# Kawamata-Viehweg vanishing for toric varieties

Hiromu Tanaka  Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, JAPAN<EMAIL_ADDRESS>
###### Abstract.

Given a boundary divisor $B$ on a projective toric variety $X$ such that
$(X,B)$ is klt, we establish the Kawamata–Viehweg vanishing theorem for
$(X,B)$.

###### Key words and phrases:

toric varieties, Kawamata–Viehweg vanishing theorem.

###### 2010 Mathematics Subject Classification:

14M25, 14E30, 14F17.

###### Contents

  1. 1 Introduction
    1. 1.1 Sketch of the proof
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 Toric minimal model program
    3. 2.3 Globally $F$-regular varieties
  3. 3 Vanishing theorems for globally F-regular varieties
  4. 4 Vanishing theorems for toric varieties

##  1\. Introduction

Many vanishing theorems have been established for toric varieties [Dem70],
[BTLM97], [Mus02], [Fuj07] (cf. [CLS11, Section 9]). Amongst them, a typical
result is the Danilov vanishing theorem, which states that given a smooth
projective toric variety $X$ and an ample Cartier divisor $A$, the equation

$H^{i}(X,\Omega^{a}_{X}\otimes\mathcal{O}_{X}(A))=0$

holds for any $i>0$ and $a\geq 0$ (cf. [CLS11, Theorem 9.3.1], [Fuj07,
Corollary 1.3]). In particular, smooth proper toric varieties satisfy the
Kodaira vanishing even in positive characteristic. As a generalisation of the
Kodaira vanishing theorem, also the Kawamata–Viehweg vanishing theorem holds
for a klt pair $(X,B)$, where $X$ is a proper toric variety and $B$ is a torus
invariant effective $\mathbb{Q}$-divisor [Fuj07, Corollary 1.7]. The main
purpose of this note is to drop the assumption that $B$ is torus invariant.

######  Theorem 1.1 (Theorem 4.3).

Let $k$ be a field. Let $\alpha:X\to S$ be a projective $k$-morphism of
schemes of finite type over $k$, where $X$ is a normal toric variety over $k$.
Let $D$ be a $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor on $X$. Assume that one
of the following conditions holds.

  1. (1)

$D\sim_{\mathbb{R}}K_{X}+B$ for some effective $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor $B$ on $X$ such that $(X,B)$ is klt and $B$ is
$\alpha$-big.

  2. (2)

$D-(K_{X}+B)$ is $\alpha$-nef and $\alpha$-big for some effective
$\mathbb{R}$-Cartier $\mathbb{R}$-divisor $B$ on $X$ such that $(X,B)$ is klt.

Then $R^{i}\alpha_{*}\mathcal{O}_{X}(D)=0$ for any $i>0$.

###  1.1. Sketch of the proof

We give a sketch of the proof of Theorem 1.1. For simplicity, we only treat
the case when $S={\operatorname{Spec}}\,k$. Taking a toric small
$\mathbb{Q}$-factorialisation, we may assume that $X$ is
$\mathbb{Q}$-factorial (cf. the proof of Theorem 4.3). Then the assumptions
(1) and (2) are known to be equivalent if $k$ is of characteristic zero.
Although we need some argument, we can show that (2) implies (1) under the
assumption that $X$ is $\mathbb{Q}$-factorial (cf. the proof of Theorem 4.3).
In what follows, we assume that (1) holds. The main idea is to run a $D$-MMP:

$X=:X_{0}\overset{f_{0}}{\dashrightarrow}X_{1}\overset{f_{1}}{\dashrightarrow}\cdots\overset{f_{N-1}}{\dashrightarrow}X_{N}$
$D=:D_{0},\qquad(f_{n})_{*}D_{n}=:D_{n+1}.$

Such a strategy is used in [CTW17] to establish the Kawamata–Viehweg vanishing
theorem for klt del Pezzo surfaces in larger characteristic. It is enough to
prove the following assertions.

  1. (a)

$H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))\simeq
H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1}))$ holds for any $i>0$ and
$n\in\\{0,...,N-1\\}$.

  2. (b)

$H^{i}(X,\mathcal{O}_{X}(D))=0$ holds for any $i>0$ under the assumption that
either $D$ is nef or $X$ has a $D$-Mori fibre space $f:X\to Y$.

(a) Recall that $f_{n}:X_{n}\dashrightarrow X_{n+1}$ is either a
$D_{n}$-divisorial contraction or a $D_{n}$-flip. If $f_{n}$ is a
$D_{n}$-divisorial contraction, then it is not difficult to check that we can
apply the relative Kawamata–Viehweg vanishing theorem for $f_{n}$ by using the
fact that the $f_{n}$-exceptional divisor is torus invariant. We then obtain
the required isomorphism $H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))\simeq
H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1}))$.

Assume that $f_{n}$ is a flip. In this case, it does not seem to be easy to
compare $H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))$ and
$H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1}))$ via $Z$, where
$\varphi:X_{n}\to Z$ denotes the corresponding flipping contraction. We use
the following diagram (1.1.1), where $Y\to X_{n}$ and $Y\to X_{n+1}$ are
suitable divisorial contractions. Such a diagram always exists as we consider
a toric MMP (Proposition 2.5). Then the situation is quite similar to the
divisorial contraction, although we need further delicate computation. For the
details on this computation, see Step 4 of the proof of Theorem 4.1.

(1.1.1)
${Y}$${{X_{n}}}$${{X_{n+1}}}$${Z}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{f_{n}}$

(b) If $D$ is nef, it is well known that the assertion (b) is true (cf.
Proposition 3.2(2)). Therefore, the problem is reduced to the case when $X$
has a $D$-Mori fibre space $f:X\to Y$. Note that $D-K_{X}$ is $f$-ample and
$f$ is a toric morphism to a normal projective toric variety $Y$. Then we have
$R^{i}f_{*}\mathcal{O}_{X}(D)=0$ for any $i>0$ (cf. Proposition 3.2(1)). Since
$-D$ is $f$-big and $\dim X>\dim Y$, we also have $f_{*}\mathcal{O}_{X}(D)=0$.
Therefore, it holds that $Rf_{*}\mathcal{O}_{X}(D)=0$. For the morphisms
$\alpha:X\xrightarrow{f}Y\xrightarrow{\beta}{\operatorname{Spec}}\,k$, we have
the following isomorphism in the derived category:

$R\alpha_{*}\mathcal{O}_{X}(D)\simeq R\beta_{*}Rf_{*}\mathcal{O}_{X}(D)=0.$

This implies that $H^{i}(X,\mathcal{O}_{X}(D))=0$ for any $i\geq 0$.

Some of the above arguments work even for globally $F$-regular varieties.
Indeed, the main theorem holds for globally $F$-regular surfaces by a similar
and easier proof.

######  Theorem 1.2 (Theorem 3.5).

Assume that $k$ is of characteristic $p>0$. Let $\alpha:X\to S$ be a proper
morphism from a normal surface $X$ to a scheme $S$ of finite type over $k$
such that $X$ is globally $F$-regular over $S$. Let $D$ be a
$\mathbb{Z}$-divisor on $X$ such that $D-(K_{X}+B)$ is $\alpha$-nef and
$\alpha$-big for some $\mathbb{R}$-divisor $B$ on $X$ whose coefficients are
contained in $[0,1)$. Then $R^{i}\alpha_{*}\mathcal{O}_{X}(D)=0$ for any
$i>0$.

######  Remark 1.3.

  1. (1)

The author does not know whether Theorem 1.1 holds after replacing “toric” by
“globally $F$-regular”.

  2. (2)

Theorem 1.1 is known if $\dim X=2$ [WX, Theorem A]. Both Theorem 1.1 and
Theorem 1.2 can be considered as generalisations of [WX, Theorem A].

Acknowledgements: The author would like to thank Professors Osamu Fujino and
Jakub Witaszek for useful comments, constructive suggestions, and answering
questions. The author was funded by JSPS KAKENHI Grant numbers JP18K13386 and
JP22H01112.

##  2\. Preliminaries

###  2.1. Notation

In this subsection, we summarise notation and terminologies used in this
paper.

  1. (1)

We will freely use the notation and terminology in [Har77] and [Kol13].

  2. (2)

Throughout this paper, we work over a field $k$.

