# Grauert-Riemenschneider multiplier ideal sheaves and the (optimal) Briançon-
Skoda number

Zhenqian Li

###### Abstract.

The goal of this note is to survey some recent results on the Grauert-
Riemenschneider multiplier ideal sheaves on any (reduced) complex space of
pure dimension. In particular, we obtain the Briançon-Skoda number for any
Noetherian ring of weakly holomorphic functions with weakly rational
singularities (_not_ essentially of finite type over $\mathbb{C}$ and Cohen-
Macaulay local rings), which will partially answer a question of Huneke.

###### 2010 Mathematics Subject Classification:

13B22, 14F18, 32B05, 32S05, 32U05

_Key words_. Integral closure of ideals, multiplier ideal sheaves, weakly
rational singularities, plurisubharmonic functions, Skoda’s $L^{2}$ division
theorem

E-mail<EMAIL_ADDRESS>
##  1\. Introduction

In [22], the author introduced a notion of Grauert-Riemenschneider multiplier
ideal sheaves on any (reduced) complex space of pure dimension (_not_
necessarily normal or $\mathbb{Q}$-Gorenstein), motivated by the definition of
Grauert-Riemenschneider canonical sheaf. Relying on the new multiplier ideals,
the author established the original Briançon-Skoda theorem for analytic local
rings with normal weakly rational singularities. In the present note, we are
going to investigate the Grauert-Riemenschneider multiplier ideals further and
prove some related properties (e.g., the weak holomorphicity and Skoda-type
division theorem, etc.). As applications, following the idea adopted in [22],
we also obtain the Briançon-Skoda number for Noetherian rings of _weakly
holomorphic functions_ (i.e., germs of locally bounded _meromorphic_ functions
on singular complex spaces) with weakly rational singularities, and give some
characterizations of validity of the original Briançon-Skoda theorem.

Throughout this note, all complex spaces are always assumed to be reduced and
paracompact, unless otherwise mentioned; we refer to [11, 32] for main
references on the theory of complex spaces.

###  1.1. Grauert-Riemenschneider multiplier ideal sheaves

Let $X$ be a complex space of pure dimension $n$ and $\varphi\in
L_{\text{loc}}^{1}(X_{\text{reg}})$ with respect to the Lebesgue measure.
Then, we can define the _Grauert-Riemenschneider multiplier ideal sheaf_
$\mathscr{I}_{\text{GR}}(\varphi)\subset\mathscr{M}_{X}$ associated to the
weight $\varphi$ on $X$ (see Definition 3.1). Moreover, we have the following:

######  Proposition 1.1.

_([22], Proposition 3.3)._ Let $\varphi\in\emph{Psh}(X)$ be a plurisubharmonic
function on $X$ such that $\varphi\not\equiv-\infty$ on every irreducible
component of $X$. Then,
$\mathscr{I}_{\emph{GR}}(\varphi)\subset\mathscr{M}_{X}$ is a coherent
fractional ideal sheaf and satisfies the strong openness property, i.e.,

$\mathscr{I}_{\emph{GR}}(\varphi)=\bigcup\limits_{\varepsilon>0}\mathscr{I}_{\emph{GR}}\big{(}(1+\varepsilon)\varphi\big{)}.$

Different from the usual multiplier ideal defined by integrability with
respect to the Lebesgue measure (cf. [8], Definition 5.4), the Grauert-
Riemenschneider multiplier ideal sheaf is in general a _fractional_ ideal
sheaf, which is an ideal sheaf when $X$ is a normal complex space and agrees
with the usual multiplier ideal if $X$ is smooth (cf. [22]). So it is
interesting for us to understand how “singular” the multiplier ideals might
be. Fortunately, the singularities encoded by Grauert-Riemenschneider
multiplier ideals are not too bad and in fact we have the following

######  Theorem 1.2.

With the same hypotheses as in Proposition 1.1, then we obtain that
$\mathscr{I}_{\emph{GR}}(\varphi)\subset\widehat{\mathcal{O}}_{X}$, the sheaf
of weakly holomorphic functions on $X$. In addition,
$\mathscr{I}_{\emph{GR}}(\varphi)$ is integrally closed in
$\widehat{\mathcal{O}}_{X}$.

Combining with a relative version of Grauert-Riemenschneider vanishing theorem
for the higher direct images, we formulate a Skoda-type division theorem for
Grauert-Riemenschneider multiplier ideal sheaves as follows (see also [22]).

######  Theorem 1.3.

Let $X$ be a complex space of pure dimension $n$ and $\varphi\in\emph{Psh}(X)$
such that $\varphi\not\equiv-\infty$ on every irreducible component of $X$.
Let $\mathfrak{a}\subset\widehat{\mathcal{O}}_{X}$ be a nonzero ideal sheaf
with $r$ (local) generators. Then, for any integer $k\geq q:=\min\\{n,r\\}$,
we have

$\mathscr{I}_{\emph{GR}}\big{(}\varphi+k\varphi_{\mathfrak{a}}\big{)}=\mathfrak{a}\cdot\mathscr{I}_{\emph{GR}}\big{(}\varphi+(k-1)\varphi_{\mathfrak{a}}\big{)},$

where $\varphi_{c\cdot\mathfrak{a}}:=\frac{c}{2}\log(\sum_{k}|g_{k}|^{2})$ for
each $c\geq 0$ and $(g_{k})$ is any local system of generators of
$\mathfrak{a}$.

######  Remark 1.4.

What we would like to mention is that the weight
$\varphi_{c\cdot\mathfrak{a}}\in L_{\text{loc}}^{1}(X_{\text{reg}})$ here is
not plurisubharmonic (_not_ even upper semi-continuous) on $X$ in general. Let
$\pi:\widetilde{X}\to X$ be any resolution of singularities of $X$. Then, we
obtain that $\varphi_{c\cdot\mathfrak{a}}\circ\pi$ is plurisubharmonic on
$\widetilde{X}$, which implies that $\varphi_{c\cdot\mathfrak{a}}$ is
plurisubharmonic if $\varphi_{c\cdot\mathfrak{a}}$ is upper semi-continuous by
Proposition 2.1 in [5] (see also [24] for more details).

###  1.2. The Briançon-Skoda theorem for weakly holomorphic functions

Let $\mathcal{I}$ be a nonzero ideal of the ring
$\mathcal{O}_{n}=\mathbb{C}\\{z_{1},...,z_{n}\\}$ of convergent power series
in $n$ variables, $g=(g_{1},...,g_{r})$ a system of generators of
$\mathcal{I}$ and $\overline{\mathcal{I}}$ the integral closure of
$\mathcal{I}$ in $\mathcal{O}_{n}$. In [4], Briançon and Skoda established the
following remarkable result on the comparison of powers of an ideal with the
integral closure of their powers, which confirmed a question posed by J.
Mather (see [8], §11.B):

Briançon-Skoda Theorem. If $q=\min\\{n,r\\}$, then
$\overline{\mathcal{I}^{k+q-1}}\subset\mathcal{I}^{k},\ k\in\mathbb{N}$.
Moreover, the number $q$ is optimal.

The original proof given by Briançon and Skoda is based on the so-called
Skoda’s $L^{2}$ division theorem for holomorphic functions (see [8], Theorem
11.13). From a purely algebraic viewpoint, the Briançon-Skoda theorem was
generalized to arbitrary regular rings by Lipman and Sathaye in [27], and
partially extended to pseudo-rational rings by Lipman and Teissier in [28].
Later on, Huneke [16] proved the following similar “uniform Briançon-Skoda”
for much more broad classes of Noetherian rings by the characteristic $p$
method relied on the theory of tight closure:

######  Theorem 1.5.

_([16], Theorem 4.13)._ Let $R$ be a Noetherian reduced ring. If $R$ satisfies
at least one of the following conditions, then there exists a positive integer
$\nu$ such that for all ideals $I$ of $R$, $\overline{I^{k+\nu}}\subset
I^{k},\ k\in\mathbb{N}$.

$(i)$ $R$ is essentially of finite type over an excellent Noetherian local
ring.

$(ii)$ $R$ is of characteristic $p$, and $R^{1/p}$ is module finite over $R$.

$(iii)$ $R$ is essentially of finite type over $\mathbb{Z}$.

Here the constant $\nu$ must be much larger than $\dim R$ in general and the
optimal such $\nu$ is called the _Briançon-Skoda number_ for the ring $R$; one
can refer to [34] and [1, 35] for more details on the above “uniform Briançon-
Skoda” as well. Specifically, in his paper, Huneke pointed that it is of great
interest to find what the Briançon-Skoda number is in Theorem 1.5 in terms of
invariants of the ring $R$ and such a number is just $\dim R-1$ for regular
$R$ by the original Briançon-Skoda theorem (cf. [16], Remark 4.14). Thus, if
we could establish the original Briançon-Skoda theorem for a class of
Noetherian rings, we would obtain the Briançon-Skoda number $\dim R-1$
immediately.

In order to establish the original Briançon-Skoda theorem for “singular”
Noetherian rings in the analytic setting, a natural idea is to generalize
Skoda’s $L^{2}$ division theorem for holomorphic functions to singular complex
spaces. Unfortunately, a trivial generalization to the singular case is not
possible due to Theorem A.1 in section A.1. In [22], the author established
the original Briançon-Skoda theorem for analytic local rings with normal
weakly rational singularities (_not_ essentially of finite type over
$\mathbb{C}$ and Cohen-Macaulay), by introducing the notion of Grauert-
Riemenschneider multiplier ideal sheaves. In this section, combining weak
holomorphicity of Grauert-Riemenschneider multiplier ideals with some
techniques adopted in [22], we are able to establish the following:

######  Theorem 1.6.

Let $X$ be an $n$-dimensional complex space with $x\in X$ a point and
$\mathcal{I}\subset\widehat{\mathcal{O}}_{X,x}$ a nonzero ideal with $r$
generators. If $x\in X$ is a weakly rational singularity, then the original
Briançon-Skoda theorem holds for the Noetherian ring
$\widehat{\mathcal{O}}_{X,x}$. In particular, the Briançon-Skoda number for
$\widehat{\mathcal{O}}_{X,x}$ is $n-1$.

######  Remark 1.7.

In [17], an algebraic analogue was established for the case that $R$ is normal
local Gorenstein and essentially of finite type over a field of characteristic
0, where the Cohen-Macaulayness of $R$ is necessary.

######  Remark 1.8.

On the other hand, relying on the Nadel-Ohsawa multiplier ideal sheaves
introduced in [23] and a version of Ohsawa-Takegoshi extension theorem
presented in [13], we can establish the following Briançon-Skoda type theorem:

Let $X$ be a complex space of pure dimension $n$ with $x\in X$ a singularity,
and $\mathcal{I}\subset\mathcal{O}_{X,x}$ a nonzero ideal with $r$ generators.
Set $q=\min\\{n,r\\}$. Then we have

$\overline{\mathcal{J}ac_{X,x}\cdot\mathcal{I}^{k+q-1}}\subset\mathcal{I}^{k},\
k\in\mathbb{N},$

where $\mathcal{J}ac_{X}$ is the Jacobian ideal of $X$.

