# Holomorphic foliations of degree four on the complex projective space

Arturo Fernández-Pérez  Department of Mathematics. Federal University of Minas
Gerais. Av. Antônio Carlos, 6627 CEP 31270-901
Pampulha - Belo Horizonte - Brazil
<EMAIL_ADDRESS>and  Vângellis Sagnori Maia  Department of Mathematics.
Federal University of Minas Gerais. Av. Antônio Carlos, 6627 CEP 31270-901
Pampulha - Belo Horizonte - Brazil
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{F}$ be a degree four codimension one holomorphic foliation on
the complex projective space $\mathbb{P}^{n}$, $n\geq 3$. We prove that either
$\mathcal{F}$ admits a rational first integral, or $\mathcal{F}$ is
transversely affine outside a compact hypersurface, or $\mathcal{F}$ is pure
transversely projective outside a compact hypersurface, or $\mathcal{F}$ is
the pull-back of a foliation on $\mathbb{P}^{2}$ by a rational map.

###### Key words and phrases:

holomorphic foliations, rational first integral, transversely affine
structure, transversely projective structure, pull-back of foliations,
Godbillon-Vey sequences.

###### 2010 Mathematics Subject Classification:

Primary 32S65 - Secondary 37F75

The first-named author was partially supported by FAPERJ-Pronex
E-26/010.001270/2016 and CNPq PQ-302790/2019-5. The second-named was supported
by CAPES-Brazil.

##  1\. Introduction and Statement of the main result

The study of holomorphic foliations has attracted the attention of many
mathematicians and has been gaining more attention in recent decades. For
instance, it is known that foliations play an important role in the study of
subvarieties of projective varieties. One beautiful example is a result given
by Bogomolov [1] about the famous Green-Griffiths-Lang conjecture. Techniques
from algebraic geometry have been extremely useful in the study of singular
holomorphic foliations, for instance, J.-P. Jouanolou, in his celebrated
Lectures Notes [12] proved that a generic polynomial vector field of degree
greater than one on the complex projective plane does not admit any invariant
algebraic curve. Among the subjects that researchers of holomorphic foliations
investigate in the theory, much effort has been given to the problem of
classification of holomorphic foliations on $\mathbb{P}^{n}$, mostly
codimension one foliations. More specifically, an example of such interest is
the following conjecture is attributed to different authors such as Marco
Brunella, Alcides Lins Neto, among others:
Conjecture 1. Any codimension one holomorphic foliation $\mathcal{F}$ on
$\mathbb{P}^{n}$, with $n\geq 3$,

  * (*)

either $\mathcal{F}$ is transversely projective outside a compact
hypersurface;

  * (**)

or $\mathcal{F}$ is the pull-back of a holomorphic foliation $\mathcal{G}$ on
$\mathbb{P}^{2}$ by a rational map
$\Phi:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{2}$.

The concepts used in the previous conjecture will be explained throughout this
paper. We emphasize that any codimension one holomorphic foliation
$\mathcal{F}$ on $\mathbb{P}^{n}$ is associated to an homogeneous 1-form
$\omega$ on $\mathbb{C}^{n+1}$ of degree $d+1$ defining
$\pi^{*}(\mathcal{F})$, with singular set of codimension at least two, here
$\pi:\mathbb{C}^{n+1}\setminus\\{0\\}\to\mathbb{P}^{n}$ the natural
projection, and $\pi^{*}(\mathcal{F})$ is the pull-back of $\mathcal{F}$ by
$\pi$. By definition the integer $d$ is the degree of the foliation
$\mathcal{F}$. We remark that the integer $d$ is exactly the number of
tangencies of $\mathcal{F}$ with a generic line $L$. We then can consider the
space of codimension one holomorphic foliations on $\mathbb{P}^{n}$ of a
specific degree. The Zariski closure of this set can be identified with an
algebraic set and therefore naturally has irreducible components.

With a focus on explaining these irreducible components to a certain degree,
several works have been done and much progress has already been made. The
first case, which can be considered is the space of codimension one foliations
on $\mathbb{P}^{n}$ of degree-zero, has already been proved that it has only
one component and it is isomorphic to the Grassmannian of lines in
$\mathbb{P}^{n}$, a proof of this fact can be found in [10]. In
$\mathbb{P}^{n}$, $n\geq 3$, the space of holomorphic foliations of degree-one
has two irreducible components, this fact was proved by Jouanolou in [12].
Much later, Dominique Cerveau and Alcides Lins Neto resumed studies on
irreducible components and published a result that brought studies on
irreducible components back into the spotlight. In [5], they proved that in
$\mathbb{P}^{n}$, $n\geq 3$, the space of codimension one foliations of
degree-two has six irreducible components, but more than that, they explicitly
show the generic element of each one of these irreducible components. These
components are called Linear pull-back foliations, Rational components,
Logarithmic components, and an Exceptional component. Cerveau-Lins Neto’s
paper was an invaluable contribution and served as a motivation for several
researchers to focus their studies on the classification of irreducible
components of the space of holomorphic foliations on $\mathbb{P}^{n}$.
Recently, M. Corrêa [8] has proved that the space of foliations of degree-two
on $\mathbb{P}^{n}$ and dimension $k\geq 2$ also has six irreducible
components.

Despite the advances in these studies, the classification problem of
irreducible components of the space of holomorphic foliations is not simple to
be solved. For instance, after the complete description of the irreducible
components of the degree-two foliations space on $\mathbb{P}^{n}$, with $n\geq
3$, it has not yet been possible to explain all the irreducible components of
the space of codimension one holomorphic foliations on $\mathbb{P}^{n}$,
$n\geq 3$, of degree $d\geq 3$. However, Cerveau and Lins Neto have proved a
structural theorem for degree-three foliations in [6]. More precisely, they
proved that if $\mathcal{F}$ is a holomorphic codimension one foliation of
degree-three on $\mathbb{P}^{n}$, with $n\geq 3$, then

  * •

either $\mathcal{F}$ admits a rational first integral,

  * •

or $\mathcal{F}$ is transversely affine outside a compact hypersurface,

  * •

or $\mathcal{F}=\Phi^{*}(\mathcal{G})$, where
$\Phi:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{2}$ is a rational map and
$\mathcal{G}$ is a foliation on $\mathbb{P}^{2}$.

Recently in [15], F. Loray, J. V. Pereira and F. Touzet proved a more accurate
structural theorem for codimension one foliations of degree-three on
$\mathbb{P}^{3}$, and in [9], R. C. da Costa, R. Lizarbe and J. V. Pereira
extend this result to $\mathbb{P}^{n}$, $n\geq 3$. Furthermore, Da Costa,
Lizarbe and Pereira [9, Theorem B] provide a complete list of the irreducible
components of the space of foliations of degree-three on $\mathbb{P}^{n}$,
$n\geq 3$, whose general elements do not admit a rational first integral.
However, even in [9], a complete classification of the irreducible components
of the space of foliations of degree-three on $\mathbb{P}^{n}$, $n\geq 3$ is
not known, but they have established that the space of codimension one
foliations of degree-three on $\mathbb{P}^{n}$, $n\geq 3$, have at least 24
distinct irreducible components.

Motivated by the above results, this paper is devoted to prove a structural
theorem for degree-four foliations on $\mathbb{P}^{n}$, $n\geq 3$. Our main
result is the following:

######  Theorem A.

Let $\mathcal{F}$ be a codimension one holomorphic foliation of degree four on
$\mathbb{P}^{n}$, $n\geq 3$. Then

  * (i)

either $\mathcal{F}$ admits a rational first integral;

  * (ii)

or $\mathcal{F}$ is transversely affine outside a compact hypersurface;

  * (iii)

or $\mathcal{F}$ is a pure transversely projective outside a compact
hypersurface;

  * (iv)

or $\mathcal{F}=\Phi^{\ast}(\mathcal{G})$, where
$\Phi:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{2}$ is a rational map and
$\mathcal{G}$ is a holomorphic foliation on $\mathbb{P}^{2}$.

Note that our main result implies that the Conjecture 1 is true for degree-
four codimension one foliations on $\mathbb{P}^{n}$, $n\geq 3$. The main
difference between Theorem A and the structural theorem of foliations of
degree three given by Cerveau-Lins Neto [6] is that pure transversely
projective foliations exist. That is, there are foliations of degree four with
projective transverse structure that is not affine. See the example given in
[7, Section 5.3].

In order to prove our main result, we will use the tools of [6] and techniques
concerning foliations admitting a finite Godbillon-Vey sequence (cf. [2] and
[7]).

The paper is organized as follows: in Section 2, we define the concept of
holomorphic foliations and state some important results about codimension one
foliations on $\mathbb{P}^{n}$. We also define the concept of the affine and
projective transverse structure of a foliation. Moreover, we also define the
notion of a Godbillon-Vey sequence and establish important results of
foliations that admit a Godbillon-Vey sequence with finite length. Section 3
is devoted to prove Theorem A. We will divide it into several lemmas. Our
proof is given step to step. Each step is according to length of a Godbillon-
Vey sequence adapted to a degree-four foliation.

##  2\. Background and knows results

###  2.1. Codimension one holomorphic foliations

Let $M$ be a complex compact connected manifold of dimension $n\geq 2$. A
codimension one singular foliation $\mathcal{F}$ on $M$ is given by a covering
by open subset $\\{U_{i}\\}_{i\in I}$ on $M$ and a collection of integrable
holomorphic 1-forms $\omega_{i}$ on $U_{i}$ satisfying:

  1. (1)

$\omega_{i}\wedge d\omega_{i}=0$.

  2. (2)

The singular set $\mathrm{Sing}(\omega_{i})=\\{p\in U_{i}:\omega_{i}(p)=0\\}$
of $\omega_{i}$ is of codimension at least two.

  3. (3)

On each non-empty intersection $U_{i}\cap U_{j}$

$\displaystyle\omega_{i}=g_{ij}\omega_{j},\,\,\,\,\,\text{with}\,\,\,\,\,g_{ij}\in\mathcal{O}^{*}(U_{i}\cap
U_{j}).$

The condition (3) implies that the singular set of $\mathcal{\mathcal{F}}$
defined by $\mathrm{Sing}(\mathcal{F}):=\cup_{i\in
I}\mathrm{Sing}(\omega_{i})$ is a complex variety of $M$ of codimension at
least two.

In the special case where $M$ is a projective manifold and $\mathcal{F}$ a
foliation as above, we can associate to $\mathcal{F}$ a meromorphic 1-form
$\omega$ in the following way. We take a rational vector field $Z$ on $M$, not
tangent to $\mathcal{F}$, that is $h_{j}=i_{Z_{|U_{j}}}\omega\not\equiv 0$;
the meromorphic 1-form $\omega$ defined on $U_{j}$ by
$\omega_{|U_{j}}=\omega_{j}/h_{j}$ is global and integrable. In this case we
will say that $\omega$ defines $\mathcal{F}$.

We specialize to the case $M=\mathbb{P}^{n}$, the $n$ dimensional complex
projective space. In that context, we can define $\mathcal{F}$ as follows: let
$\pi:\mathbb{C}^{n+1}\setminus\\{0\\}\rightarrow\mathbb{P}^{n}$ be the natural
projection, and consider $\pi^{*}(\mathcal{F})$ the pull-back of $\mathcal{F}$
by $\pi$; with the previous notations, $\pi^{*}(\mathcal{F})$ is defined by
the 1-form $\pi^{*}(\omega_{j})$ on $\pi^{-1}(U_{j})$. Recall that, for $n\geq
2$, we have $H^{1}(\mathbb{C}^{n+1}\backslash\\{0\\},\mathcal{O}^{*})=\\{1\\}$
(cf. [4]). As a consequence, there exists a global holomorphic 1-form $\omega$
on $\mathbb{C}^{n+1}\backslash\\{0\\}$ which defines $\pi^{*}(\mathcal{F})$ on
$\mathbb{C}^{n+1}\backslash\\{0\\}$. By Hartog’s prolongation theorem $\omega$
can be extended holomorphically at $0$. By construction we have
$i_{R}\omega=0$, where $R$ is the Euler (or radial) vector field:

$R=\sum^{n}_{j=0}z_{j}\partial_{z_{j}}.$

This fact and the integrability condition imply that $\omega$ is co-linear to
an integrable homogeneous 1-form $\sum_{j=0}^{n}A_{j}(z)dz_{j}$, where the
$A_{j}$’s are homogeneous polynomials of degree $d+1$,
$gcd(A_{0},\ldots,A_{n})=1$. Therefore, we can state that any codimension one
holomorphic foliation $\mathcal{F}$ on $\mathbb{P}^{n}$ has associated an
homogeneous integrable 1-form $\omega$ on $\mathbb{C}^{n+1}$ defining
$\pi^{*}(\mathcal{F})$ with $cod(\mathrm{Sing}(\omega))\geq 2$.

By definition the integer $d$ is the degree of the foliation $\mathcal{F}$.
Observe that 1-form $\omega$ is well defined up to multiplication by non-zero
complex number.