  3. (3)

We say that $X$ is a variety (over $k$) if $X$ is an integral scheme that is
separated and of finite type over $k$. We say that $X$ is a curve (resp. a
surface) if $X$ is a variety of dimension one (resp. two).

  4. (4)

Let $f:X\to Y$ be a morphism of noetherian schemes. We say that $f$ is
projective if it is projective in the sense of EGA [EGAII, Définition 5.5.2].
This definition differs from the one in [Har77, page 103]. On the other hand,
these coincide in many cases, e.g. $Y$ is affine (cf. [FGAex, Section 5.5.1]).

  5. (5)

A torus $T$ is an algebraic group which is isomorphic to $\mathbb{G}_{m}^{r}$
for some $r\geq 0$. We say that a variety $X$ is a toric if there exists a
non-empty open subscheme $T$ of $X$ such that $T$ is a torus and the left
action of $T$ on $T$ itself extends to an action on $X$: $T\times X\to X$. In
this case, $T$ is called a torus of $X$. Given toric varieties $X$ and $T$, we
say that $f:X\to Y$ is a toric morphism if there exist tori $T_{X}$ and
$T_{Y}$ of $X$ and $Y$ respectively such that $f(T_{X})\subset T_{Y}$ and the
induced morphism $f|_{T_{X}}:T_{X}\to T_{Y}$ is a homomorphism of algberaic
groups.

  6. (6)

Throughout this note, $N$ denotes a free $\mathbb{Z}$-module of finite rank.
Set $N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$. For terminologies on
toric varieties, we refer to [CLS11]. Although [CLS11] works over
$\mathbb{C}$, almost all the statements work even over arbitrary fields. For
example, if $\Sigma$ is a fan of $N_{\mathbb{R}}$, then $X_{\Sigma}$ is a
normal toric variety over $k$.

###  2.2. Toric minimal model program

In this subsection, we summarise notation and results on the toric minimal
model program. In (2.3), we recall terminologies on toric minimal model
program. In the proof of our main theorem, we need a description of toric
flips in Proposition 2.5 (cf. the proof of Theorem 4.1). This result is known
if $X$ is semi-projective [CLS11, Lemma 15.5.7]. Since we work with a slightly
generalised setting, we give a proof of it for the sake of completeness. We
start by giving the following characterisation of the quasi-projectivity for
toric varieties with convex supports.

######  Lemma 2.1.

Let $\Sigma$ be a fan of $N_{\mathbb{R}}$ and let $X_{\Sigma}$ be the toric
variety of $\Sigma$. Then the following are equivalent.

  1. (1)

$X_{\Sigma}$ is quasi-projective over $k$ and $|\Sigma|$ is a convex subset of
$N_{\mathbb{R}}$.

  2. (2)

There exist a semi-projective normal toric variety $X^{\prime}$ and a non-
negative integer $r$ such that $X_{\Sigma}\simeq
X^{\prime}\times\mathbb{G}_{m}^{r}$.

  3. (3)

There exists a projective morphism $X_{\Sigma}\to Z$ to an affine variety $Z$.

  4. (4)

There exists a toric projective morphism $X_{\Sigma}\to Z$ to a toric normal
affine variety $Z$.

###### Proof.

Let us show that (1) implies (2). Assume (1). Let $V_{\Sigma}$ be the smallest
$\mathbb{R}$-vector subspace of $N_{\mathbb{R}}$ that contains $\Sigma$. Set
$\overline{N}:=N\cap V_{\Sigma}$ and set $\overline{\Sigma}:=\Sigma$, where we
consider $\overline{\Sigma}$ as a fan in $\overline{N}_{\mathbb{R}}$. Note
that $\overline{N}$ is spanned by a subset of a basis of $N$. It follows from
[CLS11, Proposition 3.3.11] that

$X_{\Sigma}\simeq X_{\overline{\Sigma}}\times\mathbb{G}_{m}^{r}.$

Since $X_{\overline{\Sigma}}$ is quasi-projective over $k$ and
$\overline{\Sigma}$ has full dimensional convex support in
$\overline{N}_{\mathbb{R}}$, $X_{\overline{\Sigma}}$ is semi-projective
[CLS11, Proposition 7.2.9]. Hence, (2) holds.

It is clear that (2) implies (3). Let us show that (3) implies (4). Assume
(3). Let $X_{\Sigma}\xrightarrow{g}W\to Z$ be the Stein factorisation of $X\to
Z$, i.e.
$W={\operatorname{Spec}}\,\Gamma(X_{\Sigma},\mathcal{O}_{X_{\Sigma}})$. As $W$
is an affine variety with $g_{*}\mathcal{O}_{X_{\Sigma}}=\mathcal{O}_{W}$,
both $W$ and $g$ are toric ([CLS11, Exercise 7.2.3] or [Tana]). Thus, (4)
holds.

Let us show that (4) implies (1). Assume (4). It is clear that $X_{\Sigma}$ is
quasi-projective over $k$. We have the $\mathbb{Z}$-module homomorphism
$\varphi:N\to N^{\prime}$ corresponding to $X_{\Sigma}\to
Z=:X_{\Sigma^{\prime}}$. Since $X_{\Sigma}\to X_{\Sigma^{\prime}}$ is proper,
we have $\varphi_{\mathbb{R}}^{-1}(|\Sigma^{\prime}|)=|\Sigma|$ for
$\varphi_{\mathbb{R}}:N_{\mathbb{R}}\to N^{\prime}_{\mathbb{R}}$ [CLS11,
Theorem 3.4.11]. As $Z=X_{\Sigma^{\prime}}$ is affine, $\Sigma^{\prime}$ is
convex. Take
$\alpha,\beta\in|\Sigma|=\varphi_{\mathbb{R}}^{-1}(|\Sigma^{\prime}|)$. Then
$\varphi_{\mathbb{R}}(\alpha)\in|\Sigma^{\prime}|$ and
$\varphi_{\mathbb{R}}(\beta)\in|\Sigma^{\prime}|$ imply

$\varphi_{\mathbb{R}}(\alpha+\beta)=\varphi_{\mathbb{R}}(\alpha)+\varphi_{\mathbb{R}}(\beta)\in|\Sigma^{\prime}|.$

Therefore, it holds that
$\alpha+\beta\in\varphi_{\mathbb{R}}^{-1}(|\Sigma^{\prime}|)=|\Sigma|$. Thus,
$|\Sigma|$ is convex, i.e. (1) holds. ∎

######  Remark 2.2.

Let $S$ be an affine normal toric variety. Let $f:X_{\Sigma}\to
X_{\Sigma^{\prime}}$ be a toric projective birational morphism of toric normal
varieties which are projective over $S$. In this case, the associated
$\mathbb{Z}$-module homomorphism $\varphi:N\to N^{\prime}$ of their lattices
is an isomorphism. Hence, we may assume that $\Sigma$ and $\Sigma^{\prime}$
are fans of a common lattice $N$. Since $f$ is proper, we have that
$|\Sigma|=|\Sigma^{\prime}|$. By the proof of Lemma 2.1, there exists a toric
projective birational morphism of semi-projective normal toric varieties

$\overline{f}:X_{\overline{\Sigma}}\to X_{\overline{\Sigma}^{\prime}}$

such that $f=\overline{f}\times\mathbb{G}_{m}^{r}$ for some
$r\in\mathbb{Z}_{\geq 0}$.

######  2.3Terminologies on toric minimal model program.

Let $\alpha:X_{\Sigma}\to S$ be a projective toric morphism of normal toric
varieties. Assume that $X_{\Sigma}$ is $\mathbb{Q}$-factorial and $S$ is
affine. Let $D$ be an $\mathbb{R}$-divisor on $X_{\Sigma}$.

