Logarithmic Foliations

Dominique Cerveau, Alcides Lins Neto

► To cite this version:

Dominique Cerveau, Alcides Lins Neto. Logarithmic Foliations. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2021, Série 6, 30 (3), pp.561-618. 10.5802/afst.1685. hal-01762160

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# Logarithmic Foliations

Dominique Cerveau, Alcides Lins Neto

► To cite this version:

Dominique Cerveau, Alcides Lins Neto. Logarithmic Foliations. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2021, Série 6, 30 (3), pp.561-618. 10.5802/afst.1685. hal-01762160

HAL Id: hal-01762160

https://hal.science/hal-01762160v1

Submitted on 23 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
ANNALES
DE LA FACULTÉ
DES SCIENCES
DE
TOULOUSE
Mathématiques

DOMINIQUE CERVEAU AND ALCIDES LINS NETO
Logarithmic foliations

Tome XXX, n° 3 (2021), p. 561-618.

https://doi.org/10.5802/afst.1685

© Université Paul Sabatier, Toulouse, 2021.

L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.centre-mersenne.org/) implique l'accord avec les conditions générales d'utilisation (http://afst.centre-mersenne.org/legal/). Les articles sont publiés sous la license CC-BY 4.0.

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

Publication membre du centre
Mersenne pour l'édition scientifique ouverte
http://www.centre-mersenne.org/
ANNALES
DE LA FACULTÉ
DES SCIENCES
DE
TOULOUSE
Mathématiques

DOMINIQUE CERVEAU AND ALCIDES LINS NETO

Logarithmic foliations

Tome XXX, n° 3 (2021), p. 561-618.

https://doi.org/10.5802/afst.1685

© Université Paul Sabatier, Toulouse, 2021.

L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.centre-mersenne.org/) implique l'accord avec les conditions générales d'utilisation (http://afst.centre-mersenne.org/legal/). Les articles sont publiés sous la license CC-BY 4.0.

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

Publication membre du centre
Mersenne pour l'édition scientifique ouverte
http://www.centre-mersenne.org/
Logarithmic foliations (*)

DOMINIQUE CERVEAU (1) AND ALCIDES LINS NETO (2)

**ABSTRACT.** — The purpose of this paper is to study singular holomorphic foliations of arbitrary codimension defined by logarithmic forms on projective spaces.

**RÉSUMÉ.** — Nous étudions dans cet article les feuilletages holomorphes singuliers de codimension arbitraire définis par des formes logarithmiques sur les espaces projectifs.

# 1. Basic definitions and results

Recall that a logarithmic form on a complex manifold $M$ is a meromorphic $q$-form $\eta$ on $M$ such that the pole divisors of $\eta$ and $d\eta$ are reduced. It is known that a holomorphic form on a compact Kähler manifold is closed. This statement were generalized by Deligne in the context of logarithmic forms as follows:

**THEOREM 1.1.** — Let $\eta$ be a logarithmic $q$-form on a compact Kähler manifold $M$. Assume that the pole divisor $(\eta)_\infty$ of $\eta$ is an hypersurface with normal crossing singularities. Then $\eta$ is closed.

In the case of germs of closed meromorphic $1$-forms there are “normal forms” describing them in terms of the poles and residues (cf. [7]). These normal forms can be translated to the projective spaces and in the logarithmic case they are of the type

$$ \eta = \sum_j \lambda_j \frac{df_j}{f_j}, \quad \lambda_j \in \mathbb{C}^*, \quad f_j \text{ holomorphic.} $$

<footnote>(*) Reçu le 4 avril 2018, accepté le 10 septembre 2019.</footnote>

<footnote>Keywords: holomorphic foliation, logarithmic form.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification: 37F75, 32G34, 32S65, 37F75, 34M15.</footnote>

<footnote>(1) Inst. Mathématique de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France — dominique.cerveau@univ-rennes1.fr</footnote>

<footnote>(2) IMPA, Est. D. Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil — alcides@impa.br</footnote>

<footnote>Article proposé par Vincent Guedj.</footnote>
Logarithmic foliations (*)

DOMINIQUE CERVEAU (1) AND ALCIDES LINS NETO (2)

**ABSTRACT.** — The purpose of this paper is to study singular holomorphic foliations of arbitrary codimension defined by logarithmic forms on projective spaces.

**RÉSUMÉ.** — Nous étudions dans cet article les feuilletages holomorphes singuliers de codimension arbitraire définis par des formes logarithmiques sur les espaces projectifs.

# 1. Basic definitions and results

Recall that a logarithmic form on a complex manifold $M$ is a meromorphic $q$-form $\eta$ on $M$ such that the pole divisors of $\eta$ and $d\eta$ are reduced. It is known that a holomorphic form on a compact Kähler manifold is closed. This statement were generalized by Deligne in the context of logarithmic forms as follows:

**THEOREM 1.1.** — Let $\eta$ be a logarithmic $q$-form on a compact Kähler manifold $M$. Assume that the pole divisor $(\eta)_{\infty}$ of $\eta$ is an hypersurface with normal crossing singularities. Then $\eta$ is closed.

In the case of germs of closed meromorphic $1$-forms there are “normal forms” describing them in terms of the poles and residues (cf. [7]). These normal forms can be translated to the projective spaces and in the logarithmic case they are of the type

$$ \eta = \sum_j \lambda_j \frac{df_j}{f_j}, \quad \lambda_j \in \mathbb{C}^*, \quad f_j \text{ holomorphic.} $$

<footnote>(*) Reçu le 4 avril 2018, accepté le 10 septembre 2019.</footnote>

<footnote>Keywords: holomorphic foliation, logarithmic form.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification: 37F75, 32G34, 32S65, 37F75, 34M15.</footnote>

<footnote>(1) Inst. Mathématique de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France — dominique.cerveau@univ-rennes1.fr</footnote>

<footnote>(2) IMPA, Est. D. Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil — alcides@impa.br</footnote>

<footnote>Article proposé par Vincent Guedj.</footnote>
One of our purposes is to generalize the above normal form for $p$-forms,
$p \ge 2$, in a special case. We need a definition.

**DEFINITION 1.2.** — Let $X \subset (\mathbb{C}^n, 0)$ be a germ at $0 \in \mathbb{C}^n$ of holomorphic hypersurface and $f \in \mathcal{O}_n$ be a reduced germ $f = f_1 \dots f_r$, defining $X: X = (f=0)$. We say that $X$ has strictly ordinary singularities outside $0$ if $0 \in \mathbb{C}^n$ is an isolated singularity of $f_i$ (i.e. $(f_i = 0) \setminus \{0\}$ is smooth), $\forall 1 \le i \le r$, and $X$ is normal crossing outside the origin.

It is important to note that Definition 1.2 is different from the usual
definition of a germ of hypersurface having normal crossing outside 0. For
example, is $f$ is irreducible then ($f=0$) has strictly ordinary singularities
outside 0 if, and only if, $f$ has an isolated singularity at 0.

Our first main result (see Theorem 2.1) is a generalization of Cerveau–
Mattei theorem on normal forms for germs of closed meromorphic 1-forms [7].
For instance, in the case of 2-forms it says that a germ of a closed 2-form $\eta$
with poles at the strictly ordinary singularities hypersurface $X = (f_1 \dots f_r =$
0) can be written as

$$
\eta = \sum_{i<j} \lambda_{ij} \frac{\mathrm{d}f_i}{f_i} \wedge \frac{\mathrm{d}f_j}{f_j} + \sum_j \mathrm{d}g_j \wedge \frac{\mathrm{d}f_j}{f_j} + \mathrm{d}\alpha,
$$

where $\lambda_{ij} \in \mathbb{C}$, $1 \le i < j \le r$, $g_1, \dots, g_r \in \mathcal{O}_n$ and $\alpha \in \Omega^1(\mathbb{C}^n, 0)$. The numbers $\lambda_{ij}$ are the residues and can be calculated by integral formulas (see Section 2). In Section 2 we will see a precise statement of Theorem 2.1 for germs of closed logarithmic p-forms on $(\mathbb{C}^n, 0)$.

As a consequence of Theorem 2.1 in the general case we get normal forms
in the case of logarithmic p-forms on $\mathbb{P}^n$:

**COROLLARY 1.3.** — Let $\eta$ be a logarithmic $p$-form on $\mathbb{P}^n$, $p \le n-1$. Assume that the divisor of poles $(\eta)_\infty$ is given in homogeneous coordinates by $f_1 \dots f_r$, where the $f_{i'}$s are irreducible homogeneous polynomials on $\mathbb{C}^{n+1}$. Furthermore suppose that the hypersurface $X = (f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Then $r \ge p+1$ and there are numbers $\lambda_I$, $I \in S_r^p$, such that in homogeneous coordinates we have

$$
\eta = \sum_{\substack{I \in S_r^p \\ I = (i_1 < \dots < i_p)}} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}}, \quad (1.1)
$$

where $i_R \eta = 0$.

**Notation.** — Let us fix homogeneous polynomials $f_1, \dots, f_r \in \mathbb{C}[z_0, \dots, z_n]$. The projectivization of the vector space of $p$-forms $\eta$ that can be written as in (1.1) (not satisfying $i_R \eta = 0$ necessarily) will be denoted by
One of our purposes is to generalize the above normal form for $p$-forms,
$p \ge 2$, in a special case. We need a definition.

**DEFINITION 1.2.** — Let $X \subset (\mathbb{C}^n, 0)$ be a germ at $0 \in \mathbb{C}^n$ of holomorphic hypersurface and $f \in \mathcal{O}_n$ be a reduced germ $f = f_1 \dots f_r$, defining $X: X = (f=0)$. We say that $X$ has strictly ordinary singularities outside $0$ if $0 \in \mathbb{C}^n$ is an isolated singularity of $f_i$ (i.e. $(f_i = 0) \setminus \{0\}$ is smooth), $\forall 1 \le i \le r$, and $X$ is normal crossing outside the origin.

It is important to note that Definition 1.2 is different from the usual definition of a germ of hypersurface having normal crossing outside 0. For example, is $f$ is irreducible then ($f=0$) has strictly ordinary singularities outside 0 if, and only if, $f$ has an isolated singularity at 0.

Our first main result (see Theorem 2.1) is a generalization of Cerveau–Mattei theorem on normal forms for germs of closed meromorphic 1-forms [7]. For instance, in the case of 2-forms it says that a germ of a closed 2-form $\eta$ with poles at the strictly ordinary singularities hypersurface $X = (f_1 \dots f_r = 0)$ can be written as

$$ \eta = \sum_{i<j} \lambda_{ij} \frac{\mathrm{d}f_i}{f_i} \wedge \frac{\mathrm{d}f_j}{f_j} + \sum_j \mathrm{d}g_j \wedge \frac{\mathrm{d}f_j}{f_j} + \mathrm{d}\alpha, $$

where $\lambda_{ij} \in \mathbb{C}$, $1 \le i < j \le r$, $g_1, \dots, g_r \in \mathcal{O}_n$ and $\alpha \in \Omega^1(\mathbb{C}^n, 0)$. The numbers $\lambda_{ij}$ are the residues and can be calculated by integral formulas (see Section 2). In Section 2 we will see a precise statement of Theorem 2.1 for germs of closed logarithmic p-forms on $(\mathbb{C}^n, 0)$.

As a consequence of Theorem 2.1 in the general case we get normal forms in the case of logarithmic p-forms on $\mathbb{P}^n$:

**COROLLARY 1.3.** — Let $\eta$ be a logarithmic $p$-form on $\mathbb{P}^n$, $p \le n-1$. Assume that the divisor of poles $(\eta)_\infty$ is given in homogeneous coordinates by $f_1 \dots f_r$, where the $f_{i\nu}$s are irreducible homogeneous polynomials on $\mathbb{C}^{n+1}$. Furthermore suppose that the hypersurface $X = (f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Then $r \ge p+1$ and there are numbers $\lambda_I$, $I \in S_r^p$, such that in homogeneous coordinates we have

$$ \eta = \sum_{\substack{I \in S_r^p \\ I=(i_1 < \dots < i_p)}} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}}, \quad (1.1) $$

where $i_R \eta = 0$.

**Notation.** — Let us fix homogeneous polynomials $f_1, \dots, f_r \in \mathbb{C}[z_0, \dots, z_n]$. The projectivization of the vector space of $p$-forms $\eta$ that can be written as in (1.1) (not satisfying $i_R \eta = 0$ necessarily) will be denoted by
$\mathcal{L}^p(f_1, \dots, f_r)$. The subspace of forms $\eta \in \mathcal{L}^p(f_1, \dots, f_r)$ such that $i_R\eta = 0$ will be denoted by $\mathcal{L}_R^p(f_1, \dots, f_r)$. Note that $\mathcal{L}_R^p(f_1, \dots, f_r) \subsetneq \mathcal{L}^p(f_1, \dots, f_r)$.

We now turn our attention to $p$-forms defining codimension $p$ foliations. A holomorphic $p$-form $\omega$, on an open subset $U \subset \mathbb{C}^n$, defines a codimension $p$ distribution, outside its singular set $\text{Sing}(\omega) = \{z \in U | \omega(z) = 0\}$, if it is locally totally decomposable on $U \setminus \text{Sing}(\omega)$. This means that for any $z \in U \setminus \text{Sing}(\eta)$ there are holomorphic 1-forms $\omega_1, \dots, \omega_p$, defined in some neighborhood $V$ of $z$, such that $\omega|_V = \omega_1 \wedge \cdots \wedge \omega_p$. The distribution $\mathcal{D}$ is then defined on $U \setminus \text{Sing}(\omega)$ by the codimension $p$ planes

$$ \mathcal{D}_z = \ker(\omega(z)) := \{v \in T_z U \mid i_v \omega(z) = 0\} = \bigcap_{1 \le j \le p} \ker(\omega_j(z)). $$

**DEFINITION 1.4.** — A holomorphic $p$-form $\omega$ will be said integrable if it is locally totally decomposable outside its singular set and satisfies Frobenius integrability condition. In this context it means that, if $\omega|_V = \omega_1 \wedge \cdots \wedge \omega_p$ as above then $d\omega_j \wedge \omega = 0$ for all $j = 1, \dots, p$.

Remark that if $\omega$ is closed and locally totally decomposable then the Frobenius condition is automatic:

$$ \omega_j \wedge \omega = 0, \quad \forall j \implies d\omega_j \wedge \omega = d(\omega_j \wedge \omega) = 0, \quad \forall j. $$

In particular, if $\omega$ is a closed logarithmic $p$-form then it is integrable if, and only if, it is locally totally decomposable outside $(\omega)_\infty \cup \text{Sing}(\omega)$.

**Example 1.5.** — Let $f_1, \dots, f_r$ be irreducible homogeneous polynomials on $\mathbb{C}^{n+1}$. Then any 1-form $\theta \in \mathcal{L}_R^1(f_1, \dots, f_r)$ defines a logarithmic codimension one foliation on $\mathbb{P}^n$, denoted by $\mathcal{F}_\theta$. Let $\theta_1, \dots, \theta_p \in \mathcal{L}_R^1(f_1, \dots, f_r)$ and $\eta := \theta_1 \wedge \cdots \wedge \theta_p$. If $\eta \ne 0$ then $\eta \in \mathcal{L}_R^p(f_1, \dots, f_r)$ and defines a singular codimension $p$ foliation on $\mathbb{P}^n$, denoted by $\mathcal{F}_\eta$. The leaves of $\mathcal{F}_\eta$, outside the pole divisor $f_1 \dots f_r = 0$, are contained in the intersection of the leaves of $\mathcal{F}_{\theta_1}, \dots, \mathcal{F}_{\theta_p}$. By this reason, $\mathcal{F}_\eta$ is called the intersection of the foliations $\mathcal{F}_{\theta_1}, \dots, \mathcal{F}_{\theta_p}$.

**Notation 1.6.** — We will use the notation

$$ \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r) = \{\eta \in \mathcal{L}_R^p(f_1, \dots, f_r) \mid \eta \text{ is integrable}\}. $$

**Remark 1.7.** — We would like to observe that $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$ is an algebraic subset of $\mathcal{L}_R^p(f_1, \dots, f_r)$. The proof is left as an exercise to the reader.

**DEFINITION 1.8.** — We say that $\eta \in \mathcal{L}^p(f_1, \dots, f_r)$ is totally decomposable into logarithmic forms if $\eta = \theta_1 \wedge \cdots \wedge \theta_p$, where $\theta_1, \dots, \theta_p \in \mathcal{L}^1(f_1, \dots, f_r)$. We will use the notation

$$ \mathcal{L}_{td}^p(f_1, \dots, f_r) = \left\{ \eta \in \mathcal{L}_R^p(f_1, \dots, f_r) \mid \begin{array}{l} \eta \text{ is totally decomposable} \\ \text{into logarithmic forms} \end{array} \right\}. $$
$\mathcal{L}^p(f_1, \dots, f_r)$. The subspace of forms $\eta \in \mathcal{L}^p(f_1, \dots, f_r)$ such that $i_R\eta = 0$ will be denoted by $\mathcal{L}_R^p(f_1, \dots, f_r)$. Note that $\mathcal{L}_R^p(f_1, \dots, f_r) \subsetneq \mathcal{L}^p(f_1, \dots, f_r)$.

We now turn our attention to $p$-forms defining codimension $p$ foliations. A holomorphic $p$-form $\omega$, on an open subset $U \subset \mathbb{C}^n$, defines a codimension $p$ distribution, outside its singular set $\text{Sing}(\omega) = \{z \in U | \omega(z) = 0\}$, if it is locally totally decomposable on $U \setminus \text{Sing}(\omega)$. This means that for any $z \in U \setminus \text{Sing}(\eta)$ there are holomorphic 1-forms $\omega_1, \dots, \omega_p$, defined in some neighborhood $V$ of $z$, such that $\omega|_V = \omega_1 \wedge \cdots \wedge \omega_p$. The distribution $\mathcal{D}$ is then defined on $U \setminus \text{Sing}(\omega)$ by the codimension $p$ planes

$$ \mathcal{D}_z = \ker(\omega(z)) := \{v \in T_z U \mid i_v \omega(z) = 0\} = \bigcap_{1 \le j \le p} \ker(\omega_j(z)). $$

**DEFINITION 1.4.** — A holomorphic $p$-form $\omega$ will be said integrable if it is locally totally decomposable outside its singular set and satisfies Frobenius integrability condition. In this context it means that, if $\omega|_V = \omega_1 \wedge \cdots \wedge \omega_p$ as above then $d\omega_j \wedge \omega = 0$ for all $j = 1, \dots, p$.

Remark that if $\omega$ is closed and locally totally decomposable then the Frobenius condition is automatic:

$$ \omega_j \wedge \omega = 0, \quad \forall j \implies d\omega_j \wedge \omega = d(\omega_j \wedge \omega) = 0, \quad \forall j. $$

In particular, if $\omega$ is a closed logarithmic $p$-form then it is integrable if, and only if, it is locally totally decomposable outside $(\omega)_\infty \cup \text{Sing}(\omega)$.

**Example 1.5.** — Let $f_1, \dots, f_r$ be irreducible homogeneous polynomials on $\mathbb{C}^{n+1}$. Then any 1-form $\theta \in \mathcal{L}_R^1(f_1, \dots, f_r)$ defines a logarithmic codimension one foliation on $\mathbb{P}^n$, denoted by $\mathcal{F}_\theta$. Let $\theta_1, \dots, \theta_p \in \mathcal{L}_R^1(f_1, \dots, f_r)$ and $\eta := \theta_1 \wedge \cdots \wedge \theta_p$. If $\eta \ne 0$ then $\eta \in \mathcal{L}_R^p(f_1, \dots, f_r)$ and defines a singular codimension $p$ foliation on $\mathbb{P}^n$, denoted by $\mathcal{F}_\eta$. The leaves of $\mathcal{F}_\eta$, outside the pole divisor $f_1 \dots f_r = 0$, are contained in the intersection of the leaves of $\mathcal{F}_{\theta_1}, \dots, \mathcal{F}_{\theta_p}$. By this reason, $\mathcal{F}_\eta$ is called the intersection of the foliations $\mathcal{F}_{\theta_1}, \dots, \mathcal{F}_{\theta_p}$.

**Notation 1.6.** — We will use the notation

$$ \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r) = \{\eta \in \mathcal{L}_R^p(f_1, \dots, f_r) \mid \eta \text{ is integrable}\}. $$

**Remark 1.7.** — We would like to observe that $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$ is an algebraic subset of $\mathcal{L}_R^p(f_1, \dots, f_r)$. The proof is left as an exercise to the reader.

**DEFINITION 1.8.** — We say that $\eta \in \mathcal{L}^p(f_1, \dots, f_r)$ is totally decomposable into logarithmic forms if $\eta = \theta_1 \wedge \cdots \wedge \theta_p$, where $\theta_1, \dots, \theta_p \in \mathcal{L}^1(f_1, \dots, f_r)$. We will use the notation

$$ \mathcal{L}_{\text{td}}^p(f_1, \dots, f_r) = \left\{ \eta \in \mathcal{L}_R^p(f_1, \dots, f_r) \mid \begin{array}{l} \eta \text{ is totally decomposable} \\ \text{into logarithmic forms} \end{array} \right\}. $$
Observe that $\mathcal{L}_{td}^p(f_1, \dots, f_r)$ is an irreducible algebraic subset of $\mathcal{L}_F^p(f_1, \dots, f_r)$.

**PROBLEM 1.9.** — When $\mathcal{L}_{td}^p(f_1, \dots, f_r) = \mathcal{L}_F^p(f_1, \dots, f_r)$? In other words, does a foliation on $\mathbb{P}^n$ defined by a logarithmic p-form, $2 \le p < n$, is an intersection of p codimension one logarithmic foliations?

A partial answer to Problem 1.9 is given by Theorem 1.10 (see Section 3):

**THEOREM 1.10.** — Let $f_1, \dots, f_r$ be homogeneous polynomials on $\mathbb{C}^{n+1}$ and assume that $(f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Then:

(a) If $p = 2$, or $r \in \{p+1, p+2\}$ then $\mathcal{L}_{td}^p(f_1, \dots, f_r) = \mathcal{L}_F^p(f_1, \dots, f_r)$.

(b) If $2 < p \le n-2$ and $r > p+2$ then $\mathcal{L}_{td}^p(f_1, \dots, f_r)$ is an irreducible component of $\mathcal{L}_F^p(f_1, \dots, f_r)$. In particular, if $\mathcal{L}_F^p(f_1, \dots, f_r)$ is irreducible then $\mathcal{L}_{td}^p(f_1, \dots, f_r) = \mathcal{L}_F^p(f_1, \dots, f_r)$.

An interesting consequence of Theorem 1.10 is the following:

**COROLLARY 1.11.** — In the hypothesis of Theorem 1.10 if $r = p+1$ and $\eta \in \mathcal{L}_F^p(f_1, \dots, f_{p+1})$ then the foliation $\mathcal{F}_\eta$ in $\mathbb{P}^n$ is a rational fibration of codimension $p$ on $\mathbb{P}^n$. In other words, $\mathcal{F}_\eta$ has a rational first integral $F: \mathbb{P}^n \to \mathbb{P}^p$ that in homogeneous coordinates can be written as

$$F = (f_1^{k_1}, \dots, f_{p+1}^{k_{p+1}}),$$
where $k_1. \deg(f_1) = \dots = k_{p+1}. \deg(f_{p+1})$ and $\gcd(k_1, \dots, k_{p+1}) = 1$.

**Remark 1.12.** — We would like to observe that the statement of Theorem 1.10 cannot be true in the case of $p = n-1$.

In fact, if $p = n-1$ then $\mathcal{L}_F^{n-1}(f_1, \dots, f_r) = \mathcal{L}_R^{n-1}(f_1, \dots, f_r)$, because any $(n-1)$-form on $\mathbb{C}^n$ is locally decomposable outside its singular set. Moreover, if $r \ge p+2$ then $\mathcal{L}_{td}^{n-1}(f_1, \dots, f_r)$ is a proper algebraic subset of $\mathcal{L}_F^{n-1}(f_1, \dots, f_r)$. The reason is that if $\eta$ is decomposable, $\eta = \theta_1 \wedge \dots \wedge \theta_p$, where $\theta_1, \dots, \theta_p$ are logarithmic 1-forms as in Theorem 1.10, then $\eta$ cannot have isolated singularities outside its pole divisor. A specific example on $\mathbb{P}^3$ is given in homogeneous coordinates by the logarithmic 2-form

$$\eta = \sum_{1 \le i < j \le 6} \lambda_{ij} \frac{d\ell_i}{\ell_i} \wedge \frac{d\ell_j}{\ell_j},$$

where $\lambda_{ij} \in \mathbb{C}$, $1 \le i < j \le 6$, and $\ell_j \in \mathbb{C}[z_0, \dots, z_3]$ is homogeneous of degree one, $1 \le j \le 6$. If we choose the $\ell_{j's}$ and $\lambda_{ij's}$ generic then the foliation $\mathcal{F}_\eta$ defined by $\eta$ has degree three and $40 = 3^3 + 3^2 + 3 + 1$ isolated singularities. Each plane $\ell_j$ is $\mathcal{F}_\eta$-invariant and the restriction $\mathcal{F}_\eta|_{\ell_j}$ also
Observe that $\mathcal{L}_{td}^p(f_1, \dots, f_r)$ is an irreducible algebraic subset of $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$.

**PROBLEM 1.9.** — When $\mathcal{L}_{td}^p(f_1, \dots, f_r) = \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$? In other words, does a foliation on $\mathbb{P}^n$ defined by a logarithmic p-form, $2 \le p < n$, is an intersection of p codimension one logarithmic foliations?

A partial answer to Problem 1.9 is given by Theorem 1.10 (see Section 3):

**THEOREM 1.10.** — Let $f_1, \dots, f_r$ be homogeneous polynomials on $\mathbb{C}^{n+1}$ and assume that $(f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Then:

(a) If $p = 2$, or $r \in \{p+1, p+2\}$ then $\mathcal{L}_{td}^p(f_1, \dots, f_r) = \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$.

(b) If $2 < p \le n-2$ and $r > p+2$ then $\mathcal{L}_{td}^p(f_1, \dots, f_r)$ is an irreducible component of $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$. In particular, if $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$ is irreducible then $\mathcal{L}_{td}^p(f_1, \dots, f_r) = \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$.

An interesting consequence of Theorem 1.10 is the following:

**COROLLARY 1.11.** — In the hypothesis of Theorem 1.10 if $r = p+1$ and $\eta \in \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_{p+1})$ then the foliation $\mathcal{F}_{\eta}$ in $\mathbb{P}^n$ is a rational fibration of codimension $p$ on $\mathbb{P}^n$. In other words, $\mathcal{F}_{\eta}$ has a rational first integral $F: \mathbb{P}^n \to \mathbb{P}^p$ that in homogeneous coordinates can be written as

$$F = (f_1^{k_1}, \dots, f_{p+1}^{k_{p+1}}),$$
where $k_1. \deg(f_1) = \dots = k_{p+1}. \deg(f_{p+1})$ and $\gcd(k_1, \dots, k_{p+1}) = 1$.

**Remark 1.12.** — We would like to observe that the statement of Theorem 1.10 cannot be true in the case of $p = n-1$.

In fact, if $p = n-1$ then $\mathcal{L}_{\mathcal{F}}^{n-1}(f_1, \dots, f_r) = \mathcal{L}_{\mathcal{R}}^{n-1}(f_1, \dots, f_r)$, because any $(n-1)$-form on $\mathbb{C}^n$ is locally decomposable outside its singular set. Moreover, if $r \ge p+2$ then $\mathcal{L}_{td}^{n-1}(f_1, \dots, f_r)$ is a proper algebraic subset of $\mathcal{L}_{\mathcal{F}}^{n-1}(f_1, \dots, f_r)$. The reason is that if $\eta$ is decomposable, $\eta = \theta_1 \wedge \dots \wedge \theta_p$, where $\theta_1, \dots, \theta_p$ are logarithmic 1-forms as in Theorem 1.10, then $\eta$ cannot have isolated singularities outside its pole divisor. A specific example on $\mathbb{P}^3$ is given in homogeneous coordinates by the logarithmic 2-form

$$\eta = \sum_{1 \le i < j \le 6} \lambda_{ij} \frac{d\ell_i}{\ell_i} \wedge \frac{d\ell_j}{\ell_j},$$

where $\lambda_{ij} \in \mathbb{C}$, $1 \le i < j \le 6$, and $\ell_j \in \mathbb{C}[z_0, \dots, z_3]$ is homogeneous of degree one, $1 \le j \le 6$. If we choose the $\ell_{j's}$ and $\lambda_{ij's}$ generic then the foliation $\mathcal{F}_{\eta}$ defined by $\eta$ has degree three and $40 = 3^3 + 3^2 + 3 + 1$ isolated singularities. Each plane $\ell_j$ is $\mathcal{F}_{\eta}$-invariant and the restriction $\mathcal{F}_{\eta}|_{\ell_j}$ also
defines a degree three foliation and so contains $13 = 3^2 + 3 + 1$ singularities, $1 \le j \le 6$, each line $\ell_i \cap \ell_j$ contains $4 = 3 + 1$ singularities, $1 \le i < j \le 6$, and each point $\ell_i \cap \ell_j \cap \ell_k$ one singularity, $1 \le i < j < k \le 6$. In particular, there are $13 \times 6 - 4 \times \#(\ell_i \cap \ell_j) + \#(\ell_i \cap \ell_j \cap \ell_k) = 38$ singularities contained in $\bigcup_{j=1}^6 \ell_j$ and so $2 = 40 - 38$ singularities not contained in the pole divisor. If $\eta$ was decomposable as in Theorem 1.10 then these two singularities could not be isolated.

As a consequence of Theorem 1.10 we can assert that if $\mathcal{G}$ is a codimension two logarithmic foliation on $\mathbb{P}^4 \supset \mathbb{P}^3$ such that $\mathcal{G}|_{\mathbb{P}^3} = \mathcal{F}_\eta$ then $\mathcal{G}$ cannot be tangent to $\mathbb{P}^3$ outside the pole divisor $\bigcup_j \ell_j$. As a consequence $\mathcal{G}$ will be a pull-back $\Pi^*(\mathcal{F}_\eta)$, where $\Pi$ is induced by a linear map $\tilde{\Pi}: \mathbb{C}^5 \to \mathbb{C}^4$.

In fact, the example of Remark 1.12 has motivated Theorems 5.1 and 5.2 that will be proved in Section 5. These results give necessary conditions for a codimension $p$ foliation $\mathcal{F}$ to be a local or global product in terms of the codimension of the singular set of its intersection with a $(p+1)$-plane: if there is a $(p+1)$-plane $\Sigma$ such that $\text{cod}(\text{Sing}(\mathcal{F}|_\Sigma)) \ge 3$ then $\mathcal{F} = F^*(\mathcal{F}|_\Sigma)$, where $F: \mathbb{P}^n \to \mathbb{P}^{p+1}$ is induced by a linear map of maximal rank $f: \mathbb{C}^{n+1} \to \mathbb{C}^{p+2}$ (Theorem 5.1). Theorem 5.2 is a local version of Theorem 5.1.

Another kind of result that we will prove concerns the “stability” of logarithmic foliations on $\mathbb{P}^n$, $n \ge 3$. In order to precise this phrase we recall the definition of the degree of a foliation on $\mathbb{P}^n$.

**DEFINITION 1.13.** — Let $\mathcal{F}$ be a holomorphic foliation of codimension $p$ on $\mathbb{P}^n$, $1 \le p < n$. The degree of $\mathcal{F}$, $\text{deg}(\mathcal{F})$, is defined as the degree of the divisor of tangencies of $\mathcal{F}$ with a generic plane of complex dimension $p$ of $\mathbb{P}^n$.

