Search
Close
Search
Advanced Search
Search Menu
AI Discovery Assistant

Abstract

Let |$X$| be a Fano type variety and |$(X,\Delta )$| be a log Calabi–Yau pair with |$\Delta $| a Weil divisor. If |$(X,\Delta )$| admits a polarized endomorphism, then we show that |$(X,\Delta )$| is a finite quotient of a toric pair.

1 Introduction

We work over |${\mathbb{C}}$| throughout. A polarized endomorphism on a variety |$X$| is an endomorphism |$f\colon X\rightarrow X$| for which |$f^{*}A\sim mA$| for some ample divisor |$A$| on |$X$| and |$m\geqslant 2$|⁠. Although many varieties admit interesting endomorphisms, it is expected that varieties admitting polarized endomorphisms have a much more restrictive geometry. Two examples of such varieties are toric varieties and abelian varieties (see Example 6.1 and Example 6.5). Furthermore, certain finite quotients of the aforementioned examples admit polarized endomorphisms (see Example 6.2 and Example 6.6). It is a folklore conjecture that a variety |$X$| of klt type admitting a polarized endomorphism must be a finite quotient of a toric fibration over an abelian variety. In the conjecture, it is essential to impose that |$X$| has klt type singularities (see Example 6.7). Further, we know that |${\mathbb{Q}}$|-Gorenstein varieties admitting int-amplified endomorphisms are log canonical [6, 19]. In recent years, there has been a great amount of activity on this topic. For instance, the folklore conjecture is known in several cases: for surfaces [26], for smooth Fano |$3$|-folds [22], for homogeneous varieties [28], and for klt Calabi–Yau varieties [19, 31]. In [14], Kawakami and Totaro proved that varieties admitting polarized endomorphisms satisfy Bott vanishing.

A complement of a variety |$X$| is a boundary |$\Delta $| for which |$(X,\Delta )$| is log canonical and |$K_{X}+\Delta \sim _{\mathbb{Q}} 0$|⁠. More precisely, we say that |$\Delta $| is an |$N$|-complement of |$X$| if |$N(K_{X}+\Delta )\sim 0$|⁠, which in particular implies that all coefficients of |$\Delta $| are in |$\frac{1}{N}\,{\mathbb{Z}}.$| Recently, there has been a vast activity in the so-called theory of complements (see, e.g., [2, 9]). Motivated by the theory of complements on Fano varieties, we study polarized endomorphisms of Fano varieties that preserve a complement structure. The following is the main result of this article.

 

Theorem 1
(cf. Theorem 5.1).

Let |$X$| be a Fano type variety and let |$(X,\Delta )$| be a log Calabi–Yau pair with |$\Delta $| reduced. If the pair |$(X,\Delta )$| admits a polarized endomorphism, then |$(X,\Delta )$| is a finite quotient of a toric log Calabi–Yau pair.

Theorem 1 deals with Fano type varieties with a reduced complement. It is expected that every variety that has a polarized endomorphism admits a complement [5]. Note that the conditions of Theorem 1 are satisfied when |$\Delta $| is a |$1$|-complement of the Fano type variety |$X$|⁠.

In the setting of Theorem 1, the polarized endomorphism |$f\colon X\rightarrow X$| of the pair |$(X,\Delta )$| satisfies |$f^{-1}(\Delta )=\Delta $| and |$\Delta $| is said to be a completely invariant divisor. Polarized endomorphisms fixing a divisor have been extensively studied. Completely invariant divisors are expected to be defined by low degree equations in |$X$| (see, e.g., [11]). In [13], Hwang and Nakayama proved that on a Fano manifold |$X$| of Picard rank one that is different from projective space, an endomorphism |$f\colon X\rightarrow X$| that is étale outside a completely invariant divisor must be an isomorphism. In [21], Meng and Zhang proved that if |$X$| is a smooth rationally connected variety that admits a polarized endomorphism |$f\colon X\rightarrow X$| that is étale outside a completely invariant divisor |$\Delta $|⁠, then |$(X,\Delta )$| is a toric pair. In [23], Meng and Zhong proved that if |$X$| is a smooth rationally connected variety and |$\Delta $| is a reduced divisor, then |$(X,\Delta )$| is a toric pair if and only if |$X$| admits an int-amplified endomorphism |$f$| that is étale outside of |$\Delta $|⁠. In summary, Theorem 1 is already known when |$X$| is a smooth Fano type variety. It is expected that the techniques of [13, 21, 23] can prove Theorem 1 in the case that |$X$| is a terminal Fano type variety. However, we need to introduce some new ideas related to the Jordan property of fundamental groups in order to settle the klt Fano type case. In Remark 6.3, we compare our main theorem with the previous results in the literature. In Example 6.2, we show that the finite quotient in the Theorem 1 is indeed necessary. This example is based on the family of examples by Kollár and Xu [16] in which they show that a Fano variety of Picard number 1 with terminal singularities that admits a polarized endomorphism might not be rational.

We remark that Theorem 1 works for int-amplified endomorphisms (see Definition 2.1) as well. We present the statement and sketch of its proof here in terms of polarized endomorphisms for the sake of exposition.

1.1 Sketch of the proof

In this subsection, we sketch the proof of Theorem 1, which uses techniques from the Jordan property for Fano varieties [4, 24], automorphisms of log Calabi–Yau pairs [12], and singularities of |$\mathbb{T}$|-varieties [17].

Let |$f\colon (X,\Delta )\rightarrow (X,\Delta )$| be a polarized endomorphism of the pair. The proof consists of three steps. First, we show that the fundamental group |$\pi _{1}^{\textrm{alg}}(X^{\textrm{reg}}\setminus \Delta )$| is virtually abelian (see Theorem 3.3). Hence, there is a cover of log pairs |$g\colon (Y,\Delta _{Y})\rightarrow (X,\Delta )$| for which the algebraic fundamental group of |$Y^{\textrm{reg}}\setminus \Delta _{Y}$| is an abelian group. In the setting of the theorem, the covering variety |$Y$| is still Fano type (see Lemma 2.4). Secondly, we prove that some iteration of the polarized endomorphism of |$(X,\Delta )$| lifts to a polarized endomorphism of a finite cover |$(Z, \Delta _{Z})$| of the log pair |$(Y,\Delta _{Y})$| (see Theorem 3.1). For this step, we use the fact that |$\pi _{1}(X^{\textrm{reg}}\setminus \Delta )$| is a finitely presented group. Thus, we have a commutative diagram where the horizontal arrows are polarized endomorphisms:

Here, we still have |$Z$| Fano type and |$\pi _{1}^{\textrm{alg}}(Z^{\textrm{reg}} \setminus \Delta _{Z})$| abelian. Since |$\pi _{1}^{\textrm{alg}}(Z^{\textrm{reg}} \setminus \Delta _{Z})$| is abelian, we are reduced to the study of Galois polarized endomorphisms. By Theorem 2.10, we know that |$\textrm{Aut}(Z, \Delta _{Z})$| is a finite extension of an algebraic torus. In order to obtain the previous statement, we use the fact that |$Z$| is Fano type, or at least rationally connected. Then, we argue that the group |$G\leqslant \textrm{Aut}(Z,\Delta _{Z})$| corresponding to the Galois endomorphism |$f_{Z}$| is contained in the connected component |$\textrm{Aut}(Z,\Delta _{Z})^{0}$|⁠, which is an algebraic torus. Finally, we turn to use the theory of |$\mathbb{T}$|-varieties. If the smallest cardinality of a generating set for a maximal torus of |$\textrm{Aut}(Z,\Delta _{Z})$| is less than |$\dim Z$|⁠, then we argue that the quotient by |$G$| will make the singularities of |$Z$| worse. For instance, certain multiplicities must increase (see Lemma 2.8). The previous allows us to argue that either |$(Z,\Delta _{Z})$| is toric or the quotient by |$G$| is not an endomorphism (see Theorem 4.2). This finishes the proof.

2 Preliminaries

We say that two |${\mathbb{R}}$|-divisors |$A$| and |$B$| on a normal variety |$X$| are linearly equivalent and write |$A\sim B$| if |$A-B$| is the divisor of a rational function on |$X$|⁠. In particular, this implies that |$A-B$| has integer coefficients. We write |$(X;x)$| for the germ of an algebraic variety |$X$| at a closed point |$x\in X$|⁠.

In this section, we gather several well-known results about Fano type varieties, and prove some preliminary results regarding int-amplified endomorphisms and torus actions. For the singularities of the MMP, we refer the reader to [15], for Fano type varieties we refer the reader to [25], and for toric geometry, we refer the reader to [7].

 

Definition 2.1.

An endomorphism |$f\colon X\rightarrow X$| is a polarized endomorphism if |$f^{*}A\sim mA$| for some |$m\geqslant 2$| and ample divisor |$A$|⁠. An endomorphism |$f\colon X\rightarrow X$| is said to be int-amplified if |$f^{*}A-A$| is ample for some ample Cartier divisor |$A$| in |$X$|⁠.

 

Definition 2.2.

We say that a pair |$(X,\Delta )$| admits a polarized endomorphism if there is a polarized endomorphism |$f: X \rightarrow X$| such that the ramification divisor |$R_{f}$| is equal to |$f^{*}\Delta -\Delta $|⁠. We say that |$(X,\Delta )$| admits an int-amplified endomorphism if there is an int-amplified endomorphism |$f\colon X\rightarrow X$| such that |$R_{f}=f^{*}\Delta -\Delta $|⁠.

In the previous definition, the condition that |$R_{f}=f^{*}\Delta -\Delta $| is equivalent to the equality |$f^{*}(K_{X}+\Delta )=K_{X}+\Delta $| of divisors, where |$K_{X}$| is a fixed Weil divisor that represents the class of the canonical divisor.

We recall the definition of the orbifold fundamental group of a log pair.

