$ be a marked by ${\mathcal{E}}^{\diamond}$
foliation of finite type which is a generalized curve and let
${\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}$ be the marked
equisingular global family given by Theorem 2.9. If $P$ is a connected
manifold such that $H_{1}(P,{\mathbb{Z}})=0$, then:

  1. (1)

the moduli map of any marked global family
${\underline{\mathcal{G}}}^{\diamond}_{P}\in{{\mathrm{SL}}}_{P}({\mathcal{F}}^{\diamond})$
factorizes through the moduli map of
${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$, i.e. there exists a
holomorphic map $\lambda:P\to\mathbf{U}$ such that
${{\mathrm{mod}}}_{{\underline{\mathcal{G}}}^{\diamond}_{P}}={{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}}\circ\lambda$,

  2. (2)

given $t_{0}\in P$ and $({\tilde{z}},d)\in\mathbf{U}$ such that
$\mathcal{G}^{\diamond}_{P}(t_{0})$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
${\mathcal{F}}_{\mathbf{U}}^{\diamond}({\tilde{z}},d)$, there exists a unique
$\lambda:P\to\mathbf{U}$ satisfying $\lambda(t_{0})=({\tilde{z}},d)$ and
${{\mathrm{mod}}}_{{\underline{\mathcal{G}}}^{\diamond}_{P}}={{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}^{\diamond}_{\mathbf{U}}}\circ\lambda$,

  3. (3)

there exist $p\in\mathbb{N}$, a holomorphic action $\ast$ of
${\mathbb{Z}}^{p}$ on $\mathbf{U}$ and an action $\star$ of a finite product
$\mathbf{I}$ of braid groups on the quotient
$(\mathbf{U}/\ast{\mathbb{Z}}^{p})$, such that
${\mathcal{F}}_{\mathbf{U}}(u_{1})$ and ${\mathcal{F}}_{\mathbf{U}}(u_{2})$
are ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated if and only if there exists
$g\in\mathbf{I}$ such that $g\star({\mathbb{Z}}^{p}\ast
u_{1})={\mathbb{Z}}^{p}\ast u_{2}$.

###### Proof.

Assertion (1) follows from Theorem 3.5. Notice that the uniqueness of
$\lambda$ under conjugacies for each value of the parameter, stated in
assertion (2), is a stronger property than that given by Theorem 3.5. In fact,
the existence of the factorization $\lambda$ in (2) follows again from Theorem
3.5 and the uniqueness from Lemma 3.2.

To see assertion (3) we define the action

$\ast:{\mathbb{Z}}^{p}\times\mathbf{U}\to\mathbf{U},\quad(N,(z,d))\mapsto(z+\alpha(N),d),$

where $\alpha$ is given by the exact sequence (2). Notice that the map
$\mathbf{U}\to{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])$ defined by
$(z,d)\mapsto\Lambda(z)\cdot\zeta(d)=[{\mathcal{F}}_{\mathbf{U}}^{\diamond}(z,d)]$
is surjective and
$\Lambda(z)\cdot\zeta(d)=\Lambda(z^{\prime})\cdot\zeta(d^{\prime})$ if and
only if there is $N\in{\mathbb{Z}}^{p}$ such that
$N\ast(z,d)=(z^{\prime},d^{\prime})$. Thus $\mathbf{U}/\ast{\mathbb{Z}}^{p}$
is identified to
${{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])={{\mathrm{SL}}}({\mathcal{F}}^{\diamond})/\sim_{{\mathcal{C}}^{{{\mathrm{ex}}}}}$.
Let us denote by ${{\mathrm{SL}}}({\mathcal{F}})$ the set of foliations having
the same SL-type than ${\mathcal{F}}$ as was defined in the introduction by
means of properties (SL1)-(SL3). The fibers of the canonical map
${{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\to{{\mathrm{SL}}}({\mathcal{F}})/\sim_{{\mathcal{C}}^{{{\mathrm{ex}}}}}$
are the orbits of the action

$\star:\mathbf{I}\times{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\to{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}]),\quad\varphi\star[\mathcal{G},g]=[\mathcal{G},g\circ\varphi^{-1}],$

where $\mathbf{I}:={{\mathrm{MCG}}}({\mathcal{E}}^{\diamond}_{\mathcal{F}})$
is the mapping class group of the marked divisor
${\mathcal{E}}^{\diamond}_{\mathcal{F}}$ consisting of all the isotopy classes
of homeomorphisms of ${\mathcal{E}}_{\mathcal{F}}$ leaving invariant the
symmetric map $\cdot$ and the set ${{\mathrm{Sing}}}({\mathcal{F}}^{\sharp})$.
It is well known that $\mathbf{I}$ is isomorphic to a finite product of braid
groups. ∎

######  Corollary 3.8.

If $P$ is a connected compact manifold such that $H_{1}(P,{\mathbb{Z}})=0$
then any marked global family
$\mathcal{G}^{\diamond}_{P}\in{{\mathrm{SL}}}_{P}({\mathcal{F}}^{\diamond})$
is locally ${\mathcal{C}}^{{{\mathrm{ex}}}}$-trivial, and a fortiori the
topological class of $\mathcal{G}^{\diamond}_{P}(t)$, $t\in P$, is constant.

###### Proof of Theorem C.

Assertion ( ‣ C) corresponds to properties (i) and (ii) of the exact sequence
(2) stated in Section 2.3. Property (1) of Theorem 2.9 implies assertion (1)
of Theorem C, while assertion (2) of Theorem C is stated in Theorem 3.5. ∎

##  4\. Topological equivalences for families and deformations

We will compare for global families and for germs of deformations the
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy relation to a weaker conjugacy
relation defined as the topological conjugacy before reduction, on each fiber
of the family, without requiring the continuous dependence on the parameters
of the conjugating homeomorphisms.

