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\TopicFR{Analyse et géométrie complexes}
\TopicEN{Complex analysis and geometry}

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\title {A note on the weighted log canonical threshold}

\author{\lastname{Nguyen Van Phu}}
\address{Faculty of Natural Sciences, Electric Power University, Hanoi, Vietnam}
\email{ltphudk@gmail.com}

\subjclass[2010]{32U05, 32U15, 32U40, 32W20}

%~\keywords{Plurisubharmonic functions, Log canonical threshold, Radon measure, Positive measures, weighted log canonical thresholds}

\begin{abstract}
In this paper, we introduce and study a set relative to singularities of plurisubharmonic functions. We prove that this set is countable under the condition $h>0$ on $\mathbb{B}\setminus\{0\}.$
\end{abstract}


\begin{document}
\maketitle

\section{Introduction}

Let $\Omega$ be a domain in $\mathbb{C}^n, z_{0}\in \Omega$ and $\varphi$ be a plurisubharmonic function on $\Omega$ (briefly, psh). Following Demailly and Koll\'ar~\cite{De01}, we introduce the log canonical threshold of $\varphi$ at $z_{0}$:
\begin{equation}\label{eq3}
c_{\varphi}(z_{0})=\sup\{c>0:e^{-2c\varphi} \,\,\text{is}\,\, L^{1}(\dV_{2n}) \,\,\text{on a neighborhood of}\,\,z_{0}\},
\end{equation}
where $\dV_{2n}$ denotes the Lebesgue measure of $\mathbb{C}^{n}.$

It is an invariant of the singularity of $\varphi$ at $z_{0}.$ We refer the readers to~\cite{De93, De14, FEM03, FEM10, H13, H17, H18, HHT21} for further information and applications to this number.

For every non-negative Radon measures $\mu$ on a neighbourhood of $z_{0}\in \mathbb{C}^{n}$. Following Pham in~\cite{H14}, we introduce the weighted log canonical threshold of $\varphi$ with weight $\mu$ at $z_{0}$ to be:
\begin{equation}\label{eq4}
c_{\mu, \varphi}(z_{0})=\sup\Biggl\{c>0:\;\exists r>0 \,\,, \int_{\mathbb{B}(0,r)}e^{-2c\varphi (z+z_0)}\dmu (z) < +\infty\Biggr\}.
\end{equation}
In the case if $\mu=h\dV_{2n}$ where $h\in L^{1}(\dV_{2n}), h>0$ on $\mathbb{B}\setminus{\{0\}}, h\in L^{\infty}(\mathbb{B})$ then we introduce the weighted log canonical threshold of $\varphi$ with weight $\mu$ at $z_{0}$ to be:
\begin{equation}\label{eq5a}
c_{h\dV_{2n},\varphi}(z_0) =\sup\Biggl\{c>0:\:\exists r>0 \,\,, \int_{\mathbb{B}(z_{0},r)}e^{-2c\varphi(z)}h(z-z_{0})\dV_{2n}(z)<+\infty\Biggr\}.
\end{equation}
From the definition of $c_{h\dV_{2n},\varphi}(z_0)$ and $c_{\varphi}(z_0)$, we have $c_{h\dV_{2n},\varphi}(z_0) \geq c_{\varphi}(z_0).$ In the paper, we study properties of the set $E_{h,\varphi}=\{z\in\Omega: c_{h\dV_{2n},\varphi}(z) > c_{\varphi}(z) \}$. The main result of the paper prove that the set $E_{h,\varphi} $ is a countable set.


\section{Main Theorem}

\begin{theo}\label{dn1}
If $h\in L^{1}(\dV_{2n}),h>0 $ on $\mathbb{B}\setminus{\{0\}}, h\in L^{\infty}(\mathbb{B})$ then
\[
E_{h,\varphi} =\{z\in\Omega: c_{h\dV_{2n},\varphi}(z) > c_{\varphi}(z) \}
\]
is a countable set.
\end{theo}