As a direct consequence of Theorem 1.6, we can derive the main result in [22]:

######  Corollary 1.9.

_([22], Theorem 1.1)._ The original Briançon-Skoda theorem is valid for any
analytic local ring with normal weakly rational singularities.

Motivated by Theorem 1.6, a natural question is whether we could characterize
weakly rational singularities by the original Briançon-Skoda theorem.
Concretely, we raise the following

######  Question 1.1.

Let $X$ be complex space of pure dimension $n$ with $x\in X$ a point and
$\mathcal{I}\subset\widehat{\mathcal{O}}_{X,x}$ a nonzero ideal with $r$
generators.

Could we show that , $x\in X$ is a weakly rational singularity if and only if
the original Briançon-Skoda theorem holds for the Noetherian ring
$\widehat{\mathcal{O}}_{X,x}$?

When $n=1$, we will obtain

######  Theorem 1.10.

Let $X$ be a complex curve with $x\in X$ a point. Then, the original Briançon-
Skoda theorem holds for the Noetherian ring $\widehat{\mathcal{O}}_{X,x}$.
Moreover, the following statements are equivalent:

$(1)$ $x\in X$ is a weakly rational singularity.

$(2)$ $x\in X$ is a regular point.

$(3)$ The original Briançon-Skoda theorem holds for the local ring
$\mathcal{O}_{X,x}$; namely, $\overline{\mathcal{I}^{k}}=\mathcal{I}^{k}$ for
any nonzero ideal $\mathcal{I}\subset\mathcal{O}_{X,x}$ and $k\in\mathbb{N}$.

When $n=2$, the answer of Question 1.1 is positive, i.e.,

######  Theorem 1.11.

Let $X$ be a complex space of pure dimension two with $x\in X$ a point and
$\mathcal{I}\subset\widehat{\mathcal{O}}_{X,x}$ a nonzero ideal with $r$
generators. Then, $x\in X$ is a weakly rational singularity if and only if the
original Briançon-Skoda theorem holds for the Noetherian ring
$\widehat{\mathcal{O}}_{X,x}$.

Due to the Cohen-Macaulayness of normal complex surfaces, as an application of
Theorem 1.11, we have

######  Corollary 1.12.

_([22], Theorem 1.5)._ Let $X$ be a normal complex surface with $x\in X$ a
point. Then, $x\in X$ is a rational singularity if and only if the original
Briançon-Skoda theorem holds for the local ring $\mathcal{O}_{X,x}$.

######  Remark 1.13.

The above corollary does not hold when $\dim X\geq 3$; see [2] for some
examples of normal weakly rational singularities which are not rational.

##  2\. Terminologies and basic facts

Firstly, let us start by recalling some concepts and notations that will be
used throughout this paper. Various classes of the singularities of complex
spaces are given exactly as in the algebraic context (cf. [2, 19]).

######  Definition 2.1.

Let $X$ be a complex space of pure dimension $n$. The _Grothendieck canonical
sheaf_ (or _dualizing sheaf_) $\omega_{X}$ on $X$ is defined by

$\Gamma(U,\omega_{X})=\text{Ext}_{\mathcal{O}_{D}}^{N-n}(\mathcal{O}_{U},\Omega_{D}^{N}){\big{|}}_{U},$

where $U\subset X$ is an open subset isomorphic to a complex model space in a
domain $D$ of some $\mathbb{C}^{N}$. Moreover, $\omega_{X}$ is independent of
the local embedding; and $\omega_{X}\cong i_{*}(\omega_{X_{\text{reg}}})$ for
normal $X$, where $i:X_{\text{reg}}\hookrightarrow X$ is the natural
inclusion.

The _Grauert-Riemenschneider canonical sheaf_ $\omega_{X}^{\text{GR}}$ on $X$
is defined by

$\Gamma(U,\omega_{X}^{\text{GR}})=\\{\sigma\in\Gamma(U\cap
X_{\text{reg}},\omega_{X_{\text{reg}}})\ |\
(\sqrt{-1})^{n^{2}}\sigma\wedge\overline{\sigma}\in L_{\text{loc}}^{1}(U)\\}$

for any open subset $U\subset X$. Moreover, it follows that
$\omega_{X}^{\text{GR}}=\pi_{*}\omega_{\widetilde{X}}$ for any resolution of
singularities $\pi:\widetilde{X}\to X$ and
$\omega_{X}^{\text{GR}}\subset\omega_{X}$ (cf. [15]).

######  Remark 2.2.

(cf. [8], Remark 5.17). Given $\varphi\in L_{\text{loc}}^{1}(X_{\text{reg}})$,
we denote by $\omega_{X}^{\text{GR}}(\varphi)$ the $\mathcal{O}_{X}$-sheaf on
$X$ defined by

$\Gamma(U,\omega_{X}^{\text{GR}}(\varphi))=\\{\sigma\in\Gamma(U\cap
X_{\text{reg}},\omega_{X_{\text{reg}}})\ |\
(\sqrt{-1})^{n^{2}}e^{-2\varphi}\sigma\wedge\overline{\sigma}\in
L_{\text{loc}}^{1}(U)\\}$

for any open subset $U\subset X$. In particular, for any
$\varphi\in\text{Psh}(X)$, the $\mathcal{O}_{X}$-sheaf
$\omega_{X}^{\text{GR}}(\varphi)$ satisfies the functoriality property, i.e.,

$\pi_{*}\omega_{\widetilde{X}}^{\text{GR}}(\varphi\circ\pi)=\omega_{X}^{\text{GR}}(\varphi),$

where $\pi:\widetilde{X}\to X$ is any proper modification.

As a result, we have the following version of Nadel vanishing theorem on
complex spaces with singularities (cf. [8], Remark 5.17; also [31], Theorem
2.25).

######  Theorem 2.3.

Let $(X,\omega)$ be a weakly pseudoconvex Kähler space of pure dimension $n$.
Let $(L,e^{-2\varphi_{L}})$ be a holomorphic line bundle on $X$ equipped with
a singular Hermitian metric of plurisubharmonic weight $\varphi_{L}$ and
$\sqrt{-1}\partial\overline{\partial}\varphi_{L}\geq\varepsilon\omega$ for
some positive continuous function $\varepsilon$ on $X_{\text{\emph{reg}}}$.
Then,

$H^{q}\big{(}X,\omega_{X}^{\emph{GR}}(\varphi_{L})\otimes\mathcal{O}_{X}(L)\big{)}=0$

for all $q\geq 1$.

######  Definition 2.4.

(cf. [10, 18]). Let $X$ be a complex space of pure dimension. $X$ is said to
be _$\mathbb{Q}$ -Gorenstein_ if there exists a positive integer $r$ such that
$\omega_{X}^{[r]}:=(\omega_{X}^{\otimes r})^{**}$, the bidual of
$\omega_{X}^{\otimes r}$, is a locally free sheaf of rank one. We call the
minimum of such $r$ the _index of $X$_, and then $X$ is called _$r$
-Gorenstein_. $X$ is called _Gorenstein_ if it is $1$-Gorenstein and Cohen-
Macaulay.

A local generator $\sigma$ of $\omega_{X}^{[r]}$ defines a holomorphic
pluricanonical form without zeros on $X_{\text{reg}}$. In addition, if $X$ is
normal, it follows that
$\omega_{X}^{[r]}:=i_{*}(\omega_{X_{\text{reg}}}^{\otimes r})$, where
$i:X_{\text{reg}}\to X$ the natural inclusion.

######  Remark 2.5.

We need point out here that our definition of $\mathbb{Q}$-Gorenstein complex
space $X$ does not require the normality of $X$ as usual.

######  Definition 2.6.

(cf. [26]). Let $X$ be a $\mathbb{Q}$-Gorenstein complex space of dimension
$n$ and $\varphi\in L_{\text{loc}}^{1}(X_{\text{reg}})$ with respect to the
Lebesgue measure. The _multiplier ideal sheaf_ associated to $\varphi$ on $X$
is defined to be the $\mathcal{O}_{X}$-submudle
$\mathscr{I}(\varphi)\subset\mathscr{M}_{X}$ of germs of meromorphic functions
$f\in\mathscr{M}_{X,x}$ such that

$|f|^{2}e^{-2\varphi}\Big{(}(\sqrt{-1})^{rn^{2}}\sigma\wedge\overline{\sigma}\Big{)}^{\frac{1}{r}}$

is locally integrable at $x$, where $\sigma$ is a local generator of
$\omega_{X}^{[r]}$ at $x$ and $r$ is the index near $x$. Here, $\varphi$ is
regarded as the weight function of $\mathscr{I}(\varphi)$. If
$\varphi\equiv-\infty$ on $X$, we put $\mathscr{I}(\varphi)=0$.

######  Remark 2.7.

When $X$ is smooth and $\varphi\in\text{Psh}(X)$, it follows that
$\mathscr{I}(\varphi)\subset\mathcal{O}_{X}$ is nothing but the multiplier
ideal sheaf introduced by Nadel (see also [8]).

######  Definition 2.8.

(cf. [2]). Let $X$ be a complex space of pure dimension $n$ and $x\in X$ a
point. Let $\pi:\widetilde{X}\to X$ be a resolution of singularities of $X$.

The point $x$ is called a _weakly rational singularity_ of $X$ if
$\omega_{X}=\omega_{X}^{\text{GR}}$ near the point $x$, i.e., every
holomorphic section of $\omega_{X}$ on a neighborhood of $x$ is locally square
integrable around $x$.

The point $x$ is called a _rational singularity_ of $X$ if one of the
following equivalent conditions holds:

$(1)$ $\omega_{X}=\omega_{X}^{\text{GR}}$ near the point $x$ and
$\mathcal{O}_{X,x}$ is a Cohen-Macaulay local ring;

$(2)$ $x$ is a normal point of $X$ and
$(R^{q}\pi_{*}\mathcal{O}_{\widetilde{X}})_{x}=0$ for all $q>0$.

We say that $X$ has at most (weakly) rational singularities if each point in
$X$ is a (weakly) rational singularity.

######  Remark 2.9.

(cf. [2]). $(1)$ The above definition is independent of the resolution $\pi$
and every regular point of $X$ is a (weakly) rational singularity.

$(2)$ If $n=1$, then $x$ is a weakly rational singularity iff $x$ is a
rational singularity iff $x$ is a regular point of $X$.

$(3)$ If $n\geq 2$, then $x$ is a weakly rational singularity iff
$(R^{n-1}\pi_{*}\mathcal{O}_{\widetilde{X}})_{x}=0$.

######  Proposition 2.10.

Let $X$ be a complex space of pure dimension $n\ (n\geq 2)$ and let
$\xi:\widehat{X}\to X$ be a one-sheeted analytic covering of $X$. Then, $X$
has at most weakly rational singularities if and only if $\widehat{X}$ has at
most weakly rational singularities. In further, if $X$ has at most weakly
rational singularities, we have $\omega_{X}=\xi_{*}\omega_{\widehat{X}}$.