Note that ${\operatorname{NE}}(X_{\Sigma}/S)$ is generated by the proper
curves on $X_{\Sigma}$ and ${\operatorname{NE}}(X_{\Sigma}/S)$ is a polyhedral
cone [FS04, Theorem 4.1], i.e.
${\operatorname{NE}}(X_{\Sigma}/S)=\sum_{i=1}^{r}\mathbb{R}_{\geq 0}[C_{i}]$
for some proper curves $C_{1},...,C_{r}$ on $X_{\Sigma}$. Furthermore, we may
and do assume that each $\mathbb{R}_{\geq 0}[C_{i}]$ is an extremal ray of
${\operatorname{NE}}(X_{\Sigma}/S)$. Let $R=\mathbb{R}_{\geq 0}[C_{i}]$ be an
extremal ray of ${\operatorname{NE}}(X_{\Sigma}/S)$. Then there exists the
contraction of $R$ [FS04, Theorem 4.5], i.e. the projective toric morphism

$\varphi_{R}:X_{\Sigma}\to X_{\Sigma^{\prime}}$

to a normal toric variety $X_{\Sigma^{\prime}}$ projective over $S$ such that
$(\varphi_{R})_{*}\mathcal{O}_{X_{\Sigma}}=\mathcal{O}_{X_{\Sigma^{\prime}}}$
and that if $C$ is a proper curve on $X_{\Sigma}$, then $[C]\in R$ if and only
if $\varphi_{R}(C)$ is a point. We say that $R=\mathbb{R}_{\geq 0}[C_{i}]$ is
$D$-negative if $D\cdot C_{i}<0$.

  * •

We say that $\varphi_{R}$ is a $D$-divisorial contraction if $R$ is
$D$-negative, $\varphi_{R}$ is birational, and
$\dim{\operatorname{Ex}}(\varphi_{R})=\dim X_{\Sigma}-1$. In this case,
$X_{\Sigma^{\prime}}$ is $\mathbb{Q}$-factorial [FS04, Theorem 4.5].

  * •

We say that $\varphi_{R}$ is a $D$-flipping contraction if $R$ is
$D$-negative, $\varphi_{R}$ is birational, and
$\dim{\operatorname{Ex}}(\varphi_{R})<\dim X_{\Sigma}-1$.

  * •

We say that $\varphi_{R}$ is a $D$-Mori fibre space if $R$ is $D$-negative and
$\varphi_{R}$ is not birational, i.e. $\dim X_{\Sigma}>\dim
X_{\Sigma^{\prime}}$. In this case, also $X_{\Sigma}$ is called a $D$-Mori
fibre space.

If $\varphi_{R}:X_{\Sigma}\to X_{\Sigma^{\prime}}$ is a $D$-flipping
contraction, then there exists a small projective toric morphism
$\varphi^{+}:X_{\Sigma^{+}}\to X_{\Sigma^{\prime}}$ from a
$\mathbb{Q}$-factorial toric variety $X_{\Sigma^{+}}$ such that the proper
transform $D^{+}$ of $D$ on $X_{\Sigma^{+}}$ is $\varphi^{+}$-ample [FS04,
Theorem 4.8]. In this case, each of $X_{\Sigma}\dashrightarrow
X_{\Sigma^{+}}$, $\varphi^{+}$, and $X_{\Sigma^{+}}$ is called a $D$-flip of
$\varphi_{R}$.

By the same argument as in [FS04, Theorem 4.9], there exists a sequence

$X_{\Sigma}=X_{\Sigma_{0}}\overset{f_{0}}{\dashrightarrow}X_{\Sigma_{1}}\overset{f_{1}}{\dashrightarrow}\cdots\overset{f_{\ell-1}}{\dashrightarrow}X_{\Sigma_{\ell}},$

called a $D$-MMP over $S$, such that the following hold for
$D_{i+1}:=(f_{i})_{*}D_{i}$.

  1. (1)

For any $i\in\\{0,...,\ell\\}$, $\alpha_{i}:X_{i}\to S$ is a toric projective
morphism from a $\mathbb{Q}$-factorial toric variety $X_{i}$.

  2. (2)

For any $i\in\\{0,...,\ell-1\\}$, $f_{i}:X_{\Sigma_{i}}\dashrightarrow
X_{\Sigma_{i+1}}$ is either a $D_{i}$-divisorial contraction over $S$ or a
$D_{i}$-flip over $S$.

  3. (3)

Either $D_{\ell}$ is nef over $S$ or there exists a $D_{\ell}$-Mori fibre
space $X_{\Sigma_{\ell}}\to X_{\Lambda}$ over $S$.

######  Remark 2.4.

Let $\alpha:X_{\Sigma}\to S$ be a projective toric morphism of normal toric
varieties. Assume that $X_{\Sigma}$ is $\mathbb{Q}$-factorial and $S$ is
affine. Let $D$ be an $\mathbb{R}$-divisor on $X_{\Sigma}$.

  1. (1)

We have $X_{\Sigma}\simeq X_{\overline{\Sigma}}\times\mathbb{G}_{m}^{r}$ for
some $r\in\mathbb{Z}_{\geq 0}$ and a $\mathbb{Q}$-factorial semi-projective
toric variety $X_{\overline{\Sigma}}$. By ${\rm
Pic}(X_{\overline{\Sigma}}\times\mathbb{G}_{m}^{r})\simeq{\rm
Pic}(X_{\overline{\Sigma}})$, there exists an $\mathbb{R}$-divisor
$\overline{D}$ on $X_{\overline{\Sigma}}$ such that
$\pi^{*}\overline{D}\sim_{\mathbb{R}}D$, where $\pi:X_{\Sigma}\to
X_{\overline{\Sigma}}$ denotes the natural projection.

  2. (2)

Let $\varphi:X_{\Sigma}\to X_{\Sigma^{\prime}}$ be the contraction of an
extremal ray $R$ of ${\operatorname{NE}}(X_{\Sigma}/S)$. Assume that $\varphi$
is birational. Then we have
$\varphi=\overline{\varphi}\times\mathbb{G}_{m}^{r}$ for a suitable projective
birational morphism $\overline{\varphi}:X_{\overline{\Sigma}}\to
X_{\overline{\Sigma}^{\prime}}$ of semi-projective toric normal varieties
(Remark 2.2). In this case, we have
$\rho(X_{\overline{\Sigma}}/X_{\overline{\Sigma}^{\prime}})=1$ and
$-\overline{D}$ is $\overline{\varphi}$-ample, and hence $\overline{\varphi}$
is the contraction of some extremal ray $\overline{R}$ of
${\operatorname{NE}}(X_{\Sigma})$. Furthermore, $\varphi$ is a $D$-divisorial
contraction (resp. $D$-flipping contraction) if and only if
$\overline{\varphi}$ is a $\overline{D}$-divisorial contraction (resp.
$\overline{D}$-flipping contraction).

  3. (3)

Let $\varphi:X_{\Sigma}\to X_{\Sigma^{\prime}}$ be a $D$-divisorial
contraction. Then $E:={\operatorname{Ex}}(\varphi)$ is a prime divisor on
$X_{\Sigma}$. Furthermore, $E$ is torus-invariant. Indeed, this is known if
$X_{\Sigma}$ is semi-projective [CLS11, Proposition 15.4.5(b)], and hence the
general case is reduced to this case by (2).

  4. (4)

Let $\varphi:X_{\Sigma}\to X_{\Sigma^{\prime}}$ be a $D$-flipping contraction
and let $\varphi^{+}:X_{\Sigma^{+}}\to X_{\Sigma^{\prime}}$ be its flip. By
Remark 2.2, we have a flip
$\overline{\varphi}^{+}:X_{\overline{\Sigma}^{+}}\to
X_{\overline{\Sigma}^{\prime}}$ of
$\overline{\varphi}:X_{\overline{\Sigma}}\to X_{\overline{\Sigma}^{\prime}}$
such that $\varphi^{+}=\overline{\varphi}^{+}\times\mathbb{G}_{m}^{r}$.

######  Proposition 2.5.

Let $\alpha:X_{\Sigma}\to S$ be a projective toric morphism of normal toric
varieties. Assume that $X_{\Sigma}$ is $\mathbb{Q}$-factorial and $S$ is
affine. Let $D$ be an $\mathbb{R}$-divisor on $X_{\Sigma}$. Let $X_{\Sigma}\to
X_{\Pi}$ be a $D$-flipping contraction and let $X_{\Sigma^{\prime}}\to
X_{\Pi}$ be its flip. Then there exists a commutative diagram consisting of a
projective toric birational $S$-morphisms of toric normal varieties which are
projective over $S$

${X_{\Theta}}$${{X_{\Sigma}}}$${{X_{\Sigma^{\prime}}}}$${X_{\Pi}}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{f}$

such that the following hold.

  1. (1)

$X_{\Theta}$ is $\mathbb{Q}$-factorial.

  2. (2)

$E:={\operatorname{Ex}}(\psi)={\operatorname{Ex}}(\psi^{\prime})$ is a torus-
invariant prime divisor on $X_{\Theta}$.