 

Definition 2.3.
Let |$(X,\Delta )$| be a pair with standard coefficients, that is, coefficients of the form |$1 - \frac{1}{m}$| for |$m\in{\mathbb{N}}$|⁠. Let |$D_{1},\ldots , D_{l}$| be the prime components of |$\Delta $|⁠, and |$\Delta = \sum _{i=1}^{l} \left (1 - \frac{1}{m_{i}}\right ) D_{i}$|⁠. We define the orbifold fundamental group of the pair |$(X,\Delta )$| as
$$ \begin{align*} & \pi_{1}(X,\Delta):= \pi_{1}(X^{\textrm{reg}}\setminus \operatorname{supp}(\Delta))/N \end{align*} $$
where |$N$| is the normal subgroup generated by the elements |$\gamma _{i}^{m_{i}}$|⁠, where |$\gamma _{i}$| is a loop around |$D_{i}$|⁠.

The following lemma allows us to control when a finite cover of a Fano type variety is again of Fano type. This lemma is well-known to the experts.

 

Lemma 2.4.

Let |$X$| be a Fano type variety. Let |$(X,\Delta )$| be a log Calabi–Yau pair with |$\Delta $| reduced and |$K_{X}+\Delta \sim 0$|⁠. Let |$f\colon Y\rightarrow X$| be a finite cover with branched divisor contained in the support of |$\Delta $|⁠. Then, |$Y$| is a Fano type variety.

 

Proof.
Let |$D$| be a boundary divisor on |$X$| for which |$(X,D)$| is klt and |$-(K_{X}+D)$| is big and nef. For each prime divisor |$P$| on |$X$| let |$m_{P}$| be the ramification index of |$f$| at |$P$|⁠. For |$\epsilon>0$| small enough the pair
$$ \begin{align*} & (X,(1-\epsilon)\Delta+\epsilon D) \end{align*} $$
is klt and |$-(K_{X}+(1-\epsilon )\Delta +\epsilon D)$| is big and nef. Set |$\Gamma :=(1-\epsilon )\Delta +\epsilon D$|⁠. Further, for every prime divisor |$P$| on |$\operatorname{Supp}(\Gamma )$| we may assume
(2.1)
Write
$$ \begin{align*} & K_{Y}+\Gamma_{Y} = f^{*}(K_{X}+\Gamma). \end{align*} $$
By inequality (2.1), the divisor |$\Gamma _{Y}$| is effective. Thus, |$(Y,\Gamma _{Y})$| is a klt log pair and |$-(K_{Y}+\Gamma _{Y})$| is big and nef. So |$Y$| is of Fano type.

 

Definition 2.5.

Let |$(X,\Delta )$| be a log pair. We say that a finite morphism |$f\colon (X,\Delta )\rightarrow (Y,\Delta _{Y})$| is of endomorphism type if there exists an isomorphism of log pairs |$\phi \colon (Y,\Delta _{Y})\rightarrow (X,\Delta )$| such that the composition |$\phi \circ f\colon (X,\Delta )\rightarrow (X,\Delta )$| is an endomorphism of log pairs. If the composition is int-amplified, we say that |$f\colon (X,\Delta )\rightarrow (Y,\Delta _{Y})$| is of int-amplified type.

 

Lemma 2.6.

Let |$(X,\Delta )$| be a log Calabi–Yau pair with |$\Delta $| reduced and |$K_{X}+\Delta \sim 0$|⁠. Let |$f\colon (X,\Delta )\rightarrow (X,\Delta )$| be an int-amplified endomorphism. For every prime component |$S\subseteq \Delta $|⁠, there is a positive integer |$i\geqslant 1$| for which |$f^{i}(S)=S$| and |$f^{i}|_{S^\vee }\colon (S^\vee ,\Delta _{S^\vee })\rightarrow (S^\vee ,\Delta _{S^\vee })$| is an int-amplified endomorphism. Here, the log pair |$(S^\vee ,\Delta _{S^\vee })$| is induced by adjunction of |$(X,\Delta )$| to the normalization |$S^\vee $| of |$S$|⁠.

 

Proof.
The divisor |$\Delta $| has finitely many components and so |$f$| induces a permutation of these. Thus, for some |$i\geqslant 1$|⁠, we have |$f^{i}(S)=S$|⁠. Therefore, we have a commutative diagram:
In the previous diagram, the morphism |$\mu \colon S^\vee \rightarrow S$| is the normalization morphism and |$j\colon S\hookrightarrow X$| is the inclusion of |$S$|⁠. Since |$f^{i}$| is int-amplified, there is an ample divisor |$A$| on |$X$| for which |${f^{i}}^{*}A - A$| is ample. Let |$H:=\mu ^{*}j^{*}A$|⁠. Note that |$H$| is an ample divisor. By the commutativity of the previous diagram, we have
$$ \begin{align*} & (f^{i}|_{S^\vee})^{*}H - H = (f^{i}|_{S^\vee})^{*}\mu^{*}j^{*}A - \mu^{*}j^{*}A = \mu^{*}j^{*}{f^{i}}^{*}A - \mu^{*}j^{*}A = \mu^{*}j^{*}({f^{i}}^{*}A - A). \end{align*} $$
This implies that |$f^{i}|_{S^\vee }$| is int-amplified. By adjunction, we conclude that |$f^{i}|_{S^\vee }\colon (S^\vee ,\Delta _{S^\vee })\rightarrow (S^\vee ,\Delta _{S^\vee })$| is a finite morphism of log pairs. Henceforth, the morphism |$f^{i}|_{S^\vee } \colon (S^\vee ,\Delta _{S^\vee })\rightarrow (S^\vee ,\Delta _{S^\vee })$| is an int-amplified endomorphism of the log Calabi–Yau pair |$(S^\vee ,\Delta _{S^\vee })$|⁠.

The following lemma is well-known. It states that on a normal variety, only finitely many multiplicities at closed points occur.

 

Lemma 2.7.

Let |$X$| be a normal variety. Then |$\mu (X;x)$| only takes finitely many possible values on closed points |$x$| of |$X$|⁠.

 

Proof.

The multiplicity |$\mu (X;x)$| at every closed point |$x\in X$| is a positive integer. On the other hand, for every |$m$| the set |$\{x\in X \mid \mu (X;x)\geqslant m\}$| is a closed subvariety of |$X$|⁠. By Noetherian induction, for some |$m_{0}$| the set |$\{x\in X \mid \mu (X;x)\geqslant m_{0}\}$| is empty.

The following lemma states that, under some mild conditions, finite quotients of |$\mathbb{G}_{m}$|-singularities have arbitrarily large multiplicities.

 

Lemma 2.8.
let |$(X;x)$| be a normal singularity and |$\mathbb{G}_{m}\leqslant \textrm{Aut}(X;x)$| be a one-dimensional torus. Assume that there exists a |$\mathbb{G}_{m}$|-equivariant resolution |$Y\rightarrow X$| and a prime exceptional divisor |$E\subset Y$| on which |$\mathbb{G}_{m}$| acts as the identity. For each |$k\geqslant 1$|⁠, let |$\mu _{k}\leqslant \mathbb{G}_{m}$| be the subgroup of |$k$|-roots of unity and let |$(X_{k};x_{k})$| be the quotient of |$(X;x)$| by |$\mu _{k}$|⁠. Then, we have  
$$ \begin{align*} & \lim_{k\rightarrow \infty} \mu(X_{k};x_{k})=\infty. \end{align*} $$

 

Proof.
The statement is local, so we may replace |$X$| with a |$\mathbb{G}_{m}$|-equivariant affine neighborhood of |$x\in X$|⁠. By [30, Theorem 1], we may find a semiprojective variety |$Y$| and a |${\mathbb{Q}}$|-ample divisor |$D$| on |$Y$| for which
$$ \begin{align*} & X_{0}:=\textrm{Spec}\left( \bigoplus_{m\geqslant 0}H^{0}(Y,\mathcal{O}_{Y}(mD) \right) \end{align*} $$
is equivariantly isomorphic to |$X$|⁠. Then, the variety |$X_{k}$| is equivariantly isomorphic to
$$ \begin{align*} & X_{k,0}:=\textrm{Spec}\left( \bigoplus_{m\geqslant 0} H^{0}(Y,\mathcal{O}_{Y}(mkD) \right). \end{align*} $$
For each |$k$|⁠, the finite quotient |$X_{0}\rightarrow X_{k,0}$| is induced by the graded inclusion of algebras. Let |$Z:=\textrm{Spec}(H^{0}(Y,\mathcal{O}_{Y}))$| so |$Y$| is projective over |$Z$|⁠. The projective morphism |$Y\rightarrow Z$| is not the identity; otherwise, no higher equivariant birational model of |$X_{0}$| admits a divisor on which |$\mathbb{G}_{m}$| acts as the identity. Let |$\widetilde{X}_{0}$| be the relative spectrum over |$X$| of the sheaf |$\bigoplus _{m\geqslant 0}\mathcal{O}_{Y}(mD)$|⁠. Then, we have an equivariant projective birational morphism |$\widetilde{X}_{0}\rightarrow X_{0}$| that contracts the only divisor on which |$\mathbb{G}_{m}$| acts as the identity. Let |$x_{0}\in \widetilde{X}_{0}$| be the closed invariant point corresponding to |$x$|⁠. Let |$x_{k,0}$| be the image of |$x_{0}$| in |$X_{k,0}$|⁠. Let |$z_{0}$| be the image of |$x_{k,0}$| in |$Z$|⁠. Then, the maximal ideal of |$x_{k,0}$| equals
$$ \begin{align*} & \mathfrak{m}_{k}:=m_{z_{0}}\oplus \bigoplus_{m\geqslant 1} H^{0}(Y,\mathcal{O}_{Y}(kmD)), \end{align*} $$
where |$m_{z_{0}}$| is the maximal ideal of |$z_{0}$| in |$Z$|⁠. For |$k$| large enough, the homomorphism
(2.2)
is surjective and the homomorphism
(2.3)
is surjective for every |$k_{1}\geqslant 1$| (see, e.g., [18, Example 1.2.22]). By the surjectivity (2.3), we have
$$ \begin{align*} & \mathfrak{m}_{k}/\mathfrak{m}_{k}^{2} \simeq m_{z_{0}}/m_{z_{0}}^{2} \oplus H^{0}(Y,kmD)/m_{z_{0}}H^{0}(Y,kmD). \end{align*} $$
By the surjection (2.2), we have
$$ \begin{align*} & \dim_{{\mathbb{K}}} (\mathfrak{m}_{k}/\mathfrak{m}_{k}^{2}) \geqslant \dim_{{\mathbb{K}}} (H^{0}(Y_{z_{0}},kmD|_{Y_{z_{0}}})). \end{align*} $$
Thus, for |$k$| large enough, we have
$$ \begin{align*} & \mu(X_{k};x_{k}) = \mu(X_{k,0};x_{k,0}) \geqslant \dim_{{\mathbb{K}}}(H^{0}(Y,kmD)/m_{z_{0}}H^{0}(Y,kmD)) \geqslant \dim_{{\mathbb{K}}} (H^{0}(Y_{z_{0}},kmD|_{Y_{z_{0}}})). \end{align*} $$
As |$Y_{z_{0}}$| is a projective variety and |$mD|_{Y_{z_{0}}}$| an ample divisor, the right side goes to infinity when so does |$k$|⁠. This finishes the proof.

