# Lehmann-Suwa Residues of codimension one holomorphic foliations and
applications

Arturo Fernández-Pérez  Departamento de Matemática, Universidade Federal de
Minas Gerais, UFMG Av. Antônio Carlos 6627, 31270-901, Belo Horizonte-MG,
Brazil<EMAIL_ADDRESS>and  Jimmy Támara  Instituto de Matemática y
Ciencias Afines IMCA Jr. Los Biólogos 245, Lima, Perú<EMAIL_ADDRESS>
###### Abstract.

Let $\mathscr{F}$ be a singular codimension one holomorphic foliation on a
compact complex manifold $X$ of dimension at least three such that its
singular set has codimension at least two. In this paper, we determine
Lehmann-Suwa residues of $\mathscr{F}$ as multiples of complex numbers by
integration currents along irreducible complex subvarieties of $X$. We then
prove a formula that determines the Baum-Bott residue of simple almost
Liouvillian foliations of codimension one, in terms of Lehmann-Suwa residues,
generalizing a result of Marco Brunella. As an application, we give sufficient
conditions for the existence of dicritical singularities of a singular real-
analytic Levi-flat hypersurface $M\subset X$ tangent to $\mathscr{F}$.

###### Key words and phrases:

Residues formula, holomorphic foliations, Levi-flat hypersurfaces

###### 2010 Mathematics Subject Classification:

Primary 32V40 - 32S65

##  1\. Introduction

In 1999, D. Lehmann and T. Suwa [22] gave a generalization to the case of
arbitrary dimension, of the variational index, defined by Khanedani and Suwa
[20] for singular holomorphic foliations on complex surfaces. More precisely,
Lehmann and Suwa proved the following result.

######  Theorem 1.1 (Lehmann-Suwa [22]).

Let $V$ be a complex subvariety of dimension $m\geq 1$ in a complex manifold
$X$ and let $\mathscr{F}$ be a singular holomorphic foliation of dimension
$k\geq 1$ on $X$ which leaves $V$ invariant. Denote by
$\mathcal{N}_{\mathscr{F}}$ the normal sheaf of $\mathscr{F}$. Let $\varphi$
be a homogeneous symmetric polynomial of degree $d>m-k$.

  1. (1)

For each compact connected component $Z$ of the singular set
$\textsf{Sing}(\mathscr{F}|_{V})$, there exists a homology class

$\text{Res}_{\varphi}(\mathscr{F},\mathcal{N}_{\mathscr{F}}|_{V};Z)\in
H_{2m-2d}(Z;\mathbb{C}),$

which is determined by the local behavior of $\mathscr{F}$ near $Z$.

  2. (2)

If $V$ is compact,

$\sum_{Z}(i_{Z})_{*}\text{Res}_{\varphi}(\mathscr{F},\mathcal{N}_{\mathscr{F}}|_{V};Z)=\varphi(\mathcal{N}_{\mathscr{F}})\smallfrown[V]\,\,\,\,\,\,\,\,\text{in}\,\,\,\,\,\,\,H_{2m-2d}(V;\mathbb{C}),$

where $i_{Z}$ denotes the embedding $Z\hookrightarrow V$ and the sum is taken
over all the components $Z$ of $\textsf{Sing}(\mathscr{F}|_{V})$.

When $\varphi=c_{1}$, the expression
$\text{Res}_{\varphi}(\mathscr{F},\mathcal{N}_{\mathscr{F}}|_{V};Z)$ is called
the variation of $\mathscr{F}$ with respect to $V$ at $Z$. In general, the
computation and determination of these residues is a difficult problem and few
results are known. For example, if the foliation $\mathscr{F}$ is singular at
$p\in\mathbb{C}^{2}$ and $V$ is a reduced complex curve through $p$ invariant
by $\mathscr{F}$. Then the variation of $\mathscr{F}$ relative to $V$ at $p$
is given by

$\text{Res}_{c_{1}}(\mathscr{F},\mathcal{N}_{\mathscr{F}}|_{V};p)=\text{Var}(\mathscr{F},V,p)[p],$

where $\text{Var}(\mathscr{F},V,p)$ is the variational index defined by
Khanedani and Suwa in [20]. When $\mathscr{F}$ is a one-dimensional
holomorphic foliation on a complex manifold, that is, locally defined by
holomorphic vector fields, there is an explicit formula in terms of
Grothendieck residues for
$\text{Res}_{\varphi}(\mathscr{F},\mathcal{N}_{\mathscr{F}}|_{V};p)$, see for
instance [22] and [29].

This paper aims to study of residues of codimension one holomorphic foliations
on complex manifolds of dimension at least three. First, we will restrict our
attention to Lehmann-Suwa residues (or variations) of a codimension one
holomorphic foliation $\mathscr{F}$ on a compact complex manifold $X$ of
dimension at least three. In Section 3, it is shown that Lehmann-Suwa residues
localized at codimension two irreducible components of the singular set of
$\mathscr{F}$ can be determined as multiples of complex numbers by integration
currents along of these irreducible components.

In [23], Lins Neto introduced the Camacho-Sad index [8] for a codimension one
holomorphic foliation concerning a codimension one complex submanifold, and
Gmira [16] obtained a generalization of some results due to Lins Neto [23].
Recently, Corrêa and Machado [12] defined the GSV-index for holomorphic Pfaff
systems on complex manifolds generalizing the GSV-index of Gómez-
Mont–Seade–Verjovsky [17]. In Section 4, combining the Corrêa-Machado index
with the Lehmann-Suwa residues, we recover the Camacho-Sad index for a
codimension one holomorphic foliation $\mathscr{F}$ with respect to a
codimension one complex subvariety $V$ (possibly with singularities).

In [7], Brunella and Perrone determine the Baum-Bott residue [3] of a
codimension one holomorphic foliation concerning a singular component of
codimension two via integration over a 3-sphere of a certain 3-form (see for
instance Section 5). In general, the determination of Baum-Bott residues (in
terms of the Grothendieck residues) of singular holomorphic foliations of
arbitrary codimension have been obtained by Corrêa and Lourenço [13]. In
Section 6, we will prove (see Theorem 6.2) that the Baum-Bott and Lehmann-Suwa
residues are related when the codimension one foliation $\mathscr{F}$ is a
simple almost Liouvillian foliation (see Definition 6.1).

In the last part of the paper, we apply our residual formulas to prove, under
certain conditions, the existence of dicritical singularities of a real-
analytic Levi-flat hypersurface tangent to a codimension one holomorphic
foliation on a compact complex manifold of complex dimension at least three.

It is important pointing out that a general construction of residue theorems
for holomorphic foliations and Pfaff systems of any dimension can be found in
[1], [29] and [27]. In the special case of singular codimension one
holomorphic distributions, we refer the reader to [18] and the references
given there.

Acknowledgments. The authors wish to express his gratitude to Maurício Corrêa
and Miguel Rodríguez Peña for several helpful comments concerning to work. The
authors also gratefully acknowledge the anonymous referee for giving many
suggestions that helped to improve the presentation of the paper. The first
author was partially supported by CNPq Brazil grant number 427388/2016-3 and
PRONEX/FAPERJ and the second author was partially supported by Fondecyt-Perú
CG 217-2014.

##  2\. Holomorphic foliations

Let $X$ be a complex manifold and $TX$ the holomorphic tangent bundle of $M$.
Let $\Theta_{X}=\mathcal{O}(TM)$ be the sheaf of holomorphic vector fields on
$X$. A singular holomorphic foliation $\mathscr{F}$ of dimension $r$ on $X$ is
determined by a coherent subsheaf $\Theta_{\mathscr{F}}\subset\Theta_{X}$ of
rank $r$, which is involutive (or integrable), i.e., such that

$\left[\Theta_{\mathscr{F},p},\Theta_{\mathscr{F},p}\right]\subset\Theta_{\mathscr{F},p}\,\,\,\,\,\,\,\,\,\text{for
all}\,\,\,p\in X.$

We set $\mathcal{N}_{\mathscr{F}}=\Theta_{X}/\Theta_{\mathscr{F}}$ and define
$S(\mathscr{F})$ by

$S(\mathscr{F})=\textsf{Sing}(\mathcal{N}_{\mathscr{F}}).$

Note that $S(\mathscr{F})$ is an analytic subset and, from [22],
$S(\mathscr{F})$ is describe as follows: let $U$ be a sufficiently small
coordinate neighborhood with coordinates $(z_{1},\ldots,z_{n})$ and let
$v_{1},\ldots,v_{r}$ be generators of $\mathscr{F}$ on $U$. We write
$v_{i}=\displaystyle\sum^{n}_{j=1}f_{ij}(z)\frac{\partial}{\partial{z_{j}}}$.
Then