###### Proof.

Since $\xi$ is a finite holomorphic mapping, then for any $x\in X$,
$\xi^{-1}(x)$ has a fundamental system of neighborhoods consisting of open
sets of the form $\xi^{-1}(U)$ with open set $U\subset X$ containing $x$.
Moreover, if $\hat{x}_{1},...,\hat{x}_{m}$ are the distinct points of a fiber
$\xi^{-1}(x)$, then it follows that there exists sufficiently small open
neighborhood $U$ of $x$ in $X$ such that
$\xi^{-1}(U)=\bigcup_{k=1}^{m}\widehat{U}_{k}$, where
$\widehat{U}_{1},...\widehat{U}_{m}$ are pairwise disjoint open neighborhoods
of $\hat{x}_{1},...,\hat{x}_{m}$ in $\widehat{X}$.

Let $\pi:\widetilde{X}\to\widehat{X}$ be a resolution of singularities of
$\widehat{X}$ and then $\xi\circ\pi:\widetilde{X}\to X$ is a desingularization
of $X$. Note that for each point $x\in X$ and any sufficiently small
neighborhood $U$ of $x$ in $X$, we have

$\begin{split}\left(R^{n-1}(\xi\circ\pi)_{*}\mathcal{O}_{\widetilde{X}}\right)(U)&=H^{n-1}\left(\pi^{-1}\big{(}\xi^{-1}(U)\big{)},\mathcal{O}_{\widetilde{X}}\right)\\\
&=\bigoplus_{k=1}^{m}H^{n-1}\left(\pi^{-1}(\widehat{U}_{k}),\mathcal{O}_{\widetilde{X}}\right)\\\
&=\bigoplus_{k=1}^{m}\left(R^{n-1}\pi_{*}\mathcal{O}_{\widetilde{X}}\right)(\widehat{U}_{k}).\end{split}$

Thus, it follows from $(3)$ of Remark 2.9 that $x\in X$ is a weakly rational
singularity if and only if all of points $\hat{x}_{1},...,\hat{x}_{m}$ are
weakly rational singularities of $\widehat{X}$. In further, if $X$ has at most
weakly rational singularities, by the definition, we can obtain

$\omega_{X}=\omega_{X}^{\text{GR}}=(\xi\circ\pi)_{*}\omega_{\widetilde{X}}=\xi_{*}(\pi_{*}\omega_{\widetilde{X}})=\xi_{*}\omega_{\widehat{X}}.$

∎

For our proof of Theorem 1.3, we need the following relative version of
Grauert-Riemenschneider vanishing theorem for the higher direct images as
well.

######  Theorem 2.11.

_([29], Corollary 1.5)._ Let $\pi:X\to Y$ be a surjective proper (locally)
Kähler morphism from a complex manifold $X$ to a complex space $Y$, and
$(L,e^{-2\varphi_{L}})$ be a (possibly singular) Hermitian line bundle on $X$
with semi-positive curvature. Then, the higher direct image sheaf

$R^{j}\pi_{*}\Big{(}\omega_{X}\otimes\mathcal{O}_{X}(L)\otimes\mathscr{I}(\varphi_{L})\Big{)}=0,$

for every $j>\dim X-\dim Y$.

For the sake of convenience, we state the following characterization on the
slope of plurisubharmonic functions (cf. Corollary 9.3 in [3]):

######  Lemma 2.12.

Let $X$ be a locally irreducible complex space and $\varphi\in\emph{Psh}(X)$
with the slope $\nu_{x}(\varphi)=0$ for some point $x\in X$. Then, for any
resolution of singularities $\pi:\widetilde{X}\to X$ of $X$,
$\nu_{\tilde{x}}(\varphi\circ\pi)=0$ for each $\tilde{x}\in\pi^{-1}(x)$.

Here, the slope of $\varphi$ at $x$ is defined by

$\nu_{x}(\varphi):=\sup\left\\{\gamma\geq 0\ \big{|}\
\varphi\leq\gamma\log\sum_{k}|f_{k}|+O(1)\right\\}\in[0,+\infty),$

where $(f_{k})$ are local generators of the maximal ideal $\mathfrak{m}_{x}$
of $\mathcal{O}_{X,x}$.

######  Remark 2.13.

If $X$ is a complex manifold, the slope $\nu_{x}(\varphi)$ is exactly the
usual Lelong number of $\varphi$ at $x$ and the inequality
$\nu_{x}(\varphi)<1$ implies the integrability of $e^{-2\varphi}$ near $x$ by
a result of Skoda (see Lemma 5.6 in [8]).

Next, we briefly review some relevant algebraic notions and theorems; one can
refer to [34].

######  Definition 2.14.

Let $R$ be a commutative ring and $I$ an ideal of $R$. An element $h\in R$ is
said to be _integrally dependent_ on $I$ if it satisfies a relation

$h^{d}+a_{1}h^{d-1}+\cdots+a_{d}=0\quad(a_{k}\in{I}^{k},1\leq k\leq d).$

The set $\overline{I}$ consisting of all elements in $R$ which are integrally
dependent on $I$ is called the _integral closure_ of $I$ in $R$. $I$ is called
_integrally closed_ if $I=\overline{I}$. One can prove that $\overline{I}$ is
an ideal of $R$, which is the smallest integrally closed ideal in $R$
containing $I$.

######  Definition 2.15.

Let $R$ be a commutative ring with identity and let $J\subset I$ be ideals in
$R$. $J$ is said to be a _reduction_ of $I$ if there exists a nonnegative
integer $n$ such that $I^{n+1}=JI^{n}$.

A reduction $J$ of $I$ is called _minimal_ if no ideal strictly contained in
$J$ is a reduction of $I$. An ideal that has no reduction other than itself is
called _basic_.

One can prove that minimal reductions do exist in Noetherian local rings and
an ideal which is a minimal reduction of a given ideal is necessarily basic.
Moreover, if $R$ is a Noetherian ring, $J\subset I$ is a reduction of $I$ if
and only if $\overline{J}=\overline{I}$.

######  Theorem 2.16.

_([34], Proposition 8.3.7 & Corollary 8.3.9)._ Let $(R,\mathfrak{m})$ be a
Noetherian local ring with infinite residue field and $I\subset R$ an ideal.
Then every minimal reduction of $I$ can be generated by exactly $l$ (the
analytic spread of $I$) elements with $l\leq\dim R$.

In the analytic setting, we have the following characterization of integral
closure of ideals.

######  Theorem 2.17.

_(cf.[21], Théorème 2.1)._ Let $X$ be a complex space, $Y$ a proper closed
complex subspace (may be non-reduced) of $X$ defined by a coherent
$\mathcal{O}_{X}$-ideal $\mathscr{I}$ and $x\in Y$ a point. Let $\mathscr{J}$
be a coherent $\mathcal{O}_{X}$-ideal in $\mathcal{O}_{X}$, $\mathcal{I}$
(resp. $\mathcal{J}$) the germ of $\mathscr{I}$ (resp. $\mathscr{J}$) at $x$.
Then the following conditions are equivalent:

  1. (1)

$\mathcal{J}\subset\overline{\mathcal{I}}$;

  2. (2)

For every morphism $\pi:X^{\prime}\to X$ satisfying: (a) $\pi$ is a proper and
surjective, (b) $X^{\prime}$ is a normal complex space, (c)
$\mathscr{I}\cdot\mathcal{O}_{X^{\prime}}$ is an invertible
$\mathcal{O}_{X}$-module, there exists an open neighborhood $U$ of $x$ in $X$
such that

$\mathscr{J}\cdot\mathcal{O}_{X^{\prime}}|_{\pi^{-1}(U)}\subset\mathscr{I}\cdot\mathcal{O}_{X^{\prime}}|_{\pi^{-1}(U)};$

  3. (3)

If $V$ is an open neighborhood of $x$ on which $\mathscr{I}$ and $\mathscr{J}$
are generated by their global sections. For every system of generators
$g_{1},...,g_{r}\in\Gamma(V,\mathscr{I})$ and every
$f\in\Gamma(V,\mathscr{J})$, one can find an open neighborhood $V^{\prime}$ of
$x$ and a constant $C>0$ such that

$|f(y)|\leq C\cdot\sup_{k}|g_{k}(y)|,\quad\forall y\in V^{\prime}.$

Combining with the existence of universal denominator for the sheaf
$\widehat{\mathcal{O}}_{X}$ of weakly holomorphic functions on $X$, we can
deduce a characterization of the integral closures of ideals in
$\widehat{\mathcal{O}}_{X}$ as follows.

######  Corollary 2.18.

Let $X$ be a complex space with $x\in X$ a point,
$\mathcal{I}=(h_{1},...,h_{r})\cdot\widehat{\mathcal{O}}_{X,x}$ an ideal in
$\widehat{\mathcal{O}}_{X,x}$ and $\boldsymbol{\mathit{u}}$ a universal
denominator for $\widehat{\mathcal{O}}_{X}$ near $x$.

Then, for any $f\in\widehat{\mathcal{O}}_{X,x}$, it follows that
$f\in\overline{\mathcal{I}}$, the integral closure of $\mathcal{I}$ in
$\widehat{\mathcal{O}}_{X,x}$, if and only if there is a neighborhood $W$ of
$x$ in $X$ and a constant $C>0$ such that

$|\boldsymbol{\mathit{u}}\cdot f(y)|\leq
C\cdot\sup_{k}|\boldsymbol{\mathit{u}}\cdot h_{k}(y)|,\quad\forall y\in W.$

######  Remark 2.19.

Let $X$ be a complex space of pure dimension and
$\mathscr{I}\subset\mathcal{O}_{X}$ a coherent ideal sheaf. Let
$\pi:\widetilde{X}\to X$ be any proper modification from a normal complex
space $\widetilde{X}$ onto $X$ such that
$\mathscr{I}\cdot\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-D)$
for some effective Cartier divisor $D$ on $\widetilde{X}$. Then, we have
$\pi_{*}\mathcal{O}_{\widetilde{X}}(-D)=\overline{\mathscr{I}}$, the integral
closure of $\mathscr{I}$ in $\widehat{\mathcal{O}}_{X}$.

##  3\. Grauert-Riemenschneider multiplier ideal sheaves on
complex spaces with singularities

In this section, we firstly recall the concept of multiplier ideal sheaves
associated to plurisubharmonic functions on complex spaces with singularities
(_not_ necessarily normal or $\mathbb{Q}$-Gorenstein), which will play a
crucial role in this paper. Motivated by the definition of Grauert-
Riemenschneider canonical sheaf, we construct the multiplier ideals on complex
spaces of pure dimension as follows.

######  Definition 3.1.

([22], Definition 3.1). Let $X$ be a complex space of pure dimension $n$ and
$\varphi\in L_{\text{loc}}^{1}(X_{\text{reg}})$ with respect to the Lebesgue
measure. The _Grauert-Riemenschneider multiplier ideal sheaf_ associated to
the weight $\varphi$ on $X$ is defined to be the $\mathcal{O}_{X}$-submodule
$\mathscr{I}_{\text{GR}}(\varphi)\subset\mathscr{M}_{X}$ of germs of
meromorphic functions $f\in\mathscr{M}_{X,x}$ such that for any holomorphic
section $\sigma$ of $\omega_{X}$ on a neighborhood of $x$, we have

$\Big{(}\sqrt{-1}\Big{)}^{n^{2}}|f|^{2}e^{-2\varphi}\sigma\wedge\overline{\sigma}$

is locally integrable at $x$. In addition, we put
$\mathscr{I}_{\text{GR}}(\varphi)=0$ if $\varphi\equiv-\infty$ on $X$.