  3. (3)

There exists a nonzero effective $1$-cycle $\Gamma$ on $X_{\Sigma}$ such that

(2.5.1) $\psi^{*}F=\psi^{\prime*}F^{\prime}-(F\cdot\Gamma)E$

holds for any $\mathbb{R}$-divsior $F$ on $X_{\Sigma}$ and
$F^{\prime}:=f_{*}F$.

###### Proof.

By Remark 2.4(2), we also have $\overline{\varphi}:X_{\overline{\Sigma}}\to
X_{\overline{\Pi}}$ and
$\overline{\varphi}^{\prime}:X_{\overline{\Sigma}^{\prime}}\to
X_{\overline{\Pi}}$ such that
$\varphi=\overline{\varphi}\times\mathbb{G}_{m}^{r}$ and
$\varphi^{\prime}=\overline{\varphi}^{\prime}\times\mathbb{G}_{m}^{r}$. Since
$\overline{\Sigma}$ is semi-projective, it follows from [CLS11, Lemma 15.5.7]
that we have the following right square diagram. Then, by taking the direct
product $(-)\times\mathbb{G}_{m}^{r}$, we obtain the following left square
diagram (cf. Remark 2.2, e.g.
$X_{\Theta}=X_{\overline{\Theta}}\times\mathbb{G}_{m}^{r}$).

${X_{\Theta}}$${{X_{\Sigma}}}$${{X_{\Sigma^{\prime}}}}$${X_{\Pi}}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{f}$
${X_{\overline{\Theta}}}$${{X_{\overline{\Sigma}}}}$${{X_{\overline{\Sigma}^{\prime}}}}$${X_{\overline{\Pi}}}$$\scriptstyle{\overline{\psi}}$$\scriptstyle{\overline{\varphi}}$$\scriptstyle{\overline{\psi}^{\prime}}$$\scriptstyle{\overline{\varphi}^{\prime}}$$\scriptstyle{\overline{f}}$

Let us show (1). Since $X_{\overline{\Sigma}}$ and
$X_{\overline{\Sigma}^{\prime}}$ are $\mathbb{Q}$-factorial,
$X_{\overline{\Theta}}$ is $\mathbb{Q}$-factorial by [CLS11, Lemma 15.5.7(c)].
Then $\overline{\Theta}$ is simplicial, and hence also $\Theta$ is simplicial.
Therefore, $X_{\Theta}$ is $\mathbb{Q}$-factorial, i.e. (1) holds.

Let us show (2). By [CLS11, Lemma 15.5.7(c)], there exists a unique
$\psi$-exceptional prime divisor $\widetilde{E}$ on $X_{\Theta}$. Pick an
effective ample Cartier divisor $H$ on $X_{\Theta}$ whose support does not
contain any irreducible component of ${\operatorname{Ex}}(\psi)$. Since
$X_{\Sigma}$ is $\mathbb{Q}$-factorial, $\psi_{*}H$ is $\mathbb{Q}$-Cartier.
Therefore, $\psi^{*}\psi_{*}H-H$ is a $\psi$-exceptional effective divisor,
and hence we have $\psi^{*}\psi_{*}H-H=a\widetilde{E}$ for some
$a\in\mathbb{Q}_{\geq 0}$. Since $a\widetilde{E}$ is $\psi$-anti-ample, we
have $a>0$ and $\widetilde{E}\cdot C<0$ for any curve $C$ on $X_{\Theta}$ such
that $C\subset{\operatorname{Ex}}(\psi)$. This implies
${\operatorname{Ex}}(\psi)\subset\widetilde{E}$. The opposite inclusion is
clear, and hence $\widetilde{E}={\operatorname{Ex}}(\psi)$. The same argument
implies $\widetilde{E}={\operatorname{Ex}}(\psi^{\prime})$.

Let us show (3). The equation (2.5.1) holds if $\Sigma$ is semi-projective
[CLS11, Lemma 15.5.7(c)]. Hence, we have a nonzero effective $1$-cycle
$\overline{\Gamma}$ on $X_{\overline{\Sigma}}$ such that

$\overline{\psi}^{*}\overline{F}=\overline{\psi}^{*}\overline{F}^{\prime}-(\overline{F}\cdot\overline{\Gamma})\overline{E}$

holds for any $\mathbb{Q}$-divsior $\overline{F}$ on $X_{\overline{\Sigma}}$
and $\overline{F}^{\prime}:=\overline{f}_{*}\overline{F}$. For the closed
point $\\{1\\}\in\mathbb{G}_{m}^{r}$, let $\Gamma$ be the $1$-cycle on
$X\times\\{1\\}\subset X_{\Sigma}\times\mathbb{G}_{m}^{r}$ that is the
pullback of $\overline{\Gamma}$ via the composite isomorphism
$X\times\\{1\\}\hookrightarrow X_{\Sigma}\times\mathbb{G}_{m}^{r}\to
X_{\Sigma}$.

Let us show the equation (2.5.1) for any $\mathbb{R}$-divisor $F$ on
$X_{\Sigma}$. By linearity, the problem is reduced to the case when $F$ is a
Cartier divisor. In order to prove the equation (2.5.1), we may replace $F$ by
a Cartier divisor linearly equivalent to $F$. Therefore, it suffices to prove
the equation (2.5.1) for the case when $F=\pi^{*}\overline{F}$ for some
Cartier divisor $\overline{F}$ on $X_{\overline{\Sigma}}$ (cf. Remark 2.4(1)).
We then obtain

$\psi^{*}F=\psi^{*}\pi^{*}\overline{F}=\psi^{\prime*}\pi^{\prime*}\overline{F}^{\prime}-(\overline{F}\cdot\overline{\Gamma})E=\psi^{*}F^{\prime}-(F\cdot\Gamma)E,$

where $\pi:X_{\Sigma}\to X_{\overline{\Sigma}}$ and
$\pi^{\prime}:X_{\Sigma^{\prime}}\to X_{\overline{\Sigma}^{\prime}}$ denote
the natural projections. Thus (3) holds. ∎

###  2.3. Globally $F$-regular varieties

In this subsection, we first recall the definition of globally $F$-regular
pairs (Definition 2.6) and a relative version (Definition 2.7). Strictly
speaking, we do not need the relative version in our proof. However, this will
make our presentation clearer. We also give a proof of the folklore fact that
$(X_{\Sigma},(1-\epsilon)D)$ is globally $F$-regular, where $X_{\Sigma}$ is a
normal toric variety and $D$ is the sum of the torus invariant prime divisors
(Proposition 2.8).

######  Definition 2.6 (cf. Definition 3.1 of [SS10]).

Assume that $k$ is of characteristic $p>0$. Let $X$ be a normal variety and
let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$.

  1. (1)

We say that $(X,\Delta)$ is sharply globally $F$-split if there exists
$e\in\mathbb{Z}_{>0}$ such that

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}(\mathcal{O}_{X}(\ulcorner(p^{e}-1)\Delta\urcorner))$

splits as an $\mathcal{O}_{X}$-module homomorphism.

  2. (2)

We say that $(X,\Delta)$ is globally $F$-regular if given an effective
$\mathbb{Z}$-divisor $E$, there exists $e\in\mathbb{Z}_{>0}$ such that

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}(\mathcal{O}_{X}((p^{e}-1)\Delta+E))$

splits as an $\mathcal{O}_{X}$-module homomorphism.

######  Definition 2.7 (cf. Definition 2.6 of [HX15]).

Assume that $k$ is of characteristic $p>0$. Let $f:X\to Y$ be a morphism from
a normal variety $X$ to a scheme $Y$ of finite type over $k$. Let $\Delta$ be
an effective $\mathbb{R}$-divisor on $X$.

  1. (1)

We say that $(X,\Delta)$ is sharply globally $F$-split over $Y$ if there
exists an open cover $Y=\bigcup_{i\in I}Y_{i}$ such that
$(f^{-1}(Y_{i}),\Delta|_{f^{-1}(Y_{i})})$ is sharply globally $F$-split for
any $i\in I$.

  2. (2)

We say that $(X,\Delta)$ is globally $F$-regular over $Y$ if there exists an
open cover $Y=\bigcup_{i\in I}Y_{i}$ such that
$(f^{-1}(Y_{i}),\Delta|_{f^{-1}(Y_{i})})$ is globally $F$-regular for any
$i\in I$.

