---
abstract: |
    Let $X$ be a normal projective variety and $f:X\to X$ a 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 a reduced divisor $D$ such that $f^{-1}(D)=D$, $f|_{X\backslash D}$ 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.
address:
- ' <span style="font-variant:small-caps;">Department of Mathematics</span> <span style="font-variant:small-caps;">National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076</span> '
- ' <span style="font-variant:small-caps;">Department of Mathematics</span> <span style="font-variant:small-caps;">National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076</span> '
author:
- 'Sheng Meng, De-Qi Zhang'
title: Characterizations of Toric Varieties via Polarized Endomorphisms
---

Introduction
============

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

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

$X$ 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 the symmetries to characterize toric pairs.

We first give a characterization of toric varieties via polarized endomorphisms and linear group actions.

\[thm-torus\] Let $X$ be a normal projective variety and $f:X\to X$ a polarized endomorphism such that:

-   $X$ is $G$-almost homogeneous with $G$ being a linear algebraic group, i.e., a $G$-orbit $U$ is Zariski-dense open in $X$,

-   $K_X+D$ is ${\mathbb{Q}}$-Cartier, where $D$ is the codimension-$1$ part of $X \setminus U$, and

-   $f$ 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 $$c:=\inf\{n+\dim_{{\mathbb{R}}}(\sum {\mathbb{R}}[S_i])-\sum a_i\,|\, \sum a_i S_i\le \Delta, a_i\ge 0, S_i\ge 0\};$$ see Definition \[def-comp\]. Brown, ${\rm M}^c$Kernan, Svaldi and Zong ([@BMSZ]) recently give a geometric characterization of toric varieties involving the complexity; see [@BMSZ Theorem 1.2] or Theorem \[thm-bmsz\]. Their result is a special case of a conjecture of Shokurov, which is stated in the relative case (cf. [@Sho]). 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 the complexity $c(X,\Delta)\le 0$, then $(X,\Delta)$ is a toric pair*]{}; see Theorem \[thm-bmsz\] and Remark \[rmk-comp\].

We show that the complexity condition of [@BMSZ] holds true for the following case. A finite surjective morphism $h : Y \to Z$ between normal varieties is [*quasi-étale*]{} if $h$ is étale outside a codimension-$2$ subset of $Z$.

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

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

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

-   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

-   the irregularity $q(X) = h^1(X, \mathcal{O}_X) = 0$ (this holds when $X$ is rationally connected).

Then the complexity $c(X,D)\le 0$.

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 $f^{-1}(D)=D$ and $f|_{X\backslash D}$ is étale. Hwang and Nakayama show that $X=\mathbb{P}^n$ and $D$ is a simple normal crossing divisor consisting of $n + 1$ hyperplanes; see [@HN Theorem 2.1]. In particular, $(X,D)$ is a toric pair. Indeed, their argument shows that the complexity $c(X,D)\le 0$. Our Theorem \[thm-pitrivial\] 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 [@HN Proposition 2.3] and Theorem \[HN-thm\].

As applications of Theorem \[thm-pitrivial\], 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 $\ge 2$ (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.

\[smooth-cor\] 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 $f^{-1}(D)=D$ and $f|_{X\backslash D}$ is étale. Then $(X,D)$ is a toric pair.

The assumption below of $X$ being of Fano type is necessary, since a normal projective toric variety is known to be of Fano type. Recall 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.

(cf. Remark \[P\_quot\])\[toric-lift-cor\] Let $X$ be of Fano type and smooth in codimension-$2$ and $D\subset X$ a reduced divisor. Suppose $f:X\to X$ is a polarized endomorphism such that $f^{-1}(D)=D$, $f|_{X\backslash D}$ is quasi-étale and $K_X+D$ is ${\mathbb{Q}}$-Cartier. Then there is a quasi-étale cover $\pi:\widetilde{X}\to X$ and a polarized endomorphism $\widetilde{f}:\widetilde{X}\to \widetilde{X}$ such that:

-   $\pi\circ \widetilde{f}=f\circ \pi$, and

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

The following example satisfies the conditions of both Theorems \[thm-torus\] and \[thm-pitrivial\].

\[eg1\]