The rank of a group |$G$| is the least number of generators of |$G$|⁠. The following lemma states that if a finite group quotient induces an int-amplified endomorphism on a toric log Calabi–Yau pair, such group must have maximal rank.

 

Lemma 2.9.

Let |$(X,\Delta )$| be a |$n$|-dimensional toric log Calabi–Yau pair. Let |$\mathbb{T}\leqslant \textrm{Aut}(X,\Delta )$| be a maximal dimensional torus and |$G\leqslant \mathbb{T}$| be a finite group. If the finite morphism |$(X,\Delta )\rightarrow (X/G,\Delta /G)$| is of int-amplified type, then the rank of |$G$| equals |$n$|⁠.

 

Proof.

We proceed by induction. The statement is clear in dimension one. We can find a subtorus |$G\leqslant \mathbb{T}_{0} \leqslant \mathbb{T}$| such that |$\mathbb{T}_{0}$| has the same rank as |$G$|⁠. Write |$G\simeq{\mathbb{Z}}_{n_{1}}\oplus \dots \oplus{\mathbb{Z}}_{n_{k}}\leqslant \mathbb{T}_{0}$|⁠. For every |$\ell \geqslant 1$|⁠, we set |$G_\ell :={\mathbb{Z}}_{n_{1}^\ell }\oplus \dots \oplus{\mathbb{Z}}_{n_{k}^\ell }\leqslant \mathbb{T}_{0}$|⁠. It suffices to show that |$\mathbb{T}_{0}$| equals |$\mathbb{T}$|⁠. Assume this is not the case. Let |$\pi \colon (X,\Delta ) \rightarrow (X/G,\Delta /G)$| be the finite quotient and |$\psi \colon (X/G,\Delta /G)\rightarrow (X,\Delta )$| be an isomorphism such that |$\phi :=\psi \circ \pi \colon (X,\Delta )\rightarrow (X,\Delta )$| is an int-amplified endomorphism. Thus, we have a commutative diagram

Note that |$\mathbb{T}/G$| induces a maximal torus action on |$(X,\Delta )$| via |$\phi $|⁠. As all maximal tori of |$\textrm{Aut}(X,\Delta )$| are conjugate. Hence, up to replacing |$\phi $|⁠, we may assume that the action induced on |$(X,\Delta )$| by |$\mathbb{T}/G$| equals the |$\mathbb{T}$|-action. Thus, for each |$\ell \geqslant 1$|⁠, the quotient |$(X/G_\ell ,\Delta /G_\ell )$| is an int-amplified endomorphism and the quotient homomorphism |$\pi _\ell \colon (X,\Delta )\rightarrow (X/G_\ell ,\Delta /G_\ell )$| is isomorphic to |$\phi ^\ell $|⁠.

Let |$S$| be a prime component of |$\Delta $|⁠. Let |$(S,\Delta _{S})$| be the pair obtained by adjunction of |$(X,\Delta )$| to |$S$|⁠. Then, the log Calabi–Yau pair |$(S,\Delta _{S})$| is a |$(n-1)$|-dimensional toric log Calabi–Yau pair. For some |$\ell \geqslant 2$|⁠, we have |$\phi ^\ell (S)=S$|⁠. By Lemma 2.6, the quotient |$(S,\Delta _{S}) \rightarrow (S/G_\ell ,\Delta _{S}/G_\ell )$| is of int-amplified type. By induction on the dimension, we conclude that |$k\geqslant n-1$|⁠. Therefore, the algebraic torus |$\mathbb{T}_{0}$| has rank either |$n-1$| or |$n$|⁠. Furthermore, the restriction of |$\mathbb{T}_{0}$| to each prime component of |$\Delta $| has rank |$n-1$|⁠.

Assume that |$\mathbb{T}_{0}$| has rank |$n-1$|⁠. Let |$\Sigma \subset N_{\mathbb{Q}} \simeq{\mathbb{Q}}^{n}$| be the fan of |$X$| so |$X\simeq X(\Sigma )$|⁠. The algebraic torus |$\mathbb{T}_{0}$| corresponds to a surjective homomorphism |$\rho \colon N_{\mathbb{Q}} \rightarrow N^{\prime}_{{\mathbb{Q}}}\simeq{\mathbb{Q}}$|⁠. Let |$K$| be the kernel of |$\rho $|⁠. Let |$p\colon \widetilde{X}\rightarrow X$| be a projective toric morphism such that |$\widetilde{X}$| admits a quotient |$q\colon \widetilde{X}\rightarrow{\mathbb{P}}^{1}$| for the |$\mathbb{T}_{0}$|-action. The morphism |$p$| corresponds to the fan refinement |$\widetilde{\Sigma }$| of |$\Sigma $| obtained by adding the cones |$\sigma \cap K$| for each cone |$\sigma \in \Sigma $|⁠. On the other hand, the morphism |$q$| corresponds to the projection |$\pi _{0}$|⁠. Let |$Q$| be a |$\mathbb{T}_{0}$|-invariant prime divisor of |$\widetilde{X}$| that is horizontal over |${\mathbb{P}}^{1}$|⁠. We argue that |$Q$| is a log canonical place of |$(X,\Delta )$|⁠. Indeed, let |$(\widetilde{X},\widetilde{\Delta })$| be the log pull-back of |$(X,\Delta )$| to |$\widetilde{X}$|⁠. The restriction of |$(\widetilde{X},\widetilde{\Delta })$| to a general fiber of |$q$| is a toric sub-log Calabi–Yau pair. Thus, all the torus invariant components, including |$Q$|⁠, must appear with coefficient one. The image of |$Q$| on |$X$| is either a divisor or an irreducible subvariety of codimension |$2$|⁠. Indeed, the prime divisor |$Q$| corresponds to a ray |$\rho _{Q} \in \widetilde{\Sigma }(1)$|⁠; the ray |$\rho _{Q}$| is either on |$\Sigma $| or is the intersection of a |$2$|-dimensional cone of |$\Sigma $| with |$K$|⁠.

By the previous paragraph, under the assumption that |$\mathbb{T}_{0}$| has rank |$n-1$|⁠, the variety |$p(Q)$| has either codimension one or codimension two. In both cases, we will obtain a contradiction. Assume that |$p(Q)$| is a divisor. By construction, the restriction of the algebraic torus |$\mathbb{T}_{0}$| to |$p(Q)$| has rank |$n-2$|⁠. This contradicts the fact that the restriction of |$\mathbb{T}_{0}$| to each component of |$\Delta $| has rank |$n-1$|⁠. Otherwise, assume that the image of |$p(Q)$| has codimension two. Let |$\mathbb{T}_{1}\leqslant \mathbb{T}_{0}$| be the one-dimensional subtorus that acts as the identity on |$Q$| and |$\mathbb{T}_{1}^{\prime}$| be its complement torus in |$\mathbb{T}_{0}$|⁠. The torus |$\mathbb{T}_{1}^{\prime}$| acts faithfully on |$p(Q)$|⁠. Let |$x$| be a general point of |$p(Q)$|⁠. Let |$\mu _\ell \leqslant \mathbb{T}_{1}$| be the subgroup of |$\ell $| roots of unity and |$x_{0,\ell }$| be the image of |$x$| in |$X/\mu _\ell $|⁠. Let |$x_\ell $| be the image of |$x$| in |$X/G_\ell $|⁠. By construction, the germ |$(X/G_\ell ;x_\ell )$| is analytically isomorphic to |$(X/\mu _\ell ;x_{0,\ell })$|⁠. Thus, by Lemma 2.8, we conclude that
$$ \begin{align*} & \lim_{\ell \rightarrow \infty}\mu(X/G_\ell; x_\ell) = \lim_{\ell \rightarrow \infty} \mu(X/\mu_\ell; x_{0,\ell}) =\infty. \end{align*} $$
This contradicts Lemma 2.7. We conclude that the rank of |$\mathbb{T}_{0}$| equals |$n$| and so does the rank of |$G$|⁠. This finishes the proof.

The following theorem is a well-known statement about the automorphism groups of log Calabi–Yau pairs (see, e.g., [12, Theorem 1.1]).

 

Theorem 2.10.

Let |$X$| be a Fano type variety and |$(X,\Delta )$| be a log Calabi–Yau pair. Then, the automorphism group |$\textrm{Aut}(X,\Delta )$| is a finite extension of an algebraic torus.

3 Lifting Polarized Endomorphisms to Finite Covers

In this section, we develop techniques to lift polarized endomorphisms to finite covers of pairs. Then, we turn to prove some results regarding the algebraic fundamental groups of open Calabi–Yau varieties.