###  4.1. Tame foliations

Until now the only hypothesis that we have made on the germs of generalized
curve foliations is that of finite type. Under this hypothesis, which is Krull
generic [10], we have obtained, for the equivalence relation
${\mathcal{C}}^{{{\mathrm{ex}}}}$, complete families whose modular map is
surjective. In order to obtain the same result for the equivalence relation
${\mathcal{C}}^{0}$ we must add a combinatorial assumption on the exceptional
divisor ${\mathcal{E}}_{\mathcal{F}}$ and a dynamical assumption on the
transverse structure of the foliation ${\mathcal{F}}$. For that let us denote
by ${\mathcal{E}}_{\mathcal{F}}^{d}$ the union of irreducible components of
the exceptional divisor ${\mathcal{E}}_{\mathcal{F}}$ which are dicritical and
by ${\mathcal{N}}{\mathcal{C}}_{\mathcal{F}}$ the set of singular points of
${\mathcal{E}}_{\mathcal{F}}$, called nodal corners, where the Camacho-Sad
index of ${\mathcal{F}}^{\sharp}$ is a strictly positive real number. Let us
consider the following two conditions:

  1. (NC)

No Chain: the closure of each connected component of
${\mathcal{E}}_{\mathcal{F}}\setminus{\mathcal{E}}_{\mathcal{F}}^{d}$ contains
an irreducible component $D$ with
${{\mathrm{card}}}(D\cap{{\mathrm{Sing}}}({\mathcal{F}}^{\sharp}))\neq 2$,
i.e. there is no connected component of
$\overline{{\mathcal{E}}_{\mathcal{F}}\setminus{\mathcal{E}}_{\mathcal{F}}^{d}}$
as in Figure 1.

  2. (TR)

Transverse Rigidity: if the closure of a connected component of
${\mathcal{E}}_{\mathcal{F}}\setminus({\mathcal{E}}_{\mathcal{F}}^{d}\cup{\mathcal{N}}{\mathcal{C}}_{\mathcal{F}})$
contains an irreducible component with at least $3$ singular points of
${\mathcal{F}}^{\sharp}$, it also contains an irreducible component whose
holonomy group for the foliation ${\mathcal{F}}^{\sharp}$ is topologically
rigid, for instance unsolvable, cf. [11, 13].

$s^{\prime}$$s^{\prime\prime}$$\bullet$$\cdots$$\bullet$$\bullet$$\bullet$$\cdots$
Figure 1. The only situation excluded by Condition (NC). Every divisor is non-
dicritical and the elements of ${{\mathrm{Sing}}}({\mathcal{F}}^{\sharp})$ are
$s^{\prime}$, $s^{\prime\prime}$ and the intersection points of the divisors;
dicritical components may intersect any component.

Condition (NC) is technical and, as the generalized curve condition, only
depends on a finite order jet of the differential form defining
${\mathcal{F}}$. In the presence of chains ${\mathcal{C}}^{0}$-classification
must be approached differently and it will depend on open questions about the
topology of Cremer biholomorphisms in one complex variable. Property (TR) is
satisfied for a dense open set for the Krull topology of differential
$1$-forms fulfilling condition (NC), cf. [3]. The following theorem, first
proved in [6] with additional assumptions, then generalized in [8, Theorem
11.4] using results of [14], justifies these two hypothesis:

######  Theorem 4.1 ([8, Theorem A]).

Two germs of generalized curves foliations ${\mathcal{F}}$ and $\mathcal{G}$
satisfying (NC) and (TR) are ${\mathcal{C}}^{0}$-conjugated if and only if
they are ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated.

######  Definition 4.2.

A germ of singular foliation is called tame if it is a generalized curve of
finite type satisfying conditions (NC) and (TR).

######  Remark 4.3.

For a global equisingular family over a connected manifold properties (NC),
(TR) and being of finite type are satisfied by any fiber as soon as they are
satisfied by one fiber. ∎

###### Proof of Theorem A.

We mark ${\mathcal{F}}$ by ${\mathcal{E}}_{\mathcal{F}}$ using the identity
map; to obtain ${\underline{{\mathcal{F}}}}_{\mathbf{U}}$ we apply Theorem C
to
${\mathcal{F}}^{\diamond}=({\mathcal{F}},{{\mathrm{id}}}_{{\mathcal{E}}_{\mathcal{F}}})$,
that also provides a marking $(f_{u})_{u\in\mathbf{U}}$ on
${\underline{{\mathcal{F}}}}_{\mathbf{U}}$.

We begin by proving assertion (1). Since $\mathcal{G}$ has the same SL-type as
${\mathcal{F}}$ there exists a homeomorphism
$\varphi:{\mathcal{E}}_{\mathcal{F}}\to{\mathcal{E}}_{\mathcal{G}}$ satisfying
properties (SL1)-(SL3). We consider the marked by
${\mathcal{E}}_{\mathcal{F}}^{\diamond}=({\mathcal{E}}_{\mathcal{F}},{{\mathrm{Sing}}}({\mathcal{F}}^{\sharp}),\cdot)$
foliation $\mathcal{G}^{\diamond}=(\mathcal{G},\varphi)$ which has the same
marked SL-type as ${\mathcal{F}}^{\diamond}$. By assertion (1) of Theorem C
there exists $u_{0}\in\mathbf{U}$ such that $\mathcal{G}^{\diamond}$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
${\mathcal{F}}_{\mathbf{U}}^{\diamond}(u_{0})$. A fortiori, $\mathcal{G}$ is
${\mathcal{C}}^{0}$-conjugated to ${\mathcal{F}}_{\mathbf{U}}(u_{0})$.

Let us prove now assertion (2). As $\mathcal{G}_{P}(t_{0})$ is
${\mathcal{C}}^{0}$-conjugated to ${\mathcal{F}}_{\mathbf{U}}(u_{0})$, by
Theorem 4.1 there is a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy
$\phi:{\mathcal{F}}_{\mathbf{U}}(u_{0})\to\mathcal{G}_{P}(t_{0})$. The
composition $g_{t_{0}}:=\phi^{\sharp}\circ
f_{u_{0}}:{\mathcal{E}}_{\mathcal{F}}\to{\mathcal{E}}_{\mathcal{G}_{P}(t_{0})}$
of the lifting of $\phi$ through the reduction maps and the marking of
${\mathcal{F}}_{\mathbf{U}}(u_{0})$ defines a marking of
$\mathcal{G}_{P}(t_{0})$ such that $\mathcal{G}_{P}^{\diamond}(t_{0})$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
${\mathcal{F}}_{\mathbf{U}}^{\diamond}(u_{0})$ by $\phi^{-1}$. Since $P$ is
simply connected, by assertion (b) of Remark 2.5, the marking $g_{t_{0}}$
extends to a marking $(g_{t})_{t\in P}$ of the global family
${\underline{\mathcal{G}}}_{P}$. We apply assertion (2) of Theorem C to
${\underline{\mathcal{G}}}_{P}^{\diamond}=({\underline{\mathcal{G}}}_{P},(g_{t})_{t\in
P})$ and we obtain a (unique) holomorphic map $\lambda:P\to\mathbf{U}$ such
that $\lambda(t_{0})=u_{0}$ and for any $t\in P$ the germs of marked families
${\underline{\mathcal{G}}}^{\diamond}_{P,t}$ and
$\lambda^{*}{\underline{{\mathcal{F}}}}_{\mathbf{U},\lambda(t)}^{\diamond}$
over the germ of manifold $(P,t)$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated. A fortiori, the germs of
families ${\underline{\mathcal{G}}}_{P,t}$ and
$\lambda^{*}{\underline{{\mathcal{F}}}}_{\mathbf{U},\lambda(t)}$ are
${\mathcal{C}}^{0}$-conjugated.

