# Rational singularities and $q$-birational morphism

Donghyeon Kim  Department of mathematics, Yonsei University, 50 Yonsei-Ro,
Seodaemun-Gu, Seoul 03722, Korea<EMAIL_ADDRESS>
(Date: August 29, 2024)

###### Abstract.

In this paper, we generalize the notion of rational singularities for any
reflexive sheaf of rank $1$ and prove generalizations of standard facts about
rational singularities. Moreover, we introduce the notion of $(B_{q+1})$ as a
dual notion of well-known Serre’s notion of $(S_{q+1})$ and prove a theorem
about $q$-birational morphism.

###### Key words and phrases:

rational singularities, $q$-birational morphism

###### 2010 Mathematics Subject Classification:

14B05, 14E05, 14G17.

##  1\. Introduction

In this paper, $q\geq 2$ is an integer, $k$ is any algebraically closed field
and the characteristic of $k$ can be positive. Any variety is quasi-
projective, separated, finite type and integral scheme over $k$. Any divisor
is assumed to be Cartier unless otherwise stated. For convenience, in the
section, we assume that all reflexive sheaves are of rank $1$.

Throughout the paper, we will assume that for any normal variety $X$, there is
a proper birational morphism $f:X^{\prime}\to X$ in which $X^{\prime}$ is
smooth and $f$ is an isomorphism outside of the singular locus of $X$. We say
such proper birational morphism is a _resolution_ of $X$. If $k$ is of
characteristic $0$, our assumption is the well-known Hironaka’s resolution of
singularities and if $k$ is of positive characteristic, only the threefold
case is known to be positive (see [Hir64] and [Cut09]).

If $X$ is a variety, $X$ has rational singularities if and only if for any
resolution $f:X^{\prime}\to X$,
$f_{*}\mathcal{O}_{X^{\prime}}=\mathcal{O}_{X},R^{i}f_{*}\mathcal{O}_{X^{\prime}}=0$
and $R^{i}f_{*}\omega_{X^{\prime}}=0$ for $i\geq 1$ (see [Kol13], Definition
2.76).

The notion of rational singularities plays an important role in algebraic
geometry. For instance, if $X$ is a variety with rational singularities and
$f:X^{\prime}\to X$ is a resolution of $X$, we have

$H^{i}(X^{\prime},f^{*}\mathcal{E})\cong H^{i}(X,\mathcal{E})$

for any $i\geq 0$ and any vector bundle $\mathcal{E}$ on $X$ (see [KM98],
Theorem 5.10). Hence, it is worth extending the notion to various settings.

Previously, there are some works extending the notion of rational
singularities. For example, for any normal variety $X$, Schwede and Takagi in
[ST08] defined a notion of rational singularities for a pair
$(X,\mathfrak{a}^{\lambda})$ with an ideal sheaf $\mathfrak{a}$ on $X$ and
$\lambda>0$. In [Kov11] and [Kol13], Kollár and Kovács defined a notion of
rational singularities for $(X,D)$ with a reduced Weil divisor $D$ on $X$.
Both definitions are for pairs. In this paper, we will define an analogous
notion of rational singularities for a reflexive sheaf $\mathcal{F}$ on a
normal variety $X$. Our definition is different from the definitions in
[ST08], [Kov11] and [Kol13] because our notion is for a reflexive sheaf, not
for a pair.

######  Proposition 1.1 (See defi1).

idefi1 Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$. Suppose that $f:X^{\prime}\to X$ is a resolution. Then we have a natural
map

$\theta_{\mathcal{F},f}:Rf_{*}(f^{*}\mathcal{F})^{D}\to
R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega^{\bullet}_{X})[-\dim X].$

######  Definition 1.2 (See weak).

Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on $X$. We
say that $\mathcal{F}$ has _weak rational singularities_ if
$\theta_{\mathcal{F},f}$ in idefi1 is a quasi-isomorphism for the
$\mathcal{F}$ and for any resolution $f:X^{\prime}\to X$.

For the definition of the _dual_ $\mathcal{F}^{D}$ of $\mathcal{F}$, see dual.
Our definition of weak rational singularities does not require $\mathcal{F}$
to be CM. One can define a much stronger notion of rational singularities as
the following.

######  Definition 1.3 (See defi2).

idefi2 Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$ which has weak rational singularities.

  * (a)

If $\mathcal{F}$ is CM, we say $\mathcal{F}$ has _rational singularities_.

  * (b)

We say that a Weil divisor $D$ on $X$ has _rational singularities_ if
$\mathcal{O}_{X}(D)$ has rational singularities.

  * (c)

We say that $\mathcal{F}$ is $(KV_{q})$ if the support of
$R^{i}f_{*}(f^{*}\mathcal{F})^{D}$ has codimension $\geq i+q+1$ for any $i\geq
1$ and any resolution $f:X^{\prime}\to X$.

One of the fundamental properties of klt and strongly $F$-regular varieties
are that all klt varieties over a characteristic $0$ field and strongly
$F$-regular varieties over a positive characteristic field have rational
singularities (see [Elk81], [KMM87] and [HW19]). One may extend the theorems
as follows:

######  Proposition 1.4 (See ex1).

iex1 Let $X$ be any normal variety over a characteristic $0$ field, $\Delta$
any effective $\mathbb{Q}$-Weil divisor such that $(X,\Delta)$ is klt and $D$
any $\mathbb{Q}$-Cartier Weil divisor on $X$. Then $D$ has rational
singularities.

######  Proposition 1.5 (See ex11).

iex11 Let $X$ be any strongly $F$-regular variety over a positive
characteristic field and $D$ any Weil divisor such that there is an integer
$r$ such that $rD$ is linearly equivalent to a Cartier divisor and $r$ is
relatively prime to $p$. Then $D$ has rational singularities.

Our notion of rational singularities is well-behaved under finite étale
morphism.

######  Proposition 1.6 (See quo).

iquo Let $X,Y$ be normal varieties over characteristic $0$ field, $p:Y\to X$
any finite étale morphism and $\mathcal{F}$ any reflexive sheaf on $X$. If
$p^{*}\mathcal{F}$ has (resp. weak) rational singularities, then $\mathcal{F}$
has (resp. weak) rational singularities.

For a normal variety $X$, one may want to measure how far a non-rational
singularity is from being rational. For this purpose, the notion of non-
rational locus is defined in [AH12] and [Kov11]. The following definition is a
candidate for the analog of that notion.

######  Definition 1.7.

inon-RS Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$. We say that $\mathcal{F}$ is $(RS_{q})$ if for any point $x\in X$ with
$\operatorname{codim}_{X}x\leq q$ and any resolution $f:X^{\prime}\to X$,
$\theta_{\mathcal{F},f}$ is a quasi-isomorphism after localizing
$\theta_{\mathcal{F},f}$ at $x$.

By adopting the idea of proof of Lemma 3.3 in [Kov99], we can define the
notion of $(B_{q+1})$ and prove a duality theorem.

######  Definition 1.8 (See B_q+1).

iB_q+1 Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$. We say $\mathcal{F}$ is $(B_{q+1})$ if $H^{i}_{x}(X,\mathcal{F})=0$ for
any $\dim X-q<i<\dim X$ and any closed point $x\in X$.

######  Theorem 1.9 (See main).

imain Let $X$ be any normal projective variety and $\mathcal{F}$ any reflexive
sheaf on $X$, which is $(RS_{q})$ and $(KV_{q})$. Then the following are
equivalent:

  * (a)

$\mathcal{F}$ is $(S_{q+1})$.

  * (b)

For any resolution $f:X^{\prime}\to X$ such that for any $1\leq i<q$,
$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}=0$.

  * (c)

$\mathcal{F}^{D}$ is $(B_{q+1})$.

######  Corollary 1.10 (See maincoro).

imaincoro Assume that $q\geq\left\lceil\frac{\dim X+1}{2}\right\rceil$. Let
$X$ be any normal $\mathbb{Q}$-factorial variety over a characteristic $0$
field with $(R_{q})$ and any Weil divisor on $X$ is $(S_{q+1})$. Then any Weil
divisor on $X$ is CM. In particular, $X$ is CM itself.

The above theorem can be used to generalize Theorem 3.4 in [Kim22] such as the
following.

######  Definition 1.11 (See qbirational).

ibirational Let $X,X^{\prime}$ be any normal varieties and $f:X^{\prime}\to X$
any proper birational morphism.

  * (a)

The _center_ of $f$ is the reduced closed subscheme $C$ of $X$ which is the
image of exceptional locus along $f$.

  * (b)

We say $f$ is a _$q$ -birational morphism_ if the exceptional locus has
codimension $1$ and the center of $f$ has codimension $\geq q+1$.

######  Theorem 1.12 (See youaresecondapplication).

iyouaresecondapplication Let $X,X^{\prime}$ be any normal projective varieties
over a field of characteristic $0$ and $f:X^{\prime}\to X$ any $q$-birational
morphism in which $X^{\prime}$ is smooth. Suppose that $D$ is any anti $f$-nef
divisor on $X^{\prime}$ such that $f_{*}D$ is $\mathbb{Q}$-Cartier and
$(S_{q+1})$. Then $R^{i}f_{*}\mathcal{O}_{X^{\prime}}(D)=0$ for $1\leq i<q$.

######  Notation 1.13.

Given any normal varieties $X,X^{\prime}$ with any proper morphism
$f:X^{\prime}\to X$ and any (not necessarily closed) point $x\in X$, we write
$X^{\prime}_{x}:=X^{\prime}\times_{X}\operatorname{Spec}\mathcal{O}_{X,x}$,
$\dim x$ for the dimension of the closure of $x$ and
$\operatorname{codim}_{X}x:=\dim X-\dim x$.

For any coherent sheaves $\mathcal{F},\mathcal{F}^{\prime}$ on $X,X^{\prime}$
respectively and the inclusion $\jmath:X^{\prime}_{x}\to X^{\prime}$, write
$\mathcal{F}^{\prime}_{x}:=j^{*}\mathcal{F}^{\prime}$. In addition to that,
set $\mathcal{O}_{X^{\prime},x}:=(\mathcal{O}_{X^{\prime}})_{x}$ and
$\omega_{X^{\prime},x}:=(\omega_{X^{\prime}})_{x}$. Note that if
$f:X^{\prime}\to X$ is a resolution of $X$, then $X^{\prime}_{x}$ is a regular
scheme.