 

Theorem 3.1.

Let |$(X,\Delta )$| be a log Calabi–Yau pair, with |$K_{X} + \Delta \sim 0$|⁠. Let |$U:= X^{\textrm{reg}}\setminus \Delta $|⁠. Suppose that |$(X,\Delta )$| admits an int-amplified endomorphism |$f \colon (X,\Delta ) \to (X,\Delta )$|⁠. Let |$g\colon (Y,\Delta _{Y}) \to (X,\Delta )$| be a finite cover such that |$g^{-1}(U) \to U$| is étale. Then there exists |$m,n \gg 0$|⁠, a finite cover |$h\colon (Z,\Delta _{Z}) \to (Y,\Delta _{Y})$|⁠, and an int-amplified endomorphism |$f_{Z}:(Z,\Delta _{Z})\to (Z,\Delta _{Z})$| such that the following diagram commutes:

 

Proof.

Consider the commutative diagram

obtained by taking normalization of the fiber product, where |$\Delta _{n}$| is defined by taking the log pullback of |$K_{Y} + \Delta _{Y}$|⁠.

The number of irreducible components of |$Y_{n}$| is bounded by the degree of |$g$|⁠, so there exists |$m \gg 0$| such that for all |$n \geqslant m$|⁠, |$Y_{n}$| has the same number of irreducible components as |$Y_{m}$|⁠. For each |$n\geqslant m$|⁠, choose an irreducible component |$Y_{n}^{0}$| of |$Y_{n}$| such that we have a commutative diagram

where |$g^{0}_{n}$| is the restriction of |$g_{n}$| to |$Y_{n}^{0}$|⁠, and |$\Delta _{n}^{0} = \Delta _{n}|_{Y_{n}^{0}}$|⁠. The map |$g$| is finite and surjective, so each |$g_{n}^{0}$| is also finite and surjective. By the purity of the branch locus, each |$g_{n}^{0}$| is étale over |$U$|⁠.

Let us denote |$U_{n}^{0}:= (g_{n}^{0})^{-1}(U)$| and let |$\tilde{g}_{n}^{0} \colon U_{n}^{0} \to U$|⁠. As |$\tilde{g}_{n}^{0}$| is an étale finite cover of |$U$|⁠, the group |$H_{n}^{0}:=(\tilde{g}_{n}^{0})_{*}(\pi _{1}(U_{n}^{0}))$| is a subgroup of |$\pi _{1}(U)$| of index |$\deg (\tilde{g}_{n}^{0}) \leq \deg (g)$|⁠. The group |$\pi _{1}(U)$| is finitely presented, so there exist finitely many subgroups of index less than or equal to |$\deg (g)$|⁠. Therefore, for some |$ i,j \gg 0$| with |$j>i$|⁠, we have |$(\tilde{g}_{m+i}^{0})_{*}(\pi _{1}(U_{m+i}^{0})) = (\tilde{g}_{m+j}^{0})_{*}(\pi _{1}(U_{m+j}^{0}))$|⁠, which we will denote by |$H$|⁠, and so |$U_{m+i}^{0}$| is homeomorphic to |$U_{m+j}^{0}$|⁠. Furthermore, as both |$\tilde{g}_{m+i}^{0}$| and |$\tilde{g}_{m+j}^{0}$| are analytic covers [8, Theorem 3.4], there exists an isomorphism |$h\colon U_{m+i}^{0} \to U_{m+j}^{0}$| such that |$\tilde{g}_{m+j}^{0} \circ h = \tilde{g}_{n}^{i}$| (see also [10, Proposition 3.13]).

As both |${Y}_{m+i}^{0} \xrightarrow{{g}_{m+i}^{0}} X$| and |${Y}_{m+j}^{0} \xrightarrow{{g}_{m+j}^{0}} X$| are extensions of locally biholomorphic coverings corresponding to the subgroup |$H \leq \pi _{1}(U)$|⁠, |${g}_{m+i}^{0}$| and |${g}_{m+j}^{0}$| are isomorphic over |$X$|⁠, so we obtain an isomorphism |$(Y_{m+i}^{0},\Delta _{m+i}^{0}) \xrightarrow{\simeq } (Y_{m+j}^{0},\Delta _{m+j}^{0})$|⁠.

Let |$(Z,\Delta _{Z}):= (Y_{m+i}^{0},\Delta _{m+i}^{0})$|⁠, and set |$N:= j - i$|⁠, |$g^{\prime}:= g_{m+i}^{0}$| and |$h:= f_{m+i}$|⁠. Then, we have the following commutative diagram:  

Finally, we need to prove that |$f_{Z}$| is int-amplified. Let |$A$| be an ample Cartier divisor on |$X$| such that |$f^{*} A - A$| is ample. Then |$f_{Z}^{*}(g^{\prime *} A) - g^{\prime *} A = g^{\prime *}((f^{N})^{*} A - A)$| is ample, and so |$f_{Z}$| is an int-amplified endomorphism.

 

Corollary 3.2
(cf. [27, Proposition 3.5 (Step 1)]).

Let |$(X,\Delta )$| be a log Calabi–Yau pair. Let |$p\colon Y \to X$| be the index one cover of |$K_{X} + \Delta \sim _{\mathbb{Q}} 0$|⁠, and let |$K_{Y} + \Delta _{Y} = p^{*}(K_{X} + \Delta )$| If |$f\colon (X,\Delta ) \to (X,\Delta )$| is an int-amplified endomorphism, then |$(Y,\Delta _{Y})$| admits an int-amplified endomorphism |$f_{Y}$| such that the following diagram commutes:

 

Proof.

Let

be the diagram obtained by taking normalization of the fiber product.

As |$f$| is finite, so is |$p_{1}$|⁠, and |$p_{2}$| is étale in codimension 1, because |$p$| is also étale in codimension 1. Therefore, |$p_{2}^{*}(K_{X} + \Delta ) \sim K_{Z} + \Delta _{Z} \sim 0$|⁠. By [32, Lemma 4.12], |$p_{2}$| factors through |$p$|⁠, and by the minimality of the index one cover we conclude that |$p_{2}$| is isomorphic to |$p$| over |$X$|⁠.

Thus, by the end of the proof of Theorem 3.1, |$p_{1}$| induces an int-amplified endomorphism |$f_{Y}$| such that the following diagram commutes:

The following result states that if |$X$| is a Fano type variety, with |$(X,\Delta )$| a log Calabi–Yau pair of index one, then there exists a finite cover |$f\colon (Y,\Delta _{Y}) \to (X,\Delta )$| such that the algebraic fundamental group |$\pi _{1}^{\textrm{alg}}(Y\setminus \Delta _{Y})$| is abelian.

 

Theorem 3.3.

Let |$X$| be a Fano type variety of dimension |$n$| variety and |$(X,\Delta )$| be a log Calabi–Yau pair of index one. Let |$U:=X^{\textrm{reg}}\setminus \Delta $|⁠. Then, the group |$\pi _{1}^{\textrm{alg}}(U)$| is virtually abelian.

 

Proof.

Let |$P_{1},\ldots ,P_{s}$| be the prime components of the divisor |$\Delta $|⁠. Consider |${\mathbb{Z}}^{s}_{>0}$| as a directed poset, where |$\vec{m} = (m_{1},\ldots ,m_{s}) \leq \vec{n} = (n_{1},\ldots , n_{s}) $| if |$m_{i} \mid n_{i}$| for every |$1 \leq i \leq s$|⁠. Because |${\mathbb{Z}}^{s}_{>0}$| is countable, we can find an increasing sequence |$(\vec{m}_{i})_{i\in{\mathbb{N}}} \subseteq{\mathbb{Z}}^{d}_{>0}$| that is cofinal, meaning that for any |$\vec{n} \in{\mathbb{Z}}^{s}_{>0}$| there exists |$\vec{m}_{i}$| with |$\vec{n} \leq \vec{m}_{i}$|⁠.