######  Remark 3.2.

(Local description, cf. [22]). Let $X$ be a (closed) complex subspace of some
domain $\Omega$ in $\mathbb{C}^{N}$ and $Z\supset X$ is an $n$-dimensional
(reduced) complete intersection (shrinking $\Omega$ if necessary). In
particular, the dualizing sheaf $\omega_{Z}$ is free on $Z$ and so is
$\omega_{Z}|_{X}$ on $X$. Moreover, it follows that the image of the inclusion
$\omega_{X}\hookrightarrow\omega_{Z}|_{X}$ is given by

$\text{Im}\left(\omega_{X}\hookrightarrow\omega_{Z}|_{X}\right)=\mathfrak{d}_{X,Z}\otimes\omega_{Z}|_{X};$

in further, we can obtain
$\mathfrak{d}_{X,Z}=\big{(}(\mathscr{I}_{Z}:\mathscr{I}_{X})+\mathscr{I}_{X}\big{)}/\mathscr{I}_{X}$
(see [6] for more details).

Let $\pi:\widetilde{X}\to X$ be a resolution of singularities of the blow-up
of $X$ along the ideal sheaf $\mathfrak{d}_{X,Z}$ such that
$\mathfrak{d}_{X,Z}\cdot\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-D_{Z})$
for some effective divisor $D_{Z}$ on $\widetilde{X}$. Then, for any
$f\in\mathscr{I}_{\text{GR}}(\varphi)_{x}$ defined on a small enough
neighborhood $U$ of $x$ in $X$ and each holomorphic section $\sigma$ of
$\omega_{X}$ on a neighborhood of $x$, we have

$\int_{U}\left(\sqrt{-1}\right)^{n^{2}}|f|^{2}e^{-2\varphi}\sigma\wedge\overline{\sigma}=\int_{\pi^{-1}(U)}\left(\sqrt{-1}\right)^{n^{2}}|f\circ\pi|^{2}e^{-2\varphi\circ\pi}\pi^{*}\sigma\wedge\overline{\pi^{*}\sigma}<+\infty,$

which implies that

$None$
$\mathscr{I}_{\text{GR}}(\varphi)=\pi_{*}\left(\omega_{\widetilde{X}}\otimes\pi^{*}(\omega_{Z}^{-1}|_{X})\otimes\mathcal{O}_{\widetilde{X}}(D_{Z})\otimes\mathscr{I}(\varphi\circ\pi)\right).$

######  Remark 3.3.

If $X$ is a $\mathbb{Q}$-Gorenstein complex space, by a similar argument on
the discrepancies as Proposition 3.4 in [6] (see also [9], Proposition 2.21),
it follows that

$\mathscr{I}_{\text{NO}}(\varphi)\subset\mathscr{I}(\varphi)\subset\mathscr{I}_{\text{GR}}(\varphi),$

where $\mathscr{I}_{\text{NO}}(\varphi)$ is the Nadel-Ohsawa multiplier ideal
sheaf on $X$ introduced in [23] and both of inclusions may be strict; see also
[33] for an algebraic counterpart.

######  Theorem 3.4.

_(=Theorem 1.2)._ With the same hypotheses as above, then we obtain that
$\mathscr{I}_{\emph{GR}}(\varphi)\subset\widehat{\mathcal{O}}_{X}$, and
$\mathscr{I}_{\emph{GR}}(\varphi)$ is integrally closed in
$\widehat{\mathcal{O}}_{X}$.

###### Proof.

The integral closedness of $\mathscr{I}_{\text{GR}}(\varphi)$ in
$\widehat{\mathcal{O}}_{X}$ is straightforward by Corollary 2.18. Next, we
only present the inclusion
$\mathscr{I}_{\text{GR}}(\varphi)_{x}\subset\widehat{\mathcal{O}}_{X,x}$ for
each point $x\in X$. Without loss of generality, we may assume $\varphi=0$.
Let $\pi:\widetilde{X}\to X$ be a resolution of singularities of $X$ with
exceptional locus $\text{Ex}(\pi)$ and
$f\in\mathscr{I}_{\text{GR}}(\varphi)_{x}\subset\mathscr{M}_{X,x}$.

Suppose that $f\notin\widehat{\mathcal{O}}_{X,x}$, then $f\circ\pi$ is a
meromorphic function with nontrivial poles contained in $\text{Ex}(\pi)$ on
$\widetilde{X}$ after shrinking $X$ and denote by $E$ an irreducible component
of the poles of $f\circ\pi$. Let $\sigma_{1},...,\sigma_{r}$ be a system of
generators of $\omega_{X,x}$, and then the local integrability of
$|f|^{2}\sigma_{k}\wedge\overline{\sigma_{k}}\ (1\leq k\leq r)$ near $x$
implies that $\pi^{*}\sigma_{k}$ is a meromorphic $n$-form on $\widetilde{X}$,
which vanishes on $E$. Take $\pi^{*}\sigma_{k_{0}}$ to be the one which has
the least vanishing order along $E$ among all $\pi^{*}\sigma_{k}$. Thus, the
vanishing order $\text{ord}_{E}\pi^{*}(f\cdot\sigma_{k_{0})}$ of
$\pi^{*}(f\cdot\sigma_{k_{0}})$ along $E$ is at most
$\text{ord}_{E}\pi^{*}\sigma_{k_{0}}-1$.

On the other hand, note that
$f\cdot\sigma_{k_{0}}\in\omega_{X,x}^{\text{GR}}\subset\omega_{X,x}$, i.e.,
$f\cdot\sigma_{k_{0}}$ can be generated by $\sigma_{1},...,\sigma_{r}$ near
$x$, and then it implies that

$\text{ord}_{E}\pi^{*}(f\cdot\sigma_{k_{0})}\geq\text{ord}_{E}\pi^{*}\sigma_{k_{0}},$

which is a contradiction; the proof is concluded. ∎

######  Remark 3.5.

For the case that $\text{codim}\,X_{\text{sing}}\geq 2$, we can directly
obtain the above result by the Riemann extension theorem on locally pure
dimensional complex spaces (cf. [11]).

As a consequence, we can deduce the following characterization of weakly
rational singularities by the Grauert-Riemenschneider multiplier ideals.

######  Proposition 3.6.

Let $X$ be a locally irreducible complex space of dimension $n$ and
$\varphi\in\emph{Psh}(X)$ with the slope $\nu_{x}(\varphi)=0$ for every point
$x\in X$. Then, $X$ has at most weakly rational singularities if and only if
$\mathscr{I}_{\emph{GR}}(\varphi)=\widehat{\mathcal{O}}_{X}$.

###### Proof.

Let $\pi:\widetilde{X}\to X$ be a resolution of singularities. If $X$ has at
most weakly rational singularities, then for any holomorphic section $\sigma$
of $\omega_{X}$, it follows that $\pi^{*}\sigma$ is a holomorphic $n$-form on
$\widetilde{X}$. Thus, the necessity is a direct application of Lemma 2.12 and
Remark 2.13. The sufficiency is straightforward since
$\mathcal{O}_{X}\subset\widehat{\mathcal{O}}_{X}$ and $\varphi$ is upper semi-
continuous on $X$. ∎

######  Remark 3.7.

$(1)$ If $\varphi$ is locally bounded, the statement is also valid for any
complex space of pure dimension.

$(2)$ In particular, if $X$ is a normal complex space, then $X$ has at most
weakly rational singularities if and only if
$\mathscr{I}_{\text{GR}}(\varphi)=\mathcal{O}_{X}$, which implies that on
normal complex spaces with weakly rational singularities, the plurisubharmonic
weights with zero slope do not add the singularities in the sense of Grauert-
Riemenschneider multiplier ideals.

Let $X|_{U}\hookrightarrow\Omega\subset\mathbb{C}^{N}$ be any local embedding
of $X$ for some open set $U\subset X$ and $Z\supset X|_{U}$ an $n$-dimensional
(reduced) complete intersection. Note that

$\text{Im}\left(\omega_{X}\hookrightarrow\omega_{Z}|_{X}\right)=\mathfrak{d}_{X,Z}\otimes\omega_{Z}|_{X},$

and then we can deduce that
$\omega_{Z}|_{X}\otimes\mathscr{I}_{\text{GR}}(\varphi+\log|\mathfrak{d}_{X,Z}|)$
globally defines a coherent analytic sheaf on $X$ for any
$\varphi\in\text{Psh}(X)$. Thus, relying on the multiplier ideals defined
above, we will derive a form of Nadel vanishing theorem on complex spaces with
singularities as follows.

######  Proposition 3.8.

Let $(X,\omega)$ be a weakly pseudoconvex Kähler space of pure dimension $n$.
Let $(L,e^{-2\varphi_{L}})$ be a holomorphic line bundle on $X$ equipped with
a singular Hermitian metric of plurisubharmonic weight $\varphi_{L}$ and
$\sqrt{-1}\partial\overline{\partial}\varphi_{L}\geq\varepsilon\omega$ for
some positive continuous function $\varepsilon$ on $X_{\text{\emph{reg}}}$.
Then,

$H^{q}\big{(}X,\omega_{Z}|_{X}\otimes\mathscr{I}_{\emph{GR}}(\varphi_{L}+\log|\mathfrak{d}_{X,Z}|)\otimes\mathcal{O}_{X}(L)\big{)}=0$

for all $q\geq 1$, where $X$ is locally contained in an $n$-dimensional
complete intersection $Z$ and
$\omega_{X}=\mathfrak{d}_{X,Z}\otimes\omega_{Z}|_{X}$.

