Characterizations of Toric Varieties via Polarized Endomorphisms

TL;DR

This paper provides new characterizations of toric varieties using polarized endomorphisms, focusing on geometric and group action conditions, and shows under certain conditions, such varieties admit covers that are toric.

Contribution

It introduces two novel criteria for identifying toric varieties based on polarized endomorphisms and geometric properties, extending previous characterizations.

Findings

If $X$ is $Q$-factorial and $G$-almost homogeneous with $f$ $G$-equivariant, then $X$ is toric.

Under certain geometric conditions, $X$ admits a cover that is toric, and $f$ lifts to this cover.

Smoothness assumptions lead to $X$ itself being toric.

Abstract

Let $X$ be a normal projective variety and $f:X\to X$ a non-isomorphic polarized endomorphism. We give two characterizations for $X$ to be a toric variety. First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for some linear algebraic group $G$ such that $f$ is $G$-equivariant, then $X$ is a toric variety. Next we give a geometric characterization: if $X$ is of Fano type and smooth in codimension 2 and if there is an $f^{-1}$-invariant reduced divisor $D$ such that $f|_{X\backslash D}$ is quasi-\'etale and $K_X+D$ is $\mathbb{Q}$-Cartier, then $X$ admits a quasi-\'etale cover $\widetilde{X}$ such that $\widetilde{X}$ is a toric variety and $f$ lifts to $\widetilde{X}$. In particular, if $X$ is further assumed to be smooth, then $X$ is a toric variety.

Equations

$$c:=\inf \{n+{\dim }_{\mathbb{R}}(\sum \mathbb{R}[S_i])-\sum a_i|\sum a_i{S}_i\le \Delta ,a_i\ge 0,S_i\ge 0\};$$

$$f:X\to X;[X_0:\cdots :X_n]\mapsto [X_0^q:\cdots :X_n^q]$$

$$((t_1,\dots ,t_n),[X_0:\cdots :X_n])\mapsto [X_0:t_1{X}_1:\cdots :t_n{X}_n].$$

$$\phi :T\to T;(t_1,\dots ,t_n)\mapsto (t_1^q,\dots ,t_n^q)$$

$${\hat{\Omega }}_X^1(\log D)=j_{\ast }{\Omega }_U^1(\log D\cap U).$$

$$⟨\Delta ⟩:=└\Delta ┘+┌2\Delta ┐-└2\Delta ┘=\sum _{i:a_i>1/2}{D}_i.$$

$$\sum a_i{S}_i\le \Delta ,$$

$$\ell (D)\ge h^0(X,{\hat{\Omega }}_X^1(\log D))+r(D)-\tilde{q}(X).$$

$$c(X,D)\le n+\tilde{q}(X)-h^0(X,{\hat{\Omega }}_X^1(\log D)).$$

$$0\to {\Omega }_{\tilde{X}}^1\to {\Omega }_{\tilde{X}}^1(\log \tilde{D})\to {\oplus }_i{\mathcal{O}}_{{\tilde{D}}_i}\to 0,$$

$$0\to H^0(\tilde{X},{\Omega }_{\tilde{X}}^1)\to H^0(\tilde{X},{\Omega }_{\tilde{X}}^1(\log \tilde{D}))\to {\oplus }_i{H}^0({\tilde{D}}_i,{\mathcal{O}}_{{\tilde{D}}_i})\to H^1(\tilde{X},{\Omega }_{\tilde{X}}^1),$$

$$f^{\ast }{c}_i({\hat{\Omega }}_X^1(\log D))=c_i(f^{\ast }{\hat{\Omega }}_X^1(\log D))=c_i({\hat{\Omega }}_X^1(\log D)).$$

$$q^{n-i}{c}_i({\hat{\Omega }}_X^1(\log D))\cdot H^{n-i}=f^{\ast }{c}_i({\hat{\Omega }}_X^1(\log D)\cdot (f^{\ast }H{)}^{n-i}=(\mathrm{deg}f)c_i({\hat{\Omega }}_X^1(\log D)\cdot H^{n-i}$$

$${\mu }_H(\mathcal{F}):=\frac{c_1(\mathcal{F})\cdot H^{n-1}}{\mathrm{rank}\mathcal{F}}>0.$$

$$s=\sup \{{\mu }_H(\mathcal{F})|\mathcal{F}\subset {\hat{\Omega }}_X^1(\log D)\}<\infty .$$

Full text

First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for some linear algebraic group $G$ such that $f$ is $G$-equivariant, then $X$ is a toric variety.

Next we give a geometric characterization: if $X$ is of Fano type and smooth in codimension 2 and if there is an $f^{-1}$-invariant reduced divisor $D$ such that $f|_{X\backslash D}$ is quasi-étale and $K_{X}+D$ is $\mathbb{Q}$-Cartier, then $X$ admits a quasi-étale cover $\widetilde{X}$ such that $\widetilde{X}$ is a toric variety and $f$ lifts to $\widetilde{X}$. In particular, if $X$ is further assumed to be smooth, then $X$ is a toric variety.