**Remark 1.14.** — In the particular case of codimension one foliations the degree is the number of tangencies of the foliation with a generic line $\mathbb{P}^1 \subset \mathbb{P}^n$. More generally, a codimension $p$ foliation $\mathcal{F}$ on $\mathbb{P}^n$ can be defined by a meromorphic integrable $p$-form on $\mathbb{P}^n$, say $\eta$, with $\text{cod}_c(\text{Sing}(\eta)) \ge 2$. If we consider a generic $p$-plane $\Sigma \simeq \mathbb{P}^p \subset \mathbb{P}^n$ then the degree of $\mathcal{F}$ is the degree of the divisor of zeroes of $\eta|_\Sigma$.

Note that, if $\Pi: \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{P}^n$ is the canonical projection, then the foliation $\Pi^*(\mathcal{F})$ can be extended to a foliation $\mathcal{F}^*$ on $\mathbb{C}^{n+1}$. This foliation is represented by a holomorphic $p$-form $\eta$ whose coefficients are homogeneous polynomials of degree $\text{deg}(\mathcal{F})+1$ and such that $i_R\eta = 0$, where $R$ is the radial vector field on $\mathbb{C}^{n+1}$. We say that the form $\eta$ represents $\mathcal{F}$ in homogeneous coordinates.
defines a degree three foliation and so contains $13 = 3^2 + 3 + 1$ singularities, $1 \le j \le 6$, each line $\ell_i \cap \ell_j$ contains $4 = 3 + 1$ singularities, $1 \le i < j \le 6$, and each point $\ell_i \cap \ell_j \cap \ell_k$ one singularity, $1 \le i < j < k \le 6$. In particular, there are $13 \times 6 - 4 \times \#(\ell_i \cap \ell_j) + \#(\ell_i \cap \ell_j \cap \ell_k) = 38$ singularities contained in $\bigcup_{j=1}^6 \ell_j$ and so $2 = 40 - 38$ singularities not contained in the pole divisor. If $\eta$ was decomposable as in Theorem 1.10 then these two singularities could not be isolated.

As a consequence of Theorem 1.10 we can assert that if $\mathcal{G}$ is a codimension two logarithmic foliation on $\mathbb{P}^4 \supset \mathbb{P}^3$ such that $\mathcal{G}|_{\mathbb{P}^3} = \mathcal{F}_\eta$ then $\mathcal{G}$ cannot be tangent to $\mathbb{P}^3$ outside the pole divisor $\bigcup_j \ell_j$. As a consequence $\mathcal{G}$ will be a pull-back $\Pi^*(\mathcal{F}_\eta)$, where $\Pi$ is induced by a linear map $\tilde{\Pi}: \mathbb{C}^5 \to \mathbb{C}^4$.

In fact, the example of Remark 1.12 has motivated Theorems 5.1 and 5.2 that will be proved in Section 5. These results give necessary conditions for a codimension $p$ foliation $\mathcal{F}$ to be a local or global product in terms of the codimension of the singular set of its intersection with a $(p+1)$-plane: if there is a $(p+1)$-plane $\Sigma$ such that $\mathrm{cod}(\mathrm{Sing}(\mathcal{F}|_\Sigma)) \ge 3$ then $\mathcal{F} = F^*(\mathcal{F}|_\Sigma)$, where $F: \mathbb{P}^n \to \mathbb{P}^{p+1}$ is induced by a linear map of maximal rank $f: \mathbb{C}^{n+1} \to \mathbb{C}^{p+2}$ (Theorem 5.1). Theorem 5.2 is a local version of Theorem 5.1.

Another kind of result that we will prove concerns the “stability” of logarithmic foliations on $\mathbb{P}^n$, $n \ge 3$. In order to precise this phrase we recall the definition of the degree of a foliation on $\mathbb{P}^n$.

**DEFINITION 1.13.** — Let $\mathcal{F}$ be a holomorphic foliation of codimension $p$ on $\mathbb{P}^n$, $1 \le p < n$. The degree of $\mathcal{F}$, $\deg(\mathcal{F})$, is defined as the degree of the divisor of tangencies of $\mathcal{F}$ with a generic plane of complex dimension $p$ of $\mathbb{P}^n$.

**Remark 1.14.** — In the particular case of codimension one foliations the degree is the number of tangencies of the foliation with a generic line $\mathbb{P}^1 \subset \mathbb{P}^n$. More generally, a codimension $p$ foliation $\mathcal{F}$ on $\mathbb{P}^n$ can be defined by a meromorphic integrable $p$-form on $\mathbb{P}^n$, say $\eta$, with $\mathrm{cod}_c(\mathrm{Sing}(\eta)) \ge 2$. If we consider a generic $p$-plane $\Sigma \simeq \mathbb{P}^p \subset \mathbb{P}^n$ then the degree of $\mathcal{F}$ is the degree of the divisor of zeroes of $\eta|_\Sigma$.

Note that, if $\Pi: \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{P}^n$ is the canonical projection, then the foliation $\Pi^*(\mathcal{F})$ can be extended to a foliation $\mathcal{F}^*$ on $\mathbb{C}^{n+1}$. This foliation is represented by a holomorphic $p$-form $\eta$ whose coefficients are homogeneous polynomials of degree $\deg(\mathcal{F})+1$ and such that $i_R\eta = 0$, where $R$ is the radial vector field on $\mathbb{C}^{n+1}$. We say that the form $\eta$ represents $\mathcal{F}$ in homogeneous coordinates.
A consequence of the definition, is that if $T: \mathbb{P}^m - \to \mathbb{P}^n$ is a linear map of maximal rank, where $m > p$, then $\deg(T^*(\mathcal{F})) = \deg(\mathcal{F})$. In particular, if $\mathbb{P}^m \subset \mathbb{P}^n$ is a generic $m$-plane, where $m > p$, then the degree of $\mathcal{F}|_{\mathbb{P}^m}$ is equal to the degree of $\mathcal{F}$.

The space of dimension $k$ (codimension $p = n - k$) foliations on $\mathbb{P}^n$ of degree $d$ will be denoted by $\mathrm{Fol}(d; k, n)$. Note that $\mathrm{Fol}(d; k, n)$ can be identified with the subset of the projectivisation of the space of $(n - k)$-forms $\eta$ on $\mathbb{C}^{n+1}$ such that: $\eta$ is integrable, $\eta$ has homogeneous coefficients of degree $d+1$, $\mathrm{cod}_{\mathbb{C}}(\mathrm{Sing}(\eta)) \ge 2$ and $i_R\eta = 0$.

When $k = 1$ the integrability condition is automatic and $\mathrm{Fol}(d; 1, n)$ is a Zariski open and dense subset of some projective space $\mathbb{P}^N$. However, if $k \ge 2$ then the integrability condition is non-trivial and $\mathrm{Fol}(d; k, n)$ is an algebraic subset of some Zariski open and dense subset of a projective space.

**Example 1.15.** — Let $\mathcal{F}$ be the logarithmic foliation on $\mathbb{P}^n$ defined in homogeneous coordinates by an integrable $p$-form $\eta$ on $\mathbb{C}^{n+1}$ as below:

$$ \eta = \sum_{\substack{I \in S_r^p \\ I = (i_1 < \dots < i_p)}} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}}, \quad (1.2) $$

where $f_1, \dots, f_r$ are homogeneous polynomials on $\mathbb{C}^{n+1}$ with $\deg(f_j) = d_j$, $1 \le j \le r$. We assume also that $f_1, \dots, f_r$ are normal crossing outside the origin and $\lambda_I \ne 0, \forall I \in S_r^p$. With these conditions then the holomorphic form $\tilde{\eta} := f_1 \dots f_r \eta$ has singular set of codimension $\ge 2$ and so defines $\mathcal{F}$ in homogeneous coordinates. Since the degree of the coefficients of $\tilde{\eta}$ is $\sum_{j=1}^r d_j - p$ we obtain

$$ \deg(\mathcal{F}) = \sum_{j=1}^{r} d_j - p - 1 := D(d_1, \dots, d_r, p) \\ \implies \mathcal{F} \in \mathrm{Fol}(D(d_1, \dots, d_r, p); n - p, n) $$

**Notation 1.16.** — The space of dimension $k = n - p$ logarithmic foliations of $\mathbb{P}^n$ defined by a closed $p$-form as in (1.2) will be denoted by $\mathcal{L}_{\mathcal{F}}(d_1, \dots, d_r; k, n)$. Note that

$$ \mathcal{L}_{\mathcal{F}}(d_1, \dots, d_r; k, n) \subset \mathrm{Fol}(D(d_1, \dots, d_r, p); k, n). $$

The sub-space of $\mathcal{L}_{\mathcal{F}}(d_1, \dots, d_r; k, n)$ of foliations that are defined by totally decomposable into logarithmic forms $p$-forms will be denoted by $\mathcal{L}_{td}(d_1, \dots, d_r; k, n)$.

Our main result generalizes a theorem by Calvo-Andrade [3]:
A consequence of the definition, is that if $T: \mathbb{P}^m \to \mathbb{P}^n$ is a linear map of maximal rank, where $m > p$, then $\deg(T^*(\mathcal{F})) = \deg(\mathcal{F})$. In particular, if $\mathbb{P}^m \subset \mathbb{P}^n$ is a generic $m$-plane, where $m > p$, then the degree of $\mathcal{F}|_{\mathbb{P}^m}$ is equal to the degree of $\mathcal{F}$.

The space of dimension $k$ (codimension $p = n-k$) foliations on $\mathbb{P}^n$ of degree $d$ will be denoted by $\mathrm{Fol}(d; k, n)$. Note that $\mathrm{Fol}(d; k, n)$ can be identified with the subset of the projectivisation of the space of $(n-k)$-forms $\eta$ on $\mathbb{C}^{n+1}$ such that: $\eta$ is integrable, $\eta$ has homogeneous coefficients of degree $d+1$, $\mathrm{cod}_{\mathbb{C}}(\mathrm{Sing}(\eta)) \ge 2$ and $i_R\eta = 0$.

When $k=1$ the integrability condition is automatic and $\mathrm{Fol}(d; 1, n)$ is a Zariski open and dense subset of some projective space $\mathbb{P}^N$. However, if $k \ge 2$ then the integrability condition is non-trivial and $\mathrm{Fol}(d; k, n)$ is an algebraic subset of some Zariski open and dense subset of a projective space.

**Example 1.15.** — Let $\mathcal{F}$ be the logarithmic foliation on $\mathbb{P}^n$ defined in homogeneous coordinates by an integrable $p$-form $\eta$ on $\mathbb{C}^{n+1}$ as below:

$$ \eta = \sum_{\substack{I \in S_r^p \\ I = (i_1 < \dots < i_p)}} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}}, \quad (1.2) $$

where $f_1, \dots, f_r$ are homogeneous polynomials on $\mathbb{C}^{n+1}$ with $\deg(f_j) = d_j$, $1 \le j \le r$. We assume also that $f_1, \dots, f_r$ are normal crossing outside the origin and $\lambda_I \ne 0, \forall I \in S_r^p$. With these conditions then the holomorphic form $\tilde{\eta} := f_1 \dots f_r \eta$ has singular set of codimension $\ge 2$ and so defines $\mathcal{F}$ in homogeneous coordinates. Since the degree of the coefficients of $\tilde{\eta}$ is $\sum_{j=1}^r d_j - p$ we obtain

$$ \deg(\mathcal{F}) = \sum_{j=1}^{r} d_j - p - 1 := D(d_1, \dots, d_r, p) \\ \implies \mathcal{F} \in \mathrm{Fol}(D(d_1, \dots, d_r, p); n-p, n) $$

**Notation 1.16.** — The space of dimension $k = n-p$ logarithmic foliations of $\mathbb{P}^n$ defined by a closed $p$-form as in (1.2) will be denoted by $\mathcal{L}_{\mathcal{F}}(d_1, \dots, d_r; k, n)$. Note that

$$ \mathcal{L}_{\mathcal{F}}(d_1, \dots, d_r; k, n) \subset \mathrm{Fol}(D(d_1, \dots, d_r, p); k, n). $$

The sub-space of $\mathcal{L}_{\mathcal{F}}(d_1, \dots, d_r; k, n)$ of foliations that are defined by totally decomposable into logarithmic forms $p$-forms will be denoted by $\mathcal{L}_{td}(d_1, \dots, d_r; k, n)$.

Our main result generalizes a theorem by Calvo-Andrade [3]:
**THEOREM 1.17.** — If $k \ge 2$ and $r \ge p+2 = n-k+2$ then $\mathcal{L}_{td}(d_1, \dots, d_r; k, n)$ is an irreducible component of $\text{Fol}(D(d_1, \dots, d_r, p); k, n)$ for all $r > p$ and $d_1, \dots, d_r \ge 1$.

**Remark 1.18.** — The above result is also true in the case $r = p+1$. In fact, in [8] it is proven the stability of foliations induced by rational maps. On the other hand, by Corollary 1.11 the set $\mathcal{L}_F(d_1, \dots, d_{p+1}; n-p, n)$ coincides with the set of foliations induced by a rational map

$$F = (f_1^{k_1}, \dots, f_{p+1}^{k_{p+1}}) : \mathbb{C}^{n+1} \to \mathbb{C}^{p+1},$$

where $\deg(f_j) = d_j$ and $k_1.d_1 = \dots = k_{p+1}.d_{p+1}$.

Theorem 1.17 and Problem 1.9 motivate the following question:

**PROBLEM 1.19.** — When $\mathcal{L}_F(d_1, \dots, d_r; k, n) = \mathcal{L}_{td}(d_1, \dots, d_r; k, n)$?

Finally, in the appendix we give a proof of Theorem 2.10. This result, which is used in the proof of Theorem 2.1, gives sufficient conditions for the extension of forms defined in a hypersurface $X$ with a strictly ordinary singularities to the ambient space.

**Remark 1.20.** — Just before finishing this paper we have found a work by Javier Gargiulo Acea [11] in which he studies some of the problems that we have treated in our paper. For instance, he obtains the same results (decomposability and stability) of our Theorems 1.10 and 1.17 in the case $p=2$ (2-forms). He also proves the normal form for logarithmic p-forms on $\mathbb{P}^n$ if the pole divisor is normal crossing and $p \le n-1$ (our Corollary 1.3). The local case and the logarithmic foliations of codimension $\ge 3$ are not treated by him. We would like to observe that his proof of the stability of logarithmic 2-forms is purely algebraic: he computes the Zariski tangent space at a generic point.

## 2. Normal forms

The aim of this section is to prove Theorem 2.1 and its corollary (see Corollary 1.3).

**THEOREM 2.1.** — Let $\eta$ be a germ at $0 \in \mathbb{C}^n$ of closed logarithmic p-form with poles along a hypersurface $X = (f_1 \dots f_r = 0)$ with strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Assume that $n \ge p+2$. Then:

(a) If $r < p$ then $\eta$ is exact; $\eta = d\Theta$, where $\Theta$ is logarithmic non-closed and has the same pole divisor as $\eta$.
**THEOREM 1.17.** — If $k \ge 2$ and $r \ge p+2 = n-k+2$ then $\mathcal{L}_{td}(d_1, \dots, d_r; k, n)$ is an irreducible component of $\text{Fol}(D(d_1, \dots, d_r, p); k, n)$ for all $r > p$ and $d_1, \dots, d_r \ge 1$.

**Remark 1.18.** — The above result is also true in the case $r = p+1$. In fact, in [8] it is proven the stability of foliations induced by rational maps. On the other hand, by Corollary 1.11 the set $\mathcal{L}_F(d_1, \dots, d_{p+1}; n-p, n)$ coincides with the set of foliations induced by a rational map

$$F = (f_1^{k_1}, \dots, f_{p+1}^{k_{p+1}}) : \mathbb{C}^{n+1} \to \mathbb{C}^{p+1},$$

where $\deg(f_j) = d_j$ and $k_1.d_1 = \dots = k_{p+1}.d_{p+1}$.

Theorem 1.17 and Problem 1.9 motivate the following question:

**PROBLEM 1.19.** — When $\mathcal{L}_F(d_1, \dots, d_r; k, n) = \mathcal{L}_{td}(d_1, \dots, d_r; k, n)$?

Finally, in the appendix we give a proof of Theorem 2.10. This result, which is used in the proof of Theorem 2.1, gives sufficient conditions for the extension of forms defined in a hypersurface $X$ with a strictly ordinary singularities to the ambient space.

**Remark 1.20.** — Just before finishing this paper we have found a work by Javier Gargiulo Acea [11] in which he studies some of the problems that we have treated in our paper. For instance, he obtains the same results (decomposability and stability) of our Theorems 1.10 and 1.17 in the case $p=2$ (2-forms). He also proves the normal form for logarithmic p-forms on $\mathbb{P}^n$ if the pole divisor is normal crossing and $p \le n-1$ (our Corollary 1.3). The local case and the logarithmic foliations of codimension $\ge 3$ are not treated by him. We would like to observe that his proof of the stability of logarithmic 2-forms is purely algebraic: he computes the Zariski tangent space at a generic point.

## 2. Normal forms

The aim of this section is to prove Theorem 2.1 and its corollary (see Corollary 1.3).

**THEOREM 2.1.** — Let $\eta$ be a germ at $0 \in \mathbb{C}^n$ of closed logarithmic p-form with poles along a hypersurface $X = (f_1 \dots f_r = 0)$ with strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Assume that $n \ge p+2$. Then:

(a) If $r < p$ then $\eta$ is exact; $\eta = d\Theta$, where $\Theta$ is logarithmic non-closed and has the same pole divisor as $\eta$.
(b) If $r \ge p$ then there are numbers $\lambda_I \in \mathbb{C}$, $I \in S_p^r$, such that

$$ \eta = \sum_{I \in S_r^p} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}} + \mathrm{d}\Theta, \quad (2.1) $$

where, either $\Theta = 0$, or $\Theta$ is logarithmic non-closed and has pole divisor contained in X.

**Remark 2.2.** — In the above statement, if $r = 0$ then $X = \emptyset$ and $\eta$ is holomorphic and closed. In this case it can be written as $\eta = d\Theta$, where $\Theta$ is a holomorphic $(p-1)$-form, by Poincaré lemma. On the other hand, if $p = 1$ and $r \ge 1$ then $\eta$ can be written as

$$ \eta = \sum_j \lambda_j \frac{\mathrm{d}f_j}{f_j} + \mathrm{d}g, \quad g \in \mathcal{O}_n, $$

whereas when $p = 2$ and $r \ge 2$ then Theorem 2.1 implies that

$$ \eta = \sum_{i<j} \lambda_{ij} \frac{\mathrm{d}f_i}{f_i} \wedge \frac{\mathrm{d}f_j}{f_j} + \sum_j \mathrm{d}g_j \wedge \frac{\mathrm{d}f_j}{f_j} + \mathrm{d}\alpha, $$

where $g_1, \dots, g_r \in \mathcal{O}_n$ and $\alpha \in \Omega^1(\mathbb{C}^n, 0)$.

**Remark 2.3.** — The numbers $\lambda_I$ in (2.1), $I \in S_p^r$, are called the numerical residues of $\eta$ (see Section 2.1.1). Given $I = (i_1 < \cdots < i_p)$ then $\lambda_I$ can be calculated by integrating $\eta$ as follows: since $1 \le p < n$ the germ of analytic set $X_I := (f_{i_1} = \cdots = f_{i_p} = 0)$ has dimension $n - p \ge 1$. Moreover, by the normal crossing condition the set $\tilde{X}_I := X_I \setminus \bigcup_{j \notin I} (f_j = 0)$ is not empty. If we fix $m \in \tilde{X}_I$ then there are local coordinates $z = (z_1, \dots, z_n)$ such that $z(m) = 0$ and $f_{ij} = z_j$ for all $j = 1, \dots, p$. Given $\epsilon > 0$ small, consider the real $p$-dimensional torus

$$ T_{\epsilon}^{p} = \{z \mid |z_j| = \epsilon \text{ if } 1 \le j \le p, \text{ and } z_j = 0 \text{ if } j > p\}. $$

It follows from (2.1) that

$$ \lambda_I = \frac{1}{(2\pi i)^p} \int_{T_{\epsilon}^p} \eta. $$

As a consequence, a logarithmic $p$-form $\eta$ on $\mathbb{P}^n$, $p \le n - 1$, with pole divisor given in homogeneous coordinates by $f_1 \dots f_r = 0$, where the $f_{i's}$ are irreducible and the hypersurface $X = (f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$ can be written as in Corollary 1.3:

$$ \eta = \sum_{I \in S_r^p} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}} \quad \text{with } i_R \eta = 0. $$
(b) If $r \ge p$ then there are numbers $\lambda_I \in \mathbb{C}$, $I \in S_p^r$, such that

$$ \eta = \sum_{\substack{I \in S_r^p \\ I=(i_1 < \cdots < i_p)}} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}} + \mathrm{d}\Theta, \quad (2.1) $$

where, either $\Theta = 0$, or $\Theta$ is logarithmic non-closed and has pole divisor contained in $X$.

**Remark 2.2.** — In the above statement, if $r = 0$ then $X = \emptyset$ and $\eta$ is holomorphic and closed. In this case it can be written as $\eta = d\Theta$, where $\Theta$ is a holomorphic $(p-1)$-form, by Poincaré lemma. On the other hand, if $p = 1$ and $r \ge 1$ then $\eta$ can be written as

$$ \eta = \sum_j \lambda_j \frac{\mathrm{d}f_j}{f_j} + \mathrm{d}g, \quad g \in \mathcal{O}_n, $$

whereas when $p = 2$ and $r \ge 2$ then Theorem 2.1 implies that

$$ \eta = \sum_{i<j} \lambda_{ij} \frac{\mathrm{d}f_i}{f_i} \wedge \frac{\mathrm{d}f_j}{f_j} + \sum_j \mathrm{d}g_j \wedge \frac{\mathrm{d}f_j}{f_j} + \mathrm{d}\alpha, $$

where $g_1, \dots, g_r \in \mathcal{O}_n$ and $\alpha \in \Omega^1(\mathbb{C}^n, 0)$.

**Remark 2.3.** — The numbers $\lambda_I$ in (2.1), $I \in S_p^r$, are called the numerical residues of $\eta$ (see Section 2.1.1). Given $I = (i_1 < \cdots < i_p)$ then $\lambda_I$ can be calculated by integrating $\eta$ as follows: since $1 \le p < n$ the germ of analytic set $X_I := (f_{i_1} = \cdots = f_{i_p} = 0)$ has dimension $n-p \ge 1$. Moreover, by the normal crossing condition the set $\tilde{X}_I := X_I \setminus \bigcup_{j \notin I} (f_j = 0)$ is not empty. If we fix $m \in \tilde{X}_I$ then there are local coordinates $z = (z_1, \dots, z_n)$ such that $z(m) = 0$ and $f_{ij} = z_j$ for all $j = 1, \dots, p$. Given $\epsilon > 0$ small, consider the real $p$-dimensional torus

$$ T_{\epsilon}^{p} = \{z \mid |z_j| = \epsilon \text{ if } 1 \le j \le p, \text{ and } z_j = 0 \text{ if } j > p\}. $$

It follows from (2.1) that

$$ \lambda_I = \frac{1}{(2\pi i)^p} \int_{T_{\epsilon}^p} \eta. $$

As a consequence, a logarithmic $p$-form $\eta$ on $\mathbb{P}^n$, $p \le n-1$, with pole divisor given in homogeneous coordinates by $f_1 \dots f_r = 0$, where the $f_{i's}$ are irreducible and the hypersurface $X = (f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$ can be written as in Corollary 1.3:

$$ \eta = \sum_{\substack{I \in S_r^p \\ I=(i_1 < \cdots < i_p)}} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}} \quad \text{with } i_R \eta = 0. $$
Another observation is that Theorem 2.1 is false if $p = n - 1$ and $n \ge 3$ as shows the following example in $\mathbb{C}^n$, $n \ge 3$:

*Example 2.4.* — Let $P$ be an irreducible homogeneous polynomial of degree $n$ on $\mathbb{C}^n$ and set

$$ \eta = \frac{i_R(\mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_n)}{P(z_1, \ldots, z_n)} = \frac{\sum_{j=1}^{n} (-1)^{j-1} z_j \mathrm{d}z_1 \wedge \cdots \wedge \widehat{\mathrm{d}z_j} \wedge \cdots \wedge \mathrm{d}z_n}{P(z_1, \ldots, z_n)}, $$

where in the symbol $\widehat{\mathrm{d}z_j}$ means omission of $\mathrm{d}z_j$ in the product.

We would like to observe that the same example shows that Corollary 1.3 is false in $\mathbb{P}^m = \mathbb{P}^{n-1}$ if $p = m$: the form $\eta$ represents in homogeneous coordinates a closed logarithmic $m$-form on $\mathbb{P}^m$ which is not like in the statement of the corollary.

## 2.1. Preliminaries

Let $\eta$ be a germ at $0 \in \mathbb{C}^n$ of meromorphic $p$-form with reduced pole divisor $X = (f_1 \dots f_r = 0)$, $r \ge 1$. At the beginning we will not assume that $\eta$ is closed.

It follows from the definition that $\eta$ is logarithmic if, and only if, $f_1 \dots f_r \cdot d\eta$ is holomorphic. Since $(\eta)_\infty = (f_1 \dots f_r)$ we can write $\eta = \frac{1}{f_1 \dots f_r} \omega$ where $\omega \in \Omega_n^p$ is a germ of holomorphic $p$-form. We would like to observe that the following assertions are equivalent:

(a) $\eta = \frac{1}{f_1 \dots f_r} \omega$ is logarithmic.

(b) $f_j$ divides $df_j \wedge \omega$, for all $1 \le j \le r$.

In particular, we have:

(c) $\frac{1}{f_1 \dots f_s} \omega$ is logarithmic, for all $s \le r$.

In fact:

$$ \begin{align*} \text{(a)} \iff & f_1 \dots f_r \cdot d\eta = d\omega - \frac{d(f_1 \dots f_r)}{f_1 \dots f_r} \wedge \omega = \mu \text{ is holomorphic} \\ \iff & f_1 \dots f_r \cdot \sum_j \frac{df_j}{f_j} \wedge \omega = f_1 \dots f_r (d\omega - \mu) \\ \iff & f_j \text{ divides } df_j \wedge \omega, 1 \le j \le r. \end{align*} $$

The proof of Theorem 2.1 will be based in the following:
Another observation is that Theorem 2.1 is false if $p = n - 1$ and $n \ge 3$ as shows the following example in $\mathbb{C}^n$, $n \ge 3$:

*Example 2.4.* — Let $P$ be an irreducible homogeneous polynomial of degree $n$ on $\mathbb{C}^n$ and set

$$ \eta = \frac{i_R(dz_1 \wedge \cdots \wedge dz_n)}{P(z_1, \ldots, z_n)} = \frac{\sum_{j=1}^{n} (-1)^{j-1} z_j dz_1 \wedge \cdots \wedge \widehat{dz_j} \wedge \cdots \wedge dz_n}{P(z_1, \ldots, z_n)}, $$

where in the symbol $\widehat{dz_j}$ means omission of $dz_j$ in the product.

We would like to observe that the same example shows that Corollary 1.3 is false in $\mathbb{P}^m = \mathbb{P}^{n-1}$ if $p = m$: the form $\eta$ represents in homogeneous coordinates a closed logarithmic $m$-form on $\mathbb{P}^m$ which is not like in the statement of the corollary.

## 2.1. Preliminaries

Let $\eta$ be a germ at $0 \in \mathbb{C}^n$ of meromorphic $p$-form with reduced pole divisor $X = (f_1 \dots f_r = 0)$, $r \ge 1$. At the beginning we will not assume that $\eta$ is closed.

It follows from the definition that $\eta$ is logarithmic if, and only if, $f_1 \dots f_r \cdot d\eta$ is holomorphic. Since $(\eta)_\infty = (f_1 \dots f_r)$ we can write $\eta = \frac{1}{f_1 \dots f_r} \omega$ where $\omega \in \Omega_n^p$ is a germ of holomorphic $p$-form. We would like to observe that the following assertions are equivalent:

(a) $\eta = \frac{1}{f_1 \dots f_r} \omega$ is logarithmic.

(b) $f_j$ divides $df_j \wedge \omega$, for all $1 \le j \le r$.

In particular, we have:

(c) $\frac{1}{f_1 \dots f_s} \omega$ is logarithmic, for all $s \le r$.

In fact:

$$ \begin{align*} (a) \iff & f_1 \dots f_r \cdot d\eta = d\omega - \frac{d(f_1 \dots f_r)}{f_1 \dots f_r} \wedge \omega = \mu \text{ is holomorphic} \\ \iff & f_1 \dots f_r \cdot \sum_j \frac{df_j}{f_j} \wedge \omega = f_1 \dots f_r (d\omega - \mu) \\ \iff & f_j \text{ divides } df_j \wedge \omega, 1 \le j \le r. \end{align*} $$

The proof of Theorem 2.1 will be based in the following:
LEMMA 2.5. — Let $\eta = \frac{1}{f_1 \dots f_r} \omega$ be a germ of at $0 \in \mathbb{C}^n$ of logarithmic $p$-form, where $1 \le p \le n-2$. Assume that the pole divisor of $\eta$ is $X = (f_1 \dots f_r = 0)$, $r \ge 1$, has strictly ordinary singularities outside 0. Then $\eta$ can be written as

$$ \eta = \alpha_0 + \sum_{s=1}^{p-1} \left( \sum_{I \in S_r^s} \alpha_I \wedge \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{i_s}}{f_{i_s}} \right) + \sum_{J \in S_r^p} g_J \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}, \quad (2.2) $$

where $\alpha_0 \in \Omega_n^p$, $\alpha_I \in \Omega_n^{p-s}$ if $I \in S_r^s$, $s < p$, and $g_J \in \mathcal{O}_n$ if $J \in S_r^p$.

The proof of Lemma 2.5 relies in the concept of residue of a logarithmic form along an irreducible pole (cf. [10]).

### 2.1.1. Residues of a logarithmic form

Let $\eta = \frac{1}{f_1 \dots f_r} \omega$ be a germ at $0 \in \mathbb{C}^n$ of logarithmic $p$-form with pole divisor $X = (f_1 \dots f_r = 0)$. Let us define the residue along $Y_k := (f_k = 0)$, $1 \le k \le r$.

Fix representatives of $f_1, \dots, f_r$ and $\eta$, denoted by the same symbols, on some polydisc $Q$. We will assume that the $f_{j's}$ are irreducible in $Q$, and that the divisor $f_1 \dots f_r$ has strictly ordinary singularities on $Q \setminus \{0\}$. In particular, the $f_{j's}$ have isolated singularity at $0 \in Q$. We have seen that $f_k$ divides $\mathrm{d}f_k \wedge \omega$. In particular, we can write $\mathrm{d}f_k \wedge \omega = f_k \cdot \theta$ where $\theta \in \Omega_n^{p+1}$. This implies that $\mathrm{d}f_k \wedge \theta = 0$. Since $\mathrm{d}f_k$ has an isolated singularity at $0 \in Q$ and $p+1 \le n-1$, it follows from de Rham's division theorem [21] that $\theta = \mathrm{d}f_k \wedge \beta_k$, where $\beta_k \in \Omega^p(Q)$. Therefore, we can write $\mathrm{d}f_k \wedge (\omega - f_k \cdot \beta_k) = 0$ which implies, via the division theorem [21], that there exists $\alpha_k \in \Omega^{p-1}(Q)$ such that $\omega = \alpha_k \wedge \mathrm{d}f_k + f_k \cdot \beta_k$. The residue of $\frac{1}{f_k} \omega$ along $Y_k$ is the $(p-1)$-form along $Y_k$ defined as $\mathrm{Res}(\frac{1}{f_k} \omega, Y_k) := \alpha_k|_{Y_k}$. Finally, the residue of $\eta = \frac{1}{f_1 \dots f_r} \omega$ along $Y_k$ is defined as $\mathrm{Res}(\eta, Y_k) := \frac{1}{f_1 \dots f_k \dots f_r} \alpha_k|_{Y_k}$, where $\hat{f}_k$ means omission of the factor $f_k$ in the product.