Let $U_{i}:=\\{z_{i}=1\\}\subset\mathbb{P}^{n}$ be the usual affine charts
associated to the projective coordinates $(z_{0},\ldots,z_{n})$ so that
$\omega|_{U_{i}}$ is a polynomial 1-form on $U_{i}\cong\mathbb{C}^{n}$ which
can be extended meromorphically to $\mathbb{P}^{n}$. It is well known that the
integer $d=\deg(\mathcal{F})$ is exactly the number of tangencies of
$\mathcal{F}$ with a generic line $L$, that is the number of points $q\in L$
where $L$ is not transverse to $\mathcal{F}$. Therefore, a codimension one
foliation $\mathcal{F}$ on $\mathbb{P}^{n}$ of degree $d$ can be represented
in an affine coordinate system $\mathbb{C}^{n}\simeq E\subset\mathbb{P}^{n}$
by an integrable polynomial 1-form

$\omega_{E}=\displaystyle\sum_{j=0}^{d+1}\omega_{j}$

where the coefficients of the 1-forms $\omega_{j}$ are polynomials homogeneous
of degree $j$, $0\leq j\leq d+1$, and $i_{R}(\omega_{d+1})=0$. Given $p\in E$,
let $j_{p}^{k}(\omega_{E})$ be the $k^{th}$-jet of $\omega_{E}$ at $p$, and
let

$\mathcal{J}(\mathcal{F},p)=min\\{k\geq 0:j_{p}^{k}(\omega_{E})\neq 0\\}.$

Note that $\mathcal{J}(\mathcal{F},p)$ depends only on $p$ and $\mathcal{F}$
and not of $E$ and $\omega_{E}$. Moreover, the singular set of $\mathcal{F}$
is given by

$\mathrm{Sing}(\mathcal{F})=\\{p\in\mathbb{P}^{n}:\mathcal{J}(\mathcal{F},p)\geq
1\\}.$

It is well-known that $\mathrm{Sing}(\mathcal{F})$ is algebraic set and has
always irreducible components of codimension two, (cf. [13]).

######  Corollary 2.1 (Cerveau - Lins Neto [6]).

Let $\mathcal{F}$ be a codimension one holomorphic foliation on
$\mathbb{P}^{n}$, $n\geq 3$. If $\mathcal{J}(\mathcal{F},p)\leq
1,\;\forall\;p\in\mathbb{P}^{n}$, then $\mathcal{F}$ has a rational first
integral.

Let $\mathcal{F}$ be a codimension one holomorphic foliation on a projective
manifold $M$. Assume that $\omega$ is a meromorphic 1-form defining
$\mathcal{F}$. We say that $\mathcal{F}$ is transversely projective or
$\mathcal{F}$ has a transverse projective structure (cf. [16]) if there are
meromorphic 1-forms $\omega_{0}=\omega$, $\omega_{1}$ and $\omega_{2}$ on $M$
satisfying

$\left\\{\begin{array}[]{lll}d\omega_{0}=\omega_{0}\wedge\omega_{1}\\\
d\omega_{1}=\omega_{0}\wedge\omega_{2}\\\
d\omega_{2}=\omega_{1}\wedge\omega_{2}\end{array}\right.$

This means that, outside the polar and singular set of the 1-forms
$\omega_{i}$, the foliation $\mathcal{F}$ is regular and transversely
projective in the classical sense (see [11], or [7, Section 2.2]). When
$\omega_{2}\neq 0$, we say $\mathcal{F}$ is pure transversely projective. If
$\omega_{2}=0$, i.e. $d\omega_{1}=0$, we say that $\mathcal{F}$ is
transversely affine or $\mathcal{F}$ has a transverse affine structure.

To end this subsection, let us give some examples of foliations on
$\mathbb{P}^{n}$.

######  Example 2.2 (Foliations with rational first integral).

Let $P,Q:\mathbb{C}^{n+1}\rightarrow\mathbb{C}$ be a homogeneous polynomials
such that $\deg(P)=\deg(Q)=k\geq 1$. Then $\omega=QdP-PdQ$ defines a
codimension one foliation $\mathcal{F}$ on $\mathbb{P}^{n}$ with a rational
first integral $P/Q$.

######  Example 2.3 (Foliations associated to closed meromorphic 1-forms).

If $\omega$ is a closed meromorphic 1-form on $\mathbb{P}^{n}$, $n\geq 2$,
then it define a codimension one holomorphic foliation on $\mathbb{P}^{n}$. By
[14, Proposition. 1.2.5], we have $\omega$ has a decomposition

$\omega=\sum_{i}\lambda_{i}\frac{df_{i}}{f_{i}}+dh,$

where the $\lambda_{i}$’s are complex numbers and the $f_{i}$’s and $h$ are
rational functions. The leaves are (outside the singular set of the foliation)
the connected components of the level sets of the multi-valued function
$\sum_{i}\lambda_{i}\log f_{i}+h$.

###  2.2. Godbillon-Vey sequences [11]

Let $\mathcal{F}$ be a codimension one holomorphic foliation on a projective
manifold $M$. A Godbillon-Vey sequence for $\mathcal{F}$ (briefly G-V-S) is
sequence $(\omega_{0},\omega_{1},\ldots,\omega_{k},\ldots)$ of meromorphic
1-forms on $M$ such that $\mathcal{F}$ is defined by $\omega_{0}=0$ and the
formal 1-form

$\Omega=dz+\omega_{0}+z\omega_{1}+\frac{z^{2}}{2}\omega_{2}+\sum_{k\geq
3}^{\infty}\frac{z^{k}}{k!}\omega_{k},$ (1)

is integrable, that is, $\Omega\wedge d\Omega=0$. In this case, the 1-form in
(1) is meromorphic and can be extended meromorphically to
$M\times\mathbb{P}^{1}$. Since it is integrable, it defines a codimension one
foliation $\mathcal{H}$ on $M\times\mathbb{P}^{1}$ such that
$\mathcal{H}|_{M\times\\{0\\}}=\mathcal{F}$.

When there exist a meromorphic vector field $X$ on $M$ which is transversal to
$\mathcal{F}$ at a generic point, then there exist a unique meromorphic 1-form
$\omega$ defining $\mathcal{F}$ satisfying $i_{X}(\omega)=1$. According to [7,
Section 2.1], we can define a Godbillon-Vey sequence for $\mathcal{F}$ by
setting $\omega_{k}:=L_{X}^{k}(\omega)$, where $L_{X}^{k}(\omega)$ denotes the
$k^{th}$ Lie derivative along $X$ of the 1-form $\omega$.

When there exists $N\in\mathbb{N}$ such that $\omega_{N}\neq 0$ but
$\omega_{k}=0$ for all $j>N$ then we say that $\mathcal{F}$ admits a finite
G-V-S of length $N$. In general, the length is infinite. If $\mathcal{F}$
admits a G-V-S of length $N\leq 2$ then $\mathcal{F}$ is transversely
projective outside a compact hypersurface. When $N=1$ then $\mathcal{F}$ has a
transverse affine structure, see for instance [16] and [11].

For foliations that admit G-V-S of length $\geq 3$, we have the following
result:

######  Theorem 2.4 (Cerveau - Lins-Neto - Loray - Pereira - Touzet [7]).

Let $\mathcal{F}$ be a codimension one foliation on a complex manifold $M$
that admits a G-V-S of length $N\geq 3$. Then

  * •

either $\mathcal{F}$ is transversely affine;

  * •

or there is a compact Riemann surface $S$, 1-meromorphic forms
$\alpha_{0},\cdots,\alpha_{N}$ in $S$ and a rational map $\phi:M\rightarrow
S\times\mathbb{P}^{1}$ such that $\mathcal{F}$ is defined by the 1-form
$\Omega=\phi^{\ast}(\eta)$, where

$\eta=dz+\alpha_{0}+z\alpha_{1}+\cdots+z^{N}\alpha_{N}.$ (2)

When $M=\mathbb{P}^{n}$, $n\geq 3$, necessarily $S=\mathbb{P}^{1}$ and the
1-form in 2 can be written as

$\eta=dz-P(t,z)dt,$

where $P\in\mathbb{C}(t)[z]$ and $\mathcal{F}=\phi^{\ast}(\mathcal{G})$, where
$\mathcal{G}$ is defined in $\mathbb{P}^{1}\times\mathbb{P}^{1}$ by the
differential equation $\frac{dz}{dt}=P(t,z)$.

######  Remark 2.5.

Let $\mathcal{F}$ and $\mathcal{G}$ be codimension one foliations on complex
manifolds $X$ and $Y$, respectively. Suppose that $\mathcal{G}$ admits a
finite G-V-S of length $N$ and $\mathcal{F}=\Phi^{*}(\mathcal{G})$, where
$\Phi:X\dashrightarrow Y$ is a rational map. Then $\mathcal{F}$ also admits a
G-V-S of length $N$.

######  Remark 2.6.

Let $\mathcal{F}$ and $\mathcal{G}$ be codimension one foliations on complex
manifolds $X$ and $Y$, respectively. Suppose that
$\mathcal{F}=\Phi^{*}(\mathcal{G})$, where $\Phi:X\dashrightarrow Y$ is a
dominant meromorphic map. Then, $\mathcal{G}$ is transversely projective
(resp. affine) if, and only if, so is $\mathcal{F}$, (cf. [7, Theorem 2.21]).
In the case $\Phi$ is a finite ramified covering, the above statement is
equivalent to Theorem 1.6 (resp. Theorem 1.4) in [3].

##  3\. Proof of Theorem A

Let $\mathcal{F}$ be a codimension one holomorphic foliation of degree four in
$\mathbb{P}^{n}$, $n\geq 3$. In order to prove Theorem A, we consider two
possibilities:

  1. (1)

for all $p\in\mathrm{Sing}(\mathcal{F})$ such that
$\mathcal{J}(\mathcal{F},p)=1$,

  2. (2)

there exists $p\in\mathrm{Sing}(\mathcal{F})$ such that
$\mathcal{J}(\mathcal{F},p)\geq 2$.

In the first case, we have $\mathcal{F}$ admits a rational first integral by
Corollary 2.1. Hence we have the item $(i)$ of Theorem A.

Therefore, we shall assume that there exists a point $p\in\mathbb{P}^{n}$ such
that $\mathcal{J}(\mathcal{F},p)\geq 2$. Taking affine coordinates
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\subset\mathbb{P}^{n}$, where
$p=0\in\mathbb{C}^{n}$, we can assume that
$\mathcal{F}|_{\mathbb{C}^{n}}:\omega=0$, where $\omega$ is a polynomial
1-form in $\mathbb{C}^{n}$ given by

$\omega=\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5},$ (3)

where $\alpha_{j}$ are homogeneous polynomial 1-forms of degree $j$, $2\leq
j\leq 5$, and

$\displaystyle i_{R}(\alpha_{5})=0,\quad\quad\text{with}\quad
R=\sum_{i=1}^{n}z_{i}\partial z_{i}.$ (4)

We now study the pull-back of $\omega$ by the blowing-up of $\mathbb{P}^{n}$
at $0\in\mathbb{C}^{n}\subset\mathbb{P}^{n}$. Denote by
$\sigma:\tilde{\mathbb{P}}^{n}\rightarrow\mathbb{P}^{n}$ be the blow-up at
$0\in\mathbb{C}^{n}\subset\mathbb{P}^{n}$ and by $\tilde{\mathcal{F}}$ the
strict transform of $\mathcal{F}$ by $\sigma$. Let us compute
$\sigma^{\ast}(\omega)$ in the chart

$\displaystyle(\tau_{1},\ldots,\tau_{n-1},x)=(\tau,x)\in\mathbb{C}^{n-1}\times\mathbb{C}\mapsto(x\tau,x)=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\subset\mathbb{P}^{n}.$

Let us write $\displaystyle\alpha_{j}(z)=\sum_{i=1}^{n}P_{ji}(z)dz_{i}$, with
$2\leq j\leq 5$ and

$F_{j}(\tau,1)=\left(\displaystyle\sum_{i=1}^{n-1}P_{j-1i}(\tau,1)\tau_{i}\right)+P_{j-1n}(\tau,1),\quad
3\leq j\leq 6,$

where $P_{ji}$ are homogeneous polynomials of degree $j$. Then

$\displaystyle\sigma^{\ast}(\omega)$ $\displaystyle=$
$\displaystyle\sigma^{\ast}(\alpha_{2})+\sigma^{\ast}(\alpha_{3})+\sigma^{\ast}(\alpha_{4})+\sigma^{\ast}(\alpha_{5})$
$\displaystyle=$
$\displaystyle(x^{3}\theta_{2}+x^{2}F_{3}(\tau,1)dx)+(x^{4}\theta_{3}+x^{3}F_{4}(\tau,1)dx)+(x^{5}\theta_{4}+x^{4}F_{5}(\tau,1)dx)+$
$\displaystyle(x^{6}\theta_{5}+x^{5}F_{6}(\tau,1)dx)$ $\displaystyle=$
$\displaystyle
x^{2}\left(x\theta_{2}+x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+(F_{3}(\tau,1)+xF_{4}(\tau,1)+x^{2}F_{5}(\tau,1))dx\right)$

where

$\displaystyle\theta_{j}=\sum_{i=1}^{n-1}P_{ji}(\tau,1)d\tau_{i},\quad\quad
2\leq j\leq 5$

depends only $\tau$. Note that $F_{6}(\tau,1)=i_{R}(\alpha_{5})(\tau,1)=0$ by
(4). Therefore, the 1-form

$\eta=x\theta_{2}+x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+(F_{3}(\tau,1)+xF_{4}(\tau,1)+x^{2}F_{5}(\tau,1))dx$
(5)

defines the foliation $\tilde{\mathcal{F}}$ in the chart $(\tau,x)$.