$S(\mathscr{F})\cap U=\\{z\in U:rank\left(f_{ij}(z)\right)<r\\}.$

Furthermore, the foliation $\mathscr{F}$ induce an exact sequence

$0\longrightarrow\Theta_{\mathscr{F}}\longrightarrow\Theta_{X}\longrightarrow\mathcal{N}_{\mathscr{F}}\longrightarrow
0.$

In this paper, we study foliations of codimension one in $X$, i.e., foliations
of dimension $\dim(X)-1$. As is common, codimension one foliations can be
described dually utilizing differential 1-forms: a codimension one singular
holomorphic foliation $\mathscr{F}$ on $X$ is determined by a saturated
locally free subsheaf

$N^{*}_{\mathscr{F}}\subset\Omega^{1}_{X}$

of rank one, which satisfies the Frobenius integrability condition. Locally,
$N^{*}_{\mathscr{F}}$ is generated by holomorphic 1-forms
$\omega_{k}\in\Omega^{1}_{X}(U_{k})$, where $\\{U_{k}\\}_{k\in I}$ is an open
covering of $X$, such that

$\omega_{k}\wedge d\omega_{k}=0$

and

$\omega_{k}=g_{k\ell}\omega_{\ell}\,\,\,\,\,\,\,\,\,\,\text{on}\,\,\,U_{k}\cap
U_{\ell}.$

The functions $g_{k\ell}$ are nowhere vanishing, and the multiplicative
cocycle $\\{g_{k\ell}\\}$ defines a line bundle $N_{\mathscr{F}}$, called the
normal bundle of $\mathscr{F}$. The singular set $\textsf{Sing}(\mathscr{F})$
of $\mathscr{F}$ is the analytic subset of $X$ defined by

$\textsf{Sing}(\mathscr{F})\cap U_{k}=\text{zeros
of}\,\,\,\omega_{k},\,\,\,\,\,\,\,\,\,\,\,\,\,\forall\,\,k\in I.$

The saturated condition means that the zero set of every $\omega_{k}$ has
codimension at least two. Therefore, by definition,
$\textsf{Sing}(\mathscr{F})$ has codimension at least two. We will denote
$\textsf{Sing}_{2}(\mathscr{F})$ the union of all irreducible components of
$\textsf{Sing}(\mathscr{F})$ of codimension two.

Throughout this paper, we will always work with $N_{\mathscr{F}}$ and
$\textsf{Sing}(\mathscr{F})$. The relation between $S(\mathscr{F})$ and
$\textsf{Sing}(\mathscr{F})$ can be found in [30] and [29] as well. Moreover,
we will assume by hypotheses that $S(\mathscr{F})=\textsf{Sing}(\mathscr{F})$
and $\textsf{Sing}_{2}(\mathscr{F})\neq\emptyset$.

We remark that in general the characteristic classes of
$\mathcal{N}_{\mathscr{F}}$ (as in Theorem 1.1) and $N_{\mathscr{F}}$ are not
the same. However, $c_{1}(\mathcal{N}_{\mathscr{F}})$ and
$c_{1}(N_{\mathscr{F}})$ are equal in the $K$-group $K(X)$. In fact, we have
the exact sequence

$0\longrightarrow
N^{*}_{\mathscr{F}}\longrightarrow\Omega^{1}_{X}\longrightarrow\Omega^{1}_{X}/N^{*}_{\mathscr{F}}\longrightarrow
0.$

Taking the duals in the last sequence, we obtain an exact sequence

$0\longrightarrow(\Omega^{1}_{X}/N^{*}_{\mathscr{F}})^{*}\longrightarrow\Theta_{X}\longrightarrow
N_{\mathscr{F}}\longrightarrow\mathcal{E}xt^{1}(\Omega^{1}_{X}/N^{*}_{\mathscr{F}},\mathcal{O}_{X})\longrightarrow
0.$

We have $(\Omega^{1}_{X}/N^{*}_{\mathscr{F}})^{*}=\Theta_{\mathscr{F}}$ and an
exact sequence

$0\longrightarrow\mathcal{N}_{\mathscr{F}}\longrightarrow
N_{\mathscr{F}}\longrightarrow\mathcal{E}xt^{1}(\Omega^{1}_{X}/N^{*}_{\mathscr{F}},\mathcal{O}_{X})\longrightarrow
0.$

The characteristic classes of a coherent sheaf are defined by taking a
resolution of the sheaf by vector bundles and regarding it as an element in
the $K$-group $K(X)$. Since $c_{1}$ is additive on $K(X)$, we have

$c_{1}(\mathcal{N}_{\mathscr{F}})=c_{1}(\Theta_{X})-c_{1}(\Theta_{\mathscr{F}})=-c_{1}(\Omega^{1}_{X})+c_{1}(\Omega^{1}_{X})-c_{1}(N^{*}_{\mathscr{F}})=c_{1}(N_{\mathscr{F}}).$

##  3\. Lehmann-Suwa formula

Let $\mathscr{F}$ be a codimension one singular holomorphic foliation on a
compact complex manifold $X$ of dimension at least three and let $V\subset X$
be a complex hypervariety invariant by $\mathscr{F}$. Here, complex
hypervariety means codimension one complex subvariety and invariant means that
if a point of $V$ belongs to the regular part of $\mathscr{F}$, then the whole
leaf through this point is included in $V$. We shall assume furthermore that
$V$ is reduced, that is, the divisor $V$ does not contain multiple irreducible
components.

Let us denote by $\textsf{Sing}(V)$ the singular set of $V$ and set

$\textsf{Sing}_{2}(\mathscr{F},V)=\textsf{Sing}(V)\cup(\textsf{Sing}_{2}(\mathscr{F})\cap
V).$

Let $Z$ be an irreducible component of $\textsf{Sing}_{2}(\mathscr{F},V)$ such
that $Z$ has pure codimension two. Take a generic point $p\in Z$, that is, a
point where $Z$ is smooth and disjoint from the other singular components.
Take $B_{p}\subset X$ a small ball centered at $p$ such that $Z\cap B_{p}$ is
the unique irreducible component of $\textsf{Sing}_{2}(\mathscr{F},V)\cap
B_{p}$ and suppose that $\omega\in\Omega^{1}_{X}(B_{p})$ represents
$\mathscr{F}$ in $B_{p}$. Working with smooth sections of
$N^{*}_{\mathscr{F}}$, instead of holomorphic ones, the corresponding
cohomology group is trivial, and so we can certainly find a smooth (1,0)-form
$\beta\in A^{1,0}(B^{*}_{p})$ such that

(1) $d\omega=\beta\wedge\omega,$

where $B^{*}_{p}=B_{p}\setminus(\textsf{Sing}_{2}(\mathscr{F},V)\cap B_{p})$.
Since $p$ is a generic point of $Z$ and the codimension of $Z$ with respect to
$V$ is one, we may take a one-dimensional small transverse section $\sum_{p}$
to $Z$ at $p$ such that $\sum_{p}\subset V$. Then we define

(2) $\text{Var}(\mathscr{F},V,Z):=\frac{1}{2\pi i}\int_{\Gamma}\beta,$

where $\Gamma$ is a generator of $H_{1}(\sum_{p}\setminus\\{p\\},\mathbb{Z})$.
We call this complex number the Variational index of $\mathscr{F}$ concerning
$V$ along $Z$. By a connectedness argument, it does not depend on the choice
of the generic point $p\in Z$. It is the natural extension of the variational
index of Khanedani-Suwa [20].

The following result is a particular case of Theorem 1.1, the novelty will be
to obtain a proof using the Variational index given in (2).

######  Theorem 3.1.

Let $\mathscr{F}$ be a codimension one holomorphic foliation on a compact
complex manifold $X$ of dimension at least three and let $V\subset X$ be a
reduced complex hypervariety invariant by $\mathscr{F}$ such that
$\textsf{Sing}_{2}(\mathscr{F},V)\neq\emptyset$. Then

$\sum_{Z}\text{Var}(\mathscr{F},V,Z)[Z]=c_{1}(N_{\mathscr{F}}|_{V})\smallfrown[V],$

where the sum is done over all irreducible components $Z$ of
$\textsf{Sing}_{2}(\mathscr{F},V)$ and $[Z]$ denotes the integration current
associated to $Z$.