For simplicity, if $i\geq 0$, a point $x\in X$ and a coherent sheaf
$\mathcal{F}$ on $X$ is given, set

$H^{i}_{x}(X,\mathcal{F}):=H^{i}_{x}(X_{x},\mathcal{F}_{x}).$

Also, under the same conditions of the above, if a resolution $f:X^{\prime}\to
X$ and a coherent sheaf $\mathcal{F}^{\prime}$ on $X^{\prime}$ is given, let

$H^{i}_{f^{-1}(x)}(X^{\prime},\mathcal{F}^{\prime}):=H^{i}_{f^{-1}(x)}(X^{\prime}_{x},\mathcal{F}^{\prime}_{x}).$

For a noetherian scheme $X$ with a dualizing complex, $\omega^{\bullet}_{X}$
denotes the normalized dualizing complex.

The rest of the paper is organized as follows. We begin Section 2 by defining
some basic notions and by stating and proving basic theorems. Section 3 is
devoted to proving basic results about reflexive sheaves. In Section 4, we
define (weak) rational singularities and prove some basic theorems. In section
5, we define the notion of $(B_{q+1})$ and prove the main theorem about that
notion. Section 6 is devoted to defining $q$-birational morphism and proving
the main theorems about $q$-birational morphism.

######  Acknowledgments.

The author thanks his advisor Sung Rak Choi for his comments, questions, and
discussions. He is grateful for their hospitality. The author is grateful to
Sándor Kovács for his helpful comments on an earlier version of this paper.

##  2\. Preliminaries

The section is devoted to collecting basic definitions and facts used in the
paper.

######  Definition 2.1.

Let $X$ be any normal variety and $\mathcal{F}$ any torsion-free sheaf on $X$.

  * (a)

We say $\mathcal{F}$ is $(R_{q})$ if there is an open subscheme $U\subseteq X$
of $X$ such that $\operatorname{codim}_{X}(X\setminus U)\geq q+1$, $U$ is
smooth and $\mathcal{F}|_{U}$ is a vector bundle on $U$

  * (b)

We say $X$ is $(R_{q})$ if $\mathcal{O}_{X}$ is $(R_{q})$.

  * (c)

We say $\mathcal{F}$ is $(S_{q+1})$ if $H^{i}_{x}(X,\mathcal{F})=0$ for
$i<\min\\{q+1,\operatorname{codim}_{X}x\\}$ and any point $x\in X$.

  * (d)

We say $\mathcal{F}$ is _Cohen-Macaulay (CM)_ if $\mathcal{F}$ is $(S_{\dim
X})$.

  * (e)

We say $X$ is _Cohen-Macaulay (CM)_ if $\mathcal{O}_{X}$ is CM.

  * (f)

We say $\mathcal{F}$ is _reflexive_ if $\mathcal{F}$ is $(S_{2})$.

######  Remark 2.2.

If $X$ is any normal variety with $(R_{q})$, then any reflexive sheaf on $X$
of rank $1$ is also $(R_{q})$. An analogue for $(S_{q+1})$ does not hold.
Indeed, let $X:=\operatorname{Spec}k[x,y,z,w]/(xy-zw)$. Then $X$ is CM but not
any reflexive sheaf on $X$ of rank $1$ is not CM. See 3.15 in [Kol13].

We will use derived category machinery in the paper, especially in Section 4
and Section 5. Hence, it is worth stating the _local duality_ and the
_Grothendieck duality_.

######  Theorem 2.3 (See Lemma 0AAK in [Stacks]).

localduality Let $X$ be any noetherian schemes that has a dualizing complex,
$x\in X$ any closed point and $E$ the injective hull of the residue field of
$\mathcal{O}_{X,x}$. Then

$\operatorname{\mathrm{Ext}}^{-i}_{\mathcal{O}_{X,x}}(K,\omega^{\bullet}_{X})^{\wedge}_{x}\cong\operatorname{\mathrm{Hom}}_{\mathcal{O}_{X,x}}(H^{i}_{x}(X,K),E)$

for any $i$ and $K\in D(X)$, where $(-)^{\wedge}_{x}$ denotes the completion
along the maximal ideal $\mathfrak{m}_{X,x}$ of $\mathcal{O}_{X,x}$ and $D(X)$
denotes the derived category of bounded complexes of coherent sheaves on $X$.

######  Theorem 2.4 (See 0AU3, (4) in [Stacks]).

Let $X,X^{\prime}$ be any noetherian schemes that have dualizing complexes and
$f:X^{\prime}\to X$ any proper morphism of varieties. For any $K\in
D(X^{\prime})$,

$R\mathcal{H}om_{\mathcal{O}_{X}}(Rf_{*}K,\omega^{\bullet}_{X})\cong
Rf_{*}R\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}(K,\omega^{\bullet}_{X^{\prime}})\text{
in }D(X).$

The following theorem is well-known as the _relative Kawamata-Viehweg
vanishing theorem_.

######  Theorem 2.5 (See [KMM87], Theorem 1-2-3).

Let $X^{\prime}$ any smooth projective variety over a field of characteristic
$0$ and $X$ any projective variety. Let $f:X^{\prime}\to X$ be any proper
birational morphism. For any $f$-nef Cartier divisor $D$ on $X^{\prime}$,
$R^{i}f_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D)=0$ for $i\geq 1$.

Let us prove the following duality result. For a similar lemma, see
Proposition 11.6 in [Kol97].

######  Lemma 2.6.

warmup1 Let $f:X^{\prime}\to X$ be any proper birational morphism between
normal varieties and $\mathcal{E}$ any vector bundle on $X^{\prime}$. Fix any
point $x\in X$. Suppose $X^{\prime}$ is Gorenstein. Then for $i\geq 0$,
$(R^{i}f_{*}\mathcal{E})_{x}=0$ if and only if
$H^{\operatorname{codim}_{X}x-i}_{f^{-1}(x)}(X^{\prime},\omega_{X^{\prime}}\otimes\mathcal{E}^{\vee})=0$.

###### Proof.

Let $k$ be the residue field of $\mathcal{O}_{X,x}$ and $E$ the injective hull
of $k$ as an $\mathcal{O}_{X,x}$-module. We have

$\displaystyle
R\Gamma_{f^{-1}(x)}(X^{\prime},\omega_{X^{\prime}}\otimes\mathcal{E}^{\vee})$
$\displaystyle=R\Gamma_{x}(X,Rf_{*}(\omega_{X^{\prime}}\otimes\mathcal{E}^{\vee}))$
$\displaystyle=\operatorname{\mathrm{Hom}}_{k}(R\mathcal{H}om_{\mathcal{O}_{X}}(Rf_{*}(\omega_{X^{\prime}}\otimes\mathcal{E}^{\vee})_{x},\omega^{\bullet}_{X,x}),E)$
$\displaystyle=\operatorname{\mathrm{Hom}}_{k}(Rf_{*}R\mathcal{H}om_{\mathcal{O}_{X}}(\omega_{X^{\prime}}\otimes\mathcal{E}^{\vee},\omega_{X^{\prime}})_{x}[\operatorname{codim}_{X}x],E)$
$\displaystyle=\operatorname{\mathrm{Hom}}_{k}(Rf_{*}\mathcal{H}om_{\mathcal{O}_{X}}(\omega_{X^{\prime}}\otimes\mathcal{E}^{\vee},\omega_{X^{\prime}})_{x}[\operatorname{codim}_{X}x],E)$
$\displaystyle=\operatorname{\mathrm{Hom}}_{k}((Rf_{*}\mathcal{E})_{x}[\operatorname{codim}_{X}x],E),$

where we used the Leray spectral sequence in the first equality, the local
duality in the second equality, the Grothendieck duality in the third
equality, the fact that $\omega_{X^{\prime}}$ is a line bundle on $X^{\prime}$
in the forth equality. Hence, since $E$ is an injective
$\mathcal{O}_{X,x}$-module, we have the assertion. ∎

Let us prove the following corollary.

######  Corollary 2.7 (See Lemma 3.5.10 in [Nak04] and Corollary 11.7 in
[Kol97]).

warmup2 Let $X$ be any normal projective variety over a field of
characteristic $0$, $f:X^{\prime}\to X$ any resolution of $X$ and $D$ any
divisor on $X^{\prime}$, which is anti $f$-nef. Then
$f_{*}\mathcal{O}_{X^{\prime}}(D)=\mathcal{O}_{X}(f_{*}D)$.

###### Proof.

Note that by definition,
$\mathcal{O}_{X}(f_{*}D):=(f_{*}\mathcal{O}_{X^{\prime}}(D))^{\vee\vee}$.
Hence, it suffices to show that $f_{*}\mathcal{O}_{X^{\prime}}(D)$ is
reflexive. Let $x\in X$ be any point of codimension $\geq 2$.

Consider the following spectral sequence

$E^{st}_{2}=H^{s}_{x}(X,R^{t}f_{*}\mathcal{O}_{X^{\prime}}(D))\implies
H^{s+t}_{f^{-1}(x)}(X^{\prime},\mathcal{O}_{X^{\prime}}(D)).$

Inspecting the spectral sequence, we have $E^{10}_{2}=E^{10}_{\infty}$. Hence,
by considering the edge map $E^{10}_{\infty}\to E^{1}$, it suffices to show
that $H^{1}_{f^{-1}(x)}(X^{\prime},\mathcal{O}_{X^{\prime}}(D))=0$. Note that
by the relative Kawamata-Viehweg vanishing theorem,
$R^{i}f_{*}\mathcal{O}_{X}(K_{X^{\prime}}-D)=0$ for $i\geq 1$. Thus, by
warmup1, we have the assertion. ∎

######  Remark 2.8.

Note that it is a special case of [Nak04], Lemma 3.5.10. Nakayama proved the
lemma using the relative Zariski decomposition.

We may use the following lemma without any mention.

######  Lemma 2.9 (See Lemma 2.2 in [BMP+20]).

For any normal variety $X$ and any coherent sheaf $\mathcal{F}$ on $X$, we
have

$\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega^{\bullet}_{X})[-\dim
X]=\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega_{X}).$

###### Proof.

We may consider the following exact triangle

$\omega_{X}[-\dim X]\to\omega^{\bullet}_{X}\to C\to$

for some complex $C$ in $D(X)$. If we apply
$R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},-)$ to this triangle, we obtain
the following exact triangle

$R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega_{X}[-\dim X])\to
R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega^{\bullet}_{X})\to
R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},C)\to.$

Note that $C$ has cohomological degree $\geq 1$ and hence
$\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},C)=0$. Thus, the long exact
cohomology sequence gives us the assertion. ∎

##  3\. Reflexive sheaves

In this section, we collect important facts about reflexive sheaves. For a
variety $X$, the generic point $\eta\in X$ and a coherent sheaf $\mathcal{F}$
on $X$, $\mathrm{rank}\,\mathcal{F}$ denotes the dimension of
$\mathcal{F}_{\eta}$ over the function field of $X$.