Let $X = \mathbb{P}^n$ and $$f: X \, \to \, X; \,\, [X_0: \cdots : X_n] \, \mapsto \, [X_0^q : \cdots : X_n^q]$$ the power map for some $q \ge 2$. Then $f$ is polarized with $\deg f = q^n$. Let the algebraic torus $T = (k^*)^n$ act on $X$ naturally: $$T \times X \, \to \, X; \,\, ((t_1, \dots, t_n), [X_0 : \cdots : X_n]) \, \mapsto \, [X_0 : t_1 X_1 : \cdots : t_n X_n].$$ Let $D_i = \{X_i = 0\} \subseteq X$, $D = \sum_{i=0}^n D_i$ and $U = X \setminus D$. Let $$\varphi : T \, \to \, T; \,\, (t_1, \dots, t_n) \, \mapsto \, (t_1^q, \dots, t_n^q)$$ which is a surjective homomorphism.

Then $X$ 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 \[thm-torus\] are all satisfied. Of course, $(X, D)$ is a toric pair.

Note that $f^{-1}(D) = D$, $f|_{X \setminus D}$ is étale, and $\pi_1(X) = (1)$. Hence the conditions in Theorem \[thm-pitrivial\] are all satisfied. Of course, the complexity $c(X, D) \le n + 1 - (n+1) = 0$.

One may take the toric blowups or blowdowns of $X$ to get more examples satisfying all conditions in Theorems \[thm-torus\] and \[thm-pitrivial\].

\[P\_quot\] In Corollary \[toric-lift-cor\], it is not always possible to take $\pi : \widetilde{X} \to X$ to be the identity map. In other words, $(X,D)$ itself may not be a toric pair.

Indeed, let $\widetilde{f}:\widetilde{X}=\mathbb{P}^n\to \widetilde{X}$ be the power map of degree $q^n$ for some $q \ge 2$ as defined in Example \[eg1\]. 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 $G \le S_{n+1}$ such that:

-   $G$ has no non-trivial pseudo-reflections (i.e. for any non-trivial $g$ in $G$, $g$ 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

-   $X$ 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 [@KX 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 $\pi^{-1}(D) = \widetilde{D}$ and $f^{-1}(D)=D$.

We now check that $f$ and the pair $(X, D)$ satisfy the assumptions of Corollary \[toric-lift-cor\]. 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 $n+1$, $G$ permuting the $\widetilde{D}_i$ non-trivially; see Remark \[rmk-comp2\]. Of course, its quasi-étale cover $(\widetilde{X}, \widetilde{D})$ is a toric pair.

[**Acknowledgement.**]{} The second author thanks Mircea Mustata for the valuable discussion and warm hospitality during his visit to Univ. of Michigan in December 2016; he is also supported by an ARF of NUS.

Preliminary results
===================

[**Notation and terminology**]{}\[nat\].

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

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

-   $\widetilde{q}(X)=q(\widetilde{X})$ 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 ${\operatorname{codim}}(X-U)\ge 2$ 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 $$\hat{\Omega}_X^1(\log D)=j_*\Omega_U^1(\log D\cap U).$$ 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 [@MZ Proposition 2.5].

\[lem-eig\] 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 $M\equiv 0$.

Proof of Theorem \[thm-torus\]
==============================

\[toric\] 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.

Let $x\in U$. Then $U=T/T_x$, where $T_x:=\{t\in T\, |\, tx=x\}$. Since $T$ is a torus, $U=T/T_x$ is again a torus. For any $y\in U$ and $t\in T_x$, $y=t_y x$ for some $t_y\in T$ and hence $ty=tt_yx=t_ytx=y$. In particular, $T_x$ acts trivially on $U$ and hence on $X$. So the natural action of $U=T/T_x$ on itself may extend to $X$. So $X$ is a toric variety.

\[logkod\] Let $X$ be a normal projective variety and $U\subseteq X$ a Zariski-dense open 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 $D'$ be the strict transform of $D$ and $E$ the sum of $\pi$-exceptional prime divisors $E_j$ such that $Y \setminus \pi^{-1}(U) = D' + E$. Then the log Kodaira dimension $\bar{\kappa}(U) = \kappa(Y, D' + E)$ equals $\kappa(X,K_X+D)$, and $K_Y + D' + E$ is pseudo-effective (i.e., the limit of effective divisors) if and only if so is $K_X + D$.