For each |$\vec{n} = (n_{1},\ldots , n_{s}) \in{\mathbb{Z}}^{s}_{>0}$|⁠, define |$\Delta _{\vec{n}}:= \sum \left ( 1 - 1/n_{j}\right ) P_{j}$|⁠. Therefore, we can compute the following inverse limit by passing to the cofinal subset
$$ \begin{align*} & \varprojlim_{\vec{m}} \pi_{1} (X,\Delta_{\vec{m}}) = \varprojlim_{i\in{\mathbb{N}}} \pi_{1} (X,\Delta_{\vec{m}_{i}}). \end{align*} $$
For each |$\vec{m} \in{\mathbb{Z}}^{s}_{>0}$|⁠, there exists a surjective homomorphism |$\pi _{1}(X^{\textrm{reg}}\setminus \Delta ) \to \pi _{1}(X,\Delta _{\vec{n}})$|⁠. Even more, by [3, Theorem 1.2], the groups |$\pi _{1}(X,\Delta _{\vec{n}})$| are finite. Hence, for each |$i\in{\mathbb{N}}$|⁠, there exists a normal subgroup |$N_{i}$| of |$\pi _{1}(X^{\operatorname{reg}}\setminus \Delta )$| of finite index such that
(3.1)
We show now that the sequence |$(N_{i})_{i\in{\mathbb{N}}}$| is a cofinal subset of the poset of normal subgroups of |$\pi _{1}(X^{\textrm{reg}}\setminus \Delta )$| of finite index.
Let |$N \leq \pi _{1}(X^{\textrm{reg}}\setminus \Delta )$| be a normal subgroup of finite index. We need to show that there exists |$i \in{\mathbb{N}}$| such that |$N_{i} \leq N$|⁠. Let
$$ \begin{align*} & U_{Y} \to U \end{align*} $$
be the finite étale Galois cover associated to |$N$|⁠. We extend this cover to a crepant finite Galois cover
$$ \begin{align*} & (Y,\Delta_{Y}) \to (X,\Delta), \end{align*} $$
possibly ramifying along the the divisor |$\Delta $|⁠. Thus, for some |$\Delta _{\vec{n}} \leq \Delta $|⁠, there is a surjective homomorphism
$$ \begin{align*} & \pi_{1}(X,\Delta_{\vec{n}}) \to \pi_{1}(X^{\operatorname{reg}}\setminus \Delta)/N. \end{align*} $$
Let |$\vec{n} \mid \vec{m}_{i}$| for some |$i\in{\mathbb{N}}$|⁠. Then, there exists a surjective homomorphism
$$ \begin{align*} & \pi_{1}(X,\Delta_{\vec{m}_{i}}) \to \pi_{1}(X,\Delta_{\vec{n}}). \end{align*} $$
From (3.1), we obtain a surjective homomorphism
$$ \begin{align*} & \pi_{1}(X^{\operatorname{reg}}\setminus \Delta)/N_{i} \to \pi_{1}(X^{\textrm{reg}}\setminus \Delta)/N \end{align*} $$
implying that |$N_{i} \leq N$|⁠.
We conclude that
$$ \begin{align*} & \pi^{\operatorname{alg}}_{1}(U) = \varprojlim_{i\in{\mathbb{N}}} \pi_{1}(X^{\operatorname{reg}}\setminus \Delta)/N_{i} \xrightarrow{\sim} \varprojlim_{i\in{\mathbb{N}}} \pi_{1} (X,\Delta_{\vec{m}_{i}}). \end{align*} $$
By [4, Theorem 3], for each |$\vec{m}_{i}$| we have an exact sequence
$$ \begin{align*} & 1 \to A_{i} \to \pi_{1} (X,\Delta_{\vec{m}_{i}}) \to F_{i} \to 1 \end{align*} $$
where |$A_{i}$| is abelian and |$|F_{i}| \leq C$|⁠, a constant that does not depend in |$i$|⁠. We can assume, by maybe passing to a subsequence, that |$F_{i} = F$| for all |$i\in{\mathbb{N}}$|⁠. Then, for |$j\leq i$|⁠, we can induce a surjective map |$A_{i}\to A_{j}$| such that the following diagram commutes, where the rows are exact
Then, as all the groups involved are finite, we obtain the exact sequence
$$ \begin{align*} & 1 \to \varprojlim_{i\in{\mathbb{N}}} A_{i} \to \pi^{\operatorname{alg}}_{1}(U) \to F \to 1 \end{align*} $$
and because each |$A_{i}$| is abelian, the subgroup |${\varprojlim _{i\in{\mathbb{N}}} A_{i}}$| is abelian of finite index in |$\pi ^{\operatorname{alg}}_{1}(U)$|⁠, so |$\pi ^{\operatorname{alg}}_{1}(U)$| is virtually abelian.

If |$\pi _{1}^{\textrm{alg}}(Y^{\textrm{reg}}\setminus \Delta _{Y})$| is abelian, we can conclude that any finite étale cover of |$Y^{\textrm{reg}}\setminus \Delta _{Y}$| is Galois.

 

Lemma 3.4.

Let |$G$| be a group. Assume that its profinite completion |$\widehat{G}$| is an abelian group. Then, every subgroup of finite index of |$G$| is normal.

 

Proof.
Let |$H \leq G$| be a subgroup of finite index of |$G$|⁠. Consider the subgroup
$$ \begin{align*} & K = \bigcap_{g\in G} gHg^{-1} \end{align*} $$
of |$G$|⁠. By construction, |$K$| is normal in |$G$| and of finite index. In particular, as |$\widehat{G}$| is abelian, the quotient |$G/K$| is abelian. Then |$H/K$| is a normal subgroup of |$G/K$|⁠, which implies that |$H$| is normal.

4 Varieties With Torus Actions

In this section, we study varieties |$X$| for which the quotient |$X\rightarrow X/G$| with |$G \leqslant \textrm{Aut}(X)$| finite is of int-amplified type.

 

Lemma 4.1.

Let |$(X,\Delta )$| be a log Calabi–Yau pair with |$K_{X}+\Delta \sim 0$|⁠. Let |$\mathbb{G}_{m}^{k} \leqslant \textrm{Aut}^{0}(X,\Delta )$| be an algebraic torus. Let |$G_{i}:=\bigoplus _{j=1}^{k} {\mathbb{Z}}/n_{j}^{i} {\mathbb{Z}} \leqslant \mathbb{G}_{m}^{k}$| where each |$n_{j}\geqslant 2$|⁠. If each finite morphism |$(X,\Delta )\rightarrow (X/G_{i},\Delta /G_{i})$| is of endomorphism type, then |$\Delta \neq 0$|⁠.

 

Proof.
Let |$n$| be the dimension of |$X$|⁠. Assume that the statement does not hold, meaning that each finite morphism |$X \rightarrow X/G_{i}$| is of endomorphism type and |$\Delta =0$|⁠. In particular, there is an upper bound |$m_{0}$| such that for every |$i$| and every closed point |$x\in X/G_{i}$| we have |$\mu (X/G_{i};x)\leq m_{0}$|⁠. Let |$\pi \colon \widetilde{X}\rightarrow X$| be an |$\mathbb{G}_{m}^{k}$|-equivariant projective birational morphism for which |$\widetilde{X}$| admits a quotient for the |$\mathbb{G}_{m}^{k}$|-action. Thus, we have a |$\mathbb{G}_{m}^{k}$|-equivariant fibration |$\widetilde{X}\rightarrow Z$| with general fiber a |$k$|-dimensional toric variety. Let |$\pi ^{*}(K_{X})=K_{\widetilde{X}}+\widetilde{\Delta } \sim 0$|⁠. By construction, the sub-pair |$(\widetilde{X},\widetilde{\Delta })$| is a sub log Calabi–Yau pair. Further, the restriction of |$(\widetilde{X},\widetilde{\Delta })$| to a general fiber of |$\widetilde{X}\rightarrow Z$| is a sub log Calabi–Yau toric pair. In particular, there is a prime component |$S\subseteq \widetilde{\Delta }^{=1}$| that dominates |$Z$|⁠. Then, there is a one-dimensional subtorus |$\mathbb{T}_{0}:=\mathbb{G}_{m}\leqslant \mathbb{G}_{m}^{k}$| that acts as the identity on |$S$|⁠. Let |$\mathbb{T}_{1}$| be the split torus of |$\mathbb{T}_{0}$| in |$\mathbb{G}_{m}^{k}$|⁠. For each |$i$|⁠, we let |$H_{i}$| be the restriction of |$G_{i}$| to |$\mathbb{T}_{0}$|⁠. By construction, the sequence of groups |$H_{i}$| is an infinite sequence of finite subgroups of |$\mathbb{T}_{0}$|⁠. Let |$\pi (S)\subset X$| be its image on |$S$| on |$X$|⁠. By [1, Theorem 10.1], the variety |$\pi (S)\subset X$| contains a subvariety |$S_{0}$| isomorphic to |$\mathbb{G}_{m}^{k-1}$| on which |$\mathbb{T}_{1}$| acts faithfully. Let |$x\in S_{0}$| be a general point. Then, the pair |$(X;x)$| is a normal singularity with a |$\mathbb{T}_{0}$|-action. For each quotient |$X\rightarrow X_{i}:=X/G_{i}$|⁠, we let |$x_{i}$| be the image of |$x$| on |$X_{i}$|⁠. Since |$\mathbb{T}_{1}$| acts faithfully on |$S_{0}$|⁠, we conclude that |$(X_{i};x_{i})$| is locally isomorphic to |$(X/H_{i};x_{i})$|⁠. As |$\mathbb{T}_{0}$| acts as the identity on |$S$| from Lemma 2.8, we conclude that
$$ \begin{align*} & \lim_{i\rightarrow \infty} \mu(X_{i};x_{i})=\infty. \end{align*} $$
On the other hand, for each |$i$|⁠, there is an isomorphism |$\psi _{i} \colon X_{i}\rightarrow X$| so we have
$$ \begin{align*} & \mu(X_{i};x_{i}) = \mu(X;\psi_{i}(x_{i})). \end{align*} $$
Thus, there is a sequence of closed points on |$X$| for which the multiplicity diverges. This contradicts Lemma 2.7. We conclude that |$\Delta \neq 0$|⁠.

 

Theorem 4.2.

Let |$(X,\Delta )$| be a |$n$|-dimensional log Calabi–Yau pair with |$K_{X}+\Delta \sim 0$|⁠. Let |$G\leqslant \mathbb{T} \leqslant \textrm{Aut}^{0}(X,\Delta )$| where |$G$| is a finite group and |$\mathbb{T}$| is an algebraic torus. Assume that the finite morphism |$(X,\Delta ) \rightarrow (X/G,\Delta /G)$| is of int-amplified type. Then, the following conditions are satisfied:

  • (1)

    the algebraic torus |$\mathbb{T}$| has rank |$n$|⁠,

  • (2)

    the group |$G$| has rank |$n$|⁠, and

  • (3)

    the pair |$(X,\Delta )$| is a log Calabi–Yau toric pair.