**Remark 2.6.** — Let $\eta$ and $Y_k$ be as above. It is well known that $\mathrm{Res}(\eta, Y_k)$ does not depend on the particular decomposition $\omega = \alpha_k \wedge \mathrm{d}f_k + f_k \beta_k$ and on the particular equation of $Y_k$ (cf. [10]).

The above remark allow us to define the residue of a logarithmic form $\eta$ on an arbitrary complex manifold $M$ along any codimension one smooth irreducible submanifold $Y$ contained in the pole divisor of $\eta$. In particular, we can define the iterated residue. Given $I = (i_1 < \dots < i_k) \in S_r^k$, set
LEMMA 2.5. — Let $\eta = \frac{1}{f_1 \dots f_r} \omega$ be a germ of at $0 \in \mathbb{C}^n$ of logarithmic $p$-form, where $1 \le p \le n-2$. Assume that the pole divisor of $\eta$ is $X = (f_1 \dots f_r = 0)$, $r \ge 1$, has strictly ordinary singularities outside 0. Then $\eta$ can be written as

$$ \eta = \alpha_0 + \sum_{s=1}^{p-1} \left( \sum_{I \in S_r^s} \alpha_I \wedge \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{i_s}}{f_{i_s}} \right) + \sum_{J \in S_r^p} g_J \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}, \quad (2.2) $$

where $\alpha_0 \in \Omega_n^p$, $\alpha_I \in \Omega_n^{p-s}$ if $I \in S_r^s$, $s < p$, and $g_J \in \mathcal{O}_n$ if $J \in S_r^p$.

The proof of Lemma 2.5 relies in the concept of residue of a logarithmic form along an irreducible pole (cf. [10]).

### 2.1.1. Residues of a logarithmic form

Let $\eta = \frac{1}{f_1 \dots f_r} \omega$ be a germ at $0 \in \mathbb{C}^n$ of logarithmic $p$-form with pole divisor $X = (f_1 \dots f_r = 0)$. Let us define the residue along $Y_k := (f_k = 0)$, $1 \le k \le r$.

Fix representatives of $f_1, \dots, f_r$ and $\eta$, denoted by the same symbols, on some polydisc $Q$. We will assume that the $f_{j's}$ are irreducible in $Q$, and that the divisor $f_1 \dots f_r$ has strictly ordinary singularities on $Q \setminus \{0\}$. In particular, the $f_{j's}$ have isolated singularity at $0 \in Q$. We have seen that $f_k$ divides $\mathrm{d}f_k \wedge \omega$. In particular, we can write $\mathrm{d}f_k \wedge \omega = f_k \cdot \theta$ where $\theta \in \Omega_n^{p+1}$. This implies that $\mathrm{d}f_k \wedge \theta = 0$. Since $\mathrm{d}f_k$ has an isolated singularity at $0 \in Q$ and $p+1 \le n-1$, it follows from de Rham's division theorem [21] that $\theta = \mathrm{d}f_k \wedge \beta_k$, where $\beta_k \in \Omega^p(Q)$. Therefore, we can write $\mathrm{d}f_k \wedge (\omega - f_k \cdot \beta_k) = 0$ which implies, via the division theorem [21], that there exists $\alpha_k \in \Omega^{p-1}(Q)$ such that $\omega = \alpha_k \wedge \mathrm{d}f_k + f_k \cdot \beta_k$. The residue of $\frac{1}{f_k} \omega$ along $Y_k$ is the $(p-1)$-form along $Y_k$ defined as $\mathrm{Res}(\frac{1}{f_k} \omega, Y_k) := \alpha_k|_{Y_k}$. Finally, the residue of $\eta = \frac{1}{f_1 \dots f_r} \omega$ along $Y_k$ is defined as $\mathrm{Res}(\eta, Y_k) := \frac{1}{f_1 \dots f_k \dots f_r} \alpha_k|_{Y_k}$, where $\hat{f}_k$ means omission of the factor $f_k$ in the product.

**Remark 2.6.** — Let $\eta$ and $Y_k$ be as above. It is well known that $\mathrm{Res}(\eta, Y_k)$ does not depend on the particular decomposition $\omega = \alpha_k \wedge \mathrm{d}f_k + f_k \beta_k$ and on the particular equation of $Y_k$ (cf. [10]).

The above remark allow us to define the residue of a logarithmic form $\eta$ on an arbitrary complex manifold $M$ along any codimension one smooth irreducible submanifold $Y$ contained in the pole divisor of $\eta$. In particular, we can define the iterated residue. Given $I = (i_1 < \dots < i_k) \in S_r^k$, set
$X_I = (f_{i_1} = \cdots = f_{i_k} = 0)$ and $X_I^* = X_I \setminus \{0\}$. We define $\text{Res}(\eta, X_I)$ inductively. If $k=1$ then $\text{Res}(\eta, X_I) = \text{Res}(\eta, Y_{i_1})$ and for $k \ge 2$, $\text{Res}(\eta, X_I) = \text{Res}(\text{Res}(\eta, Y_{i_k}), X_I \setminus \{i_k\})$. This definition depends only of the ordering of the $f_{j's}$, that we will assume fixed.

**Example 2.7.** — If $\eta = \alpha \wedge \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_k}}{f_{i_k}}$, where $\alpha$ is holomorphic, then $\text{Res}(\eta, X_I) = \alpha|_{X_I}$, $I = (i_1 < \cdots < i_k)$. We leave the proof to the reader.

**Remark 2.8.** — Let $\eta = \frac{1}{f_1 \cdots f_k} \omega$ be logarithmic as above, $Y_k = (f_k = 0)$, where we assume $n \ge p + 2$. We would like to observe the following facts:

(a) If $\text{Res}(\eta, Y_k) = 0$ then $f_k$ divides $\omega$, or equivalently $f_k$ is not contained in the pole divisor of $\eta$.

(b) If $p=1$ then $\text{Res}(\eta, Y_k)$ is a holomorphic function on $Y_k$.

(c) If $p \ge 2$ then $\text{Res}(\eta, Y_k)$ is logarithmic on $Y_k$. Moreover, the pole divisor of $\text{Res}(\eta, Y_k)$ is $f_1 \cdots \hat{f}_k \cdots f_r|_{Y_k}$.

(d) $d\eta$ is logarithmic and $\text{Res}(d\eta, Y_k) = d\text{Res}(\eta, Y_k)$. In particular, if $\eta$ is closed then $\text{Res}(\eta, Y_k)$ is closed.

We will prove (b) and (c). The proofs of (a) and (d) will be left to the reader. Write $\omega = \alpha_k \wedge df_k + f_k \beta_k$ (resp. $\omega = g_k \wedge df_k + f_k \beta_k$ if $p=1$) as before. It is sufficient to prove that if $\ell \ne k$ then $f_\ell|_{Y_k}$ divides $\alpha_k \wedge df_\ell|_{Y_k}$ (resp. $f_\ell|_{Y_k}$ divides $g_k|_{Y_k}$ if $p=1$). Note that $\dim(Y_k \cap Y_\ell) \ge 1$, because $n \ge p+2 \ge 3$. Therefore we can fix a point $m \in Y_k \cap Y_\ell$ where $df_k(m) \wedge df_\ell(m) \ne 0$. Let $(U, z = (z_1, \dots, z_n))$ be a coordinate system around $m$ such that $f_k|_U = z_1$ and $f_\ell|_U = z_2$. Write $\omega = \omega_1 \wedge dz_1 + \omega_2 \wedge dz_2 + \omega_{12} \wedge dz_1 \wedge dz_2 + \theta$, where $\omega_1$ does not contain terms with $dz_2$, $\omega_2$ does not contain terms with $dz_1$ and $\theta$ does not contain terms in $dz_1$ or $dz_2$ (resp. $\omega = \sum_j h_j dz_j$ if $p=1$).

Let us consider the case $p > 1$. In this situation, $\omega \wedge dz_1 = \omega_2 \wedge dz_2 \wedge dz_1 + \theta \wedge dz_1$ and $\theta \wedge dz_1$ does not contain terms with $dz_1 \wedge dz_2$, so that $z_1$ divides $\omega_2$ and $\theta$. Similarly, $z_2$ divides $\omega_1$ and $\theta$. Therefore, we can write $\omega = z_2 \tilde{\omega}_1 \wedge dz_1 + z_1 \tilde{\omega}_2 + \omega_{12} \wedge dz_1 \wedge dz_2 + z_1 z_2 \tilde{\theta}$, which implies

$$ \begin{align*} \omega &= (z_2 \tilde{\omega}_1 \wedge -\omega_{12} \wedge dz_2) \wedge dz_1 + z_1 (\tilde{\omega}_2 + z_2 \tilde{\theta}) \\ &\implies \alpha_k|_{Y_k \cap U} = (z_2 \tilde{\omega}_1 \wedge -\omega_{12} \wedge dz_2)|_{Y_k \cap U} \\ &\implies \alpha_k \wedge dz_2|_{Y_k \cap U} = z_2 \tilde{\omega}_2 \wedge dz_2|_{Y_k \cap U}, \end{align*} $$

which implies (c).

In the case $p=1$, since $z_j$ divides $\omega \wedge dz_j$, $\forall j$, then $z_1$ divides $h_j$ if $j > 1$ and $z_2$ divides $h_1$, so that $\omega|_U = z_2 \tilde{h}_1 dz_1 + z_1 \sum_{j>1} \tilde{h}_j dz_j$. Hence, $g_k|_{Y_k \cap U} = z_2 \tilde{h}_1|_{Y_k \cap U}$, which implies (b).
$X_I = (f_{i_1} = \cdots = f_{i_k} = 0)$ and $X_I^* = X_I \setminus \{0\}$. We define $\text{Res}(\eta, X_I)$ inductively. If $k=1$ then $\text{Res}(\eta, X_I) = \text{Res}(\eta, Y_{i_1})$ and for $k \ge 2$, $\text{Res}(\eta, X_I) = \text{Res}(\text{Res}(\eta, Y_{i_k}), X_{I \setminus \{i_k\}})$. This definition depends only of the ordering of the $f_{j's}$, that we will assume fixed.

**Example 2.7.** — If $\eta = \alpha \wedge \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_k}}{f_{i_k}}$, where $\alpha$ is holomorphic, then $\text{Res}(\eta, X_I) = \alpha|_{X_I}$, $I = (i_1 < \cdots < i_k)$. We leave the proof to the reader.

**Remark 2.8.** — Let $\eta = \frac{1}{f_1 \cdots f_k} \omega$ be logarithmic as above, $Y_k = (f_k = 0)$, where we assume $n \ge p + 2$. We would like to observe the following facts:

(a) If $\text{Res}(\eta, Y_k) = 0$ then $f_k$ divides $\omega$, or equivalently $f_k$ is not contained in the pole divisor of $\eta$.

(b) If $p=1$ then $\text{Res}(\eta, Y_k)$ is a holomorphic function on $Y_k$.

(c) If $p \ge 2$ then $\text{Res}(\eta, Y_k)$ is logarithmic on $Y_k$. Moreover, the pole divisor of $\text{Res}(\eta, Y_k)$ is $f_1 \cdots \hat{f}_k \cdots f_r|_{Y_k}$.

(d) $d\eta$ is logarithmic and $\text{Res}(d\eta, Y_k) = d\text{Res}(\eta, Y_k)$. In particular, if $\eta$ is closed then $\text{Res}(\eta, Y_k)$ is closed.

We will prove (b) and (c). The proofs of (a) and (d) will be left to the reader. Write $\omega = \alpha_k \wedge df_k + f_k \beta_k$ (resp. $\omega = g_k \wedge df_k + f_k \beta_k$ if $p=1$) as before. It is sufficient to prove that if $\ell \ne k$ then $f_\ell|_{Y_k}$ divides $\alpha_k \wedge df_\ell|_{Y_k}$ (resp. $f_\ell|_{Y_k}$ divides $g_k|_{Y_k}$ if $p=1$). Note that $\dim(Y_k \cap Y_\ell) \ge 1$, because $n \ge p+2 \ge 3$. Therefore we can fix a point $m \in Y_k \cap Y_\ell$ where $df_k(m) \wedge df_\ell(m) \ne 0$. Let $(U, z = (z_1, \dots, z_n))$ be a coordinate system around $m$ such that $f_k|_U = z_1$ and $f_\ell|_U = z_2$. Write $\omega = \omega_1 \wedge dz_1 + \omega_2 \wedge dz_2 + \omega_{12} \wedge dz_1 \wedge dz_2 + \theta$, where $\omega_1$ does not contain terms with $dz_2$, $\omega_2$ does not contain terms with $dz_1$ and $\theta$ does not contain terms in $dz_1$ or $dz_2$ (resp. $\omega = \sum_j h_j dz_j$ if $p=1$).

Let us consider the case $p > 1$. In this situation, $\omega \wedge dz_1 = \omega_2 \wedge dz_2 \wedge dz_1 + \theta \wedge dz_1$ and $\theta \wedge dz_1$ does not contain terms with $dz_1 \wedge dz_2$, so that $z_1$ divides $\omega_2$ and $\theta$. Similarly, $z_2$ divides $\omega_1$ and $\theta$. Therefore, we can write $\omega = z_2 \tilde{\omega}_1 \wedge dz_1 + z_1 \tilde{\omega}_2 + \omega_{12} \wedge dz_1 \wedge dz_2 + z_1 z_2 \tilde{\theta}$, which implies

$$ \begin{align*} \omega &= (z_2 \tilde{\omega}_1 \wedge -\omega_{12} \wedge dz_2) \wedge dz_1 + z_1 (\tilde{\omega}_2 + z_2 \tilde{\theta}) \\ &\implies \alpha_k|_{Y_k \cap U} = (z_2 \tilde{\omega}_1 \wedge -\omega_{12} \wedge dz_2)|_{Y_k \cap U} \\ &\implies \alpha_k \wedge dz_2|_{Y_k \cap U} = z_2 \tilde{\omega}_2 \wedge dz_2|_{Y_k \cap U}, \end{align*} $$

which implies (c).

In the case $p=1$, since $z_j$ divides $\omega \wedge dz_j$, $\forall j$, then $z_1$ divides $h_j$ if $j > 1$ and $z_2$ divides $h_1$, so that $\omega|_U = z_2 \tilde{h}_1 dz_1 + z_1 \sum_{j>1} \tilde{h}_j dz_j$. Hence, $g_k|_{Y_k \cap U} = z_2 \tilde{h}_1|_{Y_k \cap U}$, which implies (b).
### 2.1.2. Proof of Lemma 2.5 in the case $p=1$

Since the proof of Lemma 2.5 is rather technical in the general case, we give first the proof in the case $p=1$ which contains essentially the idea of the general case.

As before, write $\eta = \frac{1}{f_1 \dots f_r} \omega$. The proof will be by induction on the number $r$ of components of the pole divisor.

**Formula (2.2) of Lemma 2.5 is true if $r=1$ and $p \ge 1$.** — When $\eta = \frac{1}{f_1} \omega$ is logarithmic we have seen in Section 2.1.1 that $\omega = \alpha_1 \wedge df_1 + f_1 \cdot \beta_1$ (resp. $\omega = g_1 df_1 + f_1 \beta_1$ if $p=1$). Hence $\eta = \alpha_1 \wedge \frac{df_1}{f_1} + \beta_1$ (resp. $\eta = g_1 \frac{df_1}{f_1} + \beta_1$ if $p=1$), as we wished.

If $p=1$ and (2.2) is true for $r-1 \ge 1$ then it is true for $r$. — Let $\eta = \frac{1}{f_1 \dots f_r} \omega$ and $Q \subset \mathbb{C}^n$ be a polydisc where $f_1, \dots, f_r$ and $\omega$ have representatives as before. As before we set $Y_j = (f_j = 0) \subset Q \subset \mathbb{C}^n$. We will use the following well known result in the case $n \ge 3$:

**LEMMA 2.9.** — Any holomorphic function $h \in \mathcal{O}(Y_j \setminus \{0\})$ has an extension $g \in \mathcal{O}(Q)$.

In fact, Lemma 2.9 is a particular case of Theorem 2.10 stated below and that will proved in Section 6 (see Remark 2.11).

Let $h = \text{Res}(\eta, Y_r) \in \mathcal{O}(Y_r \setminus \{0\})$. By Lemma 2.9, $h$ has an extension $g_r \in \mathcal{O}(Q)$. The form $g_r \frac{df_r}{f_r}$ is logarithmic and $\text{Res}(g_r \frac{df_r}{f_r}, Y_r) = h$. Therefore, the form $\tilde{\eta} = \eta - g_r \frac{df_r}{f_r}$ is also logarithmic and $\text{Res}(\tilde{\eta}, Y_r) = 0$. In particular, $f_r$ is not a pole of $\tilde{\eta}$ by Remark 2.8. Since the pole divisor of $\tilde{\eta}$ has $r-1$ irreducible components, by the induction hypothesis we can write

$$ \eta - g_r \frac{df_r}{f_r} = \tilde{\eta} = \alpha_0 + \sum_{j=1}^{r-1} g_j \frac{df_j}{f_j} \implies \text{Lemma 2.5 in the case } p=1. $$

### 2.1.3. Proof of Lemma 2.5 when $p \ge 2$

The case $p \ge 2$ is more involved, but the idea of the proof is the same as in the case $p=1$. Before given the details let us sketch the proof.

Given $s \in \{0, 1, \dots, r\}$ set $Y_s = (f_s = 0)$ if $s \ge 1$, $X_0 = Q$ and $X_s = Y_1 \cap \dots \cap Y_s$ if $s \ge 1$. Set also $X_s^* = X_s \setminus \{0\}$, $0 \le s \le r$. Note that $X_s = \{0\}$ if $s \ge n$. On the other hand, if $1 \le s \le n-1$ then $X_s$ is an analytic reduced germ of codimension $s$ and $X_s^*$ is a complex smooth manifold of dimension
### 2.1.2. Proof of Lemma 2.5 in the case $p=1$

Since the proof of Lemma 2.5 is rather technical in the general case, we give first the proof in the case $p=1$ which contains essentially the idea of the general case.

As before, write $\eta = \frac{1}{f_1 \dots f_r} \omega$. The proof will be by induction on the number $r$ of components of the pole divisor.

**Formula (2.2) of Lemma 2.5 is true if $r=1$ and $p \ge 1$.** — When $\eta = \frac{1}{f_1} \omega$ is logarithmic we have seen in Section 2.1.1 that $\omega = \alpha_1 \wedge df_1 + f_1 \cdot \beta_1$ (resp. $\omega = g_1 df_1 + f_1 \beta_1$ if $p=1$). Hence $\eta = \alpha_1 \wedge \frac{df_1}{f_1} + \beta_1$ (resp. $\eta = g_1 \frac{df_1}{f_1} + \beta_1$ if $p=1$), as we wished.

If $p=1$ and (2.2) is true for $r-1 \ge 1$ then it is true for $r$. — Let $\eta = \frac{1}{f_1 \dots f_r} \omega$ and $Q \subset \mathbb{C}^n$ be a polydisc where $f_1, \dots, f_r$ and $\omega$ have representatives as before. As before we set $Y_j = (f_j = 0) \subset Q \subset \mathbb{C}^n$. We will use the following well known result in the case $n \ge 3$:

**LEMMA 2.9.** — Any holomorphic function $h \in \mathcal{O}(Y_j \setminus \{0\})$ has an extension $g \in \mathcal{O}(Q)$.

In fact, Lemma 2.9 is a particular case of Theorem 2.10 stated below and that will proved in Section 6 (see Remark 2.11).

Let $h = \text{Res}(\eta, Y_r) \in \mathcal{O}(Y_r \setminus \{0\})$. By Lemma 2.9, $h$ has an extension $g_r \in \mathcal{O}(Q)$. The form $g_r \frac{df_r}{f_r}$ is logarithmic and $\text{Res}(g_r \frac{df_r}{f_r}, Y_r) = h$. Therefore, the form $\tilde{\eta} = \eta - g_r \frac{df_r}{f_r}$ is also logarithmic and $\text{Res}(\tilde{\eta}, Y_r) = 0$. In particular, $f_r$ is not a pole of $\tilde{\eta}$ by Remark 2.8. Since the pole divisor of $\tilde{\eta}$ has $r-1$ irreducible components, by the induction hypothesis we can write

$$ \eta - g_r \frac{df_r}{f_r} = \tilde{\eta} = \alpha_0 + \sum_{j=1}^{r-1} g_j \frac{df_j}{f_j} \implies \text{Lemma 2.5 in the case } p=1. $$

### 2.1.3. Proof of Lemma 2.5 when $p \ge 2$

The case $p \ge 2$ is more involved, but the idea of the proof is the same as in the case $p=1$. Before given the details let us sketch the proof.

Given $s \in \{0, 1, \dots, r\}$ set $Y_s = (f_s = 0)$ if $s \ge 1$, $X_0 = Q$ and $X_s = Y_1 \cap \dots \cap Y_s$ if $s \ge 1$. Set also $X_s^* = X_s \setminus \{0\}$, $0 \le s \le r$. Note that $X_s = \{0\}$ if $s \ge n$. On the other hand, if $1 \le s \le n-1$ then $X_s$ is an analytic reduced germ of codimension $s$ and $X_s^*$ is a complex smooth manifold of dimension
$n-s$. The proof will involve two induction arguments. In order to state properly these arguments we need a definition.

Given $1 \le s \le p \le n-2$ and $q \ge 1$, we will say that $X_s^*$ satisfies the $q$ decomposition property if any logarithmic $q$-form $\theta$ on $X_s^*$ with pole divisor on the zeroes of $f_{s+1}\dots f_r|_{X_s^*} := \tilde{f}_{s+1}\dots \tilde{f}_r$ can be decomposed as in formula (2.2):

$$ \theta = \alpha_0 + \sum_{\ell=1}^{q-1} \left( \sum_{I \in S_{r-s}^\ell} \alpha_I \wedge \frac{d\tilde{f}_{i_1}}{\tilde{f}_{i_1}} \wedge \dots \wedge \frac{d\tilde{f}_{i_\ell}}{\tilde{f}_{i_\ell}} \right) + \sum_{J \in S_{r-s}^q} g_J \cdot \frac{d\tilde{f}_{j_1}}{\tilde{f}_{j_1}} \wedge \dots \wedge \frac{d\tilde{f}_{j_q}}{\tilde{f}_{j_q}}, $$

where $\alpha_0$ is a holomorphic $q$-form, the $\alpha_{I'}$ are holomorphic $(q-\ell)$-forms on $X_s^*$ and the $g_{J'}$ are holomorphic functions on $X_s^*$. We resume below the main steps in the arguments.

1st step. — If $0 \le s \le p-1$ then $X_s^*$ satisfies the 1 decomposition property.

2nd step. — If $2 \le q \le p-s$, where $s \ge 0$, and $X_{s+1}^*$ satisfies the $q-1$ decomposition property then $X_s^*$ satisfies the $q$ decomposition property.

The 1st and 2nd steps above will be proved by induction on the number of $r \ge 1$ of factors in the pole divisor $f_1 \dots f_r$. In the proof we will use the following result:

THEOREM 2.10. — Let $X_s$ and $X_s^*$ be as above, where $s = n-k$ and $2 \le k \le n-1$ ($\dim_{\mathbb{C}}(X_s) = k$). Then there are representatives of $X_s$ and $X_s^*$ in a polydisc $Q \subset \mathbb{C}^n$, denoted by the same letters, such that:

(a) If $0 \le q \le k-2$ then any form $\tilde{\alpha} \in \Omega^q(X_s^*)$ can be extended to a form $\alpha \in \Omega^q(Q)$.

(b) If $q \ge 1$, $\ell \ge 0$ and $1 \le q + \ell \le k-2$ then $H^q(X_s^*, \Omega^\ell) = 0$.

Remark 2.11. — Note that Lemma 2.9 is a particular case of Theorem 2.10(a).

Theorem 2.10 implies that, if $X_s^*$ is as before and $0 \le q \le n-s-2$, then any holomorphic $q$-form on $X_s^*$ can be extended to a holomorphic $q$-form on $Q$. The proof of Theorem 2.10 will be done in Section 6. Let us finish the proof of Lemma 2.5 assuming Theorem 2.10.

Proof of the 1st step. — It is similar to the case $p=1$ done above with Lemma 2.9 (which corresponds to the case $q=0$ in Theorem 2.10). Therefore, we will assume $1 \le s \le p-1$. Note that the 1st step is trivially true if $r=s$, because in this case the pole divisor is empty and the 1-form is holomorphic.
$n-s$. The proof will involve two induction arguments. In order to state properly these arguments we need a definition.

Given $1 \le s \le p \le n-2$ and $q \ge 1$, we will say that $X_s^*$ satisfies the $q$ decomposition property if any logarithmic $q$-form $\theta$ on $X_s^*$ with pole divisor on the zeroes of $f_{s+1}\dots f_r|_{X_s^*} := \tilde{f}_{s+1}\dots \tilde{f}_r$ can be decomposed as in formula (2.2):

$$ \theta = \alpha_0 + \sum_{\ell=1}^{q-1} \left( \sum_{I \in S_{r-s}^\ell} \alpha_I \wedge \frac{d\tilde{f}_{i_1}}{\tilde{f}_{i_1}} \wedge \dots \wedge \frac{d\tilde{f}_{i_\ell}}{\tilde{f}_{i_\ell}} \right) + \sum_{J \in S_{r-s}^q} g_J \cdot \frac{d\tilde{f}_{j_1}}{\tilde{f}_{j_1}} \wedge \dots \wedge \frac{d\tilde{f}_{j_q}}{\tilde{f}_{j_q}}, $$

where $\alpha_0$ is a holomorphic $q$-form, the $\alpha_{I'}$ are holomorphic $(q-\ell)$-forms on $X_s^*$ and the $g_{J'}$ are holomorphic functions on $X_s^*$. We resume below the main steps in the arguments.

**1st step.** — If $0 \le s \le p-1$ then $X_s^*$ satisfies the 1 decomposition property.

**2nd step.** — If $2 \le q \le p-s$, where $s \ge 0$, and $X_{s+1}^*$ satisfies the $q-1$ decomposition property then $X_s^*$ satisfies the $q$ decomposition property.

The 1st and 2nd steps above will be proved by induction on the number of $r \ge 1$ of factors in the pole divisor $f_1 \dots f_r$. In the proof we will use the following result:

**THEOREM 2.10.** — Let $X_s$ and $X_s^*$ be as above, where $s = n-k$ and $2 \le k \le n-1$ ($\dim_{\mathbb{C}}(X_s) = k$). Then there are representatives of $X_s$ and $X_s^*$ in a polydisc $Q \subset \mathbb{C}^n$, denoted by the same letters, such that:

(a) If $0 \le q \le k-2$ then any form $\tilde{\alpha} \in \Omega^q(X_s^*)$ can be extended to a form $\tilde{\alpha} \in \Omega^q(Q)$.

(b) If $q \ge 1$, $\ell \ge 0$ and $1 \le q + \ell \le k-2$ then $H^q(X_s^*, \Omega^\ell) = 0$.

**Remark 2.11.** — Note that Lemma 2.9 is a particular case of Theorem 2.10(a).

Theorem 2.10 implies that, if $X_s^*$ is as before and $0 \le q \le n-s-2$, then any holomorphic $q$-form on $X_s^*$ can be extended to a holomorphic $q$-form on $Q$. The proof of Theorem 2.10 will be done in Section 6. Let us finish the proof of Lemma 2.5 assuming Theorem 2.10.