######  Definition 3.1 (See [HL10], Definition 1.1.7).

dual Let $X$ be any variety and $\mathcal{F}$ any coherent sheaf on $X$. We
define
$\mathcal{F}^{D}:=\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega_{X})$.

######  Remark 3.2.

Under the setting of dual, for any normal Gorenstein variety $X$ and reflexive
sheaf $\mathcal{F}$ on $X$,
$\mathcal{F}^{D}=\mathcal{F}^{\vee}\otimes\omega_{X}$. Moreover, we can
construct a natural map $\mathcal{F}\to\mathcal{F}^{DD}$. Indeed, for any
local section $a\in\mathcal{F}$, consider a map

$\varphi:\mathcal{F}\to\mathcal{F}^{DD},a\mapsto(f\mapsto f(a))\,\text{ for
}f\in\mathcal{F}^{D}:=\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega_{X}).$

Now, let us prove the following two lemmas.

######  Lemma 3.3.

Let $X$ be any normal variety and $\mathcal{F}$ any coherent sheaf on $X$.
Then the following are equivalent:

  * (a)

$\mathcal{F}$ is torsion-free.

  * (b)

The map $\mathcal{F}\to\mathcal{F}^{DD}$ is injective.

###### Proof.

It is worth noting that for any variety $X$, $\omega_{X}$ is $(S_{2})$. See
Lemma 0AWE in [Stacks]. Hence, if $X$ is normal, then $\omega_{X}$ is
reflexive.

If $\mathcal{F}$ is torsion-free, then the kernel $\mathcal{K}$ of the map is
torsion-free because any subsheaf of a torsion-free sheaf is also torsion-
free. Moreover, since $\omega_{X}$ is reflexive, $\mathcal{F}^{DD}$ is
torsion-free by Lemma 0AY4 in [Stacks]. Consider the fact that for every point
$x\in X$ of codimension $1$, $\mathcal{F}_{x}\to\mathcal{F}^{DD}_{x}$ is an
isomorphism. Then $\mathcal{K}=0$ because $\mathrm{rank}\,\mathcal{K}=0$.

For the converse, since $\omega_{X}$ is torsion-free, $\mathcal{F}^{DD}$ is
torsion-free. Hence, by our assumption, $\mathcal{F}$ is torsion-free because
any subsheaf of a torsion-free sheaf is also torsion-free. ∎

######  Lemma 3.4.

Let $X$ be any normal variety and $\mathcal{F}$ any torsion-free sheaf on $X$.
Then the following are equivalent:

  * (a)

$\mathcal{F}$ is reflexive.

  * (b)

The natural map $\mathcal{F}\to\mathcal{F}^{DD}$ is an isomorphism.

###### Proof.

Let us assume that $\mathcal{F}$ is reflexive. By Lemma 0AY4 in [Stacks], we
know that $\mathcal{F}^{DD}$ is also reflexive and
$\mathcal{F}\to\mathcal{F}^{DD}$ is an isomorphism outside codimension $\geq
2$ closed subscheme of $X$. Hence the map is an isomorphism by [Har94],
Proposition 1.11.

For the converse, let $\mathcal{E}_{1}\to\mathcal{E}_{0}\to\mathcal{F}^{D}\to
0$ be any resolution of $\mathcal{F}$ in which
$\mathcal{E}_{0},\mathcal{E}_{1}$ are vector bundles. By taking
$\mathcal{H}om_{\mathcal{O}_{X}}(-,\omega_{X})$, we have the following exact
sequence

$0\to\mathcal{F}\cong\mathcal{F}^{DD}\to\mathcal{E}^{D}_{0}\to\mathcal{E}^{D}_{1}$

and $\mathcal{E}^{D}_{i}$ are reflexive for $i=0,1$. Hence, $\mathcal{F}$ is
reflexive. ∎

######  Remark 3.5.

If $X$ is not normal, then such equivalence is false. See Remark 0AY1 in
[Stacks].

Let us note an important fact about reflexive sheaves. For any normal variety
$X$ and any reflexive sheaf $\mathcal{F}$ on $X$, $\mathcal{F}$ is a vector
bundle outside codimension $\geq 2$ closed subscheme of $X$. See Lemma 0AY6 in
[Stacks].

We will use the following lemmas without any mention. We believe that the
three following lemmas are well-known to experts but we can not find any
reference about them. Hence we include the proofs of them.

######  Lemma 3.6.

Let $X$ be any variety and $\varphi:\mathcal{F}\to\mathcal{G}$ any map between
torsion-free sheaves on $X$. Suppose that $\varphi$ is an isomorphism outside
codimension $\geq 1$ closed subscheme of $X$. Then $\varphi$ is an injection.

###### Proof.

Let $\mathcal{K}$ be the kernel of $\varphi$. Note that $\mathcal{K}$ is a
subsheaf of a torsion-free sheaf $\mathcal{F}$. By the condition on $\varphi$,
$\mathrm{rank}\,\mathcal{K}=0$ and hence $\mathcal{K}=0$, because any subsheaf
of torsion-free sheaf is also torsion-free. ∎

######  Lemma 3.7.

Let $X,X^{\prime}$ be any normal varieties and $f:X^{\prime}\to X$ any proper
birational morphism. Suppose that $\mathcal{F}$ is any reflexive sheaf on $X$.
Then $\mathcal{F}\cong f_{*}(f^{*}\mathcal{F})^{DD}$.

###### Proof.

Let us consider the adjunction property

$\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},f_{*}(f^{*}\mathcal{F})^{DD})\cong
f_{*}\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}(f^{*}\mathcal{F},(f^{*}\mathcal{F})^{DD})$

and the map $f^{*}\mathcal{F}\to(f^{*}\mathcal{F})^{DD}$. Then we can
construct the following map $\mathcal{F}\to f_{*}(f^{*}\mathcal{F})^{DD}$.
Note that the map is an injection because it is an isomorphism on the regular
locus of $X$ and $\mathcal{F},f_{*}(f^{*}\mathcal{F})^{DD}$ are torsion-free
sheaves.

Let $\mathcal{Q}$ be the cokernel. Then we have the following exact sequence

$0\to\mathcal{F}\to f_{*}(f^{*}\mathcal{F})^{DD}\to\mathcal{Q}\to 0.$

Note that the support of $\mathcal{Q}$ has codimension $\geq 2$. Let $x\in X$
be any point of codimension $\geq 2$. Then the local cohomology exact sequence
tells us that $H^{0}_{x}(X,\mathcal{Q})=0$. Hence $x$ is not an associated
point of $\mathcal{Q}$. Thus $\mathcal{Q}=0$. ∎

######  Lemma 3.8.

Let $X,Y$ be any normal varieties and $p:X^{\prime}\to X$ any proper flat
morphism. Suppose that $\mathcal{F}$ is any coherent sheaf on $X$. Then
$(p^{*}\mathcal{F})^{DD}=p^{*}(\mathcal{F}^{DD}).$

###### Proof.

Note that the pullback of a reflexive sheaf along a flat morphism is also
reflexive. See Proposition 1.8 in [Har80].

Considering a natural map $\mathcal{F}\to\mathcal{F}^{DD}$, we have a map
$\varphi:(p^{*}\mathcal{F})^{DD}\to p^{*}(\mathcal{F}^{DD})$. We know that
both $(p^{*}\mathcal{F})^{DD},p^{*}(\mathcal{F}^{DD})$ on $Y$ are reflexive
and $\varphi$ is an isomorphism outside codimension $\geq 2$ closed subscheme
of $Y$. Hence by [Har94], Proposition 1.11, we have the assertion. ∎

##  4\. Rational singularities

From this section and to the end of the paper, any reflexive sheaf is assumed
to be of rank $1$ unless otherwise stated.

######  Proposition 4.1.

defi1 Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$. Suppose that $f:X^{\prime}\to X$ is a resolution. Then we have a natural
map

$\theta_{\mathcal{F},f}:Rf_{*}(f^{*}\mathcal{F})^{D}\to
R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega^{\bullet}_{X})[-\dim X]$

Moreover, the following are equivalent:

  * (a)

$\theta_{\mathcal{F},f}$ is a quasi-isomorphism.

  * (b)

$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}=0$ for $i\geq 1$.

  * (c)

The natural map

$H^{i}_{x}(X_{x},\mathcal{F}_{x})\to
H^{i}_{f^{-1}(x)}(X^{\prime}_{x},(f^{*}\mathcal{F})^{DD}_{x})$

is an isomorphism for each $i\geq 0$ and each point $x\in X$.

###### Proof.

Note that the proof of Lemma 3.7 is inspired by the argument in the proof in
[ST08].

Let us prove $(a)\iff(b)$. Consider a map $\mathcal{F}\cong
f_{*}(f^{*}\mathcal{F})^{DD}\to Rf_{*}(f^{*}\mathcal{F})^{DD}$. We may take
$R\mathcal{H}om_{\mathcal{O}_{X}}(-,\omega^{\bullet}_{X})$ on the map and it
gives

$R\mathcal{H}om_{\mathcal{O}_{X}}(Rf_{*}(f^{*}\mathcal{F})^{DD},\omega^{\bullet}_{X})\to
R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega^{\bullet}_{X}).$

Now, we may compute the left-hand side by the Grothendieck duality. Indeed,

$\displaystyle
R\mathcal{H}om_{\mathcal{O}_{X}}(Rf_{*}(f^{*}\mathcal{F})^{DD},\omega^{\bullet}_{X})$
$\displaystyle=Rf_{*}R\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}((f^{*}\mathcal{F})^{DD},\omega_{X^{\prime}})[\dim
X]$
$\displaystyle=Rf_{*}\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}((f^{*}\mathcal{F})^{DD},\omega_{X^{\prime}})[\dim
X]$ $\displaystyle=Rf_{*}(f^{*}\mathcal{F})^{D}[\dim X],$

where we used the fact that $(f^{*}\mathcal{F})^{DD}$ is a line bundle on
$X^{\prime}$ on the second equality. Hence we have a map

$\theta_{\mathcal{F},f}:Rf_{*}(f^{*}\mathcal{F})^{D}\to
R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega^{\bullet}_{X})[-\dim X].$

If $(b)$ is true, then $\mathcal{F}\cong Rf_{*}(f^{*}\mathcal{F})^{DD}$. Thus
$\theta_{\mathcal{F},f}$ is a quasi-isomorphism and $(a)$ is true. For the
converse, we may take
$R\mathcal{H}om_{\mathcal{O}_{X}}(-,\omega^{\bullet}_{X})$ on
$\theta_{\mathcal{F},f}$. See 0AU3 (3) in [Stacks].