###### Proof.

We cover $X$ by open subsets $U_{k}$ where the foliation $\mathscr{F}$ is
defined by integrable holomorphic 1-forms $\omega_{k}$, with
$\omega_{k}=g_{k\ell}\omega_{\ell}$, where
$g_{k\ell}\in\mathcal{O}^{*}(U_{k}\cap U_{\ell})$ whenever $U_{k}\cap
U_{\ell}\neq\emptyset$. Assume that $V\cap U_{k}=\\{f_{k}=0\\}$, where
$f_{k}\in\mathcal{O}(U_{k})$. On $U_{k}\cap U_{\ell}$, we have
$f_{k}=\varphi_{k\ell}f_{\ell}$ with
$\varphi_{k\ell}\in\mathcal{O}^{*}(U_{k}\cap U_{\ell})$ and the cocycle
$\\{\varphi_{k\ell}\\}$ defines the line bundle $[V]$ on $X$.

We may find smooth $(1,0)$-forms $\gamma_{k}$ on
$U^{*}_{k}=U_{k}\setminus(\textsf{Sing}_{2}(\mathscr{F},V)\cap U_{k})$ such
that $d\omega_{k}=\gamma_{k}\wedge\omega_{k}$. We fix a small neighborhood $U$
of $\textsf{Sing}_{2}(\mathscr{F},V)$ and we regularize each $\gamma_{k}$ on
$U$, that is, we choose a smooth $(1,0)$-form $\tilde{\gamma}_{k}$ on $U_{k}$
coinciding with $\gamma_{k}$ outside of $U_{k}\cap U$. Then the smooth
$(1,0)$-forms

$\zeta_{k\ell}=\frac{dg_{k\ell}}{g_{k\ell}}-\tilde{\gamma}_{k}+\tilde{\gamma}_{\ell}$

vanish on $\mathscr{F}$ outside $U$. This cocycle can be trivialized

$\zeta_{k\ell}=\zeta_{k}-\zeta_{\ell},$

where $\zeta_{k}$ is a smooth $(1,0)$-form on $U_{k}$ vanishing on
$\mathscr{F}$ outside of $U_{k}\cap U$. Hence, after setting
$\hat{\gamma}_{k}=\tilde{\gamma}_{k}+\zeta_{k}$, we get

(3) $\frac{dg_{k\ell}}{g_{k\ell}}=\hat{\gamma}_{k}-\hat{\gamma}_{\ell}.$

Note that we still have $d\omega_{k}=\hat{\gamma}_{k}\wedge\omega_{k}$ outside
of $U_{k}\cap U$. The globally defined closed 2-form (of mixed type
$(2,0)+(1,1)$)

$\Omega=\frac{1}{2\pi i}d\hat{\gamma}_{k}$

represents, in the De Rham cohomology, the first Chern class of
$N_{\mathscr{F}}$. Moreover, outside $U$, $\Omega$ vanishes when restricted to
leaves of $\mathscr{F}$ ($\Omega\wedge\omega_{k}=0$), and in particular, when
restricted to $V$, except on small neighborhoods of
$\textsf{Sing}_{2}(\mathscr{F},V)$ in $V$. This means that

$\text{Supp}(\Omega|_{V})\subset\overline{U}.$

Let $\psi$ be a closed smooth $(2n-4)$-form on $V$ and let
$\langle\,,\,\rangle$ be a hermitian metric on $[V]$. Let $\sigma$ be the
global section of $[V]$ defined by $\sigma|_{V_{k}}=f_{k}$, where
$V_{k}:=V\cap U_{k}=\\{f_{k}=0\\}$. Set $S=\textsf{Sing}_{2}(\mathscr{F},V)$.
We consider the tubular neighborhood of $S$ in $V$ for some small number
$\epsilon>0$ as follows

$T_{S}(\epsilon)=\\{p\in V:||\sigma(p)||_{p}\leq\epsilon\\}.$

Note that $\sigma(p)=0$ if, and only if, $p\in S$, moreover

$\partial{T}_{S}(\epsilon)=\\{p\in V:||\sigma(p)||_{p}=\epsilon\\}.$

Assume that $S=\displaystyle\bigcup^{m}_{j=1}Z_{j}$. For each $Z_{j}$ choose a
small neighborhood $V_{j}$ in $U$ such that $Z_{j}\subset V_{j}$,
$\overline{V_{j}}\subset U$ and
$\text{Supp}(\Omega|_{V})=\displaystyle\bigcup^{m}_{j=1}V_{j}$. Denoting
$T_{Z_{j}}(\epsilon)=T_{S}(\epsilon)\cap\overline{V_{j}}$, we have
$Z_{j}\subset T_{Z_{j}}(\epsilon)$. Therefore,

$\displaystyle\int_{V}\Omega\wedge\psi$ $\displaystyle=$
$\displaystyle\sum^{m}_{j=1}\int_{V_{j}}\Omega\wedge\psi$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\sum^{m}_{j=1}\int_{V_{j}}d\hat{\gamma}_{j}\wedge\psi$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\sum^{m}_{j=1}\left[\int_{T_{Z_{j}}(\epsilon)}d\hat{\gamma}_{j}\wedge\psi+\int_{V_{j}-T_{Z_{j}}(\epsilon)}d\hat{\gamma}_{j}\wedge\psi\right].$

Since $\displaystyle\lim_{\epsilon\to
0}\int_{V_{j}-T_{Z_{j}}(\epsilon)}d\hat{\gamma}_{j}\wedge\psi=0$, we get

(4) $\displaystyle\int_{V}\Omega\wedge\psi$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi i}\sum^{m}_{j=1}\lim_{\epsilon\to
0}\int_{T_{Z_{j}}(\epsilon)}d\hat{\gamma}_{j}\wedge\psi$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi i}\sum^{m}_{j=1}\lim_{\epsilon\to
0}\int_{T_{Z_{j}}(\epsilon)}d(\hat{\gamma}_{j}\wedge\psi)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi i}\sum^{m}_{j=1}\lim_{\epsilon\to
0}\int_{\partial{T}_{Z_{j}}(\epsilon)}\hat{\gamma}_{j}\wedge\psi.$

Now, take a smooth point $p\in Z_{j}-\displaystyle\bigcup_{\ell\neq
j}Z_{\ell}$, then there exists a neighborhood $W_{p}\subset V_{j}$ of $p$ and
a coordinate system $(z_{1},z_{2},\ldots,z_{n-1})$ centered at $p$ such that
$Z_{j}\cap W_{p}=\\{z_{1}=0\\}$ and $\partial{T}_{Z_{j}}(\epsilon)\cap
W_{p}=\\{|z_{1}|=\epsilon,z^{\prime}\in\Delta_{\epsilon}\\}$, where
$z^{\prime}=(z_{2},\ldots,z_{n-1})$ and
$\Delta_{\epsilon}=\\{z^{\prime}\in\mathbb{C}^{n-2}:|z^{\prime}|\leq\epsilon\\}$.
Note that

$\displaystyle\lim_{\epsilon\to 0}\frac{1}{2\pi
i}\int_{\partial{T}_{Z_{j}}(\epsilon)\cap W_{p}}\hat{\gamma}_{j}\wedge\psi$
$\displaystyle=$ $\displaystyle\lim_{\epsilon\to 0}\frac{1}{2\pi
i}\int_{\Delta_{\epsilon}}\int_{|z_{1}|=\epsilon}\hat{\gamma}_{j}\wedge\psi$
$\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0}\int_{\Delta_{\epsilon}}\left[\frac{1}{2\pi
i}\int_{|z_{1}|=\epsilon}\hat{\gamma}_{j}\right]\psi$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\to
0}\int_{\Delta_{\epsilon}}\text{Var}(\mathscr{F},V,Z_{j})\psi$
$\displaystyle=$
$\displaystyle\text{Var}(\mathscr{F},V,Z_{j})\lim_{\epsilon\to
0}\int_{\Delta_{\epsilon}}\psi$ $\displaystyle=$
$\displaystyle\text{Var}(\mathscr{F},V,Z_{j})\int_{Z_{j}\cap W_{p}}\psi.$

Therefore,

(5) $\displaystyle\lim_{\epsilon\to 0}\frac{1}{2\pi
i}\int_{\partial{T}_{Z_{j}}(\epsilon)}\hat{\gamma}_{j}\wedge\psi$
$\displaystyle=$
$\displaystyle\text{Var}(\mathscr{F},V,Z_{j})\int_{Z_{j}}\psi$
$\displaystyle=$ $\displaystyle\text{Var}(\mathscr{F},V,Z_{j})[Z_{j}](\psi).$

Hence, from (4) and (5) we get

$\int_{V}\Omega\wedge\psi=\sum^{m}_{j=1}\text{Var}(\mathscr{F},V,Z_{j})[Z_{j}](\psi),$

for any closed smooth $(2n-4)$-form $\psi$ on $V$. Using Poincaré duality and
the fact that $\Omega|_{V}$ represents, in the De Rham cohomology, the Chern
class of $N_{\mathscr{F}}|_{V}$, we obtain

$\sum^{m}_{j=1}\text{Var}(\mathscr{F},V,Z_{j})[Z_{j}]=c_{1}(N_{\mathscr{F}}|_{V})\smallfrown[V].$

∎

######  Remark 3.1.

The proof above gives more, namely, the Lehmann-Suwa residues are determined
as follows:

$\text{Res}_{c_{1}}(\mathscr{F},N_{\mathscr{F}}|_{V};Z)=\text{Var}(\mathscr{F},V,Z)[Z].$

##  4\. GSV and Camacho-Sad indices for codimension one holomorphic foliations

###  4.1. Saito’s decomposition

The following lemma can be found in Saito [26, Section 1]. When $\mathscr{F}$
is a germ of holomorphic foliation at $0\in\mathbb{C}^{2}$, we refer to reader
to [24], [29].