**Proof of the 1st step.** — It is similar to the case $p=1$ done above with Lemma 2.9 (which corresponds to the case $q=0$ in Theorem 2.10). Therefore, we will assume $1 \le s \le p-1$. Note that the 1st step is trivially true if $r=s$, because in this case the pole divisor is empty and the 1-form is holomorphic.
Assume that the assertion is true for any logarithmic 1-form on $X_s^*$ with pole divisor containing $\ell - 1 \ge 0$ functions in the set $\{\tilde{f}_j = f_j|_{X_s^*} | s+1 \le j \le r\}$. Let $\theta$ be a logarithmic 1-form on $X_s^*$ with pole divisor $\tilde{f}_{s+1} \dots \tilde{f}_{s+\ell}$. By Remark 2.8, $\tilde{g}_{s+1} := \text{Res}(\theta, X_{s+1}^*) \in \mathcal{O}(X_{s+1}^*)$. By Theorem 2.10, $\tilde{g}_{s+1}$ admits an extension $g_{s+1} \in \mathcal{O}(Q)$. In particular, $\tilde{g}_{s+1} := g_{s+1}|_{X_s^*}$ is a holomorphic extension of $\tilde{g}_{s+1}$ on $X_s^*$. Let $\tilde{\theta} := \tilde{g}_{s+1} \frac{d\tilde{f}_{s+1}}{\tilde{f}_{s+1}}$. Then $\tilde{\theta}$ is logarithmic and $\text{Res}(\tilde{\theta}, X_{s+1}^*) = \text{Res}(\theta, X_{s+1}^*)$. In particular, $\tilde{f}_{s+1}$ is not contained in the pole divisor of $\theta - \tilde{\theta}$, by Remark 2.8(a). By the induction hypothesis, $\theta - \tilde{\theta}$ can be decomposed as in (2.2):

$$ \theta - \tilde{\theta} = \alpha_0 + \sum_{j=2}^{\ell} \tilde{g}_{s+j} \frac{d\tilde{f}_{s+j}}{\tilde{f}_{s+j}} \implies \text{1st step.} $$

*Proof of the 2$^{nd}$ step.* — The proof is again by induction on the number $r-s$ of factors of the pole divisor. The assertion is trivially true if $r=s$. Assume that the assertion is true for any logarithmic $q$-form, $2 \le q \le p-s$, on $X_s^*$ with pole divisor containing $\ell - 1 \ge 0$ functions in the set $\{\tilde{f}_j = f_j|_{X_s^*} | s+1 \le j \le r\}$. Let $\theta$ be a logarithmic $q$-form on $X_s^*$ with pole divisor $\tilde{f}_{s+1} \dots \tilde{f}_{s+\ell}$. By Remark 2.8 the $(q-1)$-form $\mu := \text{Res}(\theta, X_{s+1}^*)$ is logarithmic on $X_{s+1}^*$ with pole divisor $\tilde{f}_{s+2} \dots \tilde{f}_{s+\ell} := \tilde{f}_{s+2} \dots \tilde{f}_{s+\ell}|_{X_{s+1}^*}$ (or holomorphic if $\ell=1$). Since $X_{s+1}^*$ satisfies the $q-1$ decomposition property, we can write

$$ \mu = \alpha_0 + \sum_{t=1}^{q-2} \sum_{I \in S_r^t} \alpha_I \wedge \frac{d\tilde{f}_{s+i_1-1}}{\tilde{f}_{s+i_1-1}} \wedge \dots \wedge \frac{d\tilde{f}_{s+i_t-1}}{\tilde{f}_{s+i_t-1}} \\ + \sum_{J \in S_r^{q-1}} g_J \cdot \frac{d\tilde{f}_{s+j_1-1}}{\tilde{f}_{s+j_1-1}} \wedge \dots \wedge \frac{d\tilde{f}_{s+j_{(q-1)}-1}}{\tilde{f}_{s+j_{(q-1)}-1}} $$

where $\alpha_0$ and the $\alpha_{I'}$s are holomorphic forms on $X_{s+1}^*$ and the $g_{J'}$s are holomorphic functions on $X_{s+1}^*$. By Theorem 2.10 each $\alpha_I$ (resp. each $g_J$) has a holomorphic extension $\tilde{\alpha}_I$ (resp. $\tilde{g}_J$) on $X_s^*$. Therefore, $\mu$ has a logarithmic extension $\tilde{\mu}$ on $X_s^*$,

$$ \tilde{\mu} = \tilde{\alpha}_0 + \sum_{t=1}^{q-2} \sum_{I \in S_r^t} \tilde{\alpha}_I \wedge \frac{d\tilde{f}_{s+i_1-1}}{\tilde{f}_{s+i_1-1}} \wedge \dots \wedge \frac{d\tilde{f}_{s+i_t-1}}{\tilde{f}_{s+i_t-1}} \\ + \sum_{J \in S_r^{q-1}} \tilde{g}_J \cdot \frac{d\tilde{f}_{s+j_1-1}}{\tilde{f}_{s+j_1-1}} \wedge \dots \wedge \frac{d\tilde{f}_{s+j_{(q-1)}-1}}{\tilde{f}_{s+j_{(q-1)}-1}} $$
Assume that the assertion is true for any logarithmic 1-form on $X_s^*$ with pole divisor containing $\ell - 1 \ge 0$ functions in the set $\{\tilde{f}_j = f_j|_{X_s^*} | s+1 \le j \le r\}$. Let $\theta$ be a logarithmic 1-form on $X_s^*$ with pole divisor $\tilde{f}_{s+1} \dots \tilde{f}_{s+\ell}$. By Remark 2.8, $\tilde{g}_{s+1} := \text{Res}(\theta, X_{s+1}^*) \in \mathcal{O}(X_{s+1}^*)$. By Theorem 2.10, $\tilde{g}_{s+1}$ admits an extension $g_{s+1} \in \mathcal{O}(Q)$. In particular, $\tilde{g}_{s+1} := g_{s+1}|_{X_s^*}$ is a holomorphic extension of $\tilde{g}_{s+1}$ on $X_s^*$. Let $\tilde{\theta} := \tilde{g}_{s+1} \frac{d\tilde{f}_{s+1}}{\tilde{f}_{s+1}}$. Then $\tilde{\theta}$ is logarithmic and $\text{Res}(\tilde{\theta}, X_{s+1}^*) = \text{Res}(\theta, X_{s+1}^*)$. In particular, $\tilde{f}_{s+1}$ is not contained in the pole divisor of $\theta - \tilde{\theta}$, by Remark 2.8(a). By the induction hypothesis, $\theta - \tilde{\theta}$ can be decomposed as in (2.2):

$$ \theta - \tilde{\theta} = \alpha_0 + \sum_{j=2}^{\ell} \tilde{g}_{s+j} \frac{d\tilde{f}_{s+j}}{\tilde{f}_{s+j}} \implies \text{1st step.} $$

*Proof of the 2<sup>nd</sup> step.* — The proof is again by induction on the number $r-s$ of factors of the pole divisor. The assertion is trivially true if $r=s$. Assume that the assertion is true for any logarithmic $q$-form, $2 \le q \le p-s$, on $X_s^*$ with pole divisor containing $\ell - 1 \ge 0$ functions in the set $\{\tilde{f}_j = f_j|_{X_s^*} | s+1 \le j \le r\}$. Let $\theta$ be a logarithmic $q$-form on $X_s^*$ with pole divisor $\tilde{f}_{s+1} \dots \tilde{f}_{s+\ell}$. By Remark 2.8 the $(q-1)$-form $\mu := \text{Res}(\theta, X_{s+1}^*)$ is logarithmic on $X_{s+1}^*$ with pole divisor $\tilde{f}_{s+2} \dots \tilde{f}_{s+\ell} := \tilde{f}_{s+2} \dots \tilde{f}_{s+\ell}|_{X_{s+1}^*}$ (or holomorphic if $\ell=1$). Since $X_{s+1}^*$ satisfies the $q-1$ decomposition property, we can write

$$ \mu = \alpha_0 + \sum_{t=1}^{q-2} \sum_{I \in S_r^t} \alpha_I \wedge \frac{d\tilde{f}_{s+i_1-1}}{\tilde{f}_{s+i_1-1}} \wedge \dots \wedge \frac{d\tilde{f}_{s+i_t-1}}{\tilde{f}_{s+i_t-1}} \\ + \sum_{J \in S_r^{q-1}} g_J \cdot \frac{d\tilde{f}_{s+j_1-1}}{\tilde{f}_{s+j_1-1}} \wedge \dots \wedge \frac{d\tilde{f}_{s+j_{(q-1)}-1}}{\tilde{f}_{s+j_{(q-1)}-1}} $$

where $\alpha_0$ and the $\alpha_{I'}$s are holomorphic forms on $X_{s+1}^*$ and the $g_{J'}$s are holomorphic functions on $X_{s+1}^*$. By Theorem 2.10 each $\alpha_I$ (resp. each $g_J$) has a holomorphic extension $\tilde{\alpha}_I$ (resp. $\tilde{g}_J$) on $X_s^*$. Therefore, $\mu$ has a logarithmic extension $\tilde{\mu}$ on $X_s^*$,

$$ \tilde{\mu} = \tilde{\alpha}_0 + \sum_{t=1}^{q-2} \sum_{I \in S_r^t} \tilde{\alpha}_I \wedge \frac{d\tilde{f}_{s+i_1-1}}{\tilde{f}_{s+i_1-1}} \wedge \dots \wedge \frac{d\tilde{f}_{s+i_t-1}}{\tilde{f}_{s+i_t-1}} \\ + \sum_{J \in S_r^{q-1}} \tilde{g}_J \cdot \frac{d\tilde{f}_{s+j_1-1}}{\tilde{f}_{s+j_1-1}} \wedge \dots \wedge \frac{d\tilde{f}_{s+j_{(q-1)}-1}}{\tilde{f}_{s+j_{(q-1)}-1}} $$
Therefore, $\theta_1 := \tilde{\mu} \wedge \frac{d\tilde{f}_{s+1}}{f_{s+1}}$ is logarithmic on $X_s^*$ and

$$ \mathrm{Res}(\theta_1, X_{s+1}^*) = \mathrm{Res}(\theta, X_{s+1}^*) \implies \mathrm{Res}(\theta - \theta_1, X_{s+1}^*) = 0. $$

Hence, $\tilde{f}_{s+1}$ is not contained in the pole divisor of $\theta - \theta_1$. By the induction hypothesis, $\theta - \theta_1 := \theta_2$ admits a decomposition as in (2.2), and so $\theta = \theta_1 + \theta_2$ admits a decomposition as in (2.2). This finishes the proof of Lemma 2.5. $\square$

## 2.2. Proof of Theorem 2.1

In the proof of Theorem 2.1 we will use Theorem 2.10 and Hamm's generalization of Milnor's theorem (cf. [13], [14], [20] and [23]):

**THEOREM 2.12.** — Let $X = (f_1 = \cdots = f_\ell = 0)$ be a germ at $0 \in \mathbb{C}^m$ of a complete intersection with an isolated singularity at 0, so that $\dim_{\mathbb{C}}(X) = m - \ell := n$. Then there exist representatives of $f_1, \dots, f_\ell$ and $X$ defined in a ball $B_\epsilon = B(0, \epsilon)$, denoted by the same letters, such that:

(a) $X^* = X \setminus \{0\}$ is rectratable to the link $K := X \cap \mathbb{S}_\epsilon^{2m-1}$, $\mathbb{S}_\epsilon^{2m-1} = \partial B_\epsilon$.

(b) If $n \ge 3$ then $K$ is $(n-2)$-connected. In particular, $X^*$ is connected and $H_{DR}^k(X^*) = \{0\}$ if $1 \le k \le n-2$.

(c) If $n=2$ then $X^*$ is connected.

When $n=1$, $X^*$ is not necessarily connected, as shows the example $X = (x^2 + y^2 + z^2 = z = 0) \subset \mathbb{C}^3$.

Let $\eta$ be a germ at $0 \in \mathbb{C}^n$ of a closed logarithmic $p$-form, $1 \le p \le n-2$, with pole divisor $f_1 \dots f_r$ with a strictly ordinary singularity outside 0. According to Lemma 2.5 we can write $\eta$ as a sum of a holomorphic $p$-form $\alpha_0$, and “monomial” $p$-forms of the type $\alpha_I \wedge \frac{df_{f_1}}{f_{f_1}} \wedge \cdots \wedge \frac{df_{f_s}}{f_{f_s}}$, or $gJ \frac{df_{f_1}}{f_{f_1}} \wedge \cdots \wedge \frac{df_{f_p}}{f_{f_p}}$, where $I \in \mathcal{S}_r^s$ and $J \in \mathcal{S}_r^p$.

Given a monomial $\mu = \alpha_I \wedge \frac{df_{f_1}}{f_{f_1}} \wedge \cdots \wedge \frac{df_{f_s}}{f_{f_s}}$ we define the pseudo depth of $\mu$ as $\mathrm{dep}(\mu) = s$. Given $\eta = \sum_{j=1}^m \mu_j$, where the $\mu_j$ are monomials as above, we set $\widetilde{\mathrm{dep}}(\eta) = \max\{\widetilde{\mathrm{dep}}(\mu_j) | 1 \le j \le m\}$.

Observe that $\widetilde{\mathrm{dep}}$, as defined above, is not well defined. For instance, if $g \in \langle f_1, \dots, f_p \rangle$, the ideal generated by $f_1, \dots, f_p$, $g = \sum_{j=1}^p h_j f_j$, then

$$ g \frac{\mathrm{d}f_1}{f_1} \wedge \cdots \wedge \frac{\mathrm{d}f_p}{f_p} = \sum_j \alpha_j \wedge \frac{\mathrm{d}f_1}{f_1} \wedge \cdots \wedge \frac{\mathrm{d}\widetilde{f}_j}{\widetilde{f}_j} \wedge \frac{\mathrm{d}f_p}{f_p}, $$
Therefore, $\theta_1 := \tilde{\mu} \wedge \frac{d\tilde{f}_{s+1}}{f_{s+1}}$ is logarithmic on $X_s^*$ and

$$ \mathrm{Res}(\theta_1, X_{s+1}^*) = \mathrm{Res}(\theta, X_{s+1}^*) \implies \mathrm{Res}(\theta - \theta_1, X_{s+1}^*) = 0. $$

Hence, $\tilde{f}_{s+1}$ is not contained in the pole divisor of $\theta - \theta_1$. By the induction hypothesis, $\theta - \theta_1 := \theta_2$ admits a decomposition as in (2.2), and so $\theta = \theta_1 + \theta_2$ admits a decomposition as in (2.2). This finishes the proof of Lemma 2.5. $\square$

## 2.2. Proof of Theorem 2.1

In the proof of Theorem 2.1 we will use Theorem 2.10 and Hamm's generalization of Milnor's theorem (cf. [13], [14], [20] and [23]):

**THEOREM 2.12.** — Let $X = (f_1 = \cdots = f_\ell = 0)$ be a germ at $0 \in \mathbb{C}^m$ of a complete intersection with an isolated singularity at 0, so that $\dim_{\mathbb{C}}(X) = m - \ell := n$. Then there exist representatives of $f_1, \dots, f_\ell$ and $X$ defined in a ball $B_\epsilon = B(0, \epsilon)$, denoted by the same letters, such that:

(a) $X^* = X \setminus \{0\}$ is rectratable to the link $K := X \cap \mathbb{S}_\epsilon^{2m-1}$, $\mathbb{S}_\epsilon^{2m-1} = \partial B_\epsilon$.

(b) If $n \ge 3$ then $K$ is $(n-2)$-connected. In particular, $X^*$ is connected and $H_{DR}^k(X^*) = \{0\}$ if $1 \le k \le n-2$.

(c) If $n=2$ then $X^*$ is connected.

When $n=1$, $X^*$ is not necessarily connected, as shows the example $X = (x^2 + y^2 + z^2 = z = 0) \subset \mathbb{C}^3$.

Let $\eta$ be a germ at $0 \in \mathbb{C}^n$ of a closed logarithmic $p$-form, $1 \le p \le n-2$, with pole divisor $f_1 \dots f_r$ with a strictly ordinary singularity outside 0. According to Lemma 2.5 we can write $\eta$ as a sum of a holomorphic $p$-form $\alpha_0$, and “monomial” $p$-forms of the type $\alpha_I \wedge \frac{df_{f_1}}{f_{f_1}} \wedge \cdots \wedge \frac{df_{f_s}}{f_{f_s}}$, or $gJ \frac{df_{f_1}}{f_{f_1}} \wedge \cdots \wedge \frac{df_{f_p}}{f_{f_p}}$, where $I \in \mathcal{S}_r^s$ and $J \in \mathcal{S}_r^p$.

Given a monomial $\mu = \alpha_I \wedge \frac{df_{f_1}}{f_{f_1}} \wedge \cdots \wedge \frac{df_{f_s}}{f_{f_s}}$ we define the pseudo depth of $\mu$ as $\mathrm{dep}(\mu) = s$. Given $\eta = \sum_{j=1}^m \mu_j$, where the $\mu_j$ are monomials as above, we set $\widetilde{\mathrm{dep}}(\eta) = \max\{\widetilde{\mathrm{dep}}(\mu_j) | 1 \le j \le m\}$.

Observe that $\widetilde{\mathrm{dep}}$, as defined above, is not well defined. For instance, if $g \in \langle f_1, \dots, f_p \rangle$, the ideal generated by $f_1, \dots, f_p$, $g = \sum_{j=1}^p h_j f_j$, then

$$ g \frac{\mathrm{d}f_1}{f_1} \wedge \cdots \wedge \frac{\mathrm{d}f_p}{f_p} = \sum_j \alpha_j \wedge \frac{\mathrm{d}f_1}{f_1} \wedge \cdots \wedge \frac{\mathrm{d}\widetilde{f}_j}{\widetilde{f}_j} \wedge \frac{\mathrm{d}f_p}{f_p}, $$
where $\alpha_j = \pm h_j df_j$, $1 \le j \le p$. Therefore, if $\eta$ is a logarithmic form as above, then we define its depth as

$$ \operatorname{depth}(\eta) = \min \left\{ \left\langle \operatorname{dep} \left( \sum_j \mu_j \right) \right| \eta = \sum_j \mu_j, \text{ where the } \mu_{j's} \text{ are monomials} \right\}. $$

When $\eta$ is holomorphic we define $\operatorname{depth}(\eta) = 0$.

**CLAIM 2.13.** — Let $\eta$ be a germ at $0 \in \mathbb{C}^n$ of logarithmic closed $p$-form, $1 \le p \le n-2$. If $\operatorname{depth}(\eta) = p$ there exists a collection $(\lambda_J)_{J \in \mathcal{S}_r^p}$, $\lambda_J \in \mathbb{C}$, such that

$$ \operatorname{depth} \left( \eta - \sum_{J \in \mathcal{S}_r^p} \lambda_J \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}} \right) \le p-1. $$

*Proof.* — If $\operatorname{depth}(\eta) = p$ then the decomposition of $\eta$ as in (2.2) contains at least one monomial of the form $\mu_J = g_J \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$, where $g_J \notin \langle f_{j_1}, \dots, f_{j_p} \rangle$. As before, set $X_J := (f_{j_1} = \cdots = f_{j_p} = 0)$ and $X_J^* = X_J \setminus \{0\}$. We assert that $g_J|_{X_J^*}$ is a constant $\lambda_J \in \mathbb{C}^*$.

In fact, since $\dim_{\mathbb{C}} X_J = n-p \ge 2$, $X_J^*$ is connected, by Theorem 2.12. Note that $\operatorname{Res}(\mu_J, X_J) = \operatorname{Res}(\eta, X_J) = g_J|_{X_J^*}$ (see Example 2.7). Since $\eta$ is closed, we have $\operatorname{Res}(d\eta, X_J) = dg_J|_{X_J^*} = 0$. Hence, $g_J|_{X_J^*} = \lambda_J \in \mathbb{C}$. On the other hand, if $\lambda_J = 0$ then $g_J|_{X_J} = 0$ and since $X_J$ is a complete intersection we get $g_J \in \langle f_{j_1}, \dots, f_{j_p} \rangle$, a contradiction.

Let $\mu_J := \lambda_J \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$. Note that $\eta - \mu_J$ is still logarithmic, closed and does not contain terms multiples of $\frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$. By repeating this procedure finitely many times we can find the collection $(\lambda_J)_{J \in \mathcal{S}_r^p}$ as in the statement of the claim. $\square$

**CLAIM 2.14.** — Let $\eta$ be logarithmic closed $p$-form with pole divisor $f_1 \dots f_r = 0$, with a strictly ordinary singularities at $0 \in \mathbb{C}^n$. If $\operatorname{depth}(\eta) < p$ then $\eta$ is exact: $\eta = d\Theta$, where $\Theta$ is either zero, or is logarithmic with pole divisor contained in $f_1 \dots f_r = 0$.

*Proof.* — The proof will be by induction on the depth of $\eta$. If $\operatorname{depth}(\eta) = 0$ then $\eta$ is holomorphic and so it is exact by Poincaré lemma. $\square$

Assume that any closed logarithmic $p$-form $\omega$ with $\operatorname{depth}(\omega) \le q-1 < p-1$ is exact: $\omega = d\theta$ with $\theta$ logarithmic as above. Let $\eta$ be a logarithmic $p$-form with pole divisor $f_1 \dots f_r = 0$ with $\operatorname{depth}(\eta) = q < p$. By the definition of depth, when we write $\eta$ as in (2.2) then we get

$$ \eta = \sum_{I \in \mathcal{S}_r^q} \alpha_I \wedge \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{i_q}}{f_{i_q}} + \beta, $$
where $\alpha_j = \pm h_j df_j$, $1 \le j \le p$. Therefore, if $\eta$ is a logarithmic form as above, then we define its depth as

$$ \text{depth}(\eta) = \min \left\{ \left| \widetilde{\text{dep}} \left( \sum_j \mu_j \right) \right| \middle| \eta = \sum_j \mu_j, \text{ where the } \mu_{j's} \text{ are monomials} \right\}. $$

When $\eta$ is holomorphic we define $\text{depth}(\eta) = 0$.

**CLAIM 2.13.** — Let $\eta$ be a germ at $0 \in \mathbb{C}^n$ of logarithmic closed $p$-form, $1 \le p \le n-2$. If $\text{depth}(\eta) = p$ there exists a collection $(\lambda_J)_{J \in \mathcal{S}_r^p}$, $\lambda_J \in \mathbb{C}$, such that

$$ \text{depth} \left( \eta - \sum_{J \in \mathcal{S}_r^p} \lambda_J \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}} \right) \le p-1. $$

*Proof.* — If $\text{depth}(\eta) = p$ then the decomposition of $\eta$ as in (2.2) contains at least one monomial of the form $\mu_J = g_J \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$, where $g_J \notin \langle f_{j_1}, \dots, f_{j_p} \rangle$. As before, set $X_J := (f_{j_1} = \cdots = f_{j_p} = 0)$ and $X_J^* = X_J \setminus \{0\}$. We assert that $g_J|_{X_J^*}$ is a constant $\lambda_J \in \mathbb{C}^*$.

In fact, since $\dim_{\mathbb{C}} X_J = n-p \ge 2$, $X_J^*$ is connected, by Theorem 2.12. Note that $\text{Res}(\mu_J, X_J) = \text{Res}(\eta, X_J) = g_J|_{X_J^*}$ (see Example 2.7). Since $\eta$ is closed, we have $\text{Res}(d\eta, X_J) = dg_J|_{X_J^*} = 0$. Hence, $g_J|_{X_J^*} = \lambda_J \in \mathbb{C}$. On the other hand, if $\lambda_J = 0$ then $g_J|_{X_J} = 0$ and since $X_J$ is a complete intersection we get $g_J \in \langle f_{j_1}, \dots, f_{j_p} \rangle$, a contradiction.

Let $\mu_J := \lambda_J \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$. Note that $\eta - \mu_J$ is still logarithmic, closed and does not contain terms multiples of $\frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$. By repeating this procedure finitely many times we can find the collection $(\lambda_J)_{J \in \mathcal{S}_r^p}$ as in the statement of the claim. $\square$

**CLAIM 2.14.** — Let $\eta$ be logarithmic closed $p$-form with pole divisor $f_1 \dots f_r = 0$, with a strictly ordinary singularities at $0 \in \mathbb{C}^n$. If $\text{depth}(\eta) < p$ then $\eta$ is exact: $\eta = d\Theta$, where $\Theta$ is either zero, or is logarithmic with pole divisor contained in $f_1 \dots f_r = 0$.

*Proof.* — The proof will be by induction on the depth of $\eta$. If $\text{depth}(\eta) = 0$ then $\eta$ is holomorphic and so it is exact by Poincaré lemma. $\square$

Assume that any closed logarithmic $p$-form $\omega$ with $\text{depth}(\omega) \le q-1 < p-1$ is exact: $\omega = d\theta$ with $\theta$ logarithmic as above. Let $\eta$ be a logarithmic $p$-form with pole divisor $f_1 \dots f_r = 0$ with $\text{depth}(\eta) = q < p$. By the definition of depth, when we write $\eta$ as in (2.2) then we get

$$ \eta = \sum_{I \in \mathcal{S}_r^q} \alpha_I \wedge \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{\mathrm{d}f_{i_q}}{f_{i_q}} + \beta, $$
where $\beta$ is logarithmic and depth($\beta$) < $q$. Recall that, if $I = (i_1, \dots, i_q) \in S_r^q$
then $X_I = (f_{i_1} = \dots = f_{i_q} = 0)$ and $X_I^* = X_I \setminus \{0\}$. As the reader can check,
we have

$$
\mathrm{Res}(\eta, X_I) = \alpha_I|_{X_I} := \tilde{\alpha}_I \in \Omega^{p-q}(X_I^*), \quad \forall I \in S_r^q,
$$

where $\tilde{\alpha}_I$ is closed, by Remark 2.8. Now, we use Theorem 2.12 and Theorem 2.10 (b): since $\dim(X_I^*) = n-q$ we get $H_{DR}^k(X^*) = 0$ if $1 \le k \le n-q-2$. But, $p \le n-2$ and so $p-q \le n-q-2 = \dim_{\mathbb{C}}(X^*) - 2$ which implies that $\tilde{\alpha}_I \in \Omega^{p-q}(X_I^*)$ is exact: $\tilde{\alpha}_I = d\tilde{\beta}_I$, where $\tilde{\beta}_I$ is in principle a $\mathcal{C}^\infty(p-q-1)$-form. However, the fact that $H^s\partial(X_I^*) \simeq H^r(X_I^*, \Omega^s) = 0$ if $r+s=p-q-1$ implies that we can assume $\tilde{\beta}_I \in \Omega^{p-q-1}(X_I^*)$ (cf. [12]).

Therefore, there are $(p-q-1)$-forms $\tilde{\beta}_I \in \Omega^{p-q-1}(X_I^*)$ such that $\tilde{\alpha}_I = d\tilde{\beta}_I, \forall I \in S_r^q$. By Theorem 2.10(a) each form $\tilde{\beta}_I$ admits an extension $\beta_I \in \Omega^{p-q-1}(Q)$, where $Q$ is some polydisc of $\mathbb{C}^n$ where $X_I$ has a representative. Define a logarithmic form $\mu$ by

$$
\mu = \sum_{I \in S_r^q} \beta_I \wedge \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_q}}{f_{i_q}}
$$

so that

$$
d\mu = \sum_{I \in S_r^q} d\beta_I \wedge \frac{df_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{df_{i_q}}{f_{i_q}} \implies \mathrm{Res}(d\mu, X_I) = d\beta_I|_{X_I} = \mathrm{Res}(\eta, X_I),
$$

for all $I \in S_r^q$. In particular, $\mathrm{Res}(\eta - d\mu, X_I) = 0$ for all $I \in S_r^q$, and this implies that depth$(\eta - d\mu) < q$. Finally, since $\eta - d\mu$ is closed the induction hypothesis implies that $\eta - d\mu = d\theta$, where either $\theta = 0$, or $\theta$ is logarithmic with pole divisor contained in $f_1 \dots f_r = 0$. This finishes the proof of Claim 2.14 and of Theorem 2.1. $\square$

**2.3. Proof of Corollary 1.3**

Let $\eta$ be a logarithmic $p$-form on $\mathbb{P}^n$, where $p \le n-1$, with pole divisor in
homogeneous coordinates $(f_1 \dots f_r = 0)$ with strictly ordinary singularities
outside $0 \in \mathbb{C}^{n+1}$. Let $\Pi: \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{P}^n$ be the canonical projection and
$\tilde{\eta} = \Pi^*(\eta)$. We want to prove that $\tilde{\eta}$ can be written as

$$
\tilde{\eta} = \sum_{I \in S_r^p} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}}
$$

where $\lambda_I \in \mathbb{C} \forall I \in S_r^p$. We know that $\eta$ is closed (Deligne's theorem 1.1).

The pull-back $\tilde{\eta} = \Pi^*(\eta)$ can be extended to a closed logarithmic $p$-form on $\mathbb{C}^{n+1}$ which is called the expression of $\eta$ in homogeneous coordinates. The
where $\beta$ is logarithmic and depth($\beta$) < $q$. Recall that, if $I = (i_1, \dots, i_q) \in S_r^q$
then $X_I = (f_{i_1} = \dots = f_{i_q} = 0)$ and $X_I^* = X_I \setminus \{0\}$. As the reader can check,
we have

$$
\mathrm{Res}(\eta, X_I) = \alpha_I|_{X_I} := \tilde{\alpha}_I \in \Omega^{p-q}(X_I^*), \quad \forall I \in S_r^q,
$$

where $\tilde{\alpha}_I$ is closed, by Remark 2.8. Now, we use Theorem 2.12 and Theorem 2.10 (b): since $\dim(X_I^*) = n-q$ we get $H_{DR}^k(X^*) = 0$ if $1 \le k \le n-q-2$. But, $p \le n-2$ and so $p-q \le n-q-2 = \dim_{\mathbb{C}}(X^*) - 2$ which implies that $\tilde{\alpha}_I \in \Omega^{p-q}(X_I^*)$ is exact: $\tilde{\alpha}_I = d\tilde{\beta}_I$, where $\tilde{\beta}_I$ is in principle a $C^\infty(p-q-1)$-form. However, the fact that $H^s_\partial(X_I^*) \simeq H^r(X_I^*, \Omega^s) = 0$ if $r+s=p-q-1$ implies that we can assume $\tilde{\beta}_I \in \Omega^{p-q-1}(X_I^*)$ (cf. [12]).

Therefore, there are $(p-q-1)$-forms $\tilde{\beta}_I \in \Omega^{p-q-1}(X_I^*)$ such that $\tilde{\alpha}_I = d\tilde{\beta}_I, \forall I \in S_r^q$. By Theorem 2.10(a) each form $\tilde{\beta}_I$ admits an extension $\beta_I \in \Omega^{p-q-1}(Q)$, where $Q$ is some polydisc of $\mathbb{C}^n$ where $X_I$ has a representative. Define a logarithmic form $\mu$ by

$$
\mu = \sum_{I \in S_r^q} \beta_I \wedge \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_q}}{f_{i_q}}
$$

so that

$$
d\mu = \sum_{I \in S_r^q} d\beta_I \wedge \frac{df_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{df_{i_q}}{f_{i_q}} \implies \mathrm{Res}(d\mu, X_I) = d\beta_I|_{X_I} = \mathrm{Res}(\eta, X_I),
$$

for all $I \in S_r^q$. In particular, $\mathrm{Res}(\eta - d\mu, X_I) = 0$ for all $I \in S_r^q$, and this implies that depth$(\eta - d\mu) < q$. Finally, since $\eta - d\mu$ is closed the induction hypothesis implies that $\eta - d\mu = d\theta$, where either $\theta = 0$, or $\theta$ is logarithmic with pole divisor contained in $f_1 \dots f_r = 0$. This finishes the proof of Claim 2.14 and of Theorem 2.1. $\square$

2.3. Proof of Corollary 1.3

Let $\eta$ be a logarithmic $p$-form on $\mathbb{P}^n$, where $p \le n-1$, with pole divisor in
homogeneous coordinates $(f_1 \dots f_r = 0)$ with strictly ordinary singularities
outside $0 \in \mathbb{C}^{n+1}$. Let $\Pi: \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{P}^n$ be the canonical projection and
$\tilde{\eta} = \Pi^*(\eta)$. We want to prove that $\tilde{\eta}$ can be written as

$$
\tilde{\eta} = \sum_{I \in S_r^p} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}}
$$

where $\lambda_I \in \mathbb{C} \forall I \in S_r^p$. We know that $\eta$ is closed (Deligne's theorem 1.1).

The pull-back $\tilde{\eta} = \Pi^*(\eta)$ can be extended to a closed logarithmic $p$-form
on $\mathbb{C}^{n+1}$ which is called the expression of $\eta$ in homogeneous coordinates. The
pole divisor of $\tilde{\eta}$ is of course the pull-back of the pole divisor of $\eta$, and by assumption it is $f_1 \dots f_r$, where $f_j$ is a homogeneous polynomial of degree $d_j$, $1 \le j \le r$. In particular, we can write

$$ \tilde{\eta} = \frac{1}{f_1 \dots f_r} \sum g_j dz^j $$

where $dz^J = dz_{j_1} \wedge \dots \wedge dz_{j_p}$ and $g_J$ is a homogeneous polynomial. Using that $\tilde{\eta}$ is invariant by any homothety $H_t(z) = t.z, \forall t \in \mathbb{C}^*$, and with a straightforward computation we see that $g_J$ is homogeneous of degree $\deg(g_J) = \deg(f_1 \dots f_r) - p$. This implies that the coefficients $g_J/f_1 \dots f_r$ of $\tilde{\eta}$ are meromorphic homogenous of degree $-p$.

Now, the hypothesis on the pole divisor of $\eta$ implies that the pole divisor of $\tilde{\eta}$, $f_1 \dots f_r$, has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Therefore, by Theorem 2.1 we have

$$ \tilde{\eta} = \sum_{I \in S_r^p} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}} + \mathrm{d}\Theta, $$

where $\Theta$ is logarithmic with pole divisor contained in $(\eta)_\infty$. It is enough to prove that $\mathrm{d}\Theta = 0$.

The proof of Theorem 2.1 implies that the monomials of $\Theta$ have depth $< p$ and are, either of the form $\alpha \wedge \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{j_q}}{f_{j_q}}$, where $\alpha$ is a $(p-q-1)$- form, or of the form $g \cdot \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$, where $g$ is a holomorphic function. In particular, the monomials of $\mathrm{d}\Theta$ are, either of the form $\mathrm{d}\alpha \wedge \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{j_q}}{f_{j_q}}$, or of the form $\mathrm{d}g \wedge \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$. In both cases, the meromorphic degree of the coefficients of the monomial is $> -p$ and this implies that $\mathrm{d}\Theta = 0$. The proof that $i_R \tilde{\eta} = 0$ follows from the fact that $\mathrm{D}\Pi(z).R(z) = 0$ for all $z \in \mathbb{C}^{n+1} \setminus \{0\}$. Finally, $i_R \tilde{\eta} = 0$ implies that $r \ge p+1$, as the reader can check. $\square$

### 3. Decomposition of logarithmic foliations

The purpose of this section is to study the question posed in Problem 1.9: is an integrable logarithmic $p$-form on $\mathbb{P}^n$ totally decomposable into logarithmic $1$-forms?