1. Introduction

We work over an algebraically closed field $k$ of characteristic zero. Let $X$ be a normal projective variety of dimension $n\geq 1$.

A surjective endomorphism $f:X\to X$ is polarized, if there is an ample Cartier divisor $H$ on $X$ such that we have $f^{\ast}H\sim qH$ (linear equivalence) for some integer $q>1$. See [14] for some conjectures on polarised endomorphisms.

The variety $X$ is said to be toric or a toric variety if $X$ contains an algebraic torus $T=(k^{*})^{n}$ as an (affine) open dense subset such that the natural multiplication action of $T$ on itself extends to an action on the whole variety $X$. In this case, let $D:=X\backslash T$, which is a divisor; the pair $(X,D)$ is said to be a toric pair.

We observe that a toric variety (e.g. the projective space) has lots of symmetries. The purpose of this short paper is to use symmetries to characterize toric pairs.

We first give a characterization of toric varieties via polarized endomorphisms and linear algebraic group actions. Let $G$ be a linear algebraic group acting on $X$. We say $X$ is $G$-almost homogeneous if there exists an open dense $G$-orbit in $X$.

Theorem 1.1**.**

Let $f:X\to X$ be a polarized endomorphism of a $G$-almost homogeneous normal projective variety $X$ with $G$ being a linear algebraic group. Assume further the following conditions.

• (i)

Let $U$ be the open dense $G$-orbit in $X$ and $D$ the codimension-$1$* part of $X\setminus U$. The Weil divisor $K_{X}+D$ is $\mathbb{Q}$-Cartier.*

• (ii)

The endomorphism $f$ is $G$-equivariant in the sense: there is a surjective homomorphism $\varphi:G\to G$ such that $f\circ g=\varphi(g)\circ f$ for all $g$ in $G$.

Then $(X,D)$ is a toric pair.

Let $(X,\Delta)$ be a log pair. The complexity $c=c(X,\Delta)$ of $(X,\Delta)$ is defined as

[TABLE]

see Definition 4.2. Brown, ${\rm M}^{c}$Kernan, Svaldi and Zong recently gave a geometric characterization of toric varieties involving the complexity; see [4, Theorem 1.2] or Theorem 4.3. Their result is a special case of a conjecture of Shokurov, which is stated in the relative case (cf. [13]). A simple version of their result shows that if $(X,\Delta)$ is a log canonical pair such that $\Delta$ is reduced, $-(K_{X}+\Delta)$ is nef, and $c(X,\Delta)$ is non-positive, then $(X,\Delta)$ is a toric pair; see Theorem 4.3 and Remark 4.4.

A finite surjective morphism $h:Y\to Z$ between normal varieties is quasi-étale if $h$ is étale outside a codimension-$2$ subset of $Z$. We show that the complexity condition of [4, Theorem 1.2] holds true in the following case.

Theorem 1.2**.**

Let $X$ be a normal projective variety which is smooth in codimension $2$, and $D\subset X$ a reduced divisor such that

• (i)

there is a Weil $\mathbb{Q}$-divisor $\Gamma$ such that the pair $(X,\Gamma)$ has only klt singularities;

• (ii)

there is a polarized endomorphism $f:X\to X$ such that $D$ is $f^{-1}$-invariant and $f|_{X\backslash D}$ is quasi-étale;

• (iii)

the algebraic fundamental group $\pi_{1}^{\mathrm{alg}}(X_{\mathrm{reg}})$ of the smooth locus $X_{\mathrm{reg}}$ of $X$ is trivial (this holds when $X$ is smooth and rationally connected); and

• (iv)

the irregularity $q(X):=h^{1}(X,\mathcal{O}_{X})$ is zero (this holds when $X$ is rationally connected).

Then the complexity $c(X,D)$ is non-positive.

Remark 1.3**.**

Let $X$ be an $n$-dimensional smooth Fano variety of Picard number one and $D\subset X$ a reduced divisor. Assume the existence of a non-isomorphic surjective endomorphism $f:X\to X$ such that $D$ is $f^{-1}$-invariant and $f|_{X\backslash D}$ is étale. Hwang and Nakayama show that $X$ is isomorphic to $\mathbb{P}^{n}$ and $D$ is a simple normal crossing divisor consisting of $n+1$ hyperplanes; see [8, Theorem 2.1]. In particular, $(X,D)$ is a toric pair. Indeed, their argument shows that the complexity $c(X,D)$ is non-positive. Our Theorem 1.2 follows their idea and tries to generalize their result to the singular case. A key step of ours is to verify that $\hat{\Omega}_{X}^{1}(\log D)$ is free, i.e., isomorphic to $\mathcal{O}_{X}^{\oplus n}$; see [8, Proposition 2.3] and Theorem 5.4.

