%~Mouliné par MaN_auto v.0.29.1 2023-11-14 16:51:45
\documentclass[CRMATH, Unicode, XML]{cedram}

\TopicFR{Analyse fonctionnelle}
\TopicEN{Functional analysis}

\DeclareMathOperator{\sing}{Sing}
\DeclareMathOperator{\Div}{div}

\providecommand\noopsort[1]{}

\newcommand*{\dd}{\mathrm{d}}
\newcommand*{\dx}{\dd x}

\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
%  \hspace*{0pt}%
\begingroup\hsize=12.5cm\advance\hsize\abstractmarginL\raggedright
\noindent
%\hspace*{\abstractmarginL}\begin{minipage}[t]{10cm}
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\title{Transversely product singularities of foliations in projective spaces }

\author{\firstname{Rudy} \lastname{Rosas}}
\address{Departamento de Ciencias, Pontificia Universidad Cat\'olica del Per\'u, Av. Universitaria 1801, Lima, Per\'u}
\email{rudy.rosas@pucp.pe}
\thanks{The author was supported by Vicerrectorado de Investigaci\'on de la Pontificia Universidad Cat\'olica del Per\'u}

\begin{abstract}
We prove that a transversely product component of the singular set of a holomorphic foliation on $\mathbb P^n$ is necessarily a Kupka component.
\end{abstract}

\begin{document}
\maketitle

\section{Introduction}

Let $U$ be an open set of a complex manifold $M$ and let $k\in\mathbb N$. Let $\eta$ be a holomorphic $k$-form on $U$ and let $\sing \eta: =\{p\in U: \eta(p)=0\}$ denote the singular set of $\eta$. We say that $\eta$ is integrable if each point $p\in U\backslash \sing \eta$ has a neighborhood $V$ supporting holomorphic 1-forms $\xi_1,\dots, \xi_k$ with $\eta|_V=\xi_1\wedge\dots\wedge\xi_k$, such that $\dd\xi_j\wedge \eta=0$ for each $j=1,\dots, k$. In this case the distribution
\[
\mathcal D_\eta:\;\; \mathcal D_\eta (p)=\{v\in T_pM: i_v\eta (p)=0\}, \quad
p\in U\backslash \sing \eta
\]
defines a holomorphic foliation
of codimension $k$ on $U\backslash \sing \eta$. A singular holomorphic foliation $\mathcal F$ of codimension $k$ on $M$ can be defined by an open covering $(U_j)_{j\in J}$ of $M$ and a collection of integrable $k$-forms $\eta_j\in\Omega^k(U_j)$ such that $\eta_i=g_{ij}\eta_j$ for some $g_{ij}\in\mathcal{O}^*(U_i\cap U_j)$ whenever $U_i\cap U_j\neq \emptyset$. The singular set $\sing \mathcal F$ is the proper analytic subset of $M$ given by the union of the sets $\sing\eta_j$. From now on we only consider foliations $\mathcal F$ such that $\sing\mathcal F$ has no component of codimension one.

Given a singular holomorphic foliation $\mathcal F$ of codimension $k$ on $M$ as above, the Kupka singular set of $\mathcal F$, denoted by $K(\mathcal F)$, is the union of the sets
\[
K(\eta_j)=\{p\in U_j: \eta_j(p)=0, \dd\eta_j (p)\neq 0\}.
\]
This set does not depend on the collection $(\eta_j)$ of $k$-forms used to define $\mathcal F$. It is well known (see~\cite{K,Med}) that, given $p\in K(\mathcal F)$, the germ of $\mathcal F$ at $p$ is holomorphically equivalent to the product of a one-dimensional foliation with an isolated singularity by a regular foliation of dimension $(\dim \mathcal F -1)$. More precisely, if $\dim M=k+m+1$, there exist a holomorphic vector field $X=X_1{\partial_{x_1}}+ \dots +X_{k+1}\partial_{x_{k+1}}$ on $\mathbb D^{k+1}$ with a unique singularity at the origin, a neighborhood $V$ of $p$ in $M$ and a biholomorphism $\psi : V \to \mathbb D^{k+1}\times \mathbb D^{m}$, $\psi(p)=0$, which conjugates $\mathcal F$ with the foliation $\mathcal F_X$ of $\mathbb D^{k+1}\times \mathbb D^{m}$ generated by the commuting vector fields $X, \partial_{y_1},\dots, \partial_{y_m},$ where $y=(y_1,\dots,y_m)$ are the coordinates in $\mathbb D^m$. If $\mu=\dx_1\wedge\dots \wedge \dx_{k+1}$, the foliation $\mathcal F_X$ is also defined by the $k$-form $\omega= i_X\mu$ and the Kupka condition $d\omega (0)\neq 0$ is equivalent to the inequality $\Div X(0)\neq 0$.