The main theorem to be proved here gives a partial answer to the above problem:
pole divisor of $\tilde{\eta}$ is of course the pull-back of the pole divisor of $\eta$, and by assumption it is $f_1 \dots f_r$, where $f_j$ is a homogeneous polynomial of degree $d_j$, $1 \le j \le r$. In particular, we can write

$$ \tilde{\eta} = \frac{1}{f_1 \dots f_r} \sum g_J dz^J $$

where $dz^J = dz_{j_1} \wedge \dots \wedge dz_{j_p}$ and $g_J$ is a homogeneous polynomial. Using that $\tilde{\eta}$ is invariant by any homothety $H_t(z) = t.z, \forall t \in \mathbb{C}^*$, and with a straightforward computation we see that $g_J$ is homogeneous of degree $\deg(g_J) = \deg(f_1 \dots f_r) - p$. This implies that the coefficients $g_J/f_1 \dots f_r$ of $\tilde{\eta}$ are meromorphic homogenous of degree $-p$.

Now, the hypothesis on the pole divisor of $\eta$ implies that the pole divisor of $\tilde{\eta}$, $f_1 \dots f_r$, has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Therefore, by Theorem 2.1 we have

$$ \tilde{\eta} = \sum_{I \in S_r^p} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_p}}{f_{i_p}} + \mathrm{d}\Theta, $$

where $\Theta$ is logarithmic with pole divisor contained in $(\eta)_\infty$. It is enough to prove that $\mathrm{d}\Theta = 0$.

The proof of Theorem 2.1 implies that the monomials of $\Theta$ have depth $< p$ and are, either of the form $\alpha \wedge \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{j_q}}{f_{j_q}}$, where $\alpha$ is a $(p-q-1)$-form, or of the form $g \cdot \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$, where $g$ is a holomorphic function. In particular, the monomials of $\mathrm{d}\Theta$ are, either of the form $\mathrm{d}\alpha \wedge \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{j_q}}{f_{j_q}}$, or of the form $\mathrm{d}g \wedge \frac{\mathrm{d}f_{j_1}}{f_{j_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{j_p}}{f_{j_p}}$. In both cases, the meromorphic degree of the coefficients of the monomial is $> -p$ and this implies that $\mathrm{d}\Theta = 0$. The proof that $i_R \tilde{\eta} = 0$ follows from the fact that $\mathrm{D}\Pi(z).R(z) = 0$ for all $z \in \mathbb{C}^{n+1} \setminus \{0\}$. Finally, $i_R \tilde{\eta} = 0$ implies that $r \ge p+1$, as the reader can check. $\square$

### 3. Decomposition of logarithmic foliations

The purpose of this section is to study the question posed in Problem 1.9: is an integrable logarithmic $p$-form on $\mathbb{P}^n$ totally decomposable into logarithmic $1$-forms?

The main theorem to be proved here gives a partial answer to the above problem:
**THEOREM 1.10.** — Let $f_1, \dots, f_r$ be homogeneous polynomials on $\mathbb{C}^{n+1}$ and assume that the pole divisor $f_1 \dots f_r = 0$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Then:

(a) If $p=2$, or $r \in \{p+1, p+2\}$ then $\mathcal{L}_{\mathrm{td}}^p(f_1, \dots, f_r) = \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$.

(b) If $2 < p \le n-2$ and $r > p+2$ then $\mathcal{L}_{\mathrm{td}}^p(f_1, \dots, f_r)$ is an irreducible component of $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$. In particular, if $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$ is irreducible then $\mathcal{L}_{\mathrm{td}}^p(f_1, \dots, f_r) = \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$.

An interesting consequence of Theorem 1.10 is Corollary 1.11: when $r = p+1$ in Theorem 1.10 then $\mathcal{F}_\eta$ is a rational fibration (see Section 1). Since the case $r = p+1$ is the easier one, we will do it, together with the proof of Corollary 1.11, in Section 3.1.

In Section 3.2 we prove the theorem in the case $p=2$. In Section 3.3 we will see that the proof of (b) can be reduced to the case of 2-dimensional foliations (in which $p=n-2$). The proof of the case $r=p+2$ will be done in Section 3.4. We note that item (b) is an easy consequence of Theorem 1.17 and so it will be not done in this section.

**3.1. Proof of the case $r = p+1$ and of Corollary 1.11**

The proof will be based in the remark that a $p$-vector $\Omega$ in a vector space $V$ of dimension $p+1$ is always decomposable. In fact, if $\{v_1, \dots, v_{p+1}\}$ is a basis of $V$, then we can write

$$ \Omega = \sum_{j=1}^{p+1} a_j v_1 \wedge \cdots \wedge \hat{v}_j \wedge \cdots \wedge v_{p+1}, \quad a_j \in K, \quad 1 \le j \le p+1. $$

Since $\Omega \ne 0$, we can assume that $a_1 \ne 0$. Dividing $\Omega$ by $a_1$ if necessary, we can assume that $a_1 = 1$.

Let $\{g_1, \dots, g_{p+1}\}$ be dual basis of the basis $\{v_1, \dots, v_{p+1}\}$; $g_j(v_i) = \delta_{ij}$. If $X = g_1 + \sum_{j=2}^{p+1} (-1)^{j-1} a_j g_j$ then $\Omega = i_X v_1 \wedge \cdots \wedge v_{p+1}$. Now, if we set $\theta_j := v_j + (-1)^j a_j v_1$, $2 \le j \le p+1$, then $i_X \theta_j = 0$ and the reader can verify that $\Omega = \theta_2 \wedge \cdots \wedge \theta_{p+1}$.

Let $\tilde{\eta}$ be the extension of $\Pi^*(\eta)$ to $\mathbb{C}^{n+1}$, as in Corollary 1.3. Let $f_1 \dots f_{p+1}$ be the pole divisor $(\tilde{\eta})_\infty$, so that

$$ \tilde{\eta} = \sum_{j=1}^{p+1} \lambda_j \frac{\mathrm{d}f_1}{f_1} \wedge \cdots \wedge \frac{\mathrm{d}\hat{f}_j}{\hat{f}_j} \wedge \cdots \wedge \frac{\mathrm{d}f_{p+1}}{f_{p+1}}. \quad (3.1) $$
THEOREM 1.10. — Let $f_1, \dots, f_r$ be homogeneous polynomials on $\mathbb{C}^{n+1}$ and assume that the pole divisor $f_1 \dots f_r = 0$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$. Then:

(a) If $p=2$, or $r \in \{p+1, p+2\}$ then $\mathcal{L}_{\mathrm{td}}^p(f_1, \dots, f_r) = \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$.

(b) If $2 < p \le n-2$ and $r > p+2$ then $\mathcal{L}_{\mathrm{td}}^p(f_1, \dots, f_r)$ is an irreducible component of $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$. In particular, if $\mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$ is irreducible then $\mathcal{L}_{\mathrm{td}}^p(f_1, \dots, f_r) = \mathcal{L}_{\mathcal{F}}^p(f_1, \dots, f_r)$.

An interesting consequence of Theorem 1.10 is Corollary 1.11: when $r = p+1$ in Theorem 1.10 then $\mathcal{F}_\eta$ is a rational fibration (see Section 1). Since the case $r = p+1$ is the easier one, we will do it, together with the proof of Corollary 1.11, in Section 3.1.

In Section 3.2 we prove the theorem in the case $p=2$. In Section 3.3 we will see that the proof of (b) can be reduced to the case of 2-dimensional foliations (in which $p=n-2$). The proof of the case $r=p+2$ will be done in Section 3.4. We note that item (b) is an easy consequence of Theorem 1.17 and so it will be not done in this section.

## 3.1. Proof of the case $r = p+1$ and of Corollary 1.11

The proof will be based in the remark that a $p$-vector $\Omega$ in a vector space $V$ of dimension $p+1$ is always decomposable. In fact, if $\{v_1, \dots, v_{p+1}\}$ is a basis of $V$, then we can write

$$ \Omega = \sum_{j=1}^{p+1} a_j v_1 \wedge \cdots \wedge \hat{v}_j \wedge \cdots \wedge v_{p+1}, \quad a_j \in K, \quad 1 \le j \le p+1. $$

Since $\Omega \ne 0$, we can assume that $a_1 \ne 0$. Dividing $\Omega$ by $a_1$ if necessary, we can assume that $a_1 = 1$.

Let $\{g_1, \dots, g_{p+1}\}$ be dual basis of the basis $\{v_1, \dots, v_{p+1}\}$; $g_j(v_i) = \delta_{ij}$. If $X = g_1 + \sum_{j=2}^{p+1} (-1)^{j-1} a_j g_j$ then $\Omega = i_X v_1 \wedge \cdots \wedge v_{p+1}$. Now, if we set $\theta_j := v_j + (-1)^j a_j v_1$, $2 \le j \le p+1$, then $i_X \theta_j = 0$ and the reader can verify that $\Omega = \theta_2 \wedge \cdots \wedge \theta_{p+1}$.

Let $\tilde{\eta}$ be the extension of $\Pi^*(\eta)$ to $\mathbb{C}^{n+1}$, as in Corollary 1.3. Let $f_1 \dots f_{p+1}$ be the pole divisor $(\tilde{\eta})_\infty$, so that

$$ \tilde{\eta} = \sum_{j=1}^{p+1} \lambda_j \frac{\mathrm{d}f_1}{f_1} \wedge \cdots \wedge \frac{\mathrm{d}\hat{f}_j}{\hat{f}_j} \wedge \cdots \wedge \frac{\mathrm{d}f_{p+1}}{f_{p+1}}. \quad (3.1) $$
By the above remark $\tilde{\eta}$ is decomposable: if we assume $\lambda_1 \neq 0$ then there exist $\mu_2, \dots, \mu_{p+1} \in \mathbb{C}$ such that, if we set $\theta_j = \frac{df_j}{f_j} - \mu_j \frac{df_1}{f_1}$ then

$$ \tilde{\eta} = \lambda_1 \theta_2 \wedge \dots \wedge \theta_{p+1}. $$

We assert that $\mu_j \in \mathbb{Q}_+$, $2 \le j \le p+1$. In fact, from $i_R \tilde{\eta} = 0$ we get

$$ \begin{align*} i_R(\theta_1 \wedge \dots \wedge \theta_{p+1}) &= 0 \implies \sum_{j=2}^{p+1} (-1)^j i_R(\theta_j) \cdot \theta_2 \wedge \dots \wedge \hat{\theta}_j \wedge \dots \wedge \theta_{p+1} = 0 \\ &= i_R(\theta_j) = 0, \ 2 \le j \le p+1 \\ &\implies \mu_j = \frac{\deg(f_j)}{\deg(f_1)} := \frac{d_j}{d_1} \in \mathbb{Q}_+. \end{align*} $$

In particular, the rational function $f_j^{d_1}/f_1^{d_j}$ is a first integral of $\theta_j$; $d(f_j^{d_1}/f_1^{d_j}) \wedge \theta_j = 0, \ 2 \le j \le p+1$. This of course implies that $F = (f_1^{k_1}, \dots, f_{p+1}^{k_{p+1}})$ is a first integral of $\tilde{\eta}$ if $k_j := d_1 \dots d_{p+1}/d_j$. $\square$

## 3.2. Proof of Theorem 1.10 in the case $p = 2$: foliations of codimension two

Let $\mathcal{F}$ be a logarithmic foliation of codimension two on $\mathbb{P}^n$ defined by a logarithmic 2-form $\tilde{\eta} \in \mathcal{L}_{\mathcal{F}}^2(f_1, \dots, f_r)$. Note that the hypothesis $p = 2 \le n - 2$ implies that $n \ge 4$.

**Remark 3.1.** — The condition of local decomposability of $\tilde{\eta}$ outside the singular set is equivalent to $\tilde{\eta} \wedge \tilde{\eta} = 0$. This is a consequence of the fact that a two vector $\theta$ on a complex vector space is decomposable if, and only if, $\theta \wedge \theta = 0$.

In particular, we have

$$ \mathcal{L}_{\mathcal{F}}^2(f_1, \dots, f_r) = \{ \omega \in \mathcal{L}_R^2(f_1, \dots, f_r) \mid \omega \wedge \omega = 0 \}. $$

As we have seen, a form $\omega \in \mathcal{L}_R^2(f_1, \dots, f_r)$ can be written as

$$ \omega = \sum_{1 \le i < j \le r} \mu_{ij} \frac{df_i}{f_i} \wedge \frac{df_j}{f_j}. \qquad (3.2) $$

As the reader can check,

$$ \omega \wedge \omega = \sum_{1 \le i < j < k < \ell \le r} 2 \Psi(\mu_{ij}, \mu_{k\ell}, \mu_{ik}, \mu_{j\ell}, \mu_{i\ell}, \mu_{jk}) \frac{df_i}{f_i} \wedge \frac{df_j}{f_j} \wedge \frac{df_k}{f_k} \wedge \frac{df_\ell}{f_\ell}, $$
By the above remark $\tilde{\eta}$ is decomposable: if we assume $\lambda_1 \neq 0$ then there exist $\mu_2, \dots, \mu_{p+1} \in \mathbb{C}$ such that, if we set $\theta_j = \frac{df_j}{f_j} - \mu_j \frac{df_1}{f_1}$ then

$$ \tilde{\eta} = \lambda_1 \theta_2 \wedge \dots \wedge \theta_{p+1}. $$

We assert that $\mu_j \in \mathbb{Q}_+$, $2 \le j \le p+1$. In fact, from $i_R \tilde{\eta} = 0$ we get

$$ \begin{align*} i_R(\theta_1 \wedge \dots \wedge \theta_{p+1}) &= 0 \implies \sum_{j=2}^{p+1} (-1)^j i_R(\theta_j) \cdot \theta_2 \wedge \dots \wedge \hat{\theta}_j \wedge \dots \wedge \theta_{p+1} = 0 \\ &= i_R(\theta_j) = 0, 2 \le j \le p+1 \\ &\implies \mu_j = \frac{\deg(f_j)}{\deg(f_1)} := \frac{d_j}{d_1} \in \mathbb{Q}_+. \end{align*} $$

In particular, the rational function $f_j^{d_1}/f_1^{d_j}$ is a first integral of $\theta_j$; $d(f_j^{d_1}/f_1^{d_j}) \wedge \theta_j = 0, 2 \le j \le p+1$. This of course implies that $F = (f_1^{k_1}, \dots, f_{p+1}^{k_{p+1}})$ is a first integral of $\tilde{\eta}$ if $k_j := d_1 \dots d_{p+1}/d_j$. $\square$

## 3.2. Proof of Theorem 1.10 in the case $p = 2$: foliations of codimension two

Let $\mathcal{F}$ be a logarithmic foliation of codimension two on $\mathbb{P}^n$ defined by a logarithmic 2-form $\tilde{\eta} \in \mathcal{L}_{\mathcal{F}}^2(f_1, \dots, f_r)$. Note that the hypothesis $p = 2 \le n - 2$ implies that $n \ge 4$.

**Remark 3.1.** — The condition of local decomposability of $\tilde{\eta}$ outside the singular set is equivalent to $\tilde{\eta} \wedge \tilde{\eta} = 0$. This is a consequence of the fact that a two vector $\theta$ on a complex vector space is decomposable if, and only if, $\theta \wedge \theta = 0$.

In particular, we have

$$ \mathcal{L}_{\mathcal{F}}^2(f_1, \dots, f_r) = \{ \omega \in \mathcal{L}_R^2(f_1, \dots, f_r) \mid \omega \wedge \omega = 0 \}. $$

As we have seen, a form $\omega \in \mathcal{L}_R^2(f_1, \dots, f_r)$ can be written as

$$ \omega = \sum_{1 \le i < j \le r} \mu_{ij} \frac{df_i}{f_i} \wedge \frac{df_j}{f_j}. \qquad (3.2) $$

As the reader can check,

$$ \omega \wedge \omega = \sum_{1 \le i < j < k < \ell \le r} 2 \Psi(\mu_{ij}, \mu_{k\ell}, \mu_{ik}, \mu_{j\ell}, \mu_{i\ell}, \mu_{jk}) \frac{df_i}{f_i} \wedge \frac{df_j}{f_j} \wedge \frac{df_k}{f_k} \wedge \frac{df_\ell}{f_\ell}, $$
where $\Psi(a, b, c, d, e, f) = ab - cd + ef$. If $\omega \wedge \omega = 0$ their numerical residues must vanish (see Remark 2.3). This implies that $\mathcal{L}_{\mathcal{F}}^2(f_1, \dots, f_r)$ is isomorphic to the algebraic subset $\mathcal{A}$ of $\mathbb{C}^{r(r-1)/2}$ defined by

$$ \mathcal{A} = \{(\lambda_{ij})_{1 \le i < j \le r} | \Psi(\lambda_{ij}, \lambda_{k\ell}, \lambda_{ik}, \lambda_{j\ell}, \lambda_{i\ell}, \lambda_{jk}) = 0, \forall 1 \le i < j < k < \ell \le r\}, $$
where the isomorphism is given by

$$ (\lambda_{ij})_{1 \le i < j \le r} \in \mathcal{A} \mapsto \sum_{1 \le i < j \le r} \lambda_{ij} \frac{df_i}{f_i} \wedge \frac{df_j}{f_j} $$

On the other hand, if we fix a base $\{e_1, \dots, e_r\}$ of $\mathbb{C}^r$, a 2-vector $\theta$ on $\mathbb{C}^r$ can be written as

$$ \theta = \sum_{1 \le i < j \le r} a_{ij} e_i \wedge e_j. $$

Since

$$ \theta \wedge \theta = \sum_{1 \le i < j < k < \ell \le r} 2\Psi(a_{ij}, a_{k\ell}, a_{ik}, a_{j\ell}, a_{i\ell}, a_{jk}) e_i \wedge e_j \wedge e_k \wedge e_\ell, $$

we obtain $\theta \wedge \theta = 0$ if, and only if, $(a_{ij})_{1 \le i < j \le r} \in \mathcal{A}$. Now, if $\theta \wedge \theta = 0$ then $\theta$ is decomposable: $\theta = \alpha \wedge \beta$, where $\alpha, \beta \in \mathbb{C}^r$. In fact, if $\theta \neq 0$ let $u, v$ be in the dual of $\mathbb{C}^r$ and such that $\theta(u, v) \neq 0$. Then

$$ 0 = i_u(\theta \wedge \theta) = 2i_u(\theta) \wedge \theta \implies \theta = c \cdot i_u(\theta) \wedge i_v(\theta), \quad c = 1/\theta(u, v). $$

Finally, if $\omega$ is as in (3.2) and satisfies $\omega \wedge \omega = 0$ then the 2-vector $\theta = \sum_{i<j} \mu_{ij} e_i \wedge e_j$ is decomposable: $\theta = \alpha \wedge \beta$, $\alpha = \sum_i a_i e_i$ and $\beta = \sum_j b_j e_j$, so that $\omega = \omega_1 \wedge \omega_2$, $\omega_1 = \sum_i a_i \frac{df_i}{f_i}$ and $\omega_2 = \sum_j b_j \frac{df_j}{f_j}$. Moreover, if $i_R \omega = 0$ then $i_R \omega_1 = i_R \omega_2 = 0$ because

$$ 0 = i_R(\omega_1 \wedge \omega_2) = i_R \omega_1 \cdot \omega_2 - i_R \omega_2 \cdot \omega_1 \implies i_R \omega_1 = i_R \omega_2 = 0 \quad \square $$

### 3.3. Some remarks

From now on, we fix homogeneous polynomials $f_1, \dots, f_r \in \mathbb{C}[z_0, \dots, z_n]$, where $r > p + 1$, the divisor $f_1, \dots, f_r$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$ and $\deg(f_j) = d_j$, $1 \le j \le r$. Recall that $\mathcal{L}^p(f_1, \dots, f_r)$ denotes the set of logarithmic p-forms that can be written as below:

$$ \tilde{\eta} = \sum_{I \in \mathcal{S}_r^p} \lambda_I \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}}, \quad \lambda_I \in \mathbb{C}, \quad \forall I \in \mathcal{S}_r^p. \qquad (3.3) $$

Given a base $\{du_1, \dots, du_r\}$ of $\mathbb{C}^{r*}$ there exists an unique linear map $\Phi^p: \wedge^p(\mathbb{C}^{r*}) \to \mathcal{L}^p(f_1, \dots, f_r)$ such that

$$ \Phi^p(du_{i_1} \wedge \cdots \wedge du_{i_p}) = \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}}. $$
where $\Psi(a, b, c, d, e, f) = ab - cd + ef$. If $\omega \wedge \omega = 0$ their numerical residues must vanish (see Remark 2.3). This implies that $\mathcal{L}_{\mathcal{F}}^2(f_1, \dots, f_r)$ is isomorphic to the algebraic subset $\mathcal{A}$ of $\mathbb{C}^{r(r-1)/2}$ defined by

$$ \mathcal{A} = \{(\lambda_{ij})_{1 \le i < j \le r} | \Psi(\lambda_{ij}, \lambda_{k\ell}, \lambda_{ik}, \lambda_{j\ell}, \lambda_{i\ell}, \lambda_{jk}) = 0, \forall 1 \le i < j < k < \ell \le r\}, $$
where the isomorphism is given by

$$ (\lambda_{ij})_{1 \le i < j \le r} \in \mathcal{A} \mapsto \sum_{1 \le i < j \le r} \lambda_{ij} \frac{df_i}{f_i} \wedge \frac{df_j}{f_j} $$

On the other hand, if we fix a base $\{e_1, \dots, e_r\}$ of $\mathbb{C}^r$, a 2-vector $\theta$ on $\mathbb{C}^r$ can be written as

$$ \theta = \sum_{1 \le i < j \le r} a_{ij} e_i \wedge e_j. $$

Since

$$ \theta \wedge \theta = \sum_{1 \le i < j < k < \ell \le r} 2\Psi(a_{ij}, a_{k\ell}, a_{ik}, a_{j\ell}, a_{i\ell}, a_{jk}) e_i \wedge e_j \wedge e_k \wedge e_\ell, $$

we obtain $\theta \wedge \theta = 0$ if, and only if, $(a_{ij})_{1 \le i < j \le r} \in \mathcal{A}$. Now, if $\theta \wedge \theta = 0$ then $\theta$ is decomposable: $\theta = \alpha \wedge \beta$, where $\alpha, \beta \in \mathbb{C}^r$. In fact, if $\theta \neq 0$ let $u, v$ be in the dual of $\mathbb{C}^r$ and such that $\theta(u, v) \neq 0$. Then

$$ 0 = i_u(\theta \wedge \theta) = 2i_u(\theta) \wedge \theta \implies \theta = c \cdot i_u(\theta) \wedge i_v(\theta), \quad c = 1/\theta(u, v). $$

Finally, if $\omega$ is as in (3.2) and satisfies $\omega \wedge \omega = 0$ then the 2-vector $\theta = \sum_{i<j} \mu_{ij} e_i \wedge e_j$ is decomposable: $\theta = \alpha \wedge \beta$, $\alpha = \sum_i a_i e_i$ and $\beta = \sum_j b_j e_j$, so that $\omega = \omega_1 \wedge \omega_2$, $\omega_1 = \sum_i a_i \frac{df_i}{f_i}$ and $\omega_2 = \sum_j b_j \frac{df_j}{f_j}$. Moreover, if $i_R \omega = 0$ then $i_R \omega_1 = i_R \omega_2 = 0$ because

$$ 0 = i_R(\omega_1 \wedge \omega_2) = i_R \omega_1 \cdot \omega_2 - i_R \omega_2 \cdot \omega_1 \implies i_R \omega_1 = i_R \omega_2 = 0 \quad \square $$

### 3.3. Some remarks

From now on, we fix homogeneous polynomials $f_1, \dots, f_r \in \mathbb{C}[z_0, \dots, z_n]$, where $r > p + 1$, the divisor $f_1, \dots, f_r$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$ and $\deg(f_j) = d_j$, $1 \le j \le r$. Recall that $\mathcal{L}^p(f_1, \dots, f_r)$ denotes the set of logarithmic p-forms that can be written as below:

$$ \tilde{\eta} = \sum_{I \in \mathcal{S}_r^p} \lambda_I \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}}, \quad \lambda_I \in \mathbb{C}, \quad \forall I \in \mathcal{S}_r^p. \qquad (3.3) $$

Given a base $\{du_1, \dots, du_r\}$ of $\mathbb{C}^{r*}$ there exists an unique linear map $\Phi^p: \wedge^p(\mathbb{C}^{r*}) \to \mathcal{L}^p(f_1, \dots, f_r)$ such that

$$ \Phi^p(du_{i_1} \wedge \cdots \wedge du_{i_p}) = \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}}. $$
LEMMA 3.2. — $\Phi^p$ is an isomorphism for all $p \ge 1$. Moreover, if $\alpha \in \bigwedge^p(\mathbb{C}^{r*})$ and $\beta \in \bigwedge^q(\mathbb{C}^{r*})$ then

$$ \Phi^{p+q}(\alpha \wedge \beta) = \Phi^p(\alpha) \wedge \Phi^q(\beta). \qquad (3.4) $$

*Proof.* — On one hand, it is clear that $\Phi^p$ is surjective. On the other hand, if $\tilde{\eta} = \sum_{I \in S_r^p} \lambda_I \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}}$ then each numerical residue $\lambda_I$, $I \in S_r^p$, can be calculated by an integral as in Remark 2.3:

$$ \lambda_I = \frac{1}{(2\pi i)^p} \int_{T_e^p} \eta. $$

It follows that

$$ \sum_{I \in S_r^p} \lambda_I \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}} \equiv 0 \iff \lambda_I = 0, \forall I \in S_r^p $$

and so $\Phi^p$ is injective.

Finally, formula (3.4) is consequence of

$$ \begin{align*} \Phi^{p+q}((du_{i_1} \wedge \cdots \wedge du_{i_p}) \wedge (du_{j_1} \wedge \cdots \wedge du_{j_q})) \\ &= \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}} \wedge \frac{df_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{df_{j_q}}{f_{j_q}} \\ &= \Phi^p(du_{i_1} \wedge \cdots \wedge du_{i_p}) \wedge \Phi^q(du_{j_1} \wedge \cdots \wedge du_{j_q}) \quad \Box \end{align*} $$

Remark 3.3. — Given a p-form $\alpha \in \bigwedge^p(\mathbb{C}^{r*})$ its kernel is defined as
$$ \ker(\alpha) = \{v \in \mathbb{C}^r | i_v \alpha = 0\}. $$

We say that $\alpha \in \bigwedge^p(\mathbb{C}^{r*})$ is totally decomposable if there are p 1-forms $\alpha_1, \dots, \alpha_p$ such that $\alpha = \alpha_1 \wedge \cdots \wedge \alpha_p$. It is well known that:

(a) $\alpha = \alpha_1 \wedge \cdots \wedge \alpha_p$ is totally decomposable if, and only if, $\dim(\ker(\alpha)) = r - p$.

(b) If $\alpha = \alpha_1 \wedge \cdots \wedge \alpha_p$ then $\ker(\alpha) = \bigcap_{j=1}^p \ker(\alpha_j)$.

(c) The projectivization of the set of totally decomposable p-forms of $\bigwedge^p(\mathbb{C}^{r*})$ is isomorphic to the grassmanian of p planes through the origin in $\mathbb{C}^r$. In particular, it is an algebraic subset of $\mathbb{P}(\bigwedge^p(\mathbb{C}^{r*}))$.

Recall that $\tilde{\eta} \in \mathcal{L}_{td}^p(f_1, \dots, f_r)$ if it is totally decomposable into logarithmic forms (totally decomposable into logarithmic forms). An easy consequence of Lemma 3.2 and of Remark 3.3(c) is the following:

COROLLARY 3.4. — Let $p \ge 2$. A p-form $\tilde{\eta} \in \mathcal{L}^p(f_1, \dots, f_r)$ is totally decomposable into logarithmic forms if, and only if, there are 1-forms $\alpha_1, \dots, \alpha_p \in \mathbb{C}^{r*}$ such that

$$ \tilde{\eta} = \Phi^p(\alpha_1 \wedge \cdots \wedge \alpha_p). $$
LEMMA 3.2. — $\Phi^p$ is an isomorphism for all $p \ge 1$. Moreover, if $\alpha \in \bigwedge^p(\mathbb{C}^{r*})$ and $\beta \in \bigwedge^q(\mathbb{C}^{r*})$ then

$$ \Phi^{p+q}(\alpha \wedge \beta) = \Phi^p(\alpha) \wedge \Phi^q(\beta). \qquad (3.4) $$

*Proof.* — On one hand, it is clear that $\Phi^p$ is surjective. On the other hand, if $\tilde{\eta} = \sum_{I \in S_r^p} \lambda_I \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}}$ then each numerical residue $\lambda_I$, $I \in S_r^p$, can be calculated by an integral as in Remark 2.3:

$$ \lambda_I = \frac{1}{(2\pi i)^p} \int_{T_e^p} \eta. $$

It follows that

$$ \sum_{I \in S_r^p} \lambda_I \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}} \equiv 0 \iff \lambda_I = 0, \forall I \in S_r^p $$

and so $\Phi^p$ is injective.

Finally, formula (3.4) is consequence of

$$ \begin{aligned} & \Phi^{p+q}((du_{i_1} \wedge \cdots \wedge du_{i_p}) \wedge (du_{j_1} \wedge \cdots \wedge du_{j_q})) \\ &= \frac{df_{i_1}}{f_{i_1}} \wedge \cdots \wedge \frac{df_{i_p}}{f_{i_p}} \wedge \frac{df_{j_1}}{f_{j_1}} \wedge \cdots \wedge \frac{df_{j_q}}{f_{j_q}} \\ &= \Phi^p(du_{i_1} \wedge \cdots \wedge du_{i_p}) \wedge \Phi^q(du_{j_1} \wedge \cdots \wedge du_{j_q}) \quad \Box \end{aligned} $$

Remark 3.3. — Given a p-form $\alpha \in \bigwedge^p(\mathbb{C}^{r*})$ its kernel is defined as
$$ \ker(\alpha) = \{v \in \mathbb{C}^r | i_v \alpha = 0\}. $$

We say that $\alpha \in \bigwedge^p(\mathbb{C}^{r*})$ is totally decomposable if there are p 1-forms $\alpha_1, \dots, \alpha_p$ such that $\alpha = \alpha_1 \wedge \cdots \wedge \alpha_p$. It is well known that:

(a) $\alpha = \alpha_1 \wedge \cdots \wedge \alpha_p$ is totally decomposable if, and only if, $\dim(\ker(\alpha)) = r - p$.

(b) If $\alpha = \alpha_1 \wedge \cdots \wedge \alpha_p$ then $\ker(\alpha) = \bigcap_{j=1}^p \ker(\alpha_j)$.

(c) The projectivization of the set of totally decomposable p-forms of $\bigwedge^p(\mathbb{C}^{r*})$ is isomorphic to the grassmanian of p planes through the origin in $\mathbb{C}^r$. In particular, it is an algebraic subset of $\mathbb{P}(\bigwedge^p(\mathbb{C}^{r*}))$.

Recall that $\tilde{\eta} \in \mathcal{L}_{td}^p(f_1, \dots, f_r)$ if it is totally decomposable into logarithmic forms (totally decomposable into logarithmic forms). An easy consequence of Lemma 3.2 and of Remark 3.3(c) is the following:

COROLLARY 3.4. — Let $p \ge 2$. A p-form $\tilde{\eta} \in \mathcal{L}^p(f_1, \dots, f_r)$ is totally decomposable into logarithmic forms if, and only if, there are 1-forms $\alpha_1, \dots, \alpha_p \in \mathbb{C}^{r*}$ such that

$$ \tilde{\eta} = \Phi^p(\alpha_1 \wedge \cdots \wedge \alpha_p). $$
In particular, $\mathcal{L}_{td}^p(f_1, \dots, f_r)$ is an algebraic irreducible subset of $\mathcal{L}_R^p(f_1, \dots, f_r)$.