For $(a)\iff(c)$, we may take the local duality to $\theta_{\mathcal{F},f}$.
Indeed, the dual of $\theta_{\mathcal{F},f}$ is

$R\Gamma_{x}(X_{x},\mathcal{F}_{x})\to
R\Gamma_{x}(X_{x},Rf_{*}(f^{*}\mathcal{F})^{DD})$

and the right hand side is
$R\Gamma_{f^{-1}(\\{x\\})}(X^{\prime}_{x},(f^{*}\mathcal{F})^{DD})$ by the
Leray spectral sequence. ∎

######  Remark 4.2.

If we use the notions of defi1, $\mathcal{H}^{0}(\theta_{\mathcal{F},f})$ is

$\mathcal{H}^{0}(\theta_{\mathcal{F},f}):f_{*}(f^{*}\mathcal{F})^{D}\to\mathcal{F}^{D}=\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega_{X}).$

If we let $\mathcal{F}=\mathcal{O}_{X}$, then
$\mathcal{H}^{0}(\theta_{\mathcal{F},f})$ is the trace map introduced in
[KM98], Proposition 5.77.

######  Definition 4.3.

weak Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$. We say that $\mathcal{F}$ has _weak rational singularities_ if one of the
conditions in idefi1 is true for the $\mathcal{F}$ and for any resolution
$f:X^{\prime}\to X$.

It is worth saying that for characteristic $0$ case, the choice of resolution
is not important to the definition of weak rational singularities.

######  Proposition 4.4.

Let $X$ be any normal variety over a characteristic $0$ field and
$\mathcal{F}$ any reflexive sheaf on $X$. Then $\mathcal{F}$ has weak rational
singularities if and only if for some resolution $f:X^{\prime}\to X$,
$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}=0$ for $i\geq 1$.

###### Proof.

For the if direction, let $g:X^{\prime\prime}\to X$ be a resolution. Our goal
is to show $R^{i}g_{*}(g^{*}\mathcal{F})^{DD}=0$ for $i\geq 1$. Let us choose
any common resolution

${X^{\prime\prime\prime}}$${X^{\prime}}$${X^{\prime\prime}}$${X.}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$

We may consider the following maps

$(f\circ
f^{\prime})^{*}\mathcal{F}\to(f^{\prime})^{*}(f^{*}\mathcal{F})^{DD}\to((f\circ
f^{\prime})^{*}\mathcal{F})^{DD}.$

By taking double dual on the maps, we have maps

$((f\circ
f^{\prime})^{*}\mathcal{F})^{DD}\to(f^{\prime})^{*}(f^{*}\mathcal{F})^{DD}\to((f\circ
f^{\prime})^{*}\mathcal{F})^{DD}.$

Note that the composition is the identity. Since the above maps are
injections, we have $(f^{\prime})^{*}(f^{*}\mathcal{F})^{DD}=((f\circ
f^{\prime})^{*}\mathcal{F})^{DD}$.

Consider the following spectral sequence

$E^{st}_{2}=R^{s}f_{*}R^{t}(f^{\prime})_{*}(f^{\prime})^{*}(f^{*}\mathcal{F})^{DD}\implies
R^{s+t}(f\circ f^{\prime})(f^{*}((f^{\prime})^{*}\mathcal{F})^{DD}),$

by the fact that $R^{i}(f^{\prime})_{*}\mathcal{O}_{X^{\prime\prime\prime}}=0$
for $i\geq 1$ and the projection formula, we have

$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}=R^{i}(f\circ f^{\prime})_{*}((f\circ
f^{\prime})^{*}\mathcal{F})^{DD}\text{ for }i\geq 0.$

In the same way, we have

$R^{i}g_{*}(g^{*}\mathcal{F})^{DD}=R^{i}(g\circ g^{\prime})_{*}((g\circ
g^{\prime})^{*}\mathcal{F})^{DD}\text{ for }i\geq 0.$

Hence, we have the assertion. The only if direction is trivial. ∎

######  Remark 4.5.

We can not prove the above proposition on a positive characteristic field
because the existence of a common resolution is largely unknown in such a
setting. If $X$ is threefold, we can prove the above proposition.

######  Lemma 4.6.

tr Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on $X$
which has weak rational singularities. Then for any resolution
$f:X^{\prime}\to X$, the following are equivalent:

  * (a)

$\mathcal{F}$ is $(S_{q+1})$.

  * (b)

The support of $R^{i}f_{*}(f^{*}\mathcal{F})^{D}$ has codimension $\leq i+q+1$
for any $i\geq 1$ and for some resolution $f:X^{\prime}\to X$ of $X$.

  * (c)

The support of $R^{i}f_{*}(f^{*}\mathcal{F})^{D}$ has codimension $\leq i+q+1$
for any $i\geq 1$ and any resolution $f:X^{\prime}\to X$ of $X$.

###### Proof.

For $(a)\implies(c)$, by our assumption, there is an isomorphism

$\theta_{\mathcal{F},f}:Rf_{*}(f^{*}\mathcal{F})^{D}\cong
R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\omega^{\bullet}_{X})[-\dim X].$

Moreover, for any point $x\in X$ with $\operatorname{codim}_{X}x\leq i+q$,
$H^{\operatorname{codim}_{X}x-i}_{x}(X,\mathcal{F})=0$ holds, because
$\mathcal{F}$ is $(S_{q+1})$. Hence, by the local duality,
$\operatorname{\mathrm{Ext}}^{-\operatorname{codim}_{X}x+i}_{\mathcal{O}_{X,x}}(\mathcal{F}_{x},\omega^{\bullet}_{X,x})=0$
for such $i$. Thus, $R^{i}f_{*}(f^{*}\mathcal{F})^{D}_{x}=0$ and the support
of $R^{i}f_{*}(f^{*}\mathcal{F})^{D}$ does not contain $x$. $(c)\implies(b)$
is trivial.

For $(b)\implies(a)$, let us consider a point $x\in X$. If
$\operatorname{codim}_{X}x\leq q$, then
$R^{i}f_{*}(f^{*}\mathcal{F})^{D}_{x}=0$ and
$\operatorname{\mathrm{Ext}}^{-\operatorname{codim}_{X}x+i}_{\mathcal{O}_{X,x}}(\mathcal{F}_{x},\omega^{\bullet}_{X,x})=0$
hold for any $i\geq 1$. Hence, the local duality gives

$H^{\operatorname{codim}_{X}x-i}_{x}(X,\mathcal{F})=0\text{ for any }i\geq 1.$

If $\operatorname{codim}_{X}x\geq q+1$ and
$R^{i}f_{*}(f^{*}\mathcal{F})^{D}_{x}=0$, by the local duality,
$H^{\operatorname{codim}_{X}x-i}_{x}(X,\mathcal{F})=0$. Moreover, if
$\operatorname{codim}_{X}x-i<q$, then
$R^{i}f_{*}(f^{*}\mathcal{F})^{D}_{x}=0$. Hence, for $0\leq j<q$,
$H^{j}_{x}(X,\mathcal{F})=0$. ∎

######  Corollary 4.7.

defi2 Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$ which has weak rational singularities. Then the following are equivalent:

  * (a)

$\mathcal{F}$ is CM.

  * (b)

For a resolution $f:X^{\prime}\to X$, $R^{i}f_{*}(f^{*}\mathcal{F})^{D}=0$ for
$i\geq 1$.

  * (c)

For any resolution $f:X^{\prime}\to X$, $R^{i}f_{*}(f^{*}\mathcal{F})^{D}=0$
for $i\geq 1$.

Now, we can define the notion of _rational singularities_ for any reflexive
sheaf of rank $1$.

######  Definition 4.8.

defi3 Let $X$ be any normal variety, $\mathcal{F}$ any reflexive sheaf on $X$.

  * (a)

We say that $\mathcal{F}$ has _rational singularities_ if $\mathcal{F}$ has
weak rational singularities and one of the conditions in defi2 is true for any
resolution $f:X^{\prime}\to X$.

  * (b)

We say that a Weil divisor $D$ on $X$ has _rational singularities_ if
$\mathcal{O}_{X}(D)$ has rational singularities.

  * (c)

We say that $\mathcal{F}$ is $(KV_{q})$ if for any resolution $f:X^{\prime}\to
X$, the support of $R^{i}f_{*}(f^{*}\mathcal{F})^{D}$ has dimension $\geq
i+q+1$ for any $i\geq 1$.

######  Remark 4.9.

The above definitions are similar to Definition 2.5 in [Kov11] and Definition
2.78 in [Kol13]. However, there are four main differences.

  * •

We used double dual to define the notion of rational singularities instead of
strict transform. Thus the notion of normal pair in [Kov11] is unnecessary.

  * •

It is obvious that some Cartier divisor on $X$ has rational singularities in
our sense if and only if $X$ has rational singularities. In contrast to our
notion, it is far from obvious that for some reduced, effective and Cartier
divisor $D$ on $X$, $(X,D)$ has rational singularities in the sense of [Kov11]
if and only if $X$ has rational singularities. For the only if assertion, see
Corollary 2.13 in [Kov11]. Note that the proof of Corollary 2.13 in [Kov11]
implicitly used our notion of rational singularities.

  * •

Moreover, we defined the notion of rational singularities for any reflexive
sheaf of rank $1$, not only for reduced Weil divisor.

  * •

It seems hard to believe a good restriction of a reflexive sheaf with rational
singularities also has rational singularities unlike Remark 2.85 in [Kol13]
because our notion is for reflexive sheaves. For treating this problem, one
may consider a combination of our notion and the notion of [Kov11].

######  Remark 4.10.

If $k$ is of characteristic $0$ and $\mathcal{F}\cong\mathcal{O}_{X}(D)$ for
some $\mathbb{Q}$-Cartier Weil divisor $D$, we have
$(f^{*}\mathcal{O}_{X}(D))^{D}\cong\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}-L)$
for some Cartier divisor $L$ on $X^{\prime}$ and for such $L$,
$f^{*}D\sim_{\mathbb{Q}}L$. In that setting, $-L$ is $f$-nef and the relative
Kawamata-Viehweg vanishing theorem tells us that
$R^{i}f_{*}(f^{*}\mathcal{F})^{D}=0$ for $i\geq 1$. Hence, for such
$\mathcal{F}$, having weak rational singularities and having rational
singularities are the same.

