 X_{\mathcal{F}}$, 
there exist a neighborhood  $U  \subset X_{\mathcal{F}}$ of $x$ and 
local generators $v_1, \ldots, v_n$ of $T_X$ on $U$
such that $\mathcal{F}|_{U}$ is generated by $v_1, \ldots, v_r$.
Then  
$\omega_{\mathcal{F}}|_{U} $ is identified with $ \alpha_{r+1} \wedge \cdots \wedge \alpha_{n}$, where $\alpha_{i}$ is a dual of  $v_i$ for $ r+1 \le i \le n$. 
\end{rem}


\begin{lem}[{\textit{cf.} \cite[Lemma 6.7]{AD14}}]
\label{decent}
Let $f : X \rightarrow Y$ be a smooth morphism with connected fibers between smooth projective varieties and $\mathcal{F}$ be a rank $r$ regular foliation on $X$. 
Assume that all fibers of $f$ are contained in the leaves of $\mathcal{F}$.
Then there exists a regular foliation $\mathcal{G}$ on $Y$ such that 
$\mathcal{F}=f^{-1}\mathcal{G}$ and 
\begin{equation*}
\xymatrix@C=25pt@R=20pt{
0\ar@{->}[r]& T_{X/Y}  \ar@{->}[r]&\mathcal{F}\ar@{->}[r
 &f^{*}\mathcal{G}\ar@{->}[r]&0.\\}
\end{equation*}
\end{lem}
\begin{proof}

Set $\dim Y := m$, $k := \dim X - \dim Y $, and $n := \dim X = m+k $.
Fix $y \in Y$.
Let $V$ be a neighborhood of $y$ and 
$(w_{1}, \ldots, w_{m})$ be  local coordinates on $V$.
We may regard $y \in V$ as an origin.
We can take Euclidean open sets $U_{1}, \ldots, U_{N}$ of $X$
which satisfy the following conditions:
\begin{enumerate}
\item $X_y \subset \cup_{\lambda=1}^{N} U_{\lambda}$.
\item For any $1 \le \lambda \le N$, we can choose local coordinates $(z_{1}^{\lambda}, \ldots, z_{n}^{\lambda})$ on $U_{\lambda}$ such that
$$
\begin{array}{cccc}
f|_{U_{\lambda} } :&  U_{\lambda}    & \rightarrow& V       \\
&(z_{1}^{\lambda}, \ldots, z_{n}^{\lambda})                &    \mapsto  & (z_{1}^{\lambda}, \ldots, z_{m}^{\lambda}).
\end{array}
$$
In particular, $T_{X/Y} |_{U_{\lambda}}$ is generated by $\frac{\partial}{\partial z_{m+1}^{\lambda}}, \ldots, \frac{\partial}{\partial z_{n}^{\lambda}}$ on $U_{\lambda}$.
\item For any $1 \le \lambda \le N$,
there exists a holomorphic morphism $p_{\lambda}$ on $U_{\lambda}$ such that 
$$
\begin{array}{cccc}
p_{\lambda}  :&  U_{\lambda}    & \rightarrow& \mathbb{C}^{n-r}      \\
&(z_{1}^{\lambda}, \ldots, z_{n}^{\lambda})                &    \mapsto  & (z_{1}^{\lambda}, \ldots,  z_{n-r}^{\lambda})
\end{array}
$$
and $\ker (dp_{\lambda}) = \mathcal{F}|_{U_{\lambda}}$. 
In particular, $\mathcal{F}|_{U_{\lambda}}$ is generated by
$\frac{\partial}{\partial z_{n-r+1}^{\lambda}}, \ldots, \frac{\partial}{\partial z_{n}^{\lambda}}$ on $U_{\lambda}$.
\end{enumerate}
Then there exists a local section  $s_{\lambda} : V \rightarrow U_{\lambda}$ such that
$$\begin{array}{cccc}s_{\lambda}  :&  V & \rightarrow& U_{\lambda}      \\&(w_{1}, \ldots, w_{m})                &    \mapsto  & (w_{1}, \ldots, w_{m}, 0, \ldots, 0)\end{array}
$$
for any $1 \le \lambda \le N$.