Following~\cite{LN2021}, we say that $\mathcal F$ is a transversely product at $p\in\sing\mathcal F$ if as above there exist a holomorphic vector field $X$ and a biholomorphism $\psi : V \to \mathbb D^{k+1}\times \mathbb D^{m}$ conjugating $\mathcal F$ with $\mathcal F_X$, except that it is not assumed that $\Div X(0)\neq 0$. We say that $\Gamma$ is a local transversely product component of $\sing \mathcal F$ if $\Gamma$ is a compact irreducible component of $\sing \mathcal F$ and $\mathcal F$ is a transversely product at each $p\in \Gamma$. In particular, if $\Gamma\subset K(\mathcal F)$ we say that $\Gamma$ is a Kupka component --- for more information about Kupka singularities and Kupka components we refer the reader to~\cite{K,CLN, B, CA99, CA09, CA16, CCF}. If $\Gamma$ is a transversely product component of $\sing \mathcal F$, we can cover $\Gamma$ by finitely many normal coordinates like $\psi$, with the same vector field $X$: that is, there exist a holomorphic vector field $X$ on $\mathbb D^{k+1}$ with a unique singularity at the origin and a covering of $\Gamma$ by open sets $(V_{\alpha})_{\alpha\in A}$ such that each $V_\alpha$ supports a biholomorphism $\psi_\alpha: V_\alpha \to \mathbb D^{k+1}\times \mathbb D^{m}$ that maps $\Gamma\cap V_\alpha$ onto $\{0\}\times\mathbb D^{m}$ and conjugates $\mathcal F$ with the foliation $\mathcal F_X$. The sets $(V_\alpha)$ can be chosen arbitrarily close to $\Gamma$.


In~\cite{LN2021}, the author proves that a local transversely product component of a codimension one foliation on $\mathbb P^n$ is necessarily a Kupka component. The goal of the present paper is to generalize this theorem to foliations of any codimension.

\begin{theo}\label{teopro}
Let $\mathcal F$ a holomorphic foliation of dimension $\ge 2$ and codimension $\ge 1$ on $\mathbb P^n $. Let $\Gamma$ be a transversely product component of $\sing \mathcal F$. Then $\Gamma$ is a Kupka component.
\end{theo}

This theorem is a corollary of the following result.

\begin{theo}\label{teotubo}
Let $\mathcal F$ a holomorphic foliation of dimension $\ge 2$ and codimension $k\ge 1$ on a complex manifold $M$. Suppose that $\mathcal F$ is defined by an open covering $(U_j)_{j\in J}$ of $M$ and a collection of $k$-forms $\eta_j\in\Omega^k(U_j)$. Let $L$ be the line bundle defined by the cocycle $(g_{ij})$ such that $\eta_i=g_{ij}\eta_j$, $g_{ij}\in\mathcal{O}^*(U_i\cap U_j)$. Let $\Gamma$ be a transversely product component of $\sing \mathcal F$ that is not a Kupka component. Then $c_1(L|_\Gamma)=0$.
\end{theo}