Now, we will use the following notation $F_{i}:=F_{i}(\tau,1)$. Under the
conditions described above we have the following possibilities for $F_{i}$:

  * (1)

$F_{3}\equiv F_{4}\equiv F_{5}\equiv 0$;

  * (2)

$F_{4}\equiv F_{5}\equiv 0$ and $F_{3}\not\equiv 0$;

  * (3)

$F_{3}\equiv F_{4}\equiv 0$ and $F_{5}\not\equiv 0$;

  * (4)

Possibilities solved in an analogous way:

(a) $F_{3}\equiv F_{5}\equiv 0$ and $F_{4}\not\equiv 0$;

(b) $F_{3}\equiv 0$ and $F_{4}\not\equiv 0\not\equiv F_{5}$;

  * (5)

$F_{5}\equiv 0$ and $F_{3}\not\equiv 0\not\equiv F_{4}$;

  * (6)

$F_{4}\equiv 0$ and $F_{3}\not\equiv 0\not\equiv F_{5}$;

  * (7)

$F_{3}\not\equiv 0$, $F_{4}\not\equiv 0$ and $F_{5}\not\equiv 0$.

We will explore these possibilities through the following lemmas considering
$\omega$ and $\mathcal{F}$ under the conditions described above unless
otherwise noted.

######  Lemma 3.1 (Case 1).

If $F_{3}\equiv F_{4}\equiv F_{5}\equiv 0$. Then $\mathcal{F}$ is a pull-back
of a foliation on $\mathbb{P}^{n-1}$ by a linear map.

###### Proof.

Since
$i_{R}(\omega)(z)=i_{R}(\omega)(x\tau,x)=x^{3}F_{3}(\tau,1)+x^{4}F_{4}(\tau,1)+x^{5}F_{5}(\tau,1),$
the assumption $F_{3}\equiv F_{4}\equiv F_{5}\equiv 0$ implies that
$i_{R}(\omega)=0$. It follows from (3) that $\alpha_{5}\not\equiv 0$,
otherwise $\mathcal{F}$ would be not of degree four. On the other hand, the
integrability of $\omega$ implies that

$\displaystyle\omega\wedge d\omega=0$ $\displaystyle\Rightarrow
i_{R}(\omega\wedge d\omega)=0$ $\displaystyle\Rightarrow i_{R}(\omega)\wedge
d\omega-\omega\wedge i_{R}(d\omega)=0.$

Since $i_{R}(\omega)=0$ we have

$\omega\wedge i_{R}(d\omega)=0.$ (6)

Up applying Cartan’s magic formula, we get

$\displaystyle L_{R}(\omega)$ $\displaystyle=$ $\displaystyle
di_{R}(\omega)+i_{R}(d\omega)$ $\displaystyle L_{R}(\omega)$ $\displaystyle=$
$\displaystyle i_{R}(d\omega)$ (7)

and applying Euler’s formula to $L_{R}(\omega)$ yields

$\displaystyle L_{R}(\omega)$ $\displaystyle=$ $\displaystyle
L_{R}(\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5})$ $\displaystyle=$
$\displaystyle
L_{R}(\alpha_{2})+L_{R}(\alpha_{3})+L_{R}(\alpha_{4})+L_{R}(\alpha_{5})$
$\displaystyle=$ $\displaystyle
3\alpha_{2}+4\alpha_{3}+5\alpha_{4}+6\alpha_{5}.$

It follows from (3) that
$i_{R}(d\omega)=3\alpha_{2}+4\alpha_{3}+5\alpha_{4}+6\alpha_{5}$ and from (6),
we deduce

$\displaystyle 0=\omega\wedge i_{R}(d\omega)$ $\displaystyle=$ $\displaystyle
4\alpha_{2}\wedge\alpha_{3}+5\alpha_{2}\wedge\alpha_{4}+6\alpha_{2}\wedge\alpha_{5}+$
$\displaystyle
3\alpha_{3}\wedge\alpha_{2}+5\alpha_{3}\wedge\alpha_{4}+6\alpha_{3}\wedge\alpha_{5}+$
$\displaystyle
3\alpha_{4}\wedge\alpha_{2}+4\alpha_{4}\wedge\alpha_{3}+6\alpha_{4}\wedge\alpha_{5}+$
$\displaystyle
3\alpha_{5}\wedge\alpha_{2}+4\alpha_{5}\wedge\alpha_{3}+5\alpha_{5}\wedge\alpha_{4}$
$\displaystyle=$
$\displaystyle\alpha_{2}\wedge\alpha_{3}+2\alpha_{2}\wedge\alpha_{4}+3\alpha_{2}\wedge\alpha_{5}+$
$\displaystyle\alpha_{3}\wedge\alpha_{4}+2\alpha_{3}\wedge\alpha_{5}+\alpha_{4}\wedge\alpha_{5}$

Since the coefficients of $\alpha_{j}$ are homogeneous polynomials of degree
$j$, each of these exterior products generates a homogeneous 2-form polynomial
of a degree different from each other, so none of them can be a combination of
the others, we obtain

$\alpha_{2}\wedge\alpha_{3}=\alpha_{2}\wedge\alpha_{4}=\alpha_{2}\wedge\alpha_{5}=\alpha_{3}\wedge\alpha_{4}=\alpha_{3}\wedge\alpha_{5}=\alpha_{4}\wedge\alpha_{5}=0.$

Since $\alpha_{5}\not\equiv 0$ and
$\alpha_{2}\wedge\alpha_{5}=\alpha_{3}\wedge\alpha_{5}=\alpha_{4}\wedge\alpha_{5}=0$,
we have that there exists meromorphic functions $f_{j}$, $j=2,3,4$, such that
$\alpha_{j}=f_{j}\alpha_{5}$. We assert that $f_{j}\equiv 0$ for all
$j=2,3,4$. Indeed, suppose by contradiction that some $f_{j}\not\equiv 0$.
Then

$\omega=f_{2}\alpha_{5}+f_{3}\alpha_{5}+f_{4}\alpha_{5}+\alpha_{5}=(f_{2}+f_{3}+f_{4}+1)\alpha_{5},$

we have that the coefficients of $(f_{2}+f_{3}+f_{4}+1)\alpha_{5}$ would be of
a degree different than 5, an absurd. Hence, the assertion is proved.

Therefore, all $f_{j}=0$ and hence $\alpha_{j}=0$, for all $j=2,3,4$. In
particular, we have $\omega=\alpha_{5}$. Since $\alpha_{5}$ is integrable, it
defines a foliation of degree four, say $\mathcal{F}_{n-1}$, on
$\mathbb{P}^{n-1}$. If we consider $\mathbb{P}^{n-1}$ as the set of lines
through $0\in\mathbb{C}^{n}\subset\mathbb{P}^{n}$ and
$\Phi:\mathbb{C}^{n}\setminus\\{0\\}\to\mathbb{P}^{n-1}$ the natural
projection then $\mathcal{F}=\Phi^{*}(\mathcal{F}_{n-1})$. This finishes the
proof of lemma. ∎

######  Lemma 3.2 (Case 2).

Suppose that $F_{4}\equiv F_{5}\equiv 0$ and $F_{3}\not\equiv 0$. Then, either
$\mathcal{F}$ is transversely affine, or $\mathcal{F}$ is a pull-back by a
rational map of a foliation on $\mathbb{P}^{2}$.

###### Proof.

Let $\displaystyle\beta_{j}=\frac{\theta_{j+1}}{F_{3}}$, for $1\leq j\leq 4$.
It follows from (5) that

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
x\theta_{2}+x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+F_{3}dx$
$\displaystyle\tilde{\eta}$ $\displaystyle=$
$\displaystyle\frac{\eta}{F_{3}}=x\beta_{1}+x^{2}\beta_{2}+x^{3}\beta_{3}+x^{4}\beta_{4}+dx.$
(8)

We have that $\tilde{\eta}$ defines $\tilde{\mathcal{F}}$. Note that
$\beta_{j}$ only depends on $\tau$, for all $1\leq j\leq 4$, because
$\theta_{j}$ and $F_{3}$ only depend on $\tau$. Then, $\tilde{\eta}$ admits
the G-V-S

$(\eta_{0}=\tilde{\eta},\eta_{1},\eta_{2},\eta_{3},\eta_{4}),$

where $\tilde{\eta}(\partial_{x})=1$,
$\eta_{j}=L_{\partial_{x}}^{j}(\tilde{\eta})$, $1\leq j\leq 4$. Here
$L_{\partial_{x}}^{j}(\tilde{\eta})$ denotes the $j^{th}$ Lie derivative along
$\partial_{x}$ of the form $\tilde{\eta}$, see for instance [7, Section 2.1].
We have two subcases:
Subcase 1. ($\beta_{4}\equiv 0$ and $\beta_{3}\not\equiv 0$) or
($\beta_{4}\not\equiv 0$). In this subcase, $\mathcal{F}$ admits a finite
G-V-S of length $\geq 3$. Therefore, either $\mathcal{F}$ is transversely
affine, or $\mathcal{F}$ is a pull-back by a rational map of a foliation on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ by Theorem 2.4. Since
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ is birrational to $\mathbb{P}^{2}$, we
get that $\mathcal{F}$ is a pull-back of a foliation on $\mathbb{P}^{2}$ by a
rational map.
Subcase 2. $\beta_{4}\equiv\beta_{3}\equiv 0$. In this case, $\mathcal{F}$
admits a finite G-V-S of length $\leq 2$, where

$\tilde{\eta}=dx+x\beta_{1}+x^{2}\beta_{2}$

Taking the change of coordinates $x=\frac{1}{w}$, we get
$\tilde{\eta}=-\frac{\Omega}{w^{2}}$, where $\Omega=dw-\beta_{2}-w\beta_{1}.$
We assert that $\beta_{1}$ is closed, that is, $d\beta_{1}=0$. Indeed, from
the integrability condition of $\Omega$, we have

$\displaystyle 0=\Omega\wedge d\Omega$ $\displaystyle=$
$\displaystyle(dw-\beta_{2}-w\beta_{1})\wedge(-d\beta_{2}-dw\wedge\beta_{1}-wd\beta_{1})$
(9) $\displaystyle=$ $\displaystyle-dw\wedge d\beta_{2}-wdw\wedge
d\beta_{1}+\beta_{2}\wedge d\beta_{2}-\beta_{2}\wedge\beta_{1}\wedge dw+$
$\displaystyle w\beta_{2}\wedge d\beta_{1}+w\beta_{1}\wedge
d\beta_{2}+w^{2}\beta_{1}\wedge d\beta_{1}$

Using $2^{th}$ Lie derivative $L_{\partial_{w}}^{2}(\Omega\wedge
d\Omega)=2\beta_{1}\wedge d\beta_{1}=0\Rightarrow w^{2}\beta_{1}\wedge
d\beta_{1}=0$. Again using Lie derivative $L_{\partial_{w}}$ in (9), we get

$\displaystyle L_{\partial_{w}}(\Omega\wedge d\Omega)=-dw\wedge
d\beta_{1}+\beta_{2}\wedge d\beta_{1}+\beta_{1}\wedge d\beta_{2}=0$ (10)

From (10), we obtain that $dw\wedge d\beta_{1}=0$, because $\beta_{2}\wedge
d\beta_{1}+\beta_{1}\wedge d\beta_{2}$ does not depend on $dw$. Since $dw\neq
0$, and $d\beta_{1}$ does not depend on $w$, we deduce $d\beta_{1}=0$ and the
assertion is proved. Differentiating $\Omega$, we have

$\displaystyle d\Omega=$ $\displaystyle-d\beta_{2}-dw\wedge\beta_{1}$ (11)

and thus

$\displaystyle\beta_{1}\wedge\Omega=\beta_{1}\wedge
dw-\beta_{1}\wedge\beta_{2}.$ (12)

Let us show that equations $(\ref{eq8})$ and $(\ref{eq9})$ are equals, which
implies that $\Omega$ has a transversely affine structure, and therefore
$\mathcal{F}$ also has. To show the equality is sufficient to prove that

$d\beta_{2}=\beta_{1}\wedge\beta_{2}.$

It follows from (10) that $\beta_{2}\wedge d\beta_{1}+\beta_{1}\wedge
d\beta_{2}=0\Rightarrow w\beta_{2}\wedge d\beta_{1}+w\beta_{1}\wedge
d\beta_{2}=0$. Then we summarize $\Omega\wedge d\Omega$ in

$\Omega\wedge d\Omega=-dw\wedge d\beta_{2}+\beta_{2}\wedge
d\beta_{2}-\beta_{2}\wedge\beta_{1}\wedge dw=0$

of the above equal $\beta_{2}\wedge d\beta_{2}=0$, because the other parcels
depend on $dw$. Therefore

$\displaystyle(-d\beta_{2}-\beta_{2}\wedge\beta_{1})\wedge dw=0\Rightarrow
d\beta_{2}+\beta_{2}\wedge\beta_{1}=0\Rightarrow
d\beta_{2}=\beta_{1}\wedge\beta_{2}.$

As consequence from (11) and (12), we get

$d\Omega=\beta_{1}\wedge\Omega.$

This implies that $\tilde{\eta}$ has a transversely affine structure and thus
$\mathcal{F}$ also has. ∎

######  Lemma 3.3 (Case 3).

Suppose that $F_{4}\equiv F_{3}\equiv 0$ and $F_{5}\not\equiv 0$. Then either
$\mathcal{F}$ is transversely affine, or $\mathcal{F}$ is a pull-back by a
rational map of a foliation on $\mathbb{P}^{2}$, or $\mathcal{F}$ is pure
transversely projective.