Redundancy property (3) in Theorem A follows from assertion (3) in Theorem 3.7
taking into account that ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy and
${\mathcal{C}}^{0}$-conjugacy are equivalent for tame foliations, see Theorem
4.1. ∎

###  4.2. Weak and strong conjugacies of families

In this section we will prove Theorem B of the introduction. Before that we
state a marked version of that result in which the hypothesis are weaker.

######  Theorem 4.4.

Let ${\underline{{\mathcal{F}}}}^{\diamond}_{Q}$ and
${\underline{\mathcal{G}}}_{Q}^{\diamond}$ be marked by
${\mathcal{E}}^{\diamond}$ equisingular global families of foliations over a
complex manifold $Q$, whose fibers are generalized curves of finite type. The
following properties are equivalent

  1. (1)

for any $u\in Q$ the marked foliations ${\mathcal{F}}^{\diamond}_{Q}(u)$ and
$\mathcal{G}^{\diamond}_{Q}(u)$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated,

  2. (2)

the marked global families ${\underline{{\mathcal{F}}}}^{\diamond}_{Q}$ and
${\underline{\mathcal{G}}}^{\diamond}_{Q}$ are locally
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated.

###### Proof.

The implication $(2)\Rightarrow(1)$ is trivial. To prove the converse we can
assume that $Q$ is connected and simply connected. Let us fix a fiber
${\mathcal{F}}^{\diamond}:={\mathcal{F}}_{Q}^{\diamond}({\tilde{u}})$,
${\tilde{u}}\in Q$. According to the connectedness of $Q$ and Remark 2.7, each
$\mathcal{G}^{\diamond}_{Q}(u)$, $u\in Q$, belongs to
${{\mathrm{SL}}}({\mathcal{F}}^{\diamond})$, see Definition 2.6. By assertion
(1) we have the equality

${{\mathrm{mod}}}_{{\underline{\mathcal{G}}}_{Q}^{\diamond}}={{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}^{\diamond}_{Q}}:Q\longrightarrow{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\,.$

Let ${\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ be the marked global
family given by Theorem 2.9. Let us consider $({\tilde{z}},d)\in\mathbf{U}$
such that ${\mathcal{F}}_{Q}^{\diamond}({\tilde{u}})$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
${\mathcal{F}}_{\mathbf{U}}^{\diamond}({\tilde{z}},d)$. Since
${\mathcal{F}}_{Q}^{\diamond}({\tilde{u}})$ and
$\mathcal{G}_{Q}^{\diamond}({\tilde{u}})$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated, Theorem 3.5 provides holomorphic
maps $\lambda,\mu:Q\to{\mathbb{C}}^{\tau}\times\\{d\\}\subset\mathbf{U}$
satisfying $\lambda({\tilde{u}})=\mu({\tilde{u}})=({\tilde{z}},d)$, such that
${\underline{{\mathcal{F}}}}^{\diamond}_{Q}$ is locally
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
$\lambda^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$ and
${\underline{\mathcal{G}}}^{\diamond}_{Q}$ is locally
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
$\mu^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$. We thus have:

${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}\circ\lambda={{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}^{\diamond}_{Q}}={{\mathrm{mod}}}_{{\underline{\mathcal{G}}}^{\diamond}_{Q}}={{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}\circ\mu\,.$

Consequently $\lambda$ and $\mu$ are two liftings of the map
${{\mathrm{mod}}}_{{\underline{\mathcal{G}}}^{\diamond}_{Q}}={{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}^{\diamond}_{Q}}$
through the map
${{\mathrm{mod}}}_{{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}}$,
which coincide at the point ${\tilde{u}}$. It follows from assertion (2) of
Theorem 3.7 that $\lambda=\mu$. Therefore
${\underline{{\mathcal{F}}}}_{Q}^{\diamond}$ and
${\underline{\mathcal{G}}}^{\diamond}_{Q}$ are locally
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated, since they are both locally
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
$\lambda^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}=\mu^{\ast}{\underline{{\mathcal{F}}}}_{\mathbf{U}}^{\diamond}$.
∎

Now we will use Theorem 4.4 to prove Theorem B of the introduction.

###### Proof of Theorem B.

Thanks to Theorem 4.1, assertions (1) and (2) are equivalent. The implication
$(\ref{strongconj})\Longrightarrow(\ref{weakconj})$ is trivial. To prove
$(\ref{exconj})\Longrightarrow(\ref{strongconj})$ let us fix a point
${\tilde{u}}$ in $Q$ and a marking
$f_{{\tilde{u}}}:{\mathcal{E}}{\overset{\sim}{\longrightarrow}}{\mathcal{E}}_{{\mathcal{F}}_{Q}({\tilde{u}})}$
of the fiber ${\mathcal{F}}_{Q}({\tilde{u}})$ by an appropriate marked divisor
${\mathcal{E}}^{\diamond}$. By restricting both families to a suitable
neighborhood of ${\tilde{u}}$ we may assume that $Q$ is connected and simply
connected. Thanks to (b) in Remark 2.5, $f_{\tilde{u}}$ extends to a marking
$(f_{u})_{u\in Q}$ of the global family ${\underline{{\mathcal{F}}}}_{Q}$ and
we will write:

${\underline{{\mathcal{F}}}}_{Q}^{\diamond}:=({\underline{{\mathcal{F}}}}_{Q},(f_{u})_{u\in
Q})\quad\text{and}\quad{\mathcal{F}}^{\diamond}:=({\mathcal{F}}_{Q}({\tilde{u}}),f_{{\tilde{u}}})\,.$

According to Theorem 4.4, in order to obtain assertion (3) it only remains to
prove the existence of a marking $(G_{u})_{u\in Q}$ of
${\underline{\mathcal{G}}}_{Q}$ such that for each $u\in Q$ the marked
foliation $\mathcal{G}_{Q}^{\diamond}(u):=(\mathcal{G}_{Q}(u),G_{u})$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
${\mathcal{F}}^{\diamond}_{Q}(u)$:

$[(\mathcal{G}_{Q}(u),G_{u})]=[({\mathcal{F}}_{Q}(u),f_{u})]\in{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\,,\quad
u\in Q\,.$ (11)

For this, we choose for each $u\in Q$ a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy

$\varphi_{u}:(M(u),\theta(u)){\overset{\sim}{\longrightarrow}}(N(u),\vartheta(u))\,,\quad\varphi_{u}({\mathcal{F}}_{Q}(u))=\mathcal{G}_{Q}(u)\,,$

and we denote by
$\varphi^{\sharp}_{u}:(M_{{\mathcal{F}}_{Q}(u)},{\mathcal{E}}_{{\mathcal{F}}_{Q}(u)})\to(M_{\mathcal{G}_{Q}(u)},{\mathcal{E}}_{\mathcal{G}_{Q}(u)})$
the germ of homeomorphism obtained by lifting it through the reduction of
singularities of ${\mathcal{F}}_{Q}(u)$ and $\mathcal{G}_{Q}(u)$. We endow
${\underline{\mathcal{G}}}_{Q}$ with a marking by ${\mathcal{E}}^{\diamond}$

$g_{u}:{\mathcal{E}}\to{\mathcal{E}}_{\mathcal{G}_{Q}(u)}\,,\quad u\in Q\,,$

obtained thanks to Remark 2.5 by extending the marking
$\varphi_{\tilde{u}}^{\sharp}\circ f_{{\tilde{u}}}$ of
$\mathcal{G}_{Q}({\tilde{u}})$. We also consider the following pre-marking of
${\underline{\mathcal{G}}}_{Q}$:

$\varphi^{\sharp}_{u}\circ
f_{u}:{\mathcal{E}}\to{\mathcal{E}}_{\mathcal{G}_{Q}(u)}\,,\quad u\in Q\,.$

Since $\varphi_{u}$ is a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy from
${\mathcal{F}}_{Q}^{\diamond}(u)$ to
$(\mathcal{G}_{Q}(u),\,\varphi^{\sharp}_{u}\circ f_{u})$ and
${\mathcal{F}}^{\diamond}_{Q}(u)$ belongs to
${{\mathrm{SL}}}({\mathcal{F}}^{\diamond})$, this pre-marking satisfies

$(\mathcal{G}_{Q}(u),\,\varphi^{\sharp}_{u}\circ
f_{u})\in{{\mathrm{SL}}}({\mathcal{F}}^{\diamond})$

for each $u\in Q$.

Let us denote by ${{\mathrm{MCG}}}({\mathcal{E}}^{\diamond})$ the mapping
class group of ${\mathcal{E}}^{\diamond}=({\mathcal{E}},\Sigma,\cdot)$, that
is the group of isotopy classes $[\phi]$ of homeomorphisms
$\phi:{\mathcal{E}}{\overset{\sim}{\longrightarrow}}{\mathcal{E}}$ leaving
invariant the symmetric map $\cdot$ and the set $\Sigma$. For each
$[\phi]\in{{\mathrm{MCG}}}({\mathcal{E}}^{\diamond})$ let us consider the set

$K_{[\phi]}:=\left\\{u\in Q\;;\;[g_{u}^{-1}\circ\varphi^{\sharp}_{u}\circ
f_{u}]=[\phi]\right\\}.$

Since ${{\mathrm{MCG}}}({\mathcal{E}}^{\diamond})$ is countable and

$\bigcup_{[\phi]\in{{\mathrm{MCG}}}({\mathcal{E}}^{\diamond})}K_{[\phi]}=Q\,,$

there exists an element $[\Phi]\in{{\mathrm{MCG}}}({\mathcal{E}}^{\diamond})$
such that $K_{[\Phi]}$ is not contained in any countable union of proper
closed analytic subsets of $Q$. Let us consider the marked by
${\mathcal{E}}^{\diamond}$ equisingular global family

${\underline{\mathcal{G}}}_{Q}^{\diamond}:=\left({\underline{\mathcal{G}}}_{Q},\,(G_{u})_{u\in
Q}\right)\,,\quad G_{u}:=g_{u}\circ\Phi\,.$

We highlight that

$\varphi_{u}({\mathcal{F}}_{Q}^{\diamond}(u))=\mathcal{G}_{Q}^{\diamond}(u)\,,\quad\hbox{
if }\quad u\in K_{[\Phi]}\,,$ (12)

as in this case $g_{u}\circ\Phi$ is isotopic to $\varphi^{\sharp}_{u}\circ
f_{u}$. Therefore $\mathcal{G}_{Q}^{\diamond}(u)$ belongs to
${{\mathrm{SL}}}({\mathcal{F}}^{\diamond})$ when $u\in K_{[\Phi]}$. It follows
from Remark 2.7 that ${\underline{\mathcal{G}}}_{Q}^{\diamond}$ belongs to
${{\mathrm{SL}}}_{Q}({\mathcal{F}}^{\diamond})$ and we can consider the map
$Q\ni
u\mapsto[\mathcal{G}^{\diamond}_{Q}(u)]\in{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])$.

Let us now consider the map
$\beta:{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])\to\mathbf{D}$ in the exact
sequence (2). By Corollary 3.3 there is $d\in\mathbf{D}$ such that
$\beta([{\mathcal{F}}^{\diamond}_{Q}(u)])=d$, for every $u\in Q$. From (12)
and Corollary 3.3 we also have $\beta([\mathcal{G}^{\diamond}_{Q}(u)])=d$, for
every $u\in Q$. Let us fix $u_{1}\in K_{[\Phi]}$ and
$z_{1}\in{\mathbb{C}}^{\tau}$ satisfying

$[{\mathcal{F}}_{Q}^{\diamond}(u_{1})]=[\mathcal{G}_{Q}^{\diamond}(u_{1})]=\Lambda(z_{1})\cdot\zeta(d)$

By Theorem 3.5 there exist two holomorphic maps

$\lambda:Q\longrightarrow{\mathbb{C}}^{\tau}\,,\qquad\lambda^{\prime}:Q\longrightarrow{\mathbb{C}}^{\tau}\,,$

satisfying $\lambda(u_{1})=\lambda^{\prime}(u_{1})=z_{1}$ and

$\Lambda(\lambda(u))\cdot\zeta(d)=[{\mathcal{F}}_{Q}^{\diamond}(u)]\,,\quad\Lambda(\lambda^{\prime}(u))\cdot\zeta(d)=[\mathcal{G}_{Q}^{\diamond}(u)]\,,\quad
u\in Q\,,$ (13)

where
$\Lambda:{\mathbb{C}}^{\tau}\to{{\mathrm{Mod}}}([{\mathcal{F}}^{\diamond}])$
is the map in (2). From (12) for $u\in K_{[\Phi]}$ we have:

