On Levi-flat hypersurfaces tangent to

holomorphic webs

Arturo Fernández-Pérez 111Work supported by CNPq-Brazil.
Keywords: Levi-flat hypersurfaces - Holomorphic webs
Mathematical subjet classification: 32V40 - 14C21

Abstract. We investigate germs of real analytic Levi-flat hypersurfaces
tangent to germs of codimension one holomorphic webs. We introduce the notion
of first integrals for local webs. In particular, we prove that a $k$-web with
finitely many invariant analytic subvarieties through the origin tangent to a
Levi-flat hypersurface has a holomorphic first integral.

Résumé. Nous étudions les germes d’hypersurfaces réelles analytiques Levi-
plate tangente à les germes d’webs holomorphes codimension un. Nous
introduisons la notion des intégrales premières des webs locales. En
particulier, nous montrons que une $k$-web avec un nombre fini de feuilles
invariant analytique par l’origine, tangente à une hypersurface Levi-plate
possède une intégrale première holomorphe.

##  1 Introduction

In very general terms, a germ of codimension one $k$-web is a collection of
$k$ germs of codimension one holomorphic foliations in “general position”. The
study of webs was initiated by Blaschke and his school in the late 1920 s. For
a recent account of the theory, we refer the reader to [12].

For instance, take $\omega\in Sym^{k}\Omega^{1}(\mathbb{C}^{2},0)$ defined by

$\omega=(dy)^{k}+a_{k-1}(dy)^{k-1}dx+\ldots+a_{0}(dx)^{k},$

where $a_{j}\in\mathcal{O}_{2}$ for all $0\leq j\leq k-1$. Then
$\mathcal{W}:\omega=0$, define a non-trivial $k$-web on $(\mathbb{C}^{2},0)$.
In this paper we study webs and its relation with Levi-flat hypersurfaces.

Let $M$ be a germ at $0\in\mathbb{C}^{n}$ of a real codimension one analytic
irreducible analytic set. Since $M$ is real analytic of codimension one, it
can be decomposed into $M_{reg}$ and $\textsf{Sing}(M)$, where $M_{reg}$ is a
germ of smooth real analytic hypersurface in $\mathbb{C}^{n}$ and
$\textsf{Sing}(M)$, the singular locus, is contained in a proper analytic
subvariety of lower dimension. We shall say that $M$ is Levi-flat if the
complex distribution $L$ on $M_{reg}$

$\displaystyle L_{p}:=T_{p}M\cap iT_{p}M\subset T_{p}M,\,\,\,\,\text{for
any}\,\,p\in M_{reg}$ (1.1)

is integrable, in Frobenius sense. It follows that $M_{reg}$ is smoothly
foliated by immersed complex manifolds of complex dimension $n-1$. The
foliation defined by $L$ is called the Levi foliation and will be denoted by
$\mathcal{L}_{M}$.

If $M$ is a real analytic smooth Levi-flat hypersurface, by a classic result
of E. Cartan there exists a local holomorphic coordinates
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ such that $M$ can be represented by
$M=\\{\mathcal{I}m(z_{n})=0\\}$. The situation if different if the
hypersurface have singularities. Singular Levi-flat real analytic
hypersurfaces have been studied by Burns and Gong [1], Brunella [2], Lebl [9],
the author [6], [7] and many others.

Recently D.Cerveau and A. Lins Neto [5] have studied codimension one
holomorphic foliations tangent to singular Levi-flat hypersurfaces. A
codimension one holomorphic foliation $\mathcal{F}$ is tangent to $M$, if any
leaf of $\mathcal{L}_{M}$ is also a leaf of $\mathcal{F}$. In [5] it is proved
that a germ of codimension one holomorphic foliation tangent to a real
analytic Levi-flat hypersurface has a non-constant meromorphic first integral.
In the same spirit, the authors propose a problem for webs, which is as
follows:

Problem.- Let $M$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 2$, of real
analytic hypersurface Levi-flat. Assume that there exists a singular
codimension one $k$-web, $k\geq 2$, such that any leaf of the Levi foliation
$\mathcal{L}_{M}$ on $M_{reg}$ is also a leaf of the web. Does the web has a
non-constant meromorphic first integral?.

By a meromorphic first integral we mean something like
$f_{0}(x)+z.f_{1}(x)+\ldots+z^{k}.f_{k}(x)=0$, where
$f_{0},f_{1},\ldots,f_{k}\in\mathcal{O}_{n}$. In this situation, the web is
obtained by the elimination of $z$ in the system given by

$\left\\{\begin{array}[]{ll}f_{0}+z.f_{1}+z^{2}.f_{2}+\ldots+z^{k}.f_{k}=0&\\\
df_{0}+z.df_{1}+z^{2}.df_{2}+\ldots+z^{k}.df_{k}=0.&\end{array}\right.$

In this work, we organize some results on singular Levi-flat hypersurfaces and
holomorphic foliations which provide a best approach to study of webs and
Levi-flats. Concerning the problem, we obtain an interesting result in a case
very special (Theorem 1), the problem remains open in general.

###  1.1 Local singular webs

It is customary to define a germ of singular holomorphic foliation as an
equivalence class $[\omega]$ of germs of holomorphic 1-forms in
$\Omega^{1}(\mathbb{C}^{n},0)$ modulo multiplication by elements of
$\mathcal{O}^{*}(\mathbb{C}^{n},0)$ such that any representative $\omega$ is
integrable ( $\omega\wedge d\omega=0$ ) and with singular set
$\textsf{Sing}(\omega)=\\{p\in(\mathbb{C}^{n},0):\omega(p)=0\\}$ of
codimension at least two.

An analogous definition can be made for codimension one $k$-webs. A germ at
$(\mathbb{C}^{n},0)$, $n\geq 2$ of codimension one $k$-web $\mathcal{W}$ is an
equivalence class $[\omega]$ of germs of $k$-symmetric 1-forms, that is
sections of $Sym^{k}\Omega^{1}(\mathbb{C}^{n},0)$, modulo multiplication by
$\mathcal{O}^{*}(\mathbb{C}^{n},0)$ such that a suitable representative
$\omega$ defined in a connected neighborhood $U$ of the origin satisfies the
following conditions:

  1. 1.

The zero set of $\omega$ has codimension at least two.

  2. 2.

The 1-form $\omega$, seen as a homogeneous polynomial of degree $k$ in the
ring $\mathcal{O}_{n}[dx_{1},\ldots,dx_{n}]$, is square-free.

  3. 3.

(Brill’s condition) For a generic $p\in U$, $\omega(p)$ is a product of $k$
linear forms.