###### Proof.

It follows from (5) that

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
x\theta_{2}+x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+x^{2}F_{5}dx$ (13)
$\displaystyle\tilde{\eta}$ $\displaystyle=$
$\displaystyle\frac{\eta}{xF_{5}}=\frac{\theta_{2}}{F_{5}}+\frac{x\theta_{3}}{F_{5}}+\frac{x^{2}\theta_{4}}{F_{5}}+\frac{x^{3}\theta_{5}}{F_{5}}+xdx$

Consider the birrational map $\psi(\tau,z)=(\tau,\frac{1}{z})=(\tau,x)$, we
get

$\displaystyle\psi^{\ast}(\tilde{\eta})$ $\displaystyle=$
$\displaystyle\Bigg{[}\frac{\theta_{2}}{F_{5}}+\frac{\theta_{3}}{zF_{5}}+\frac{\theta_{4}}{z^{2}F_{5}}+\frac{\theta_{5}}{z^{3}F_{5}}-\frac{1}{z^{3}}dz\Bigg{]}$
$\displaystyle\psi^{\ast}(\tilde{\eta})$ $\displaystyle=$
$\displaystyle-\frac{1}{z^{3}}\Bigg{[}-\frac{z^{3}\theta_{2}}{F_{5}}-\frac{z^{2}\theta_{3}}{F_{5}}-\frac{z\theta_{4}}{F_{5}}-\frac{\theta_{5}}{F_{5}}+dz\Bigg{]}$
$\displaystyle-z^{3}\psi^{\ast}(\tilde{\eta})$ $\displaystyle=$
$\displaystyle\Bigg{[}z^{3}\left(\frac{-\theta_{2}}{F_{5}}\right)+z^{2}\left(\frac{-\theta_{3}}{F_{5}}\right)+z\left(\frac{-\theta_{4}}{F_{5}}\right)+\left(\frac{-\theta_{5}}{F_{5}}\right)+dz\Bigg{]}$
$\displaystyle\zeta$ $\displaystyle:=$
$\displaystyle-z^{3}\psi^{\ast}(\tilde{\eta})=\beta_{0}+z\beta_{1}+z^{2}\beta_{2}+z^{3}\beta_{3}+dz,$

where $\displaystyle\beta_{0}=-\frac{\theta_{5}}{F_{5}}$,
$\displaystyle\beta_{1}=-\frac{\theta_{4}}{F_{5}}$,
$\displaystyle\beta_{2}=-\frac{\theta_{3}}{F_{5}}$ and
$\displaystyle\beta_{3}=-\frac{\theta_{2}}{F_{5}}$. Note that $\zeta$ admits a
finite G-V-S of length $\geq 3$ if $\beta_{3}\not\equiv 0$. Therefore
$\mathcal{F}$ also has. Hence, in this case, we have that either $\mathcal{F}$
is transversely affine, or $\mathcal{F}$ is a pull-back by a rational map of a
foliation on $\mathbb{P}^{2}$, by Theorem 2.4.

On the other hand, suppose that $\beta_{3}\equiv 0$, that is,
$\theta_{2}\equiv 0$. It follows from (13) that

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+x^{2}F_{5}dx$
$\displaystyle\tilde{\eta}$ $\displaystyle=$
$\displaystyle\frac{\eta}{x^{2}F_{5}}=\frac{\theta_{3}}{F_{5}}+x\frac{\theta_{4}}{F_{5}}+x^{2}\frac{\theta_{5}}{F_{5}}+dx$

thus

$\displaystyle\tilde{\eta}=\beta_{0}+x\beta_{1}+x^{2}\beta_{2}+dx$ (14)

where $\displaystyle\beta_{j}=\frac{\theta_{j+3}}{F_{5}}$ for all $0\leq j\leq
2$. Since $i_{\partial_{w}}(\beta_{j})=0$ and $L_{\partial_{w}}(\beta_{j})=0$,
for all $0\leq j\leq 2$, the integrability of $\tilde{\eta}$ implies

$\left\\{\begin{array}[]{lll}d\beta_{0}=\beta_{0}\wedge\beta_{1}\\\
d\beta_{1}=2\beta_{0}\wedge\beta_{2}\\\
d\beta_{2}=\beta_{1}\wedge\beta_{2}\end{array}\right.$

We shall now show $(\eta,\Omega,\xi)$ which defines a projective transverse
structure for $\eta$ satisfying

$\left\\{\begin{array}[]{lll}d\eta=\Omega\wedge\eta\\\
d\Omega=\eta\wedge\xi\\\ d\xi=\xi\wedge\Omega\end{array}\right.$

By taking $\Omega=-L_{\partial_{x}}(\eta)=-\beta_{1}-2x\beta_{2}$, we have

$\displaystyle\Omega\wedge\eta$ $\displaystyle=$
$\displaystyle(-\beta_{1}-2x\beta_{2})\wedge(\beta_{0}+x\beta_{1}+x^{2}\beta_{2}+dx)$
(15) $\displaystyle=$
$\displaystyle-\beta_{1}\wedge\beta_{0}-x^{2}\beta_{1}\wedge\beta_{2}-\beta_{1}\wedge
dx-2x\beta_{2}\wedge\beta_{0}-2x^{2}\beta_{2}\wedge\beta_{1}-2x\beta_{2}\wedge
dx$ $\displaystyle=$ $\displaystyle d\beta_{0}-x^{2}d\beta_{2}-\beta_{1}\wedge
dx+xd\beta_{1}+2x^{2}d\beta_{2}-2x\beta_{2}\wedge dx$ $\displaystyle=$
$\displaystyle
d\beta_{0}+dx\wedge\beta_{1}+xd\beta_{1}+2xdx\wedge\beta_{2}+x^{2}d\beta_{2}$
$\displaystyle=$ $\displaystyle d\eta.$

On the other hand, by calculating $d\Omega$, we have

$\displaystyle d\Omega$ $\displaystyle=$
$\displaystyle-d\beta_{1}-2dx\wedge\beta_{2}-2xd\beta_{2}$ (16)
$\displaystyle=$
$\displaystyle-2\beta_{0}\wedge\beta_{2}-2dx\wedge\beta_{2}-2x\beta_{1}\wedge\beta_{2}$
$\displaystyle=$
$\displaystyle\beta_{0}\wedge(-2\beta_{2})+dx\wedge(-2\beta_{2})+x\beta_{1}\wedge(-2\beta_{2})$
$\displaystyle=$ $\displaystyle(\beta_{0}+dx+x\beta_{1})\wedge(-2\beta_{2})$
$\displaystyle=$
$\displaystyle(\beta_{0}+dx+x\beta_{1}+x^{2}\beta_{2})\wedge(-2\beta_{2})$
$\displaystyle=$ $\displaystyle\eta\wedge(-2\beta_{2}).$

Now, we take $\xi=-L_{\partial_{x}}^{2}(\eta)=-2\beta_{2}$, which satisfies
the following equality

$\displaystyle d\xi$ $\displaystyle=$
$\displaystyle-2d\beta_{2}=-2\beta_{1}\wedge\beta_{2}$ (17) $\displaystyle=$
$\displaystyle(-2\beta_{2})\wedge(-\beta_{1})$ $\displaystyle=$
$\displaystyle(-2\beta_{2})\wedge(-\beta_{1}-2x\beta_{2})$ $\displaystyle=$
$\displaystyle\xi\wedge\Omega.$

Note that $\Omega$ is closed if, and only if $\beta_{2}\equiv 0$ which,
consequently, we will have $\xi\equiv 0$ and the transverse structure of
$\tilde{\eta}$ will be affine. On the other hand, if $\beta_{2}\not\equiv 0$,
then $\tilde{\eta}$ has a pure transversely projective structure. Since
$\tilde{\mathcal{F}}$ is defined by $\tilde{\eta}$, we deduce that either
$\mathcal{F}$ is transversely affine, or $\mathcal{F}$ is pure transversely
projective. ∎

######  Lemma 3.4 (Case 4).

Suppose that $F_{3}\equiv 0$ and $F_{4}\not\equiv 0$. Then, either
$\mathcal{F}$ is transversely affine, or $\mathcal{F}$ is a pull-back by a
rational map of a foliation on $\mathbb{P}^{2}$.

###### Proof.

It follows from (5) that
$\eta=x\theta_{2}+x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+(xF_{4}+x^{2}F_{5})dx$.
We will divide the proof in two subcases and in each of them we shall get a
1-form similar to

$\tilde{\eta}=\beta_{0}+x\beta_{1}+x^{2}\beta_{2}+dx$ (18)

satisfying $L_{\partial_{x}}(\beta_{j})=0$ and $i_{\partial_{x}}(\beta_{j})=0$
for all $0\leq j\leq 2$. With this in hand, we shall thus obtain some of the
expected results.
Subcase 1: Suppose that $F_{5}\equiv 0$. Let
$\tilde{\eta}=\frac{\eta}{xF_{4}}$, then

$\displaystyle\tilde{\eta}$ $\displaystyle=$
$\displaystyle\frac{\theta_{2}}{F_{4}}+x\frac{\theta_{3}}{F_{4}}+x^{2}\frac{\theta_{4}}{F_{4}}+x^{3}\frac{\theta_{5}}{F_{4}}+dx$
$\displaystyle\tilde{\eta}$ $\displaystyle=$
$\displaystyle\beta_{0}+x\beta_{1}+x^{2}\beta_{2}+x^{3}\beta_{3}+dx$ (19)

where $\beta_{j}=\frac{\theta_{j+2}}{F_{4}}$, $0\leq j\leq 3$. We can assume
that $\beta_{3}\equiv 0$, because otherwise we would get a finite G-V-S of
length $\geq 3$, and in this case, by applying Theorem 2.4, either
$\mathcal{F}$ is transversely affine, or it is a pull-back by a rational map
of a foliation on $\mathbb{P}^{2}$.
Subcase 2: Suppose that $F_{5}\not\equiv 0$. Let
$\tilde{\eta}=\frac{\eta}{x}$, then

$\tilde{\eta}=\theta_{2}+x\theta_{3}+x^{2}\theta_{4}+x^{3}\theta_{5}+(F_{4}+xF_{5})dx$

Consider the birrational map
$\psi(\tau,z)=\left(\tau,\frac{\frac{F_{4}(\tau,1)}{F_{5}(\tau,1)}z}{1-z}\right)=(\tau,x)$
with inverse
$\psi^{-1}(\tau,x)=\left(\tau,\frac{x}{x+\frac{F_{4}(\tau,1)}{F_{5}(\tau,1)}}\right)$.
A straightforward computation gives
$\psi^{\ast}(\tilde{\eta})=\frac{F_{4}^{2}}{F_{5}(1-z)^{3}}\bar{\eta}$, where

$\bar{\eta}=\beta_{0}+z\beta_{1}+z^{2}\beta_{2}+z^{3}\beta_{3}+dz$

with

$\displaystyle\beta_{0}$ $\displaystyle=$
$\displaystyle\frac{F_{5}\theta_{2}}{F_{4}^{2}}$ $\displaystyle\beta_{1}$
$\displaystyle=$
$\displaystyle-3\frac{F_{5}\theta_{2}}{F_{4}^{2}}+\frac{\theta_{3}}{F_{4}}+\frac{dF_{4}}{F_{4}}-\frac{dF_{5}}{F_{5}}$
$\displaystyle\beta_{2}$ $\displaystyle=$ $\displaystyle
3\frac{F_{5}\theta_{2}}{F_{4}^{2}}-2\frac{\theta_{3}}{F_{4}}+\frac{\theta_{4}}{F_{5}}-\frac{dF_{4}}{F_{4}}+\frac{dF_{5}}{F_{5}}$
$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle-\frac{F_{5}\theta_{2}}{F_{4}^{2}}+\frac{\theta_{3}}{F_{4}}-\frac{\theta_{4}}{F_{5}}+\frac{F_{4}\theta_{5}}{F_{5}^{2}}$

As in the previous subcase we can assume that $\beta_{3}\equiv 0$ by Theorem
2.4.