######  Example 4.11.

ex If $k$ is of characteristic $0$ and $\mathcal{F}\cong\mathcal{O}_{X}(D)$
where $D$ is a Weil divisor on $X$ which is not $\mathbb{Q}$-Cartier, then
having weak rational singularities does not ensure having rational
singularities.

For example, take $X:=\mathrm{Spec}\;k[x,y,z,w]/(xy-zw)$, $A:=\\{x=z=0\\}$,
the blowup $f:X^{\prime}\to X$ along $A$ and the exceptional divisor $E$ on
$X^{\prime}$. We know that $f$ is a small resolution and
$(f^{*}\mathcal{O}_{X}(-mA))^{DD}=\mathcal{O}_{X^{\prime}}(-mE)$. Then for
$m\gg 0$, $R^{i}f_{*}(f^{*}\mathcal{O}_{X^{\prime}}(-mA))^{DD}=0$ for $i\geq
1$. Thus $-mA$ has weak rational singularities. However, for $m\geq 2$, $-mA$
is not CM as in (3.15) in [Kol13] and hence $-mA$ does not have rational
singularities.

Let us prove the following lemma. Similar statements are proved in Lemma 3.2
in [Ale08] and Theorem 7.1.1 in [Fuj17].

######  Lemma 4.12.

twospectral Let $f:X^{\prime}\to X$ be any proper birational morphism of
varieties and $\mathcal{F},\mathcal{F}^{\prime}$ any coherent sheaves on
$X^{\prime}$. For any given positive integer $n\geq 1$, suppose the three
conditions hold:

  * (1)

There is an injection $\imath:\mathcal{F}\to\mathcal{F^{\prime}}$,

  * (2)

$\imath$ induces an isomorphism $f_{*}\mathcal{F}\cong
f_{*}\mathcal{F}^{\prime}$, and

  * (3)

$R^{i}f_{*}\mathcal{F}^{\prime}=0$ for $1\leq i<n$.

Then we have $R^{i}f_{*}\mathcal{F}=0$ for $1\leq i<n$.

###### Proof.

Fix any point $x\in X$ and consider the following two spectral sequences

$\displaystyle{}^{1}E^{st}_{2}=H^{s}_{x}(X,R^{t}f_{*}\mathcal{F})\implies
H^{i+j}_{f^{-1}(x)}(X^{\prime},\mathcal{F}),$
$\displaystyle{}^{2}E^{st}_{2}=H^{s}_{x}(X,R^{t}f_{*}\mathcal{F}^{\prime})\implies
H^{i+j}_{f^{-1}(x)}(X^{\prime},\mathcal{F^{\prime}}).$

By (3), we have ${}^{2}E^{i0}_{2}={}^{2}E^{i0}_{\infty}$ and hence the edge
map ${}^{2}E^{i0}_{2}\to{}^{2}E^{i}$ is an injection for any $0\leq i\leq n$.
If we use (1), then there is a diagram

${{}^{1}E^{i}=H^{i}_{f^{-1}(x)}(X^{\prime},\mathcal{F})}$${{}^{2}E^{i}=H^{i}_{f^{-1}(x)}(X^{\prime},\mathcal{F}^{\prime})}$${{}^{1}E^{i0}_{2}=H^{i}_{x}(X,f_{*}\mathcal{F})}$${{}^{2}E^{i0}_{2}=H^{i}_{x}(X,f_{*}\mathcal{F}^{\prime})}$$\scriptstyle{\gamma_{i}}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{\beta_{i}}$

By (2), $\gamma_{i}$ is an isomorphism for any $i$. Since $\beta_{i}$ is an
injection for $1\leq i\leq n$, $\alpha_{i}$ is also an injection for such $i$.

Now, we may use induction. Indeed, suppose for a positive integer
$1<n^{\prime}\leq n$, $R^{i}f_{*}\mathcal{F}=0$ for $1\leq i<n^{\prime}$. Our
goal is to show $R^{n^{\prime}}f_{*}\mathcal{F}=0$. By the induction
hypothesis, we have ${}^{1}E^{0n^{\prime}}_{2}={}^{1}E^{0n^{\prime}}_{n}$ and
${}^{1}E^{(n^{\prime}+1)0}_{2}={}^{1}E^{(n^{\prime}+1)0}_{n}$. Hence, there is
an exact sequence

$0\to{}^{1}E^{0n^{\prime}}_{2}\to{}^{1}E^{(n^{\prime}+1)0}_{2}\overset{\beta_{n^{\prime}+1}}{\to}{}^{1}E^{n^{\prime}+1}.$

Since $\beta_{n^{\prime}+1}$ is an isomorphism, we have
$H^{0}_{x}(X,R^{n^{\prime}+1}f_{*}\mathcal{F})={}^{1}E^{(n^{\prime}+1)0}_{2}=0$
and hence $R^{n^{\prime}+1}f_{*}\mathcal{F}$ does not have an associated point
as $x$. Thus $R^{n^{\prime}+1}f_{*}\mathcal{F}=0$ and we have the assertion. ∎

Now, we have an alternative description of the definition of rational
singularities.

######  Proposition 4.13.

stronger Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf
on $X$. Then $\mathcal{F}$ has rational singularities if and only if for any
resolution $f:X^{\prime}\to X$, there is a line bundle
$\mathcal{L}_{\mathcal{F},f}$ on $X^{\prime}$ such that the three following
conditions hold:

  * (i)

$f_{*}\mathcal{L}_{\mathcal{F},f}=\mathcal{F}$,

  * (ii)

$R^{i}f_{*}\mathcal{L}_{\mathcal{F},f}=0$ for $i\geq 1$, and

  * (iii)

$R^{i}f_{*}\mathcal{L}^{D}_{\mathcal{F},f}=0$ for $i\geq 1$.

###### Proof.

For the only if direction, set
$\mathcal{L}_{\mathcal{F},f}:=(f^{*}\mathcal{F})^{DD}$.

For the if direction, let us consider the counit map
$f^{*}f_{*}\mathcal{L}_{\mathcal{F},f}\to\mathcal{L}_{\mathcal{F},f}$. By
taking double dual, we have an injection
$(f^{*}f_{*}\mathcal{L}_{\mathcal{F},f})^{DD}\to\mathcal{L}_{\mathcal{F},f}.$
Since (i) holds, $f^{*}f_{*}\mathcal{L}_{\mathcal{F},f}=f^{*}\mathcal{F}$ and
hence $(f^{*}f_{*}\mathcal{L}_{\mathcal{F},f})=(f^{*}\mathcal{F})^{DD}$. Thus,
there is an injection $(f^{*}\mathcal{F})^{DD}\to\mathcal{L}_{\mathcal{F},f}$.

Considering twospectral, (i) and (ii), we have
$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}=0$ for $i\geq 1$. Let us use the notations
in twospectral. By (iii) and warmup1, we have

$H^{i}_{f^{-1}(x)}(X,\mathcal{L}_{\mathcal{F},f})=0\text{ for
}i<\operatorname{codim}_{X}x.$

Since $\beta_{i}$ is an isomorphism for $i\geq 1$, we have

$H^{i}_{x}(X,\mathcal{F})=H^{i}_{f^{-1}(x)}(X^{\prime},\mathcal{L}_{\mathcal{F},f})=0\text{
for }i<\operatorname{codim}_{X}x$

and hence $\mathcal{F}$ is CM. Thus $\mathcal{F}$ has rational singularities.
∎

######  Remark 4.14.

stronger means that for any normal variety $X$ and any reduced Weil divisor
$D$ on $X$, if $(X,D)$ has rational singularities in the sense of [Kov11],
then $-D$ has rational singularities in our sense.

Now, we may consider the following two properties.

######  Proposition 4.15.

ex1 Let $X$ any normal variety over a characteristic $0$ field, $\Delta$ any
effective $\mathbb{Q}$-Weil divisor such that $(X,\Delta)$ is klt and $D$ any
$\mathbb{Q}$-Cartier Weil divisor on $X$. Then $D$ has rational singularities.

###### Proof.

Indeed, suppose $f:X^{\prime}\to X$ is any resolution of $X$ and
$K_{X^{\prime}}\sim_{\mathbb{Q}}f^{*}(K_{X}+\Delta)+F-F^{\prime}$, where $F$
is an effective, $f$-exceptional Cartier divisor and $F^{\prime}$ is a simple
normal crossing divisor with $\left\lfloor F^{\prime}\right\rfloor=0$. Then
$F\sim_{\mathbb{Q}}K_{X^{\prime}}-f^{*}(K_{X}+\Delta)+F^{\prime}$. If we let a
divisor $L$ on $X^{\prime}$ in which
$\mathcal{O}_{X^{\prime}}(L)=(f^{*}\mathcal{O}_{X^{\prime}}(D))^{DD}$ holds,
we have

$F+L\sim_{\mathbb{Q}}K_{X^{\prime}}+F^{\prime}+(f^{*}D-f^{*}(K_{X}+\Delta)).$

Hence by the relative Kawamata-Viehweg vanishing theorem,
$R^{i}f_{*}\mathcal{O}_{X}(F+L)=0$. If we consider the injection
$\mathcal{O}_{X}(L)\to\mathcal{O}_{X}(F+L)$, we have
$R^{i}f_{*}\mathcal{O}_{X}(L)=0$ by twospectral and hence $D$ has rational
singularities. ∎

######  Remark 4.16.

Note that ex1 is a generalization of Corollary 5.25 in [KM98] and the proof
does not require the cyclic covering theory.

######  Proposition 4.17.

ex11 Let $X$ any strongly $F$-regular variety over a field of characteristic
$p>0$ and $D$ any Weil divisor such that there is an integer $r$ such that
$rD$ is linearly equivalent to a Cartier divisor and $r$ is relatively prime
to $p$. Then $D$ has rational singularities.

###### Proof.

Since having rational singularities is a local property, we may assume
$X:=\operatorname{Spec}R$ is an affine scheme and $rD\sim 0$. Moreover, by
multiplying some integer on $r$, we may take $r=p^{e}-1$ for sufficiently
large $e$. Let $x\in X$ be a point of $X$ and $f:X^{\prime}\to X$ any
resolution. We follow the proof of Theorem 3.1 in [PS14].

We may equip an endomorphism
$F^{e}_{*}:H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))\to
H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))$ on
$H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))$ as an endomorphism
of an abelian group. Indeed, if we consider the inclusion
$\mathcal{O}_{X}\subseteq F^{e}_{*}\mathcal{O}_{X}$, by tensoring
$\mathcal{O}_{X}(D)$ and reflexing, we have a map $\mathcal{O}_{X}(D)\to
F^{e}_{*}\mathcal{O}_{X}(p^{e}D)$. Since $(p^{e}-1)D\sim 0$ holds, we have a
map $\mathcal{O}_{X}(D)\to F^{e}_{*}\mathcal{O}_{X}(D)$. Taking local
cohomology gives the desired map. Let us call a submodule $K\subseteq
H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))$ _$F$ -stable_ if
$F^{e}_{*}(K)\subseteq K$.