 

Proof.
The algebraic group |$\textrm{Aut}^{0}(X,\Delta )$| fits in an exact sequence
$$ \begin{align*} & 1\rightarrow \textrm{Aut}^{0}_{L}(X,\Delta) \rightarrow \textrm{Aut}^{0}(X,\Delta) \rightarrow A(X,\Delta) \rightarrow 1, \end{align*} $$
where |$A(X,\Delta )$| is a complex torus over the base field and |$\textrm{Aut}^{0}_{L}(X,\Delta )$| is a linear algebraic group. It is clear that |$\mathbb{T} \leqslant \textrm{Aut}^{0}_{L}(X,\Delta )$| and we may assume it is a maximal algebraic torus of this linear algebraic group.
There exists an isomorphism |$\psi \colon (X/G,\Delta /G)\rightarrow (X,\Delta )$| of pairs making the following diagram commute:
 
(4.1)
where |$f$| is an int-amplified endomorphism of the log pair |$(X,\Delta )$|⁠. Since |$\mathbb{T}/G \simeq \mathbb{T}$|⁠, the previous commutative diagram induces, via |$f$|⁠, a maximal torus action on |$(X,\Delta )$|⁠. Thus, there is a finite subgroup |$H \leqslant \mathbb{T}/G$| for which |$(X,\Delta )\rightarrow (X/H,\Delta /H)$| is of int-amplified type. Furthermore, the group |$H$| is isomorphic to |$G$|⁠.
Set |$G_{1}:=G$| and define |$G_{i}$| to be the preimage of |$G_{1}$| via |$\pi _{i}$| in the following short exact sequence:
Thus, if |$G\simeq \bigoplus _{j=1}^{r} {\mathbb{Z}}/n_{j}{\mathbb{Z}}$|⁠, then |$G_{i}\simeq \bigoplus _{j=1}^{r} {\mathbb{Z}}/n_{j}^{i}{\mathbb{Z}}$| for each |$i$|⁠. Due to the commutative diagram (4.1), for each |$k$|⁠, we have a commutative diagram as follows:
 
(4.2)
In the previous diagram, all the vertical morphisms are finite quotients, |$\psi _{1}:=\psi $|⁠, and |$\phi _{k-1}:=\psi _{1}\circ \dots \circ \psi _{k-1}$|⁠. Hence, due to the commutative diagram (4.2), for every |$k$|⁠, we have a commutative diagram:
 
(4.3)
Up to passing to a sub-torus of |$\mathbb{T}$|⁠, we are in the situation of Lemma 4.1. Thus, we conclude that |$\Delta \neq 0$|⁠. Let |$S\subseteq \Delta = \lfloor \Delta \rfloor $| be a prime component. By Lemma 2.6, there exists |$i$| for which the finite morphism |$(S^\vee ,\Delta _{S^\vee })\rightarrow (S^\vee /G_{i},\Delta _{S^\vee }/G_{i})$| is of int-amplified type. Observe that |$(S^\vee ,\Delta _{S^\vee })$| is a |$(n-1)$|-dimensional log Calabi–Yau pair with |$K_{S^\vee }+\Delta _{S^\vee }\sim 0$|⁠. Further, we have |$H_{i}\leqslant \mathbb{T}_{S} \leqslant \textrm{Aut}^{0}(S^\vee ,\Delta _{S^\vee })$| where |$\mathbb{T}_{S}:=\mathbb{T}|_{S^\vee }$| and |$H_{i}:=G_{i}|_{S^\vee }$|⁠. Thus, by induction on the dimension, the following conditions are satisfied:
  • (i)

    we have |$\operatorname{rank}(\mathbb{T}_{S})=n-1$|⁠,

  • (ii)

    the group |$H_{i}$| has rank |$n-1$|⁠, and

  • (iii)

    the pair |$(S^\vee ,\Delta _{S^\vee })$| is a log Calabi–Yau toric pair.

As the torus |$\mathbb{T}_{S}$| is a homomorphic image of |$\mathbb{T}$|⁠, we conclude that the rank of |$\mathbb{T}$| is at least |$n-1$|⁠. From the previous argument, we deduce that every prime component of |$\lfloor \Delta \rfloor $| is a, possibly non-normal, toric variety. Further, the restriction of |$\mathbb{T}$| to every prime component |$S$| of |$\lfloor \Delta \rfloor $| is a maximal torus of |$S$|⁠.

We aim to show that |$\mathbb{T}$| has rank |$n$|⁠. This would show |$(1)$|⁠. For the sake of contradiction, assume that |$\mathbb{T}$| has rank |$n-1$|⁠. Thus, the variety |$X$| is a normal variety of complexity one (see, e.g., [1]). Let |$q\colon (\widetilde{X},\widetilde{\Delta })\rightarrow (X,\Delta )$| be a |$\mathbb{T}$|-equivariant projective birational morphism such that |$(\widetilde{X},\widetilde{\Delta })$| admits a |$\mathbb{T}$|-quotient to a smooth projective curve |$C$|⁠. By [1, Theorem 10.1], all the divisors contracted by |$q\colon \widetilde{X}\rightarrow X$| are prime components of |$\lfloor \widetilde{\Delta }\rfloor $| that are horizontal over |$C$|⁠. In particular, the log pair |$(\widetilde{X},\widetilde{\Delta })$| is a log Calabi–Yau pair. We write |$\lfloor \widetilde{\Delta }\rfloor _{\textrm{hor}}$| for the sum of the components of |$\lfloor \widetilde{\Delta }\rfloor $| that are horizontal over |$C$|⁠. On the other hand, since |$\mathbb{T}$| acts fiberwise over |$C$|⁠, then the restriction of |$\mathbb{T}$| to any prime component of |$\lfloor \widetilde{\Delta }\rfloor _{\textrm{hor}}$| has rank at most |$n-2$|⁠. Thus, we conclude that |$\lfloor \widetilde{\Delta }\rfloor _{\textrm{hor}}$| is contracted by the projective birational morphism |$\pi \colon \widetilde{X}\rightarrow X$|⁠. Let |$S$| be a prime component of |$\lfloor \widetilde{\Delta }\rfloor _{\textrm{hor}}$|⁠. Let |$\mathbb{T}_{0}$| be the torus that acts as the identity on |$S$| and |$\mathbb{T}_{1}$| be its complement torus on |$\mathbb{T}$|⁠. The image of |$S$| on |$X$| is a subvariety of codimension |$2$|⁠. Let |$x$| be the general point of |$q(S)$|⁠. For every |$i$|⁠, we let |$x_{i}$| to be the image of |$x$| on |$X/G_{i}$|⁠. For each |$i$|⁠, there is an isomorphism |$\psi \colon X/G_{i}\rightarrow X$| that induces an equality
$$ \begin{align*} & \mu(X/G_{i};x_{i}) = \mu(X;\psi_{i}(x_{i})). \end{align*} $$
Let |$\mu _{i}$| be the subgroup of |$i$|-roots of unity of |$\mathbb{T}_{0}$|⁠. Let |$r_{i}\colon X\rightarrow X/\mu _{i}$|⁠. Since |$\mathbb{T}_{1}$| acts faithfully on |$q(S)$|⁠, we have
$$ \begin{align*} & \mu(X/G_{i};x_{i}) = \mu(X/\mu_{i};r_{i}(x)). \end{align*} $$
By Lemma 2.8 the left-hand side diverges. Hence, we have a sequence of points |$\psi _{i}(x_{i}))$| of |$X$| for which the sequence |$\mu (X;\psi _{i}(x_{i}))$| diverges. This contradicts Lemma 2.7. Hence, we have concluded that |$\mathbb{T}$| has rank |$n$|⁠. This implies |$(1)$| in the statement of the theorem.

Now, we turn to prove |$(3)$|⁠. We have |$\mathbb{T}\leqslant \textrm{Aut}^{0}(X)$| where |$n$| is the dimension of |$X$|⁠. In particular, we have |$\mathbb{T}\leqslant \textrm{Aut}^{0}_{L}(X)$|⁠, so |$\mathbb{T}$| has a regular effective action on |$X$|⁠. Hence, |$X$| is a toric variety. As |$\mathbb{T}\leqslant \textrm{Aut}(X,\Delta )$|⁠, we conclude that |$\Delta $| is a |$\mathbb{T}$|-invariant divisor, so the pair |$(X,\Delta )$| is toric. Thus, the log pair |$(X,\Delta )$| is a toric log Calabi–Yau pair. Finally, statement |$(2)$| follows from |$(3)$| and Lemma 2.9.

5 Proof of the Main Theorem

In this section, we prove the main theorem of the article.

 

Theorem 5.1.

Let |$X$| be a Fano type variety and let |$(X,\Delta )$| be a log Calabi–Yau pair with |$\Delta $| reduced. If the pair |$(X,\Delta )$| admits an int-amplified endomorphism, then |$(X,\Delta )$| is a finite quotient of a toric log Calabi–Yau pair.

First, we introduce a lemma regarding Galois endomorphisms.

 

Lemma 5.2.

Let |$f\colon (X,\Delta ) \to (X,\Delta )$| be an endomorphism with |$f^{n}$| Galois for all |$n\geqslant 1$|⁠. Assume that |$\textrm{Aut}(X,\Delta )$| is a finite extension of an algebraic torus |$\mathbb{T}$|⁠. Let |$G_{n} \leqslant \textrm{Aut}(X,\Delta )$| be the subgroup associated to |$f^{n}$|⁠. Then, for some |$n \geqslant 1$|⁠, we have |$G_{n} \leqslant \mathbb{T}$|⁠.

 

Proof.
For each |$m,n\geqslant 1$|⁠, there is a short exact sequence
(5.1)
By assumption, we have
$$ \begin{align*} &1\rightarrow \mathbb{T} \rightarrow \textrm{Aut}(X,\Delta) \xrightarrow{\pi} F \rightarrow 1,\end{align*} $$
where |$F$| is a finite group. Let |$m\geqslant 1$| and take |$n\gg 0$| such that |$\pi (G_{n}) = \pi (G_{n+m})$|⁠. Then |$G_{n+m}/G_{n}$| is abelian, and by (5.1) the group |$G_{m}$| is also abelian. Thus, |$G_{m}$| is an abelian group for each |$m\in{\mathbb{Z}}_{\geqslant 1}$|⁠.
Let |$Z_{m}$| be the centralizer of |$G_{m}$| in |$\textrm{Aut}(X,\Delta )$|⁠. As |$G_{m}$| is abelian, the group |$G_{m}$| is contained in |$Z_{m}$|⁠. We have the following chains of subgroups in |$\textrm{Aut}(X,\Delta )$|⁠:
$$ \begin{align*} & G_{1} \leq G_{2} \leq \cdots \leq G_{n} \leq \cdots \end{align*} $$
 
$$ \begin{align*} & Z_{1} \geqslant Z_{2} \geqslant \cdots \geqslant Z_{n} \geqslant \cdots \end{align*} $$
We want to show that there exists a subgroup |$Z_\infty \leq \textrm{Aut}(X,\Delta )$| such that for some |$N \geqslant 1$|⁠, we have |$Z_{n} = Z_\infty $| for all |$n\geqslant N$|⁠.