We will now work both cases at the same time as they both fell into the form
(19). Remembering that in the initial blow-up we have
$\pi^{\ast}(\alpha_{2})=x^{2}[x\theta_{2}+F_{3}(\tau,1)dx]=x^{3}\theta_{2}$.
When $\alpha_{2}\equiv 0$ we have $\beta_{0}\equiv 0$ and
$\eta=x\beta_{1}+x^{2}\beta_{2}+dx$ where we fall into the Lemma 3.2, so let
us assume $\alpha_{2}\not\equiv 0$. The integrability of
$\omega=\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}$ gives us

$\displaystyle 0=\omega\wedge d\omega$ $\displaystyle=$
$\displaystyle\alpha_{2}\wedge d\alpha_{2}+\alpha_{2}\wedge
d\alpha_{3}+\alpha_{2}\wedge d\alpha_{4}+\alpha_{2}\wedge d\alpha_{5}+$ (20)
$\displaystyle\alpha_{3}\wedge d\alpha_{2}+\alpha_{3}\wedge
d\alpha_{3}+\alpha_{3}\wedge d\alpha_{4}+\alpha_{3}\wedge d\alpha_{5}+$
$\displaystyle\alpha_{4}\wedge d\alpha_{2}+\alpha_{4}\wedge
d\alpha_{3}+\alpha_{4}\wedge d\alpha_{4}+\alpha_{4}\wedge d\alpha_{5}+$
$\displaystyle\alpha_{5}\wedge d\alpha_{2}+\alpha_{5}\wedge
d\alpha_{3}+\alpha_{5}\wedge d\alpha_{4}+\alpha_{5}\wedge d\alpha_{5}$

The coefficients of $\alpha_{j}$ are homogeneous polynomials of degree $j$,
then the coefficients of $\alpha_{2}\wedge d\alpha_{2}$ are homogeneous
polynomials of degree $3$ and none of the other 2-forms in the other parcels
have coefficients of degree 3, then $\alpha_{2}\wedge d\alpha_{2}=0$. In
particular, either $cod(Sing(\alpha_{2}))\geq 2$ and $\alpha_{2}$ define a
degree one foliation in $\mathbb{P}^{n-1}$; or $\alpha_{2}=h\alpha_{1}$, where
$\alpha_{1}$ defines a degree zero foliation in $\mathbb{P}^{n-1}$
($\alpha_{2}=h\alpha_{1}$, in this case $cod(Sing(\alpha_{2}))<2$, and the
foliation needs to be saturated and therefore what is left is a 1-form
polynomial of degree 1). In both cases $\alpha_{2}$ has an integrating factor,
that is, there exists a function $f$ such that $d(f^{-1}\alpha_{2})=0$.

In Subcase 1, we have

$\displaystyle\pi^{\ast}\Bigg{(}\frac{\alpha_{2}}{f}\Bigg{)}=\frac{\theta_{2}}{f(\tau,1)}\Rightarrow
d\Bigg{(}\frac{\theta_{2}}{f(\tau,1)}\Bigg{)}=0\Rightarrow
d\Bigg{(}\frac{F_{4}(\tau,1)}{f(\tau,1)}\beta_{0}\Bigg{)}=0$ (21)

We put $\displaystyle F_{1}(\tau):=\frac{f(\tau,1)}{F_{4}(\tau,1)}$ for this
case.

In Subcase 2, we have

$\displaystyle\pi^{\ast}\Bigg{(}\frac{\alpha_{2}}{f}\Bigg{)}=\frac{\theta_{2}}{f(\tau,1)}\Rightarrow
d\Bigg{(}\frac{\theta_{2}}{f(\tau,1)}\Bigg{)}=0\Rightarrow
d\Bigg{(}\frac{F_{4}^{2}(\tau,1)}{F_{5}(\tau,1)f(\tau,1)}\beta_{0}\Bigg{)}=0$
(22)

We put $\displaystyle
F_{2}(\tau):=\frac{F_{5}(\tau,1)f(\tau,1)}{F_{4}^{2}(\tau,1)}$ for this case.

Now consider the birrational map
$\Phi_{i}(\tau,z)=(\tau,F_{i}(\tau)z)=(\tau,x)$, $i=1,2$. If $\tilde{\eta}$ is
the one that appears in Subcase 1, just use $\Phi_{1}$, if it is
$\tilde{\eta}$ in Subcase 2, just use $\Phi_{2}$, so we will omit the indices
$i$ of $\Phi_{i}$ and $F_{i}$ because the calculation we will make with these
applications in one case is identical to the other. If $\tilde{\eta}$ is like
in $(\ref{eq_case4})$, a straightforward calculation gives
$\Phi^{*}(\tilde{\eta})=F\bar{\eta}$, where

$\bar{\eta}=dz+\tilde{\beta}_{0}+z\tilde{\beta}_{1}+z^{2}\tilde{\beta}_{2}$

with $\tilde{\beta}_{0}=\frac{\beta_{0}}{F}$,
$\tilde{\beta}_{1}=\beta_{1}+\frac{dF}{F}$, and
$\tilde{\beta}_{2}=F\beta_{2}$. Let us consider the birational map
$\phi(\tau,w)=(\tau,\frac{1}{w})=(\tau,z)$. Then
$\phi^{\ast}(\bar{\eta})=-w^{2}\hat{\eta}$, where
$\hat{\eta}=dw-\tilde{\beta_{2}}-w\tilde{\beta_{1}}-w^{2}\tilde{\beta_{0}}$.
Since $i_{\partial_{w}}(\tilde{\beta}_{j})=0$ and
$L_{\partial_{w}}(\tilde{\beta}_{j})=0$, for all $0\leq j\leq 2$, the
integrability of $\hat{\eta}$ implies

$\left\\{\begin{array}[]{lll}d\tilde{\beta_{0}}=\tilde{\beta_{0}}\wedge\tilde{\beta_{1}}\\\
d\tilde{\beta_{1}}=2\tilde{\beta_{0}}\wedge\tilde{\beta_{2}}\\\
d\tilde{\beta_{2}}=\tilde{\beta_{1}}\wedge\tilde{\beta_{2}}\end{array}\right.$
(23)

From (21) and (22) we get $d\tilde{\beta_{0}}=0$ and by the first equation in
(23) we obtain $\tilde{\beta_{0}}\wedge\tilde{\beta_{1}}=0$. Denoting by
$\mathcal{M}_{k}$ the set of meromorphic functions in $\mathbb{P}^{k}$. It
follows from $\tilde{\beta_{0}}\wedge\tilde{\beta_{1}}=0$ that there exists
$g\in\mathcal{M}_{n-1}$ such that $\tilde{\beta_{1}}=g\tilde{\beta_{0}}$. The
second relation in (23) gives us

$\displaystyle
d\tilde{\beta_{1}}=dg\wedge\tilde{\beta_{0}}=2\tilde{\beta_{0}}\wedge\tilde{\beta_{2}}$
$\displaystyle\Rightarrow
dg\wedge\tilde{\beta_{0}}+2\tilde{\beta_{2}}\wedge\tilde{\beta_{0}}=0$
$\displaystyle\Rightarrow(dg+2\tilde{\beta_{2}})\wedge\tilde{\beta_{0}}=0.$

Therefore, there exists $h\in\mathcal{M}_{n-1}$ such that

$\displaystyle\frac{dg}{2}+\tilde{\beta_{2}}=h\tilde{\beta_{0}}\Rightarrow\tilde{\beta_{2}}=h\tilde{\beta_{0}}-\frac{dg}{2}.$

The third relation in (23) implies

$\displaystyle d\tilde{\beta_{2}}=dh\wedge\tilde{\beta_{0}}=$
$\displaystyle\tilde{\beta_{1}}\wedge\tilde{\beta_{2}}=g\tilde{\beta_{0}}\wedge\Bigg{(}h\tilde{\beta_{0}}-\frac{dg}{2}\Bigg{)}=g\frac{dg}{2}\wedge\tilde{\beta_{0}}$
$\displaystyle\Rightarrow$ $\displaystyle
dh\wedge\tilde{\beta_{0}}=g\frac{dg}{2}\wedge\tilde{\beta_{0}}\Rightarrow\Bigg{(}dh-g\frac{dg}{2}\Bigg{)}\wedge\tilde{\beta_{0}}=0$
$\displaystyle\Rightarrow$ $\displaystyle
d\Bigg{(}h-\frac{g^{2}}{4}\Bigg{)}\wedge\tilde{\beta_{0}}=0.$ (24)

Let us denote by $\mathcal{G}$ the foliation generated by $\tilde{\beta_{0}}$
in $\mathbb{P}^{n-1}$. Note that $\tilde{\beta_{0}}$ generates a foliation in
$\mathbb{P}^{n-1}$ because it is defined in $\mathbb{C}^{n-1}$ and moreover it
is closed.

We have two possibilities:
Possibility I: $\mathcal{G}$ has no non-constant first integral. We claim that
$\omega$ has an integral factor. In fact, by (24)

$d\left(h-\frac{g^{2}}{4}\right)\wedge\tilde{\beta_{0}}=0\Rightarrow
h=\frac{g^{2}}{4}+c,$

where $c\in\mathbb{C}$, otherwise $\mathcal{G}$ would have non-constant first
integral. We can then write

$\displaystyle\hat{\eta}$ $\displaystyle=$ $\displaystyle
dw-\tilde{\beta_{2}}-w\tilde{\beta_{1}}-w^{2}\tilde{\beta_{0}}=dw-\left(\frac{g^{2}}{4}+c\right)\tilde{\beta_{0}}-\frac{dg}{2}-g\tilde{\beta_{0}}w-\tilde{\beta_{0}}w^{2}$
(25) $\displaystyle=$ $\displaystyle
dw-\frac{dg}{2}-\left(\frac{g}{4}+c+gw+w^{2}\right)\tilde{\beta_{0}}$
$\displaystyle=$ $\displaystyle
d\left(w-\frac{g}{2}\right)-\left(\left[w+\frac{g}{2}\right]^{2}+c\right)\tilde{\beta_{0}},$

in particular, if we set
$\mathcal{P}:=\left(\left[w+\frac{g}{2}\right]^{2}+c\right)^{-1}\hat{\eta}$,
then

$\mathcal{P}:=\frac{d(w-\frac{g}{2})}{\left[w+\frac{g}{2}\right]^{2}+c}-\tilde{\beta_{0}}\Rightarrow
d\tilde{\beta_{0}}=0.$

Therefore $\hat{\eta}$ has an integral factor and thus $\omega$ as well. Then
there exists $h$ such that

$\displaystyle d(h\omega)=0$ $\displaystyle\Rightarrow
dh\wedge\omega+hd\omega=0$
$\displaystyle\Rightarrow-\frac{dh}{h}\wedge\omega=dw.$ (26)

Thus $\mathcal{F}$ is transversely affine.
Possibility II: $\mathcal{G}$ has non-constant first integral. We assert that
$\mathcal{F}$ is the pull-back of a Riccati equation on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ by a birational map. Indeed, by Stein’s
Factorization Theorem $\mathcal{G}$ has a meromorphic first integral, say $f$,
with connected fibers: if $\phi\in\mathcal{M}_{n-1}$ and $d\phi\wedge df=0$
then there exists $\psi\in\mathcal{M}_{1}$ such that $\phi=\psi(f)$ where
$\psi(f)=\psi\circ f$.
On the other hand, the relation (24) implies that there exists
$\phi_{2}\in\mathcal{M}_{1}$ such that

$d\left(h-\frac{g^{2}}{4}\right)\wedge\tilde{\beta_{0}}=0\Rightarrow
d\left(h-\frac{g^{2}}{4}\right)\wedge\phi_{1}df=0\Rightarrow
d\left(h-\frac{g^{2}}{4}\right)\wedge df=0$

and by Stein’s Factorization Theorem

$h-\frac{g^{2}}{4}=\psi_{2}(f)\Rightarrow h=\psi_{2}(f)+\frac{g^{2}}{4},$

replacing in $\hat{\eta}$ we have

$\displaystyle\hat{\eta}$ $\displaystyle=$ $\displaystyle
dw-\left(h\tilde{\beta_{0}}-\frac{dg}{2}\right)-w(g\tilde{\beta_{0}})-\tilde{\beta_{0}}w^{2}$
(27) $\displaystyle=$ $\displaystyle
d\left(w-\frac{g}{2}\right)+(-h-wg-w^{2})\tilde{\beta_{0}}$ $\displaystyle=$
$\displaystyle
d\left(w-\frac{g}{2}\right)-\left(\frac{g^{2}}{4}+\psi_{2}(f)+wg+w^{2}\right)\psi_{1}(f)df$
$\displaystyle=$ $\displaystyle
d\left(w-\frac{g}{2}\right)-\left(\left[w+\frac{g}{2}\right]^{2}+\psi_{2}(f)\right)\psi_{1}(f)df$

Consider the rational map
$\Phi_{1}:\mathbb{P}^{n-1}\times\mathbb{P}^{1}\dashrightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$
given by

$\Phi_{1}(\tau,w)=\left(f(\tau),w-\frac{g(\tau)}{2}\right)=:(x,y)$

Then $\hat{\eta}=\Phi^{\ast}(\theta)$, where
$\theta=dy-(y^{2}+\psi_{2}(x))\psi_{1}(x)dx$. Note that $\theta$ is an
integrable 1-form that defines a Riccati equation in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ and furthermore, it has a transversely
affine structure as we wanted, then $\mathcal{F}$ is transversely affine. ∎

######  Lemma 3.5 (Case 5).

Suppose that $F_{5}\equiv 0$ and $F_{3}\not\equiv 0\not\equiv F_{4}$. Then,
either $\mathcal{F}$ is transversely affine, or $\mathcal{F}$ is pure
transversely projective.