  4. 4.

(Frobenius’s condition) For a generic $p\in U$, the germ of $\omega$ at $p$ is
the product of $k$ germs of integrable 1-forms.

Both conditions $(3)$ and $(4)$ are automatic for germs at
$(\mathbb{C}^{2},0)$ of webs and non-trivial for germs at $(\mathbb{C}^{n},0)$
when $n\geq 3$.

We can think $k$-webs as first order differential equations of degree $k$. The
idea is to consider the germ of web as a meromorphic section of the
projectivization of the cotangent bundle of $(\mathbb{C}^{n},0)$. This is a
classical point view in the theory of differential equations, which has been
recently explored in Web-geometry. For instance see [3], [4], [14].

###  1.2 The contact distribution

Let us denote $\mathbb{P}:=\mathbb{P}T^{*}(\mathbb{C}^{n},0)$ the
projectivization of the cotangent bundle of $(\mathbb{C}^{n},0)$ and
$\pi:\mathbb{P}T^{*}(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ the
natural projection. Over a point $p$ the fiber $\pi^{-1}(p)$ parametrizes the
one-dimensional subspaces of $T^{*}_{p}(\mathbb{C}^{n},0)$. On $\mathbb{P}$
there is a canonical codimension one distribution, the so called contact
distribution $\mathcal{D}$. Its description in terms of a system of
coordinates $x=(x_{1},\ldots,x_{n})$ of $(\mathbb{C}^{n},0)$ goes as follows:
let $dx_{1},\ldots,dx_{n}$ be the basis of $T^{*}(\mathbb{C}^{n},0)$
associated to the coordinate system $(x_{1},\ldots,x_{n})$. Given a point
$(x,y)\in T^{*}(\mathbb{C}^{n},0)$, we can write
$y=\sum_{j=1}^{n}y_{j}dx_{j}$, $(y_{1},\ldots,y_{n})\in\mathbb{C}^{n}$. In
this way, if $(y_{1},\ldots,y_{n})\neq 0$ then we set
$[y]=[y_{1},\ldots.y_{n}]\in\mathbb{P}^{n-1}$ and
$(x,[y])\in(\mathbb{C}^{n},0)\times\mathbb{P}^{n-1}\cong\mathbb{P}$. In the
affine coordinate system $y_{n}\neq 0$ of $\mathbb{P}$, the distribution
$\mathcal{D}$ is defined by $\alpha=0$, where

$\displaystyle\alpha=dx_{n}-\sum_{j=1}^{n-1}p_{j}dx_{j},\,\,\,\,\,\,\,\,p_{j}=-\frac{y_{j}}{y_{n}}\,\,\,\,\,\,\,\,\,(1\leq
j\leq n-1).$ (1.2)

The 1-form $\alpha$ is called the contact form.

###  1.3 Webs as closures of meromorphic multi-sections

Let us consider $X\subset\mathbb{P}$ a subvariety, not necessarily
irreducible, but of pure dimension $n$. Let
$\pi_{X}:X\rightarrow(\mathbb{C}^{n},0)$ be the restriction to $X$ of the
projection $\pi$. Suppose also that $X$ satisfies the following conditions:

  1. 1.

The image under $\pi$ of every irreducible component of $X$ has dimension $n$.

  2. 2.

The generic fiber of $\pi$ intersects $X$ in $k$ distinct smooth points and at
these the differential $d\pi_{X}:T_{p}X\rightarrow
T_{\pi(p)}(\mathbb{C}^{n},0)$ is surjective. Note that $k=\deg(\pi_{X})$.

  3. 3.

The restriction of the contact form $\alpha$ to the smooth part of every
irreducible component of $X$ is integrable. We denote $\mathcal{F}_{X}$ the
foliation defined by $\alpha|_{X}=0$.

We can define a germ $\mathcal{W}$ at $0\in\mathbb{C}^{n}$ of $k$-web as a
triple $(X,\pi_{X},\mathcal{F}_{X})$. This definition is equivalent to one
given in Section 1.1. In the sequel, $X$ will always be the variety associated
to $\mathcal{W}$, the singular set of $X$ will be denoted by
$\textsf{Sing}(X)$ and its the smooth part will be denoted by $X_{reg}$.

######  Definition 1.1.

Let $R$ be the set of points $p\in X$ where

  * •

either $X$ is singular,

  * •

or the differential $d\pi_{X}:T_{p}X_{reg}\rightarrow
T_{\pi(p)}(\mathbb{C}^{n},0)$ is not an isomorphism.

The analytic set $R$ is called the criminant set of $\mathcal{W}$ and
$\Delta_{\mathcal{W}}=\pi(R)$ the discriminant of $\mathcal{W}$. Note that
$\textsf{dim}(R)\leq n-1$.

######  Remark 1.2.

Let $\omega\in Sym^{k}\Omega_{1}(\mathbb{C}^{n},0)$ and assume that it defines
a $k$-web $\mathcal{W}$ with variety $X$. Then $X$ is irreducible if, and only
if, $\omega$ is irreducible in the ring
$\mathcal{O}_{n}[dx_{1},\ldots,dx_{n}]$. In this case we say that the web is
irreducible.

Let $M$ be a germ at $0\in\mathbb{C}^{n}$ of a real analytic Levi-flat
hypersurface.

######  Definition 1.3.

We say that $M$ is tangent to $\mathcal{W}$ if any leaf of the Levi foliation
$\mathcal{L}_{M}$ on $M_{reg}$ is also a leaf of $\mathcal{W}$.

###  1.4 First integrals for webs

######  Definition 1.4.

We say that $\mathcal{W}$ a $k$-web has a meromorphic first integral if, and
only if, there exists

$P(z)=f_{0}+z.f_{1}+\ldots+z^{k}.f_{k}\in\mathcal{O}_{n}[z],$

where $f_{0},\ldots,f_{k}\in\mathcal{O}_{n}$, such that every irreducible
component of the hypersurface $(P(z_{0})=0)$ is a leaf of $\mathcal{W}$, for
all $z_{0}\in(\mathbb{C},0)$.

######  Definition 1.5.

We say that $\mathcal{W}$ a $k$-web has a holomorphic first integral if, and
only if, there exists

$P(z)=f_{0}+z.f_{1}+\ldots+z^{k-1}.f_{k-1}+z^{k}\in\mathcal{O}_{n}[z],$

where $f_{0},\ldots,f_{k-1}\in\mathcal{O}_{n}$, such that every irreducible
component of the hypersurface $(P(z_{0})=0)$ is a leaf of $\mathcal{W}$, for
all $z_{0}\in(\mathbb{C},0)$.