$\Lambda(\lambda(u))\cdot\zeta(d)=\Lambda(\lambda^{\prime}(u))\cdot\zeta(d)\quad\hbox{if}\quad
u\in K_{[\Phi]}$

and $\lambda(u)-\lambda^{\prime}(u)$ belongs to $\ker(\Lambda)$ for $u\in
K_{[\Phi]}$. Let us choose $N_{u}\in{\mathbb{Z}}^{p}$ such that

$\lambda(u)=\alpha(N_{u})+\lambda^{\prime}(u)\,,\quad u\in K_{[\Phi]}\,,$

and for each $N\in{\mathbb{Z}}^{p}$ let us consider the following sets:

$L_{N}:=\\{u\in K_{[\Phi]}\;;\;N_{u}=N\\}\subset L^{\prime}_{N}=\\{u\in
Q\;;\;\lambda(u)-\lambda^{\prime}(u)=\alpha(N)\\}\subset Q\,.$

Clearly we have:

$K_{[\Phi]}\;=\;\bigcup_{N\in{\mathbb{Z}}^{p}}\,L_{N}\subset\bigcup_{N\in{\mathbb{Z}}^{p}}L^{\prime}_{N}\,.$

Since each $L^{\prime}_{N}$ is a closed analytic subset of $Q$ and
$K_{[\Phi]}$ is not contained in a countable union of proper such sets, there
exists ${\widetilde{N}}\in{\mathbb{Z}}^{p}$ such that
$L^{\prime}_{{\widetilde{N}}}=Q$. Consequently:

$\lambda(u)=\alpha({\widetilde{N}})+\lambda^{\prime}(u)\quad\hbox{for
every}\quad u\in Q\,.$

Then equalities (13) give the required equalities (11); that ends the proof. ∎

###  4.3. Conjugacies of families versus conjugacies of deformations

According to Remark 2.5, any deformation of a marked foliation may be
canonically endowed with a marking. We will see that under finite type
assumptions, the notion of conjugacy of deformations is equivalent to that of
the conjugacy of their marked associated families.

######  Theorem 4.5.

Let us consider a finite type foliation ${\mathcal{F}}$ which is a generalized
curve, $f:{\mathcal{E}}\to{\mathcal{E}}_{\mathcal{F}}$ a marking of
${\mathcal{F}}$, $({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$
and $({\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}},\delta)$ two equisingular
deformations of ${\mathcal{F}}$ over a germ of manifold
$Q^{\mathbf{\cdot}}:=(Q,{\tilde{u}})$. Let us denote by
${\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}^{\diamond}$ and
${\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}}^{\diamond}$ the families
${\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$ resp.
${\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}}$ endowed with the markings
induced by the markings $\iota^{\sharp}\circ f$ and $\delta^{\sharp}\circ f$
of their special fibers. Then the following properties are equivalent:

  1. (1)

there is a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy $\phi$ between the
germs of families of ${\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$ and
${\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}}$ such that the lifting of
$\delta^{-1}\circ\phi\circ\iota$ through the reduction of singularities of
${\mathcal{F}}$ leaves invariant each irreducible component of
${\mathcal{E}}_{\mathcal{F}}$,

  2. (2)

the marked families
${\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}^{\diamond}$ and
${\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}}^{\diamond}$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated,

  3. (3)

the deformations $({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$
and $({\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}},\delta)$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated.

This theorem will be easily obtained using the following extension result.

######  Theorem 4.6.

Let ${\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$ be an equisingular
family over a germ of manifold $Q^{\mathbf{\cdot}}=(Q,{\tilde{u}})$ whose
fibers are finite type generalized curves. Let $\phi$ be a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-automorphism of the special fiber
${\mathcal{F}}_{Q^{\mathbf{\cdot}}}({\tilde{u}})$ and
$\phi^{\sharp}:M_{{\mathcal{F}}_{Q^{\mathbf{\cdot}}}({\tilde{u}})}{\overset{\sim}{\longrightarrow}}M_{{\mathcal{F}}_{Q^{\mathbf{\cdot}}}({\tilde{u}})}$
the lifting of $\phi$ through the reduction of singularities of
${\mathcal{F}}_{Q^{\mathbf{\cdot}}}({\tilde{u}})$. If $\phi^{\sharp}$ leaves
invariant each irreducible component of the exceptional divisor
${\mathcal{E}}_{{\mathcal{F}}_{Q^{\mathbf{\cdot}}}({\tilde{u}})}$, then $\phi$
extends to a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-automorphism of the family
${\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$.

###### Proof of Theorem 4.5 .

The implications
$(\ref{conjdeformations})\Rightarrow(\ref{conjassfam})\Rightarrow(\ref{conjcomp})$
are trivial. To see that
$(\ref{conjcomp})\Rightarrow(\ref{conjdeformations})$, we denote by
$\phi_{{\tilde{u}}}$ the restriction of $\phi$ to the fibers over
${\tilde{u}}$ and we consider the
${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism
$\psi:=\iota\circ\delta^{-1}\circ\phi_{{\tilde{u}}}$ which is an automorphism
of ${\mathcal{F}}_{Q^{\mathbf{\cdot}}}({\tilde{u}})$. Notice that $\psi$ is
conjugated by $\iota$ to $\delta^{-1}\circ\phi\circ\iota$ and consequently its
lifting $\psi^{\sharp}$ through the reduction of singularities of
${\mathcal{F}}_{Q^{\mathbf{\cdot}}}({\tilde{u}})$ leaves invariant each
irreducible component of the exceptional divisor
${\mathcal{E}}_{{\mathcal{F}}_{Q^{\mathbf{\cdot}}}({\tilde{u}})}$. By Theorem
4.6 $\psi$ extends to a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-automorphism $\Psi$
of the family ${\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$. Then the
${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism $\Phi:=\phi\circ\Psi^{-1}$
satisfies
$\Phi({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}})={\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$
and

$\Phi\circ\iota=\Phi_{{\tilde{u}}}\circ\iota=\phi_{{\tilde{u}}}\circ\Psi_{{\tilde{u}}}^{-1}\circ\iota=\phi_{{\tilde{u}}}\circ\psi^{-1}\circ\iota=\phi_{\tilde{u}}\circ\phi_{\tilde{u}}^{-1}\circ\delta\circ\iota^{-1}\circ\iota=\delta.$

∎

The proof of Theorem 4.6 is based on the following property of the pull-back
map $\phi^{*}$ introduced in Theorem 2.4.