Let $K\subsetneq H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))$ be
any $F$-stable submodule of
$H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))$ and let
$c\in\mathrm{Ann}_{R}K$ be any nontrivial element of $\mathrm{Ann}_{R}K$. Note
that such an element exists because of the Matlis duality. Indeed, if
$R^{\wedge}$ is the completion of $R$ along $x$, $N^{\wedge}:=R^{\wedge}N$ for
any $R$-module $N$ and $N^{\prime}$ is the $R$-module corresponding to
$\mathcal{O}_{X}(D)$, the inclusion $K\subsetneq
H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))$ gives us a surjection

$\mathrm{Hom}_{R^{\wedge}}((N^{\prime})^{\wedge},\omega_{R^{\wedge}})\to\mathrm{Hom}_{R^{\wedge}}(K^{\wedge},E),$

where $E$ is the injective hull of $R^{\wedge}$ (see Theorem 3.5.8 in [BH98]).
Since $\omega_{R^{\wedge}},(N^{\prime})^{\wedge}$ are torsion-free,

$0\neq\mathrm{Ann}_{R^{\wedge}}(\mathrm{Hom}_{R^{\wedge}}(K^{\wedge},E))=\mathrm{Ann}_{R^{\wedge}}(K^{\wedge}).$

Now, consider Lemma 07T8 in [Stacks] (note that the natural map $R\to
R^{\wedge}$ is a flat map).

Since $X$ is strongly $F$-regular, there is a positive integer $e^{\prime}$
and a map $\varphi:F^{e^{\prime}}_{*}\mathcal{O}_{X}\to\mathcal{O}_{X}$ such
that the composition

(4.1) $\mathcal{O}_{X}\to
F^{e^{\prime}}_{*}\mathcal{O}_{X}\overset{F^{e^{\prime}}_{*}(\cdot
c)}{\to}F^{e^{\prime}}_{*}\mathcal{O}_{X}\overset{\varphi}{\to}\mathcal{O}_{X}$

is the identity. We may assume $e|e^{\prime}$ and hence
$(p^{e}-1)|(p^{e^{\prime}}-1)$. Twisting by $\mathcal{O}_{X}(D)$ and reflexing
on (4.1), we have the following composition which is the identity

$\mathcal{O}_{X}(D)\to F^{e^{\prime}}_{*}\mathcal{O}_{X}(p^{e^{\prime}}D)\cong
F^{e^{\prime}}_{*}\mathcal{O}_{X}(D)\to
F^{e^{\prime}}_{*}\mathcal{O}_{X}(p^{e^{\prime}}D)\cong
F^{e^{\prime}}_{X}\mathcal{O}_{X}(D)\to\mathcal{O}_{X}(D).$

Now, taking local cohomology gives us

$\displaystyle H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))$
$\displaystyle\to$ $\displaystyle
H^{\operatorname{codim}_{X}x}_{x}(X,F^{e^{\prime}}_{*}\mathcal{O}_{X}(D))$
$\displaystyle\overset{F^{e^{\prime}}_{*}(\cdot c)}{\to}$ $\displaystyle
H^{\operatorname{codim}_{X}x}_{x}(X,F^{e^{\prime}}_{*}\mathcal{O}_{X}(D))$
$\displaystyle\to$ $\displaystyle
H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D)).$

Since $K$ is $F$-stable and $cK=0$ holds, $K=0$.

We may mimic the strategy of Smith (see [Smi97], Theorem 3.1). Now, consider
the Leray spectral sequence

(4.2)
$E^{st}_{2}=H^{s}_{x}(X,R^{t}f_{*}(f^{*}\mathcal{O}_{X}(D))^{DD})\implies
H^{s+t}_{f^{-1}(x)}(X^{\prime},(f^{*}\mathcal{O}_{X}(D))^{DD})$

and its edge map
$\delta_{D}:H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))\to
H^{\operatorname{codim}_{X}x}_{f^{-1}(x)}(X,(f^{*}\mathcal{O}_{X}(D))^{DD})$.
Let $K_{D}$ be the kernel of $\delta_{D}$. If we consider Proposition 1.12 in
[Smi97] and set $\psi$ by

$\psi:(f^{*}\mathcal{O}_{X}(D))^{DD}\to
F^{e}_{*}(f^{*}\mathcal{O}_{X}(p^{e}D))^{DD},$

we have a diagram

${H^{\operatorname{codim}_{X}x}_{f^{-1}(x)}(X’,(f^{*}\mathcal{O}_{X}(D))^{DD})}$${H^{\operatorname{codim}_{X}x}_{f^{-1}(x)}(X’,(f^{*}\mathcal{O}_{X}(D))^{DD})}$${H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))}$${H^{\operatorname{codim}_{X}x}_{x}(X,\mathcal{O}_{X}(D))}$$\scriptstyle{F^{e}_{*}}$$\scriptstyle{F^{e}_{*}}$$\scriptstyle{\delta_{D}}$$\scriptstyle{\delta_{D}}$

by considering the argument in the proof of Theorem 3.1 in [Smi97] almost
unchangingly. For any $c\in K_{D}$,
$\delta_{D}(F^{e}_{*}(c))=F^{e}_{*}(\delta_{D}(c))=0$ and hence $K_{D}$ is
$F$-stable. Therefore $K_{D}=0$. It means $\delta_{D}$ is an injection.

We may use induction. Suppose $n\geq 2$ is an integer and assume
$R^{i}f_{*}(f^{*}\mathcal{O}_{X}(D))^{DD}=0$ for $1\leq i<n-1$. Then by
inspecting (4.2), we have an exact sequence

$0\to H^{0}_{x}(X,R^{n-1}f_{*}(f^{*}\mathcal{O}_{X}(D))^{DD})\to
H^{n}_{x}(X,\mathcal{O}_{X}(D))\overset{\delta_{n,D}}{\to}H^{n}_{f^{-1}(x)}(X^{\prime},(f^{*}\mathcal{O}_{X}(D))^{DD}).$

For $n<\operatorname{codim}_{X}x$, we know $\mathcal{O}_{X}(D)$ is CM by
Theorem 3.2 in [PS14] and hence $\delta_{n,D}$ is an injection. Moreover, we
have proven that $\delta_{\operatorname{codim}_{X}x,D}$ is an injection. Thus,
$H^{0}_{x}(X,R^{n}f_{*}(f^{*}\mathcal{O}_{X}(D))^{DD})=0$ and
$R^{n}f_{*}(f^{*}\mathcal{O}_{X}(D))^{DD}$ does not have an associated point
as $x$. This argument works for any point $x\in X$ and therefore
$R^{n-1}f_{*}(f^{*}\mathcal{O}_{X}(D))^{DD}=0$. By considering defi1, $D$ has
weak rational singularities.

The remaining is to show $R^{i}f_{*}(f^{*}\mathcal{O}_{X}(D))^{D}=0$ for
$i\geq 1$. By warmup1, it suffices to show
$H^{\operatorname{codim}_{X}x-i}_{f^{-1}(x)}(X^{\prime},(f^{*}\mathcal{O}_{X}(D))^{DD})=0$
for $i\geq 1$. If we consider (4.2), $E^{st}_{2}=0$ for
$s+t<\operatorname{codim}_{X}x$ and hence $E^{\operatorname{codim}_{X}x-i}=0$
and that is the assertion. ∎

######  Example 4.18.

Let us consider what happens to klt varieties over a positive characteristic
field. On the positive side, suppose $X$ is any normal threefold over a field
of characteristic $p>5$. Assume that there is an effective
$\mathbb{Q}$-divisor $\Delta$ on $X$ such that $(X,\Delta)$ is klt. If we
consider Theorem 3 in [BK21], in the same way as ex1, we can prove that $X$
has rational singularities. There is a question of whether any
$\mathbb{Q}$-Cartier Weil divisor on $X$ has rational singularities.

On the negative side, if the characteristic of $k$ is $3$, then there is an
example of klt $\mathbb{Q}$-factorial threefold $X$ such that $X$ does not
have rational singularities (see Theorem 1.2 in [Ber21]). Moreover, for any
prime number $p>2$ and any field $k$ with characteristic $p$, there is an
example of variety with terminal singularities which does not have rational
singularities (see Corollary 2.2 in [Tot19]).

Let us prove that having rational singularities is stable under finite étale
morphism. We believe it might be a partial answer to Remark 2.81 (3) in
[Kol13].

######  Proposition 4.19.

quo Let $X,Y$ be normal varieties over characteristic $0$ field, $p:Y\to X$
any finite étale morphism and $\mathcal{F}$ any reflexive sheaf on $X$. If
$p^{*}\mathcal{F}$ has (resp. weak) rational singularities, then $\mathcal{F}$
has (resp. weak) rational singularities.

###### Proof.

By the functoriality of resolution of singularities (see Theorem 3.36 in
[Kol07]), for some resolution $f:X^{\prime}\to X$, there is a resolution
$f^{\prime}:Y^{\prime}\to Y$ such that $f,f^{\prime}$ fit in the following
Cartesian diagram:

${Y^{\prime}\cong
X^{\prime}\times_{X}Y}$${Y}$${X^{\prime}}$${X.}$$\scriptstyle{p^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{p}$$\scriptstyle{f}$

Note that $p^{\prime}$ is finite étale because being finite étale is stable
under base change (see Lemma 01TS and Lemma 02GO in [Stacks]). Moreover, for
any coherent sheaf $\mathcal{H}$ on $X^{\prime}$, we have a split injection
$\mathcal{H}\to(p^{\prime})_{*}(p^{\prime})^{*}\mathcal{H}$. Indeed, let $n$
be the degree of $p$. If we let
$(p^{\prime})^{!}\mathcal{H}:=\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}(p_{*}\mathcal{O}_{Y^{\prime}},\mathcal{H})$
as in Exercises 3.6.10 in [Har77], by the trace
$\frac{1}{n}\mathrm{Tr}:(p^{\prime})_{*}(p^{\prime})^{*}\mathcal{O}_{Y^{\prime}}\to\mathcal{O}_{X^{\prime}}$
which is a right inverse of the natural map
$\varphi:\mathcal{O}_{X^{\prime}}\to p_{*}\mathcal{O}_{Y^{\prime}}$, we have a
split injection
$\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}\left(\frac{1}{n}\mathrm{Tr},\mathcal{H}\right):\mathcal{H}\to(p^{\prime})_{*}(p^{\prime})^{!}\mathcal{H}$
whose right inverse is
$\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}(\varphi,\mathcal{H})$. Now, by Lemma
0FWI in [Stacks], $p^{*}=p^{!}$ and we are done.