Another consequence of Lemma 3.2 is that Theorem 1.10(b) can be reduced to the case of 2-dimensional foliations. Let $\Sigma \simeq \mathbb{P}^q$ be a $q$-plane linearly embedded in $\mathbb{P}^n$. We say that $\Sigma$ is in general position with respect to the divisor $f_1 \dots f_r$ if for all $J = \{j_1, \dots, j_k\} \subset \{1, \dots, r\}$ then $\Sigma$ is transverse to $\bigcap_{j \in J} \Pi(f_j = 0)$. By transversality theory, the set of $q$-planes of $\mathbb{P}^n$ in general position with respect to $f_1 \dots f_r$ is a Zariski open and dense subset of the grassmanian of $q$-planes on $\mathbb{P}^n$.

**Remark 3.5.** — Let $\tilde{\eta} \in \mathcal{L}_R^p(f_1, \dots, f_r)$. Let $\Sigma$ be $(p+k)$-plane of $\mathbb{P}^n$ in general position with respect to $f_1 \dots f_r, p < p+k < n$, and $\tilde{\Sigma}$ be a $p+k+1$ plane through $0 \in \mathbb{C}^{n+1}$ such that $\Pi(\tilde{\Sigma}) = \Sigma$. Then $\tilde{\eta}|_{\tilde{\Sigma}}$ is a logarithmic $p$-form on $\tilde{\Sigma}$. An easy consequence of Lemma 3.2 and Corollary 3.4 is the following:

**COROLLARY 3.6.** — Let $\tilde{\eta}$, $\Sigma$ and $\tilde{\Sigma}$ be as in Remark 3.5. Then $\tilde{\eta}$ is totally decomposable into logarithmic forms if, and only if, $\tilde{\eta}|_{\tilde{\Sigma}}$ is totally decomposable into logarithmic forms.

Taking $k=2$ in the above statement, we reduce the proofs of the case $r=p+2$ and of Theorem 1.10(b) to the case of 2-dimensional foliations. From now on, we will assume that $\tilde{\eta} = \Pi^*(\eta) \in \mathcal{L}_F^p(f_1, \dots, f_r)$ and that $n=p+2$. By Section 3.1 we will assume also that $r \ge p+2$. As we have seen, we can write

$$ \tilde{\eta} = \sum_{I \in \mathcal{S}_r^p} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_{n-2}}}{f_{i_{n-2}}}. \quad (3.5) $$

The foliation $\mathcal{F}_\eta$ is defined in homogeneous coordinates by the (n-2)-form $\omega = f_1 \dots f_r \tilde{\eta}$. As a consequence, the part of $\mathrm{Sing}(\mathcal{F}_{\tilde{\eta}})$ contained in the pole divisor contains an union of curves: given $J = (j_1, \dots, j_{n-1}) \in \mathcal{S}_r^{n-1}$ let $S_J = \Pi(f_{j_1} = \dots = f_{j_{n-1}} = 0)$. By the assumption on the pole divisor $f_1 \dots f_r$, $S_J$ is a smooth complex curve and

$$ \mathrm{Sing}(\mathcal{F}_\eta) \cap \Pi(f_1 \dots f_j \dots f_r = 0) \supset \bigcup_{J \in \mathcal{S}_r^{n-1}} S_J. $$

A point $q = \Pi(p) \in S_J$, for a fixed $J \in \mathcal{S}_r^{n-1}$, will be said *generic* if for all $i \notin J$ then $f_i(p) \ne 0$. Otherwise, $q$ will be said *non-generic*. By the assumption on the pole divisor, if $q = \Pi(p)$ is non-generic and $f_i(p) = 0$ then $f_\ell(p) \ne 0$ for all $\ell \notin J \cup \{i\}$.

Let us fix $J = (j_1 < \dots < j_{n-1}) \in \mathcal{S}_r^{n-1}$ and a point $q = \Pi(q) \in S_J$. After an automorphism of $\mathbb{P}^n$ we can assume that $q = (0, \dots, 0)$ in the affine
In particular, $\mathcal{L}_{td}^p(f_1, \dots, f_r)$ is an algebraic irreducible subset of $\mathcal{L}_R^p(f_1, \dots, f_r)$.

Another consequence of Lemma 3.2 is that Theorem 1.10(b) can be reduced to the case of 2-dimensional foliations. Let $\Sigma \simeq \mathbb{P}^q$ be a $q$-plane linearly embedded in $\mathbb{P}^n$. We say that $\Sigma$ is in general position with respect to the divisor $f_1 \dots f_r$ if for all $J = \{j_1, \dots, j_k\} \subset \{1, \dots, r\}$ then $\Sigma$ is transverse to $\bigcap_{j \in J} \Pi(f_j = 0)$. By transversality theory, the set of $q$-planes of $\mathbb{P}^n$ in general position with respect to $f_1 \dots f_r$ is a Zariski open and dense subset of the grassmanian of $q$-planes on $\mathbb{P}^n$.

**Remark 3.5.** — Let $\tilde{\eta} \in \mathcal{L}_R^p(f_1, \dots, f_r)$. Let $\Sigma$ be $(p+k)$-plane of $\mathbb{P}^n$ in general position with respect to $f_1 \dots f_r, p < p+k < n$, and $\tilde{\Sigma}$ be a $p+k+1$ plane through $0 \in \mathbb{C}^{n+1}$ such that $\Pi(\tilde{\Sigma}) = \Sigma$. Then $\tilde{\eta}|_{\tilde{\Sigma}}$ is a logarithmic $p$-form on $\tilde{\Sigma}$. An easy consequence of Lemma 3.2 and Corollary 3.4 is the following:

**COROLLARY 3.6.** — Let $\tilde{\eta}$, $\Sigma$ and $\tilde{\Sigma}$ be as in Remark 3.5. Then $\tilde{\eta}$ is totally decomposable into logarithmic forms if, and only if, $\tilde{\eta}|_{\tilde{\Sigma}}$ is totally decomposable into logarithmic forms.

Taking $k=2$ in the above statement, we reduce the proofs of the case $r=p+2$ and of Theorem 1.10(b) to the case of 2-dimensional foliations. From now on, we will assume that $\tilde{\eta} = \Pi^*(\eta) \in \mathcal{L}_F^p(f_1, \dots, f_r)$ and that $n=p+2$. By Section 3.1 we will assume also that $r \ge p+2$. As we have seen, we can write

$$ \tilde{\eta} = \sum_{I \in \mathcal{S}_r^p} \lambda_I \frac{\mathrm{d}f_{i_1}}{f_{i_1}} \wedge \dots \wedge \frac{\mathrm{d}f_{i_{n-2}}}{f_{i_{n-2}}}. \quad (3.5) $$

The foliation $\mathcal{F}_\eta$ is defined in homogeneous coordinates by the (n-2)-form $\omega = f_1 \dots f_r \tilde{\eta}$. As a consequence, the part of $\mathrm{Sing}(\mathcal{F}_{\tilde{\eta}})$ contained in the pole divisor contains an union of curves: given $J = (j_1, \dots, j_{n-1}) \in \mathcal{S}_r^{n-1}$ let $S_J = \Pi(f_{j_1} = \dots = f_{j_{n-1}} = 0)$. By the assumption on the pole divisor $f_1 \dots f_r$, $S_J$ is a smooth complex curve and

$$ \mathrm{Sing}(\mathcal{F}_\eta) \cap \Pi(f_1 \dots f_j \dots f_r = 0) \supset \bigcup_{J \in \mathcal{S}_r^{n-1}} S_J. $$

A point $q = \Pi(p) \in S_J$, for a fixed $J \in \mathcal{S}_r^{n-1}$, will be said *generic* if for all $i \notin J$ then $f_i(p) \ne 0$. Otherwise, $q$ will be said *non-generic*. By the assumption on the pole divisor, if $q = \Pi(p)$ is non-generic and $f_i(p) = 0$ then $f_\ell(p) \ne 0$ for all $\ell \notin J \cup \{i\}$.

Let us fix $J = (j_1 < \dots < j_{n-1}) \in \mathcal{S}_r^{n-1}$ and a point $q = \Pi(q) \in S_J$. After an automorphism of $\mathbb{P}^n$ we can assume that $q = (0, \dots, 0)$ in the affine
chart ($x_0 = 1) \simeq \mathbb{C}^n$. In this chart, the pole divisor of $\eta$ is $g_1 \dots g_r$, where
$g_j(x) = g_j(x_1, \dots, x_n) = f_j(1, x_1, \dots, x_n)$. Since the equation of the curve
$S_J$ is $(g_{j_1} = \dots = g_{j_{n-1}} = 0)$, there exists a holomorphic coordinate system
$(U, z = (z_1, \dots, z_n))$ around $q$ such that $g_{j_i}|_U = z_i$, $1 \le i \le n-1$.

Remark 3.7. — Let $q \in S_J$ and $(U, z)$ be as above. We would like to
observe that:

(a) If $q$ is a generic point of $S_J$ then we can write

$$
\eta|_U = \sum_{j=1}^{n-1} \mu_j \frac{dz_1}{z_1} \wedge \cdots \wedge \frac{\widehat{dz_j}}{z_j} \wedge \cdots \wedge \frac{dz_{n-1}}{z_{n-1}} + d\Theta, \quad (3.6)
$$

where, either $\Theta = 0$, or $\Theta$ is a non-closed logarithmic (n-3)-form with
pole divisor contained in $x = z_1 \dots z_{n-1}$, and $\mu_i = \lambda_{I_i}$, $I_i = J \setminus \{j_i\}$.

(b) If $q \in S_J$ is a non-generic point then there exists $j \notin J$ such that $g_j(q) = 0$ and $g_i(q) \neq 0$ if $i \notin J \cup \{j\}$. In this case, we can assume that $g_j|_U = z_n$. Moreover, we can write

$$
\eta|_U = \sum_{1 \le k < \ell \le n} \mu_{k\ell} \frac{dz_1}{z_1} \wedge \dots \wedge \frac{\widehat{dz_k}}{z_k} \wedge \dots \wedge \frac{\widehat{dz_\ell}}{z_\ell} \wedge \dots \wedge \frac{dz_n}{z_n} + d\Theta, \quad (3.7)
$$

where $\Theta$ is as in (a) and $\mu_{k\ell} = \lambda_{I_{k\ell}}$, $I_{k\ell} = J \cup \{j\} \setminus \{j_k, j_\ell\}$ if $\ell < n$,
$\mu_{kn} = J \setminus \{j_k\}$.

The proof can be done directly by using (3.5) or Theorem 2.1.

**3.4. Proof of the case r = p + 2**

In this case $r = p + 2 = n$ and the non generic points of $\text{Sing}(\mathcal{F}_\eta) \cap \Pi(f_1 \dots f_n = 0)$ are in the finite set $\Pi(f_1 = \dots = f_n = 0)$. In particular, if we fix a non-generic point $q \in \Pi(f_1 = \dots = f_n = 0)$ there exists a local coordinate system $(U, z = (z_1, \dots, z_n))$ around $q$ such that $g_j|_U = z_j$, $1 \le j \le n$. In particular, by (3.5) we have

$$
\eta|_U = \sum_{1 \le k < \ell \le n} \mu_{k\ell} \frac{dz_1}{z_1} \wedge \dots \wedge \frac{\widehat{dz_k}}{z_k} \wedge \dots \wedge \frac{\widehat{dz_\ell}}{z_\ell} \wedge \dots \wedge \frac{dz_n}{z_n}.
$$

Since $\eta \in \mathcal{L}_{\mathcal{F}}^{n-2}(f_1, \dots, f_n)$ then $\eta|_U$ is locally decomposable outside the
polar set $z_1 \dots z_n = 0$. The foliation $\mathcal{F}_\eta$ is defined in $U$ by the holomorphic
form

$$
\omega := z_1 \dots z_n \eta|_U = \sum_{1 \le k < \ell \le n} \mu_{k\ell} z_k z_\ell \mathrm{d}z_1 \wedge \dots \wedge \widehat{\mathrm{d}z_k} \wedge \dots \wedge \widehat{\mathrm{d}z_\ell} \wedge \dots \wedge \mathrm{d}z_n. (*)
$$
chart ($x_0 = 1) \simeq \mathbb{C}^n$. In this chart, the pole divisor of $\eta$ is $g_1 \dots g_r$, where $g_j(x) = g_j(x_1, \dots, x_n) = f_j(1, x_1, \dots, x_n)$. Since the equation of the curve $S_J$ is $(g_{j_1} = \dots = g_{j_{n-1}} = 0)$, there exists a holomorphic coordinate system $(U, z = (z_1, \dots, z_n))$ around $q$ such that $g_{j_i}|_U = z_i$, $1 \le i \le n-1$.

Remark 3.7. — Let $q \in S_J$ and $(U, z)$ be as above. We would like to observe that:

(a) If $q$ is a generic point of $S_J$ then we can write

$$ \eta|_U = \sum_{j=1}^{n-1} \mu_j \frac{dz_1}{z_1} \wedge \dots \wedge \frac{\widehat{dz_j}}{z_j} \wedge \dots \wedge \frac{dz_{n-1}}{z_{n-1}} + d\Theta, \quad (3.6) $$

where, either $\Theta = 0$, or $\Theta$ is a non-closed logarithmic (n-3)-form with pole divisor contained in $x = z_1 \dots z_{n-1}$, and $\mu_i = \lambda_{I_i}$, $I_i = J \setminus \{j_i\}$.

(b) If $q \in S_J$ is a non-generic point then there exists $j \notin J$ such that $g_j(q) = 0$ and $g_i(q) \ne 0$ if $i \notin J \cup \{j\}$. In this case, we can assume that $g_j|_U = z_n$. Moreover, we can write

$$ \eta|_U = \sum_{1 \le k < \ell \le n} \mu_{k\ell} \frac{dz_1}{z_1} \wedge \dots \wedge \frac{\widehat{dz_k}}{z_k} \wedge \dots \wedge \frac{\widehat{dz_\ell}}{z_\ell} \wedge \dots \wedge \frac{dz_{n-1}}{z_{n-1}} + d\Theta, \quad (3.7) $$

where $\Theta$ is as in (a) and $\mu_{k\ell} = \lambda_{I_{k\ell}}$, $I_{k\ell} = J \cup \{j\} \setminus \{j_k, j_\ell\}$ if $\ell < n$, $\mu_{kn} = J \setminus \{j_k\}$.

The proof can be done directly by using (3.5) or Theorem 2.1.

## 3.4. Proof of the case $r = p + 2$

In this case $r = p + 2 = n$ and the non generic points of $\text{Sing}(\mathcal{F}_\eta) \cap \Pi(f_1 \dots f_n = 0)$ are in the finite set $\Pi(f_1 = \dots = f_n = 0)$. In particular, if we fix a non-generic point $q \in \Pi(f_1 = \dots = f_n = 0)$ there exists a local coordinate system $(U, z = (z_1, \dots, z_n))$ around $q$ such that $g_j|_U = z_j$, $1 \le j \le n$. In particular, by (3.5) we have

$$ \eta|_U = \sum_{1 \le k < \ell \le n} \mu_{k\ell} \frac{dz_1}{z_1} \wedge \dots \wedge \frac{\widehat{dz_k}}{z_k} \wedge \dots \wedge \frac{\widehat{dz_\ell}}{z_\ell} \wedge \dots \wedge \frac{dz_{n-1}}{z_{n-1}} + d\Theta. $$

Since $\eta \in \mathcal{L}_{\mathcal{F}}^{n-2}(f_1, \dots, f_n)$ then $\eta|_U$ is locally decomposable outside the polar set $z_1 \dots z_n = 0$. The foliation $\mathcal{F}_\eta$ is defined in $U$ by the holomorphic form

$$ \omega := z_1 \dots z_n \eta|_U = \sum_{1 \le k < \ell \le n} \mu_{k\ell} z_k z_\ell dz_1 \wedge \dots \wedge \widehat{dz_k} \wedge \dots \wedge \widehat{dz_\ell} \wedge \dots \wedge dz_n. \quad (*) $$
Remark 3.8. — Let $\alpha$ be a holomorphic (n-2)-form on an open subset $V \subset \mathbb{C}^n$. Given $1 \le j \le n$ and $p \in V$ such that $\alpha|_{(z_j=z_j(p))} \ne 0$ we can define a vector field $X_p^j$, tangent to the hyperplane $(z_j = z_j(p))$, by

$$ \alpha|_{(z_j=z_j(p))} = i_{X_p^j} \nu_j, \nu_j = \mathrm{d}z_1 \wedge \cdots \wedge \widehat{\mathrm{d}z_j} \wedge \cdots \wedge \mathrm{d}z_n. $$

This process defines a holomorphic vector field $X^j$ on $V$, tangent to the fibration $(z_j = cte)$, by

$$ X^j(p) = X_p^j(p), p \in V. $$

Although $i_{X_p^j} \alpha|_{(z_j=z_j(p))} = 0$, in general $i_{X_j} \alpha \ne 0$. However, if the form $\alpha$ is locally decomposable outside its singular set then $i_{X_j} \alpha = 0$, so that $X^j$ is tangent to the distribution defined by $\alpha$. The proof is straightforward and is left to the reader.

If we apply Remark 3.8 to the (n-2)-form $\omega$ in (*) we obtain $X^j = z_j.Y^j$, where

$$ Y^j = \sum_{k \ne j} \rho_k^j z_k \frac{\partial}{\partial z_k}, $$

and $\rho_k^j = (-1)^{k-1}\mu_{kj}$, with the convention $\mu_{rs} = -\mu_{sr}$, $\forall r, s$. Since $\mathcal{F}_\eta$ has dimension two, at least two of the linear vector fields above, that we can suppose to be $Y^1$ and $Y^2$, are not identically zero and generically linearly independent. In this case, the form $\tilde{\omega} = i_{Y^1}i_{Y^2}\nu$, $\nu = \mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_n$, defines the same distribution as $\omega$. The reader can check that

$$ \tilde{\omega} = \sum_{1 \le k < \ell \le n} (-1)^{k+\ell} (\rho_k^1 \rho_\ell^2 - \rho_\ell^1 \rho_k^2) z_k z_\ell \mathrm{d}z_1 \wedge \cdots \wedge \widehat{\mathrm{d}z_k} \wedge \cdots \wedge \widehat{\mathrm{d}z_\ell} \wedge \cdots \wedge \mathrm{d}z_n. $$

Since the coefficients of $\omega$ and $\tilde{\omega}$ are homogeneous of degree two it follows that $\tilde{\omega} = c.\omega$, where $c \in \mathbb{C}^*$. From $\rho_k^j = (-1)^{k-1}\mu_{kj}$ and the above expression, we get

$$ \mu_{k1}\mu_{\ell2} - \mu_{\ell1}\mu_{k2} = c.\mu_{k\ell} \underset{k=2}{\implies} c = \mu_{12}. $$

Now, consider the n-2 closed logarithmic 1-forms $\tilde{\theta}_3, \dots, \tilde{\theta}_n$ defined by

$$ \tilde{\theta}_j = \rho_j^2 \frac{\mathrm{d}z_1}{z_1} - \rho_j^1 \frac{\mathrm{d}z_2}{z_2} + \rho_2^1 \frac{\mathrm{d}z_j}{z_j}. $$

Using that $\rho_1^2 = -\rho_2^1$ we get $i_{Y^1}\tilde{\theta}_j = i_{Y^2}\tilde{\theta}_j = 0$, $3 \le j \le n$, and this implies that $\tilde{\theta}_3 \wedge \cdots \wedge \tilde{\theta}_n = k.\eta|_U$, $k \in \mathbb{C}^*$. Comparing the coefficients of $\frac{\mathrm{d}z_3}{z_3} \wedge \cdots \wedge \frac{\mathrm{d}z_n}{z_n}$ of the two members of the relation we get $k = (\rho_2^1)^{n-3} = \mu_{12}^{n-3}$. Finally, if we define $\theta_j = \rho_j^2 \frac{\mathrm{d}f_1}{f_1} - \rho_j^1 \frac{\mathrm{d}f_2}{f_2} + \rho_2^1 \frac{\mathrm{d}f_j}{f_j}$ then $\theta_3 \wedge \cdots \wedge \theta_n$ then

$$ \theta_3 \wedge \cdots \wedge \theta_n = \mu_{12}^{n-3} \tilde{\eta}, $$

which proves that $\mathcal{L}_{\mathcal{F}}^{n-2}(f_1, \dots, f_n) = \mathcal{L}_{td}^{n-2}(f_1, \dots, f_n)$.
**Remark 3.8.** — Let $\alpha$ be a holomorphic (n-2)-form on an open subset $V \subset \mathbb{C}^n$. Given $1 \le j \le n$ and $p \in V$ such that $\alpha|_{(z_j=z_j(p))} \ne 0$ we can define a vector field $X_p^j$, tangent to the hyperplane $(z_j = z_j(p))$, by

$$ \alpha|_{(z_j=z_j(p))} = i_{X_p^j} \nu_j, \nu_j = \mathrm{d}z_1 \wedge \cdots \wedge \widehat{\mathrm{d}z_j} \wedge \cdots \wedge \mathrm{d}z_n. $$

This process defines a holomorphic vector field $X^j$ on $V$, tangent to the fibration $(z_j = cte)$, by

$$ X^j(p) = X_p^j(p), p \in V. $$

Although $i_{X_p^j} \alpha|_{(z_j=z_j(p))} = 0$, in general $i_{X_j} \alpha \ne 0$. However, if the form $\alpha$ is locally decomposable outside its singular set then $i_{X_j} \alpha = 0$, so that $X^j$ is tangent to the distribution defined by $\alpha$. The proof is straightforward and is left to the reader.

If we apply Remark 3.8 to the (n-2)-form $\omega$ in (*) we obtain $X^j = z_j.Y^j$, where

$$ Y^j = \sum_{k \ne j} \rho_k^j z_k \frac{\partial}{\partial z_k}, $$

and $\rho_k^j = (-1)^{k-1}\mu_{kj}$, with the convention $\mu_{rs} = -\mu_{sr}$, $\forall r, s$. Since $\mathcal{F}_\eta$ has dimension two, at least two of the linear vector fields above, that we can suppose to be $Y^1$ and $Y^2$, are not identically zero and generically linearly independent. In this case, the form $\tilde{\omega} = i_{Y^1}i_{Y^2}\nu$, $\nu = \mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_n$, defines the same distribution as $\omega$. The reader can check that

$$ \tilde{\omega} = \sum_{1 \le k < \ell \le n} (-1)^{k+\ell} (\rho_k^1 \rho_\ell^2 - \rho_\ell^1 \rho_k^2) z_k z_\ell \mathrm{d}z_1 \wedge \cdots \wedge \widehat{\mathrm{d}z_k} \wedge \cdots \wedge \widehat{\mathrm{d}z_\ell} \wedge \cdots \wedge \mathrm{d}z_n. $$

Since the coefficients of $\omega$ and $\tilde{\omega}$ are homogeneous of degree two it follows that $\tilde{\omega} = c.\omega$, where $c \in \mathbb{C}^*$. From $\rho_k^j = (-1)^{k-1}\mu_{kj}$ and the above expression, we get

$$ \mu_{k1}\mu_{\ell2} - \mu_{\ell1}\mu_{k2} = c.\mu_{k\ell} \underset{k=2}{\implies} c = \mu_{12}. $$

Now, consider the n-2 closed logarithmic 1-forms $\tilde{\theta}_3, \dots, \tilde{\theta}_n$ defined by

$$ \tilde{\theta}_j = \rho_j^2 \frac{\mathrm{d}z_1}{z_1} - \rho_j^1 \frac{\mathrm{d}z_2}{z_2} + \rho_2^1 \frac{\mathrm{d}z_j}{z_j}. $$

Using that $\rho_1^2 = -\rho_2^1$ we get $i_{Y^1}\tilde{\theta}_j = i_{Y^2}\tilde{\theta}_j = 0$, $3 \le j \le n$, and this implies that $\tilde{\theta}_3 \wedge \cdots \wedge \tilde{\theta}_n = k.\eta|_U$, $k \in \mathbb{C}^*$. Comparing the coefficients of $\frac{\mathrm{d}z_3}{z_3} \wedge \cdots \wedge \frac{\mathrm{d}z_n}{z_n}$ of the two members of the relation we get $k = (\rho_2^1)^{n-3} = \mu_{12}^{n-3}$. Finally, if we define $\theta_j = \rho_j^2 \frac{\mathrm{d}f_1}{f_1} - \rho_j^1 \frac{\mathrm{d}f_2}{f_2} + \rho_2^1 \frac{\mathrm{d}f_j}{f_j}$ then $\theta_3 \wedge \cdots \wedge \theta_n$ then

$$ \theta_3 \wedge \cdots \wedge \theta_n = \mu_{12}^{n-3} \tilde{\eta}, $$

which proves that $\mathcal{L}_{\mathcal{F}}^{n-2}(f_1, \dots, f_n) = \mathcal{L}_{td}^{n-2}(f_1, \dots, f_n)$.
4. **Proof of Theorem 1.17**

The purpose of this section is to prove the following:

THEOREM 1.17. — If $k \ge 2$ and $r \ge p + 2 = n - k + 2$ then
$\mathcal{L}_{td}(d_1, \dots, d_r; k, n)$ is an irreducible component of $\mathrm{Fol}(D(d_1, \dots, d_r, p); k, n)$
for all $r > p$ and $d_1, \dots, d_r \ge 1$.

The proof of Theorem 1.17 will be done first in the case of foliations of
dimension two. The general case will be reduced to this one by using the
following result:

THEOREM 4.1. — Let $\mathcal{F}$ be a codimension $p$ holomorphic foliation on $\mathbb{P}^n$,
$n > p + 1$. Assume that there is an algebraic smooth submanifold $M \subset \mathbb{P}^n$,
$\dim_{\mathbb{C}}(M) = m$, where $p + 1 \le m < n$, such that:

• The set of tangencies of $\mathcal{F}$ with $M$ has codimension $\ge 2$ on $M$.

• $\mathcal{F}|_M$ can be defined by a closed meromorphic $p$-form on $M$, say $\eta$.

Then $\eta$ can be extended to a closed meromorphic $p$-form $\tilde{\eta}$ on $\mathbb{P}^n$ defining
$\mathcal{F}$. Moreover, if $\eta$ is logarithmic so is $\tilde{\eta}$.

In fact, Theorem 4.1 is a generalization of a result in [4] (see also [16]).

The proof of Theorem 1.17 in the two dimensional case will be reduced
to a result that we state next (Theorem 4.3). In order to state it properly
let us recall the definition of a Kupka or generalized Kupka singularity for
two dimensional foliations (see also [17]).

Let $\omega$ be a germ at $p \in \mathbb{C}^n$ of integrable $(n-2)$-form with $\omega(p) = 0$. Recall
that the rotational of $\omega$ is the vector field $X = \operatorname{rot}(\omega)$ (cf. [17]) defined by

$$
d\omega = i_X \nu, \nu = dz_1 \wedge \cdots \wedge dz_n.
$$

The singularity $p$ of $\omega$ is of Kupka type if $X(p) \neq 0$ and it is of generalized Kupka type (briefly g.K) if $X(p) = 0$ and $p$ is an isolated singularity of $X$. When $X(p)=0$ and the linear part of $X$ at $p$ is non singular ($\det(DX(p)) \neq 0$) we say that $p$ is non degenerated g.K.

The Kupka set of $\mathcal{F}_\omega$ is the set of Kupka singularities of $\mathcal{F}_\omega$.

If $p$ is of Kupka or g.K type then the division theorem [21] implies that
there exists another germ of holomorphic vector field, say $Y$, such that $\omega =
i_Y i_X \nu$.

Remark 4.2. — If $p$ is of Kupka type then there exists a local coordinate
system $z = (z_1, \dots, z_n)$ around $p$ such that $z(p) = 0$, $X = \frac{\partial}{\partial z_n}$ and $Y =$
4. **Proof of Theorem 1.17**

The purpose of this section is to prove the following:

THEOREM 1.17. — If $k \ge 2$ and $r \ge p + 2 = n - k + 2$ then
$\mathcal{L}_{td}(d_1, \dots, d_r; k, n)$ is an irreducible component of $\mathrm{Fol}(D(d_1, \dots, d_r, p); k, n)$
for all $r > p$ and $d_1, \dots, d_r \ge 1$.

The proof of Theorem 1.17 will be done first in the case of foliations of
dimension two. The general case will be reduced to this one by using the
following result:

THEOREM 4.1. — Let $\mathcal{F}$ be a codimension $p$ holomorphic foliation on $\mathbb{P}^n$,
$n > p + 1$. Assume that there is an algebraic smooth submanifold $M \subset \mathbb{P}^n$,
$\dim_{\mathbb{C}}(M) = m$, where $p + 1 \le m < n$, such that:

• The set of tangencies of $\mathcal{F}$ with $M$ has codimension $\ge 2$ on $M$.

• $\mathcal{F}|_M$ can be defined by a closed meromorphic $p$-form on $M$, say $\eta$.

Then $\eta$ can be extended to a closed meromorphic $p$-form $\tilde{\eta}$ on $\mathbb{P}^n$ defining
$\mathcal{F}$. Moreover, if $\eta$ is logarithmic so is $\tilde{\eta}$.

In fact, Theorem 4.1 is a generalization of a result in [4] (see also [16]).

The proof of Theorem 1.17 in the two dimensional case will be reduced
to a result that we state next (Theorem 4.3). In order to state it properly
let us recall the definition of a Kupka or generalized Kupka singularity for
two dimensional foliations (see also [17]).

Let $\omega$ be a germ at $p \in \mathbb{C}^n$ of integrable $(n-2)$-form with $\omega(p) = 0$. Recall
that the rotational of $\omega$ is the vector field $X = \operatorname{rot}(\omega)$ (cf. [17]) defined by

$$
d\omega = i_X \nu, \nu = dz_1 \wedge \cdots \wedge dz_n.
$$

The singularity $p$ of $\omega$ is of Kupka type if $X(p) \neq 0$ and it is of generalized Kupka type (briefly g.K) if $X(p) = 0$ and $p$ is an isolated singularity of $X$. When $X(p)=0$ and the linear part of $X$ at $p$ is non singular ($\det(DX(p)) \neq 0$) we say that $p$ is non degenerated g.K.

The Kupka set of $\mathcal{F}_\omega$ is the set of Kupka singularities of $\mathcal{F}_\omega$.

If $p$ is of Kupka or g.K type then the division theorem [21] implies that
there exists another germ of holomorphic vector field, say $Y$, such that $\omega =
i_Y i_X \nu$.

Remark 4.2. — If $p$ is of Kupka type then there exists a local coordinate
system $z = (z_1, \dots, z_n)$ around $p$ such that $z(p) = 0$, $X = \frac{\partial}{\partial z_n}$ and $Y =$
$\sum_{j=1}^{n-1} Y_j(z_1, \dots, z_{n-1}) \frac{\partial}{\partial z_j}$, $Y(0) = 0$. In particular, the foliation $\mathcal{F}_\omega$ has the structure of a local product, the germ of curve $\gamma = (z_1 = \dots = z_{n-1} = 0)$ is contained in the Kupka set of $\mathcal{F}_\omega$ and the vector field $Y$ defines the normal type of $\mathcal{F}_\omega$ along $\gamma$ [17].