######  Lemma 4.7.

Let ${\mathcal{F}}$ be a germ of foliation and let $\phi$ be a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism that conjugates
${\mathcal{F}}$ to itself. Assume that ${\mathcal{F}}$ is a finite type
generalized curve and that the lifting $\phi^{\sharp}$ of $\phi$ through the
reduction of singularities of ${\mathcal{F}}$ leaves invariant each
irreducible component of the exceptional divisor
${\mathcal{E}}_{\mathcal{F}}$. Then for any pointed manifold
$P^{\mathbf{\cdot}}$ the pull-back map
$\phi^{\ast}:{{\mathrm{Def}}}^{P^{\mathbf{\cdot}}}_{{\mathcal{F}}}{\overset{\sim}{\longrightarrow}}{{\mathrm{Def}}}^{P^{\mathbf{\cdot}}}_{{\mathcal{F}}}$
is the identity map.

###### Proof of Theorem 4.6.

Let $\iota$ be the canonical embedding of the special fiber
${\mathcal{F}}_{Q^{\mathbf{\cdot}}}(\tilde{u})$ in the family
${\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$, so that
$({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ is a deformation of
${\mathcal{F}}_{Q^{\mathbf{\cdot}}}(\tilde{u})$. By definition, there is an
element $({\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}},\delta)$ of
$\phi^{*}([{\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota])$ that is
conjugated to $({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$ by a
${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism germ
$\Phi:{\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}}\to{\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}}$
such that $\Phi\circ\delta=\iota\circ\phi$. By Lemma 4.7 we have
$\phi^{\ast}([{\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota])=[{\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota]$.
This means that
$({\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}},\delta)\sim_{{\mathcal{C}}^{{\mathrm{ex}}}}({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}},\iota)$,
and there is a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-homeomorphism germ $\Psi$
such that
$\Psi({\underline{{\mathcal{F}}}}_{Q^{\mathbf{\cdot}}})={\underline{\mathcal{G}}}_{Q^{\mathbf{\cdot}}}$
and $\Psi\circ\iota=\delta$. Hence $\Phi\circ\Psi\circ\iota=\iota\circ\phi$,
i.e. $\Phi\circ\Psi$ extends $\phi$. ∎

It remains to prove Lemma 4.7 but before we will recall some representation
results proved in [9]. Let $\mathbf{Fol}$ be the category of germs of
generalized curves on $({\mathbb{C}}^{2},0)$ and excellent conjugacies. Let
$\mathbf{Fol}_{\mathbf{ft}}\subset\mathbf{Fol}$ be the full subcategory
consisting in finite type foliations. Let
$\mathbf{Man}^{\mathbf{\cdot}}\subset\mathbf{Set}^{\mathbf{\cdot}}$ be the
categories of pointed complex manifolds and sets. In [9, §5.3 and §1.2] we
have introduced the contravariant functor
$\mathbf{Fol_{ft}}\to\mathbf{Man}^{\mathbf{\cdot}}$, ${\mathcal{F}}\mapsto
H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})$. Any
excellent conjugacy $\phi:\mathcal{G}\to{\mathcal{F}}$ induces the graph
morphism
${\mathsf{A}}_{\phi}:{\mathsf{A}}_{\mathcal{G}}\to{\mathsf{A}}_{\mathcal{F}}$,
$\star\mapsto\phi^{\sharp}(\star)$, which allows to define a morphism
$[\phi^{*}]:H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})\to
H^{1}({\mathsf{A}}_{\mathcal{G}},\mathcal{T}_{\mathcal{G}})$ in the following
way:

$[\phi^{*}]([X_{D,\langle D,D^{\prime}\rangle}])=[Y_{D,\langle
D,D^{\prime}\rangle}],\quad Y_{D,\langle
D,D^{\prime}\rangle}=(\phi^{\sharp})^{*}X_{{\mathsf{A}}_{\phi}(D,\langle
D,D^{\prime}\rangle)}\,.$

We also considered the contravariant functor
${{\mathrm{Fac}}}:\mathbf{Man}^{\mathbf{\cdot}}\times\mathbf{Fol_{ft}}\to\mathbf{Set}^{\mathbf{\cdot}}$
defined by

${{\mathrm{Fac}}}:(P^{\mathbf{\cdot}},{\mathcal{F}})\mapsto{\mathcal{O}}(P^{\mathbf{\cdot}},H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})^{\mathbf{\cdot}})\,,$

and if $\mu:Q^{\mathbf{\cdot}}\to P^{\mathbf{\cdot}}$ and
$\phi:\mathcal{G}\to{\mathcal{F}}$ are morphisms in
$\mathbf{Man}^{\mathbf{\cdot}}$ and $\mathbf{Fol_{ft}}$ respectively, then
${{\mathrm{Fac}}}_{\phi}^{\mu}$ sends
$\lambda\in{\mathcal{O}}(P^{\mathbf{\cdot}},H^{1}({\mathsf{A}}_{{\mathcal{F}}},\mathcal{T}_{{\mathcal{F}}})^{\mathbf{\cdot}})$
into
$[\phi^{*}]\circ\lambda\circ\mu\in{\mathcal{O}}(Q^{\mathbf{\cdot}},H^{1}({\mathsf{A}}_{\mathcal{G}},\mathcal{T}_{\mathcal{G}})^{\mathbf{\cdot}})$.

On the other hand, Theorem 2.4 allows us to consider the correspondence

${{\mathrm{Def}}}:(P^{\mathbf{\cdot}},{\mathcal{F}})\mapsto{{\mathrm{Def}}}_{\mathcal{F}}^{P^{\mathbf{\cdot}}}:=\\{\text{equisingular
deformations of ${\mathcal{F}}$ over
$P^{\mathbf{\cdot}}$}\\}/\sim_{{\mathcal{C}}^{{{\mathrm{ex}}}}}$

together with the morphisms
${{\mathrm{Def}}}_{\phi}^{\mu}=\phi^{*}\circ\mu^{*}=\mu^{*}\circ\phi^{*}$.
According to [9, Theorem 3.11]
${{\mathrm{Def}}}:\mathbf{Man}^{\mathbf{\cdot}}\times\mathbf{Fol_{ft}}\to\mathbf{Set}^{\mathbf{\cdot}}$
is a contravariant functor.

######  Theorem 4.8 ([9, Theorem 6.3]).

There is an isomorphism of functors
$\xi:{{\mathrm{Def}}}{\overset{\sim}{\longrightarrow}}{{\mathrm{Fac}}}$.