Note that for any coherent sheaf $\mathcal{G}$ on $Y$,
$R^{i}p_{*}\mathcal{G}=0$ for $i\geq 1$ by Lemma 02OE in [Stacks]. Now, we
consider the following two Grothendieck spectral sequences

$\displaystyle{}^{1}E^{st}_{2}$
$\displaystyle=R^{s}p_{*}(R^{t}(f^{\prime})_{*}((f^{\prime})^{*}(p^{*}\mathcal{F}))^{DD})\implies
R^{s+t}(p\circ f^{\prime})_{*}((f^{\prime})^{*}(p^{*}\mathcal{F}))^{DD}$
$\displaystyle{}^{2}E^{st}_{2}$
$\displaystyle=R^{s}f_{*}(R^{t}(p^{\prime})_{*}((p^{\prime})^{*}(f^{*}\mathcal{F})^{DD}))\implies
R^{s+t}(f\circ p^{\prime})_{*}((p^{\prime})^{*}(f^{*}\mathcal{F})^{DD}).$

Note that

$((f^{\prime})^{*}(p^{*}\mathcal{F}))^{DD}=((p\circ
f^{\prime})^{*}\mathcal{F})^{DD}=((f\circ
p^{\prime})^{*}\mathcal{F})^{DD}=((p^{\prime})^{*}(f^{*}\mathcal{F}))^{DD}=(p^{\prime})^{*}(f^{*}\mathcal{F})^{DD}.$

Hence, by inspecting the spectral sequences, we have

$p_{*}(R^{i}(f^{\prime})_{*}((f^{\prime})^{*}(p^{*}\mathcal{F}))^{DD})=R^{i}f_{*}((p^{\prime})_{*}((p^{\prime})^{*}(f^{*}\mathcal{F})^{DD}))$

for any $i\geq 0$.

Suppose $p^{*}\mathcal{F}$ has weak rational singularities that
$R^{i}(f^{\prime})_{*}((f^{\prime})^{*}(p^{*}\mathcal{F}))^{DD}=0$ for $i\geq
1$. Then
$R^{i}f_{*}((p^{\prime})_{*}((p^{\prime})^{*}(f^{*}\mathcal{F})^{DD}))=0$ for
$i\geq 1$. By the splitting

(4.3)
$(f^{*}\mathcal{F})^{DD}\to(p^{\prime})_{*}(p^{\prime})^{*}(f^{*}\mathcal{F})^{DD}\to(f^{*}\mathcal{F})^{DD}$

and taking $R^{i}f_{*}$ on (4.3), we also have
$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}=0$ for $i\geq 1$. Therefore $\mathcal{F}$
has weak rational singularities.

Let us prove that if $p^{*}\mathcal{F}$ is CM, then $\mathcal{F}$ is CM. Note
that $p$ is finite and faithfully flat. By Lemma 00LM in [Stacks], for any
point $x\in X$, any regular sequence of $\mathcal{F}_{x}$ comes from a regular
sequence of $\mathcal{F}_{y}$, where $p(y)=x$. Thus, we have the assertion. ∎

######  Remark 4.20.

We hope quo is true for any field $k$ which can be of positive characteristic.
The difficulty is there may be no splitting
$\mathcal{H}\to(p^{\prime})_{*}(p^{\prime})^{*}\mathcal{H}\to\mathcal{H}$ if
the degree of $p$ is coprime to the characteristic of $k$.

If $p$ is a quotient by a finite group which may not be étale, the major
problem for proving quo is we cannot ensure the equality
$(p^{\prime})^{*}(f^{*}\mathcal{F})^{DD}=(p^{\prime})^{*}(f^{*}\mathcal{F})^{DD}$.

Now, we define the following notion.

######  Definition 4.21.

non-RS Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$. We say that $\mathcal{F}$ is $(RS_{q})$ if for any point $x\in X$ with
$\operatorname{codim}_{X}x\leq q$ and any resolution $f:X^{\prime}\to X$,
$\theta_{\mathcal{F},f}$ is a quasi-isomorphism after localizing
$\theta_{\mathcal{F},f}$ at $x$.

######  Remark 4.22.

In non-RS, $\mathcal{F}$ is $(RS_{\dim X})$ if and only if $\mathcal{F}$ has
rational singularities and it is defi2. Also in non-RS, condition (b) is
equivalent to the following: For any $i\geq 1$,
$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}$ has the support of codimension $\geq q+1$.

######  Lemma 4.23.

Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on $X$. If
$X$ is $(R_{q})$, then $\mathcal{F}$ is $(RS_{q})$.

###### Proof.

Let $f:X^{\prime}\to X$ be any resolution. By dualizing
$\theta_{\mathcal{F},f}$, it suffices to show that
$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}$ has the support of codimension $\geq q+1$
for $i\geq 1$. For any point $x\in X$ with $\operatorname{codim}_{X}x\leq q$,
$\mathcal{F}_{x}=\mathcal{O}_{X,x}$ and hence
$(R^{i}f_{*}(f^{*}\mathcal{F}))^{DD}_{x}=(R^{i}f_{*}\mathcal{O}_{X})_{x}$ for
$i\geq 0$. Thus, by Theorem 1.1 in [CR15], we have the assertion. ∎

######  Example 4.24.

ex2 We can give examples of reflexive sheaf with $(RS_{q})$. Indeed, $X$ be
any normal variety over $\mathbb{C}$ and suppose $\Delta$ is an effective Weil
divisor such that $(X,\Delta)$ is log canonical. If $D$ is a
$\mathbb{Q}$-Cartier Weil divisor on $X$, then $D$ is $(RS_{q+1})$, where $q$
is the codimension of the union of the non-klt centers of $(X,\Delta)$.

It would be interesting to know whether any closed subvariety defined by the
points $x\in X$ in which $(\theta_{\mathcal{F},f})_{x}$ is not a quasi-
isomorphism for some resolution $f:X^{\prime}\to X$ is a non-klt center of
$(X,\Delta)$ or not. Note that if $D=0$, there is an affirmative answer. See
Theorem 1.2 in [AH12].

##  5\. Notion of $(B_{q+1})$

This section introduces the notion of $(B_{q+1})$.

######  Definition 5.1.

B_q+1 Let $X$ be any normal variety and $\mathcal{F}$ any reflexive sheaf on
$X$. We say $\mathcal{F}$ is $(B_{q+1})$ if $H^{i}_{x}(X,\mathcal{F})=0$ for
any $\dim X-q<i<\dim X$ and closed point $x\in X$.

For any normal variety $X$, $X$ is CM if and only if $\omega_{X}$ is CM by
Lemma 0AWS in [Stacks]. However, for $q<\dim X-1$, it is not true that $X$ is
$(S_{q+1})$ if and only if $\omega_{X}$ is $(S_{q+1})$. It seems to us that
the correct notion for $\omega_{X}$ is $(B_{q+1})$, not $(S_{q+1})$ (for a
more discussion, see (4.3) in [Kol11]).

######  Theorem 5.2.

main Let $X$ be any normal projective variety and $\mathcal{F}$ any reflexive
sheaf on $X$, which is $(RS_{q})$ and $(KV_{q})$. Then the following are
equivalent:

  * (a)

$\mathcal{F}$ is $(S_{q+1})$.

  * (b)

For any resolution $f:X^{\prime}\to X$ such that for any $1\leq i<q$,
$R^{i}f_{*}(f^{*}\mathcal{F})^{DD}=0$.

  * (c)

$\mathcal{F}^{D}$ is $(B_{q+1})$.

###### Proof.

Let $f:X^{\prime}\to X$ be any resolution of $X$. For $(a)\implies(b)$, as
ex1, we may consider the following spectral sequence

(5.1) $E^{st}_{2}=H^{s}_{x}(X,R^{t}f_{*}(f^{*}\mathcal{F})^{DD})\implies
H^{s+t}_{f^{-1}(x)}(X^{\prime},(f^{*}\mathcal{F})^{DD}).$

We may use induction. Indeed, let us fix any positive integer $2\leq n\leq q$
and assume that $R^{i}f_{*}(f^{*}\mathcal{F})^{DD}=0$ for $1\leq i<n-1$.

Given any point $x\in X$ with $\operatorname{codim}_{X}x\geq q+1$, since
$\mathcal{F}$ is $(KV_{q})$ and warmup1 holds, we have
$H^{i}_{f^{-1}(x)}(X^{\prime},(f^{*}\mathcal{F})^{DD})=0$ for $i\leq q$. By
the spectral sequence (5.1), we have the following exact sequence

$0\to H^{0}_{x}(X,R^{n-1}f_{*}(f^{*}\mathcal{F})^{DD})\to
H^{n}_{x}(X,\mathcal{F})\overset{\alpha_{n}}{\to}H^{n}_{f^{-1}(x)}(X,(f^{*}\mathcal{F})^{DD}).$

From the above exact sequence,
$H^{0}_{x}(X,R^{n-1}f_{*}(f^{*}\mathcal{F})^{DD})\cong
H^{n}_{x}(X,\mathcal{F})$ and if we use (a),
$H^{0}_{x}(X,R^{n-1}f_{*}(f^{*}\mathcal{F})^{DD})=0$ holds. Thus,
$R^{n-1}f_{*}(f^{*}\mathcal{F})^{DD}$ does not have associated point as $x$.
If $\operatorname{codim}_{X}x\leq q$, since $\mathcal{F}$ is $(RS_{q})$,
$(R^{i}f_{*}(f^{*}\mathcal{F}))_{x}=0$ for $i\geq 1$ and hence we have
$R^{n-1}f_{*}(f^{*}\mathcal{F})^{DD}$.

For $(b)\implies(a)$, since $\theta_{\mathcal{F},f}$ is a quasi-isomorphism at
$x\in X$ with $\operatorname{codim}_{X}x\leq q$, we have
$(R^{i}f_{*}(f^{*}\mathcal{F})^{D})_{x}\cong\mathcal{E}xt^{-\operatorname{codim}_{X}x+i}_{\mathcal{O}_{X,x}}((f^{*}\mathcal{F})^{DD}_{x},\omega^{\bullet}_{X,x})$.
Hence the fact that $\mathcal{F}$ is $(KV_{q})$ and the local duality gives us
$H^{i}_{x}(X,\mathcal{F})=0$ for $i<\operatorname{codim}_{X}x$.