######  Lemma 4.1 (Saito [26]).

Let $\mathscr{F}$ be a germ of codimension one singular holomorphic foliation
at $0\in\mathbb{C}^{n}$, $n\geq 2$, defined by a germ of an integrable
holomorphic 1-form $\omega$. Suppose $V=\\{f=0\\}$ is a germ at
$0\in\mathbb{C}^{n}$ of reduced complex hypervariety invariant by
$\mathscr{F}$. Then there exist germs of holomorphic functions $g$, $h$ and a
germ of holomorphic 1-form $\eta$ at $0\in\mathbb{C}^{n}$ such that

(6) $g\omega=hdf+f\eta,$

where $h$ and $f$ have no common factors. Moreover, $g$ and $f$ also have no
common factors.

Using the Saito’s decomposition, we can now state a similar result to Brunella
[6, Proposition 5].

######  Proposition 4.2.

Let $\mathscr{F}$, $V$ and $Z$ be as in Lemma 4.1. Then

$\text{Var}(\mathscr{F},V,Z)=\frac{1}{2\pi
i}\int_{\Gamma}\left(\frac{g}{h}d\left(\frac{h}{g}\right)-\frac{\eta}{h}\right),$

where $\Gamma$ is as in equation (2).

###### Proof.

By Lemma 4.1 we have

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\frac{h}{g}df+f\frac{\eta}{g}.$

Therefore,

$d\omega=d\left(\frac{h}{g}\right)\wedge
df+df\wedge\frac{\eta}{g}+fd\left(\frac{\eta}{g}\right).$

Restringing to $V$, we get

(7) $\displaystyle d\omega$ $\displaystyle=$ $\displaystyle
d\left(\frac{h}{g}\right)\wedge df+df\wedge\frac{\eta}{g}$

and

(8) $\displaystyle\omega$ $\displaystyle=$ $\displaystyle\frac{h}{g}df.$

From (7) and (8) it follows that

$\displaystyle d\omega$ $\displaystyle=$
$\displaystyle\left(d\left(\frac{h}{g}\right)-\frac{\eta}{g}\right)\wedge df$
$\displaystyle=$
$\displaystyle\left(\frac{g}{h}d\left(\frac{h}{g}\right)-\frac{\eta}{h}\right)\wedge\omega.$

Hence

$\text{Var}(\mathscr{F},V,Z)=\frac{1}{2\pi
i}\int_{\Gamma}\left(\frac{g}{h}d\left(\frac{h}{g}\right)-\frac{\eta}{h}\right),$

where $\Gamma$ is a curve as in (2). ∎

###  4.2. GSV-index for codimension one holomorphic foliations

A. G. Aleksandrov in [2] introduced the concept of multiple residues of
logarithmic differentials forms and generalize the Saito’s decomposition
theorem [26]. Using Aleksandrov’s decomposition theorem, Corrêa and Machado
defined in [12] the GSV-index for holomorphic Pfaff systems. In this
subsection, we particularizing this definition for codimension one holomorphic
foliations.

Let $\mathscr{F}$ be a germ of codimension one singular holomorphic foliation
at $0\in\mathbb{C}^{n}$, $n\geq 3$, defined by a germ of an integrable
holomorphic 1-form $\omega$. Suppose $V=\\{f=0\\}$ is a germ at
$0\in\mathbb{C}^{n}$ of reduced complex hypervariety invariant by
$\mathscr{F}$. Then by Lemma 4.1 we have

$g\omega=hdf+f\eta.$

For each irreducible component $Z$ of $\textsf{Sing}_{2}(\mathscr{F},V)$,
Corrêa and Machado [12] defined the GSV-index as follows:

(9)
$\text{GSV}(\mathscr{F},V,Z):=\textsf{ord}_{Z}\left(\frac{h}{g}\Big{|}_{V}\right).$

According to Corrêa-Machado [12, Theorem 3.2] we can formulate:

######  Theorem 4.3 (Corrêa-Machado [12]).

Let $\mathscr{F}$ be a codimension one holomorphic foliation on a compact
complex manifold $X$ of dimension at least three and let $V\subset X$ be a
reduced complex hypervariety invariant by $\mathscr{F}$ such that
$\textsf{Sing}_{2}(\mathscr{F},V)\neq\emptyset$. Denote by $N_{V/X}$ the
normal bundle of $V$ in $X$. Then

(10)
$\sum_{Z}\text{GSV}(\mathscr{F},V,Z)[Z]=c_{1}(N_{\mathscr{F}}|_{V}\otimes(N_{V/X})^{-1})\smallfrown[V],$

where the sum is done over all irreducible components $Z$ of
$\textsf{Sing}_{2}(\mathscr{F},V)$ and $[Z]$ denotes the integration current
associated to $Z$.

###  4.3. Camacho-Sad index for codimension one holomorphic foliations

Define the Camacho-Sad index as follows:

(11)
$\text{CS}(\mathscr{F},V,Z):=\text{Var}(\mathscr{F},V,Z)-\text{GSV}(\mathscr{F},V,Z).$

When $V$ is smooth, $CS(\mathscr{F},V,Z)$ coincide with the index defined by
Lins Neto [23], see also [16]. Note that Theorem 3.1 and Theorem 4.3 implies
the following result of global nature.

######  Theorem 4.4.

Let $\mathscr{F}$ be a codimension one holomorphic foliation on a compact
complex manifold $X$ of dimension at least three and let $V\subset X$ be a
reduced complex hypervariety invariant by $\mathscr{F}$ such that
$\textsf{Sing}_{2}(\mathscr{F},V)\neq\emptyset$. Then

$\sum_{Z}\text{CS}(\mathscr{F},V,Z)[Z]=c_{1}(N_{V/X})\smallfrown[V]$

where the sum is done over all irreducible components $Z$ of
$\textsf{Sing}_{2}(\mathscr{F},V)$ and $[Z]$ denotes the integration current
associated to $Z$.

##  5\. Baum-Bott index

In this section, we define the Baum-Bott index, following [7]. Similarly to
above section, we work with smooth sections of $N^{*}_{\mathscr{F}}$, instead
of holomorphic ones, then there exists a smooth (1,0)-form $\beta\in
A^{1,0}(B^{*}_{p})$ such that $d\omega=\beta\wedge\omega$, where $\omega$ is a
local generator of $N^{*}_{\mathscr{F}}$. The smooth 3-form (of mixed type
$(3,0)+(2,1)$)

(12) $\frac{1}{(2\pi i)^{2}}\beta\wedge d\beta$

is closed, and it has a De Rham cohomology class in
$H^{3}(B^{*}_{p},\mathbb{C})$, which does not depend on the choice of $\omega$
and $\beta$.

Let $Z$ be an irreducible component of $\textsf{Sing}_{2}(\mathscr{F})$. Take
a generic point $p\in Z$ and pick $B_{p}$ sufficiently small ball, so that
$S(B_{p}):=\textsf{Sing}_{2}(\mathscr{F})\cap B_{p}$ is a codimension two
subball of $B_{p}$. Then the above De Rham class can be integrated over an
oriented 3-sphere $L_{p}\subset B^{*}_{p}$ positively linked with $S(B_{p})$:

$\text{BB}(\mathscr{F},Z)=\frac{1}{(2\pi i)^{2}}\int_{L_{p}}\beta\wedge
d\beta.$

This complex number is called Baum-Bott residue of $\mathscr{F}$ along $Z$.
Again by a connectedness argument, it does not depend on the choice of the
generic point $p\in Z$.

Let us recall that every irreducible component $Z$ of
$\textsf{Sing}_{2}(\mathscr{F})$ has a class $[Z]\in H^{4}(X,\mathbb{C})$
(conveniently defined via the integration current over $Z$). Therefore, we
have the following result.

######  Theorem 5.1 (Baum-Bott [3], Brunella-Perrone [7]).

$\sum_{Z}\text{BB}(\mathscr{F},Z)[Z]=c_{1}^{2}(N_{\mathscr{F}})$

where the sum is done over all irreducible components of
$\textsf{Sing}_{2}(\mathscr{F})$.