As applications of Theorem 1.2, we have the following characterizations for toric pairs. A well known conjecture asserts that projective spaces are the only smooth projective Fano varieties of Picard number one admitting an endomorphism $f$ of degree $\geq 2$ (this kind of $f$ is automatically polarized). One may generalize it to the case of arbitrary Picard number. Below is a partial solution to it.

Corollary 1.4**.**

Let $X$ be a rationally connected smooth projective variety and $D\subset X$ a reduced divisor. Suppose $f:X\to X$ is a polarized endomorphism such that $D$ is $f^{-1}$-invariant and $f|_{X\backslash D}$ is étale. Then $(X,D)$ is a toric pair.

We say that a normal projective variety $X$ is of Fano type if there is a Weil $\mathbb{Q}$-divisor $\Delta$ such that the pair $(X,\Delta)$ has only klt singularities and $-(K_{X}+\Delta)$ is an ample $\mathbb{Q}$-Cartier divisor. The assumption below of $X$ being of Fano type is necessary, since a normal projective toric variety is known to be of Fano type.

Corollary 1.5**.**

(cf. Remark 1.7) Let $f:X\to X$ be a polarized endomorphism of a normal projective variety $X$ of Fano type which is smooth in codimension $2$. Let $D\subset X$ be an $f^{-1}$-invariant reduced divisor such that $f|_{X\backslash D}$ is quasi-étale and $K_{X}+D$ is $\mathbb{Q}$-Cartier. Then there exist a quasi-étale cover $\pi:\widetilde{X}\to X$ and a polarized endomorphism $\widetilde{f}:\widetilde{X}\to\widetilde{X}$ such that

• (1)

the endomorphism $f$ lifts to $\tilde{f}$, i.e., $\pi\circ\widetilde{f}=f\circ\pi$, and

• (2)

the pair $(\widetilde{X},\widetilde{D})$ is toric, where $\widetilde{D}=\pi^{-1}(D)$.

The following example satisfies the conditions of both Theorems 1.1 and 1.2.

Example 1.6**.**

Let $X:=\mathbb{P}^{n}$ and

[TABLE]

the power map for some $q\geq 2$. Then $f$ is polarized with $\deg f=q^{n}$. Let the algebraic torus $T=(k^{*})^{n}$ act on $X$ naturally: $T\times X\,\to\,X$ via

[TABLE]

Let $D_{i}:=\{X_{i}=0\}\subseteq X$. Denote by $D:=\sum_{i=0}^{n}D_{i}$ and $U:=X\setminus D$. Let

[TABLE]

which is a surjective homomorphism.

One can check easily that $X$ is $T$-almost homogeneous with $U$ the big open orbit and $f$ is $T$-equivariant in the sense that $f\circ g=\varphi(g)\circ f$ for all $g$ in $T$. Hence the conditions in Theorem 1.1 are all satisfied. Of course, $(X,D)$ is a toric pair.

Note that $D$ is $f^{-1}$-invariant, the restriction $f|_{X\setminus D}$ is étale, and $\pi_{1}(X)$ is trivial. So the conditions in Theorem 1.2 are all satisfied. Clearly, we have $c(X,D)\leq n+1-(n+1)=0$.

One may take the toric blowups of $X$ to get more examples satisfying all conditions in Theorems 1.1 and 1.2. **

Remark 1.7**.**

In Corollary 1.5, it is not always possible to take $\pi:\widetilde{X}\to X$ to be the identity map. In other words, the pair $(X,D)$ itself may not be toric.

Indeed, let $\widetilde{f}:\widetilde{X}=\mathbb{P}^{n}\to\widetilde{X}$ be the power map of degree $q^{n}$ for some $q\geq 2$ as defined in Example 1.6. The symmetric group $S_{n+1}$ in $(n+1)$-letters acts naturally on $\mathbb{P}^{n}$ as (coordinates) permutations. Let $\widetilde{D}_{i}:=\{X_{i}=0\}\,\subset\,\widetilde{X}$ and $\widetilde{D}:=\sum\widetilde{D}_{i}$. Then $S_{n+1}$ fixes $\widetilde{D}$ (as a set). Choose a non-trivial subgroup $G\leq S_{n+1}$ such that

• (i)

the group $G$ has no non-trivial pseudo-reflections (i.e. each non-trivial $g\in G$ fixes at most a codimension-$2$ subset) and hence the quotient map $\pi:\widetilde{X}\,\to\,X:=\mathbb{P}^{n}/G$ is quasi-étale; and

• (ii)

the variety $X$ has only terminal singularities and hence is smooth in codimension $2$.