If $\operatorname{codim}_{X}x\geq q+1$, by inspecting the spectral sequence
(5.1), $E^{s0}_{2}=E^{s0}_{\infty}$ for $s\leq q$. Considering the edge map
$E^{s0}_{\infty}\to
E^{s}=H^{s}_{f^{-1}(x)}(X^{\prime},(f^{*}\mathcal{F})^{DD})$ for such $s$, if
we use warmup1, then we have the assertion.

For $(a)\iff(c)$, let us consider the following exact sequence

${0}$${f_{*}(f^{*}\mathcal{F})^{D}}$${\mathcal{F}^{D}}$${\mathcal{Q}}$${0}$$\scriptstyle{\mathcal{H}^{0}(\theta_{\mathcal{F},f})}$

for a coherent sheaf $\mathcal{Q}$ on $X$. Since $\mathcal{F}$ is $(RS_{q})$,
the support of $\mathcal{Q}$ has codimension $\geq q+1$ and hence
$H^{i}_{x}(X,\mathcal{Q})=0$ for $i>\dim X-q$.

By taking local cohomology on the above exact sequence, we have

$H^{i}_{x}(X,f_{*}(f^{*}\mathcal{F})^{D})\cong
H^{i}_{x}(X,\mathcal{F}^{D})\text{ for any }\dim X-q<i<q\text{ and any closed
point }x\in X.$

Hence, $\mathcal{F}^{D}$ is $(B_{q+1})$ if and only if
$H^{i}_{x}(X,f_{*}(f^{*}\mathcal{F})^{D})=0$ for any $\dim X-q<i<i$ and any
closed point $x\in X$.

Let $x\in X$ be any closed point and consider the Leray spectral sequence

$E^{st}_{2}=H^{s}_{x}(X,R^{t}f_{*}(f^{*}\mathcal{F})^{D})\implies
H^{s+t}_{f^{-1}(x)}(X^{\prime},(f^{*}\mathcal{F})^{D}).$

Then, by the fact that $\mathcal{F}$ is $(KV_{q})$ and the dimension counting,
$E^{st}_{2}=0$ for $s+t\geq\dim X-q$. Hence, by inspecting the spectral
sequence, $E^{s0}_{2}=E^{s0}_{\infty}$ for $s>\dim X-q$. Moreover,
$E^{s0}_{\infty}=E^{s}$ for $s>\dim X-q$. Thus, $\mathcal{F}^{D}$ is
$(B_{q+1})$ if and only if
$H^{i}_{f^{-1}(x)}(X^{\prime},(f^{*}\mathcal{F})^{D})=0$ for any $\dim
X-q<i<\dim X$ and any closed point $x\in X$. If we use warmup1, then we have
the assertion. ∎

######  Remark 5.3.

In main, the strategy of the proof of $(a)\iff(b)$ is similar to the strategy
of the proof of Lemma 3.1 in [Ale08] and Proposition 7.1.7 in [Fuj17].
Moreover, the idea of the proof of $(a)\iff(c)$ is inspired by the proof of
Lemma 3.3 in [Kov99].

There is an interesting corollary of main.

######  Corollary 5.4.

maincoro Assume that $q\geq\left\lceil\frac{\dim X+1}{2}\right\rceil$. Let $X$
be any normal $\mathbb{Q}$-factorial variety over a characteristic $0$ field
and any Weil divisor on $X$ is $(RS_{q})$ and $(S_{q+1})$. Then any Weil
divisor on $X$ is CM. In particular, $X$ is CM itself.

###### Proof.

Note that any Weil divisor $D$ on $X$ is $(KV_{q})$ for any $q$. Since
$(\mathcal{O}_{X}(D))^{D}$ is $(S_{q})$ and
$\mathcal{O}_{X}(D)\cong(\mathcal{O}_{X}(D))^{DD}$ is $(B_{q})$ by main,
$\mathcal{O}_{X}(D)$ is CM. ∎

######  Remark 5.5.

maincoro proves that in the setting of $\mathrm{char}\,k=0$, for any factorial
variety $X$, if $X$ is $(RS_{\left\lceil\frac{\dim X+1}{2}\right\rceil})$ and
$(S_{\left\lceil\frac{\dim X+3}{2}\right\rceil})$, then $X$ is CM.

One may think that Theorem 1.6 in [HO74] is similar to maincoro. Hence, if we
assume $X$ is $(R_{q})$, one can believe that there is a simple proof of
maincoro using the local duality only.

##  6\. $q$-birational morphisms

In this section, we always assume that the field $k$ is of characteristic $0$.
Let us define the notion of _$q$ -birational morphism_.

######  Definition 6.1 (See Definition 3.1, [Kim22]).

qbirational Let $X,X^{\prime}$ be any normal varieties and $f:X^{\prime}\to X$
any proper birational morphism.

  * (a)

The _center_ of $f$ is the reduced closed subscheme $C$ of $X$ which is the
image of exceptional locus along $f$.

  * (b)

We say $f$ is a _$q$ -birational morphism_ if the exceptional locus has
codimension $1$ and the center of $f$ has codimension $\geq q+1$.

######  Remark 6.2.

Let us remark that for any $q$-birational morphism $f:X^{\prime}\to X$ between
normal varieties, if $X^{\prime}$ is smooth, then $X$ is $(R_{q})$.

Let us prove the following lemma which can be regarded as a partial converse
of twospectral.

######  Lemma 6.3.

firstapplicationdead Let $X$ be any normal variety and $\mathcal{F}$ any
reflexive sheaf on $X$ which is $(RS_{q})$, $(KV_{q})$ and $(S_{q+1})$.
Suppose $f:X^{\prime}\to X$ is any resolution and $\mathcal{L}$ is a line
bundle on $X^{\prime}$ such that there is an isomorphism $\mathcal{F}\cong
f_{*}\mathcal{L}$. If $f$ is $q$-birational, then $R^{i}f_{*}\mathcal{L}=0$
for $1\leq i<q$.

###### Proof.

We may use a similar argument as in the proof of $(a)\iff(c)$ in main.

By the same argument as in the proof of stronger, we have an injection
$(f^{*}\mathcal{F})^{DD}\to\mathcal{L}$ and hence
$\varphi:\mathcal{L}^{D}\to(f^{*}\mathcal{F})^{D}$. Let $\mathcal{Q}$ be the
cokernel of $f_{*}\varphi$. Then we have the following exact sequence

(6.1) $0\to f_{*}\mathcal{L}^{D}\to
f_{*}(f^{*}\mathcal{F})^{D}\to\mathcal{Q}\to 0.$

Since $f$ is $q$-birational, the support of $\mathcal{Q}$ has codimension
$\geq q+1$.

By taking local cohomology on (6.1), we have isomorphisms

$H^{i}_{x}(X,\mathcal{L}^{D})\cong
H^{i}_{x}(X,f_{*}(f^{*}\mathcal{F})^{D})\text{ for any closed point }x\in
X\text{ and any }\dim X-q<i<\dim X.$

As in the proof of main, we have $H^{i}_{x}(X,f_{*}(f^{*}\mathcal{F})^{D})=0$
for any closed point $x\in X$ and $\dim X-q<i<\dim X$. Hence,
$H^{i}_{x}(X,\mathcal{L}^{D})=0$ for such $i$ and $x\in X$.

Consider the Leray spectral sequence

$E^{st}_{2}=H^{s}_{x}(X,R^{t}f_{*}\mathcal{L}^{D})\implies
H^{s+t}_{f^{-1}(x)}(X,\mathcal{L}^{D}).$

By inspecting the spectral sequence, we have
$H^{i}_{f^{-1}(x)}(X^{\prime},\mathcal{L}^{D})=0$ for any $\dim X-q<i<\dim X$
and any closed point $x\in X$. Now, we may apply warmup1 and obtain the
assertion. ∎

Using firstapplicationdead, we have the following theorem.

######  Theorem 6.4.

youaresecondapplication Let $X,X^{\prime}$ be any normal projective varieties,
$X^{\prime}$ smooth and $f:X^{\prime}\to X$ any $q$-birational morphism.
Suppose that $D$ is any anti $f$-nef divisor on $X^{\prime}$ such that
$f_{*}D$ is $\mathbb{Q}$-Cartier and $(S_{q+1})$. Then
$R^{i}f_{*}\mathcal{O}_{X^{\prime}}(D)=0$ for $1\leq i<q$.

###### Proof.

Since $f_{*}D$ is $\mathbb{Q}$-Cartier, we may consider a $\mathbb{Q}$-divisor
$f^{*}f_{*}D$. If we define a divisor $L$ on $X^{\prime}$ in which
$\mathcal{O}_{X^{\prime}}(L)=(f^{*}\mathcal{O}_{X}(f_{*}D))^{DD}$ holds, we
have $f^{*}f_{*}D\sim_{\mathbb{Q}}L$ and hence
$D-L\sim_{\mathbb{Q}}D-f^{*}f_{*}D$ is anti $f$-nef. Thus $D-L$ is an
effective $f$-exceptional divisor on $X^{\prime}$ by the Negativity lemma.
Therefore, by firstapplicationdead, we have the assertion. ∎

We can write youaresecondapplication as the absolute cohomology vanishing. For
a similar result, see Corollary 3.8 in [Kim22].

######  Corollary 6.5.

thirdapplication Let $X,X^{\prime}$ be any normal projective varieties,
$X^{\prime}$ smooth and $f:X^{\prime}\to X$ any $q$-birational morphism.
Suppose that $D$ is any anti $f$-nef divisor on $X^{\prime}$ such that
$f_{*}D$ is a $\mathbb{Q}$-divisor and $(S_{q+1})$ on $X$ and $E$ is any
effective $f$-exceptional divisor on $X^{\prime}$. Then

$H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(D+E))=H^{i}(X,\mathcal{O}_{X}(f_{*}D))=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(D))$

for $0\leq i<q$.

###### Proof.

We may use the Leray spectral sequence

$E^{st}_{2}=H^{s}(X,R^{t}f_{*}\mathcal{O}_{X^{\prime}}(D+E))\implies
H^{s+t}(X^{\prime},\mathcal{O}_{X^{\prime}}(D+E))$

and hence
$H^{i}(X,\mathcal{O}_{X}(f_{*}D))=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(D+E))$
for $0\leq i<q$ by youaresecondapplication. Note that
$f_{*}\mathcal{O}_{X^{\prime}}(D+E)=\mathcal{O}_{X}(f_{*}D)$ because of
warmup2 and the fact that $D+E$ is anti $f$-nef. ∎

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