##  6\. Almost Liouvillian foliations

In this section, we consider germs at $0\in\mathbb{C}^{n}$, $n\geq 3$, of
singular holomorphic foliations of codimension one. Let $\mathscr{F}$ be a
germ at $0\in\mathbb{C}^{n}$ of a codimension one holomorphic foliation such
that $0\in\textsf{Sing}(\mathscr{F})$ and suppose that $\mathscr{F}$ is
defined by a germ of an integrable holomorphic 1-form
$\omega\in\Omega^{1}(\mathbb{C}^{n},0)$. Let $\textsf{Sing}_{2}(\mathscr{F})$
be the germ at $0\in\mathbb{C}^{n}$ defined by the union of the germs at
$0\in\mathbb{C}^{n}$ of irreducible components of the singular set of
$\mathscr{F}$ whose codimension is precisely two. In this section, we assume
that $\textsf{Sing}_{2}(\mathscr{F})$ is not empty.

######  Definition 6.1.

We say that the germ $\mathscr{F}$ is an almost Liouvillian foliation at
$0\in\mathbb{C}^{n}$ if there exists a germ of closed meromorphic 1-form
$\gamma_{0}$ and a germ of holomorphic 1-form $\gamma_{1}$ at
$0\in\mathbb{C}^{n}$ such that

(13) $d\omega=(\gamma_{0}+\gamma_{1})\wedge\omega.$

We say that $\mathscr{F}$ is a simple almost Liouvillian foliation at
$0\in\mathbb{C}^{n}$ if we can choose $\gamma_{0}$ having only first-order
poles.

The next lemma was proved by Brunella [6] in the two-dimensional case. We
extend this fact for high dimension.

######  Lemma 6.1.

If $\mathscr{F}$ is almost Liouvillian foliation at $0\in\mathbb{C}^{n}$
defined by $\omega\in\Omega^{1}(\mathbb{C}^{n},0)$, $n\geq 3$. Then the poles
divisor of $\gamma=\gamma_{0}+\gamma_{1}$ is invariant by $\mathscr{F}$.

###### Proof.

Let $V=(\gamma)_{\infty}$ be the poles divisor of $\gamma$. If $p$ is a smooth
point of $V$ such that $p\not\in\textsf{Sing}(\mathscr{F})$, then there exists
a coordinate system $(z_{1},\ldots,z_{n})$ at $p$ such that
$z_{1}(p)=\ldots=z_{n}(p)=0$ and $\omega=a(z_{1},\ldots,z_{n})dz_{n}$, where
$a\in\mathcal{O}^{*}(\mathbb{C}^{n},0)$. Let
$\gamma=b_{1}dz_{1}+\ldots+b_{n}dz_{n}$, where $b_{1},\ldots,b_{n}$ are germs
of meromorphic functions at $0\in\mathbb{C}^{n}$. It follows from (13) that
$b_{1},\ldots,b_{n-1}\in\mathcal{O}(\mathbb{C}^{n},0)$ and therefore
$\gamma_{0}=b_{n}dz_{n}$. Since $\gamma_{0}$ is closed, we get
$b_{n}=b_{n}(z_{n})$ and $\gamma_{0}$ may be written as

$\gamma_{0}=\frac{h(z_{n})}{z^{k}_{n}}dz_{n},$

where $h(z_{n})$ is a holomorphic function and $k\geq 1$. Consequently, the
germ of $V$ at $p$ is given by $\\{z_{n}=0\\}$, which implies that $V$ is
invariant by $\omega$. ∎

######  Remark 6.1.

Let $\mathscr{F}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 3$, of a
codimension one holomorphic foliation such that $Z$ is a germ at
$0\in\mathbb{C}^{n}$ of an irreducible component of
$\textsf{Sing}_{2}(\mathscr{F})$. Suppose that there exists a germ at
$0\in\mathbb{C}^{n}$ of a complex hypervariety $V$ invariant by $\mathscr{F}$
such that $V$ does not contain $Z$. Then it is not difficult to see that the
definition of index variational $\text{Var}(\mathscr{F},V,Z)$ (see for
instance (2)) may be extended to an irreducible component $Z$ of
$\textsf{Sing}_{2}(\mathscr{F})$ that is not contained in $V$. In this case,
we have $\text{Var}(\mathscr{F},V,Z)=0$.

The next theorem extends a result due to Brunella [6, Proposition 8]. This
result provides an effective way of computing Baum-Bott residues of
codimension one holomorphic foliations in high dimension. We remark that germs
at $0\in\mathbb{C}^{3}$ of codimension one holomorphic foliations with reduced
singularities (see for instance Cano [9]), logarithmic foliations and some
transversally affine foliations are examples of simple almost Liouvillian
foliations.

######  Theorem 6.2.

Let $\mathscr{F}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 3$, of a simple
almost Liouvillian foliation defined by
$\omega\in\Omega^{1}(\mathbb{C}^{n},0)$ such that

$d\omega=(\gamma_{0}+\gamma_{1})\wedge\omega.$

Let $V$ be the divisor of poles of $\gamma=\gamma_{0}+\gamma_{1}$ and
$V_{1},\ldots,V_{\ell}$ the irreducible components of $V$. Let $Z$ be an
irreducible component of $\textsf{Sing}_{2}(\mathscr{F})$. Then

$\text{BB}(\mathscr{F},Z)=\sum^{k}_{j=1}\text{Res}(\gamma_{0},V_{j})\text{Var}(\mathscr{F},V_{j},Z),$

where $V_{1},\ldots,V_{k}$ are the irreducible components of $V$ that contains
$Z$.

###### Proof.

Take a generic point $p\in Z$ and pick $B_{p}$ a sufficiently small ball such
that $S(B_{p})$ is a codimension two subball of $B_{p}$ at $Z$ (as in Section
5). Let $S^{3}_{\epsilon}\subset B^{*}_{p}$ be an oriented 3-sphere positively
linked with $S(B_{p})$. Let $\partial{V}_{j}:=S^{3}_{\epsilon}\cap V_{j}$ and
let $W_{j}$ be a tubular neighborhood of $\partial{V}_{j}$ such that
$W_{j}\cap Z=\emptyset$. Then there exists holomorphic 1-form
$\gamma^{\prime}_{j}$ in a neighborhood of $W_{j}$ such that

(14) $d\omega=\gamma^{\prime}_{j}\wedge\omega.$

Note that $W:=\displaystyle\bigcup^{N}_{j=1}W_{j}$ is a tubular neighborhood
of $\partial{V}=S^{3}_{\epsilon}\cap V$ and there is an partition of unity
$\rho=\\{\rho_{j}\\}$ for $W$ subordinate to open cover $\\{W_{j}\\}$. With
this we can define

$\gamma^{\prime}=\rho_{1}\gamma^{\prime}_{1}+\ldots+\rho_{N}\gamma^{\prime}_{N}.$

It is easily seen that $\gamma^{\prime}$ is a holomorphic 1-form in a
neighborhood of $W$.

Let $\phi\in C^{\infty}_{c}(W)$ be equal to 1 on a smaller neighborhood of
$\partial{V}$. Then $\beta=\phi\gamma^{\prime}+(1-\phi)\gamma^{\prime}$ is a
smooth $(1,0)$-form on a neighborhood of $S^{3}_{\epsilon}$. Note that

$d\omega=\beta\wedge\omega.$

Denoting $\beta_{j}=\beta|_{S^{3}_{\epsilon}\cap W_{j}}$ and
$\phi_{j}=\phi|_{S^{3}_{\epsilon}\cap W_{j}}$ for each $j=1,\ldots,N$, we get

$\beta_{j}=\phi_{j}\gamma^{\prime}_{j}+(1-\phi_{j})\gamma^{\prime}_{j},\,\,\,\,\,\,\,\,\,\text{on}\,\,\,\,\,\,S^{3}_{\epsilon}\cap
W_{j}.$

Note also that $S^{3}_{\epsilon}\cap
W=\displaystyle\bigcup^{N}_{j=1}(S^{3}_{\epsilon}\cap W_{j})$. To continue, we
may choose holomorphic coordinates $(z_{1},\ldots,z_{n})$ near each
$\partial{V}_{j}=S^{3}_{\epsilon}\cap V_{j}$, with $z_{1}$ varying on a
neighborhood of the unitary circle and $(z_{2},\ldots,z_{n})$ on a
neighborhood of the origin of $\mathbb{C}^{n-1}$ such that