###### Proof.

It follows from (5) that
$\eta=x\theta_{2}+x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+(F_{3}+xF_{4})dx$.
We will do some pull-backs by rational maps on $\eta$ until we get the
expected result. First, we consider the pull-back by the application
$\psi_{z}(\tau,z)=(\tau,\frac{1}{z})=(\tau,x)$. Then
$\psi_{z}^{\ast}(\eta)=\frac{\eta_{z}}{z^{4}}$, where

$\eta_{z}=\theta_{5}+z\theta_{4}+z^{2}\theta_{3}+z^{3}\theta_{2}-(zF_{4}+z^{2}F_{3})dz.$

Consider the pull-back of $\eta_{z}$ by the birrational map
$\displaystyle\psi_{t}(\tau,t)=\left(\tau,\frac{\frac{F_{4}(\tau,1)}{F_{3}(\tau,1)}t}{(1-t)}\right)=(\tau,z)$.
A straightforward computation gives
$\psi^{*}_{t}(\eta_{z})=\frac{F_{4}^{3}}{(1-t)^{4}F_{3}^{2}}\eta_{t}$, where

$\eta_{t}=\beta_{0}+t\beta_{1}+t^{2}\beta_{2}+t^{3}\beta_{3}+t^{4}\beta_{4}-tdt$

with

$\displaystyle\beta_{0}$ $\displaystyle=$
$\displaystyle\frac{F_{3}^{2}}{F_{4}^{3}}\theta_{5}$ $\displaystyle\beta_{1}$
$\displaystyle=$
$\displaystyle-4\frac{F_{3}^{2}}{F_{4}^{3}}\theta_{5}+\frac{F_{3}}{F_{4}^{2}}\theta_{4}$
$\displaystyle\beta_{2}$ $\displaystyle=$ $\displaystyle
6\frac{F_{3}^{2}}{F_{4}^{3}}\theta_{5}-3\frac{F_{3}}{F_{4}^{2}}\theta_{4}+\frac{\theta_{3}}{F_{4}}-\frac{dF_{4}}{F_{4}}+\frac{dF_{3}}{F_{3}}$
$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle-4\frac{F_{3}^{2}}{F_{4}^{3}}\theta_{5}+3\frac{F_{3}}{F_{4}^{2}}\theta_{4}-2\frac{\theta_{3}}{F_{4}}+2\frac{dF_{4}}{F_{4}}+\frac{dF_{3}}{F_{3}}+\frac{\theta_{2}}{F_{3}}$
$\displaystyle\beta_{4}$ $\displaystyle=$
$\displaystyle\frac{F_{3}^{2}}{F_{4}^{3}}\theta_{5}-\frac{F_{3}}{F_{4}^{2}}\theta_{4}+\frac{\theta_{3}}{F_{4}}-\frac{\theta_{2}}{F_{3}}$

Note that $\beta_{i}$ only depend on $\tau$ for all $1\leq j\leq 4$. Now we
will make a lifting of $\eta_{t}$ for the finite branched covering

$E=\\{(\tau,t,s):\tau\in\mathbb{C}^{n-1}\;\;\text{and}\;\;2s+t^{2}=0\\}$

through the projection natural $\pi_{E}:E\rightarrow\mathbb{C}^{n}$ given by
$\pi_{E}(\tau,t,s)=(\tau,t)$, therefore

$\displaystyle\pi_{E}^{\ast}(\eta_{t})$ $\displaystyle=$
$\displaystyle\pi_{E}^{\ast}\Bigg{(}\beta_{0}+t\beta_{1}+t^{2}\beta_{2}+t^{3}\beta_{3}+t^{4}\beta_{4}+d\left(-\frac{t^{2}}{2}\right)\Bigg{)}$
$\displaystyle=$
$\displaystyle\beta_{0}+t\beta_{1}+s\left(-2\beta_{2}\right)+t^{3}\beta_{3}+s^{2}\left(4\beta_{4}\right)+ds$
$\displaystyle=$
$\displaystyle\left(\beta_{0}+t\beta_{1}+t^{3}\beta_{3}\right)+s\left(-2\beta_{2}\right)+s^{2}\left(4\beta_{4}\right)+ds$

Let
$\widetilde{\eta_{t}}:=\pi_{E}^{\ast}(\eta_{t})=\tilde{\beta_{0}}+s\tilde{\beta_{1}}+s^{2}\tilde{\beta_{2}}+ds$,
where

$\displaystyle\tilde{\beta_{0}}$ $\displaystyle=$
$\displaystyle\beta_{0}+t\beta_{1}+t^{3}\beta_{3}$
$\displaystyle\tilde{\beta_{1}}$ $\displaystyle=$ $\displaystyle-2\beta_{2}$
$\displaystyle\tilde{\beta_{2}}$ $\displaystyle=$ $\displaystyle 4\beta_{4}$

Thus $\widetilde{\eta_{t}}$ has a form similar to (14) and the rest of the
proof is analogous to Lemma 3.3. Therefore, either $\widetilde{\eta_{t}}$ is a
transversely affine structure or $\mathcal{F}$ is a pure transversely
projective structure. Hence, we conclude that either $\mathcal{F}$ has a
transversely affine structure, or it has a pure transversely projective
structure. ∎

######  Lemma 3.6 (Case 6).

Suppose that $F_{4}\equiv 0$ and $F_{3}\not\equiv 0\not\equiv F_{5}$. Then,
either $\mathcal{F}$ is transversely affine, or $\mathcal{F}$ is pure
transversely projective.

###### Proof.

It follows from (5) that

$\displaystyle\eta=x\theta_{2}+x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+(F_{3}+x^{2}F_{5})dx.$

In a similar way to Lemma 3.5 we will lifting $\eta$ through the map
$\pi_{E}:E\rightarrow\mathbb{C}^{n}$ given by $\pi_{E}(\tau,x,z)=(\tau,x)$ to
the finite branched covering

$E=\\{(\tau,x,z):\tau\in\mathbb{C}^{n-1}\;\;\text{and}\;\;x^{2}-z=0\\}$ (28)

Note that

$\displaystyle x^{2}=z$ $\displaystyle\Rightarrow$ $\displaystyle 2xdx=dz$
$\displaystyle\Rightarrow$ $\displaystyle dx=\frac{dz}{2x}$

We get $\pi_{E}^{\ast}(\eta)=\frac{1}{2x}\tilde{\eta}$, where

$\widetilde{\eta}=\tilde{\beta_{0}}+z\tilde{\beta_{1}}+z^{2}\tilde{\beta_{2}}+(F_{3}+zF_{5})dz$

with

$\displaystyle\tilde{\beta_{0}}$ $\displaystyle=$ $\displaystyle
2x^{2}\theta_{2}+2x^{4}\theta_{4}$ $\displaystyle\tilde{\beta_{1}}$
$\displaystyle=$ $\displaystyle 2x\theta_{3}$ $\displaystyle\tilde{\beta_{3}}$
$\displaystyle=$ $\displaystyle 2x\theta_{5}$

Consider the birrational map
$\psi(\tau,x,t)=\left(\tau,x,\frac{\frac{F_{3}(\tau,1)}{F_{5}(\tau,1)}t}{(1-t)}\right)=(\tau,x,z)$.
A straightforward computation gives
$\psi^{*}(\tilde{\eta})=\frac{F_{3}^{2}}{(1-t)^{3}F_{5}}\widetilde{\eta}^{\prime}$,
where

$\widetilde{\eta}^{\prime}=\tilde{\beta_{0}}^{\prime}+t\tilde{\beta_{1}}^{\prime}+t^{2}\tilde{\beta_{2}}^{\prime}+dt$

with

$\displaystyle\tilde{\beta_{0}}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{F_{5}}{F_{3}^{2}}\tilde{\beta_{0}}$
$\displaystyle\tilde{\beta_{1}}^{\prime}$ $\displaystyle=$
$\displaystyle-3\frac{F_{5}}{F_{3}^{2}}\tilde{\beta_{0}}+\frac{\tilde{\beta_{1}}}{F_{3}}+\frac{dF_{3}}{F_{3}}-\frac{dF_{5}}{F_{5}}$
$\displaystyle\tilde{\beta_{2}}^{\prime}$ $\displaystyle=$ $\displaystyle
3\frac{F_{5}}{F_{3}^{2}}\tilde{\beta_{0}}-2\frac{\tilde{\beta_{1}}}{F_{3}}+\frac{\tilde{\beta_{2}}}{F_{5}}-\frac{dF_{3}}{F_{3}}+\frac{dF_{5}}{F_{5}}$

Thus, we fall into the same case as the Lemma 3.5, therefore, either
$\widetilde{\eta}^{\prime}$ has a transversely affine structure or it has a
pure transversely projective structure. Again, we conclude that either
$\mathcal{F}$ is transversely affine, or $\mathcal{F}$ is pure transversely
projective. ∎

######  Lemma 3.7 (Case 7).

Suppose that $F_{3}\not\equiv 0$, $F_{4}\not\equiv 0$, and $F_{5}\not\equiv
0$. Then, either $\mathcal{F}$ is transversely affine, or $\mathcal{F}$ is
pure transversely projective.

###### Proof.

It follows from (5) that

$\displaystyle\eta=x\theta_{2}+x^{2}\theta_{3}+x^{3}\theta_{4}+x^{4}\theta_{5}+(F_{3}+xF_{4}+x^{2}F_{5})dx.$

Let $E$ be the finite branched covering given in (28), and consider the
natural projection $\pi_{E}:E\rightarrow\mathbb{C}^{n}$, where
$\pi_{E}(\tau,x,z)=(\tau,x)$. Then
$\pi_{E}^{\ast}(\eta)=\frac{1}{2x}\tilde{\eta}$, where

$\widetilde{\eta}=\tilde{\beta_{0}}+z\tilde{\beta_{1}}+z^{2}\tilde{\beta_{2}}+(G+zF_{5})dz$

with $\tilde{\beta_{0}}=2x^{2}\theta_{2}+2x^{4}\theta_{4}$,
$\tilde{\beta_{1}}=2x\theta_{3}$, $\tilde{\beta_{2}}=2x\theta_{5}$, and
$G(\tau,x)=F_{3}+xF_{4}$. Consider the birrational map

$\psi_{z}(\tau,x,t)=\left(\tau,x,\frac{\frac{G(\tau,x)}{F_{5}(\tau,1)}t}{1-t}\right)=(\tau,x,z).$

A straightforward computation gives
$\psi^{*}_{z}(\tilde{\eta})=\frac{G^{2}}{(1-t)^{3}F_{5}}\tilde{\eta}^{\prime}$,
where

$\widetilde{\eta}^{\prime}=\tilde{\beta_{0}}^{\prime}+t\tilde{\beta_{1}}^{\prime}+t^{2}\tilde{\beta_{2}}^{\prime}+dt$

with

$\displaystyle\tilde{\beta_{0}}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{F_{5}}{G^{2}}\tilde{\beta_{0}}$
$\displaystyle\tilde{\beta_{1}}^{\prime}$ $\displaystyle=$
$\displaystyle-3\frac{F_{5}}{G^{2}}\tilde{\beta_{0}}+\frac{\tilde{\beta_{1}}}{G}+\frac{dG}{G}-\frac{dF_{5}}{F_{5}}$
$\displaystyle\tilde{\beta_{2}}^{\prime}$ $\displaystyle=$ $\displaystyle
3\frac{F_{5}}{G^{2}}\tilde{\beta_{0}}-2\frac{\tilde{\beta_{1}}}{G}+\frac{\tilde{\beta_{2}}}{F_{5}}-\frac{dG}{G}+\frac{dF_{5}}{F_{5}}$

Therefore, the conclusion is the same as in Lemma 3.6. ∎

The following table illustrates all the possible types of foliations of degree
four found in the above lemmas.

Cases | Pull-back | Transversely affine | Transversely projective
---|---|---|---
$F_{3}\equiv F_{4}\equiv F_{5}\equiv 0$ | $\checkmark$ |  |
$F_{4}\equiv F_{5}\equiv 0$, $F_{3}\not\equiv 0$ | $\checkmark$ | $\checkmark$ |
$F_{3}\equiv F_{4}\equiv 0$, $F_{5}\not\equiv 0$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$F_{3}\equiv F_{5}\equiv 0$, $F_{4}\not\equiv 0$ or $F_{3}\equiv 0$, $F_{4}\not\equiv 0\not\equiv F_{5}$ | $\checkmark$ | $\checkmark$ |
$F_{5}\equiv 0$, $F_{3}\not\equiv 0\not\equiv F_{4}$ |  | $\checkmark$ | $\checkmark$
$F_{4}\equiv 0$, $F_{3}\not\equiv 0\not\equiv F_{5}$ |  | $\checkmark$ | $\checkmark$
$F_{3}\not\equiv 0$, $F_{4}\not\equiv 0$, $F_{5}\not\equiv 0$. |  | $\checkmark$ | $\checkmark$

We can condense the results obtained in the above lemmas in the following
proposition:

######  Proposition 3.8.

In the above situation we have the four possibilities:

  * (i)

either $\mathcal{F}$ is transversely affine outside a compact hypersurface;

  * (ii)

or $\mathcal{F}$ is pure transversely projective outside a compact
hypersurface;

  * (iii)

or $\mathcal{F}$ is a pull-back by a rational map of a foliation on
$\mathbb{P}^{2}$;

  * (iv)

or $\mathcal{F}$ is a pull-back by a linear map
$\pi:\mathbb{P}^{n}\to\mathbb{P}^{n-1}$ of a foliation of degree four on
$\mathbb{P}^{n-1}$.

In particular, if $n=3$ then $\mathcal{F}$ satisfies $(i)$, $(ii)$, or
$(iii)$.

###  3.1. End of the proof of Theorem A.

We give the proof by induction on the dimension $n\geq 3$. If $n=3$, then
Theorem A follows from Corollary 2.1 and Proposition 3.8. Let us assume that
Theorem A is true for $n-1\geq 3$ and prove that it is true for $n$.