For instance, take $n>2$ and $G=\langle(1,2,\dots,n+1)\rangle\cong\mathbb{Z}/(n+1)\mathbb{Z}$; see [11, Lemma 3].

Note that $g\circ\widetilde{f}=\widetilde{f}\circ g$ for any $g$ in $G$. Hence $\widetilde{f}$ descends to a polarized endomorphism $f$ on $X$. Let $D:=\pi(\widetilde{D})$. Then we have $\pi^{-1}(D)=\widetilde{D}$ and $f^{-1}(D)=D$.

We now check that $f$ and the pair $(X,D)$ satisfy the assumptions of Corollary 1.5. Since both $\widetilde{f}|_{\widetilde{X}\backslash\widetilde{D}}$ and $\pi$ are quasi-étale, so is $f|_{X\backslash D}$. Clearly, $X$ is $\mathbb{Q}$-factorial. Since $K_{\widetilde{X}}=\pi^{*}K_{X}$ is anti-ample, so is $K_{X}$. Thus, $X$ is a Fano variety and hence of Fano type.

The pair $(X,D)$ is not toric because the number of the irreducible components of $D$ is less than $n+1$, noting that $G$ permutes the $\widetilde{D}_{i}$ non-trivially; see Remark 4.6. Of course, its quasi-étale cover $(\widetilde{X},\widetilde{D})$ is a toric pair.

Acknowledgement. The second author thanks Mircea Mustata for valuable discussions and warm hospitality during his visit to Univ. of Michigan in December 2016; he is also supported by an ARF of National University of Singapore. The authors thank the referee for suggestions to improve the paper.

2. Preliminary results

2.1**.**

Notation and terminology.

Let $X$ be a normal projective variety of dimension $n\geq 1$. Define:

• (1)

$q(X)=h^{1}(X,\mathcal{O}_{X})=\dim H^{1}(X,\mathcal{O}_{X})$ (the irregularity); and

• (2)

$\widetilde{q}(X)=q(\widetilde{X})$ with $\widetilde{X}$ a smooth projective model of $X$.

Let $D_{1}$ and $D_{2}$ be two Cartier $\mathbb{R}$-divisors on $X$. Denote by $D_{1}\equiv D_{2}$ if $D_{1}$ is numerically equivalent to $D_{2}$, i.e., if $(D_{1}-D_{2})\cdot C=0$ for every curve $C$ on $X$.

Denote by $X_{\mathrm{reg}}$ the smooth locus of $X$. Let $U\subseteq X_{\mathrm{reg}}$ be an open dense subset. Let $\pi:Y\to X$ be a log resolution such that $\pi$ is isomorphic over $U$ and $B=Y\setminus\pi^{-1}(U)$ is a simple normal crossing (SNC) divisor. Define the log Kodaira dimension of $U$ as $\bar{\kappa}(U)=\kappa(Y,K_{Y}+B)$ (Iitaka’s $D$-dimension), which is independent of the choice of the compactification $Y$ of $U$.

Given a reduced divisor $D$ on $X$, we define the sheaf $\hat{\Omega}_{X}^{1}(\log D)$ as follows. Let $U\subseteq X$ be an open subset with $\operatorname{codim}(X-U)\geq 2$ and $D\cap U$ being a normal crossing divisor. Denote by $\Omega_{U}^{1}(\log D\cap U)$ the locally free sheaf of germs of logarithmic 1-forms on $U$ with poles only along $U\setminus D$. Using the open immersion $j:U\hookrightarrow X$, we define

[TABLE]

This is a reflexive coherent sheaf on $X$.

Throughout this paper, for a pair $(X,\Delta)$, the coefficients of $\Delta$ lie in $[0,1]$.

The result below is frequently used and part of [12, Proposition 2.9].

Lemma 2.2**.**

Let $f:X\to X$ be a polarized endomorphism of a normal projective variety $X$. Suppose $f^{*}M\equiv M$ (numerical equivalence) for some Cartier $\mathbb{R}$-divisor $M$. Then $M\equiv 0$.

3. Proof of Theorem 1.1

Lemma 3.1**.**

Let $X$ be a normal projective variety with an algebraic torus $T$-action. Suppose $T$ has a Zariski-dense open orbit $U$ in $X$. Then $X$ is a toric variety.

Proof.