$\displaystyle V_{j}$ $\displaystyle=$ $\displaystyle\\{z_{n}=0\\},$
$\displaystyle\partial{V}_{j}$ $\displaystyle=$
$\displaystyle\\{|z_{1}|=1,z_{2}=z_{3}=\ldots=z_{n}=0\\},$ $\displaystyle
S^{3}_{\epsilon}\cap W_{j}$ $\displaystyle=$
$\displaystyle\\{|z_{1}|=1,|z_{n}|\leq\epsilon,z_{2}=z_{3}=\ldots=z_{n-1}=0\\},$
$\displaystyle\partial(S^{3}_{\epsilon}\cap W_{j})$ $\displaystyle=$
$\displaystyle\\{|z_{1}|=1,|z_{n}|=\epsilon,z_{2}=z_{3}=\ldots=z_{n-1}=0\\}.$

We claim $Supp(\beta\wedge d\beta)\subset S^{3}_{\epsilon}\cap W$. In fact, by
construction we have

(15) $\beta\wedge d\beta=\beta_{j}\wedge
d\beta_{j}\,\,\,\,\,\,\,\text{in}\,\,\,\,\,\,S^{3}_{\epsilon}\cap W_{j}.$

On the other hand, since

$\beta_{j}\wedge d\beta_{j}=\phi^{2}\gamma^{\prime}_{j}\wedge
d\gamma^{\prime}_{j}-\gamma^{\prime}_{j}\wedge
d\phi\wedge\gamma_{j}+\phi(1-\phi)\gamma^{\prime}_{j}\wedge
d\gamma_{j}+(1-\phi)\phi\gamma_{j}\wedge
d\gamma^{\prime}_{j}+(1-\phi)^{2}\gamma_{j}\wedge d\gamma_{j},$

and $\gamma^{\prime}_{j}\wedge d\gamma^{\prime}_{j}=\gamma_{j}\wedge
d\gamma^{\prime}_{j}=\gamma^{\prime}_{j}\wedge d\gamma_{j}=\gamma_{j}\wedge
d\gamma_{j}=0$ in $S^{3}_{\epsilon}\cap W_{j}$, we get

$\beta\wedge d\beta=\beta_{j}\wedge
d\beta_{j}=d\phi\wedge\gamma^{\prime}_{j}\wedge\gamma_{j}.$

Therefore, $Supp(\beta\wedge d\beta)\subset S^{3}_{\epsilon}\cap W$ and the
assertion is proved.

Now

(16) $\int_{S^{3}_{\epsilon}}\beta\wedge d\beta=\int_{S^{3}_{\epsilon}\cap
W}\beta\wedge d\beta=\sum^{N}_{j=1}\int_{S^{3}_{\epsilon}\cap
W_{j}}\beta_{j}\wedge d\beta_{j}.$

Since $\beta_{j}\wedge
d\beta_{j}=d\phi\wedge\gamma^{\prime}_{j}\wedge\gamma_{j}$ in
$S^{3}_{\epsilon}\cap W_{j}$, we obtain
$d((1-\phi)\gamma\wedge\gamma^{\prime}_{j})=\beta_{j}\wedge d\beta_{j}$. Then

(17) $\displaystyle\int_{S^{3}_{\epsilon}\cap W_{j}}\beta_{j}\wedge
d\beta_{j}$ $\displaystyle=$ $\displaystyle\int_{S^{3}_{\epsilon}\cap
W_{j}}d((1-\phi)\gamma\wedge\gamma^{\prime}_{j})$ $\displaystyle=$
$\displaystyle\int_{\partial{(S^{3}_{\epsilon}\cap
W_{j})}}(1-\phi)\gamma\wedge\gamma^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\int_{\partial{(S^{3}_{\epsilon}\cap
W_{j})}}\gamma\wedge\gamma^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\int_{\partial{(S^{3}_{\epsilon}\cap
W_{j})}}(\gamma_{0}+\gamma_{1})\wedge\gamma^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\int_{\partial{(S^{3}_{\epsilon}\cap
W_{j})}}\gamma_{0}\wedge\gamma^{\prime}_{j}.$

In the coordinate system $(z_{1},\ldots,z_{n})$, we have

$\gamma=b_{1}dz_{1}+\ldots+b_{n-1}dz_{n-1}+b_{n}dz_{n}$

where $\gamma_{1}=b_{1}dz_{1}+\ldots+b_{n-1}dz_{n-1}$ and
$\gamma_{0}=b_{n}dz_{n}$. Furthermore, since $\mathscr{F}$ is a simple almost
Liouvillian foliation, we have

$\gamma_{0}=\lambda_{j}\frac{dz_{n}}{z_{n}}+\gamma_{0j},$

where $\lambda_{j}=\text{Res}(\gamma_{0},V_{j})$ and $\gamma_{0j}$ is a
suitable holomorphic 1-form. On the other hand,
$\gamma^{\prime}_{j}=a_{1}dz_{1}+\ldots+a_{n}dz_{n},$ with
$a_{j}\in\mathcal{O}(W_{j})$ for all $i=1,\ldots,n,$ and in particular
$\gamma^{\prime}_{j}|_{\partial{V_{j}}}=a_{1}(z_{1},0,\ldots,0)dz_{1}$, where
$\partial{V}_{j}=\\{|z_{1}|=1,z_{2}=\ldots=z_{n}=0\\}$. Then

(18) $\displaystyle\int_{\partial(S^{3}_{\epsilon}\cap
W_{j})}\gamma_{0}\wedge\gamma^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\int_{|z_{1}|=1,|z_{n}|=\epsilon}\left(\lambda_{j}\frac{dz_{n}}{z_{n}}+\gamma_{0j}\right)\wedge(a_{1}dz_{1}+a_{n}dz_{n})$
$\displaystyle=$
$\displaystyle\int_{|z_{1}|=1,|z_{n}|=\epsilon}\left(\lambda_{j}a_{1}\frac{dz_{n}}{z_{n}}\wedge
dz_{1}\right)$ $\displaystyle=$ $\displaystyle(2\pi
i)\lambda_{j}\int_{|z_{1}|=1}\left(\frac{1}{2\pi
i}\int_{|z_{n}|=\epsilon}\frac{a_{1}(z_{1},0,\ldots,z_{n})}{z_{n}}dz_{n}\right)dz_{1}$
$\displaystyle=$ $\displaystyle(2\pi
i)\lambda_{j}\int_{|z_{1}|=1}a_{1}(z_{1},0,\ldots,0)dz_{1}$ $\displaystyle=$
$\displaystyle(2\pi
i)\lambda_{j}\int_{\partial{V_{j}}}\gamma^{\prime}_{j}|_{\partial{V_{j}}}.$

If $V_{j}$ contains $Z$ then it follows from (14) that

$\frac{1}{2\pi
i}\int_{\partial{V_{j}}}\gamma^{\prime}_{j}|_{\partial{V_{j}}}=\text{Var}(\mathscr{F},V_{j},Z).$

Thus from (18) we get

$\frac{1}{(2\pi i)^{2}}\int_{\partial(S^{3}_{\epsilon}\cap
W_{j})}\gamma_{0}\wedge\gamma^{\prime}_{j}=\lambda_{j}\text{Var}(\mathscr{F},V_{j},Z).$

If $V_{j}$ does not contain $Z$ then $\text{Var}(\mathscr{F},V_{j},Z)=0$ by
Remark 6.1. Finally, from (16) and (17), we obtain

$\text{BB}(\mathscr{F},Z)=\sum^{k}_{j=1}\text{Res}(\gamma_{0},V_{j})\text{Var}(\mathscr{F},V_{j},Z),$

where $V_{1},\ldots,V_{k}$ are the irreducible components of $V$ that contains
$Z$. ∎

To end this section we give an example where Theorem 6.2 applies.

######  Example 6.1.

Let $\mathscr{F}$ be the germ at $0\in\mathbb{C}^{3}$ of a holomorphic
foliation defined by

$\omega=2yzdx+3xzdy+4xydz.$

We have

$d\omega=\gamma_{0}\wedge\omega,\quad\text{where}\quad\gamma_{0}=-\frac{dx}{x}-2\frac{dy}{y}-3\frac{dz}{z}.$

In particular, $\mathscr{F}$ is a codimension one Liouvillian foliation at
$0\in\mathbb{C}^{3}$. Let
$V=(\gamma_{0})_{\infty}=\displaystyle\bigcup^{3}_{j=1}V_{j}$, where
$V_{1}=\\{x=0\\}$, $V_{2}=\\{y=0\\}$ and $V_{3}=\\{z=0\\}$. Note that

$\text{Res}(\gamma_{0},V_{1})=-1,\,\,\,\,\,\,\,\,\text{Res}(\gamma_{0},V_{2})=-2,\,\,\,\,\,\,\,\,\,\text{Res}(\gamma_{0},V_{3})=-3.$

Let $Z=\\{y=z=0\\}$, it is evident that
$Z\subset\textsf{Sing}_{2}(\mathscr{F})$. Furthermore $Z\subset V_{2}$ and
$Z\subset V_{3}$. To compute $\text{Var}(\mathscr{F},V_{3},Z)$, we pick
$p=(1,0,0)\in Z$ and the transverse section $\sum_{p}=\\{|y|\leq 1,x=1,z=0\\}$
to $Z$ in $V_{3}$. By Proposition 4.2, we get

$\text{Var}(\mathscr{F},V_{3},Z)=\frac{1}{2\pi
i}\int_{\Gamma_{3}}\left(\frac{d(4xy)}{4xy}-\frac{2ydx+3xdy}{4xy}\right)|_{x=1}=\frac{1}{4},$

where $\Gamma_{3}\in H_{1}(\sum_{p}\setminus\\{p\\},\mathbb{Z})$.