Set $\mathcal{G}_{V} := \ker (d(p_{\lambda}\circ s_{\lambda}))$, then $\mathcal{G}_{V}$ does not depend on  $\lambda$, and $\mathcal{G}_{V}$ glue and extend to an $r-k$ rank foliation $\mathcal{G}$ on $Y$
(see \cite[Lemma 6.7]{AD14}).
$\mathcal{G}_{V}$
is generated by $\frac{\partial}{\partial w_{n-r+1}}, \ldots, \frac{\partial}{\partial w_{m}}$ on $V$. 
Therefore $\mathcal{G}|_{V}$ is a subbundle of $T_V$, 
$f^{*} \mathcal{G}|_{U_{\lambda}}$ is locally free, and
\begin{equation*}
\xymatrix@C=25pt@R=20pt{
0\ar@{->}[r]& T_{X/Y}|_{U_{\lambda}}  \ar@{->}[r]& \mathcal{F} |_{U_{\lambda}}\ar@{->}[r
 &f^{*}\mathcal{G}|_{U_{\lambda}}\ar@{->}[r]&0, \\}
\end{equation*}
for any $1 \le \lambda \le N$.
From $\omega_{\mathcal{G}} |_{V} = dw_{1} \wedge \cdots \wedge dw_{n-r}$ and $\omega_{\mathcal{F}} |_{U_{\lambda}} = dz_{1}^{\lambda} \wedge \cdots \wedge dz_{n-r}^{\lambda}$,
we obtain $\mathcal{F}|_{U_{\lambda} } = f^{-1}\mathcal{G}|_{U_{\lambda} } $.
\end{proof}
\begin{lem}[{\cite[Lemma 6.7]{AD14}}]
\label{rationaldecent}
Let $f: X \dashrightarrow Y$ be an essentially equidimensional \footnote{A rational map $f: X \dashrightarrow Y$ is \textit{essentially equidimensional} if there exists a Zariski open set $U \subset X$ such that ${\rm codim}(X \setminus U) \ge 2$ and $f|_{U}$ is an equidimensional morphism. We refer the reader to \cite[Definition 2.22]{CKT16}.}
 dominant rational map with connected fibers between normal projective varieties, 

 $\mathcal{H}$ be a foliation induced by $f$, and $\mathcal{F}$ be a foliation on $X$. 
Assume that a general fiber of $f$ is contained in a leaf of $\mathcal{F}$.
Then there exists a foliation $\mathcal{G}$ on $Y$ such that 
$\mathcal{F}=f^{-1}\mathcal{G}$ and 
\begin{equation*}
\xymatrix@C=25pt@R=20pt{
0\ar@{->}[r]& \mathcal{H}  \ar@{->}[r]& \mathcal{F}\ar@{->}[r
&(f^{*}\mathcal{G})^{\vee\vee}. \\}
\end{equation*}
\end{lem}
\begin{proof}
We take Zariski open sets $X_1 \subset X_{\mathcal{F}}$ and $Y_1 \subset Y_{reg}$ such that $\mathcal{H}|_{X_1}$ is a subbundle of $ \mathcal{F}|_{X_1}$ and $f|_{X_1}: X_1 \rightarrow Y_1$ is a morphism.
We may assume that ${\rm codim}(X \setminus X_1) \ge 2$.
Let $X_0 \subset X_1$ be the set of smooth points of $f$ and  $Y_0 :=f(X_0)$.
By the argument of Lemma \ref{decent}, there exists a foliation $\mathcal{G}_{0}$ on $Y_0$ such that $\mathcal{F}|_{X_0} = (f|_{X_0})^{-1}\mathcal{G}_{0}$ and 
\begin{equation*}
\xymatrix@C=25pt@R=20pt{
0\ar@{->}[r]& \mathcal{H}|_{X_0}  \ar@{->}[r]&  \mathcal{F} |_{X_0}\ar@{->}[r
 &(f|_{X_0})^{*}\mathcal{G}_0|_{X_0} \ar@{->}[r]&0. \\}
\end{equation*}

Let $\mathcal{G}$ denote a saturation of $\mathcal{G}_{0}$ in $T_Y$. 
By \cite[Lemma