Since $(X,D)$ is log canonical, $K_Y+D'+E=\pi^\ast(K_X+D)+\sum\limits_{j} b_j E_j$ with $b_j\ge 0$ Hence the lemma follows by the projection formula.

Write $\deg f=q^{\dim(X)}$ with $q>1$. We may assume $G$ is connected and acts faithfully on $X$. Let $U = Gx_0$ be the open dense $G$-orbit in $X$. Note that $f(U) = f(Gx_0) = \varphi(G)f(x_0) = G f(x_0)$ is an orbit of dimension equal to $\dim U = \dim X$, and hence $f(U) = U$. So $f^{-1}(U) \supseteq U$. Further, we claim that $f^{-1}(U)=U$. Indeed, for any $x\not\in U$, the orbit $Gx$ has $\dim (Gx) < \dim (X)$. Hence $Gf(x) = \varphi(G) f(x) = f(Gx)$ has dimension equal to $\dim (Gx)$ ($< \dim (X)$). Thus $f(x)$ is not in $U$. The claim is proved. Hence $f^{-1}(X \setminus U) = X \setminus U$. So $f^{-1}(D) = D$ where $D$ is the divisorial part of $X \setminus U$.

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 \[lem-eig\]. Since $K_X+D$ is ${\mathbb{Q}}$-Cartier, $(X,D)$ is log canonical by [@BH Theorem 1.4]. Then, in notation of Lemma \[logkod\], the log canonical divisor $K_Y + D' + E$ of the smooth compactification $(Y, D' + E)$ of $U$ is pseudo-effective. Fix some $u$ in $U$. Let $H:=\{g\in G\, |\,gu=u\}$. Then $U=G/H$. Since $G$ is linear and acts faithfully on $X$, $G$ is an algebraic torus by [@BZ Theorem 1.1]. So $X$ is a toric variety by Lemma \[toric\]. Since the big torus $U$ is an affine variety, $X \setminus U$ is of pure co-dimension one and hence equal to $D$. Thus $(X, D)$ is a toric pair.

The Complexity {#sec-comp}
==============

[**Some notations.**]{}\[setup2\]

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 $D_i$ distinct irreducible divisors. Denote by $$\langle\Delta\rangle:=\llcorner \Delta \lrcorner+\ulcorner 2\Delta\urcorner-\llcorner 2\Delta\lrcorner = \sum_{i: a_i > 1/2} \, D_i .$$

\[def-comp\] A [*decomposition*]{} of $\Delta$ is an expression of the form $$\sum a_i S_i\le \Delta,$$ where $S_i \ge 0$ are ${\mathbb{Z}}$-divisors and $a_i \ge 0$, $1 \le i \le k$. 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; see [@BMSZ Theorem 1.2].

(cf. [@BMSZ]) \[thm-bmsz\] 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 $\sum a_iS_i$ is a decomposition of complexity $c$ less than one then there is a divisor $D$ such that $(X,D)$ is a toric pair, where $D \ge \langle\Delta\rangle$ and all but $\llcorner 2c \lrcorner$ components of $D$ are elements of the set $\{S_i\, |\, 1 \le i \le k \}$.

\[rmk-comp\] (1) If the $\Delta$ in Theorem \[thm-bmsz\] is a reduced divisor with the complexity $n+r(\Delta)-\ell(\Delta)\le 0$, then Theorem \[thm-bmsz\] implies that $(X,\Delta)$ itself is a toric pair.

\(2) Let $f:X\to X$ be a polarized endomorphism and $D$ is a reduced divisor such that $f^{-1}(D)=D$, $f|_{X\backslash D}$ 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 \[lem-eig\]. In particular, $-(K_X+D)$ is nef; further, $(X, D)$ is log canonical (cf. [@BH Theorem 1.4]).