We will prove a result concerning the situation of definitions 1.3 and 1.5.

######  Theorem 1.

Let $\mathcal{W}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq{2}$ of $k$-web
defined by

$\omega=\underset{i_{1},\ldots,i_{n}\geq
0}{\sum_{i_{1}+\ldots+i_{n}=k}}a_{i_{1},\ldots,i_{n}}(z)dz_{1}^{i_{1}}\ldots
dz_{n}^{i_{n}},$

where $a_{i_{1},\ldots,i_{n}}\in\mathcal{O}_{n}$ and
$a_{0,0,\ldots,0,k}(0)\neq 0$. Suppose that $\mathcal{W}$ is tangent to a germ
at $0\in\mathbb{C}^{n}$ of an irreducible real-analytic Levi-flat hypersurface
$M$. Furthermore, assume that $\mathcal{W}$ is irreducible and has finitely
many invariant analytic subvarieties through the origin. Let $X$ be the
variety associated to $\mathcal{W}$. Then $\mathcal{W}$ has a non-constant
holomorphic first integral, if one of the following conditions is fulfilled :

  1. 1.

If $n=2$.

  2. 2.

If $n\geq 3$ and $cod_{X_{reg}}(\textsf{Sing}(X))\geq 2$.

Moreover, if
$P(z)=f_{0}+z.f_{1}+\ldots+z^{k-1}.f_{k-1}+z^{k}\in\mathcal{O}_{n}[z]$ is a
holomorphic first integral for $\mathcal{W}$, then $M=(F=0)$, where $F$ is
obtained by the elimination of $z$ in the system given by

$\left\\{\begin{array}[]{ll}f_{0}+z.f_{1}+z^{2}.f_{2}+\ldots+z^{k-1}.f_{k-1}+z^{k}=0&\\\
\bar{f}_{0}+z.\bar{f}_{1}+z^{2}.\bar{f}_{2}+\ldots+z^{k-1}.\bar{f}_{k-1}+z^{k}=0.&\end{array}\right.$

######  Remark 1.6.

Under the hypotheses of Theorem 1, if $n=2$ and $k=1$, $\mathcal{W}$ is a non-
dicritical holomorphic foliation at $(\mathbb{C}^{2},0)$ tangent to a germ of
an irreducible real analytic Levi-flat hypersurface $M$, then Theorem 1 given
by Cerveau and Lins Neto [5] assures that $\mathcal{W}$ has a non-constant
holomorphic first integral. In this sense, our theorem is a generalization of
result of Cerveau and Lins Neto.

######  Remark 1.7.

Let $\mathcal{W}$ a germ at $0\in\mathbb{C}^{n}$, $n\geq{2}$, of a smooth
$k$-web tangent to a germ at $0\in\mathbb{C}^{n}$ of an irreducible real
codimension one submanifold $M$. In other words,
$\mathcal{W}=\mathcal{F}_{1}\boxtimes\ldots\boxtimes\mathcal{F}_{k}$ is a
generic superposition of $k$ germs at $0\in\mathbb{C}^{n}$ of smooth
foliations $\mathcal{F}_{1},\ldots,\mathcal{F}_{k}$. In this case the
irreducibility and tangency conditions to $M$ implies the existence of a
unique $i\in\\{1,\ldots,k\\}$ such that $\mathcal{F}_{i}$ is tangent to $M$.
Therefore we can find a coordinates system $z_{1},\ldots,z_{n}$ of
$\mathbb{C}^{n}$ such that $\mathcal{F}_{i}$ is defined by $dz_{n}=0$ and
$M=(\mathcal{I}m(z_{n})=0)$.

##  2 The foliation associated to a web

In this section, we prove a key lemma which will be used in the proof of main
theorem.

Since the restriction of $\mathcal{D}$ to $X_{reg}$ is integrable, it defines
a foliation $\mathcal{F}_{X}$, which in general is a singular foliation. Given
$p\in(\mathbb{C}^{n},0)\backslash\Delta_{\mathcal{W}}$,
$\pi_{X}^{-1}(p)=\\{q_{1},\ldots,q_{k}\\}$, where $q_{i}\neq q_{j}$, if $i\neq
j$, ($\deg(\pi_{X})=k$), denote by $\mathcal{F}_{X}^{i}$ the germ of
$\mathcal{F}_{X}$ at $q_{i}$, $i=1,\ldots,k$.

The projections $\pi_{*}(\mathcal{F}_{X}^{i}):=\mathcal{F}_{p}^{i}$ define $k$
germs of codimension one foliations at $p$.

######  Definition 2.1.

A leaf of the web $\mathcal{W}$ is, by definition, the projection on
$(\mathbb{C}^{n},0)$ of a leaf of $\mathcal{F}_{X}$.

######  Remark 2.2.

Given $p\in(\mathbb{C}^{n},0)\backslash\Delta_{\mathcal{W}}$, and
$q_{i}\in\pi_{X}^{-1}(p)$, the projection $\pi_{X}(L_{i})$ of the leaf $L_{i}$
of $\mathcal{F}_{X}$ through $q_{i}$, gives rise to a leaf of $\mathcal{W}$
through $p$. In particular, $\mathcal{W}$ has at most $k$ leaves through $p$.

We will use the following proposition (cf. [8] Th. 5, pg. 32). Let
$\mathcal{O}(X)$ denote the ring of holomorphic functions on $X$.

######  Proposition 2.3.

Let $V$ be an analytic variety. If $\pi:V\rightarrow W$ is a finite branched
holomorphic covering of pure order $k$ over an open subset
$W\subseteq\mathbb{C}^{n}$, then to each holomorphic function
$f\in\mathcal{O}(V)$ there is a canonically associated monic polynomial
$P_{f}(z)\in\mathcal{O}_{n}[z]\subseteq\mathcal{O}(V)[z]$ of degree $k$ such
that $P_{f}(f)=0$ in $\mathcal{O}(V)$.

We have now the following lemma.

######  Lemma 2.4.

Suppose that $(X,\pi_{X},\mathcal{F}_{X})$ defines a $k$-web $\mathcal{W}$ on
$(\mathbb{C}^{n},0)$, $n\geq 2$, where $X$ is an irreducible subvariety of
$\mathbb{P}$. If $\mathcal{F}_{X}$ has a non-constant holomorphic first
integral then $\mathcal{W}$ also has a holomorphic first integral.