\begin{proof}
We have
\[
E_{h,\varphi} = \cup_{c\in \mathbb{Q}}\{z\in\Omega:c_{\varphi}(z) < c < c_{h\dV_{2n},\varphi}(z)\}.
\]
It means that we need to prove the following set
\[
E_{c,h,\varphi}=\{z\in\Omega:c_{\varphi}(z) < c < c_{h\dV_{2n},\varphi}(z)\}
\]
is a countable set. Indeed, let $z_{0}\in E_{c,h,\varphi}$. We have
\[
c_{\varphi}(z_0) < c <c_{h\dV_{2n},\varphi}(z_0).
\]
Since $c <c_{h\dV_{2n},\varphi}(z_0) $ we can find $r>0$ such that
\[
\int_{\mathbb{B}(z_{0},r)}e^{-2c\varphi(z)}h(z-z_{0})\dV=\int_{\mathbb{B}(0,r)}e^{-2c\varphi(z+z_0)}h(z)\dV_{2n}<+\infty.
\]
Since $ h>0$ on $\mathbb{B}(0,r)\setminus{\{0\}}$ ($h(0)=0$), we have $\forall w\in \mathbb{B}(z_0,r)\setminus\{z_0\}$
there exists $\delta>0$ such that $\mathbb{B}(w,\delta)\subset \mathbb{B}(z_{0},r)\setminus \{z_{0}\}$ and
\[
\int_{\mathbb{B}(0,\delta)}e^{-2c\varphi(z+w)}\dV_{2n}(z)<+\infty.
\]
We obtain that $c_{\varphi}(z)>c, \,\,\,\forall w\in \mathbb{B}(z_0,r)\setminus\{z_0\}.$ Thus, if $z_{0}\in E_{c,h,\varphi}$ then
\[
E_{c,h,\varphi}\cap \mathbb{B}(z_0,r)\setminus\{z_0\}=\varnothing.
\]
So we have $E_{c,h,\varphi}$ is a countable set. From
\[
E_{h,\varphi} = \bigcup_{c\in\mathbb{Q}}E_{c,h,\varphi}
\]
we have $E_{h,\varphi}$ is a countable set.
\end{proof}

The following proposition shows a corollary of main theorem.

\begin{coro}\label{cor1}
Let $\Omega$ be a domain of\/ $\mathbb{C}^{n}$ and $f: \Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Assuming that $h\in L^{1}(\dV_{2n}),h>0 $ on $\mathbb{B}\setminus{\{0\}}, h\in L^{\infty}(\mathbb{B})$. Then
\[
E_{ h,\log |f| }\subset\{z\in\Omega: f =0\}_{\sing},
\]
where $\{z\in\Omega: f =0\}_{\sing}$ is the singularities of the hypersurface $\{z\in\Omega: f =0\}$.
\end{coro}

\begin{proof}
By Theorem~\ref{dn1}, we have $E_{ h,\log |f| }$ is a countable subset of $\{ z\in\Omega: f = 0 \}$. Take $z_0\in \{z\in\Omega: f =0\}_{\reg}$. We only need to prove that $z_0\not\in E_{ h,\log |f| }$. Indeed, we have $f(z) = h^m$ in a neighborhood of the point $z_0$, where $\{z\in\Omega: f =0\}$ defined locally at the point $z_0$ by $h$. On the other hand, from the proof of Theorem~\ref{dn1} we have
\[
c_{\varphi} (z_0) \leq c_{h\dV_{2n},\varphi} (z_0) \leq \lim_{r\to 0} \min\{ c_{\varphi} (z) :\ z\in\mathbb B(z_0,r)\backslash \{ z_0 \} \}.
\]
Therefore
\[
c_{\log |f|} (z_0) = c_{h\dV_{2n},\log |f|} (z_0) =\frac 1 m.
\]
This implies that $z_0\not\in E_{ h,\log |f| }$.
\end{proof}

\begin{exam*}
We choose $f(z) = z_1^{m_1}+\dots+z_n^{m_n}$ and $h(z) = \|z\|^{2t}$ ($t>0$). Then
\begin{enumerate}%\romanenumi
\item $E_{ h,\log |f| } = \{0\}$ if $\sum_{j=1}^n \frac 1 {m_j} < 1$.
\item $E_{ h,\log |f| } =\emptyset$ if $\sum_{j=1}^n \frac 1 {m_j} \geq 1$.
\end{enumerate}
\end{exam*}

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