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

\[thm-complexity\] Let $X$ be a normal projective variety and $D$ a reduced divisor of $X$. Then, in notations of \[nat\] and \[setup2\], we have $$\ell(D)\ge h^0(X,\hat{\Omega}_X^1(\log D))+r(D)-\widetilde{q}(X) .$$ In particular, the complexity $$c(X,D)\le n+\widetilde{q}(X)-h^0(X,\hat{\Omega}_X^1(\log D)) .$$

Let $\pi:\widetilde{X}\to X$ be a log resolution of the pair $(X,D)$. Set $\widetilde{D}:=$ largest reduced divisor contained in ${\operatorname{Supp}}\pi^{-1}(\text{non-klt locus}$ of $(X, D)$). Write $\widetilde{D}=\sum \widetilde{D}_i=D'+E$, where the $\widetilde{D}_i$ are the irreducible components of $\widetilde{D}$, $D'$ is the strict transform of $D$, and $E$ is the $\pi$-exceptional part.

From the exact sequence $$0\to \Omega_{\widetilde{X}}^1\to \Omega_{\widetilde{X}}^1(\log \widetilde{D})\to \oplus_i \mathcal{O}_{\widetilde{D}_i}\to 0,$$ we get $$0\to H^0(\widetilde{X},\Omega_{\widetilde{X}}^1)\to H^0(\widetilde{X},\Omega_{\widetilde{X}}^1(\log \widetilde{D}))\to \oplus_i H^0(\widetilde{D}_i,\mathcal{O}_{\widetilde{D}_i})\to H^1(\widetilde{X},\Omega_{\widetilde{X}}^1),$$ 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 [@GKKP 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 ${\operatorname{codim}}(X\backslash U)\ge 2$ and $D\cap U$ is a smooth divisor. Then $H^0(\widetilde{X},\Omega_{\widetilde{X}}^1(\log \widetilde{D}))=H^0(X,\mathcal{F})=H^0(U,\mathcal{F}|_U)=H^0(X,\hat{\Omega}_X^1(\log D))$.

By the negativity lemma (cf. [@BCHM Lemma 3.6.2]), we have $r(E)=\ell(E)$ and hence $r(D)\le r(\widetilde{D})-\ell(E)$. So the Lemma is proved.

\[rmk-comp2\] In Theorem \[thm-complexity\], 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 [@Ful 4.3, page 87]. Since $X$ is rationally connected, $\widetilde{q}(X)=0$. Therefore, $\ell(D)\ge n+r(D)$ (with equality holding true when $X$ is ${\mathbb{Q}}$-factorial).

Proof of Theorem \[thm-pitrivial\]
==================================

\[uqc\] 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 $\widetilde{X}$.

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 [@HN Proposition 2.4] gives the following.

\[codim3-prop\] 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 ${\operatorname{codim}}(X \backslash U) \ge 3$. In particular, $\hat{\Omega}_X^1(\log D)$ is locally free over $U$.

The following slightly extends [@HN Propositions 2.2 and 2.3].

\[mix-prop\] Let $X$ be a normal projective variety which is of dimension $n \ge 2$ 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 such that $f^*H\sim qH$ for some $q>1$. Then the following hold.

-   $c_1(\hat{\Omega}_X^1(\log D))\cdot H^{n-1}= c_1(\hat{\Omega}_X^1(\log D))^2\cdot H^{n-2}= c_2(\hat{\Omega}_X^1(\log D))\cdot H^{n-2}= 0$.

-   $\hat{\Omega}_X^1(\log D)$ is an $H$-slope semistable reflexive sheaf.

By Proposition \[codim3-prop\], there is a smooth open subset $U\subseteq X$ such that $D \cap U$ is a normal crossing divisor and ${\operatorname{codim}}(X \backslash U) \ge 3$. 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 $i=1,2$, $f^*c_i(\hat{\Omega}_X^1(\log D))=c_i(f^*\hat{\Omega}_X^1(\log D))= c_i(\hat{\Omega}_X^1(\log D))$. Then $$q^{n-i}c_i(\hat{\Omega}_X^1(\log D))\cdot H^{n-i}=f^*c_i(\hat{\Omega}_X^1(\log D)\cdot (f^*H)^{n-i}=(\deg f)c_i(\hat{\Omega}_X^1(\log D)\cdot H^{n-i}$$ implies $c_i(\hat{\Omega}_X^1(\log D))\cdot H^{n-i}= 0$ for $i=1,2$, 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 $$\mu_H(\mathcal{F}):=\frac{c_1(\mathcal{F})\cdot H^{n-1}}{{\operatorname{rank}}\mathcal{F}}>0.$$ Note that $$s=\sup\{\mu_H(\mathcal{F})\,|\, \mathcal{F}\subset \hat{\Omega}_X^1(\log D)\}<\infty.$$ 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 $i_*$ is left exact and ${\operatorname{codim}}(X \backslash g^{-1}(U)) \ge 3$, $\mathcal{G}$ is a coherent subsheaf of the reflexive sheaf $\hat{\Omega}_X^1(\log D)$ . So we get a contradiction and (2) is proved.