On the other hand, to compute $\text{Var}(\mathscr{F},V_{2},Z)$, take again
$p=(1,0,0)\in Z$ and the transverse section $\sum_{p}=\\{|z|\leq 1,x=1,y=0\\}$
to $Z$ in $V_{2}$. Again by Proposition 4.2, we get

$\text{Var}(\mathscr{F},V_{2},Z)=\frac{1}{2\pi
i}\int_{\Gamma_{2}}\left(\frac{d(3xz)}{3xz}-\frac{2zdx+4xdz}{3xz}\right)|_{x=1}=-\frac{1}{3},$

where $\Gamma_{2}\in H_{1}(\sum_{p}\setminus\\{p\\},\mathbb{Z})$. Hence,
applying Theorem 6.2, we conclude

$\text{BB}(\mathscr{F},Z)=(-2)\left(-\frac{1}{3}\right)+(-3)\left(\frac{1}{4}\right)=-\frac{1}{12}.$

Using a recently result of Corrêa-Lourenço [13], we can verify that the above
computations are correct. In fact, as in [13, Example 4.1], let us consider
$p=(1,0,0)\in Z$, $D=\\{|(y,z)|\leq 1,x=1\\}$ and

$\omega|_{D}=3zdy+4ydz.$

The dual vector field of $\omega|_{D}$ is
$X=4y\frac{\partial}{\partial{y}}-3z\frac{\partial}{\partial{z}}$. A
straightforward calculation shows that

$JX(0,0)=\begin{bmatrix}4&0\\\ 0&-3\end{bmatrix}.$

Thus

$\text{BB}(\mathscr{F},Z)=\frac{\text{Tr}(JX(0,0))^{2}}{\det(JX(0,0))}=-\frac{1}{12}.$

##  7\. Singular holomorphic foliations tangent to singular Levi-flat
hypersurfaces

Motived by [4] and [15], we study singular codimension one holomorphic
foliations tangent to singular real-analytic Levi-flat hypersurfaces in
compact complex manifolds with emphasis on the type of singularities of them.

Let us clarify these terms. A closed set $M$ of a complex manifold $X$ is a
_real-analytic subvariety_ if it is defined, in some neighborhood of each
point of $M$, by the vanishing of finitely many real-analytic functions with
real values. We say that a real-analytic subvariety $M$ is _irreducible_ if it
cannot be written as the union of two real-analytic subvarieties properly
contained in it. If $M$ is irreducible, it has a well-defined dimension
$\dim_{\mathbb{R}}M$. A _hypervariety_ is a subvariety of real codimension
one.

If $M\subset X$ is a real-analytic submanifold of real codimension one. For
each $p\in M$, there is a unique complex hyperplane $L_{p}$ contained in the
tangent space $T_{p}M\subset T_{p}X$. This defines a real-analytic
distribution $p\mapsto L_{p}$ of complex hyperplanes in $TM$. When this
distribution is integrable in the sense of Frobenius, we say that $M$ is a
Levi-flat hypersurface. In this case, $M$ is foliated by immersed complex
manifolds of dimension $n-1$. This foliation, denoted by $\mathcal{L}$, is
known as Levi foliation. A normal form for such an object was given by E.
Cartan [10, Theorem IV]: at each $p\in M$, there are holomorphic coordinates
$(z_{1},\ldots,z_{n})$ in a neighborhood $U$ of $p$ such that

(19) $M\cap U=\\{\text{Im}(z_{n})=0\\}.$

As a consequence, the leaves of $\mathcal{L}$ have local equations $z_{n}=c$,
for $c\in\mathbb{R}$.

In the singular case, an irreducible real-analytic hypervariety $M\subset X$
is said to be _Levi-flat_ if its _regular part_ is a Levi-flat hypersurface.
We denote by $M_{reg}$ its _regular part_ — the points near which $M$ is a
real-analytic manifold of dimension equal to $\dim_{\mathbb{R}}M$. Let
$\textsf{Sing}(M)$ be the singular points of $M$, points near which $M$ is not
a real-analytic submanifold (of any dimension). Because we are working with
real-analytic sets, the set $\textsf{Sing}(M)$ is not in general equal to the
complement of $M_{reg}$ as defined above, and is only a semi-analytic set (see
for instance [21]). If $M\subset X$ is a real-analytic Levi-flat hypervariety,
Cartan’s local trivialization allows the extension of the Levi foliation to a
non-singular holomorphic foliation in a neighborhood of $M_{reg}$ in $X$,
which is unique as a germ around $M_{reg}$. In general, it is not possible to
extend $\mathcal{L}$ to a singular holomorphic foliation in a neighborhood of
$M$. There are examples of Levi-flat hypervarieties whose Levi foliations
extend to singular $k$-webs in the ambient space [14]. However, there is an
extension in some “holomorphic lifting” of $M$ (see for instance [5]). If a
singular holomorphic foliation $\mathscr{F}$ in the ambient space $X$
coincides with the Levi foliation on $M_{reg}$, we say either that $M$ is
_invariant_ by $\mathscr{F}$ or that $\mathscr{F}$ is _tangent_ to $M$.

######  Definition 7.1.

A singular point $p\in\textsf{Sing}(M)$ is called dicritical if, for every
neighborhood $U$ of $p$, infinitely many leaves of the Levi-foliation on
$M^{*}\cap U$ have $p$ in their closure.

Recently dicritical singularities of singular real-analytic Levi-flat
hypersurfaces have been characterized in terms of the Segre varieties, see for
instance Pinchuk-Shafikov-Sukhov [25].

We recall the definition of meromorphic and holomorphic first integral for
holomorphic foliations. Let $\mathscr{F}$ be a singular holomorphic foliation
on $X$. Recall that $\mathscr{F}$ admits a meromorphic (holomorphic) first
integral at $p\in X$, if there exists a neighborhood $U$ of $p$ and a
meromorphic (holomorphic) function $h$ defined in $U$ such that its
indeterminacy (zeros) set is contained in $\textsf{Sing}(\mathscr{F})\cap U$
and its level curves contain the leaves of $\mathscr{F}$ in $U$.

To prove the main result of this section, we need the following result.

######  Theorem 7.1 (Cerveau-Lins Neto [11]).

Let $\mathscr{F}$ be a germ of codimension one holomorphic foliation at
$0\in\mathbb{C}^{n}$, $n\geq{2}$, tangent to a germ of an irreducible real-
analytic hypersurface $M$. Then $\mathscr{F}$ has a non-constant meromorphic
first integral. In the case of dimension two, we can precise more:

  1. (1)

If $\mathscr{F}$ is dicritical then it has a non-constant meromorphic first
integral.

  2. (2)

If $\mathscr{F}$ is non-dicritical then it has a non-constant holomorphic
first integral.

We now prove a generalization of [4, Lemma 3.2]. To prove this we use Theorem
6.2 and Theorem 7.1.

######  Proposition 7.2.

Let $\mathscr{F}$ be a germ of a codimension one holomorphic foliation at
$0\in\mathbb{C}^{n}$, $n\geq 3$. Suppose that
$\textsf{Sing}_{2}(\mathscr{F})\neq\emptyset$ and that $\mathscr{F}$ has a
non-constant holomorphic first integral, then for every irreducible component
$Z$ of $\textsf{Sing}_{2}(\mathscr{F})$, we have

$\text{BB}(\mathscr{F},Z)\leq 0.$

###### Proof.

Let $g=g_{1}^{m_{1}}g_{2}^{m_{2}}\ldots g_{k}^{m_{k}}$ be a germ at
$0\in\mathbb{C}^{n}$ of a holomorphic first integral for $\mathscr{F}$, where
$g_{1},\ldots,g_{k}$ are irreducible germs at $0\in\mathbb{C}^{n}$ and
$m_{1},\ldots,m_{k}$ are non-negative integers. Then the germ
$\omega=m_{1}g_{2}\ldots g_{k}dg_{1}+\ldots+m_{k}g_{1}\ldots g_{k-1}dg_{k}$ at
$0\in\mathbb{C}^{n}$ defines $\mathscr{F}$. Since $dg=h\omega$ with
$h=g_{1}^{m_{1}-1}g_{2}^{m_{2}-1}\ldots g_{k}^{m_{k}-1}$ we get

(20) $\displaystyle d\omega=-\frac{dh}{h}\wedge\omega,$

where
$\frac{dh}{h}=(m_{1}-1)\frac{dg_{1}}{g_{1}}+\ldots+(m_{k}-1)\frac{dg_{k}}{g_{k}}$.
In particular, $\mathscr{F}$ is a simple Liouvillian foliation at
$0\in\mathbb{C}^{n}$.