######  Proposition 2.8.

Assume that $k$ is of characteristic $p>0$. Let $\Sigma$ be a fan of
$N_{\mathbb{R}}$ and let $X_{\Sigma}$ be the toric variety of $\Sigma$. Set
$D:=\sum_{\rho\in\Sigma(1)}D_{\rho}$ (recall that $D_{\rho}$ denotes the prime
divisor that is the closure of the orbit $O(\rho)$ of $\rho\in\Sigma(1)$).
Then the following hold.

  1. (1)

$(X,D)$ is globally sharply $F$-split.

  2. (2)

$(X,(1-\epsilon)D)$ is globally $F$-regular for any $\epsilon\in\mathbb{R}$
with $0<\epsilon\leq 1$.

###### Proof.

Set $X:=X_{\Sigma}$. For the algebraic closure $\overline{k}$ of $k$, if the
base change $(X_{\overline{k}},\Delta_{\overline{k}})$ is globally sharply
$F$-split (resp. globally $F$-regular), then so is $(X,\Delta)$. Hence, the
problem is reduced to the case when $k$ is an algebraically closed field.

Let us show (1). For the relative Frobenius $\overline{F}:X\to X^{\prime}$ and
the absolute Frobenius morphisms $F:X\to X$ and
$F_{k}:{\operatorname{Spec}}\,k\to{\operatorname{Spec}}\,k$, we have the
following commutative diagram in which the square diagram is cartesian:

${X}$${{X^{\prime}}}$${X}$${{\operatorname{Spec}}\,k}$${{\operatorname{Spec}}\,k.}$$\scriptstyle{\overline{F}}$$\scriptstyle{\alpha}$$\scriptstyle{F_{k}}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{F=F_{X}}$$\scriptstyle{\pi}$

Note that $\alpha$ and $F_{k}$ are isomorphisms of schemes because $k$ is a
perfect field. For $D^{\prime}:=\alpha^{*}D$, it follows from [Fuj07, 2.6]
that

$\mathcal{O}_{X}(-D)\to\overline{F}_{*}\mathcal{O}_{X^{\prime}}(-D^{\prime}).$

splits. Since $\alpha$ is an isomorphism, also

$F:\mathcal{O}_{X}(-D)\to F_{*}\mathcal{O}_{X}(-D)$

splits. Applying $(-)\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}(D)$, we see that

$F:\mathcal{O}_{X}\to F_{*}\mathcal{O}_{X}((p-1)D)$

splits. Thus, (1) holds.

Let us show (2). Fix a rational number $\epsilon$ with $0<\epsilon<1$. Set
$\Delta:=(1-\epsilon)D$. By [SS10, Theorem 3.9], it suffices to show the
following assertions (i) and (ii).

  1. (i)

There exists $e\in\mathbb{Z}_{>0}$ such that

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\ulcorner(p^{e}-1)\Delta\urcorner+D)$

splits as an $\mathcal{O}_{X}$-module homomorphism.

  2. (ii)

$(X\setminus{\operatorname{Supp}}\,D,\Delta|_{X\setminus{\operatorname{Supp}}\,D})$
is globally $F$-regular.

Let us show (i). By $\Delta=(1-\epsilon)D$ and $0<\epsilon<1$, we have

$(p^{e}-1)\Delta+D\leq(p^{e}-1)D$

for $e\gg 0$. Since

$\mathcal{O}_{X}\to F_{*}^{e}\mathcal{O}_{X}((p^{e}-1)D)$

splits by (1), we see that (i) holds. The assertion (ii) follows from the fact
that $\Delta|_{X\setminus{\operatorname{Supp}}\,D}=0$ and
$X\setminus{\operatorname{Supp}}\,D$ is isomorphic to the torus
$\mathbb{G}_{m}^{\dim X}$ (note that $(Z,0)$ is globally $F$-regular for any
smooth affine variety $Z$). ∎

##  3\. Vanishing theorems for globally F-regular varieties

In this section, we establish some vanishing theorems for globally $F$-regular
varieties. In particular, we establish an analogue of our main theorem
(Theorem 1.1) for globally $F$-regular surfaces (Theorem 3.5).

######  Lemma 3.1.

Let $\alpha:X\to S$ be a proper morphism of normal varieties such that
$\alpha_{*}\mathcal{O}_{X}=\mathcal{O}_{S}$ and $\dim X>\dim S$. Let $D$ be a
$\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor on $X$ such that $-D$ is
$\alpha$-big. Then $\alpha_{*}\mathcal{O}_{X}(D)=0$.

###### Proof.

We may assume that $S$ is affine. It is enough to show that
$H^{0}(X,\mathcal{O}_{X}(D))=0$. Suppose that $H^{0}(X,\mathcal{O}_{X}(D))\neq
0$. Then there exists an effective $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor
$E$ on $X$ such that $D\sim E$. For the generic fibre $X_{\xi}$ of
$\alpha:X\to S$, we have that $E|_{X_{\xi}}$ is effective and $-E_{X_{\xi}}$
is big. This contradicts $\dim X_{\xi}=\dim X-\dim S>0$. ∎

Although the following proposition is well known to experts, we give a proof
for the sake of completeness.

######  Proposition 3.2.

Assume that $k$ is of characteristic $p>0$. Let $\alpha:X\to S$ be a
projective morphism from a normal variety $X$ to a scheme $S$ of finite type
over $k$. Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$(X,\Delta)$ is globally $F$-regular over $S$.

  1. (1)

If $D$ is a $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor on $X$ such that
$D-(K_{X}+\Delta)$ is $\alpha$-nef and $\alpha$-big, then
$R^{i}f_{*}\mathcal{O}_{X}(D)=0$ for $i>0$.

  2. (2)

If $D$ is an $\alpha$-nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor on $X$,
then $R^{i}f_{*}\mathcal{O}_{X}(D)=0$ for any $i>0$.

###### Proof.

Taking the Stein factorisation of $S$ and an affine cover of $S$, we may
assume that $\alpha_{*}\mathcal{O}_{X}=\mathcal{O}_{S}$, $S$ is affine, and
$(X,\Delta)$ is globally $F$-regular. Furthermore, the problem is reduced to
the case when $k$ is $F$-finite by taking a suitable subfield of $k$ which is
finitely generated over $\mathbb{F}_{p}$.

Let us show (1). Since $D-(K_{X}+\Delta)$ is $\alpha$-nef and $\alpha$-big,
there exist $\epsilon_{1}\in\mathbb{Q}_{>0}$ and an effective
$\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $E$ such that

$D-(K_{X}+\Delta)-\epsilon E$

is $\alpha$-ample for any $\epsilon\in\mathbb{Q}$ with
$0<\epsilon<\epsilon_{1}$. There also exists $\epsilon_{2}\in\mathbb{Q}_{>0}$
such that $(X,\Delta+\epsilon E)$ is globally $F$-regular if
$0\leq\epsilon<\epsilon_{2}$ [SS10, Corollary 6.1]. After replacing
$(X,\Delta)$ by $(X,\Delta+\epsilon E)$ for
$0<\epsilon<\min\\{\epsilon_{1},\epsilon_{2}\\}$, we may assume that
$D-(K_{X}+\Delta)$ is $\alpha$-ample.

We have a split surjection:

$F_{*}^{e}\mathcal{O}_{X}(-\ulcorner(p^{e}-1)(K_{X}+\Delta)\urcorner)\to\mathcal{O}_{X}.$

Taking the tensor product with $\mathcal{O}_{X}(D)$ and applying the double
dual, we obtain another split surjection:

$F_{*}^{e}\mathcal{O}_{X}(p^{e}D-\ulcorner(p^{e}-1)(K_{X}+\Delta)\urcorner)\to\mathcal{O}_{X}(D).$

For $i>0$ and $e\gg 0$, we have

$\displaystyle
H^{i}(X,F_{*}^{e}\mathcal{O}_{X}(p^{e}D-\ulcorner(p^{e}-1)(K_{X}+\Delta)\urcorner))$
$\displaystyle\simeq$ $\displaystyle
H^{i}(X,\mathcal{O}_{X}(p^{e}D-\ulcorner(p^{e}-1)(K_{X}+\Delta)\urcorner))$
$\displaystyle=$ $\displaystyle
H^{i}(X,\mathcal{O}_{X}(D+\llcorner(p^{e}-1)(D-(K_{X}+\Delta))\lrcorner))$
$\displaystyle=$ $\displaystyle 0,$

where the last equality follows from the Serre vanishing theorem. Therefore,
it holds that

$H^{i}(X,\mathcal{O}_{X}(D))=0,$

which completes the proof of (1).