\section{Proof of the results}

\begin{proof}[Proof of Theorem~\ref{teotubo}]
Let $\dim M=k+m+1$. As explained in the introduction, there exist a holomorphic vector field $X$ on $\mathbb D^{k+1}$ with a unique singularity at the origin and a covering of $\Gamma$ by open sets $(V_{\alpha})_{\alpha\in A}$ such that each $V_\alpha$ is contained in $V$ and supports a biholomorphism $\psi_\alpha: V_\alpha \to \mathbb D^{k+1}\times \mathbb D^{m}$ that maps $\Gamma\cap V_\alpha$ onto $\{0\}\times\mathbb D^{m}$ and conjugates $\mathcal F$ with the foliation $\mathcal F_X$ generated by the commuting vector fields $X, \partial_{y_1},\dots, \partial_{y_m}$. Notice that $\Div(X)(0)=0$, because $\Gamma$ is not a Kupka component. Since $\mathcal F_X$ is defined by the $k$-form $\omega= i_X\mu$, where $\mu=\dx_1\wedge\dots\wedge \dx_{k+1}$, we have that $\mathcal F|_{V_\alpha}$ is defined by the $k$-form $\psi^*_\alpha(\omega)$. If $V_\alpha\cap V_\beta\neq\emptyset$, there exists $\theta_{\alpha\beta}\in\mathcal{O}^*(V_\alpha\cap V_\beta)$ such that
\begin{align}\label{cociclo}
\psi^*_\alpha(\omega)=\theta_{\alpha\beta}\psi^*_\beta(\omega).
\end{align}
Therefore the cocycle $(\theta_{\alpha\beta})$ define the line bundle $L$ restricted to some neighborhood of $\Gamma$. Thus, in order to prove that $c_1(L|_\Gamma)=0$ it is enough to show that each $\theta_{\alpha\beta}|_\Gamma$ is locally constant. Fix some $\alpha, \beta\in A$ such that $V_\alpha\cap V_\beta\neq \emptyset$. If we set $\psi=\psi_\alpha\circ\psi_\beta^{-1}$ and $\theta=\theta_{\alpha\beta}\circ\psi_\beta^{-1}$, from~\eqref{cociclo} we have that $\psi^*(\omega)=\theta \omega, $ which means that $\psi$ preserves the foliation $\mathcal F_X$. It suffices to prove that the derivatives $\theta_{y_1}(p),\dots, \theta_{y_m}(p)$ vanish if $p\in \{0\}\times \mathbb D^m$. Since $\partial_{y_1}$ is tangent to $\mathcal F_X$, then the vector field ${\psi_*({\partial_{y_1}})}$ is tangent to $\mathcal F_X$ and so we can express
\[
\psi_*({\partial_{y_1}})=\lambda X +\lambda_1\partial_{y_1}+\dots + \lambda_m\partial_{y_m},
\]
where $\lambda,\lambda_1,\dots,\lambda_m$ are holomorphic. Then
\begin{align*}\label{ecua1}
\mathcal{L}_{\psi_*({\partial_{y_1}})}\omega=\mathcal{L}_{\lambda X}\omega=
\lambda\mathcal{L}_{X}\omega+\dd\lambda\wedge i_X \omega= \lambda\mathcal{L}_{X}\omega =\lambda \Div (X)\omega,
\end{align*}
where the last equality follows from the identity $\omega=i_X\mu$. Thus, since
\[
\psi^*\left(\mathcal{L}_{\psi_*({\partial_{y_1}})}\omega\right)=
\mathcal{L}_{\partial_{y_1}}\psi^*\omega=\mathcal{L}_{\partial_{y_1}}(\theta\omega)
=\theta_{y_1}\omega,
\]
we obtain that
\[
\theta_{y_1}\omega=\psi^*\left(\lambda \Div (X)\omega\right)=
\lambda (\psi) \Div (X)(\psi) \theta\omega
\]
and therefore $\theta_{y_1}(p)=0$ if $p\in \{0\}\times \mathbb D^m$, because $\Div (X)$ vanishes along $\{0\}\times \mathbb D^m$. In the same way we prove that $\theta_{y_2}(p)=\dots =\theta_{y_m}(p)=0$ if $p\in \{0\}\times \mathbb D^m$, which finishes the proof.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{teopro}]
Suppose that $\Gamma$ is not a Kupka component. Let $L$ be the line bundle associated to $\mathcal F$ as in the statement of Theorem~\ref{teotubo}. We notice that $c_1(L)\neq 0$, otherwise $\mathcal F$ will be defined by a global $k$-form on $\mathbb P^n$, which is impossible. Then, if we take an algebraic curve $\mathcal C \subset \Gamma$, we have $c_1(L)\cdot\mathcal C\neq 0$. Therefore, if $\Omega$ is a 2-form on $\mathbb P^n$ in the class $c_1(L)$ and $V$ is a tubular neighborhood of $\Gamma$,
\[
c_1(L|_\Gamma)\cdot\mathcal C =\int_{\mathcal C}\Omega|_\Gamma=\int_{\mathcal C}\Omega=c_1(L)\cdot
\mathcal C\neq 0,
\]
which contradicts Theorem~\ref{teotubo}.
\end{proof}

\bibliographystyle{crplain}
\bibliography{CRMATH_Rosas_20230443}
\end{document}