Let $x\in U$. Then we have $U=T/T_{x}$, where $T_{x}:=\{t\in T\,|\,tx=x\}$. Since $T$ is a torus, $U$ is again a torus. For any $y\in U$ and $t\in T_{x}$, we have $y=t_{y}x$ for some $t_{y}\in T$. Then $ty=tt_{y}x=t_{y}tx=y$. In particular, $T_{x}$ acts trivially on $U$ and hence on $X$. So the natural action of $U=T/T_{x}$ on itself extends to $X$. ∎

Lemma 3.2**.**

Let $X$ be a normal projective variety and $U\subseteq X$ an open dense subset such that $U$ is contained in the smooth locus of $X$. Let $D$ be the sum of all the prime divisors $D_{i}$ contained in $X\setminus U$. Assume $(X,D)$ is log canonical. Let $\pi:Y\to X$ be a resolution such that $\pi$ is isomorphic over $U$ and $Y\setminus\pi^{-1}(U)$ is an SNC divisor. Let $D^{\prime}$ be the strict transform of $D$ and $E$ the sum of $\pi$-exceptional prime divisors $E_{j}$ such that $Y\setminus\pi^{-1}(U)=D^{\prime}+E$. Then the log Kodaira dimension $\bar{\kappa}(U):=\kappa(Y,D^{\prime}+E)$ equals $\kappa(X,K_{X}+D)$, and $K_{Y}+D^{\prime}+E$ is pseudo-effective (i.e., the limit of effective $\mathbb{R}$-Cartier divisors) if and only if so is $K_{X}+D$.

Proof.

Since $(X,D)$ is log canonical, $K_{Y}+D^{\prime}+E=\pi^{\ast}(K_{X}+D)+\sum\limits_{j}b_{j}E_{j}$ with $b_{j}\geq 0$. Hence the lemma follows by the projection formula. ∎

Proof of Theorem 1.1.

Write $\deg f=q^{\dim(X)}$ with $q>1$. We may assume $G$ is connected and acts faithfully on $X$. Let $x_{0}\in U$. Then $U=Gx_{0}$ and $f(U)=f(Gx_{0})=\varphi(G)f(x_{0})=Gf(x_{0})$. So $f(U)$ is an open dense $G$-orbit in $X$ and hence $f(U)=U$. Further, we claim that $f^{-1}(U)=U$. Indeed, for any $x\not\in U$, the orbit $Gx$ has $\dim(Gx)<\dim(X)$. Since $Gf(x)=\varphi(G)f(x)=f(Gx)$, we have $\dim(Gf(x))=\dim(Gx)<\dim(X)$. In particular, $f(x)$ is not in $U$. So the claim is proved. Let $D$ be the divisorial part of $X\setminus U$. Then $f^{-1}(D)=D$.

Since $U$ is $G$-transitive, $f|_{U}:f^{-1}(U)=U\to U$ is étale. Thus, by the logarithmic ramification divisor formula, we have $K_{X}+D=f^{*}(K_{X}+D)$. Hence, $K_{X}+D\equiv 0$ by Lemma 2.2. Since $K_{X}+D$ is $\mathbb{Q}$-Cartier, the pair $(X,D)$ is log canonical by [3, Theorem 1.4]. Let $\pi:Y\to X$ be a $G$-equivariant log resolution of $X$ such that $\pi|_{\pi^{-1}(U)}$ is isomorphic and $Y\backslash\pi^{-1}(U)$ is an SNC divisor (cf. [9]). Applying Lemma 3.2 and using the notation there, we have that $K_{Y}+D^{\prime}+E$ is pseudo-effective. Note that $G$ is linear, $\pi^{-1}(U)$ is an open dense $G$-orbit in $Y$, and $G$ acts faithfully on $Y$. By [2, Theorem 1.1], $G$ is an algebraic torus. So $X$ is a toric variety by Lemma 3.1. Since the big torus $U$ is an affine variety, $X\setminus U$ is of pure codimension one and hence equal to $D$. Thus $(X,D)$ is a toric pair. ∎

4. The Complexity

4.1**.**

Some notation. Let $X$ be an $n$-dimensional normal projective variety and $D=\sum\limits_{i=1}^{\ell}D_{i}$ a reduced divisor on $X$. Denote by $\ell(D):=\ell$, the number of irreducible components in $D$; and $r(D)$ the rank of the vector space spanned by $D_{1},\cdots,D_{\ell}$ in the space of Weil $\mathbb{R}$-divisors modulo algebraic equivalence.

Let $(X,\Delta)$ be a log pair. Write $\Delta=\sum_{i}\,a_{i}D_{i}$ with $a_{i}>0$ and $D_{i}$ distinct irreducible divisors. Denote by

[TABLE]

Definition 4.2**.**

A decomposition of $\Delta$ is an expression of the form

[TABLE]

where $S_{i}\geq 0$ are $\mathbb{Z}$-divisors and $a_{i}\geq 0$, $1\leq i\leq k$. The complexity of this decomposition is $n+r-d$, where $r$ is the rank of the vector space spanned by $S_{1},S_{2},\cdots,S_{k}$ in the space of Weil $\mathbb{R}$-divisors modulo algebraic equivalence and $d=\sum a_{i}$. The complexity $c=c(X,\Delta)$ of $(X,\Delta)$ is the infimum of the complexity of any decomposition of $\Delta$.