Let us show (2). Since $(X,\Delta)$ is globally $F$-regular, after possibly
taking $S$ by a suitable open cover, there exists an effective
$\mathbb{R}$-divisor $\Delta^{\prime}$ on $X$ such that $-(K_{X}+\Delta)$ is
$\alpha$-ample [SS10, Theorem 4.3(i)]. In particular,
$D-(K_{X}+\Delta^{\prime})$ is $\alpha$-ample, hence the assertion (2) follows
from (1). ∎

######  Proposition 3.3.

Assume that $k$ is of characteristic $p>0$. Let
$\alpha:X\xrightarrow{f}Y\xrightarrow{\beta}S$ be projective morphisms of
normal varieties with $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$. Let $\Delta$ be
an effective $\mathbb{R}$-divisor such that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier and $(X,\Delta)$ is globally $F$-regular over $Y$. Let
$D$ be a $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor such that

  1. (1)

$-D$ is $f$-big,

  2. (2)

$D-(K_{X}+\Delta)$ is $f$-nef and $f$-big, and

  3. (3)

$\dim X>\dim Y$.

Then $R^{i}\alpha_{*}\mathcal{O}_{X}(D)=0$ for any $i\geq 0$.

###### Proof.

We have the following isomorphism in the derived category:

$R\alpha_{*}\mathcal{O}_{X}(D)\simeq R\beta_{*}Rf_{*}\mathcal{O}_{X}(D).$

Hence, it suffices to show that $Rf_{*}\mathcal{O}_{X}(D)=0$, i.e.
$R^{i}f_{*}\mathcal{O}_{X}(D)=0$ for any $i\geq 0$. We have
$f_{*}\mathcal{O}_{X}(D)=0$ by (1), (3), and Lemma 3.1. By (2), it follows
from Proposition 3.2 that $R^{i}f_{*}\mathcal{O}_{X}(D)=0$ holds for any
$i>0$. ∎

######  Corollary 3.4.

Let $\alpha:X\to S$ be a projective morphism from a normal
$\mathbb{Q}$-factorial variety $X$ to a scheme $S$ of finite type over $k$.
Let $D$ be a $\mathbb{Z}$-divisor on $X$ such that $D-K_{X}$ is $\alpha$-big.
Assume that

  1. (1)

$X$ is globally $F$-regular over $S$, and

  2. (2)

either $D$ is $\alpha$-nef or there exists a $D$-Mori fibre space $f:X\to Y$
over $S$, i.e. $f$ is a projective $S$-morphism to a normal variety $Y$
projective over $S$ to a normal variety $Y$ which is projective over $S$ such
that $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$, $\dim X>\dim Y$, $\rho(X/Y)=1$,
and $-D$ is $f$-ample.

Then $R^{i}\alpha_{*}\mathcal{O}_{X}(D)=0$ for any $i>0$.

###### Proof.

If $D$ is $\alpha$-nef, then the assertion follows from Proposition 3.2(2).
Hence, we may assume that there exists a $D$-Mori fibre space $f:X\to Y$ over
$S$. Then we have the induced morphisms
$\alpha:X\xrightarrow{f}Y\xrightarrow{\beta}S$. Since $D-K_{X}$ is
$\alpha$-big, $D-K_{X}$ is $f$-big. Furthermore, it follows from $\dim X>\dim
Y$ and $\rho(X/Y)=1$ that any effective $\mathbb{Q}$-divisor on $X$ is
$f$-nef. Therefore, $D-K_{X}$ is $f$-nef and $f$-big. Since $-D$ is $f$-ample,
it holds by Proposition 3.3 that $R^{i}\alpha_{*}\mathcal{O}_{X}(D)=0$ for any
$i>0$. ∎

######  Theorem 3.5.

Assume that $k$ is of characteristic $p>0$. Let $\alpha:X\to S$ be a proper
morphism from a normal surface $X$ to a scheme $S$ of finite type over $k$
such that $X$ is globally $F$-regular over $S$. Let $D$ be a
$\mathbb{Z}$-divisor on $X$ such that $D-(K_{X}+B)$ is $\alpha$-nef and
$\alpha$-big for some $\mathbb{R}$-divisor $B$ on $X$ whose coefficients are
contained in $[0,1)$. Then $R^{i}\alpha_{*}\mathcal{O}_{X}(D)=0$ for any
$i>0$.

###### Proof.

Note that $X$ is $\mathbb{Q}$-factorial [Tan18, Corollary 4.11]. By the same
argument as in [Fuj12, Lemma 2.2], $X$ is quasi-projective over $k$, and hence
$f$ is projective. Taking a suitable affine open cover of $S$, we may assume
that $S$ is affine and $X$ is globally $F$-regular. We run a $D$-MMP over $S$
[Tan18, Theorem 1.1]:

$X=X_{0}\xrightarrow{f_{0}}X_{1}\xrightarrow{f_{1}}\cdots\xrightarrow{f_{N-1}}X_{N}.$

Set $D_{n}$ and $B_{n}$ to be the push-forwards of $D$ and $B$ to $X_{n}$,
respectively. Note that either $D_{N}$ is $\alpha_{N}$-nef or there exists a
$D_{N}$-Mori fibre space $X_{N}\to Y$ over $S$. By Corollary 3.4, we get

$H^{i}(X_{N},\mathcal{O}_{X_{N}}(D_{N}))=0$

for any $i>0$. Fix $n\in\\{0,1,...,N-1\\}$. It is enough to show that

$H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))\simeq
H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1})).$

For the $f_{n}$-exceptional prime divisor $E:={\operatorname{Ex}}(f_{n})$, we
have $D_{n}=f_{n}^{*}D_{n+1}+aE$ for some $a\in\mathbb{Q}_{>0}$. Therefore, we
obtain

$(f_{n})_{*}\mathcal{O}_{X_{n}}(D_{n})=\mathcal{O}_{X_{n+1}}(D_{n+1}).$

We have the following spectral sequence:

$E_{2}^{i,j}=H^{i}(X_{n+1},R^{j}(f_{n})_{*}\mathcal{O}_{X_{n}}(D_{n}))\Rightarrow
H^{i+j}(X_{n},\mathcal{O}_{X_{n}}(D_{n})).$

By [Tan18, Theorem 3.3], we obtain
$R^{j}(f_{n})_{*}\mathcal{O}_{X_{n}}(D_{n})=0$ for $j>0$. Hence,

$H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1}))=H^{i}(X_{n+1},(f_{n})_{*}\mathcal{O}_{X_{n}}(D_{n}))$
$=E_{2}^{i,0}\simeq E^{i}=H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n})),$

which completes the proof. ∎

##  4\. Vanishing theorems for toric varieties

In this section, we prove the main theorem of this paper (Theorem 4.3). To
this end, we first treat a special but essential case (Theorem 4.1). In order
to conclude Theorem 4.3 from Theorem 4.1, we also establish an auxiliary
result (Lemma 4.2).

######  Theorem 4.1.

Let $\alpha:X\to S$ be a projective morphism from a $\mathbb{Q}$-factorial
normal toric variety $X$ to a separated scheme $S$ of finite type over $k$.
Let $D$ be a $\mathbb{Z}$-divisor on $X$ such that $D\sim_{\mathbb{R}}K_{X}+B$
for some klt pair $(X,B)$, where $B$ is an $\alpha$-big effective
$\mathbb{R}$-divisor. Then $R^{i}\alpha_{*}\mathcal{O}_{X}(D)=0$ for any
$i>0$.

###### Proof.

We may assume that $k$ is of characteristic $p>0$, as otherwise the assertion
is well known. We divide the proof into four steps.

######  Step 1.

In order to prove Theorem 4.1, we may assume that the following conditions
hold.

  1. (i)

$\alpha_{*}\mathcal{O}_{X}=\mathcal{O}_{S}$.

  2. (ii)

$S$ is an affine normal toric variety.

  3. (iii)

$f$ is a toric morphism.