Let $\mathcal{F}$ be a codimension one foliation of degree four on
$\mathbb{P}^{n}$, $n\geq 4$. It follows from Corollary 2.1 and Proposition 3.8
that, either $\mathcal{F}$ satisfies one of the conclusions of Theorem A, or
$\mathcal{F}$ is the pull-back by a linear map
$\pi:\mathbb{P}^{n}\to\mathbb{P}^{n-1}$ of a foliation of degree four on
$\mathbb{P}^{n-1}$. In this last case, since Theorem A is true for $n-1$, then
one the four possibilities below is true:

  1. (i)

$\mathcal{F}_{n-1}$ has a rational first integral, say
$F:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{1}$. In this case, $F\circ\pi$
is a rational first integral of $\mathcal{F}$ and we are done.

  2. (ii)

$\mathcal{F}_{n-1}$ is transversely affine. In this case, $\mathcal{F}_{n-1}$
admits a G-V-S of length one. Hence, $\mathcal{F}$ also admits a G-V-S of
length one by Remark 2.5.

  3. (iii)

$\mathcal{F}_{n-1}$ is transversely projective. In this case,
$\mathcal{F}_{n-1}$ admits a G-V-S of length two. Hence, $\mathcal{F}$ also
admits a G-V-S of length two by Remark 2.5.

  4. (iv)

$\mathcal{F}_{n-1}=\Phi^{*}(\mathcal{G})$, where $\mathcal{G}$ is a foliation
on $\mathbb{P}^{2}$ and $\Phi:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{2}$ a
rational map. In this case, we get
$\mathcal{F}=(\Phi\circ\pi)^{*}(\mathcal{G})$ and we are done.

This finishes the proof of Theorem A.

Acknowledgement. We would like to thank Miguel Rodríguez Peña, Maurício
Corrêa, Omegar Calvo Andrade, and Jorge Vitório Pereira for their comments and
references that helped to improve the results of the paper. This paper is a
compilation of the results of the second author’s doctoral thesis at the
Federal University of Minas Gerais, Brazil.

## References

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# Self-supervised text-independent speaker verification using prototypical
momentum contrastive learning

###### Abstract

In this study, we investigate self-supervised representation learning for
speaker verification (SV). First, we examine a simple contrastive learning
approach (SimCLR) with a momentum contrastive (MoCo) learning framework, where
the MoCo speaker embedding system utilizes a queue to maintain a large set of
negative examples. We show that better speaker embeddings can be learned by
momentum contrastive learning. Next, alternative augmentation strategies are
explored to normalize extrinsic speaker variabilities of two random segments
from the same speech utterance. Specifically, augmentation in the waveform
largely improves the speaker representations for SV tasks. The proposed MoCo
speaker embedding is further improved when a prototypical memory bank is
introduced, which encourages the speaker embeddings to be closer to their
assigned prototypes with an intermediate clustering step. In addition, we
generalize the self-supervised framework to a semi-supervised scenario where
only a small portion of the data is labeled. Comprehensive experiments on the
Voxceleb dataset demonstrate that our proposed self-supervised approach
achieves competitive performance compared with existing techniques, and can
approach fully supervised results with partially labeled data.

Index Terms—  Contrastive learning, self-supervised learning, speaker
embedding, data augmentation, semi-supervised learning

##  1 Introduction

Speaker verification (SV) is a task to verify a person’s identity from audio
streams. It provides a natural and efficient way for biometric identity
authentication. SV can be directly employed in audio forensics, computer
access control, and telephone voice authentication for long-distance calling.
The learned speaker representation is also a key component in speaker
diarization, text to speech synthesis, voice conversion, automatic speech
recognition as well as targeted speech enhancement and separation systems [1,
2, 3, 4].

State-of-the-art approaches for learning speaker representations usually rely
on supervised deep neural networks and large labeled speaker corpora. Various
neural network architectures [5, 6, 7] were explored as a speaker embedding
extractor. Novel loss functions [8, 9, 10, 11] were investigated to learn more
discriminative speaker embeddings. New temporal pooling methods [12, 13] were
presented to aggregate a variable-length audio input to a fixed length
utterance level representation. Noise and language robust speaker recognition
models and training paradigms have been proposed and improve SV systems’
performance [14, 15].

Though impressive progress has been achieved with fully supervised models,
there remain many challenges for SV systems. For example, fully supervised
models with a limited amount of labels usually do not perform well in a domain
mismatched condition [16, 17]. In many practical cases, it is very costly and
difficult to obtain sufficient data annotations. Self-supervised learning
(SSL) that only requires a large amount of unlabeled data provides an
alternative solution for learning representations from speech. This study
mainly focuses on self-supervised contrastive learning approaches, where the
goal is to discriminate positive and negative examples. The popular
contrastive learning frameworks, such as SimCLR or MoCo, are usually
constructed by an InfoNCE loss, neural network-based encoders and a data
augmentation module for providing different views of the same sample [18, 19,
20, 21]. The InfoNCE loss guides the network training, so the distance of a
positive pair in the latent space (i.e., embeddings) is forced to be small,
and distances of negative pairs are large. Recently, in [22, 23, 24, 25],
researchers explored self-supervised contrastive learning for speaker
verification as an unsupervised learning framework or a pre-training method
for subsequent tasks. They have shown the effectiveness and the potential of
self-supervised contrastive learning-based speaker embeddings.

In this work, we focus on effectively utilizing unlabeled and partially
labeled speech data in the constrastive learning frameworks for speaker
representation learning. First, we investigate and compare SimCLR and MoCo as
our contrastive speaker representation learning frameworks. To remove
extrinsic speaker variabilities (channel, noise or spectrum distortions) of
two random chunks from an utterance, two kinds of augmentation strategies -
WavAugment and SpecAugment are investigated. We find that the SpecAug module
does not help with our proposed contrastive speaker embedding approaches,
while the WavAug module improves the speaker verification performance by a
large margin. Moreover, we introduce a prototypical memory bank to alleviate
the class collision problem [26]. Since instance discrimination in both SimCLR
and MoCo treats every other example as a negative sample, the positive
examples may be categorized as negative wrongly. We perform clustering on all
samples every epoch and store the prototypes in a memory bank. Negatives
prototypes are only sampled from different disjoint classes. Experimental
results on the Voxceleb corpus show that with the WavAug module and the
prototype memory bank, the Equal Error Rate (EER) of the MoCo based SV system
decreases from 15.11% to 8.23%, a relative 45.5% reduction. Finally, we
generalize our proposed self-supervised speaker embedding system to a semi-
supervised condition, with partially labeled speakers involved in the
training. The results show that our proposed semi-supervised speaker embedding
system can approach fully supervised results of the whole labeled set with
limited labeled speakers.

##  2 Contrastive Speaker representation learning

Fig. 1: Left: SimCLR approach; Right: Momentum Contrastive learning method for
speaker representation learning.

To learn a good speaker representation, we want the learned embeddings to have
the property of inter-class separation and intra-class compactness. This can
be achieved by a contrastive learning framework. It comprises three major
components: (1) A data augmentation module that transforms a data example into
two different views, which is a positive pair. (2) A neural encoder module
that encodes the input features into a fixed dimensional embedding. (3) A
projection head that maps embeddings to the latent space where a contrastive
loss is applied. In the following sections, we investigate two contrastive
speaker representation frameworks and introduce a novel prototypical
contrastive speaker embedding method.

###  2.1 SimCLR based speaker embedding

SimCLR directly maximizes the similarity between augmented positive pairs and
minimizes the similarity of negative pairs via a contrastive InfoNCE loss [18]
in the latent space. For a mini-batch of $N$ samples, we can get $2N$ samples
after augmentation. SimCLR doesn’t sample negative examples explicitly. The
$2(N-1)$ samples different from the positive pair are treated as negative
examples. The loss function is defined as,

$\displaystyle\mathcal{L}_{SimCLR}=\frac{1}{N}\sum_{i=1}^{N}-\log\frac{\exp\left(\bm{\mathbf{v}}_{i}\cdot\bm{\mathbf{v}}^{\prime}_{i}/\tau\right)}{\sum_{j=1}^{2N}\mathbbm{1}_{[i\neq
j]}\exp\left(\bm{\mathbf{v}}_{i}\cdot\bm{\mathbf{v}}_{j}/\tau\right)}$ (1)

where $\bm{\mathbf{v}}_{i}$ and $\bm{\mathbf{v}}^{\prime}_{i}$ are the two
augmentations of one sample, $\bm{\mathbf{v}}_{j}$ is from the rest $2N-1$
augmented samples, $\mathbbm{1}_{[i\neq j]}$ is an indicator function that
equals to 1 when $i\neq j$. $\tau$ is the temperature parameter. Our above
definition is slightly different from the original SimCLR implementation,
which also swaps sample $\bm{\mathbf{v}}_{i}$ and
$\bm{\mathbf{v}}^{\prime}_{i}$ and calculates an average loss.

###  2.2 Momentum contrastive speaker embedding

SimCLR uses samples in the current mini-batch for negative example mining. The
negative sample size is constrained with the mini-batch size $N$, which is
limited by the GPU memory. To prevent this problem and introduce more negative
samples in the InfoNCE loss, we also explore a momentum contrastive based
speaker embedding system. We mainly follow the MoCo framework for visual
representation learning [20]. MoCo uses a dynamic queue to sample a large
number of negative samples to calculate the loss with an assumption that
robust features can be learned by a large dictionary that covers a rich set of
negative samples. Illustrated in Fig. 1, one major difference between MoCo and
SimCLR is that MoCo uses one query encoder (left) to encode one copy of the
augmented samples, and employs a momentum encoder (right) to encode the other
copy. So the InfoNCE loss employed in the momentum contrastive speaker
embedding learning is defined as,

$\displaystyle\mathcal{L}_{MoCo}=\frac{1}{N}\sum_{i=1}^{N}-\log\frac{\exp\left(\bm{\mathbf{v}}_{qi}\cdot\bm{\mathbf{v}}^{\prime}_{{ki}}/\tau\right)}{\sum_{j=0}^{K}\exp\left(\bm{\mathbf{v}}_{qi}\cdot\bm{\mathbf{v}}_{kj}/\tau\right)}$
(2)

where $\bm{\mathbf{v}}_{qi}$ and $\bm{\mathbf{v}}^{\prime}_{{ki}}$ is one of
the query samples (i.e., $anchor$) and corresponding key samples (i.e.,
$positive$) in the mini-batch. $\bm{\mathbf{v}}_{{kj}}$ are the key samples
encoded with the key encoder (momentum encoder), the sum in Eq. 2 is over one
$positive$ pair and $K$ $negative$ pairs. The introduction of a queue with the
dictionary size $K$ is able to enlarge the number of negative samples compared
with the original InfoNCE. At the same time, samples in the dictionary are
progressively replaced. $K$ is set to $10000$ in our experiments. When the
current mini-batch is enqueued to the dictionary, the oldest mini-batch will
be dequeued.

To make the key embeddings in the negative queue consistent, the key encoder
$f_{k}$ is updated as an Exponential Moving Average (EMA) of the query encoder
$f_{q}$. Denoting the parameters of $f_{k}$ as $\theta_{k}$ and those of
$f_{q}$ as $\theta_{q}$, we update $\theta_{k}$ by
$\theta_{\mathrm{k}}\leftarrow m\theta_{\mathrm{k}}+(1-m)\theta_{\mathrm{q}}$,
where $m$ = 0.999 is the momentum coefficient. Only the parameters
$\theta_{q}$ are updated by back-propagation during training. The momentum
update makes the key encoder evolve more smoothly than the query encoder. With
this design, though the keys in the queue are encoded by different encoders
(in different mini-batches), the difference among these encoders is small.
Therefore we can maintain a sizeable negative sample queue with a stable
training process.

###  2.3 Prototypical memory bank

MoCo’s negative samples in the queue are randomly sampled. It may contain
wrong $(anchor,negative)$ pairs since we treat all the other samples as
negative. This is known as the class collision problem in self-supervised
contrastive learning [26]. To alleviate this problem, a prototype memory bank
is introduced. As shown in Fig. 2, we perform clustering on all samples every
epoch, and negatives prototypes are only sampled from different disjoint
classes. Positives are from the same class prototypes. Cluster density is
dynamically estimated in order to use a more accurate concentration
coefficient ($\tau$ in Eq. 2) for each cluster. We can formulate this as an
iterative self-labeling process:

  1. 1.

Train the MoCo with InfoNCE loss for certain epochs in order to make the
learned speaker embedding robust and stable for clustering.

  2. 2.

Cluster the whole dataset into $M$ clusters every epoch and store the
prototypes (centroids of clusters) in a memory bank.

  3. 3.

Use the pseudo-labels to train a prototypical NCE loss. For an anchor, we pick
a positive prototype from the prototype memory bank. From the rest $M-1$
classes, we randomly sample $R$ negative prototypes w.r.t. their pseudo-
labels.