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

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

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

-   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 $n := \dim X$.

The case $n = 1$ is clear. If $n \ge 2$, apply Proposition \[mix-prop\] and [@GKP Theorem 1.20].

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 \[rmk-comp\]. So $c(X,D)=0$. Now we may assume $\dim(X)>1$. By Theorem \[HN-thm\], $\hat{\Omega}_{X}^1(\log D)$ is free. Since $(X,\Gamma)$ is klt, $X$ has rational singularities. So we have $\widetilde{q}(X) = q(X) = 0$ by the assumption. Thus $c(X,D)\le\widetilde{q}(X)=0$ (cf. Theorem \[thm-complexity\]).

By Theorem \[thm-pitrivial\], $c(X,D)\le 0$. So $(X,D)$ is a toric pair by Remark \[rmk-comp\] and [@BMSZ Theorem 1.2] or Theorem \[thm-bmsz\].

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 [@GKP Theorem 1.13], $\pi_1^{{\mathrm{alg}}}(X_{{\mathrm{reg}}})$ is finite. By Lemma \[uqc\], $X$ 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 \[rmk-comp\]). 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. [@KM Proposition 5.20]). Hence $\widetilde{X}$ is also of Fano type, so $q(\widetilde{X}) = 0$ 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 \[thm-pitrivial\] is applicable: $c(\widetilde{X},\widetilde{D})\le 0$. So $(\widetilde{X},\widetilde{D})$ is a toric pair by Remark \[rmk-comp\] and [@BMSZ Theorem 1.2] or Theorem \[thm-bmsz\].

[99]{} C. Birkar, P. Cascini, C. D. Hacon and J. ${\rm M}^c$Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., **23**(2):405-468, 2010.

M. Brion and D. -Q. Zhang, Log Kodaira dimension of homogeneous varieties, Advanced Studies in Pure Mathematics (to appear), 2015, [arXiv:1511.01274](http://arXiv.org/abs/1511.01274).

A. Broustet and A. Höring, Singularities of varieties admitting an endomorphism, Math. Ann. **360** (2014), no. 1-2, 439-456.

M. Brown, J. ${\rm M}^c$Kernan, R. Svaldi and H. Zong, A geometric characterisation of toric varieties, [arXiv:1605.08911](http://arXiv.org/abs/1605.08911).

W. Fulton, Introduction to toric varieties, No. **131**. Princeton University Press, 1993. D. Greb, S. Kebekus, S. J. Kovács and T. Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. No. **114** (2011), 87-169.

D. Greb, S. Kebekus and T. Peternell, Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties, Duke Math. J. **165** (2016), no. 10, 1965-2004.

J. M. Hwang and N. Nakayama, On endomorphisms of Fano manifolds of Picard number one, Pure Appl. Math. Q, **7**(4), pp.1407-1426, 2011.

J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., **134** Cambridge Univ. Press, 1998.

J. Kollár and C. Xu, Fano varieties with large degree endomorphisms, [arXiv:0901.1692](http://arXiv.org/abs/0901.1692).

S. Meng and D. -Q. Zhang, Building blocks of polarized endomorphisms of normal projective varieties, [arXiv:1606.01345v2](http://arXiv.org/abs/1606.01345v2).

V. V. Shokurov, Complements on surfaces, J. Math. Sci. (New York) **102** (2000), no. 2, 3876-3932, Algebraic geometry, **10**.

S. W. Zhang, Distributions in algebraic dynamics, Surveys in differential geometry, Vol. X, 381-430, Surv. Differ. Geom., **10**, Int. Press, Somerville, MA, 2006.