It suffices to show that |$Z_{i} \cap \mathbb{T}$| stabilizes for some |$i\gg 0$|⁠. Indeed, |$Z_{i}$| stabilizes if the images of |$Z_{i}$| in |$F$| and |$Z_{i} \cap \mathbb{T}$| stabilize. The former follows from the finiteness of |$F$|⁠, so it is enough to prove the latter.

Note that we have
$$ \begin{align*} &Z_{i}\cap \mathbb{T} = \bigcap\limits_{g\in G_{i}} \left(Z(g)\cap \mathbb{T} \right)\end{align*} $$
for each |$i$|⁠, where |$Z(g)$| denotes the centralizer of |$g$| in |$\textrm{Aut}(X,\Delta )$|⁠. Even more, |$Z(g) \cap \mathbb{T} = Z(gt) \cap \mathbb{T}$| for all |$t\in \mathbb{T}$|⁠. As |$\textrm{Aut}(X,\Delta )/\mathbb{T}$| is finite, there exist finitely many |$g_{1},\ldots , g_{k} \in \textrm{Aut}(X,\Delta )$| such that for any |$g\in \textrm{Aut}(X,\Delta )$|⁠, |$g = g_{j}t$| for some |$1\leq j \leq k$| and |$t\in \mathbb{T}$|⁠. Therefore, we can write
$$ \begin{align*} & Z_{i} \cap \mathbb{T} = \bigcap\limits_{j\in I} (Z(g_{j})\cap \mathbb{T}) \end{align*} $$
for some |$I\subseteq \{1,\dots ,k\}$|⁠. Thus |$Z_{i}\cap \mathbb{T}$| stabilize for |$i\gg 0$|⁠.
Let |$Z_\infty = \bigcap Z_{i}$|⁠, and let |$Z_\infty ^{0}$| be the connected component of |$Z_\infty $|⁠. Then, we have an exact sequence
 
(5.2)
where |$F_\infty $| is a finite group, being |$Z_\infty $| a linear algebraic group. Using (5.2), for any |$G_{n}$| we obtain an exact sequence
(5.3)
For all |$i\gg 0$|⁠, we have |$G_{i}\leq Z_\infty \leq N_{i}$|⁠, where |$N_{i}$| is the normalizer of |$G_{i}$| in |$\textrm{Aut}(X,\Delta )$|⁠. For every integers |$m>n$|⁠, we have a commutative diagram
 
(5.4)
The endomorphism |$f^{m-n}$| is isomorphic to the quotient in the bottom of the commutative diagram (5.4). Thus, it suffices to show that |$G_{m}/G_{n}$| is contained in the connected component |$\textrm{Aut}(X/G_{n},\Delta /G_{n})^{0}$|⁠.

Take |$m> n \gg 0$| such that the images of |$G_{m}$| and |$G_{n}$| in |$F_\infty $| via |$\pi _\infty $| agree. For each |$m \geqslant 1$|⁠, there exists a homomorphism |$N_{n}/G_{n} \to \textrm{Aut}(X/G_{n},\Delta /G_{n})$|⁠. Therefore, we obtain a commutative diagram:

Since |$Z_\infty ^{0}$| is a connected linear algebraic group its image in |$\textrm{Aut}(X/G_{m},\Delta /G_{m})$| lies in the connected component. On the other hand, due to (5.3), we have a commutative diagram
Since |$\pi _\infty (G_{m})/\pi _\infty (G_{n})=1$|⁠, we conclude that the image of |$G_{m}/G_{n}$| in |$\textrm{Aut}(X/G_{m},\Delta /G_{m})$| is contained in the image of |$Z_\infty ^{0}/(Z_\infty ^{0}\cap G_{m})$| in |$\textrm{Aut}(X/G_{m},\Delta /G_{m})$|⁠. Hence, the image of |$G_{m}/G_{n}$| is contained in the connected component |$\textrm{Aut}(X/G_{n},\Delta /G_{n})^{0}$|⁠. This finishes the proof of the lemma.

Now, we turn to prove the main theorem of the article.

 

Proof of Theorem 5.1.

Let |$p_{0}\colon X_{0}\rightarrow X$| be the index one cover of |$K_{X}+\Delta \sim _{\mathbb{Q}} 0$|⁠. As |$K_{X}+\Delta $| is a Weil divisor, the finite morphism |$p_{0}\colon X_{0}\rightarrow X$| is unramified in codimension one. Let |$p_{0}^{*}(K_{X}+\Delta )=K_{X_{0}}+\Delta _{X_{0}}$|⁠. By Lemma 2.4, we conclude that |$X_{0}$| is a Fano type variety and |$\Delta _{X_{0}}$| is a |$1$|-complement of |$X_{0}$|⁠. By Corollary 3.2, we have a commutative diagram:

In the previous commutative diagram, the endomorphism |$f_{0}$| is an int-amplified endomorphism of the pair |$(X_{0},\Delta _{0})$|⁠. Replacing |$(X,\Delta )$| with |$(X_{0},\Delta _{0})$|⁠, we may assume that |$\Delta $| is a |$1$|-complement of the Fano type variety |$X$|⁠.

From now on, we assume that |$X$| is a Fano type variety and |$(X,\Delta )$| is a log Calabi–Yau pair of index one. Let |$f\colon (X,\Delta )\rightarrow (X,\Delta )$| be an int-amplified endomorphism. Let |$U=X^{\textrm{reg}}\setminus \Delta $|⁠. By Theorem 3.3, we know that |$\pi _{1}^{\textrm{alg}}(U)$| is a virtually abelian group. Let |$N\leqslant \pi _{1}^{\textrm{alg}}(U)$| be a normal abelian subgroup of finite index. Let |$U^{\prime}\rightarrow U$| be the corresponding finite cover, and let |$g\colon (Y,\Delta _{Y})\rightarrow (X,\Delta )$| be the induced finite cover of log pairs. By Theorem 3.1, for some |$n, m\gg 1$|⁠, we have a commutative diagram

where |$f_{Z}$| is an int-amplified endomorphism for the log pair |$(Z,\Delta _{Z})$|⁠, and |$h$| is a finite cover, étale over |$U_{Y}$|⁠.

Set |$U_{Y}=Y^{\textrm{reg}}\setminus \Delta _{Y}$| and |$U_{Z} = Z^{\textrm{reg}}\setminus \Delta _{Z}$|⁠. The group |$\pi _{1}^{\textrm{alg}}(U_{Y})$| is abelian, and therefore so is |$\pi _{1}^{\textrm{alg}}(U_{Z})$|⁠. Indeed, |$U_{Y}$| is smooth and |$U^{\prime}$| is a big open subset of |$U_{Y}$|⁠. The finite morphism |$f_{Z}\colon (Z,\Delta _{Z})\rightarrow (Z,\Delta _{Z})$| corresponds to a subgroup of finite index of |$\pi _{1}(U_{Z})$|⁠, and the profinite completion of |$\pi _{1}(U_{Z})$| is |$\pi _{1}^{\textrm{alg}}(U_{Z})$|⁠. Thus, by Lemma 3.4 the finite morphism |$f_{Z}\colon (Z,\Delta _{Z})\rightarrow (Z,\Delta _{Z})$| corresponds to a normal subgroup of finite index of |$\pi _{1}(U_{Z})$|⁠. Henceforth, the finite morphism |$f_{Z}\colon (Z,\Delta _{Z})\rightarrow (Z,\Delta _{Z})$|⁠, and all iterations |$f_{Z}^{n}$|⁠, are Galois. Therefore, there exists a finite group |$G\leqslant \textrm{Aut}(Z,\Delta _{Z})$| and an isomorphism of log pairs |$\psi \colon (Z/G,\Delta _{Z}/G)\rightarrow (Z,\Delta _{Z})$| making the following diagram commutative:

In particular, the quotient |$(Z,\Delta _{Z})\rightarrow (Z/G,\Delta _{Z}/G)$| is of int-amplified type. By Lemma 2.4, the variety |$Z$| is of Fano type. Hence, by Theorem 2.10, we know that |$\textrm{Aut}(Z,\Delta _{Z})$| is a finite extension of an algebraic torus. By Lemma 5.2, up to replacing |$f_{Z}$| with an iteration |$f_{Z}^{n}$|⁠, we may assume that |$G$| is a finite subgroup of a maximal algebraic torus of |$\textrm{Aut}(Z,\Delta _{Z})$|⁠. By Theorem 4.2, we conclude that |$(Z,\Delta _{Z})$| is a log Calabi–Yau toric pair.

As the finite map |$(Z,\Delta _{Z}) \xrightarrow{g\,\circ \, h} (X,\Delta )$| is Galois, we conclude that the pair |$(X,\Delta )$| is the finite quotient of a toric log Calabi–Yau pair |$(Y,\Delta _{Y})$|⁠. This finishes the proof.

6 Examples and Questions

In this section, we provide some examples related to the main theorem of the paper and questions for further research. The first example is the most well-known in this direction.