In the next result we will consider the following situation: let $\mathcal{F}$ be a two-dimensional foliation on $\mathbb{P}^n$, $n \ge 4$. Assume that $\text{Sing}(\mathcal{F})$ contains a smooth irreducible curve, say $S$, with the following properties:

(I) There is a finite subset $F = \{p_1, \dots, p_k\} \subset S$ such that $S \setminus F \subset K(\mathcal{F})$, the Kupka set of $\mathcal{F}$. Since $S \setminus F$ is connected, the normal type of $\mathcal{F}$ is the same at all points of $S \setminus F$. We will denote by $Y$ a germ at $0 \in \mathbb{C}^{n-1}$ of holomorphic vector representing this normal type.

(II) The eigenvalues of the linear part of $Y$, say $\rho_1, \dots, \rho_{n-1}$, are in the Poincaré domain and satisfy the following non-resonant conditions
(\star) $\rho_j \neq \sum_{i \neq j} m_i \rho_i$ for all $m = (m_1, \dots, \widehat{m_j}, \dots, m_{n-1}) \in \mathbb{Z}_{\ge 0}^{n-2}$
with $\sum_i m_i \ge 1$.

In particular, we have $\rho_i \neq \rho_j$ if $i \neq j$. Recall that $\rho_1, \dots, \rho_{n-1}$ are in the Poincaré domain if there exists $a \neq 0$ such that $\text{Re}(a \cdot \rho_j) > 0$, $1 \le j \le n-1$. With these conditions the germ of vector field $Y$ is linearizable and semi-simple (cf. [1] and [19]).

(III) Given $p \in F$ let $\omega$ be a germ of (n-2)-form defining the germ of $\mathcal{F}$ at $p$. We will assume that there is a local coordinate system $(U, z = (z_1, \dots, z_n))$ around $p$ with the following properties:

(i) $z(p) = 0$ and $S \cap U = (z_1 = \dots = z_{n-1} = 0)$.

(ii) Set $X = \text{rot}(\omega)$, so that $d\omega = i_X \nu$, $\nu = dz_1 \wedge \dots \wedge dz_n$. Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of the linear part $DX(p)$. We will assume that there exists $a \neq 0$ such that $\text{Re}(a \cdot \lambda_n) < 0$ and $\text{Re}(a \cdot \lambda_j) > 0$, $\forall 1 \le j \le n-1$. Moreover, we will assume that the eingenspace of $DX(p)$ associated to the eigenvalue $\lambda_n$ is the tangent space $T_pS$.

(iii) Setting $\rho_n = 0$, we will assume that $\lambda_i \rho_j - \lambda_j \rho_i \neq 0$, $\forall 1 \le i < j \le n$.

**THEOREM 4.3.** — If $\mathcal{F}$ satisfies conditions (I), (II) and (III) above then there exist homogeneous polynomials $g_1, \dots, g_r$ such that $\mathcal{F} \in \mathcal{L}_{td}^{n-2}(g_1, \dots, g_r)$.

A crucial fact that will be used in the proof of Theorem 4.3 is that, if $r \ge n$ then there are foliations $\mathcal{F} \in \mathcal{L}_{td}(d_1, \dots, d_r; 2, n)$ with a curve $S \subset \text{Sing}(\mathcal{F})$ that satisfies conditions (I), (II) and (III) above:
$\sum_{j=1}^{n-1} Y_j(z_1, \dots, z_{n-1}) \frac{\partial}{\partial z_j}$, $Y(0) = 0$. In particular, the foliation $\mathcal{F}_\omega$ has the structure of a local product, the germ of curve $\gamma = (z_1 = \dots = z_{n-1} = 0)$ is contained in the Kupka set of $\mathcal{F}_\omega$ and the vector field $Y$ defines the normal type of $\mathcal{F}_\omega$ along $\gamma$ [17].

In the next result we will consider the following situation: let $\mathcal{F}$ be a two-dimensional foliation on $\mathbb{P}^n$, $n \ge 4$. Assume that $\text{Sing}(\mathcal{F})$ contains a smooth irreducible curve, say $S$, with the following properties:

(I) There is a finite subset $F = \{p_1, \dots, p_k\} \subset S$ such that $S \setminus F \subset K(\mathcal{F})$, the Kupka set of $\mathcal{F}$. Since $S \setminus F$ is connected, the normal type of $\mathcal{F}$ is the same at all points of $S \setminus F$. We will denote by $Y$ a germ at $0 \in \mathbb{C}^{n-1}$ of holomorphic vector representing this normal type.

(II) The eigenvalues of the linear part of $Y$, say $\rho_1, \dots, \rho_{n-1}$, are in the Poincaré domain and satisfy the following non-resonant conditions
(\star) $\rho_j \neq \sum_{i \neq j} m_i \rho_i$ for all $m = (m_1, \dots, \widehat{m_j}, \dots, m_{n-1}) \in \mathbb{Z}_{\ge 0}^{n-2}$
with $\sum_i m_i \ge 1$.

In particular, we have $\rho_i \neq \rho_j$ if $i \neq j$. Recall that $\rho_1, \dots, \rho_{n-1}$ are in the Poincaré domain if there exists $a \neq 0$ such that $\text{Re}(a.\rho_j) > 0$, $1 \le j \le n-1$. With these conditions the germ of vector field $Y$ is linearizable and semi-simple (cf. [1] and [19]).

(III) Given $p \in F$ let $\omega$ be a germ of (n-2)-form defining the germ of $\mathcal{F}$ at $p$. We will assume that there is a local coordinate system $(U, z = (z_1, \dots, z_n))$ around $p$ with the following properties:

(i) $z(p) = 0$ and $S \cap U = (z_1 = \dots = z_{n-1} = 0)$.

(ii) Set $X = \text{rot}(\omega)$, so that $d\omega = i_X \nu$, $\nu = dz_1 \wedge \dots \wedge dz_n$. Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of the linear part $DX(p)$. We will assume that there exists $a \neq 0$ such that $\text{Re}(a.\lambda_n) < 0$ and $\text{Re}(a.\lambda_j) > 0$, $\forall 1 \le j \le n-1$. Moreover, we will assume that the eingenspace of $DX(p)$ associated to the eigenvalue $\lambda_n$ is the tangent space $T_pS$.

(iii) Setting $\rho_n = 0$, we will assume that $\lambda_i\rho_j - \lambda_j\rho_i \neq 0$, $\forall 1 \le i < j \le n$.

**THEOREM 4.3.** — If $\mathcal{F}$ satisfies conditions (I), (II) and (III) above then there exist homogeneous polynomials $g_1, \dots, g_r$ such that $\mathcal{F} \in \mathcal{L}_{td}^{n-2}(g_1, \dots, g_r)$.

A crucial fact that will be used in the proof of Theorem 4.3 is that, if $r \ge n$ then there are foliations $\mathcal{F} \in \mathcal{L}_{td}(d_1, \dots, d_r; 2, n)$ with a curve $S \subset \text{Sing}(\mathcal{F})$ that satisfies conditions (I), (II) and (III) above:
LEMMA 4.4. — If $r \ge n$ then there are $\theta_2, \dots, \theta_{n-1} \in \mathcal{L}_F^1(d_1, \dots, d_r; n-1, n)$ such that, if $\eta_o := \theta_2 \wedge \dots \wedge \theta_{n-1}$, then $\mathcal{F}_{\eta_o}$ satisfies (I), (II) and (III) along the curve $S = \Pi(f_1 = \dots = f_{n-1} = 0)$.

Since the proof of Lemma 4.4 is not difficult we begin by it.

## 4.1. Proof of Lemma 4.4

Let $f_1, \dots, f_r$ be homogeneous polynomials on $\mathbb{C}^{n+1}$ with $\deg(f_j) = d_j$, $1 \le j \le r$, and such that $(f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$.

When $r = n-1$ and $S = \Pi(f_1 = \dots = f_{n-1} = 0)$ we have seen in the proof of Corollary 1.3 in Section 2.3 that the foliation $\mathcal{F}_0 \in \mathcal{L}_F(f_1, \dots, f_{n-1})$ is unique and defined by $\theta_0^2 \wedge \dots \wedge \theta_0^{n-1}$, where

$$ \theta_0^j = \frac{\mathrm{d}f_j}{f_j} - A_j \frac{\mathrm{d}f_1}{f_1}, \quad A_j = \frac{d_j}{d_1}, \quad 2 \le j \le n-1, \qquad (4.1) $$

with $d_j = \deg(f_j)$. In a neighborhood $U$ of any point $p \in S$ we can find local coordinates $z = (z_1, z_2, \dots, z_n)$ such that $f_j|_U = z_j$, $1 \le j \le n-1$, $\implies \theta_0^j = \frac{dz_j}{z_j} - A_j \frac{dz_1}{z_1}$ and $\mathcal{F}_{\eta_0}$ is defined by $\omega = z_1 \dots z_{n-1} \theta_0^2 \wedge \dots \wedge \theta_0^{n-1}|_U$. Since the $\theta_{j's}^0$ are closed we get

$$ d\omega = z_1 \dots z_{n-1} \left( \frac{dz_1}{z_1} + \dots + \frac{dz_{n-1}}{z_{n-1}} \right) \wedge \theta_0^2 \wedge \dots \wedge \theta_0^{n-1}|_U = \rho dz_1 \wedge \dots \wedge dz_{n-1}, $$
where $\rho = \frac{1}{d_1} \sum_j d_j \ne 0$, as the reader can check. Therefore, all points in $S$ are of Kupka type and $S$ is a Kupka component of $\mathcal{F}_{\eta_0}$. The normal type of $\mathcal{F}_{\eta_0}$ at $p$ can be defined by the linear vector field $Y = \sum_{j=1}^{n-1} dz_j \frac{\partial}{\partial z_j}$ because it satisfies $i_Y \theta_0^j = 0, \forall 2 \le j \le n-1$. In particular, the eigenvalues $\rho_j = d_j$, $1 \le j \le n-1$, are in the Poincaré domain: $\mathrm{Re}(\rho_j) > 0, \forall 1 \le j \le n-1$.

When $r \ge n$ we will consider small deformations of the forms $\theta_0^j$ above. For instance, if $r = n$ then the non-generic points of $S$ are the points of the set $F = S \cap \Pi(f_n = 0)$.

Let us consider the case $r = n$. Given $\tau = (t_2, \dots, t_{n-1}) \in \mathbb{C}^{n-2}$ consider the family of 1-forms

$$ \theta_{\tau}^{j} = \frac{\mathrm{d}f_{j}}{f_{j}} - A_{j}(\tau)\frac{\mathrm{d}f_{1}}{f_{1}} - B_{j}(\tau)\frac{\mathrm{d}f_{n}}{f_{n}}, \quad 2 \le j \le n-1, $$

where $A_j(\tau) = \frac{d_j}{d_1} - t_j$ and $B_j(\tau) = \frac{t_j d_1}{d_n}$. Note that $i_R \theta_{\tau}^j = 0, \forall 2 \le j \le n-1$, so that $\theta_{\tau}^j \in \mathcal{L}_F(f_1, \dots, f_n)$, $\forall \tau \in \mathbb{C}^{n-2}, \forall 2 \le j \le n-1$. Let $\mathcal{F}_{\tau}$ be the foliation defined by $\eta_{\tau} = \theta_{\tau}^2 \wedge \dots \wedge \theta_{\tau}^{n-1}$.
LEMMA 4.4. — If $r \ge n$ then there are $\theta_2, \dots, \theta_{n-1} \in \mathcal{L}_F^1(d_1, \dots, d_r; n-1, n)$ such that, if $\eta_o := \theta_2 \wedge \dots \wedge \theta_{n-1}$, then $\mathcal{F}_{\eta_o}$ satisfies (I), (II) and (III) along the curve $S = \Pi(f_1 = \dots = f_{n-1} = 0)$.

Since the proof of Lemma 4.4 is not difficult we begin by it.

## 4.1. Proof of Lemma 4.4

Let $f_1, \dots, f_r$ be homogeneous polynomials on $\mathbb{C}^{n+1}$ with $\deg(f_j) = d_j$, $1 \le j \le r$, and such that $(f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$.

When $r = n-1$ and $S = \Pi(f_1 = \dots = f_{n-1} = 0)$ we have seen in the proof of Corollary 1.3 in Section 2.3 that the foliation $\mathcal{F}_0 \in \mathcal{L}_F(f_1, \dots, f_{n-1})$ is unique and defined by $\theta_0^2 \wedge \dots \wedge \theta_0^{n-1}$, where

$$ \theta_0^j = \frac{\mathrm{d}f_j}{f_j} - A_j \frac{\mathrm{d}f_1}{f_1}, \quad A_j = \frac{d_j}{d_1}, \quad 2 \le j \le n-1, \qquad (4.1) $$

with $d_j = \deg(f_j)$. In a neighborhood $U$ of any point $p \in S$ we can find local coordinates $z = (z_1, z_2, \dots, z_n)$ such that $f_j|_U = z_j$, $1 \le j \le n-1$, $\implies \theta_0^j = \frac{dz_j}{z_j} - A_j \frac{dz_1}{z_1}$ and $\mathcal{F}_{\eta_0}$ is defined by $\omega = z_1 \dots z_{n-1} \theta_0^2 \wedge \dots \wedge \theta_0^{n-1}|_U$. Since the $\theta_{j's}^0$ are closed we get

$$ d\omega = z_1 \dots z_{n-1} \left( \frac{dz_1}{z_1} + \dots + \frac{dz_{n-1}}{z_{n-1}} \right) \wedge \theta_0^2 \wedge \dots \wedge \theta_0^{n-1}|_U = \rho dz_1 \wedge \dots \wedge dz_{n-1}, $$
where $\rho = \frac{1}{d_1} \sum_j d_j \ne 0$, as the reader can check. Therefore, all points in $S$ are of Kupka type and $S$ is a Kupka component of $\mathcal{F}_{\eta_0}$. The normal type of $\mathcal{F}_{\eta_0}$ at $p$ can be defined by the linear vector field $Y = \sum_{j=1}^{n-1} dz_j \frac{\partial}{\partial z_j}$ because it satisfies $i_Y \theta_0^j = 0, \forall 2 \le j \le n-1$. In particular, the eigenvalues $\rho_j = d_j$, $1 \le j \le n-1$, are in the Poincaré domain: $\mathrm{Re}(\rho_j) > 0, \forall 1 \le j \le n-1$.

When $r \ge n$ we will consider small deformations of the forms $\theta_0^j$ above. For instance, if $r = n$ then the non-generic points of $S$ are the points of the set $F = S \cap \Pi(f_n = 0)$.

Let us consider the case $r = n$. Given $\tau = (t_2, \dots, t_{n-1}) \in \mathbb{C}^{n-2}$ consider the family of 1-forms

$$ \theta_{\tau}^{j} = \frac{\mathrm{d}f_{j}}{f_{j}} - A_{j}(\tau)\frac{\mathrm{d}f_{1}}{f_{1}} - B_{j}(\tau)\frac{\mathrm{d}f_{n}}{f_{n}}, \quad 2 \le j \le n-1, $$

where $A_j(\tau) = \frac{d_j}{d_1} - t_j$ and $B_j(\tau) = \frac{t_j d_1}{d_n}$. Note that $i_R \theta_{\tau}^j = 0, \forall 2 \le j \le n-1$, so that $\theta_{\tau}^j \in \mathcal{L}_F(f_1, \dots, f_n)$, $\forall \tau \in \mathbb{C}^{n-2}, \forall 2 \le j \le n-1$. Let $\mathcal{F}_{\tau}$ be the foliation defined by $\eta_{\tau} = \theta_{\tau}^2 \wedge \dots \wedge \theta_{\tau}^{n-1}$.
If $p \in S$ is a generic point then $f_n(p) \neq 0$ and there are local coordinates at $p$, $(U, z = (z_1, \dots, z_{n-1}, z_n))$, such that $f_j|_U = z_j$, $1 \le j \le n-1$. In these coordinates we have $S \cap U = (z_1 = \dots = z_{n-1} = 0)$ and the normal type can be calculated by considering the restriction of $\mathcal{F}_\tau$ to a normal section, for instance $\Sigma := (f_n = f_n(p)) \cap U$. The foliation $\mathcal{F}_\tau|_\Sigma$ is defined by the (n-2)-form

$$ z_1 \dots z_{n-1} \eta_{\tau} |_{\Sigma} \\ = z_1 \dots z_{n-1} \left( \frac{dz_2}{z_2} - A_2(\tau) \frac{dz_1}{z_1} \right) \wedge \dots \wedge \left( \frac{dz_{n-1}}{z_{n-1}} - A_{n-1}(\tau) \frac{dz_1}{z_1} \right). $$

In particular, the normal type can be defined by the vector field

$$ Y_{\tau} = \sum_{j=1}^{n-1} \rho_{j}(\tau) z_{j} \frac{\partial}{\partial z_{j}}, $$

where $\rho_1(\tau) = d_1$ and $\rho_j(\tau) = d_j - t_j d_1$, because $i_{Y_\tau} \theta_\tau^i = 0, \forall 2 \le i \le n-1$.

If $|\tau|$ is small enough then the generic points of $S$ are of Kupka type and the eigenvalues of the normal type are in the Poincaré domain (these are open conditions). Moreover, the parameter $\tau$ can be chosen in such a way that the eigenvalues $\rho_1(\tau), \dots, \rho_{n-1}(\tau)$ satisfy the non-resonance conditions (⋆) of (II). This is a consequence of the fact that the set $\{(\rho_2, \dots, \rho_{n-1}) \in \mathbb{C}^{n-2} | d_1 = \rho_1, \dots, \rho_{n-1} \text{ satisfy conditions } (\star)\}$ is dense in $\mathbb{C}^{n-2}$.

At a point $p \in F$ we can find local coordinates $(U, z = (z_1, \dots, z_n))$ such that $z(p) = 0$ and the foliation is defined by the form $\omega_\tau = z_1 \dots z_n \tilde{\theta}_\tau^2 \wedge \dots \wedge \tilde{\theta}_\tau^{n-1}$, where

$$ \tilde{\theta}_{\tau}^{j} = \frac{dz_{j}}{z_{j}} - A_{j}(\tau) \frac{dz_{1}}{z_{1}} - B_{j}(\tau) \frac{dz_{n}}{z_{n}}. $$

Since the forms $\tilde{\theta}_\tau^j$ are closed, we get

$$ d\omega_{\tau} = \left( \frac{dz_1}{z_1} + \dots + \frac{dz_n}{z_n} \right) \wedge \omega_{\tau}. $$

The rotational $X_\tau$ of $\omega_\tau$ is defined by $d\omega_\tau = i_{X_\tau} dz_1 \wedge \dots \wedge dz_n$ and so $X_\tau = \sum_{j=1}^n \lambda_j(\tau) z_j \frac{\partial}{\partial z_j}$ is linear and must satisfy $i_{X_\tau} \left( \frac{dz_1}{z_1} + \dots + \frac{dz_n}{z_n} \right) = 0$ and $i_{X_\tau} \tilde{\theta}_\tau^j = 0, \forall 2 \le j \le n-1$. It follows that the eigenvalues $\lambda_1(\tau), \dots, \lambda_n(\tau)$ must satisfy the homogeneous system

$$ \begin{cases} x_1 + \dots + x_n = 0 \\ x_j - A_j(\tau)x_1 - B_j(\tau)x_n = 0, \quad 1 \le j \le n-2 \end{cases} \quad (4.2) $$

When $\tau = 0$ we are in the situation of the case $r = n-1$ and a solution of (4.2) is $x_j = d_j > 0$, if $1 \le j \le n-1$, and $x_n = -(d_1 + \dots + d_{n-1}) < 0$. Therefore, $\lambda_1(0), \dots, \lambda_n(0)$ satisfy condition Re$(a.\lambda_n(0)) < 0$
If $p \in S$ is a generic point then $f_n(p) \neq 0$ and there are local coordinates at $p$, $(U, z = (z_1, \dots, z_{n-1}, z_n))$, such that $f_j|_U = z_j$, $1 \le j \le n-1$. In these coordinates we have $S \cap U = (z_1 = \dots = z_{n-1} = 0)$ and the normal type can be calculated by considering the restriction of $\mathcal{F}_\tau$ to a normal section, for instance $\Sigma := (f_n = f_n(p)) \cap U$. The foliation $\mathcal{F}_\tau|_\Sigma$ is defined by the (n-2)-form

$$ z_1 \dots z_{n-1} \eta_\tau |_\Sigma \\ = z_1 \dots z_{n-1} \left( \frac{dz_2}{z_2} - A_2(\tau) \frac{dz_1}{z_1} \right) \wedge \dots \wedge \left( \frac{dz_{n-1}}{z_{n-1}} - A_{n-1}(\tau) \frac{dz_1}{z_1} \right). $$

In particular, the normal type can be defined by the vector field

$$ Y_{\tau} = \sum_{j=1}^{n-1} \rho_{j}(\tau) z_{j} \frac{\partial}{\partial z_{j}}, $$

where $\rho_1(\tau) = d_1$ and $\rho_j(\tau) = d_j - t_j d_1$, because $i_{Y_\tau} \theta_\tau^i = 0, \forall 2 \le i \le n-1$.

If $|\tau|$ is small enough then the generic points of $S$ are of Kupka type and the eigenvalues of the normal type are in the Poincaré domain (these are open conditions). Moreover, the parameter $\tau$ can be chosen in such a way that the eigenvalues $\rho_1(\tau), \dots, \rho_{n-1}(\tau)$ satisfy the non-resonance conditions (⋆) of (II). This is a consequence of the fact that the set $\{(\rho_2, \dots, \rho_{n-1}) \in \mathbb{C}^{n-2} | d_1 = \rho_1, \rho_2, \dots, \rho_{n-1} \text{ satisfy conditions } (\star)\}$ is dense in $\mathbb{C}^{n-2}$.

At a point $p \in F$ we can find local coordinates $(U, z = (z_1, \dots, z_n))$ such that $z(p) = 0$ and the foliation is defined by the form $\omega_\tau = z_1 \dots z_n \tilde{\theta}_\tau^2 \wedge \dots \wedge \tilde{\theta}_\tau^{n-1}$, where

$$ \tilde{\theta}_{\tau}^{j} = \frac{dz_{j}}{z_{j}} - A_{j}(\tau) \frac{dz_{1}}{z_{1}} - B_{j}(\tau) \frac{dz_{n}}{z_{n}}. $$

Since the forms $\tilde{\theta}_\tau^j$ are closed, we get

$$ d\omega_{\tau} = \left( \frac{dz_1}{z_1} + \dots + \frac{dz_n}{z_n} \right) \wedge \omega_{\tau}. $$

The rotational $X_\tau$ of $\omega_\tau$ is defined by $d\omega_\tau = i_{X_\tau} dz_1 \wedge \dots \wedge dz_n$ and so $X_\tau = \sum_{j=1}^n \lambda_j(\tau) z_j \frac{\partial}{\partial z_j}$ is linear and must satisfy $i_{X_\tau} \left( \frac{dz_1}{z_1} + \dots + \frac{dz_n}{z_n} \right) = 0$ and $i_{X_\tau} \tilde{\theta}_\tau^j = 0, \forall 2 \le j \le n-1$. It follows that the eigenvalues $\lambda_1(\tau), \dots, \lambda_n(\tau)$ must satisfy the homogeneous system

$$ \begin{cases} x_1 + \dots + x_n = 0 \\ x_j - A_j(\tau)x_1 - B_j(\tau)x_n = 0, \quad 1 \le j \le n-2 \end{cases} \quad (4.2) $$

When $\tau = 0$ we are in the situation of the case $r = n-1$ and a solution of (4.2) is $x_j = d_j > 0$, if $1 \le j \le n-1$, and $x_n = -(d_1 + \dots + d_{n-1}) < 0$. Therefore, $\lambda_1(0), \dots, \lambda_n(0)$ satisfy condition Re$(a.\lambda_n(0)) < 0$
and $\mathrm{Re}(a.\lambda_j(0)) > 0$, $1 \le j \le n-1$, for some $a \ne 0$. Of course, this implies that for small $|\tau|$ the eigenvalues of $X_\tau$ has eigenvalues that satisfy $\mathrm{Re}(a_\tau.\lambda_n(\tau)) < 0$ and $\mathrm{Re}(a_\tau.\lambda_j(\tau)) > 0$, $1 \le j \le n-1$, for some $a_\tau \ne 0$. It remains to verify that $\mathcal{F}_\tau$ satisfies condition (iii) of (III) near $p$.

First of all, recall that $X_\tau$ and $Z_\tau = Y_\tau = \sum_{j=1}^{n-1} \rho_j(\tau)z_j \frac{\partial}{\partial z_j}$ are tangent to $\mathcal{F}_\tau$. Moreover, since $X_\tau \wedge Z_\tau \ne 0$ these vector fields generate the foliation in a neighborhood of $p=0$. In particular, we must have $\omega_\tau = b.i_{X_\tau} i_{Z_\tau} \nu$ for some $b \ne 0$. If we set $\rho_n(\tau) = 0$ then the coefficient of $\frac{\partial}{\partial z_i} \wedge \frac{\partial}{\partial z_j}$ in $X_\tau \wedge Z_\tau$ is $\lambda_i(\tau)\rho_j(\tau) - \lambda_j(\tau)\rho_i(\tau) \ne 0$ if $i < j$. Therefore (iii) is equivalent to prove that all coefficients of $\omega_\tau$ are not zero.

Set $\alpha = z_1 \dots z_{n-1} \tilde{\theta}_0^2 \wedge \dots \wedge \tilde{\theta}_0^{n-1}$. By the case $r=n-1$ we have $\mathrm{Sing}(\alpha) = S \cap U = (z_1 = \dots = z_{n-1} = 0)$. Since $\omega_0 = z_n.\alpha$ we have

$$ \mathrm{Sing}(\omega_0) = (z_n = 0) \cup (z_1 = \dots = z_{n-1} = 0). $$

On the other hand, if $\tau \ne 0$ then the form $\omega_\tau$ can be written as

$$ \omega_\tau = z_n.\alpha_\tau + \mathrm{d}z_n \wedge \beta_\tau $$

where $\alpha_0 = \alpha$, $\alpha_\tau$ has linear coefficients and

$$ \beta_\tau = \sum_{1 \le i < j \le n-1} A_{ij}(\tau) z_i z_j \mathrm{d}z_1 \wedge \dots \wedge \widehat{\mathrm{d}z_i} \wedge \dots \wedge \widehat{\mathrm{d}z_j} \wedge \dots \wedge \mathrm{d}z_{n-1}, $$

where $A_{ij}(\tau) = \pm(A_i(\tau)B_j(\tau) - A_j(\tau)B_i(\tau))$, if $i,j > 1$ and $A_{1j}(\tau) = \pm B_j(\tau)$, if $2 \le j \le n-1$. We leave this computation for the reader. If $t_j \ne 0$ then $A_{1j}(\tau) = \pm B_j(\tau) = \pm t_j.d_1/d_n \ne 0$. If $i,j > 1$ then

$$ A_{ij}(\tau) = \pm \frac{d_1}{d_n} (t_j d_i - t_i d_j). $$

Hence, we can choose $\tau$ small so that $t_j d_i - t_i d_j \ne 0$, $\forall 1 < i < j \le n-1$.

In the case $r > n$ we consider the parameter $\tau = (t_{ji})_{\substack{2 \le j \le n-1 \\ n \le i \le r}}$ and

$$ \theta_{\tau}^{rj} = \frac{\mathrm{d}f_j}{f_j} - A_j(\tau) \frac{\mathrm{d}f_1}{f_1} - \sum_{i=n}^{r} B_{ji}(\tau) \frac{\mathrm{d}f_i}{f_i}, $$

where $A_j(\tau) = \frac{d_j}{d_1} - \sum_{i=n}^r t_{ji}$ and $B_{ji}(\tau) = \frac{d_i}{d_1} t_{ji}$, $2 \le j \le n-1$. It can be checked directly that $i_R \theta_\tau^{rj} = 0, \forall j$. The proof of the lemma in this case can be done by induction on $r \ge n$. We leave the details for the reader. $\square$

## 4.2. Proof of Theorem 1.17 in the case of two dimensional foliations

In the proof we will use Lemma 4.4 and Theorem 4.3. Theorem 4.3 will be proved in the next section.
and $\text{Re}(a.\lambda_j(0)) > 0$, $1 \le j \le n-1$, for some $a \ne 0$. Of course, this implies that for small $|\tau|$ the eigenvalues of $X_\tau$ has eigenvalues that satisfy $\text{Re}(a_\tau.\lambda_n(\tau)) < 0$ and $\text{Re}(a_\tau.\lambda_j(\tau)) > 0$, $1 \le j \le n-1$, for some $a_\tau \ne 0$. It remains to verify that $\mathcal{F}_\tau$ satisfies condition (iii) of (III) near $p$.

First of all, recall that $X_\tau$ and $Z_\tau = Y_\tau = \sum_{j=1}^{n-1} \rho_j(\tau)z_j \frac{\partial}{\partial z_j}$ are tangent to $\mathcal{F}_\tau$. Moreover, since $X_\tau \wedge Z_\tau \ne 0$ these vector fields generate the foliation in a neighborhood of $p=0$. In particular, we must have $\omega_\tau = b.i_{X_\tau} i_{Z_\tau} \nu$ for some $b \ne 0$. If we set $\rho_n(\tau) = 0$ then the coefficient of $\frac{\partial}{\partial z_i} \wedge \frac{\partial}{\partial z_j}$ in $X_\tau \wedge Z_\tau$ is $\lambda_i(\tau)\rho_j(\tau) - \lambda_j(\tau)\rho_i(\tau) \ne 0$ if $i < j$. Therefore (iii) is equivalent to prove that all coefficients of $\omega_\tau$ are not zero.

Set $\alpha = z_1 \dots z_{n-1} \tilde{\theta}_0^2 \wedge \dots \wedge \tilde{\theta}_0^{n-1}$. By the case $r=n-1$ we have $\text{Sing}(\alpha) = S \cap U = (z_1 = \dots = z_{n-1} = 0)$. Since $\omega_0 = z_n.\alpha$ we have

$$ \text{Sing}(\omega_0) = (z_n = 0) \cup (z_1 = \dots = z_{n-1} = 0). $$

On the other hand, if $\tau \ne 0$ then the form $\omega_\tau$ can be written as

$$ \omega_\tau = z_n.\alpha_\tau + \mathrm{d}z_n \wedge \beta_\tau $$

where $\alpha_0 = \alpha$, $\alpha_\tau$ has linear coefficients and

$$ \beta_\tau = \sum_{1 \le i < j \le n-1} A_{ij}(\tau) z_i z_j \mathrm{d}z_1 \wedge \dots \wedge \widehat{\mathrm{d}z_i} \wedge \dots \wedge \widehat{\mathrm{d}z_j} \wedge \dots \wedge \mathrm{d}z_{n-1}, $$

where $A_{ij}(\tau) = \pm(A_i(\tau)B_j(\tau) - A_j(\tau)B_i(\tau))$, if $i,j > 1$ and $A_{1j}(\tau) = \pm B_j(\tau)$, if $2 \le j \le n-1$. We leave this computation for the reader. If $t_j \ne 0$ then $A_{1j}(\tau) = \pm B_j(\tau) = \pm t_j.d_1/d_n \ne 0$. If $i,j > 1$ then

$$ A_{ij}(\tau) = \pm \frac{d_1}{d_n} (t_j d_i - t_i d_j). $$

Hence, we can choose $\tau$ small so that $t_j d_i - t_i d_j \ne 0$, $\forall 1 < i < j \le n-1$.