###### Proof.

Let $g\in\mathcal{O}(X)$ be the first integral for $\mathcal{F}_{X}$. By
Proposition 2.3, there exists a monic polynomial
$P_{g}(z)\in\mathcal{O}_{n}[z]$ of degree $k$ such that $P_{g}(g)=0$ in
$\mathcal{O}(X)$. Write

$P_{g}(z)=g_{0}+z.g_{1}+\ldots+z^{k-1}.g_{k-1}+z^{k},$

where $g_{0},\ldots,g_{k-1}\in\mathcal{O}_{n}$.

Assertion.– $P_{g}$ define a holomorphic first integral for $\mathcal{W}$.

Let $U\subseteq(\mathbb{C}^{n},0)\backslash\Delta_{\mathcal{W}}$ be an open
subset and let $\varphi:X\rightarrow(\mathbb{C}^{n},0)\times\mathbb{C}$ be
defined by $\varphi=(\pi_{X},g)$. Take a leaf $L$ of $\mathcal{W}|_{U}$. Then
there is $z\in\mathbb{C}$ such that the following diagram

---
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\pi^{-1}_{X}(U)\cap\varphi^{-1}(L\times\\{z\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\varphi}$$\textstyle{L\times\\{z\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{1}}$$\textstyle{L}$

is commutative, where $pr_{1}$ is the projection on the first coordinate. It
follows that $L$ is a leaf of $\mathcal{W}$ if and only if $g$ is constant
along of each connected component of $\pi_{X}^{-1}(L)$ contained in
$\varphi^{-1}(L\times\\{z\\})$.

Consider now the hypersurface
$G=\varphi(X)\subset(\mathbb{C}^{n},0)\times\mathbb{C}$ which is the closure
of set

$\\{(x,s)\in
U\times\mathbb{C}:g_{0}(x)+s.g_{1}(x)+\ldots+s^{k-1}.g_{k-1}(x)+s^{k}=0\\}.$

Let $\psi:(\mathbb{C}^{n},0)\times\mathbb{C}\rightarrow(\mathbb{C}^{n},0)$ be
the usual projection and denote by $Z\subset(\mathbb{C}^{n},0)$ the analytic
subset such that the restriction to $G$ of $\psi$ not is a finite branched
covering. Notice that for all $x_{0}\in(\mathbb{C}^{n},0)\backslash Z$, the
equation

$g_{0}(x)+s.g_{1}(x)+\ldots+s^{k-1}.g_{k-1}(x)+s^{k}=0$

defines $k$ analytic hypersurfaces pairwise transverse in $x_{0}$ and
therefore correspond to leaves of $\mathcal{W}$. ∎

##  3 Examples

This section is devoted to give some examples of Levi-flat hypersurfaces
tangent to holomorphic foliations or webs.

######  Example 3.1.

Take a non constant holomorphic function
$f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)$ and set
$M=(\mathcal{I}m(f)=0)$. Then $M$ is Levi-flat and $M_{sing}$ is the set of
critical points of $f$ lying on $M$. Leaves of the Levi foliation on $M_{reg}$
are given by $\\{f=c\\}$, $c\in\mathbb{R}$. Of course, $M$ is tangent to a
singular holomorphic foliation generated by the kernel of $df$.

######  Example 3.2.

([5]) Let $f_{0},f_{1},\ldots,f_{k}\in\mathcal{O}_{n}$, $n\geq{2}$, be
irreducible germs of holomorphic functions, where $k\geq{2}$. Consider the
family of hypersurfaces

$G:=\\{G_{s}:=f_{0}+sf_{1}+\ldots+s^{k}f_{k}/s\in\mathbb{R}\\}.$

By eliminating the real variable $s$ in the system $G_{s}=\bar{G}_{s}=0$, we
obtain a real analytic germ $F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$
such that any complex hypersurface $(G_{s}=0)$ is contained in the real
hypersurface $(F=0)$. For instance, in the case $k=2$, we obtain

$F=det\left(\begin{array}[]{cccc}f_{0}&f_{1}&f_{2}&0\\\ 0&f_{0}&f_{1}&f_{2}\\\
\bar{f}_{0}&\bar{f}_{1}&\bar{f}_{2}&0\\\
0&\bar{f}_{0}&\bar{f}_{1}&\bar{f}_{2}\end{array}\right)=$
$=f^{2}_{0}.\bar{f}^{2}_{2}+\bar{f}^{2}_{0}.f^{2}_{2}+f_{0}.f_{2}.\bar{f}^{2}_{1}+\bar{f}_{0}.\bar{f}_{2}.f_{1}^{2}-|f_{1}|^{2}(f_{0}.\bar{f}_{2}+\bar{f}_{0}.f_{2})-2|f_{0}|^{2}.|f_{2}|^{2}.$
(3.1)

which comes from the elimination of $s$ in the system

$f_{0}+s.f_{1}+s^{2}.f_{2}=\bar{f}_{0}+s.\bar{f}_{1}+s^{2}.\bar{f}_{2}=0.$

We would like to observe that the examples of this type are tangent to
singular webs. The web is obtained by the elimination of $s$ in the system
given by

$\left\\{\begin{array}[]{ll}f_{0}+s.f_{1}+s^{2}.f_{2}+\ldots+s^{k}.f_{k}=0&\\\
df_{0}+s.df_{1}+s^{2}.df_{2}\ldots+s^{k}.df_{k}=0&\end{array}\right.$

In the case we get a $2$-web given by the implicit differential equation
$\Omega=0$, where

$\Omega=det\left(\begin{array}[]{cccc}f_{0}&f_{1}&f_{2}&0\\\
0&f_{0}&f_{1}&f_{2}\\\ df_{0}&df_{1}&df_{2}&0\\\
0&df_{0}&df_{1}&df_{2}\end{array}\right)$

This example shows that, although $\mathcal{L}_{M}$ is a foliation on
$M_{reg}\subset M=(F=0)$, in general it is not tangent to a germ of
holomorphic foliation at $(\mathbb{C}^{n},0)$.