###### Proof.

Thanks to Theorem 2.3, it is sufficient to prove

$\omega_{X}^{\text{GR}}(\varphi_{L})=\omega_{Z}|_{X}\otimes\mathscr{I}_{\text{GR}}(\varphi_{L}+\log|\mathfrak{d}_{X,Z}|).$

Let $\pi:\widetilde{X}\to X$ be a resolution of singularities of the blow-up
of $X$ along the ideal sheaf $\mathfrak{d}_{X,Z}$, and
$\mathfrak{d}_{X,Z}\cdot\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-D_{Z})$
for some effective divisor $D_{Z}$ (not necessarily SNC) on $\widetilde{X}$.
Then, it follows from the projection formula and $(\spadesuit)$ that

$\begin{split}\omega_{X}^{\text{GR}}(\varphi_{L})&=\pi_{*}\left(\omega_{\widetilde{X}}\otimes\mathscr{I}(\varphi_{L}\circ\pi)\right)\\\
&=\pi_{*}\left(\pi^{*}(\omega_{Z}|_{X})\otimes\mathcal{O}_{\widetilde{X}}(-D_{Z})\otimes\omega_{\widetilde{X}}\otimes\pi^{*}(\omega_{Z}^{-1}|_{X})\otimes\mathcal{O}_{\widetilde{X}}(D_{Z})\otimes\mathscr{I}(\varphi_{L}\circ\pi)\right)\\\
&=\pi_{*}\left(\pi^{*}(\omega_{Z}|_{X})\otimes\omega_{\widetilde{X}}\otimes\pi^{*}(\omega_{Z}^{-1}|_{X})\otimes\mathcal{O}_{\widetilde{X}}(D_{Z})\otimes\mathscr{I}(\varphi_{L}\circ\pi+\log|\mathfrak{d}_{X,Z}|\circ\pi)\right)\\\
&=\omega_{Z}|_{X}\otimes\pi_{*}\left(\omega_{\widetilde{X}}\otimes\pi^{*}(\omega_{Z}^{-1}|_{X})\otimes\mathcal{O}_{\widetilde{X}}(D_{Z})\otimes\mathscr{I}(\varphi_{L}\circ\pi+\log|\mathfrak{d}_{X,Z}|\circ\pi)\right)\\\
&=\omega_{Z}|_{X}\otimes\mathscr{I}_{\text{GR}}(\varphi_{L}+\log|\mathfrak{d}_{X,Z}|).\end{split}$

∎

Note when $X$ is 1-Gorenstein, one can infer that

$\omega_{Z}|_{X}\otimes\mathscr{I}_{\text{GR}}(\varphi_{L}+\log|\mathfrak{d}_{X,Z}|)=\omega_{X}\otimes\mathscr{I}_{\text{GR}}(\varphi_{L}).$

Thus, the above result immediately implies that

######  Corollary 3.9.

With the same hypotheses as above, we in addition assume that $X$ is a
1-Gorenstein complex space. Then, we have

$H^{q}\big{(}X,\omega_{X}\otimes\mathcal{O}_{X}(L)\otimes\mathscr{I}_{\emph{GR}}(\varphi_{L})\big{)}=0$

for all $q\geq 1$.

As a result, by combining with Richberg’s result [32], we will deduce a
solution to the Levi problem on $1$-Gorenstein complex spaces with weakly
rational singularities.

######  Corollary 3.10.

Let $X$ be a $1$-Gorenstein complex space of dimension $n$ with weakly
rational singularities. Then, the following statements are equivalent:

$(1)$ There is a continuous strictly plurisubharmonic exhaustion function on
$X$.

$(2)$ $X$ is a Stein space.

###### Proof.

"$(2)\Rightarrow(1)$". Since $X$ is a Stein space, it carries a real analytic
strictly plurisubharmonic exhaustion function (cf. [30]).

"$(1)\Rightarrow(2)$". By Richberg’s result, there exists a smooth strictly
plurisubharmonic exhaustion function $\psi$ on $X$, which implies $X$ is a
Kähler space. Moreover, since $X$ is $1$-Gorenstein, we have
$\mathscr{I}_{\text{GR}}(\varphi)\subset\mathcal{O}_{X}$ for any
$\varphi\in\text{Psh}(X)$.

Let $(x_{k})\subset X$ be any discrete sequence of points and $u$ a quasi-
plurisubharmonic function on $X$ such that

$u\sim c_{k}\log(|f_{k1}|+\cdots+|f_{kl_{k}}|)$

near each $x_{k}$, where $(f_{k1},...,f_{kl_{k}})$ are generators of the
maximal ideal $\mathfrak{m}_{x_{k}}$ and $c_{k}\gg 0$ such that

$\text{Supp}(\mathcal{O}_{X}/\mathscr{I}_{\text{GR}}(u))=\\{x_{k}\\}$

(in fact, $c_{k}\geq n$ is enough); here, the weak rationality of
singularities of $X$ is necessary. Take a holomorphic line bundle $L$ on $X$
with $\mathcal{O}_{X}(L^{-1})=\omega_{X}$ and a singular Hermitian metric on
$L$ of the form $h_{L}=h_{0}e^{-2(u+\chi\circ\psi)}$, where $h_{0}$ is a
smooth metric on $L$ and $\chi$ is a convex increasing function of arbitrary
fast growth at infinity such that

$\sqrt{-1}\Theta_{L,h_{L}}=\sqrt{-1}\left(\Theta_{L,h_{0}}+2\partial\overline{\partial}(u+\chi\circ\psi)\right)\geq\varepsilon\omega$

for some positive continuous function $\varepsilon$ on $X_{\text{reg}}$. Then,
it follows that $\mathscr{I}_{\text{GR}}(h_{L})=\mathscr{I}_{\text{GR}}(u)$.

Consider the long exact sequence of cohomology associated to the short exact
sequence

$0\rightarrow\mathscr{I}_{\text{GR}}(h_{L})\rightarrow\mathcal{O}_{X}\rightarrow\mathcal{O}_{X}/\mathscr{I}_{\text{GR}}(h_{L})\rightarrow
0$

twisted by $\omega_{X}\otimes\mathcal{O}_{X}(L)$. Applying Corollary 3.9, we
obtain the surjectivity

$H^{0}\big{(}X,\mathcal{O}_{X}\big{)}\twoheadrightarrow
H^{0}\big{(}X,\mathcal{O}_{X}/\mathscr{I}_{\text{GR}}(u)\big{)}$

by our choice of $L$, which implies that $X$ is holomorphically separable and
convex, i.e., $X$ is a Stein space. ∎

##  4\. Proofs of main results

###  4.1. Proof of Theorem 1.3

Firstly, we state a result on the existence of reductions for coherent
(fractional) ideal sheaves on complex spaces.

######  Lemma 4.1.

Let $X$ be a complex space of dimension $n$ and
$\mathfrak{a}\subset\widehat{\mathcal{O}}_{X}$ a non-zero ideal. Then there
exists an open covering $\\{U_{\alpha}\\}_{\alpha\in\mathbb{N}}$ of $X$ such
that $\mathfrak{a}|_{U_{\alpha}}$ has a reduction $\mathfrak{b}_{\alpha}$
generated by at most $n$ elements.

Thanks to Corollary 2.18, the proof is similar to the argument of Example
9.6.19 in [20] by considering the normalized blow-up of the normalization of
$X$, where an analytic Bertini-type theorem is necessary.

Proof of Theorem 1.3. By the definition of Grauert-Riemenschneider multiplier
ideals, we can deduce the inclusion

$\mathfrak{a}\cdot\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-1)\varphi_{\mathfrak{a}}\big{)}\subset\mathscr{I}_{\text{GR}}\big{(}\varphi+k\varphi_{\mathfrak{a}}\big{)}.$

Thus, in order to prove Theorem 1.3, it is sufficient to show the reverse
inclusion.

As the question is local, without loss of generality, we may assume that $X$
is a (closed) complex subspace of some domain $\Omega$ in $\mathbb{C}^{N}$ and
$Z\supset X$ is an $n$-dimensional (reduced) complete intersection.

Case (i). When $r\leq n$, i.e., $q=r$.

Let $g=(g_{1},\dots,g_{r})$ be a system of generators of $\mathfrak{a}$. Let
$\pi:\widetilde{X}\to\widehat{X}\stackrel{{\scriptstyle\xi}}{{\to}}X$ be a log
resolution of $\mathfrak{d}_{X,Z}$ and $\mathfrak{a}$ such that
$\mathfrak{d}_{X,Z}\cdot\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-D_{Z})$
and
$\mathfrak{a}\cdot\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)$
for some effective divisors $D_{Z}$ and $F$ on $\widetilde{X}$, where
$\xi:\widehat{X}\to X$ is the normalization of $X$. Denote by

$\begin{split}\mathscr{A}_{m}:=&\
\omega_{\widetilde{X}}\otimes\pi^{*}(\omega_{Z}^{-1}|_{X})\otimes\mathcal{O}_{\widetilde{X}}(D_{Z})\otimes\mathscr{I}(\varphi\circ\pi+m\varphi_{\mathfrak{a}}\circ\pi)\\\
=&\
\omega_{\widetilde{X}}\otimes\pi^{*}(\omega_{Z}^{-1}|_{X})\otimes\mathcal{O}_{\widetilde{X}}(D_{Z})\otimes\mathscr{I}(\varphi\circ\pi)\otimes\mathcal{O}_{\widetilde{X}}(-mF)\end{split}$

for any $m\in\mathbb{N}$, and consider the Koszul complex determined by
$g_{1},\dots,g_{r}$:

$0\to\Lambda^{r}V\otimes\mathcal{O}_{\widetilde{X}}(rF)\to\cdots\to\Lambda^{2}V\otimes\mathcal{O}_{\widetilde{X}}(2F)\to
V\otimes\mathcal{O}_{\widetilde{X}}(F)\to\mathcal{O}_{\widetilde{X}}\to 0,$

where $V$ is the vector space spanned by $g_{1},\dots,g_{r}$. Note that the
Koszul complex is locally split and its syzygies are locally free, so twisting
through by any coherent sheaf will preserve the exactness. Then, by twisting
with $\mathscr{A}_{k}\ (k\geq r=q)$, we obtain the following long exact
sequence

$None$
$0\to\Lambda^{r}V\otimes\mathscr{A}_{k-r}\to\cdots\to\Lambda^{2}V\otimes\mathscr{A}_{k-2}\to
V\otimes\mathscr{A}_{k-1}\to\mathscr{A}_{k}\to 0.$

On the other hand, for any $m\in\mathbb{N}$, we can derive the vanishing of
higher direct images $R^{j}\pi_{*}\mathscr{A}_{m}=0\ (\forall j>0)$ by Theorem
2.11. Note that

$\mathscr{I}_{\text{GR}}\big{(}\varphi+m\varphi_{\mathfrak{a}}\big{)}=\pi_{*}\mathscr{A}_{m}$

by $(\spadesuit)$, and then by taking direct images of $(\star)$ we will
deduce the following so-called exact Skoda complex (cf. [20], p. 228):

$0\to\Lambda^{r}V\otimes\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-r)\varphi_{\mathfrak{a}}\big{)}\to\cdots\to
V\otimes\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-1)\varphi_{\mathfrak{a}}\big{)}\to\mathscr{I}_{\text{GR}}\big{(}\varphi+k\varphi_{\mathfrak{a}}\big{)}\to
0.$

In particular, the map

$V\otimes\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-1)\varphi_{\mathfrak{a}}\big{)}\to\mathscr{I}_{\text{GR}}\big{(}\varphi+k\varphi_{\mathfrak{a}}\big{)}$

is surjective, by which we can infer that

$\mathscr{I}_{\text{GR}}\big{(}\varphi+k\varphi_{\mathfrak{a}}\big{)}\subset\mathfrak{a}\cdot\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-1)\varphi_{\mathfrak{a}}\big{)}.$

Case (ii). When $r>n$, i.e., $q=n$.

