[proofplan]
We prove equality of sheaves by checking equality on every stalk. The inclusion from right to left follows immediately from monotonicity of the exponential weight, since plurisubharmonic functions are locally bounded above after shrinking coordinates. The reverse inclusion is precisely the local Guan-Zhou strong openness theorem: local square-integrability with weight $e^{-\varphi}$ improves to local square-integrability with weight $e^{-(1+\varepsilon)\varphi}$ for some $\varepsilon>0$. Applying this local result at each point gives equality of multiplier ideal stalks, hence equality of sheaves.
[/proofplan]
[step:Reduce the sheaf equality to equality of stalks]
For a point $x\in X$, let $\mathcal O_{X,x}$ denote the [local ring](/page/Local%20Ring) of [holomorphic function](/page/Holomorphic%20Function) germs at $x$, and let $\mathcal I(\varphi)_x$ denote the [stalk](/page/Stalk) at $x$ of the [multiplier ideal sheaf](/page/Multiplier%20Ideal%20Sheaf) associated to the [plurisubharmonic function](/page/Plurisubharmonic%20Function) $\varphi$. The stalk is
\begin{align*}
\mathcal I(\varphi)_x
=
\left\{ f_x\in \mathcal O_{X,x}: |f|^2 e^{-\varphi}\in L^1_{\mathrm{loc}}(U,\mathcal L^{2n})\text{ for one representative }f:U\to\mathbb C\right\},
\end{align*}
where $(U,z)$ is a holomorphic coordinate neighbourhood of $x$, $z:U\to z(U)\subseteq\mathbb C^n\cong\mathbb R^{2n}$ is the coordinate map, and local integrability is computed with respect to Lebesgue measure $\mathcal L^{2n}$ in these coordinates. This condition is independent of shrinking $U$: if the weighted square is integrable on compact subsets of one neighbourhood of $x$, then it remains so on every smaller neighbourhood, and conversely a representative on a smaller neighbourhood defines the same germ. The corresponding stalk of the right-hand side is
\begin{align*}
\left(\bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)\right)_x
=
\bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)_x,
\end{align*}
because stalks commute with filtered unions of subsheaves. Therefore it suffices to prove, for every $x\in X$,
\begin{align*}
\mathcal I(\varphi)_x
=
\bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)_x.
\end{align*}
[/step]
[step:Prove the immediate inclusion by monotonicity of the weight]
Fix $x\in X$, fix $\varepsilon>0$, and let $f_x\in \mathcal I((1+\varepsilon)\varphi)_x$. Choose a holomorphic representative
\begin{align*}
f:U&\to\mathbb C
\end{align*}
on a coordinate neighbourhood $U\subset X$ of $x$ such that $|f|^2e^{-(1+\varepsilon)\varphi}\in L^1_{\mathrm{loc}}(U,\mathcal L^{2n})$.
Since $\varphi$ is plurisubharmonic, it is upper semicontinuous and hence locally bounded above. After replacing $U$ by a smaller coordinate neighbourhood of $x$, choose $M\in\mathbb R$ such that $\varphi(y)\le M$ for all $y\in U$. Then
\begin{align*}
e^{-\varphi(y)}
&=e^{\varepsilon\varphi(y)}e^{-(1+\varepsilon)\varphi(y)}
\le e^{\varepsilon M}e^{-(1+\varepsilon)\varphi(y)}
\end{align*}
for all $y\in U$. Thus, for every compact set $K\subset U$,
\begin{align*}
\int_K |f(y)|^2e^{-\varphi(y)}\,d\mathcal L^{2n}(y)
&\le e^{\varepsilon M}\int_K |f(y)|^2e^{-(1+\varepsilon)\varphi(y)}\,d\mathcal L^{2n}(y)<\infty.
\end{align*}
Hence $f_x\in\mathcal I(\varphi)_x$, and therefore
\begin{align*}
\bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)_x\subseteq \mathcal I(\varphi)_x.