Let $Z$ be an irreducible component of $\textsf{Sing}_{2}(\mathscr{F})$, and
let $V_{j}=\\{g_{j}=0\\}$. Note that $Z\subset V_{j}$ for all $1\leq j\leq k$.
To compute $\text{BB}(\mathscr{F},Z)$, we need to compute
$\text{Var}(\mathscr{F},V_{j},Z)$. By Proposition 4.2 we get

$\displaystyle\text{Var}(\mathscr{F},V_{j},Z)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi i}\int_{\partial{V}_{j}}\left[\sum^{k}_{\ell\neq
j}\frac{dg_{\ell}}{g_{\ell}}-\sum^{k}_{\ell\neq
j}\frac{m_{\ell}}{m_{j}}\frac{dg_{\ell}}{g_{\ell}}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi i}\int_{\partial{V}_{j}}\sum^{k}_{\ell\neq
j}\left(1-\frac{m_{\ell}}{m_{j}}\right)\frac{dg_{\ell}}{g_{\ell}}$
$\displaystyle=$ $\displaystyle\sum^{k}_{\ell\neq
j}\left(1-\frac{m_{\ell}}{m_{j}}\right)\textsf{ord}_{Z}(g_{\ell}|_{V_{j}}).$

On the other hand,
$\text{Res}\left(-\frac{dh}{h},V_{j}\right)=-(m_{j}-1)=1-m_{j}$. According to
Theorem 6.2, we obtain

$\displaystyle\text{BB}(\mathscr{F},Z)$ $\displaystyle=$
$\displaystyle\sum^{k}_{j=1}\text{Res}\left(-\frac{dh}{h},V_{j}\right)\text{Var}(\mathscr{F},V_{j},Z)$
$\displaystyle=$ $\displaystyle\sum^{k}_{j=1}(1-m_{j})\left[\sum^{k}_{\ell\neq
j}\left(1-\frac{m_{\ell}}{m_{j}}\right)\textsf{ord}_{Z}(g_{\ell}|_{V_{j}})\right].$

Since
$\textsf{ord}_{Z}(g_{\ell}|_{V_{j}})=\textsf{ord}_{Z}(g_{j}|_{V_{\ell}})$ for
$\ell\neq j$, we get

(21) $\displaystyle\text{BB}(\mathscr{F},Z)=-\sum_{1\leq\ell<j\leq
k}\frac{(m_{\ell}-m_{j})^{2}}{m_{\ell}m_{j}}\textsf{ord}_{Z}(g_{j}|_{V_{\ell}})\leq
0.$

∎

Now we state the main result of this section.

######  Theorem 7.3.

Let $\mathscr{F}$ be a codimension one singular holomorphic foliation on a
compact complex manifold $X$ of dimension at least three, tangent to an
irreducible real-analytic Levi-flat hypervariety $M\subset X$. Suppose that:

  1. (1)

$Sing_{2}(\mathscr{F})$ is not empty and
$\textsf{Sing}_{2}(\mathscr{F})\subset M$,

  2. (2)

$h^{4}(X,\mathbb{C})=1$ and denote by $\zeta$ the generator of
$H^{4}(X,\mathbb{C})$,

  3. (3)

for every fundamental class $[W]\in H^{4}(X,\mathbb{C})$ of an irreducible
complex subvariety $W\subset X$ of codimension two, there exists $\alpha>0$
such that $[W]=\alpha\zeta$,

  4. (4)

for the Chern class $c_{1}^{2}(N_{\mathscr{F}})\in H^{4}(X,\mathbb{C})$, there
exists $\alpha_{0}>0$ such that

$c_{1}^{2}(N_{\mathscr{F}})=\alpha_{0}\zeta.$

Then there exists an irreducible component $Z$ of
$\textsf{Sing}_{2}(\mathscr{F})$ such that it contains some dicritical point
$p\in\textsf{Sing}(M)$. Moreover, $\mathscr{F}$ has a non-constant meromorphic
first integral at $p$.

###### Proof.

Suppose by contradiction that $\textsf{Sing}_{2}(\mathscr{F})$ consists of
irreducible components with only non-dicritical singularities of $M$. Take an
irreducible component $Z$ of $\textsf{Sing}_{2}(\mathscr{F})$ and a generic
point $q\in Z$. By hypothesis (1) we have $Z\subset M$. Let $U$ be a small
neighborhood of $q$ in $X$ such that $\mathscr{F}$ is represented by a
holomorphic 1-form $\omega$ on $U$ and $Z\cap U$ is the unique singular
component of $\omega$. Then, since $\mathscr{F}$ and $M$ are tangent in $U$ we
have $\mathscr{F}|_{U}$ admits a meromorphic first integral $g$ on $U$, by
Theorem 7.1. But since $q\in U$ is a non-dicritical singularity, $g$ must be a
holomorphic first integral.

Applying Proposition 7.2 to $\mathscr{F}|_{U}$, we get
$\text{BB}(\mathscr{F},Z)\leq 0$, for any
$Z\subset\textsf{Sing}_{2}(\mathscr{F})$. Assume that
$\textsf{Sing}_{2}(\mathscr{F})=\displaystyle\bigcup^{k}_{j=1}Z_{j}$. Then
Baum-Bott’s formula (cf. Theorem 5.1) implies that

$\displaystyle c^{2}_{1}(N_{\mathcal{F}})$ $\displaystyle=$
$\displaystyle\sum^{k}_{j=1}\text{BB}(\mathscr{F},Z_{j})[Z_{j}],\quad\text{in}\quad
H^{4}(X,\mathbb{C})$ $\displaystyle=$
$\displaystyle\left(\sum^{k}_{j=1}\text{BB}(\mathscr{F},Z_{j})\alpha_{j}\right)\zeta,\quad\text{for
some}\quad\alpha_{j}>0$

which is absurd with hypothesis $(4)$, because
$\alpha_{0}=\displaystyle\sum^{k}_{j=1}\text{BB}(\mathscr{F},Z_{j})\alpha_{j}\leq
0$. Therefore, there exists an irreducible component $Z$ of
$\textsf{Sing}_{2}(\mathscr{F})$ such that it contains some dicritical point
$p\in M$. Applying again Theorem 7.1, we obtain a non-constant meromorphic
first integral for $\mathscr{F}$ in a neighborhood of $p$. ∎

When $X=\mathbb{P}^{n}$, the complex projective space, $n\geq 3$, we recall
the singular set of any codimension one holomorphic foliation on
$\mathbb{P}^{n}$, $n\geq 3$, has an irreducible component of codimenion two,
see for instance [19, Proposition 2.6, page 95].

Let $h$ be the hyperplane class in $\mathbb{P}^{n}$. Then
$H^{4}(\mathbb{P}^{n},\mathbb{C})$ is generated by $h^{2}$. Thus for every
codimension two irreducible component $W$ in $\mathbb{P}^{n}$ we have
$[W]=\deg(W)h^{2}$. Moreover, for a codimension one foliation $\mathscr{F}$ on
$\mathbb{P}^{n}$, $n\geq 3$, of degree $d\geq 0$, we have

$c^{2}_{1}(N_{\mathscr{F}})=(d+2)^{2}h^{2}.$

This implies that the hypotheses (2), (3) and (4) of Theorem 7.3 are satisfied
for codimension one foliations of $\mathbb{P}^{n}$. Hence, we can state the
following corollary.

######  Corollary 7.4.

Let $\mathscr{F}$ be a codimension one singular holomorphic foliation on
$\mathbb{P}^{n}$, $n\geq 3$, tangent to an irreducible real-analytic Levi-flat
hypervariety $M\subset\mathbb{P}^{n}$. Suppose that
$\textsf{Sing}_{2}(\mathscr{F})\subset M$. Then there exists an irreducible
component $Z$ of $\textsf{Sing}_{2}(\mathscr{F})$ such that it contains some
dicritical point $p\in\textsf{Sing}(M)$. Moreover, $\mathscr{F}$ has a non-
constant meromorphic first integral at $p$.

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