The overall objective function $\mathcal{L}_{joint}$ is formulated as,

$\displaystyle-\frac{1}{N}\sum_{i=1}^{N}\log\frac{\exp\left(\bm{\mathbf{v}}_{qi}\cdot\bm{\mathbf{v}}^{\prime}_{ki}/\tau\right)}{\sum_{j=0}^{K}\exp\left(\bm{\mathbf{v}}_{qi}\cdot\bm{\mathbf{v}}_{kj}/\tau\right)}+\alpha\log\frac{\exp\left(\bm{\mathbf{v}}_{qi}\cdot\bm{\mathbf{c}}_{s}/\phi_{s}\right)}{\sum_{j=0}^{R}\exp\left(\bm{\mathbf{v}}_{qi}\cdot\bm{\mathbf{c}}_{j}/\phi_{j}\right)}$
(3)

The first term InfoNCE loss in Eq. 3 is used to retain the property of local
smoothness and help bootstrap clustering. The second term is denoted as the
ProtoNCE loss. $\alpha$ is a weight coefficient to balance two losses. It is
set to 0.25 in our experiments. Embedding $\bm{\mathbf{c}}_{s}$ is the
prototype of class $s$ and a positive example of the query embedding
$\bm{\mathbf{v}}_{qi}$. Different class prototypes
$\\{\bm{\mathbf{c}}_{j}\\}_{j=1}^{R}$ are the negative examples. The
dynamically estimated temperature coefficient,
$\phi=\frac{\sum_{i=1}^{Z}\left\|\bm{\mathbf{v}}_{i}-\bm{\mathbf{c}}\right\|_{2}}{Z\log(Z+\epsilon)}$,
indicates the concentration level of embeddings
$\\{\bm{\mathbf{v}}_{i}\\}_{i=1}^{Z}$ around their class centroid
$\bm{\mathbf{c}}$. $Z$ is total number of embeddings in that class and
$\epsilon$ is a small constant for numerical stability.

###  2.4 Generalization to semi-supervised contrastive learning

In practice, we hope to approach the fully supervised results with a limited
amount of labeled data. The self-supervised contrastive loss is generalized to
the semi-supervised condition in the following. First, we explore a
generalized supervised contrastive loss (SupCon) [27] to utilize arbitrary
numbers of positives belonging to the same class. For a sample $i$, the loss
$\mathcal{L}_{SupCon}^{i}$ is presented as,

$\displaystyle\frac{-1}{2N_{y_{i}}-1}\sum_{p=1}^{2N}\mathbbm{1}_{[i\neq
p,y_{i}=y_{p}]}\log\frac{\exp\left(\bm{\mathbf{v}}_{i}\cdot\bm{\mathbf{v}}_{p}/\tau\right)}{\sum_{j=1}^{2N}\mathbbm{1}_{[i\neq
j]}\exp\left(\bm{\mathbf{v}}_{i}\cdot\bm{\mathbf{v}}_{j}/\tau\right)}$ (4)

For each example, we still generate two augmentations. Since the labels are
known, all positives $\bm{\mathbf{v}}_{p}$ in a mini-batch (the augmented one
and any of the remaining samples with the same class label $y_{i}$) are used
in the loss. $N_{y_{i}}$ is the total number of samples with the same label
$y_{i}$ in the mini-batch. In a partially labeled condition, we add the SupCon
loss to the self-supervised MoCo loss and train the model jointly with the
following semi-supervised loss,

$\displaystyle\mathcal{L}_{semi}=\mathcal{L}^{v}_{SupCon}+\lambda\mathcal{L}^{u+v}_{MoCo}$
(5)

where $u$ and $v$ denote the unlabeled and labeled data in a mini-batch,
respectively. Parameter $\lambda$ is a weight coefficient to adjust the loss.
In a mini-batch, self-supervised MoCo loss is trained on both unlabeled and
labeled sets and SupCon loss is trained on the labeled set.

Fig. 2: Prototypical momentum contrastive speaker representation learning
pipeline.

##  3 Experimental Setup

###  3.1 Dataset and feature extraction

Experiments are performed on the Voxceleb corpus to study the effectiveness of
our presented methods. The proposed speaker model is trained only on the
VoxCeleb2 dev set which contains over 1 million utterances from 5,994
speakers. We evaluate our model on the Voxceleb1 test set. There are 40
speakers in the test set with 4874 utterances. The performance is reported in
terms of Equal Error Rate (EER) and minimum Detection Cost Function (DCF) with
$P_{target}=0.01$.

We compute 30-dimensional MFCC on the frame level as input features. A Hamming
window of length 25 ms with a 10 ms frame-shift is used to extract the MFCC
from input audio signals. We use a random chunk of 200-400 frame features of
each audio file as the input to the network. The input feature is mean
normalized on the frame level. An energy-based VAD is used to remove silent
frames.

###  3.2 Data augmentation

In the self-supervised learning frameworks we proposed above, the learned
representation is expected to be robust against to the extrinsic
variabilities. The data augmentation module is essential as it provides
multiple views of the raw signal. In this study, we intend to factor out other
information like background noise, interference speakers, or channel
variabilities of two random chunks from an utterance as much as possible. Two
kinds of augmentation strategies are investigated: WavAug and SpecAug, on the
time and frequency domain respectively.

WavAug. Conventionally in speaker recognition, only one kind of extrinsic
variabilities is augmented to the raw sample. In our WavAug module, we find
that progressive data augmentation is more effective for self-supervised
training. Specifically, we first augment two chunks from the same utterance
with reverberation by randomly selecting RIRs in the OpenSLR Room Impulse
Response and Noise Database. 200 rooms are sampled, and 100 RIRs are picked in
each room based on the speaker and receiver position. We set a 80% probability
that the sample to be reverb-augmented, so about 20% of the whole data is kept
unchanged. After the reverb-augmentation, we further add one of the three
kinds of noises with equal probability: 1) Noise, 2) Music, and 3) Babble.

In each augmentation, one noise file is randomly selected in the MUSAN
database and added to the recording with [0, 5, 10, 15] dB SNR. Alternatively,
one music file is randomly selected from the same database and added to the
recording with [5, 8, 10, 15] dB SNR. Otherwise, one human babble speech is
selected and added to the recording with [13, 15, 17, 20] dB SNR. We repeat
the progressive data augmentation process with a different random seed to make
each copy of the augmentation contain about 2M chunks.

SpecAug. SpecAug is also investigated after feature extraction [28]. We apply
time warping and time frequency masking at the feature level. The time warping
window length is 10. The maximum time mask width is 5 and the maximum
frequency mask is 3. Each time we apply two randomly selected time masks and
two frequency masks on the features to generate two augmentations.

###  3.3 Model configuration and training

The encoding model is a standard TDNN [5] model. It has five layers of 1-D CNN
and uses a statistical pooling layer to encode the variable-length feature
input to a fixed 3000 dimension vector. The MLP projection head contains two
512-d layers. After the first layer, BatchNorm and ReLU is applied. L2 norm is
applied at the last hidden layer to normalize the embedding to the unit
length. The model is trained on the VoxCeleb2 dev split for 150 epochs with a
batch size of 4096. We use the SGD optimizer with 0.9 momentum and initialize
the learning rate as $10^{-1}$. The learning rate is reduced to $10^{-4}$ with
a cosine learning rate scheduler. The extracted utterance-level embedding size
is 512. A dot-product scoring function is applied to compute the similarity
between two embeddings.

For ProtoNCE, we pre-train the model with a MoCo InfoNCE loss for the first 60
epochs. After the 60th epoch, clustering is performed on the whole set using
the FAISS [29] GPU accelerated k-means algorithm. The number of classes $M$ is
set to 5000. The final performance is not very sensitive to the cluster
number. For each sample $\bm{\mathbf{v_{i}}}$, we choose the same class
prototype as the positive, and sample $R=10000$ negative prototypes with
replacement from the rest disjoint prototypes. We jointly train the model for
another 90 epochs with the loss $\mathcal{L}_{joint}$ in Eq.(3). For the semi-
supervised setting, $\lambda$ is set to $9$ in our experiments. In each mini-
batch, 10% of samples are labeled, and the others are unlabeled.

##  4 Results and Discussions

###  4.1 Unlabeled condition

To thoroughly evaluate our proposed self-supervised methods, we conduct
experiments and describe the results and analysis in this section. In Table 1,
we show the speaker verification evaluation results on the Voxceleb1 test set.
Cosine Distance Scoring (CDS) is applied to evaluate the performance.

Table 1: Self-supervised SV results on the VoxCeleb1 test set using proposed
contrastive methods.

Model CDS EER (%) DCF Disent [30] 22.09 N/A CDDL [31] 17.52 N/A GCL [23] 15.26
N/A i-vector [25] 15.28 0.63 (p=0.05) AP [25] 9.56 0.51 (p=0.05) AP+AAT [25]
8.65 0.45 (p=0.05) SimCLR 18.14 0.81 MoCo 15.11 0.80 MoCo + SpecAug 15.50 0.81
MoCo + WavAug 8.63 0.64 MoCo + WavAug (ProtoNCE) 8.23 0.59

Methods [23, 25, 30, 31] are some recent unsupervised neural methods for
speaker representation learning. As shown in the table, we can obtain a
$15.28\%$ EER using the i-Vector method and $15.11\%$ EER using the MoCo
speaker embedding system. It indicates the potential effectiveness of
contrastive speaker embeddings. Moreover, MoCo achieves better results than
SimCLR with a large memory queue. There is a relative $16.70\%$ reduction in
EER. It meets our assumption that a large amount of negative samples is
beneficial to self-supervised contrastive learning. Consistent embeddings in
the queue with a momentum update mechanism are also essential to the MoCo
method.

To investigate the effects of data augmentation, we first compare the MoCo
method with a WavAug and a SpecAug module, respectively. We find that our
SpecAug module does not help and make the verification results worse. It can
be removed in our experiments. The SpecAug module is like a dropout operation
on the feature level to improve the model generalization. The Voxceleb test
data may not contain lots of variabilities in the spectral domain. On the
contrary, with our progressive WavAug module, the results are improved by a
large margin. There is a relative $42.89\%$ improvement in CDS EER from MoCo
to the MoCo with WavAug approach. It shows the importance of reducing
extrinsic information, such as channel and noise characteristics. With
ProtoNCE, there is another relative $4.63\%$ improvement in CDS EER compared
with WavAug. Though the benefit is limited due to the inherent difficulty of
speaker clustering on a large dataset, the ProtoNCE loss improves the MoCo
training process by alleviating the class collision problem. The EER is also
relatively $4.86\%$ lower than the current best reported AP+AAT method without
an adversarial augmentation module.

###  4.2 Visualization of speaker embeddings

(a) MoCo

(b) WavAug+ProtoNCE

(c) Spk-id10290

Fig. 3: Visualizations of learned speaker embeddings using t-SNE

In Fig. 3, we visualize the learned self-supervised speaker embeddings of 6
random speakers in the Voxceleb test set. It can be observed that MoCo speaker
embeddings without data augmentation (Fig. 3a) do not distribute well in the
embedding space. Utterances from different speakers are grouped together and
not well separated.

MoCo+WavAug+ProtoNCE embedding (Fig. 3b) has much better intra-speaker
compactness and inter-speaker separability, which is essential for the speaker
verification task. It meets our assumption that WavAug can help significantly
reduce channel and noise variabilities. Also, the test data contains each
recording’s session id, reflecting the acoustic condition to some extent. We
pick one speaker, “id10290”, from the randomly selected speakers and visualize
the embeddings belonging to 10 different sessions. We find that different
sessions of the same speaker still tend to be separated. It suggests that
there might still be plenty of room to improve the performance if we can
further normalize extrinsic variabilities.

###  4.3 Partially labelled condition

We then investigate the performance when there is only limited amount of
labeled data. Using the same setting in papers [23, 24], we randomly sampled
$899$ speakers (15%) from the Voxceleb2 dev set as labeled data. The rest data
are used as unlabeled.

Table 2: Semi-supervised SV results on the VoxCeleb1 test set. Data
augmentations are applied to all approaches.

Model CDS EER (%) DCF PLDA EER (%) DCF x-vector (100% label) 6.69 0.55 3.48
0.37 899-spk x-vector (15% label) 7.91 0.70 6.39 0.60 MoCo + 899-spk PLDA 8.63
0.64 5.21 0.49 GCL [23] N/A N/A 6.01 N/A SSL embedding [24] N/A N/A 6.31 0.53
SupCon + MoCo (15% label) 6.85 0.55 4.34 0.44

From Table 2, we observe a relative $30.34\%$ improvement from a supervised
x-vector trained on $899$ labeled speakers to our proposed semi-supervised
SupCon+MoCo approach. It shows that with an extra-large amount of unlabeled
data, our proposed semi-supervised approach can achieve a better result than a
supervised approach trained on partially labeled data. Moreover, compared with
a self-supervised MoCo model trained on all data with a 899-speaker PLDA, our
proposed semi-supervised approach obtains a relative $16.70\%$ improvement in
EER. In the partially labeled condition, we outperform the semi-supervised GCL
and SSL embedding and can approach fully supervised results of the whole
labeled set.

##  5 Conclusions

In this study, we introduced a momentum contrastive speaker representation
framework with carefully designed augmentation strategies for text independent
speaker verification. To alleviate the class collision problem in self-
supervised contrastive learning, we further used clustering to obtain a
prototype memory bank and formulated the contrastive learning as an iterative
self-labeling process. The proposed methods were evaluated with two
augmentation modules. WavAug was particularly crucial in our proposed self-
supervised framework. Moreover, we generalized the self-supervised contrastive
loss to the semi-supervised condition and obtained further performance gain.
Additional analysis in the learned speaker embedding space demonstrated the
effectiveness of our proposed method. At the same time, we still observed the
existence of session variability, which restricted the performance of current
systems and provided a future direction to be exploited for finer analysis.

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