###### Proof of Lemma 4.7.

Thanks to Theorem 4.8 it suffices to prove that the morphisms

${{\mathrm{Fac}}}^{{{\mathrm{id}}}_{P^{\mathbf{\cdot}}}}_{\phi}:{\mathcal{O}}(P^{\mathbf{\cdot}},H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})^{\mathbf{\cdot}})\to{\mathcal{O}}(P^{\mathbf{\cdot}},H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})^{\mathbf{\cdot}})\,,\quad\lambda\mapsto[\phi^{\ast}]\circ\lambda\,,$

are the identity maps. Indeed the naturality of $\xi$ gives the following
commutative diagrams

$\textstyle{{{\mathrm{Def}}}^{P^{\mathbf{\cdot}}}_{{\mathcal{F}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\ast}}$

$\sim$

$\scriptstyle{\xi^{P^{\mathbf{\cdot}}}_{{\mathcal{F}}}}$$\textstyle{{{\mathrm{Def}}}^{P^{\mathbf{\cdot}}}_{{\mathcal{F}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\scriptstyle{\xi^{P^{\mathbf{\cdot}}}_{{\mathcal{F}}}}$$\textstyle{{\mathcal{O}}(P^{\mathbf{\cdot}},H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})^{\mathbf{\cdot}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{{\mathrm{Fac}}}^{{{\mathrm{id}}}_{P^{\mathbf{\cdot}}}}_{\phi}}$$\textstyle{{\mathcal{O}}(P^{\mathbf{\cdot}},H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})^{\mathbf{\cdot}})\,.}$

The map ${{\mathrm{Fac}}}_{\phi}^{{{\mathrm{id}}}_{P^{\mathbf{\cdot}}}}$ is
the identity if and only if the pull-back map
$[\phi^{\ast}]:H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})\longrightarrow
H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})$ is the
identity. Since each irreducible component
$D\in{\mathsf{Ve}}_{{\mathsf{A}}_{\mathcal{F}}}$ is fixed by $\phi^{\sharp}$,
the induced graph morphism
${\mathsf{A}}_{\phi}:{\mathsf{A}}_{\mathcal{F}}\to{\mathsf{A}}_{\mathcal{F}}$
is the identity map and

$[\phi^{\ast}]:H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})\longrightarrow
H^{1}({\mathsf{A}}_{\mathcal{F}},{\mathcal{T}}_{{\mathcal{F}}})\,,\quad[(X_{D,\mathsf{e}})_{D\in\mathsf{e}\in{\mathsf{Ed}}_{{\mathsf{A}}_{\mathcal{F}}}}]\mapsto\big{[}(\phi^{\sharp\ast}(X_{D,\mathsf{e}}))_{D\in\mathsf{e}\in{\mathsf{Ed}}_{{\mathsf{A}}_{\mathcal{F}}}}\big{]}\;.$

Thanks to Theorem 2.11, it suffices to see that
$\phi^{\sharp*}:\mathcal{T}_{{\mathcal{F}}}(\mathsf{e})\to\mathcal{T}_{{\mathcal{F}}}(\mathsf{e})$
is the identity for each $\mathsf{e}=\langle
D,D^{\prime}\rangle\in{\mathsf{Ed}}_{{\mathsf{A}}_{\mathcal{F}}}$ such that
$\mathcal{T}_{\mathcal{F}}(\mathsf{e})$ is one-dimensional. The germ at
$\\{m\\}=D\cap D^{\prime}$ of the foliation ${\mathcal{F}}^{\sharp}$ is either
linearizable non-resonant or resonant normalizable and non-linearizable. Let
us fix a local chart
$(z_{1},z_{2}):\Omega{\overset{\sim}{\longrightarrow}}\overline{{\mathbb{D}}}_{r}\times{\overline{{\mathbb{D}}}}_{r}$
centered at the point $m$, satisfying

$r>1,\quad z(m)=(0,0),\quad D\cup D^{\prime}=\\{z_{1}z_{2}=0\\}$

such that the foliation ${\mathcal{F}}^{\sharp}$ is defined on $\Omega$ either
by the $1$-form $\omega=\omega_{L}$ or by $\omega=\omega_{N}$, with

  * •

$\omega_{L}=az_{1}dz_{2}+bz_{2}dz_{1}$, with $a,\,b,\in{\mathbb{C}}$,
$b/a\notin{\mathbb{Q}}\,$,

  * •

$\omega_{N}=bz_{1}(1+\zeta(z_{1}^{a}z_{2}^{b})^{k})dz_{2}+az_{1}(1+(\zeta-1)(z_{1}^{a}z_{2}^{b})^{k})dz_{2}$,
$a,b,k\in\mathbb{N}^{\ast}$, $\zeta\in{\mathbb{C}}$.

According to [9, Lemma 5.4] in both cases, $\omega=\omega_{L}$ or
$\omega=\omega_{N}$, there exists an explicit holomorphic vector field $Z$ on
$\Omega$, that is tangent to $\\{z_{1}=1\\}$ such that $[Z]$ generates
$\mathcal{T}_{{\mathcal{F}}}(\mathsf{e})$:

$Z=z_{2}\frac{\partial}{\partial z_{2}}\text{ if }\omega=\omega_{L},\qquad
Z=\frac{(z_{1}^{a}z_{2}^{b})^{k}}{1+\zeta(z_{1}^{a}z_{2}^{b})^{k}}z_{2}\frac{\partial}{\partial
z_{2}}\text{ if }\omega=\omega_{N}\,.$

Let us fix a point $p\in\Omega$ with coordinates $z_{1}=\varepsilon$,
$z_{2}=0$, where $\varepsilon\in{\mathbb{R}}_{>0}$ is sufficiently small so
that $\phi^{\sharp}$ is holomorphic on
$\Omega_{\varepsilon}:=\\{|z_{1}|,|z_{2}|\leq\varepsilon\\}\subset\Omega$ and
$\phi^{\sharp}(\Omega_{\varepsilon})\subset\Omega$. For $q\in\Omega$ we will
denote by $Z_{q}$ the germ of $Z$ at $q$. We must prove the equality
$[\phi^{\sharp\,\ast}Z_{m}]=[Z_{m}]$ in
$\mathcal{T}_{\mathcal{F}}(\mathsf{e})$, or equivalently that
$Z_{m}-(D\phi^{\sharp}\cdot Z_{m})\circ\phi^{\sharp\,-1}$ is tangent to
${\mathcal{F}}^{\sharp}$. We will use the following fact about the quotient
sheaf
${\underline{{\mathcal{B}}}}_{{\mathcal{F}}}/{\underline{{\mathcal{X}}}}_{\mathcal{F}}$
of basic and tangent vector fields of ${\mathcal{F}}^{\sharp}$, cf. §2.3:

  1. -

Let $X$ be a section of the sheaf
${\underline{{\mathcal{B}}}}_{{\mathcal{F}}}/{\underline{{\mathcal{X}}}}_{\mathcal{F}}$
restricted to a connected open subset $V$ of an invariant irreducible
component of ${\mathcal{E}}_{\mathcal{F}}$. 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 $V$. 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
similarly.