By Lemma 4.1, we may assume that $\mathfrak{b}$ is a reduction of
$\mathfrak{a}$ generated by $n$ elements
$\widetilde{g}_{1},...,\widetilde{g}_{n}$. Consider a log resolution
$\pi:\widetilde{X}\to\widehat{X}\stackrel{{\scriptstyle\xi}}{{\to}}X$ of
$\mathfrak{d}_{X,Z},\ \mathfrak{a}$ and $\mathfrak{b}$ such that
$\mathfrak{d}_{X,Z}\cdot\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-D_{Z})$
and
$\mathfrak{a}\cdot\mathcal{O}_{\widetilde{X}}=\mathfrak{b}\cdot\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)$
for some effective divisors $D_{Z}$ and $F$ on $\widetilde{X}$. Then, by the
same argument as above, we can deduce the following exact Skoda complex:

$0\to\Lambda^{n}V\otimes\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-n)\varphi_{\mathfrak{a}}\big{)}\to\cdots\to
V\otimes\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-1)\varphi_{\mathfrak{a}}\big{)}\to\mathscr{I}_{\text{GR}}\big{(}\varphi+k\varphi_{\mathfrak{a}}\big{)}\to
0.$

for any $k\geq n=q$, where $V$ is the vector space spanned by
$\widetilde{g}_{1},...,\widetilde{g}_{n}$. Therefore, it follows that

$\mathscr{I}_{\text{GR}}\big{(}\varphi+k\varphi_{\mathfrak{a}}\big{)}\subset\mathfrak{b}\cdot\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-1)\varphi_{\mathfrak{a}}\big{)}\subset\mathfrak{a}\cdot\mathscr{I}_{\text{GR}}\big{(}\varphi+(k-1)\varphi_{\mathfrak{a}}\big{)};$

the proof is concluded. $\Box$

###  4.2. Proof of Theorem 1.6

Firstly, by Theorem 2.16, it is sufficient to prove the desired result for the
case $r\leq n$, i.e., $q=r$. In fact, we then can replace $\mathcal{I}$ by a
minimal reduction $\mathcal{J}$ of $\mathcal{I}$, generated by at most $n$
elements, and in further obtain

$\overline{\mathcal{I}^{k+q-1}}=\overline{\mathcal{J}^{k+q-1}}\subset\mathcal{J}^{k}\subset\mathcal{I}^{k}.$

Let $g=(g_{1},\dots,g_{r})$ be a system of generators of
$\mathcal{I}\subset\widehat{\mathcal{O}}_{X,x}$. As the statement is local, we
may assume that all functions $g_{1},...,g_{r}\in\widehat{\mathcal{O}}(X)$ and
put $\mathfrak{a}=(g_{1},...,g_{r})\cdot\widehat{\mathcal{O}}_{X}$. For any
$f\in\overline{\mathcal{I}^{k+q-1}}$, it follows from Corollary 2.18 that
$|f|\leq C\cdot|g|^{k+q-1}$ for some constant $C\geq 0$ near $x$ on $X$, and
then by weak rationality of the point $x$, we deduce that
$f\in\mathscr{I}_{\text{GR}}(\mathfrak{a}^{k+q-1})_{x}$. Therefore, in order
to prove Theorem 1.6, by Theorem 1.2 it is sufficient to show that

$\mathscr{I}_{\text{GR}}(\mathfrak{a}^{p})=\mathfrak{a}\cdot\mathscr{I}_{\text{GR}}(\mathfrak{a}^{p-1})$

for any integer $p\geq r$ on $X$, which is an immediate consequence of Theorem
1.3. $\Box$

###  4.3. Proof of Theorem 1.10

Relying on the reduction argument as above, we may consider a principal ideal
$\mathcal{J}=(g)\cdot\widehat{\mathcal{O}}_{X,x}\subset\widehat{\mathcal{O}}_{X,x}$
and $f\in\overline{\mathcal{J}}$. Then, it follows from Corollary 2.18 that
$f/g\in\widehat{\mathcal{O}}_{X,x}$, which implies that
$\overline{\mathcal{J}}\subset\mathcal{J}$, i.e., the original Briançon-Skoda
theorem holds for the Noetherian ring $\widehat{\mathcal{O}}_{X,x}$.

Note that each reduced complex space of dimension one is Cohen-Macaulay, and
then it follows that $x\in X$ is a weakly rational singularity if and only if
$x\in X$ is a rational singularity if and only if $x\in X$ is a regular point.

“$(2)\Longrightarrow(3)$”. If $x\in X$ is a regular point, then
$\mathcal{O}_{X,x}$ is a principal ideal domain. Then, by Proposition 1.5.2 in
[34] (or the original result of Briançon and Skoda), it follows that
$\overline{\mathcal{I}^{k}}=\mathcal{I}^{k}$ for any $k\in\mathbb{N}$.

“$(3)\Longrightarrow(2)$”. Suppose that $x\in X$ is a singularity. As the
question is local, we may assume that $X$ is a (closed) complex subspace of
some domain $\Omega$ in $\mathbb{C}^{N}$, where $N$ is the embedding dimension
of $X$ at $x$ (= the origin $o$ of $\mathbb{C}^{N}$). In particular, we have

$\text{ord}\mathscr{I}_{X,o}:=\min\\{\,\text{ord}_{o}(f)\,|\,f\in\mathscr{I}_{X,o}\\}\geq
2;$

otherwise, the embedding dimension of $X$ at $o$ is at most $N-1$.

By the Weierstrass Preparation Theorem, in some local coordinates
$(z^{\prime};z^{\prime\prime})=(z_{1};z_{2},...,z_{N})$ near $o$, there exist
Weierstrass polynomials

$None$
$P_{k}(z)=z_{k}^{m_{k}}+a_{1k}z_{k}^{m_{k}-1}+\cdots+a_{m_{k}k}\in\mathcal{O}_{k-1}[z_{k}]\cap\mathscr{I}_{X,o},\
k=2,...,N,$

with $m_{k}=\mbox{ord}_{o}P_{k}$. Hence, we have

$None$ $a_{jk}(z_{1},...,z_{k-1})\in\mathfrak{m}_{k-1}^{j},\ 2\leq k\leq N,\
1\leq j\leq m_{k}.$

Denote by $\mathcal{I}$ the ideal in $\mathcal{O}_{X,o}$ generated by the germ
of holomorphic function $\widehat{z}_{1}\in\mathcal{O}_{X,o}$, where
$\widehat{z}_{k}$ are the residue classes of $z_{k}$ in $\mathcal{O}_{X,o}$.
Then, combining $(*)$ with $(**)$, we obtain that the integral closure
$\overline{\mathcal{I}}$ of $\mathcal{I}$ in $\mathcal{O}_{X,o}$ is
$\mathfrak{m}_{X,o}=(\widehat{z}_{1},...,\widehat{z}_{N})\cdot\mathcal{O}_{X,o}$,
the maximal ideal of $\mathcal{O}_{X,o}$. Moreover, since
ord$\mathscr{I}_{X,o}\geq 2$, we have $\widehat{z}_{k}\not\in\mathcal{I},\
2\leq k\leq n$. Then, we have $\overline{\mathcal{I}}\not\subset\mathcal{I}$,
which contradicts to validity of the original Briançon-Skoda theorem for
$\mathcal{O}_{X,o}$; i.e., $x\in X$ is a regular point. $\Box$

###  4.4. Proof of Theorem 1.11

_Necessity._ It is a direct application of Theorem 1.6.

_Sufficiency._ Firstly, we show that the desired result holds for the case
when $x\in X$ is a normal point. As the question is local, by Artin’s
algebraization theorem, we may assume that $X$ is a normal, affine algebraic
variety over $\mathbb{C}$.

Note the Cohen-Macaulayness of normal complex surfaces, and then we can deduce
from Theorem 1.1 in [33] that if the original Briançon-Skoda theorem holds for
$\mathcal{O}_{X,x}$, essentially of finite type over $\mathbb{C}$, then $x\in
X$ is an algebraic rational singularity, which is actually equivalent to the
fact that $x\in X$ is an analytic rational singularity (cf. [19], Corollary
5.11).

In general, we consider the normalization $\xi:\widehat{X}\to X$ of $X$. By
Proposition 2.10, it is enough to present that $\hat{x}\in\widehat{X}$ is a
weakly rational singularity for any $\hat{x}\in\xi^{-1}(x)$. Let
$\hat{\mathcal{I}}\subset\mathcal{O}_{\widehat{X},\hat{x}}$ be a nonzero ideal
with $r$ generators $\hat{g}_{1},...,\hat{g}_{r}$. Note that
$\widehat{\mathcal{O}}_{X,x}\cong\bigoplus_{j=1}^{m}\mathcal{O}_{\widehat{X},\hat{x}_{j}}$,
where $\hat{x}_{1},...,\hat{x}_{m}$ are the distinct points of a fiber
$\xi^{-1}(x)$ and we take $\hat{x}=\hat{x}_{1}$. Then, there exist
$g_{1},...,g_{r}\in\widehat{\mathcal{O}}_{X,x}$ such that for any $1\leq j\leq
m$, $g_{j}\circ\xi$ equals to $\hat{g}_{j}$ near $\hat{x}_{1}$ and equals to
one near $\hat{x}_{2},...,\hat{x}_{m}$. Denote by $\mathcal{I}$ the ideal in
$\widehat{\mathcal{O}}_{X,x}$ generated by $g_{1},...,g_{r}$. Then, by the
assumption, we have $\overline{\mathcal{I}^{k+q-1}}\subset\mathcal{I}^{k}$ for
any $k\in\mathbb{N}$. Thanks to Corollary 2.18, it follows that
$\overline{\hat{\mathcal{I}}^{k+q-1}}\subset\hat{\mathcal{I}}^{k}$, which
implies that $\hat{x}\in X$ is a rational singularity by the initial argument;
the proof is complete. $\Box$

##  Appendix A A Some remarks on Skoda’s $L^{2}$ division theorem

###  A.1. Skoda’s $L^{2}$ division theorem: a trivial generalization

For the readers’ convenience, here we restate some facts on a trivial
generalization of Skoda’s $L^{2}$ division theorem (for holomorphic functions)
on singular complex spaces in [22]. The approach in the arguments is the same
as in the author’s joint paper [12] (see also [25, 31]).

######  Theorem A.1.

_([22], Theorem 4.1)._ Let $\Omega\subset\mathbb{C}^{n}\ (n\geq 2)$ be a
domain, $A\subset\Omega$ an analytic set of pure dimension $d$ through the
origin $o$. Then, for small enough ball $B_{r}(o)\subset\Omega$, the $L^{2}$
division theorem holds on $B_{r}(o)\cap A$ if and only if $o$ is a regular
point of $A$.

######  Remark A.2.

In addition, via an argument on the integral closure of ideals as in the proof
of Theorem 1.3 in [12], we are able to deduce the following:

Let $\Omega\subset\mathbb{C}^{n}\ (n\geq 2)$ be a domain and $A\subset\Omega$
an analytic set of pure dimension $d$. Then, the $L^{2}$ division of bounded
holomorphic sections will imply that

$\text{ord}\mathscr{I}_{A,x}:=\min\\{\,\mbox{\text{ord}}_{x}(f)\,|\,f\in\mathscr{I}_{A,x}\,\\}\leq
d,$

for every point $x\in A$.