\end{align*}
[guided]
We first prove the direction that uses only elementary monotonicity. Fix $x\in X$, a number $\varepsilon>0$, and a germ $f_x\in\mathcal I((1+\varepsilon)\varphi)_x$. By the definition of the stalk, this means that there is a coordinate neighbourhood $U\subset X$ of $x$ and a holomorphic representative
\begin{align*}
f:U&\to\mathbb C
\end{align*}
such that $|f|^2e^{-(1+\varepsilon)\varphi}$ is locally integrable with respect to $\mathcal L^{2n}$ in the chosen complex coordinates.
The only point to check is that the weaker weight $e^{-\varphi}$ is controlled by the stronger weight $e^{-(1+\varepsilon)\varphi}$ on a sufficiently small neighbourhood. Plurisubharmonic functions are upper semicontinuous, so after shrinking $U$ around $x$ there is a real number $M\in\mathbb R$ satisfying $\varphi(y)\le M$ for every $y\in U$. Hence
\begin{align*}
e^{-\varphi(y)}
&=e^{\varepsilon\varphi(y)}e^{-(1+\varepsilon)\varphi(y)}
\le e^{\varepsilon M}e^{-(1+\varepsilon)\varphi(y)}.
\end{align*}
Multiplying by $|f(y)|^2$ and integrating over an arbitrary compact set $K\subset U$ gives
\begin{align*}
\int_K |f(y)|^2e^{-\varphi(y)}\,d\mathcal L^{2n}(y)
&\le e^{\varepsilon M}\int_K |f(y)|^2e^{-(1+\varepsilon)\varphi(y)}\,d\mathcal L^{2n}(y)<\infty.
\end{align*}
This verifies the defining local integrability condition for $f_x\in\mathcal I(\varphi)_x$. Since the argument works for every $\varepsilon>0$, we obtain
\begin{align*}
\bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)_x\subseteq \mathcal I(\varphi)_x.
\end{align*}
[/guided]
[/step]
[step:Apply the local Guan-Zhou strong openness theorem to obtain the reverse inclusion]
Fix $x\in X$ and let $f_x\in\mathcal I(\varphi)_x$. Choose a coordinate neighbourhood $(U,z)$ of $x$ and a holomorphic representative
\begin{align*}
f:U&\to\mathbb C
\end{align*}
such that $|f|^2e^{-\varphi}\in L^1_{\mathrm{loc}}(U,\mathcal L^{2n})$.
We apply the established local form of the [Guan-Zhou strong openness theorem](https://doi.org/10.4007/annals.2015.182.2.3) in the coordinate domain $z(U)\subseteq\mathbb C^n$: if $\Omega\subseteq\mathbb C^n$ is a domain, $\psi:\Omega\to[-\infty,\infty)$ is plurisubharmonic, $g:\Omega\to\mathbb C$ is holomorphic, and $|g|^2e^{-\psi}$ is locally integrable near a point $w_0\in\Omega$, then there are a number $\varepsilon>0$ and a neighbourhood $W$ of $w_0$ such that $|g|^2e^{-(1+\varepsilon)\psi}$ is integrable on compact subsets of $W$. Its hypotheses are satisfied here: $z(U)$ is a complex [Euclidean domain](/page/Euclidean%20Domain), $\varphi\circ z^{-1}:z(U)\to[-\infty,\infty)$ is plurisubharmonic because plurisubharmonicity is invariant under holomorphic coordinate changes, and $f\circ z^{-1}:z(U)\to\mathbb C$ is holomorphic with
\begin{align*}
\int_{z(K)} |f(z^{-1}(w))|^2 e^{-\varphi(z^{-1}(w))}\,d\mathcal L^{2n}(w)<\infty
\end{align*}
for every compact $K\subset U$, after shrinking $U$ so that the coordinate Jacobian is bounded above and below on compact subsets. The theorem yields a number $\varepsilon_x>0$ and a neighbourhood $V\subset U$ of $x$ such that
\begin{align*}
\int_K |f(y)|^2e^{-(1+\varepsilon_x)\varphi(y)}\,d\mathcal L^{2n}(y)<\infty
\end{align*}
for every compact set $K\subset V$. Therefore $f_x\in\mathcal I((1+\varepsilon_x)\varphi)_x$, and consequently
\begin{align*}
\mathcal I(\varphi)_x\subseteq \bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)_x.