######  Example 3.3.

[Clairaut’s equations] Clairaut’s equations are tangent to Levi-flat
hypersurfaces. Consider the first-order implicit differential equation

$y=xp+f(p),$ (3.2)

where $(x,y)\in\mathbb{C}^{2}$, $p=\frac{dy}{dx}$ and $f\in\mathbb{C}[p]$ is a
polynomial of degree $k$, the equation (3.2) define a $k$-web $\mathcal{W}$ on
$(\mathbb{C}^{2},0)$. The variety $S$ associated to $\mathcal{W}$ is given by
$(y-xp-f(p)=0)$ and the foliation $\mathcal{F}_{S}$ is defined by
$\alpha|_{S}=0$, where $\alpha=dy-pdx$. In the chart $(x,p)$ of $S$, we get
$\alpha|_{S}=(x+f^{\prime}(p))dp$. The criminant set of $\mathcal{W}$ is given
by

$R=(y-xp-f(p)=x+f^{\prime}(p)=0).$

Observe that $\mathcal{F}_{S}$ is tangent to $S$ along $R$ and has a non-
constant first integral $g(x,p)=p$. Denote by
$\pi_{S}:S\rightarrow(\mathbb{C}^{2},0)$ the restriction to $S$ of the usual
projection $\pi:\mathbb{P}\rightarrow(\mathbb{C}^{2},0)$, then the leaves of
$\mathcal{F}_{S}$ project by $\pi_{S}$ in leaves of $\mathcal{W}$. Those
leaves are as follows

$\displaystyle-y+s.x+f(s)=0,$ (3.3)

where $s$ is a constant. By the elimination of the variable $s\in\mathbb{R}$
in the system

$\left\\{\begin{array}[]{ll}-y+s.x+f(s)=0&\\\
-\bar{y}+s.\bar{x}+\overline{f(s)}=0,&\end{array}\right.$

we obtain a Levi-flat hypersurface tangent to $\mathcal{W}$. In particular,
Clairaut’s equation has a holomorphic first integral.

##  4 Lifting of Levi-flat hypersurfaces to the cotangent bundle

In this section we give some remarks about the lifting of a Levi-flat
hypersurface to the cotangent bundle of $(\mathbb{C}^{n},0)$.

Let $\mathbb{P}$ be as before, the projectivized cotangent bundle of
$(\mathbb{C}^{n},0)$ and $M$ an irreducible real analytic Levi-flat at
$(\mathbb{C}^{n},0)$, $n\geq 2$. Note that $\mathbb{P}$ is a
$\mathbb{P}^{n-1}$-bundle over $(\mathbb{C}^{n},0)$, whose fiber
$\mathbb{P}T_{z}^{*}\mathbb{C}^{n}$ over $z\in\mathbb{C}^{n}$ will be thought
of as the set of complex hyperplanes in $T_{z}^{*}\mathbb{C}^{n}$. Let
$\pi:\mathbb{P}\rightarrow(\mathbb{C}^{n},0)$ be the usual projection.

The regular part $M_{reg}$ of $M$ can be lifted to $\mathbb{P}$: just take,
for every $z\in M_{reg}$, the complex hyperplane

$T_{z}^{\mathbb{C}}M_{reg}=T_{z}M_{reg}\cap i(T_{z}M_{reg})\subset
T_{z}\mathbb{C}^{n}.$ (4.1)

We call

$M^{\prime}_{reg}\subset\mathbb{P}$ (4.2)

this lifting of $M_{reg}$. We remark that it is no more a hypersurface: its
(real) dimension $2n-1$ is half of the real dimension of
$\mathbb{P}T^{*}\mathbb{C}^{n}$. However, it is still “Levi-flat”, in a sense
which will be precised below.

Take now a point $y$ in the closure $\overline{M^{\prime}_{reg}}$ projecting
on $\mathbb{C}^{n}$ to a point $x\in\overline{M}$. Now, we shall consider the
following results, which are adapted from [2].

######  Lemma 4.1.

There exist, in a germ of neighborhood
$U_{y}\subset\mathbb{P}T^{*}(\mathbb{C}^{n},0)$ of $y$, a germ of real
analytic subset $N_{y}$ of dimension $2n-1$ containing $M^{\prime}_{reg}\cap
U_{y}$.

######  Proposition 4.2.

Under the above conditions, in a germ of neighborhood $V_{y}\subset U_{y}$ of
$y$, there exists a germ of complex analytic subset $Y_{y}$ of (complex)
dimension $n$ containing $N_{y}\cap V_{y}$.

##  5 Proof of Theorem 1

The proof will be divided in two parts. First, we give the proof for $n=2$.
The proof in dimension $n\geq 3$ will be done by reduction to the case of
dimension two.

First of all, we recall some results (cf. [5]) about foliations and Levi-
flats. Let $M$ and $\mathcal{F}$ be germs at $(\mathbb{C}^{2},0)$ of a real
analytic Levi-flat hypersurface and of a holomorphic foliation, respectively,
where $\mathcal{F}$ is tangent to $M$. Assume that:

  1. (i)

$\mathcal{F}$ is defined by a germ at $0\in\mathbb{C}^{2}$ of holomorphic
vector field $X$ with an isolated singularity at $0$.

  2. (ii)

$M$ is irreducible.

Let us assume that $0$ is a reduced singularity of $X$, in the sense of
Seidenberg [13]. Denote the eigenvalues of $DX(0)$ by
$\lambda_{1},\lambda_{2}$.

######  Proposition 5.1.

Suppose that $X$ has a reduced singularity at $0\in\mathbb{C}^{2}$ and is
tangent to a real analytic Levi-flat hypersurface $M$. Then
$\lambda_{1},\lambda_{2}\neq 0$, $\lambda_{2}/\lambda_{1}\in\mathbb{Q}_{-}$
and $X$ has a holomorphic first integral.
In particular, in a suitable coordinates system $(x,y)$ around
$0\in\mathbb{C}^{2}$, $X=\phi.Y$, where $\phi(0)\neq 0$ and

$\displaystyle Y=q.x\partial_{x}-p.y\partial_{y}\,\,,g.c.d(p,q)=1.$ (5.1)

In this coordinate system, $f(x,y):=x^{p}.y^{q}$ is a first integral of $X$.

We call this type of singularity of $\mathcal{F}$ a saddle with first
integral, (cf. [10], pg. 162). Now we have the following lemma.

######  Lemma 5.2.

For any $z_{0}\in M_{reg}$, the leaf $L_{z_{0}}$ of $\mathcal{L}_{M}$ through
$z_{0}$ is closed in $M_{reg}$.