###### Proof.

(of Step 1) Taking the Stein factorisation of $f$, we may assume (i). Then the
problem is reduced to the case when $f$ is a toric morphism to a normal toric
variety $S$ [Tana, Theorem 1.1]. In particular, (iii) holds. Replacing $(X,S)$
by $(f^{-1}(S_{i}),S_{i})$ for a toric open affine cover $S=\bigcup_{i\in
I}S_{i}$, we may further assume that (ii) holds. This completes the proof of
Step 1. ∎

In what follows, we assume that (i)–(iii) hold. We run a $D$-MMP (i.e. a
$(K_{X}+B)$-MMP) over $S$ (cf. (2.3)):

$X=X_{0}\overset{f_{0}}{\dashrightarrow}X_{1}\overset{f_{1}}{\dashrightarrow}\cdots\overset{f_{N-1}}{\dashrightarrow}X_{N}$
$B_{n+1}:=(f_{n})_{*}B_{n},\qquad D_{n+1}:=(f_{n})_{*}D_{n},$

where $X_{N}$ denotes the end result, so that either $D_{N}$ is nef over $S$
or there exists a $D_{N}$-Mori fibre space $X_{N}\to W$ over $S$. Note that
the assumptions are stable under this MMP, i.e. for any $n\in\\{0,1,...,N\\}$,
it holds that

  1. (iv)

the induced morphism $\alpha_{n}:X_{n}\to S$ is a projective morphism,

  2. (v)

$X_{n}$ is a $\mathbb{Q}$-factorial toric variety,

  3. (vi)

$(X_{n},B_{n})$ is klt,

  4. (vii)

$B_{n}$ is an $\alpha_{n}$-big effective $\mathbb{R}$-divisor, and

  5. (viii)

$D_{n}$ is a $\mathbb{Z}$-divisor such that
$D_{n}\sim_{\mathbb{R}}K_{X_{n}}+B_{n}$.

######  Step 2.

The equation $H^{i}(X_{N},\mathcal{O}_{X_{N}}(D_{N}))=0$ holds for any $i>0$.

###### Proof.

(of Step 2) Since $X$ is a toric variety, $X$ is globally $F$-regular
(Proposition 2.8). In particular, $X$ is globally $F$-regular over $S$. It
follows from Corollary 3.4 that $H^{i}(X,\mathcal{O}_{X}(D))=0$ for any $i>0$.
This completes the proof of Step 2. ∎

######  Step 3.

Fix $n\in\\{0,1,...,N-1\\}$. Assume that $f_{n}:X_{n}\dashrightarrow X_{n+1}$
is a $D_{n}$-divisorial contraction. Then

$H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))\simeq
H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1}))$

for any $i\geq 0$.

###### Proof.

(of Step 3) Since $f_{n}:X_{n}\to X_{n+1}$ is a $D_{n}$-divisorial
contraction, $E:={\operatorname{Ex}}(f_{n})$ is a torus invariant primer
divisor (Remark 2.4(3)). Then we have

$D_{n}=f_{n}^{*}D_{n+1}+aE$

for some $a\in\mathbb{Q}_{>0}$. In particular, we obtain
$(f_{n})_{*}\mathcal{O}_{X_{n}}(D_{n})=\mathcal{O}_{X_{n+1}}(D_{n+1})$. Let us
prove that $R^{i}(f_{n})_{*}\mathcal{O}_{X_{n}}(D_{n})=0$ for any $i>0$. For
the coefficient $b$ of $E$ in $B_{n}$, we can write $B_{n}=bE+A$, where $A$ is
$f_{n}$-nef and $f_{n}$-big. We have that

$D\sim_{\mathbb{R}}K_{X_{n}}+B_{n}=K_{X_{n}}+bE+A_{n}$

Since $(X_{n},bE)$ is globally $F$-regular (Proposition 2.8), we have that
$R^{i}(f_{n})_{*}\mathcal{O}_{X_{n}}(D)=0$ for $i>0$ (Proposition 3.2(1)).
Therefore it holds that

$H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))\simeq
H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1})).$

This completes the proof of Step 3. ∎

######  Step 4.

Fix $n\in\\{0,1,...,N-1\\}$. Assume that $f_{n}:X_{n}\dashrightarrow X_{n+1}$
is a $D_{n}$-flip. Then

$H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))\simeq
H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1}))$

for any $i\geq 0$.

###### Proof.

(of Step 4) We have the $D_{n}$-flipping contraction $\varphi:X_{n}\to Z$ and
its flip $\varphi^{\prime}:X_{n+1}\to Z$. In this case, by Proposition 2.5, we
have the following square diagram of projective birational toric morphisms of
quasi-projective normal toric varieties

${Y}$${{X_{n}}}$${{X_{n+1}}}$${Z}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{f_{n}}$

such that

  1. (1)

$Y$ is $\mathbb{Q}$-factorial,

  2. (2)

$E:={\operatorname{Ex}}(\psi)={\operatorname{Ex}}(\psi^{\prime})$ is a torus
invariant prime divisor on $Y$, and

  3. (3)

there exists a nonzero effective $1$-cycle $\Gamma$ on $X_{n}$ satisfying the
equation

$\psi^{*}F=\psi^{\prime*}((f_{n})_{*}F)-(F\cdot\Gamma)E$

for any $\mathbb{R}$-divisor $F$ on $X_{n}$.

We have

(4.1.1) $\psi^{*}D_{n}\sim_{\mathbb{R}}\psi^{*}(K_{X_{n}}+B_{n})$

and

(4.1.2) $K_{Y}+\psi_{*}^{-1}B_{n}=\psi^{*}(K_{X_{n}}+B_{n})+aE$

for some rational number $a>-1$. By (4.1.1) and (4.1.2), we obtain

(4.1.3) $\psi^{*}D_{n}\sim_{\mathbb{R}}K_{Y}+\psi_{*}^{-1}B_{n}-aE.$

On the other hand, we have

(4.1.4) $\ulcorner\psi^{*}D_{n}\urcorner=\psi^{*}D_{n}+bE$

for some rational number $b\in[0,1)$. Since $\varphi:X_{n}\to Z$ is the
contraction of a $D_{n}$-negative extremal ray, it holds by (3) that

(4.1.5) $\psi^{*}D_{n}=\psi^{\prime*}D_{n+1}+cE$

for some rational number $c>0$. By (4.1.3) and (4.1.4), we get

(4.1.6)
$\ulcorner\psi^{*}D_{n}\urcorner\sim_{\mathbb{R}}K_{Y}+\psi_{*}^{-1}B_{n}+(-a+b)E.$

We separately treat the following two cases: $-a+b<1$ and $-a+b\geq 1$.

Assume that $-a+b<1$. Then there uniquely exists $m\in\mathbb{Z}_{\geq 0}$
such that $-a+b+m\in[0,1)$. Set

$D_{Y}:=\ulcorner\psi^{*}D_{n}\urcorner+mE\sim_{\mathbb{R}}K_{Y}+\psi_{*}^{-1}B_{n}+(-a+b+m)E.$

Since $E$ is torus invariant, $(Y,(-a+b+m)E)$ is globally $F$-regular
(Proposition 2.8). Note that $\psi_{*}^{-1}B_{n}$ is $\psi$-nef and
$\psi^{\prime}$-nef. It follows from Proposition 3.2(1) that

$R^{i}\psi_{*}\mathcal{O}_{Y}(D_{Y})=R^{i}\psi^{\prime}_{*}\mathcal{O}_{Y}(D_{Y})=0$

for any $i>0$. Hence, we have

$H^{i}(X_{n},\psi_{*}\mathcal{O}_{Y}(D_{Y}))\simeq
H^{i}(X_{n+1},\psi^{\prime}_{*}\mathcal{O}_{Y}(D_{Y}))$

for any $i\geq 0$. By $c>0$ and

$D_{Y}=\ulcorner\psi^{*}D_{n}\urcorner+mE=\ulcorner\psi^{\prime*}D_{n+1}+cE\urcorner+mE,$

we have $\psi_{*}\mathcal{O}_{Y}(D_{Y})=\mathcal{O}_{X_{n}}(D_{n})$ and
$\psi^{\prime}_{*}\mathcal{O}_{Y}(D_{Y})=\mathcal{O}_{X_{n+1}}(D_{n+1})$, so
that we get

$H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))\simeq
H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1}))$

for any $i\geq 0$. This completes the proof for the case when $-a+b<1$.