\end{align*}
[guided]
Now we prove the substantive direction. Fix $x\in X$ and a germ $f_x\in\mathcal I(\varphi)_x$. By definition, there is a coordinate neighbourhood $(U,z)$ of $x$ and a holomorphic representative
\begin{align*}
f:U&\to\mathbb C
\end{align*}
for which $|f|^2e^{-\varphi}$ is locally integrable with respect to $\mathcal L^{2n}$.
The established local form of the [Guan-Zhou strong openness theorem](https://doi.org/10.4007/annals.2015.182.2.3) is the analytic input. It says that if $\Omega\subseteq\mathbb C^n$ is a domain, $\psi:\Omega\to[-\infty,\infty)$ is plurisubharmonic, $g:\Omega\to\mathbb C$ is holomorphic, and $|g|^2e^{-\psi}$ is locally integrable near a point $w_0\in\Omega$, then there exist a number $\varepsilon>0$ and a neighbourhood $W\subset\Omega$ of $w_0$ such that $|g|^2e^{-(1+\varepsilon)\psi}$ is integrable on compact subsets of $W$. This is a local Euclidean theorem already proved independently of the present sheaf-level statement, so invoking it here is not circular.
We verify that this local theorem applies. The coordinate image $z(U)$ is an open subset of $\mathbb C^n$. Define the coordinate representatives
\begin{align*}
\psi:z(U)&\to[-\infty,\infty), & w&\mapsto \varphi(z^{-1}(w)),\\
g:z(U)&\to\mathbb C, & w&\mapsto f(z^{-1}(w)).
\end{align*}
The function $\psi$ is plurisubharmonic because plurisubharmonicity is preserved by holomorphic coordinate changes, and $g$ is holomorphic because both $f$ and $z^{-1}$ are holomorphic. The local integrability hypothesis transfers through the coordinate map: after shrinking $U$ if needed, the real Jacobian determinant of the coordinate map and its inverse is bounded above and below on compact subsets, so local finiteness of
\begin{align*}
\int_K |f(y)|^2e^{-\varphi(y)}\,d\mathcal L^{2n}(y)
\end{align*}
for compact $K\subset U$ is equivalent to local finiteness of
\begin{align*}
\int_{z(K)} |g(w)|^2e^{-\psi(w)}\,d\mathcal L^{2n}(w).
\end{align*}
Thus all hypotheses of the local Guan-Zhou theorem are satisfied. It gives a number $\varepsilon_x>0$ and a neighbourhood $W\subset z(U)$ of $z(x)$ such that
\begin{align*}
\int_L |g(w)|^2e^{-(1+\varepsilon_x)\psi(w)}\,d\mathcal L^{2n}(w)<\infty
\end{align*}
for every compact set $L\subset W$. Returning to $V:=z^{-1}(W)\subset U$ and using the same coordinate-change comparison of Lebesgue measure gives
\begin{align*}
\int_K |f(y)|^2e^{-(1+\varepsilon_x)\varphi(y)}\,d\mathcal L^{2n}(y)<\infty
\end{align*}
for every compact set $K\subset V$. Hence $f_x\in\mathcal I((1+\varepsilon_x)\varphi)_x$, which proves
\begin{align*}
\mathcal I(\varphi)_x\subseteq \bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)_x.
\end{align*}
[/guided]
[/step]
[step:Conclude equality of multiplier ideal sheaves from equality of stalks]
For every point $x\in X$, the two inclusions proved above give
\begin{align*}
\mathcal I(\varphi)_x
=
\bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)_x.
\end{align*}
A morphism of [sheaves](/page/Sheaf) of ideals is an equality if and only if it is an equality on all stalks, because sections of a sheaf are determined locally by their germs. Therefore
\begin{align*}
\mathcal I(\varphi)=\bigcup_{\varepsilon>0}\mathcal I((1+\varepsilon)\varphi)
\end{align*}
as sheaves of ideals on $X$.
[/step]