Firstly we state a basic estimate of the volume of an analytic subset as
follows.

######  Lemma A.3.

_([12], Lemma 2.3)._ Let $\Omega\subset\mathbb{C}^{n}\ (n\geq 2)$ be a domain
and $A\subset\Omega$ an analytic set of pure dimension $d$ through the origin
$o$. Then, there exists a neighborhood $U$ of $o$ such that, for any
$0\leq\varepsilon<1$,

$\int_{U{\cap}A}\frac{1}{(|z_{1}|^{2}+\cdots+|z_{n}|^{2})^{d+\varepsilon-1}}dV_{A}<+\infty,$

where $dV_{A}=\frac{1}{d!}\omega^{d}|_{A_{\emph{reg}}}\ \text{and}\
\omega=\frac{\sqrt{-1}}{2}\sum\limits_{k=1}^{n}dz_{k}{\wedge}d\bar{z}_{k}$.

The above statement is in fact slightly stronger than Lemma 2.3 in [12], but
it is proved in the same way. As a result, with such a minor modification, it
turns out that it is not necessary to use the strong openness of multiplier
ideal sheaves associated to plurisubharmonic functions in the proof of the
main result, Theorem 1.2, in [12].

_Proof of Theorem_ A.1. The sufficiency is a straightforward application of
Skoda’s $L^{2}$ division theorem (see [8], Theorem 11.13) on $d$-dimensional
balls centered at the origin in $\mathbb{C}^{d}$. Thus, we only need to prove
the necessity.

Suppose that $o$ is a singular point of $A$ with $1\leq\dim_{o}A=d\leq n-1$.
Thanks to the local parametrization theorem for analytic sets, we can find a
local coordinate system

$(z^{\prime};z^{\prime\prime})=(z_{1},...,z_{d};z_{d+1},...,z_{n})$

near $o$ such that for some constant $C>0$, we have $|z^{\prime\prime}|\leq
C\cdot|z^{\prime}|$ for any $z\in A$ near $o$.

Let $\mathcal{I}\subset\mathcal{O}_{A,o}$ be the ideal generated by
holomorphic functions
$\widehat{z}_{1},...,\widehat{z}_{d}\in\mathcal{O}_{A,o}$, where
$\mathcal{O}_{A}=\mathcal{O}_{\Omega}/\mathscr{I}_{A}\big{|}_{A}$ and
$\widehat{z}_{k}$ are the residue classes of $z_{k}$ in $\mathcal{O}_{A,o}$.
From the non-smoothness of $A$ at the origin $o$, we deduce that the embedding
dimension $\dim_{\mathbb{C}}(\mathfrak{m}_{A,o}/\mathfrak{m}^{2}_{A,o})\geq
d+1$ of $A$ at $o$, which implies that there exists $d+1\leq k_{0}\leq n$ such
that $\widehat{z}_{k_{0}}\not\in\mathcal{I}$.

On the other hand, since $|z^{\prime\prime}|\leq C\cdot|z^{\prime}|$ for any
$z\in A$ near $o$, we infer that

$|z_{k_{0}}|^{2}\leq C^{2}\cdot|z^{\prime}|^{2}$ and
$\frac{|z|^{2}}{1+C^{2}}\leq|z^{\prime}|^{2}$

on $U\cap A$ for some neighborhood $U$ of $o$. Thus, by Lemma A.3, for some
smaller neighborhood $U$ of $o$ and any $0<\varepsilon<1$, it follows that

$\int_{U{\cap}A}\frac{|{z}_{k_{0}}|^{2}}{|z^{\prime}|^{2(d+\varepsilon)}}dV_{A}\leq
C^{2}(1+C^{2})^{d+\varepsilon-1}\cdot\int_{U{\cap}A}|z|^{-2(d+\varepsilon-1)}dV_{A}<+\infty.$

Take a small ball $B_{r}(o)\subset U$ and apply the $L^{2}$ division theorem
with $\varphi=0$ on $B_{r}(o)\cap A$. Then, there exist holomorphic functions
$h_{k}\in\mathcal{O}(B_{r}(o)\cap A)$ such that

$\widehat{z}_{k_{0}}=h_{1}{\cdot}\widehat{z}_{1}+\cdots+h_{d}{\cdot}\widehat{z}_{d}$

on $B_{r}(o)\cap A$, which contradicts that $\widehat{z}_{k_{0}}$ is not in
$\mathcal{I}$. Thus the proof is concluded. $\Box$

###  A.2. An alternative argument on the proof of Theorem 1.6

Analogous to the arguments as in [22], we can also give an alternative proof
of Theorem 1.6 for $1$-Gorenstein case, based on a version of Skoda’s $L^{2}$
division theorem for holomorphic sections on complete Kähler manifolds (see
[8], Theorem 11.8), i.e.,

######  Theorem A.4.

Let $X$ be a complete Kähler manifold equipped with a Kähler metric $\omega$
on $X$, $\texttt{g}:E\to Q$ a surjective morphism of hermitian vector bundles
and $L\to X$ a hermitian line bundle. Set

$r=\emph{rk}\ E,\ q=\emph{rk}\ Q,\ m=\min\\{n-k,r-q\\}$

and suppose $E\geq_{m}0$,

$\sqrt{-1}\Theta(L)-(m+\varepsilon)\sqrt{-1}\Theta(\det\ Q)\geq 0$

for some $\varepsilon>0$. Then for every $D^{\prime\prime}$-closed form $f$ of
type $(n,k)$ with values in $Q\otimes L$ such that

$I=\int_{X}\langle\widetilde{\texttt{gg}^{\star}}f,f\rangle(\det\texttt{gg}^{\star})^{-m-1-\varepsilon}dV_{\omega}<+\infty,$

there exists a $D^{\prime\prime}$-closed form $h$ of type $(n,k)$ with values
in $E\otimes L$ such that $f=\texttt{g}\cdot h$ and

$\int_{X}|h|^{2}(\det\texttt{gg}^{\star})^{-m-\varepsilon}dV_{\omega}\leq(1+m/\varepsilon)I.$

######  Theorem A.5.

Let $X$ be a complex space of pure dimension $n$ with $x\in X$ a singularity,
and $\mathcal{I}\subset\widehat{\mathcal{O}}_{X,x}$ a nonzero ideal with $r$
generators. If $x\in X$ is a weakly rational singularity and $X$ is
$1$-Gorenstein near $x$, then the original Briançon-Skoda theorem holds for
the Noetherian ring $\widehat{\mathcal{O}}_{X,x}$.

Acknowledgements. The author would like to sincerely thank Professor Xiangyu
Zhou for guiding to the subject on Briançon-Skoda theorem, as well as his
generous support and encouragements.

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# Discrete Natural Evolution Strategies

Ahmad Ayaz Amin
Department of Computer Science
Toronto Metropolitan University
Toronto, Canada
<EMAIL_ADDRESS>
###### Abstract

Natural evolution strategies are a class of approximate-gradient black-box
optimizers that have been successfully used for continuous parameter spaces.
In this paper, we derive NES algorithms for discrete parameter spaces and
demonstrate their effectiveness in tasks involving discrete parameters.

##  1 Introduction

Stochastic gradient descent exploits gradient information to smartly steer
parameters towards an optimal solution. However, not all objectives have an
easy-to-compute gradient through differentiation, particularly models with
discrete parameters. Natural Evolution Strategies (NES) (Wierstra et al.,
2011) are a class of black-box optimizers that alleviate the need to compute
derivatives, while still retaining gradient-based parameter updates.

At a high level, NES algorithms estimate a gradient by Monte Carlo sampling
parameters from a probability distribution known as the search distribution.
The sampled parameters are evaluated by a fitness function $f(x_{k})$ and then
weighted by the score of the search distribution. The expectation of the
scores is the approximate gradient of the search distribution:

$\nabla_{\theta}J(\theta)\approx\frac{1}{\lambda}\sum_{k=1}^{\lambda}f(x_{k})\nabla_{\theta}\log\pi(x_{k}|\theta)$

With regular search gradients, vanilla stochastic gradient descent
$\theta\leftarrow\theta+\eta\nabla_{\theta}J(\theta)$ updates the parameters
of the search distribution. While very effective for complex tasks (Salimans
et al., 2017), natural evolution strategies employs the natural gradient to
its advantage, which involves multiplying the gradient with the inverse Fisher
information matrix:

$\theta\leftarrow\theta+\eta\textbf{F}^{-1}\nabla_{\theta}J(\theta)$

The FIM is a measure of the amount of information that a random variable $x$
carries about the parameters of the distribution from which it was sampled
from; it is the variance of the score of the probability distribution.

Using the natural gradient is advantageous, but its costly computational
requirements necessitates approximations like Adam (Kingma & Ba, 2017;
Martens, 2020) for large parameter spaces. Thankfully, not only is the FIM for
discrete parameter spaces rather easy to compute, its effects are implicit in
regular search gradients, making explicit computation redundant (see Appendix
A.3 for details). Nonetheless, we provide the NES updates with the explicit
FIM for reference (see Appendix A.1).

The reason for this is that the natural gradient automatically adjust the
variance of the search distribution to accelerate convergence. With discrete
search gradients, the entropy (the discrete counterpart of variance) readily
changes by virtue of updating its parameters after each update.

NES is very similar to variational optimization (Staines & Barber, 2012) as
both estimate the gradient as a Monte Carlo expectation. The major difference
lies in the gradient of the search distribution: VO uses the gradient of the
probability, while NES uses the score. This subtle difference results in VO
computing a lower bound on the true objective, while NES computes an exact
gradient in the limit of infinite samples.

##  2 Experiment and Results

A program induction problem is solved using sketching (Solar-Lezama, 2008) to
demonstrate discrete NES as a practical programming aid. In sketching, a high-
level program is specified containing ”holes”, or incomplete parts, that are
completed by a solver. NES is used to complete these holes.

The program is a very simple if-else piecewise function whose discrete parts
can be solved easily using brute force. Therefore, holes for coefficients are
introduced into the program to make it intractable for brute force
computation. These continuous holes are optimized using Gaussian NES (Wierstra
et al., 2011). Discrete holes for operators (e.g. + and *) are solved using
discrete NES.

A ground-truth program is created in order to generate input-output pairs
(specification) as well as to compare the induced program with.

The sketch is trained to minimize the MSE between the specification output and
the inferred output for 10,000 iterations with a learning rate of 0.1. A
baseline sketch using VO to solve for the discrete holes is trained for
comparison. The training graph is shown below (lower is better):

The final outputs of the NES program given the specification inputs [1.0, 2.0,
4.0, 5.0] are [3.9859228, 7.9718456, 15.943691, 19.929615], which are very
close to the true outputs [2.1, 4.2, 16.8, 21.0]. The VO sketch had similar
results, with its outputs being [2.1309583, 4.2619166, 16.501104, 20.62638].
However, the VO optimizer had notable spikes in the training graph, indicating
instability. The low training loss, faster convergence and better stability
than VO shows practical utility of discrete NES as a solver for discrete
parameter spaces.