###  5.1 Planar webs

A $k$-web $\mathcal{W}$ on $(\mathbb{C}^{2},0)$ can be written in coordinates
$(x,y)\in\mathbb{C}^{2}$ by

$\omega=a_{0}(x,y)(dy)^{k}+a_{1}(x,y)(dy)^{k-1}(dx)+\ldots+a_{k}(x,y)(dx)^{k}=0,$

where the coefficients $a_{j}\in\mathcal{O}_{2}$, $j=1,\ldots,k$. We set

$U=\\{(x,y,[adx+bdy])\in\mathbb{P}T^{*}(\mathbb{C}^{2},0):a\neq 0\\}$

and

$V=\\{(x,y,[adx+bdy])\in\mathbb{P}T^{*}(\mathbb{C}^{2},0):b\neq 0\\}.$

Note that $\mathbb{P}T^{*}(\mathbb{C}^{2},0)=U\cup V$. Suppose that
$(S,\pi_{S},\mathcal{F}_{S})$ define $\mathcal{W}$, in the coordinates
$(x,y,p)\in U$, where $p=\frac{dy}{dx}$, we have

  1. 1.

$S\cap U=\\{(x,y,p)\in\mathbb{P}T^{*}(\mathbb{C}^{2},0):F(x,y,p)=0\\},$ where

$F(x,y,p)=a_{0}(x,y)p^{k}+a_{1}(x,y)p^{k-1}+\ldots+a_{k}(x,y).$

Note that $S$ is possibly singular at $0$.

  2. 2.

$\mathcal{F}_{S}$ is defined by $\alpha|_{S}=0$, where $\alpha=dy-pdx$.

  3. 3.

The criminant set $R$ is defined by the equations

$F(x,y,p)=F_{p}(x,y,p)=0.$

In $V$ the coordinate system is $(x,y,q)\in\mathbb{C}^{3}$, where
$q=\frac{1}{p}$, the equations are similar.

###  5.2 Proof in dimension two

Let $\mathcal{W}$ be a $k$-web tangent to $M$ Levi-flat and let us consider
$S$, $\pi$ be as before. The idea is to use Lemma 2.4, assume that
$\mathcal{W}$ is defined by

$\omega=a_{0}(x,y)(dy)^{k}+a_{1}(x,y)(dy)^{k-1}dx+\ldots+a_{k}(x,y)(dx)^{k}=0,$
(5.2)

where the coefficients $a_{j}\in\mathcal{O}_{2}$, $j=1,\ldots,k$ and
$a_{0}(0,0)=1$.

######  Lemma 5.3.

Under the hypotheses of Theorem 1 and above conditions, the surface $S$ is
irreducible and $S\cap\pi^{-1}(0)$ contains just a number finite of points.
See figure 1.

###### Proof.

Since $\mathcal{W}$ is irreducible so is $S$. On the other hand,
$S\cap\pi^{-1}(0)$ is finite because $\mathcal{W}$ has a finite number of
invariant analytic leaves through the origin and is defined as in 5.2. ∎

Figure 1: $S\cap\pi^{-1}(0)$

We can assume without lost of generality that $S\cap\pi^{-1}(0)$ contains just
one point, in the case general, the idea of the proof is the same. Then in the
coordinate system $(x,y,p)\in\mathbb{C}^{3}$, where $p=\frac{dy}{dx}$, we have
$\pi^{-1}(0)\cap S=\\{p_{0}=(0,0,0)\\}$, which implies that $S$ must be
singular at $p_{0}\in\mathbb{P}T^{*}(\mathbb{C}^{2},0)$. In particular,
$(S,p_{0})$ the germ of $S$ at $p_{0}$ is defined by $F^{-1}(0)$, where

$F(x,y,p)=p^{k}+a_{1}(x,y)p^{k-1}+\ldots+a_{k}(x,y),$

and $a_{1},\ldots,a_{k}\in\mathcal{O}_{2}$. Let $\mathcal{F}_{S}$ be the
foliation defined by $\alpha|_{S}=0$. The assumptions implies that
$\mathcal{F}_{S}$ is a non-dicritical foliation with an isolated singularity
at $p_{0}$.

Recall that a germ of foliation $\mathcal{F}$ at $p_{0}\in S$ is dicritical if
it has infinitely many analytic separatrices through $p_{0}$. Otherwise it is
called non-dicritical.

Let $M^{\prime}_{reg}$ be the lifting of $M_{reg}$ by $\pi_{S}$, and denote by
$\sigma:(\tilde{S},D)\rightarrow(S,p_{0})$ the resolution of singularities of
$S$ at $p_{0}$. Let $\tilde{\mathcal{F}}=\sigma^{*}(\mathcal{F}_{S})$ be the
pull-back of $\mathcal{F}_{S}$ under $\sigma$. See figure 2.

######  Lemma 5.4.

In the above situation. The foliation $\tilde{\mathcal{F}}$ has only
singularities of saddle with first integral type in $D$.

###### Proof.

Let $y\in\overline{M^{\prime}_{reg}}$, it follows from Lemma 4.1 the
existence, in a neighborhood $U_{y}\subset\mathbb{P}T^{*}(\mathbb{C}^{2},0)$
containing $y$, of a real analytic subset $N_{y}$ of dimension $3$ containing
$M^{\prime}_{reg}\cap U_{y}$. Then by Proposition 4.2, there exists, in a
neighborhood $V_{y}\subset U_{y}$ of $y$, a complex analytic subset $Y_{y}$ of
(complex) dimension $2$ containing $N_{y}\cap V_{y}$. As germs at $y$, we get
$Y_{y}=S_{y}$ then $N_{y}\cap V_{y}\subset S_{y}$, we have that $N_{y}\cap
V_{y}$ is a real analytic hypersurface in $S_{y}$, and it is Levi-flat because
each irreducible component contains a Levi-flat piece (cf. [1], Lemma 2.2).

Let us denote $M^{\prime}_{y}=N_{y}\cap V_{y}$. The hypotheses implies that
$\mathcal{F}_{S}$ is tangent to $M^{\prime}_{y}$. These local constructions
are sufficiently canonical to be patched together, when $y$ varies on
$\overline{M^{\prime}_{reg}}$: if $S_{y_{1}}\subset V_{y_{1}}$ and
$S_{y_{2}}\subset V_{y_{2}}$ are as above, with $M^{\prime}_{reg}\cap
V_{y_{1}}\cap V_{y_{2}}\neq\emptyset$, then $S_{y_{2}}\cap(V_{y_{1}}\cap
V_{y_{2}})$ and $S_{y_{1}}\cap(V_{y_{1}}\cap V_{y_{2}})$ have some common
irreducible components containing $M^{\prime}_{reg}\cap V_{y_{1}}\cap
V_{y_{2}}$, so that $M^{\prime}_{y_{1}}$, $M^{\prime}_{y_{2}}$ can be glued by
identifying those components. In this way, we obtain a Levi-flat hypersurface
$N$ on $S$ tangent to $\mathcal{F}_{S}$.