The following theorem gives a geometric characterization of toric varieties involving the complexity by Brown, ${\rm M}^{c}$Kernan, Svaldi and Zong.

Theorem 4.3**.**

(cf. [4, Theorem 1.2]) Let $X$ be a proper variety of dimension $n$ and let $(X,\Delta)$ be a log canonical pair such that $-(K_{X}+\Delta)$ is nef. If $\sum a_{i}S_{i}$ is a decomposition of complexity $c$ less than one then there is a divisor $D$ such that $(X,D)$ is a toric pair, where $D\geq\langle\Delta\rangle$ and all but $\llcorner 2c\lrcorner$ components of $D$ are elements of the set $\{S_{i}\,|\,1\leq i\leq k\}$.

Remark 4.4**.**

(1) If the $\Delta$ in Theorem 4.3 is a reduced divisor with the complexity $n+r(\Delta)-\ell(\Delta)\leq 0$, then Theorem 4.3 implies that $(X,\Delta)$ itself is a toric pair.

(2) Let $f:X\to X$ be a polarized endomorphism and let $D$ be an $f^{-1}$-invariant reduced divisor such that $f|_{X\backslash D}$ is quasi-étale and $K_{X}+D$ is $\mathbb{Q}$-Cartier. Then $K_{X}+D\equiv 0$ by the logarithmic ramification divisor formula and Lemma 2.2. In particular, $-(K_{X}+D)$ is nef; further, the pair $(X,D)$ is log canonical (cf. [3, Theorem 1.4]).

The following theorem provides us with a useful upper bound of the complexity.

Theorem 4.5**.**

Let $X$ be a normal projective variety and $D$ a reduced divisor of $X$. Then, in notation of 2.1 and 4.1,

[TABLE]

In particular,

[TABLE]

Proof.

Let $\pi:\widetilde{X}\to X$ be a log resolution of the pair $(X,D)$ with $E$ being the reduced $\pi$-exceptional divisor. Denote by $\widetilde{D}$ the largest reduced divisor contained in $\operatorname{Supp}\pi^{-1}(\text{non-klt locus}$ of $(X,D)$). Let $D^{\prime}$ be the strict transform of $D$ and $\widetilde{D}_{i}$ the irreducible component of $\tilde{D}$. Then we may write $\widetilde{D}=\sum_{i}\widetilde{D}_{i}=D^{\prime}+E$.

From the exact sequence

[TABLE]

we get

[TABLE]

where the connecting homomorphism essentially sends a generator 1 of $\mathcal{O}_{\widetilde{D}_{i}}$ for each component $\widetilde{D}_{i}$ to the first Chern class $c_{1}(\widetilde{D}_{i})$. So $\ell(\widetilde{D})=h^{0}(\widetilde{X},\Omega_{\widetilde{X}}^{1}(\log\widetilde{D}))+r(\widetilde{D})-q(\widetilde{X})$.

By [6, Theorem 1.5], $\mathcal{F}:=\pi_{*}\Omega_{\widetilde{X}}^{1}(\log\widetilde{D})$ is reflexive. Note that $\mathcal{F}|_{U}=\Omega_{U}^{1}(\log D\cap U)$, where $U\subseteq X_{\mathrm{reg}}$ such that $\operatorname{codim}(X\backslash U)\geq 2$ and $D\cap U$ is a smooth divisor. Then we have $H^{0}(\widetilde{X},\Omega_{\widetilde{X}}^{1}(\log\widetilde{D}))=H^{0}(X,\mathcal{F})=H^{0}(U,\mathcal{F}|_{U})=H^{0}(X,\hat{\Omega}_{X}^{1}(\log D))$.

By the negativity lemma (cf. [1, Lemma 3.6.2]), $r(E)=\ell(E)$ and hence $r(D)\leq r(\widetilde{D})-\ell(E)$. So the theorem is proved. ∎

Remark 4.6**.**

In Theorem 4.5, if $X$ is assumed to be $\mathbb{Q}$-factorial, then the negativity lemma implies $r(D)=r(\widetilde{D})-\ell(E)$ at the end of the proof. In particular, we will have $\ell(D)=h^{0}(X,\hat{\Omega}_{X}^{1}(\log D))+r(D)-\widetilde{q}(X)$ and $c(X,D)=n+\widetilde{q}(X)-h^{0}(X,\hat{\Omega}_{X}^{1}(\log D))$.

If $(X,D)$ is assumed to be a normal projective toric pair, then it is known that $\hat{\Omega}_{X}^{1}(\log D)$ is free; see [5, 4.3, page 87]. Since $X$ is rationally connected, $\widetilde{q}(X)=0$. Therefore, $\ell(D)\geq n+r(D)$ (with equality holding true when $X$ is $\mathbb{Q}$-factorial).