We may assume that $-a+b\geq 1$. In particular, we obtain $0<-a<1$ and
$0<b<1$. Hence, $0\leq-a+b-1<1$. Set $D_{Y}:=\llcorner\psi^{*}D_{n}\lrcorner$.
We obtain

$D_{Y}=\llcorner\psi^{*}D_{n}\lrcorner=\ulcorner\psi^{*}D_{n}\urcorner-E=\psi^{*}D_{n}-(1-b)E$
$\sim_{\mathbb{R}}K_{Y}+\psi_{*}^{-1}B_{n}+(-a+b-1)E.$

where the second equality holds by (4.1.4) and $0<b<1$, the third one by
(4.1.4), and the last $\mathbb{R}$-linear equivalence by (4.1.3). Since $E$ is
torus invariant, $(Y,(-a+b-1)E)$ is globally $F$-regular (Proposition 2.8). As
$\psi_{*}^{-1}B_{n}$ is $\psi$-nef and $\psi^{\prime}$-nef, it follows from
Proposition 3.2 that

$R^{j}\psi_{*}\mathcal{O}_{Y}(D_{Y})=R^{j}\psi^{\prime}_{*}\mathcal{O}_{Y}(D_{Y})=0$

for any $j>0$, which in turn implies

$H^{i}(X_{n},\psi_{*}\mathcal{O}_{Y}(D_{Y}))\simeq
H^{i}(X_{n+1},\psi^{\prime}_{*}\mathcal{O}_{Y}(D_{Y}))$

for any $i\geq 0$. By
$D_{Y}=\llcorner\psi^{*}D_{n}\lrcorner=\llcorner\psi^{\prime*}D_{n+1}+cE\lrcorner$
and $c>0$, we have

$\psi_{*}\mathcal{O}_{Y}(D_{Y})=\mathcal{O}_{X_{n}}(D_{n})\qquad{\rm
and}\qquad\psi^{\prime}_{*}\mathcal{O}_{Y}(D_{Y})=\mathcal{O}_{X_{n+1}}(D_{n+1}).$

Hence, we obtain

$H^{i}(X_{n},\mathcal{O}_{X_{n}}(D_{n}))\simeq
H^{i}(X_{n+1},\mathcal{O}_{X_{n+1}}(D_{n+1})),$

This completes the proof of Step 4. ∎

Step 2, Step 3, and Step 4 complete the proof of Theorem 4.1. ∎

######  Lemma 4.2.

Let $\alpha:X\to S$ be a proper morphism from a normal toric variety to a
scheme of finite type over $k$. Let $D$ be an $\alpha$-nef
$\mathbb{R}$-Cartier $\mathbb{R}$-divisor. Then $D$ is $\alpha$-semi-ample.

###### Proof.

We may assume that $S$ is affine. By toric Chow’s lemma, the problem is
reduced to the case when $\alpha$ is projective. Since ${\rm NE}(X/S)$ is a
rational polyhedtral cone [FS04, Theorem 4.1], its dual ${\rm Nef}(X/S)$ is a
rational polyhedral cone, where ${\rm Nef}(X/S)$ consists of the classes of
the $\alpha$-nef $\mathbb{R}$-divisors on $X$. Then we can write
$D=\sum_{i=1}^{r}a_{i}D_{i}$ for some $a_{1},...,a_{r}\in\mathbb{R}_{>0}$ and
nef $\mathbb{Q}$-divisors $D_{1},...,D_{r}$. Since each $D_{i}$ is
$\alpha$-semi-ample [FS04, Proposition 4.6], also $D$ is $\alpha$-semi-ample
(note that [FS04, Proposition 4.6] imposes the assumption that $D_{i}$ is
torus invariant, however we can drop this assumption because any Cartier
divisor is linearly equivalent to a torus invariant Cartier divisor [CLS11,
Theorem 4.2.1]). ∎

######  Theorem 4.3.

Let $\alpha:X\to S$ be a projective morphism from a normal toric variety $X$
to a separated scheme $S$ of finite type over $k$. Let $D$ be a
$\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor on $X$. Assume that one of the
following conditions holds.

  1. (1)

$D\sim_{\mathbb{R}}K_{X}+B_{1}+B_{2}$ for some effective $\mathbb{R}$-divisors
$B_{1}$ and $B_{2}$ on $X$ such that $(X,B_{1}+B_{2})$ is klt and $B_{2}$ is
an $\alpha$-big $\mathbb{R}$-Cartier $\mathbb{R}$-divisor.

  2. (2)

$D-(K_{X}+B)$ is $\alpha$-nef and $\alpha$-big for some effective
$\mathbb{R}$-divisor $B$ on $X$ such that $(X,B)$ is klt.

Then $R^{i}\alpha_{*}\mathcal{O}_{X}(D)=0$ for any $i>0$.

###### Proof.

We may assume that $S$ is affine, $\alpha_{*}\mathcal{O}_{X}=\mathcal{O}_{S}$,
and $\alpha$ is a toric morphism to a normal toric variety $S$ [Tana, Theorem
1.1].

Assume (1). Let $\mu:X^{\prime}\to X$ be a projective toric small birational
morphism from a $\mathbb{Q}$-factorial toric variety $X^{\prime}$ [CLS11,
Proposition 11.1.7]:

$\alpha^{\prime}:X^{\prime}\xrightarrow{\mu}X\xrightarrow{\alpha}S.$

We define a $\mathbb{Z}$-divisor $D^{\prime}$ and $\mathbb{R}$-divisors
$B^{\prime}_{1}$ and $B^{\prime}_{2}$ on $X^{\prime}$ by

$\mu_{*}B^{\prime}_{1}=B_{1},\quad\mu_{*}B^{\prime}_{2}=B_{2},\quad\mu_{*}D^{\prime}=D.$

Note that $D^{\prime}$ is a $\mathbb{Z}$-divisor. Set
$B^{\prime}:=B^{\prime}_{1}+B^{\prime}_{2}$, which is an effective
$\mathbb{R}$-divisor on $X^{\prime}$. We then have

  1. (a)

$B^{\prime}_{2}=\mu^{*}B_{2}$,

  2. (b)

$K_{X^{\prime}}+B^{\prime}=\mu^{*}(K_{X}+B_{1}+B_{2})$, and
/

  3. (c)

$D^{\prime}=\mu^{*}D\sim_{\mathbb{R}}K_{X^{\prime}}+B^{\prime}$.

Since $(X,B_{1}+B_{2})$ is klt, also $(X^{\prime},B^{\prime})$ is klt by (b).
As $B_{2}$ is $\alpha$-big, $B^{\prime}_{2}$ is $\alpha^{\prime}$-big by (a).
Hence, $B^{\prime}=B^{\prime}_{1}+B^{\prime}_{2}$ is $\alpha^{\prime}$-big. By
Theorem 4.1, we obtain

$R^{i}\alpha^{\prime}_{*}\mathcal{O}_{X^{\prime}}(D^{\prime})=0$

for any $i>0$. Since $B^{\prime}$ is $\mu$-big, it follows again from Theorem
4.1 that

$R^{j}\mu_{*}\mathcal{O}_{X^{\prime}}(D^{\prime})=0$

for any $j>0$. By the Leray spectral sequence

$E_{2}^{i,j}:=H^{i}(X,R^{j}\mu_{*}\mathcal{O}_{X^{\prime}}(D^{\prime}))\Rightarrow
H^{i+j}(X^{\prime},\mathcal{O}_{X^{\prime}}(D^{\prime}))=:E^{i+j},$

we obtain

$H^{i}(X,\mathcal{O}_{X}(D))=H^{i}(X,\mu_{*}\mathcal{O}_{X^{\prime}}(D^{\prime}))=E_{2}^{i,0}\simeq
E^{i}=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(D^{\prime}))=0$

for any $i>0$, where the first equality is guaranteed by
$D^{\prime}=\mu^{*}D$. This completes the proof for the case when (1) holds.

Assume (2). It suffices to show that (1) holds. Set $A:=D-(K_{X}+B)$. Note
that $A$ is $\alpha$-semi-ample (Lemma 4.2). In particular, we may assume that
$A$ is semi-ample, i.e. $|mA|$ is base point free for some
$m\in\mathbb{Z}_{>0}$. It follows from [Tanb, Theorem 1.4] that after taking a
suitable base change of the base field $k$, there exists an effective
$\mathbb{R}$-Cartier $\mathbb{R}$-divisor $B_{2}$ such that
$A\sim_{\mathbb{R}}B_{2}$ and $(X,B+B_{2})$ is klt. We see that the condition
(1) holds for $B_{1}:=B$. ∎

## References