By doing additional blowing-ups if necessary, we can suppose that
$\tilde{\mathcal{F}}$ has reduced singularities. Since $\mathcal{F}_{S}$ is
non-dicritical, all irreducible components of $D$ are
$\tilde{\mathcal{F}}$-invariants. Let $\tilde{N}$ be the strict transform of
$N$ under $\sigma$, then $\tilde{N}\supset D$. In particular, $\tilde{N}$
contains all singularities of $\tilde{\mathcal{F}}$ in $D$. It follows from
Proposition 5.1 that all singularities of $\tilde{\mathcal{F}}$ are saddle
with first integral. ∎

Figure 2: Resolution of singularities of $S$ at $p_{0}$.

###  5.3 End of the proof of Theorem 1 in dimension two

The idea is to prove that $\mathcal{F}_{S}$ has a holomorphic first integral.
Since $D$ is invariant by $\tilde{\mathcal{F}}$, i.e., it is the union of
leaves and singularities of $\tilde{\mathcal{F}}$, we have
$S:=D\backslash\textsf{Sing}(\tilde{\mathcal{F}})$ is a leaf of
$\tilde{\mathcal{F}}$. Now, fix $p\in S$ and a transverse section $\sum$
through $p$. By Lemma 5.4, the singularities of $\tilde{\mathcal{F}}$ in $D$
are saddle with first integral types. Therefore the transverse section $\sum$
is complete, (see [10], pg. 162). Let $G\subset\textsf{Diff}(\sum,p)$ be the
Holonomy group of the leaf $S$ of $\tilde{\mathcal{F}}$. It follows from Lemma
5.2 that all leaves of $\mathcal{F}_{S}$ through points of $N_{reg}$ are
closed in $N_{reg}$. This implies that all transformations of $G$ have finite
order and $G$ is linearizable. According to [11], $\mathcal{F}_{S}$ has a non-
constant holomorphic first integral. Finally from Lemma 2.4, $\mathcal{W}$ has
a first integral as follows:

$P(z)=f_{0}(x,y)+z.f_{1}(x,y)+\ldots+z^{k-1}.f_{k-1}(x,y)+z^{k},$

where $f_{0},f_{1},\ldots,f_{k-1}\in\mathcal{O}_{2}$.

###  5.4 Proof in dimension $n\geq 3$

Let us give an idea of the proof. First of all, we will prove that there is a
holomorphic embedding $i:(\mathbb{C}^{2},0)\rightarrow(\mathbb{C}^{n},0)$ with
the following properties:

  1. (i)

$i^{-1}(M)$ has real codimension one on $(\mathbb{C}^{2},0)$.

  2. (ii)

$i^{*}(\mathcal{W})$ is a $k$-web on $(\mathbb{C}^{2},0)$ tangent to
$i^{-1}(M)$.

Set $E:=i(\mathbb{C}^{2},0)$. The above conditions and Theorem 1 in dimension
two imply that $\mathcal{W}|_{E}$ has a non-constant holomorphic first
integral, say $g=f_{0}+z.f_{1}+\ldots+z^{k-1}.f_{k-1}+z^{k}$, where
$f_{0},\ldots,f_{k-1}\in\mathcal{O}_{2}$. After that we will use a lemma to
prove that $g$ can be extended to a holomorphic germ $g_{1}$, which is a first
integral of $\mathcal{W}$.

Let $\mathcal{F}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 3$, of a
holomorphic codimension one foliation, tangent to a real analytic hypersurface
$M$. Let us suppose that $\mathcal{F}$ is defined by $\omega=0$, where
$\omega$ is a germ at $0\in\mathbb{C}^{n}$ of an integrable holomorphic 1-form
with $cod_{\mathbb{C}^{n}}(\textsf{Sing}(\omega))\geq 2$. We say that a
holomorphic embedding $i:(\mathbb{C}^{2},0)\rightarrow(\mathbb{C}^{n},0)$ is
transverse to $\omega$ if $cod_{\mathbb{C}^{n}}(\textsf{Sing}(\omega))=2$,
which means in fact that, as a germ of set, we have
$\textsf{Sing}(i^{*}(\omega))=\\{0\\}$. Note that the definition is
independent of the particular germ of holomorphic 1-form which represents
$\mathcal{F}$. Therefore, we will say that the embedding $i$ is transverse to
$\mathcal{F}$ if it is transverse to some holomorphic 1-form $\omega$
representing $\mathcal{F}$.

We will use the following lemma of [5].

######  Lemma 5.5.

In the above situation, there exists a 2-plane $E\subset\mathbb{C}^{n}$,
transverse to $\mathcal{F}$, such that the germ at $0\in E$ of $M\cap E$ has
real codimension one.

We say that a embedding $i$ is transverse to $\mathcal{W}$ if it is transverse
to all $k$-foliations which defines $\mathcal{W}$. Now, one deduces the
following

######  Lemma 5.6.

There exists a 2-plane $E\subset\mathbb{C}^{n}$, transverse to $\mathcal{W}$,
such that the germ at $0\in E$ of $M\cap E$ has real codimension one.

###### Proof.

First of all, note that outside of the discriminant set of $\mathcal{W}$, we
can suppose that
$\mathcal{W}=\mathcal{F}_{1}\boxtimes\ldots\boxtimes\mathcal{F}_{k}$, where
$\mathcal{F}_{1},\ldots,\mathcal{F}_{k}$ are germs of codimension one smooth
foliations. Since $\mathcal{W}$ is tangent to $M$, there is a foliation
$\mathcal{F}_{j}$ such that is tangent to a Levi foliation $\mathcal{L}_{M}$
on $M_{reg}$. Lemma 5.5 implies that we can find a 2-plane $E_{0}$ transverse
to $M$ and to $\mathcal{F}_{j}$. Clearly the set of linear mappings transverse
to $\mathcal{F}_{1},\ldots,\mathcal{F}_{k}$ simultaneously is open and dense
in the set of linear mappings from $\mathbb{C}^{2}$ to $\mathbb{C}^{n}$, by
Transversality theory, there exists a linear embedding $i$ such that
$E=i(\mathbb{C}^{2},0)$ is transverse to $M_{reg}$ and to $\mathcal{W}$
simultaneously. ∎

Let $E$ be a 2-plane as in Lemma 5.6. It easy to check that $\mathcal{W}|_{E}$
satisfies the hypotheses of Theorem 1. By the two dimensional case
$\mathcal{W}|_{E}$ has a non-constant first integral:

$g_{0}+z.g_{1}+\ldots+z^{k-1}.g_{k-1}+z^{k},$ (5.3)

where $g_{0},\ldots,g_{k-1}\in\mathcal{O}_{2}$.