5. Proof of Theorem 1.2

Lemma 5.1**.**

Let $X$ be a normal projective variety with finite algebraic fundamental group $\pi_{1}^{\mathrm{alg}}(X_{\mathrm{reg}})$. Then $X$ admits a universal quasi-étale cover $\pi:\widetilde{X}\to X$, such that $\pi_{1}^{\mathrm{alg}}(\widetilde{X}_{\mathrm{reg}})$ is trivial and any surjective endomorphism $f$ of $X$ lifts to $\widetilde{X}$.

Proof.

Since $\pi_{1}^{\mathrm{alg}}(X_{\mathrm{reg}})$ is finite, there is a universal quasi-étale cover $\pi:\widetilde{X}\to X$ such that $\pi_{1}^{\mathrm{alg}}(\widetilde{X}_{\mathrm{reg}})$ is trivial. Let $W$ be the normalization of the fibre product of $f$ and $\pi$ and $W_{0}$ a dominant irreducible component of $W$. Then $W_{0}\to X$ is also a quasi-étale cover. Taking the universal quasi-étale cover of $W_{0}$ which is $\widetilde{X}$, we are done. ∎

The same argument of [8, Proposition 2.4] gives the following.

Proposition 5.2**.**

Let $X$ be a normal projective variety smooth in codimension $2$ and $D\subset X$ a reduced divisor. Suppose $f:X\to X$ is a polarized endomorphism such that $f^{-1}(D)=D$ and $f|_{X\backslash D}$ is quasi-étale. Then there is a smooth open subset $U\subseteq X$ such that $D\cap U$ is a normal crossing divisor and $\operatorname{codim}(X\backslash U)\geq 3$. In particular, $\hat{\Omega}_{X}^{1}(\log D)$ is locally free over $U$.

The following slightly extends [8, Propositions 2.2 and 2.3].

Proposition 5.3**.**

Let $X$ be a normal projective variety which is of dimension $n\geq 2$ and smooth in codimension $2$, and $D\subset X$ a reduced divisor. Suppose $f:X\to X$ is a polarized endomorphism such that $f^{-1}(D)=D$ and $f|_{X\backslash D}$ is quasi-étale. Let $H$ be an ample divisor on $X$ such that $f^{*}H\sim qH$ for some $q>1$. Then the following hold.

• (1)

The intersection numbers vanish: $c_{1}(\hat{\Omega}_{X}^{1}(\log D))\cdot H^{n-1}=c_{1}(\hat{\Omega}_{X}^{1}(\log D))^{2}\cdot H^{n-2}=c_{2}(\hat{\Omega}_{X}^{1}(\log D))\cdot H^{n-2}=0$.

• (2)

The sheaf $\hat{\Omega}_{X}^{1}(\log D)$ is reflexive and $H$-slope semistable.

Proof.

By Proposition 5.2, there is a smooth open subset $U\subseteq X$ such that $D\cap U$ is a normal crossing divisor and $\operatorname{codim}(X\backslash U)\geq 3$. Since $f|_{X\backslash D}$ is quasi-étale, $f|_{f^{-1}(U)\backslash D}$ is étale by the purity of branch loci.

There is a natural morphism $\varphi:f^{*}\hat{\Omega}_{X}^{1}(\log D)\to\hat{\Omega}_{X}^{1}(\log D)$ and $\varphi|_{f^{-1}(U)}$ is an isomorphism. So for $1\leq i\leq 2$, we have

[TABLE]

Then

[TABLE]

implies $c_{i}(\hat{\Omega}_{X}^{1}(\log D))\cdot H^{n-i}=0$ for $1\leq i\leq 2$, noting that $\deg f=q^{n}>1$. The proof for $c_{1}(\hat{\Omega}_{X}^{1}(\log D))^{2}\cdot H^{n-2}=0$, is similar.

For (2), suppose the contrary that $\hat{\Omega}_{X}^{1}(\log D)$ is not $H$-slope semistable. Then there is a coherent subsheaf $\mathcal{F}\subset\hat{\Omega}_{X}^{1}(\log D)$ such that

[TABLE]

Note that

[TABLE]