In the case $r > n$ we consider the parameter $\tau = (t_{ji})_{\substack{2 \le j \le n-1 \\ n \le i \le r}}$ and

$$ \theta_{\tau}^{rj} = \frac{\mathrm{d}f_j}{f_j} - A_j(\tau) \frac{\mathrm{d}f_1}{f_1} - \sum_{i=n}^{r} B_{ji}(\tau) \frac{\mathrm{d}f_i}{f_i}, $$

where $A_j(\tau) = \frac{d_j}{d_1} - \sum_{i=n}^r t_{ji}$ and $B_{ji}(\tau) = \frac{d_i}{d_1}t_{ji}$, $2 \le j \le n-1$. It can be checked directly that $i_R \theta_\tau^{rj} = 0, \forall j$. The proof of the lemma in this case can be done by induction on $r \ge n$. We leave the details for the reader. $\square$

## 4.2. Proof of Theorem 1.17 in the case of two dimensional foliations

In the proof we will use Lemma 4.4 and Theorem 4.3. Theorem 4.3 will be proved in the next section.
We want to prove that for all $r \ge n$ and $d_1, \dots, d_r \ge 1$ then $\mathcal{L}_{td}(d_1, \dots, d_r; 2, n)$ is an irreducible component of $\mathrm{Fol}(D; 2, n)$, where $D = \sum_j d_j - n + 1$. To avoid confusion we assume $d_1 \le d_2 \le \dots \le d_r$. Recall that the definition implies

$$ \mathcal{L}_{td}(d_1, \dots, d_r; 2, n) = \bigcup_{\substack{dg(f_j)=d_j \\ 1 \le j \le r}} \mathcal{L}_{td}^{n-2}(f_1, \dots, f_r). $$

Since $\overline{\mathcal{L}_{td}^{n-2}(f_1, \dots, f_r)}$ is irreducible for all polynomials $f_1, \dots, f_r$ with $\deg(f_j) = d_j$, $1 \le j \le r$, it is clear that $\mathcal{L}_{td}(d_1, \dots, d_r; 2, n)$ is an irreducible algebraic subset of $\mathrm{Fol}(D; 2, n)$. The idea is to exhibit a foliation $\mathcal{F}_0 \in \mathcal{L}_{td}(d_1, \dots, d_r; 2, n)$ such that for any germ of holomorphic deformation $t \in (\mathbb{C}, 0) \mapsto \mathcal{F}_t \in \mathrm{Fol}(D; 2, n)$, with $\mathcal{F}_t|_{t=0} = \mathcal{F}_0$, then $\mathcal{F}_t \in \mathcal{L}_{td}(d_1, \dots, d_n; 2, n) \forall t \in (\mathbb{C}, 0)$.

In order to do that, first of all let us fix homogeneous polynomials $f_1, \dots, f_r$ in $\mathbb{C}^{n+1}$ with $\deg(f_j) = d_j$, $1 \le j \le r$, such that the hypersurface $(f_1 \dots f_r = 0) \subset \mathbb{C}^{n+1}$ has a strictly ordinary singularity outside $0 \in \mathbb{C}^{n+1}$. In particular, for any $J = (1 \le j_1 < \dots < j_{n-1} \le r)$ then the curve $S_J = \Pi(f_{j_1} = \dots = f_{j_{n-1}} = 0) \subset \mathbb{P}^n$ is a smooth complete intersection.

From now on we fix $J = (1, 2, \dots, n-1)$ and set $S_J = S$. By Lemma 4.4 there exists $\eta_\tau = \theta_\tau^2 \wedge \dots \wedge \theta_\tau^{n-1} \in \mathcal{L}_{td}^{n-2}(f_1, \dots, f_r)$ such that the foliation $\mathcal{F}_{\eta_\tau}$ defined by $\eta_\tau$ satisfies (I), (II) and (III) along the curve $S$. The finite set of (I) is $F = S \cap \bigcup_{j \ge n} \Pi(f_j = 0)$.

**Remark 4.5.** — The parameter $\tau = (t_{ji})_{n \le i \le r}^{2 \le j \le n-1}$ in Lemma 4.4 can be chosen in such a way that if $\mathcal{F}_{\eta_\tau} \in \mathcal{L}_{td}(d'_1, \dots, d'_s; 2, n)$, where $d'_1 \le d'_2 \le \dots \le d'_s$ then $d'_i = d_i$, $\forall i$. Recalling the definition of the $\theta_\tau^{j'_s}$, an example in which $\mathcal{F}_{\eta_\tau}$ belongs to two different $\mathcal{L}_{td'}$ is when $B_{jn}(\tau) = B_{jn+1}(\tau)$ for all $2 \le j \le n-1$. In this case, in the sum that defines $\theta_\tau^j$ there are terms as below

$$ B_{jn}(\tau) \frac{\mathrm{d}f_n}{f_n} + B_{jn+1}(\tau) \frac{\mathrm{d}f_{n+1}}{f_{n+1}} = B_{jn}(\tau) \frac{\mathrm{d}(f_n f_{n+1})}{f_n f_{n+1}}. $$

In this case

$$ \mathcal{F}_{\eta_\tau} \in \mathcal{L}_{td}(d_1, \dots, d_r : 2, D) \cap \mathcal{L}_{td}(d_1, \dots, d_{n-1}, d_n + d_{n+1}, \dots, d_r; 2, D). $$

On the other hand, if we choose the parameters $t_{ji}$ linearly independent over $\mathbb{Z}$ then the required property is true. From now on, we will assume this property.

Since $\mathcal{F}_{\eta_o}$ satisfies property (III) along the curve $S$, all points of the finite set $F$ are non degenerated g.K singularities of $\mathcal{F}_{\eta_o}$. Fix any holomorphic germ of deformation $t \in (\mathbb{C}, 0) \mapsto \mathcal{F}_t \in \mathrm{Fol}(D; 2, n)$. The main fact that we will
We want to prove that for all $r \ge n$ and $d_1, \dots, d_r \ge 1$ then $\mathcal{L}_{td}(d_1, \dots, d_r; 2, n)$ is an irreducible component of $\mathrm{Fol}(D; 2, n)$, where $D = \sum_j d_j - n + 1$. To avoid confusion we assume $d_1 \le d_2 \le \dots \le d_r$. Recall that the definition implies

$$ \mathcal{L}_{td}(d_1, \dots, d_r; 2, n) = \bigcup_{\substack{dg(f_j)=d_j \\ 1 \le j \le r}} \mathcal{L}_{td}^{n-2}(f_1, \dots, f_r). $$

Since $\overline{\mathcal{L}_{td}^{n-2}(f_1, \dots, f_r)}$ is irreducible for all polynomials $f_1, \dots, f_r$ with $\deg(f_j) = d_j$, $1 \le j \le r$, it is clear that $\mathcal{L}_{td}(d_1, \dots, d_r; 2, n)$ is an irreducible algebraic subset of $\mathrm{Fol}(D; 2, n)$. The idea is to exhibit a foliation $\mathcal{F}_0 \in \mathcal{L}_{td}(d_1, \dots, d_r; 2, n)$ such that for any germ of holomorphic deformation $t \in (\mathbb{C}, 0) \mapsto \mathcal{F}_t \in \mathrm{Fol}(D; 2, n)$, with $\mathcal{F}_t|_{t=0} = \mathcal{F}_0$, then $\mathcal{F}_t \in \mathcal{L}_{td}(d_1, \dots, d_n; 2, n) \forall t \in (\mathbb{C}, 0)$.

In order to do that, first of all let us fix homogeneous polynomials $f_1, \dots, f_r$ in $\mathbb{C}^{n+1}$ with $\deg(f_j) = d_j$, $1 \le j \le r$, such that the hypersurface $(f_1 \dots f_r = 0) \subset \mathbb{C}^{n+1}$ has a strictly ordinary singularity outside $0 \in \mathbb{C}^{n+1}$. In particular, for any $J = (1 \le j_1 < \dots < j_{n-1} \le r)$ then the curve $S_J = \Pi(f_{j_1} = \dots = f_{j_{n-1}} = 0) \subset \mathbb{P}^n$ is a smooth complete intersection.

From now on we fix $J = (1, 2, \dots, n-1)$ and set $S_J = S$. By Lemma 4.4 there exists $\eta_\tau = \theta_\tau^2 \wedge \dots \wedge \theta_\tau^{n-1} \in \mathcal{L}_{td}^{n-2}(f_1, \dots, f_r)$ such that the foliation $\mathcal{F}_{\eta_\tau}$ defined by $\eta_\tau$ satisfies (I), (II) and (III) along the curve $S$. The finite set of (I) is $F = S \cap \bigcup_{j \ge n} \Pi(f_j = 0)$.

**Remark 4.5.** — The parameter $\tau = (t_{ji})_{n \le i \le r}^{2 \le j \le n-1}$ in Lemma 4.4 can be chosen in such a way that if $\mathcal{F}_{\eta_\tau} \in \mathcal{L}_{td}(d'_1, \dots, d'_s; 2, n)$, where $d'_1 \le d'_2 \le \dots \le d'_s$ then $d'_i = d_i$, $\forall i$. Recalling the definition of the $\theta_\tau^{j'_s}$, an example in which $\mathcal{F}_{\eta_\tau}$ belongs to two different $\mathcal{L}_{td'}$ is when $B_{jn}(\tau) = B_{jn+1}(\tau)$ for all $2 \le j \le n-1$. In this case, in the sum that defines $\theta_\tau^j$ there are terms as below

$$ B_{jn}(\tau) \frac{\mathrm{d}f_n}{f_n} + B_{jn+1}(\tau) \frac{\mathrm{d}f_{n+1}}{f_{n+1}} = B_{jn}(\tau) \frac{\mathrm{d}(f_n f_{n+1})}{f_n f_{n+1}}. $$

In this case

$$ \mathcal{F}_{\eta_{\tau}} \in \mathcal{L}_{td}(d_1, \dots, d_r : 2, D) \cap \mathcal{L}_{td}(d_1, \dots, d_{n-1}, d_n + d_{n+1}, \dots, d_r; 2, D). $$

On the other hand, if we choose the parameters $t_{ji}$ linearly independent over $\mathbb{Z}$ then the required property is true. From now on, we will assume this property.

Since $\mathcal{F}_{\eta_o}$ satisfies property (III) along the curve $S$, all points of the finite set $F$ are non degenerated g.K singularities of $\mathcal{F}_{\eta_o}$. Fix any holomorphic germ of deformation $t \in (\mathbb{C}, 0) \mapsto \mathcal{F}_t \in \mathrm{Fol}(D; 2, n)$. The main fact that we will
use is that the curve $S$ admits a $C^\infty$ deformation $t \in (\mathbb{C}, 0) \mapsto S(t)$ such that $S(t) \subset \text{Sing}(\mathcal{F}_t)$ and the foliation $\mathcal{F}_t$ satisfies properties (I), (II), (III) along $S(t)$.

LEMMA 4.6. — There exists a germ of $C^\infty$ isotopy $\Phi: (\mathbb{C}, 0) \times S \mapsto \mathbb{P}^n$, such that, if we denote $S(t) := \phi(t, S)$, then:

(a) $S(0) = S$ and $S(t) \subset \text{Sing}(\mathcal{F}_t)$ is smooth $\forall s \in (\mathbb{C}, 0)$. In particular, $S(t)$ is an algebraic complete intersection, $\forall t \in (\mathbb{C}, 0)$.

(b) If $r = n - 1$ then all points of $S(t)$ are of Kupka type.

(c) If $r > n - 1$ then any point $p \in F = S \cap \bigcup_{k \ge n} \Pi(f_k = 0)$, $n - 1 < k \le r$, has a holomorphic deformation $t \in (\mathbb{C}, 0) \mapsto P_p(t)$ such that $P_p(t) \in S(t)$ is a non degenerated g.K singularity of $\mathcal{F}_t$. Set $F(t) := \{P_p(t) | p \in F\}$.

(d) The points of $S(t) \setminus F(t)$ are in the Kupka set of $\mathcal{F}(t)$. Moreover, if we denote by $Y_t$ the normal type of $\mathcal{F}_t$ along $S(t) \setminus F(t)$ then the correspondence $t \in (\mathbb{C}, 0) \mapsto Y_t$ is holomorphic.

*Proof*. — The argument for the proof of (c) uses the stability under deformations of the non degenerated g.K points [17, Theorem 6]. The argument for the existence of the isotopy $\Phi$ is similar to [16, Lemma 2.3.3, p. 83] and uses essentially the local stability under deformations of the Kupka set [9] and of the non degenerated g.K singular points [17]. The fact that the deformed curve $S(t)$ satisfies (I), (II) and (III) for the foliation $\mathcal{F}(t)$ is a consequence of the fact that these conditions are open. We leave the details for the reader. $\square$

Let us finish the proof. We will assume that $\mathcal{F}_{\eta_0}$ satisfies Remark 4.5. Lemma 4.6 implies that the foliation $\mathcal{F}(t)$ has a curve $S(t)$ in the singular set that satisfies (I), (II) and (III). In particular, there are homogeneous polynomials $g_1(t), \dots, g_{s(t)}(t)$ such that $\mathcal{F}_t \in \mathcal{L}_{td}^{n-2}(g_1(t), \dots, g_{s(t)}(t))$. Set $\text{deg}(g_j(t)) = d_j(t)$. We assert that $s(t) = r$ and that we can assume $d_j(t) = d_j, 1 \le j \le r$.

In fact, since $D = \sum_{j=1}^{s(t)} d_j(t) - n + 1$ we have $s(t) \le D + n - 1$ and the number of possibilities for the degrees $d_j(t)$, $1 \le j \le s(t)$ is finite. In particular, there is a germ of non-contable set $A \subset (\mathbb{C}, 0)$ such that the functions $t \in A \mapsto s(t)$ and $t \in A \mapsto d_j(t)$, $1 \le j \le s(t)$, are all constants, say $s|_A = r'$ and $d_j|_A = d'_j$. In particular, $\mathcal{F}(t) \in \mathcal{L}_{td}(d'_1, \dots, d'_{r'}; 2; D)$ for all $t \in A$. Since 0 is in the adherence of $A$ we get $\mathcal{F}_{\eta_0} \in \mathcal{L}_{td}(d'_1, \dots, d'_{r'}; 2, D)$. Hence, $r' = r$ and $d'_j = d_j$, $1 \le j \le r$, and $\mathcal{F}(t) \in \mathcal{L}_{td}(d_1, \dots, d_r; 2, D)$. $\square$
use is that the curve $S$ admits a $C^\infty$ deformation $t \in (\mathbb{C}, 0) \mapsto S(t)$ such that $S(t) \subset \text{Sing}(\mathcal{F}_t)$ and the foliation $\mathcal{F}_t$ satisfies properties (I), (II), (III) along $S(t)$.

LEMMA 4.6. — There exists a germ of $C^\infty$ isotopy $\Phi: (\mathbb{C}, 0) \times S \mapsto \mathbb{P}^n$, such that, if we denote $S(t) := \phi(t, S)$, then:

(a) $S(0) = S$ and $S(t) \subset \text{Sing}(\mathcal{F}_t)$ is smooth $\forall s \in (\mathbb{C}, 0)$. In particular, $S(t)$ is an algebraic complete intersection, $\forall t \in (\mathbb{C}, 0)$.

(b) If $r = n - 1$ then all points of $S(t)$ are of Kupka type.

(c) If $r > n - 1$ then any point $p \in F = S \cap \bigcup_{k \ge n} \Pi(f_k = 0)$, $n - 1 < k \le r$, has a holomorphic deformation $t \in (\mathbb{C}, 0) \mapsto P_p(t)$ such that $P_p(t) \in S(t)$ is a non degenerated g.K singularity of $\mathcal{F}_t$. Set $F(t) := \{P_p(t) | p \in F\}$.

(d) The points of $S(t) \setminus F(t)$ are in the Kupka set of $\mathcal{F}(t)$. Moreover, if we denote by $Y_t$ the normal type of $\mathcal{F}_t$ along $S(t) \setminus F(t)$ then the correspondence $t \in (\mathbb{C}, 0) \mapsto Y_t$ is holomorphic.

*Proof*. — The argument for the proof of (c) uses the stability under deformations of the non degenerated g.K points [17, Theorem 6]. The argument for the existence of the isotopy $\Phi$ is similar to [16, Lemma 2.3.3, p. 83] and uses essentially the local stability under deformations of the Kupka set [9] and of the non degenerated g.K singular points [17]. The fact that the deformed curve $S(t)$ satisfies (I), (II) and (III) for the foliation $\mathcal{F}(t)$ is a consequence of the fact that these conditions are open. We leave the details for the reader. $\square$

Let us finish the proof. We will assume that $\mathcal{F}_{\eta_0}$ satisfies Remark 4.5. Lemma 4.6 implies that the foliation $\mathcal{F}(t)$ has a curve $S(t)$ in the singular set that satisfies (I), (II) and (III). In particular, there are homogeneous polynomials $g_1(t), \dots, g_{s(t)}(t)$ such that $\mathcal{F}_t \in \mathcal{L}_{td}^{n-2}(g_1(t), \dots, g_{s(t)}(t))$. Set $\text{deg}(g_j(t)) = d_j(t)$. We assert that $s(t) = r$ and that we can assume $d_j(t) = d_j, 1 \le j \le r$.

In fact, since $D = \sum_{j=1}^{s(t)} d_j(t) - n + 1$ we have $s(t) \le D + n - 1$ and the number of possibilities for the degrees $d_j(t)$, $1 \le j \le s(t)$ is finite. In particular, there is a germ of non-contable set $A \subset (\mathbb{C}, 0)$ such that the functions $t \in A \mapsto s(t)$ and $t \in A \mapsto d_j(t)$, $1 \le j \le s(t)$, are all constants, say $s|_A = r'$ and $d_j|_A = d'_j$. In particular, $\mathcal{F}(t) \in \mathcal{L}_{td}(d'_1, \dots, d'_{r'}; 2; D)$ for all $t \in A$. Since 0 is in the adherence of $A$ we get $\mathcal{F}_{\eta_0} \in \mathcal{L}_{td}(d'_1, \dots, d'_{r'}; 2, D)$. Hence, $r' = r$ and $d'_j = d_j$, $1 \le j \le r$, and $\mathcal{F}(t) \in \mathcal{L}_{td}(d_1, \dots, d_r; 2, D)$. $\square$
4.3. *Proof of Theorem 4.3*

Let $\mathcal{F}$ be a two dimensional foliation on $\mathbb{P}^n$, $n \ge 4$, having a curve $S$ in the singular set and that satisfies (I), (II) and (III). The idea is to construct closed logarithmic 1-forms $\theta_2, \dots, \theta_{n-1}$, defined in a neighborhood $U$ of the curve $S$, such that $\theta_2 \wedge \dots \wedge \theta_{n-1}$ defines the foliation $\mathcal{F}|_U$. By using an extension theorem of meromorphic functions (cf. [2] and [22]), each form $\theta_j$ can be extended to a global closed meromorphic 1-form on $\mathbb{P}^n$, denoted again by $\theta_j$, $2 \le j \le n-1$. The fact that $\theta_j|_U$ is logarithmic implies that $\theta_j$ is also logarithmic: there are homogeneous polynomials in $\mathbb{C}^{n+1}$, say $g_1, \dots, g_r$, such that $\theta_j \in \mathcal{L}_R^1(g_1, \dots, g_r)$, $\forall 1 \le j \le n-2$, $\implies \mathcal{F} \in \mathcal{L}_{td}^{n-2}(g_1, \dots, g_r)$. The following result will be usefull:

**THEOREM 4.7 (Parametric linearization).** — Let $(W_\tau)_{\tau \in (\mathbb{C}^k, 0)}$ be a germ at $0 \in \mathbb{C}^k$ of a holomorphic family of germs of holomorphic vector fields at $0 \in \mathbb{C}^m$. Assume that:

(a) The linear part $L_\tau = DW_\tau(0)$ is diagonal of the form $L_\tau = \sum_{j=1}^m \rho_j(\tau) z_j \frac{\partial}{\partial z_j}$ in some local coordinate system $z = (z_1, \dots, z_m)$ around $0 \in \mathbb{C}^m$.

(b) $\rho_1(0), \dots, \rho_m(0)$ are in the Poincaré domain and satisfy the non-resonance condition $(\star)$ in (II).

Then there exists a holomorphic family of germs of biholomorphisms $(\Psi_\tau)_{\tau \in (\mathbb{C}^k, 0)}$ such that $D\Psi_\tau(0) = I$ and

$$ \Psi_{\tau}^{*}(W_{\tau}) = L_{\tau} = \sum_{j=1}^{m} \rho_{j}(\tau) w_{j} \frac{\partial}{\partial w_{j}}. $$

Theorem 4.7 is a parametric version of Poincaré's linearization theorem. Its proof can be found in [1] or [19].

Let us continue the proof of Theorem 4.3. First of all, we will prove that there are n-2 closed logarithmic 1-forms $\theta_2, \dots, \theta_{n-1}$, defined in some neighborhood $W$ of $S \setminus F$, such that $\eta = \theta_2 \wedge \dots \wedge \theta_{n-1}$ defines $\mathcal{F}|_W$.

Fix $p \in S \setminus F$. Since $p \in K(\mathcal{F})$ there are local coordinates $(V, z = (z_1, \dots, z_n))$, with $p \in V$, such that

(i) $z(p) = 0$ and $S \cap V = (z_1 = \dots = z_{n-1} = 0)$.

(ii) $\mathcal{F}|_V$ is defined by a holomorphic (n-2)-form $\omega$ that can be written as $\omega = i_Y i_X \nu$, where $X = \frac{\partial}{\partial z_n}$, $Y = \sum_{j=1}^{n-1} Y_j(z_1, \dots, z_{n-1}) \frac{\partial}{\partial z_j}$ is the normal type and $\nu = dz_n \wedge dz_1 \wedge \dots \wedge dz_{n-1}$.
4.3. *Proof of Theorem 4.3*

Let $\mathcal{F}$ be a two dimensional foliation on $\mathbb{P}^n$, $n \ge 4$, having a curve $S$ in the singular set and that satisfies (I), (II) and (III). The idea is to construct closed logarithmic 1-forms $\theta_2, \dots, \theta_{n-1}$, defined in a neighborhood $U$ of the curve $S$, such that $\theta_2 \wedge \dots \wedge \theta_{n-1}$ defines the foliation $\mathcal{F}|_U$. By using an extension theorem of meromorphic functions (cf. [2] and [22]), each form $\theta_j$ can be extended to a global closed meromorphic 1-form on $\mathbb{P}^n$, denoted again by $\theta_j$, $2 \le j \le n-1$. The fact that $\theta_j|_U$ is logarithmic implies that $\theta_j$ is also logarithmic: there are homogeneous polynomials in $\mathbb{C}^{n+1}$, say $g_1, \dots, g_r$, such that $\theta_j \in \mathcal{L}_R^1(g_1, \dots, g_r)$, $\forall 1 \le j \le n-2$, $\implies \mathcal{F} \in \mathcal{L}_{td}^{n-2}(g_1, \dots, g_r)$. The following result will be usefull:

**THEOREM 4.7 (Parametric linearization).** — Let $(W_\tau)_{\tau \in (\mathbb{C}^k, 0)}$ be a germ at $0 \in \mathbb{C}^k$ of a holomorphic family of germs of holomorphic vector fields at $0 \in \mathbb{C}^m$. Assume that:

(a) The linear part $L_\tau = DW_\tau(0)$ is diagonal of the form $L_\tau = \sum_{j=1}^m \rho_j(\tau) z_j \frac{\partial}{\partial z_j}$ in some local coordinate system $z = (z_1, \dots, z_m)$ around $0 \in \mathbb{C}^m$.

(b) $\rho_1(0), \dots, \rho_m(0)$ are in the Poincaré domain and satisfy the non-resonance condition $(\star)$ in (II).

Then there exists a holomorphic family of germs of biholomorphisms $(\Psi_\tau)_{\tau \in (\mathbb{C}^k, 0)}$ such that $D\Psi_\tau(0) = I$ and

$$ \Psi_{\tau}^{*}(W_{\tau}) = L_{\tau} = \sum_{j=1}^{m} \rho_{j}(\tau) w_{j} \frac{\partial}{\partial w_{j}}. $$

Theorem 4.7 is a parametric version of Poincaré's linearization theorem. Its proof can be found in [1] or [19].

Let us continue the proof of Theorem 4.3. First of all, we will prove that there are n-2 closed logarithmic 1-forms $\theta_2, \dots, \theta_{n-1}$, defined in some neighborhood $W$ of $S \setminus F$, such that $\eta = \theta_2 \wedge \dots \wedge \theta_{n-1}$ defines $\mathcal{F}|_W$.

Fix $p \in S \setminus F$. Since $p \in K(\mathcal{F})$ there are local coordinates $(V, z = (z_1, \dots, z_n))$, with $p \in V$, such that

(i) $z(p) = 0$ and $S \cap V = (z_1 = \dots = z_{n-1} = 0)$.

(ii) $\mathcal{F}|_V$ is defined by a holomorphic (n-2)-form $\omega$ that can be written as $\omega = i_Y i_X \nu$, where $X = \frac{\partial}{\partial z_n}$, $Y = \sum_{j=1}^{n-1} Y_j(z_1, \dots, z_{n-1}) \frac{\partial}{\partial z_j}$ is the normal type and $\nu = dz_n \wedge dz_1 \wedge \dots \wedge dz_{n-1}$.
Since the eigenvalues $\rho_1, \dots, \rho_{n-1}$ of $DY(0)$ satisfy the non-resonance conditions $(\star)$, by Theorem 4.7 (without parameters) we can assume that $Y$ is linear

$$Y = \sum_{j=1}^{n-1} \rho_j z_j \frac{\partial}{\partial z_j},$$

which implies

$$\omega = \sum_{j=1}^{n-1} (-1)^{j-1} \rho_j z_j dz_1 \wedge \dots \wedge \widehat{dz_j} \wedge \dots \wedge dz_{n-1}.$$

In particular, the form $\eta_V := \frac{1}{z_1 \dots z_{n-1}} \omega$ is logarithmic

$$\eta_V = \sum_{j=1}^{n-1} (-1)^{j-1} \rho_j \frac{dz_j}{z_1} \wedge \dots \wedge \frac{\widehat{dz_j}}{z_j} \wedge \dots \wedge \frac{dz_{n-1}}{z_{n-1}}.$$

Note that $\eta_V$ can be decomposed as $\eta_V = \rho_1 \theta_V^2 \wedge \dots \wedge \theta_V^{n-1}$, where

$$\theta_V^j = \frac{dz_j}{z_j} - \frac{\rho_j}{\rho_1} \frac{dz_1}{z_1}, \quad (4.3)$$

because $i_Y \theta_V^j = i_X \theta_V^j = 0, \forall 2 \le j \le n-1$.

The above argument implies that there exists a covering $\mathcal{V}$ of $S \setminus F$, by open sets, such that

(iii) For each $V \in \mathcal{V}$ there exists a coordinate system $z_V = (z_1, \dots, z_n) : V \to \mathbb{C}^n$ with $V = \{z || z_j| < 1, 1 \le j \le n\}$ and $S \cap V = (z_1 = \dots = z_{n-1} = 0)$.

(iv) If $\theta_V^j$ is as (4.3), $2 \le j \le n-1$, then the logarithmic form $\theta_V^2 \wedge \dots \wedge \theta_V^{n-1}$ defines $\mathcal{F}|_V$.

(v) The vector fields $X_V = \frac{\partial}{\partial z_n}$ and $Y_V = \sum_{j=1}^{n-1} \rho_j z_j \frac{\partial}{\partial z_j}$ generate $\mathcal{F}|_V$.

We assert that if $V, \tilde{V} \in \mathcal{V}$ are such that $V \cap \tilde{V} \neq \emptyset$ then $\theta_V^j = \theta_{\tilde{V}}^j$ on $V \cap \tilde{V}$.

In fact, first of all let us remark that

(vi) For all $j \in \{1, \dots, n-1\}$ the hypersurface $\Sigma_V^j := (z_j = 0) \subset V$ is invariant by $\mathcal{F}|_V$. Moreover, if $\Sigma^j \subset V$ is another smooth hypersurface which is $\mathcal{F}|_V$ invariant and is tangent to $\Sigma_V^j$ along $S$ then $\Sigma^j \subset \Sigma_V^j$.

Note that (vi) above is equivalent to the fact that the hyperplane $(z_j = 0)$ is $Y$-invariant, $1 \le j \le n-1$. Moreover, it is the unique smooth hypersurface which is $Y$-invariant and tangent to $(z_j = 0)$. This is well-known and is a consequence of the fact that $\rho_1, \dots, \rho_{n-1}$ satisfy $(\star)$ (see [1]).
Since the eigenvalues $\rho_1, \dots, \rho_{n-1}$ of $DY(0)$ satisfy the non-resonance conditions $(\star)$, by Theorem 4.7 (without parameters) we can assume that $Y$ is linear

$$Y = \sum_{j=1}^{n-1} \rho_j z_j \frac{\partial}{\partial z_j},$$

which implies

$$\omega = \sum_{j=1}^{n-1} (-1)^{j-1} \rho_j z_j dz_1 \wedge \dots \wedge \widehat{dz_j} \wedge \dots \wedge dz_{n-1}.$$

In particular, the form $\eta_V := \frac{1}{z_1 \dots z_{n-1}} \omega$ is logarithmic

$$\eta_V = \sum_{j=1}^{n-1} (-1)^{j-1} \rho_j \frac{dz_1}{z_1} \wedge \dots \wedge \frac{dz_j}{z_j} \wedge \dots \wedge \frac{dz_{n-1}}{z_{n-1}}.$$

Note that $\eta_V$ can be decomposed as $\eta_V = \rho_1 \theta_V^2 \wedge \dots \wedge \theta_V^{n-1}$, where

$$\theta_V^j = \frac{dz_j}{z_j} - \frac{\rho_j}{\rho_1} \frac{dz_1}{z_1}, \quad (4.3)$$

because $i_Y \theta_V^j = i_X \theta_V^j = 0, \forall 2 \le j \le n-1$.

The above argument implies that there exists a covering $\mathcal{V}$ of $S \setminus F$, by open sets, such that

(iii) For each $V \in \mathcal{V}$ there exists a coordinate system $z_V = (z_1, \dots, z_n) : V \to \mathbb{C}^n$ with $V = \{z || z_j| < 1, 1 \le j \le n\}$ and $S \cap V = (z_1 = \dots = z_{n-1} = 0)$.

(iv) If $\theta_V^j$ is as (4.3), $2 \le j \le n-1$, then the logarithmic form $\theta_V^2 \wedge \dots \wedge \theta_V^{n-1}$ defines $\mathcal{F}|_V$.

(v) The vector fields $X_V = \frac{\partial}{\partial z_n}$ and $Y_V = \sum_{j=1}^{n-1} \rho_j z_j \frac{\partial}{\partial z_j}$ generate $\mathcal{F}|_V$.

We assert that if $V, \tilde{V} \in \mathcal{V}$ are such that $V \cap \tilde{V} \neq \emptyset$ then $\theta_V^j = \theta_{\tilde{V}}^j$ on $V \cap \tilde{V}$.

In fact, first of all let us remark that

(vi) For all $j \in \{1, \dots, n-1\}$ the hypersurface $\Sigma_V^j := (z_j = 0) \subset V$ is invariant by $\mathcal{F}|_V$. Moreover, if $\Sigma^j \subset V$ is another smooth hypersurface which is $\mathcal{F}|_V$ invariant and is tangent to $\Sigma_V^j$ along $S$ then $\Sigma^j \subset \Sigma_V^j$.

Note that (vi) above is equivalent to the fact that the hyperplane $(z_j = 0)$ is $Y$-invariant, $1 \le j \le n-1$. Moreover, it is the unique smooth hypersurface which is $Y$-invariant and tangent to $(z_j = 0)$. This is well-known and is a consequence of the fact that $\rho_1, \dots, \rho_{n-1}$ satisfy $(\star)$ (see [1]).