Let $X$ be the variety associated to $\mathcal{W}$ and set $S$ be the surface
associated to $\mathcal{W}|_{E}$. Observe that $\mathcal{F}_{S}$ has a non-
constant holomorphic first integral $g$ defined on $S$.

######  Lemma 5.7.

In the above situation, we have $\mathcal{F}_{X}|_{S}=\mathcal{F}_{S}$ and
$\mathcal{F}_{X}$ has a non-constant holomorphic first integral $g_{1}$ on X,
such that $g_{1}|_{S}=g$.

###### Proof.

It is easily seen that $S\subset X$ which implies that
$\mathcal{F}_{X}|_{S}=\mathcal{F}_{S}$. Let us extend $g$ to $X$. Fix $p\in
X_{reg}\backslash\textsf{Sing}(\mathcal{F}_{X})$. It is possible to find a
small neighborhood $W_{p}\subset X$ of $p$ and a holomorphic coordinate chart
$\varphi:W_{p}\rightarrow\triangle$, where $\triangle\subset\mathbb{C}^{n}$ is
a polydisc, such that:

  1. (i)

$\varphi(S\cap W_{p})=\\{z_{3}=\ldots=z_{n}=0\\}\cap\triangle$.

  2. (ii)

$\varphi_{*}(\mathcal{F}_{X})$ is given by $dz_{n}|_{\triangle}=0$.

Let $\pi_{n}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{2}$ be the projection
defined by $\pi_{n}(z_{1},\ldots,z_{n})=(z_{1},z_{2})$ and set
$\tilde{g}_{p}:=g\circ\varphi^{-1}\circ\pi_{n}|_{\triangle}$. We obtain that
$\tilde{g}$ is a holomorphic function defined in $\triangle$ and is a first
integral of $\varphi_{*}(\mathcal{F}_{X})$. Let
$g_{p}=\tilde{g}_{p}\circ\varphi$. Notice that, if $W_{p}\cap
W_{q}\neq\emptyset$, $p$ and $q$ being regular points for $\mathcal{F}_{X}$,
then we have $g_{p}|_{W_{p}\cap W_{q}}=g_{q}|_{W_{p}\cap W_{q}}$. This follows
easily form the identity principle for holomorphic functions. In particular,
$g$ can be extended to

$W=\bigcup_{p\in X_{reg}\backslash\textsf{Sing}(\mathcal{F}_{X})}W_{p},$

which is a neighborhood of $X_{reg}\backslash\textsf{Sing}(\mathcal{F}_{X})$.
Call $g_{W}$ this extension.

Since $cod_{X_{reg}}\textsf{Sing}(\mathcal{F}_{X})\geq 2$, by a theorem of
Levi (cf. [15]), $g_{W}$ can be extended to $X_{reg}$, as
$cod_{X_{reg}}(\textsf{Sing}(X))\geq 2$ this allows us to extend $g_{W}$ to
$g_{1}$ as holomorphic first integral for $\mathcal{F}_{X}$, in whole $X$. ∎

###  5.5 End of the proof of Theorem 1 in dimension $n\geq 3$

Since $\mathcal{F}_{X}$ has a non-constant holomorphic first integral on $X$,
Lemma 2.4 imply that $\mathcal{W}$ has a non-constant holomorphic first
integral.

Acknowledgments.– The author is greatly indebted to Alcides Lins Neto for
suggesting the problem and for many stimulating conversations. This is part of
the author’s Ph.D. thesis, written at IMPA. I also want to thank the referee
for his suggestions.

## References

  * [1] D. Burns, X. Gong: Singular Levi-flat real analytic hypersurfaces, Amer. J. Math. 121, $(1999)$, no. 1, pp. 23-53.
  * [2] M. Brunella: Singular Levi-flat hypersurfaces and codimension one foliations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 4, pp. 661-672.
  * [3] V. Cavalier, D. Lehmann: Introduction à l’étude globale des tissus sur une surface holomorphe. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 4, pp. 1095-1133.
  * [4] V. Cavalier, D. Lehmann: Global structure of holomorphic webs on surfaces. Geometry and topology of caustics–CAUSTICS ’06, pp. 35-44, Banach Center Publ., 82, Polish Acad. Sci. Inst. Math., Warsaw, 2008.
  * [5] D. Cerveau, A. Lins Neto: Local Levi-flat hypersurfaces invariants by a codimension one holomorphic foliation. Amer. J. Math. 133, $(2011)$, no. 3, pp. 677-716.
  * [6] A. Fernández-Pérez: On normal forms of singular Levi-flat real analytic hypersurfaces. Bull. Braz. Math. Soc. (N.S.) 42, (2011), no. 1, pp. 75-85.
  * [7] ——–: Singular Levi-flat hypersurfaces. An approach through holomorphic foliations. Ph.D Thesis IMPA, Brazil 2010.
  * [8] R. Gunning: Introduction to holomorphic functions of several variables. Vol. II. Local theory. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990.
  * [9] J. Lebl: Singularities and complexity in CR geometry. Ph.D. Thesis, University of California at San Diego, Spring 2007.
  * [10] F. Loray: Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. ; http://hal.archives-ouvertures.fr/ccsd-00016434
  * [11] J.F. Mattei, R. Moussu: Holonomie et intégrales premières, Ann. Ec. Norm. Sup. 13, $(1980)$, pp. 469–523.
  * [12] J.V. Pereira, L. Pirio: “An invitation to web geometry”. From Abel’s addition theorem to the algebraization of codimension one webs. Publicações Matemáticas do IMPA, $27^{\circ}$ Cóloquio Brasileiro de Matemática, Instituto Nacional de Matemática Pura e Aplicada IMPA. Rio de Janeiro, 2009.
  * [13] A. Seidenberg: Reduction of singularities of the differential equation $A\,dy=B\,dx$. Amer. J. Math. 90, $(1968)$, pp. 248–269.
  * [14] J.N.A Yartey: Number of singularities of a generic web on the complex projective plane. J. Dyn. Control Syst. 11 $(2005)$, no. 2, pp. 281-296.
  * [15] Y.T. Siu: Techniques of extension of analytic objects. Lecture Notes in Pure and Applied Mathematics, Vol. 8. Marcel Dekker, Inc., New York, 1974.

Arturo Fernández-Pérez

Departamento de Matemática, UFMG
Av. Antônio Carlos, 6627 C.P. 702,
30123-970 - Belo Horizonte - MG, BRAZIL.
E-mail<EMAIL_ADDRESS>