So for some $k\gg 1$, $\mu_{H}(g^{*}\mathcal{F})=q^{k}\mu_{H}(\mathcal{F})>s$ with $g=f^{k}$. Let $i:f^{-1}(U)\hookrightarrow X$ be the inclusion map and let $\mathcal{G}:=i_{*}((g^{*}\mathcal{F})|_{g^{-1}(U)})$. Then $\mu_{H}(\mathcal{G})=\mu_{H}(g^{*}\mathcal{F})>s$. Note that $(g^{*}\mathcal{F})|_{g^{-1}(U)}$ is a subsheaf of the locally free sheaf $(g^{*}\hat{\Omega}_{X}^{1}(\log D))|_{g^{-1}(U)}\cong\hat{\Omega}_{X}^{1}(\log D)|_{g^{-1}(U)}$. Since $\operatorname{codim}(X\backslash g^{-1}(U))\geq 3$ and $i_{*}$ is left exact, $\mathcal{G}$ is a coherent subsheaf of the reflexive sheaf $\hat{\Omega}_{X}^{1}(\log D)$ . So we get a contradiction and (2) is proved. ∎

Theorem 5.4**.**

Let $X$ be a normal projective variety which is smooth in codimension $2$, and $D\subset X$ a reduced divisor such that:

• (i)

there is a Weil $\mathbb{Q}$-divisor $\Gamma$ such that the pair $(X,\Gamma)$ has only klt singularities;

• (ii)

there is a polarized endomorphism $f:X\to X$ such that $f^{-1}(D)=D$ and $f|_{X\backslash D}$ is quasi-étale; and

• (iii)

the algebraic fundamental group $\pi_{1}^{\mathrm{alg}}(X_{\mathrm{reg}})$ of the smooth locus $X_{\mathrm{reg}}$ of $X$ is trivial.

Then $\hat{\Omega}_{X}^{1}(\log D)$ is free, i.e., isomorphic to $\mathcal{O}_{X}^{\oplus n}$, where $n:=\dim X$.

Proof.

The case $n=1$ is clear. If $n\geq 2$, apply Proposition 5.3 and [7, Theorem 1.20]. ∎

Proof of Theorem 1.2.

If $\dim(X)=1$, then $X\cong\mathbb{P}^{1}$ since $\pi_{1}^{\mathrm{alg}}(X_{\mathrm{reg}})$ is trivial and $\deg f>1$. Clearly, $\ell(D)=\deg D=2$ and $r(D)=1$; see Remark 4.4. So $c(X,D)=0$. Now we may assume $\dim(X)>1$. By Theorem 5.4, $\hat{\Omega}_{X}^{1}(\log D)$ is free. Since $(X,\Gamma)$ is klt, $X$ has rational singularities. So we have $\widetilde{q}(X)=q(X)=0$ by the assumption. Thus $c(X,D)\leq\widetilde{q}(X)=0$ (cf. Theorem 4.5). ∎

Proof of Corollary 1.4.

By Theorem 1.2, $c(X,D)\leq 0$. So $(X,D)$ is a toric pair by Remark 4.4 and [4, Theorem 1.2] or Theorem 4.3. ∎

Proof of Corollary 1.5.

Since $X$ is of Fano type, there is a Weil $\mathbb{Q}$-divisor $\Delta$ such that the pair $(X,\Delta)$ is klt and $-(K_{X}+\Delta)$ is ample. By [7, Theorem 1.13], $\pi_{1}^{\mathrm{alg}}(X_{\mathrm{reg}})$ is finite. By Lemma 5.1, $X$ admits a universal quasi-étale cover $\pi:\widetilde{X}\to X$, such that $\pi_{1}^{\mathrm{alg}}(\widetilde{X}_{\mathrm{reg}})$ is trivial and $f$ lifts to $\widetilde{f}$ on $\widetilde{X}$. Note that $\widetilde{X}$, like $X$, is still smooth in codimension $2$ by the purity of branch loci.

Let $\widetilde{D}:=\pi^{-1}(D)$ and $\widetilde{\Delta}:=\pi^{*}(\Delta)$. Then $K_{\widetilde{X}}+\widetilde{D}$ equals $\pi^{*}(K_{X}+D)$ and hence is $\mathbb{Q}$-Cartier (and numerically trivial; see Remark 4.4). Also $K_{\widetilde{X}}+\widetilde{\Delta}$ equals $\pi^{*}(K_{X}+\Delta)$ and is hence anti-ample. Note that $(\widetilde{X},\widetilde{\Delta})$ is also klt (cf. [10, Proposition 5.20]). Therefore, $\widetilde{X}$ is also of Fano type and $q(\widetilde{X})=0$ by the Kawamata-Viehweg vanishing. Since $\widetilde{f}$ is the lifting of $f$, it is polarized and $\widetilde{f}^{-1}(\widetilde{D})=\widetilde{D}$. Since both $\pi:\widetilde{X}\to X$ and $f|_{X\setminus D}$ are quasi-étale, so is $\widetilde{f}|_{\widetilde{X}\backslash\widetilde{D}}$.

Thus Theorem 1.2 is applicable: $c(\widetilde{X},\widetilde{D})\leq 0$. So $(\widetilde{X},\widetilde{D})$ is a toric pair by Remark 4.4 and [4, Theorem 1.2] or Theorem 4.3. ∎