##  3 Conclusion and Discussion

In this paper, the natural evolution strategies algorithms for discrete search
distributions is derived, allowing optimization of discrete parameters using
standard gradient descent optimizers. Like its continuous counterpart,
discrete NES demonstrates practical performance on problems.

One quality of discrete NES demonstrated in this paper is that it can
effectively work in tandem with continuous NES. However, the experiment is
very basic, so it doesn’t demonstrate how well it performs on real-life
problems with many parameters.

One future avenue is to apply discrete NES to more rigorous problem sets, such
as policies with discrete parameters for reinforcement learning (Silver et
al., 2019).

#### URM Statement

The authors acknowledge that at least one key author of this work meets the
URM criteria of ICLR 2024 Tiny Papers Track.

## References

  * Kingma & Ba (2017) Diederik P. Kingma and Jimmy Ba.  Adam: A method for stochastic optimization, 2017.
  * Martens (2020) James Martens.  New insights and perspectives on the natural gradient method.  _Journal of Machine Learning Research_ , 21(146):1–76, 2020.  URL http://jmlr.org/papers/v21/17-678.html.
  * Salimans et al. (2017) Tim Salimans, Jonathan Ho, Xi Chen, Szymon Sidor, and Ilya Sutskever.  Evolution strategies as a scalable alternative to reinforcement learning, 2017.
  * Silver et al. (2019) Tom Silver, Kelsey R. Allen, Alex K. Lew, Leslie Pack Kaelbling, and Josh Tenenbaum.  Few-shot bayesian imitation learning with logical program policies, 2019.
  * Solar-Lezama (2008) Armando Solar-Lezama.  _Program Synthesis by Sketching_.  PhD thesis, USA, 2008.  AAI3353225.
  * Staines & Barber (2012) Joe Staines and David Barber.  Variational optimization, 2012.
  * Tomczak (2012) Jakub M. Tomczak.  Fisher information matrix for Gaussian and categorical distributions.  https://www.ii.pwr.edu.pl/~tomczak/PDF/[JMT]Fisher_inf.pdf, 2012.  [Accessed 03-12-2023].
  * Wierstra et al. (2011) Daan Wierstra, Tom Schaul, Tobias Glasmachers, Yi Sun, and Jürgen Schmidhuber.  Natural evolution strategies, 2011.

##  Appendix A Appendix

###  A.1 Derivation of Discrete NES

####  A.1.1 Bernoulli NES

The log probability mass function $\log\pi(x|\theta)$ of the Bernoulli
distribution is $\log\pi(x|\theta)=x\log(\theta)+(1-x)\log(1-\theta)$ for
parameters $\theta\in(0,1)$ and observations $x\in\\{0,1\\}$. The score
$\nabla_{\theta}\log\pi(x|\theta)$ is thus
$\frac{x-\theta}{\theta(1-\theta)}$. The Fisher information of the Bernoulli
distribution is $\textbf{F}=\frac{1}{\theta(1-\theta)}$, or the natural
gradient of the Bernoulli distribution is:

$\nabla_{\theta}\tilde{J}(\theta)\approx\frac{1}{\lambda}\sum_{k=1}^{\lambda}f(x_{k})(x_{k}-\theta)$
(1)

####  A.1.2 Categorical NES

Let $\log\pi(x|\theta)$ be the log probability mass function of the
categorical distribution:

$\log\pi(x|\theta)=\sum_{k=1}^{K}x_{k}\log\theta_{k}$ (2)

Then, the partial derivatives are:

$\frac{\partial}{\partial\theta_{k}}\log\pi(x|\theta)=\frac{x_{k}}{\theta_{k}}$
(3)

Which gives us the following Fisher score:

$g(x,\theta)=\begin{bmatrix}\frac{x_{1}}{\theta_{1}}\\\ \vdots\\\
\frac{x_{K}}{\theta_{K}}\end{bmatrix}$ (4)

The product of the Fisher score and its transpose is:

$g(x,\theta)g(x,\theta)^{T}=\begin{bmatrix}(\frac{x_{1}}{\theta_{1}})^{2}&\frac{x_{1}x_{2}}{\theta_{1}\theta_{2}}&\ldots&\frac{x_{1}x_{K}}{\theta_{1}\theta_{K}}\\\
\vdots&\vdots&\ldots&\vdots\\\
(\frac{x_{K}x_{1}}{\theta_{K}\theta_{1}})^{2}&\frac{x_{K}x_{2}}{\theta_{K}\theta_{2}}&\ldots&(\frac{x_{K}}{\theta_{K}})^{2}\end{bmatrix}=\begin{bmatrix}g_{11}&g_{12}&\ldots
g_{1K}\\\ \vdots&\vdots&\ldots\vdots\\\ g_{K1}&g_{K2}&\ldots g_{KK}\\\
\end{bmatrix}$ (5)

For $g_{kk}$,
$\mathbb{E}_{x}[g_{kk}]=\mathbb{E}_{x}[(\frac{x_{k}}{\theta_{k}})^{2}]=\frac{1}{\theta_{k}}$.
For $g_{ij}$, $i\neq j$,
$\mathbb{E}_{x}[g_{ij}]=\mathbb{E}_{x}[\frac{x_{i}x_{j}}{\theta_{i}\theta_{j}}]=0$.
Therefore, the Fisher information matrix for the categorical distribution is
$\textbf{F}=\operatorname{diag}\\{\frac{1}{\theta_{1}},...,\frac{1}{\theta_{K}}\\}$
(Tomczak, 2012). Similarly, the inverse FIM is
$\textbf{F}^{-1}=\operatorname{diag}\\{\theta_{1},...,\theta_{K}\\}$, or the
probability vector of the search distribution itself.

The score of the categorical distribution
$\nabla_{\theta}\log\pi(x_{k}|\theta)$ is as follows:

$\nabla_{\theta}\log\pi(x_{k}|\theta)=\begin{cases}1-\pi(x_{k}|\theta_{i})&\mbox{if
$k=i$}\\\ -\pi(x_{k}|\theta_{i})&\mbox{if $k\neq i$}\end{cases}$ (6)

Putting everything together, the natural gradient of the categorical search
distribution can be calculated as such:

$\nabla_{\theta}\tilde{J}(\theta)\approx\frac{1}{\lambda}\operatorname{diag}\\{\theta_{1},...,\theta_{K}\\}\sum_{k=1}^{\lambda}\left[f(x_{k})\begin{cases}1-\pi(x_{k}|\theta_{i})&\mbox{if
$k=i$}\\\ -\pi(x_{k}|\theta_{i})&\mbox{if $k\neq i$}\end{cases}\right]$ (7)

###  A.2 Program Induction

The input-output specification was generated using the following Rust program:

⬇

1fn prog_true(x: f32) -> f32

2{

3 if x > 3.5

4 {

5 return 4.2 * x;

6 }

7

8 return x * 2.1;

9}

The program sketch that was optimized has the same general layout:

⬇

1fn prog_sketch(x: f32) -> f32

2{

3 if x [COND] [REAL]

4 {

5 return [REAL] [OP] x;

6 }

7

8 return [OP] * [REAL];

9}

The [COND] and [OP] fields indicate conditionals (e.g. ==) and operations
(e.g. +) respectively. The [REAL] fields indicate real-valued numbers. The
[COND] and [OP] fields are optimized by discrete NES, while the [REAL] fields
are optimized by Gaussian NES. At each iteration, the parameters in the fields
are sampled from the search distributions and then inserted into the program
to evaluate the specification inputs.

The program resulting from the sketch after 10,000 iterations is as follows:

⬇

1fn prog_output(x: f32) -> f32

2{

3 if x < -1.5677981

4 {

5 return 1.1321394 * x;

6 }

7

8 return x * 3.9859228;

9}

With the exception of the order of the conditions, the learned program is
functionally similar in both its realization and its outputs to the ground-
truth program, indicated by the coefficients and the operators in the return
statements. This demonstrates that categorical NES has practical performance,
and in turn potential for construction of more expressive models.

###  A.3 Ablation Study

In this section, the effect of the FIM in training performance is investigated
to evaluate whether FIM is necessary for better convergence behaviour.

The experiment is a slightly more sophisticated program induction problem
involving two inputs (i.e. a multi-variable piecewise function) in order to
demonstrate performance under increased complexity. The same sketch is used
for both the explicit NES as well as the search gradients (no FIM) optimizer.

Similar to the main experiment, discrete search distributions are used for
operators and continuous search distributions are used for real-valued
constants. The experiment is performed with a series of different learning
rates ranging from 0.1 to 0.001 to investigate the behaviour of the two
optimizers. Each run is performed for 10,000 iterations.

The inputs for the program are the pairs [(5.8, 2.5), (5.0, 6.2), (7.4, 6.1),
(5.5, 9.4)]. The outputs produced by the ground-truth program are [14.1,
-4.677419, 20.9, -5.287234].

Effect of FIM on Convergence After 10,000 Iterations
---
Learning Rate |  NES Loss |  NES Outputs |  SG Loss |  SG Outputs
$0.1$ |  $36.94874$ |  [$19.529205$, $-3.3582695$, $10.21171$, $-6.8166084$] |  $36.948715$ |  [$19.529203$, $-3.3576767$, $10.211709$, $-6.8160696$]
$0.05$ |  $36.949665$ |  [$19.515165$, $-3.3503892$, $10.204369$, $-6.8094444$] |  $36.949516$ |  [$19.515043$, $-3.3440251$, $10.204305$, $-6.803659$]
$0.01$ |  $38.834015$ |  [$19.653248$, $-5.3543134$, $10.27657$, $-8.631194$] |  $37.132545$ |  [$19.62102$, $-3.9629102$, $10.259719$, $-7.3662815$]
$0.005$ |  $53.801556$ |  [$15.546269$, $0.0117374435$, $8.129055$, $0.0070379255$] |  $39.12318$ |  [$19.64944$, $-5.504375$, $10.27458$, $-8.767613$]
$0.001$ |  $170.9544$ |  [$0.047796037$, $0.022356212$, $0.0153531805$, $0.013405079$] |  $133.90349$ |  [$3.8961499$, $1.3543346$, $2.0372744$, $0.98261297$]

Strangely enough, it seems as though the FIM has a negative impact on the
discrete NES, as indicated by the gradually increasing loss and diverging
outputs with respect to learning rate. In contrast, search gradients maintains
lower and more stable loss values under these changes.

This behaviour is not entirely unexpected, as the FIM is only applicable to
continuous search distributions where the variance of the distribution
requires continuous adjustment with respect to the optima (Wierstra et al.,
2011).

In the case of a Gaussian search distribution, the variance parameter
$\sigma^{2}$ needs to be large when the parameters are far from the optimal
solution, and small when the parameters are within a close radius. This is to
prevent potential training instabilities and faster convergence.

In the case of the categorical search distribution, the entropy (the
categorical equivalent of variance) readily changes with gradient updates. In
other words, the behaviour of the natural gradient is implicitly built into
the search gradient update. Thus, adding the extra FIM is not only redundant
and computationally wasteful, it is actually detrimental to model performance.
This is further emphasized by the fact that the inverse FIM for the
categorical distribution is the probability vector of the distribution itself.