Thanks to this property it suffices to show that at the point
$\phi^{\sharp}(p)$ the vector field germ
$Z_{\phi^{\sharp}(p)}-(D\phi^{\sharp}\cdot Z_{p})\circ\phi^{\sharp\,-1}$ is
tangent to ${\mathcal{F}}^{\sharp}$. Let us choose a simple path

$\gamma:[0,1]\to\\{z_{1}\neq
0\,,\,z_{2}=0\\}\,,\quad\gamma(0)=p\,,\quad\gamma(1)=\phi^{\sharp}(p)\,,$

and a germ at $p$ of holomorphic submersion
$I_{p}:(\Omega,p)\to({\mathbb{C}},0)$ constant on the leaves of
${\mathcal{F}}^{\sharp}$ whose restriction to $\\{z_{1}=\varepsilon\\}$ is
equal to $z_{2}$. Let us denote by
$I_{\phi^{\sharp}(p)}:(\Omega,\phi^{\sharp}(p))\to({\mathbb{C}},0)$ the
analytic extension of $I_{p}$ along $\gamma$, which coincides with its
extension as first integral of ${\mathcal{F}}^{\sharp}$. The vector field $Z$
being basic, the germ of holomorphic vector field on $({\mathbb{C}},0)$

$Z^{\flat}=z\frac{\partial\phantom{z}}{\partial
z}\quad\hbox{if}\quad\omega=\omega_{L}\,,\qquad\hbox{or}\qquad
Z^{\flat}=\frac{z^{bk}}{1+\zeta z^{bk}}z\frac{\partial\phantom{z}}{\partial
z}\quad\hbox{if}\quad\omega=\omega_{N}\,,$

satisfies the relations

$DI_{p}\cdot Z_{p}=Z^{\flat}\circ I_{p}\quad\hbox{and}\quad
DI_{\phi^{\sharp}(p)}\cdot Z_{\phi^{\sharp}(p)}=Z^{\flat}\circ
I_{\phi^{\sharp}(p)}\,,$ (14)

the second equality resulting from the first one by analytic extension. On the
other hand, the germ of $\phi^{\sharp}$ at $p$ factorizes through the first
integrals, inducing a biholomophism germ
$\phi^{\flat}:({\mathbb{C}},0)\to({\mathbb{C}},0)$ such that

$I_{\phi^{\sharp}(p)}\circ\phi^{\sharp}=\phi^{\flat}\circ I_{p}\,.$ (15)

Using the chain rule we have:

$\displaystyle DI_{\phi^{\sharp}(p)}\cdot$
$\displaystyle\big{(}(D\phi^{\sharp}\cdot
Z_{p})\circ\phi^{\sharp\,-1}\big{)}=\big{(}(DI_{\phi^{\sharp}(p)}\circ\phi^{\sharp})\cdot
D\phi^{\sharp}\cdot Z_{p}\big{)}\circ\phi^{\sharp\,-1}$
$\displaystyle=\big{(}D(I_{\phi^{\sharp}(p)}\circ\phi^{\sharp})\cdot
Z_{p}\big{)}\circ\phi^{\sharp\,-1}\stackrel{{\scriptstyle(\ref{defvarphiflat})}}{{=}}\big{(}D(\phi^{\flat}\circ
I_{p})\cdot Z_{p}\big{)}\circ\phi^{\sharp\,-1}$
$\displaystyle=\big{(}(D\phi^{\flat}\circ I_{p})\cdot DI_{p}\cdot
Z_{p}\big{)}\circ\phi^{\sharp\,-1}\stackrel{{\scriptstyle(\ref{DIZ})}}{{=}}\big{(}(D\phi^{\flat}\circ
I_{p})\cdot(Z^{\flat}\circ I_{p})\big{)}\circ\phi^{\sharp\,-1}$
$\displaystyle=(D\phi^{\flat}\cdot Z^{\flat})\circ
I_{p}\circ\phi^{\sharp\,-1}\,.$ (16)

Since $\phi^{\sharp}$ is defined on a neighborhood of the singular point $m$,
$\phi^{\flat}$ commutes with the biholomorphism of holonomy of
${\mathcal{F}}^{\sharp}$ along $D$ and around $m$. According to [8,
Proposition 6.10] (cases (L) and (R)) there is some $t_{1}\in{\mathbb{C}}$ and
a linear periodic map $\ell:{\mathbb{C}}\to{\mathbb{C}}$ such that:

$\phi^{\flat}=\ell\circ\exp(Z^{\flat})[t_{1}]\,,\quad\ell^{\ast}(Z^{\flat})=Z^{\flat}\,;$

therefore $\phi^{\flat}$ leaves $Z^{\flat}$ invariant. Hence the equality
$D\phi^{\flat}\cdot Z^{\flat}=Z^{\flat}\circ\phi^{\flat}$ holds and using it
we obtain

$\displaystyle DI_{\phi^{\sharp}(p)}\cdot\big{(}(D\phi^{\sharp}\cdot
Z_{p})\circ\phi^{\sharp\,-1}\big{)}$
$\displaystyle\stackrel{{\scriptstyle(\ref{DIZ2})}}{{=}}(D\phi^{\flat}\cdot
Z^{\flat})\circ I_{p}\circ\phi^{\sharp\,-1}=Z^{\flat}\circ\phi^{\flat}\circ
I_{p}\circ\phi^{\sharp\,-1}$
$\displaystyle\stackrel{{\scriptstyle(\ref{defvarphiflat})}}{{=}}Z^{\flat}\circ
I_{\phi^{\sharp}(p)}\circ\phi^{\sharp}\circ\phi^{\sharp\,-1}\stackrel{{\scriptstyle(\ref{DIZ})}}{{=}}DI_{\phi^{\sharp}(p)}\cdot
Z_{\phi^{\sharp}(p)}\,.$

We finally have the equality

$DI_{\phi^{\sharp}(p)}\cdot\big{(}Z_{\phi^{\sharp}(p)}-(D\phi^{\sharp}\cdot
Z_{p})\circ\phi^{\sharp\,-1}\big{)}=0\,,$

that shows that the vector field germ
$Z_{\phi^{\sharp}(p)}-(D\phi^{\sharp}\cdot Z_{p})\circ\phi^{\sharp\,-1}$ is
tangent to ${\mathcal{F}}^{\sharp}$. ∎

## References

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