 

Example 6.1.
Consider the endomorphism
$$ \begin{align*} f_{m}: ({\mathbb{P}}^{n},H_{0}+\dots+H_{n}) &\rightarrow ({\mathbb{P}}^{n},H_{0}+\dots+H_{n})\\ f_{m}([x_{0}:\dots:x_{n}])&= [x_{0}^{m}:\dots:x_{n}^{m}]. \end{align*} $$
Then, |$f_{m}$| is a polarized endomorphism as |$f_{m}^{*}H_{i}=mH_{i}$| for every hyperplane coordinate. By Riemann–Hurwitz, we have
$$ \begin{align*} & f_{m}^{*}\left(K_{{\mathbb{P}}^{n}}+\sum_{i=0}^{n}\left(1-\frac{1}{m}\right)H_{i}\right) = K_{{\mathbb{P}}^{n}}. \end{align*} $$
Thus, we conclude that
$$ \begin{align*} & f_{m}^{*}\left( K_{{\mathbb{P}}^{n}}+\sum_{i=0}^{n} H_{i}\right) = K_{{\mathbb{P}}^{n}}+\sum_{i=0}^{n} H_{i}. \end{align*} $$
Thus, the log Calabi–Yau pair |$({\mathbb{P}}^{n},H_{0}+\dots +H_{n})$| admits a polarized endomorphism. This gives an example of Theorem 1.
The previous example can be generalized in the following way. For every |$n$|-dimensional projective toric variety |$X$|⁠, we consider the log Calabi–Yau pair |$(X,\Delta )$| where |$\Delta $| is the reduced torus invariant divisor. Consider the polarized endomorphism
$$ \begin{align*} & f_{m}\colon X \rightarrow X \end{align*} $$
corresponding to the inclusion of lattices |$m{\mathbb{Z}}^{n} \subseteq{\mathbb{Z}}^{n}$| in the fan |$\Sigma $| of |$X$|⁠. It is clear that |$f_{m}^{*}\Delta =m\Delta $| so we have |$f_{m}^{*}(K_{X}+\Delta )=K_{X}+\Delta $| as above. If |$X$| is projective, then it admits a torus invariant ample divisor |$H$|⁠, for which |$f_{m}^{*}H=mH$|⁠. The previous statement implies that |$f_{m}\colon (X,\Delta )\rightarrow (X,\Delta )$| is a polarized endomorphism.

The second example shows that the quotient in Theorem 1 is necessary.

 

Example 6.2.

Consider the pair |$({\mathbb{P}}^{n},\Delta ^{\prime})$| where |$\Delta ^{\prime}$| is the reduced torus invariant boundary divisor and |$n\geqslant 3$|⁠, which admits a polarized endomorphism |$f_{m}$| as in Example 6.1. Let |$S_{n+1}$| be acting on |${\mathbb{P}}^{n}$| by permutations of the components. Let |$(X,\Delta ):=({\mathbb{P}}^{n}/S_{n+1},\Delta ^{\prime}/S_{n+1})$|⁠. The variety |$X$| is a klt Fano variety and |$K_{X}+\Delta \sim 0$|⁠. Further, |$X$| is not a toric variety as |$\pi _{1}(X^{\operatorname{reg}}\setminus \Delta )$| surjects onto |$S_{n+1}$|⁠, while the smooth locus of a toric variety contains |$\mathbb{G}_{m}^{n}$| so its fundamental group is abelian.

The polarized endomorphism |$f_{m}$| descends to an endomorphism |$g_{m}$| of |$(X,\Delta )$| as follows:
By construction, we have |$g_{m}^{*}(K_{X}+\Delta )=K_{X}+\Delta .$| Since |$q^{*}\Delta =\Delta ^{\prime}$| and |$f_{m}^{*}\Delta ^{\prime}=m\Delta ^{\prime},$| we have
$$ \begin{align*} & g_{m}^{*}\Delta=q_{*}f_{m}^{*}q^{*}\Delta=q_{*}f_{m}^{*}\Delta^{\prime}=q_{*}(m\Delta^{\prime})=m\Delta. \end{align*} $$
Hence, |$g_{m}$| is a polarized endomorphism.

This example is based on [16]. In fact, as is referred in that article, in [29] one can find an example of a nonabelian subgroup |$G\leq S_{|G|+1}$| of order |$p^{9}$|⁠, for some prime number |$p$|⁠, for which the quotient |${\mathbb{P}}^{n}/G$| is not rational, and hence not toric.

 

Remark 6.3.

In [20, Corollary 1.4], Meng and Zhang show that if |$X$| is a smooth rationally connected and |$D$| a reduced divisor that is |$f^{-1}$|-invariant for a polarized endomorphism |$f$|⁠, where |$f|_{X\setminus D}$| is étale, then |$(X,D)$| is a toric pair. As in Example 6.2, for |$n\geqslant 3$|⁠, |$(X,D)$| is not a toric pair and it is smooth in codimension 2, with at worst canonical singularities, we see that either |$f|_{X\setminus D}$| must be étale, or the smoothness condition is essential for [20, Corollary 1.4].

The next two examples go in a somewhat orthogonal direction, and are related to the following result of Yoshikawa [31] about polarized endomorphisms of klt Calabi–Yau pairs with standard coefficients.

 

Theorem 6.4.

(Yoshikawa) Let |$(X,\Delta )$| be a klt log Calabi–Yau pair with standard coefficients. If |$(X,\Delta )$| admits an int-amplified endomorphism, then |$(X,\Delta )$| is a finite quotient of an abelian variety.

To summarize Theorem 1 and Yoshikawa’s Theorem 6.4, we can understand polarized endomorphisms of Fano varieties with complements in two cases: when the complement is reduced, and when the complement is klt and has standard coefficients.

 

Example 6.5.
Let |$A$| be an abelian variety of dimension |$n$| with corresponding rank |$2n$| lattice |$\Lambda \subset{\mathbb{C}}^{n}.$| For each integer |$m$|⁠, the map |$g_{m}\colon{\mathbb{C}}^{n}\rightarrow{\mathbb{C}}^{n}$| given by |$z\mapsto mz$| preserves the lattice, so it induces an endomorphism
(6.1)
which is polarized when |$m> 1$| since |$f_{m}^{*}H\sim m^{2}H$| for every symmetric ample divisor |$H$|⁠.

The following example shows that the finite quotient in Yoshikawa’s Theorem 6.4 is necessary.

 

Example 6.6.
Let |$E$| be an elliptic curve with corresponding lattice |$\Lambda \subset{\mathbb{C}}.$| Consider its quotient by the involution map |$f_{-1}$| from (6.1), which gives
$$ \begin{align*} &q:E \xrightarrow{/f_{-1}} ({\mathbb{P}}_{1},\Delta),\end{align*} $$
where |$\Delta =\frac{1}{2}\left \{0\right \}+\frac{1}{2}\left \{1\right \}+\frac{1}{2}\left \{\lambda \right \}+\frac{1}{2}\left \{\infty \right \}.$| The polarized endomorphism |$f_{m}$| from (6.1) descends to an endomorphism |$g_{m}$| of |$({\mathbb{P}}^{1},\Delta )$| as follows:
From the commutativity of the diagram, it follows that |$g_{m}^{*}(K_{{\mathbb{P}}^{1}}+\Delta )=K_{{\mathbb{P}}^{1}}+\Delta $| and |$g_{m}^{*}\Delta \sim m^{2} \Delta $|⁠, so |$g_{m}$| is a polarized endomorphism of the pair |$({\mathbb{P}}^{1},\Delta )$|⁠.

The following example shows that log canonical varieties with polarized endomorphisms may not be finite quotients of toric fibrations over abelian varieties. Thus, in the folklore conjecture, it is necessary to impose that the variety has klt singularities.

 

Example 6.7.

Consider the projectivized cone |$X$| over an elliptic curve |$E$|⁠, which admits polarized endomorphisms induced by polarized endomorphisms on |$E$| as in Example 6.5. At the cone vertex, |$X$| has a singularity that is log canonical but not klt type. However, |$X$| is not a finite quotient of a toric fibration over an abelian variety since such varieties must be klt type.

Two natural questions emanate from this article.

 

Question 6.8.

Let |$X$| be a Fano variety and |$\Delta $| be a complement. Assume that |$(X,\Delta )$| admits a polarized endomorphisms. Is |$(X,\Delta )$| a finite quotient of a log Calabi–Yau toric fibration over an abelian variety?

Our techniques so far can prove the previous statement when |$K_{X}+\Delta \sim _{\mathbb{Q}} 0$| with |$\Delta $| reduced. We remark that our techniques can also be applied with some additional argument needed when |$(X,\Delta )$| is klt log Calabi–Yau with standard coefficients. We expect that the previous question has a positive answer, but it goes beyond the scope of this article. Some new ideas are required to conclude. This question will be addressed in an upcoming paper by the authors. Let’s say that a polarized endomorphism |$f\colon X \rightarrow X$| admits a complement if there exists a complement |$(X,\Delta )$| for which |$f^{*}(K_{X}+\Delta )=K_{X}+\Delta $|⁠. Not every polarized endomorphism admits a complement. The hardest task seems to be the following:

 

Question 6.9.

Let |$X$| be a normal projective variety admitting a polarized endomorphism |$f$|⁠. Can we construct a polarized endomorphism on |$X$| that admits a complement?

We expect that a positive answer to the previous question would settle the main conjecture on the topic. However, nowadays, it is not even clear how to construct more polarized endomorphisms from a given one.

Acknowledgments

The authors would like to thank Rohan Joshi, Sheng Meng, and Burt Totaro for many useful discussions and comments. The authors are also grateful to the referee for their useful comments and corrections.

Communicated by Prof. Chenyang Xu

References

1.

Altmann
,
K.
and
J.
 
Hausen
. “
Polyhedral divisors and algebraic torus actions
.”
Math. Ann.
 
334
, no.
3
(
2006
):
557
–
607
.
10.1007/s00208-005-0705-8
.

Google Scholar

Crossref
Search ADS

 
2.

Birkar
,
C.
“
Anti-pluricanonical systems on Fano varieties
.”
Ann. Math. (2)
 
190
, no.
2
(
2019
):
345
–
463
.
10.4007/annals.2019.190.2.1
.

Google Scholar

Crossref
Search ADS

 
3.

Braun
,
L.
“
The local fundamental group of a Kawamata log terminal singularity is finite
.”
Invent. Math.
 
226
, no.
3
(
2021
):
845
–
96
.
10.1007/s00222-021-01062-0
.

Google Scholar

Crossref
Search ADS

 
4.

Braun
,
L.
,
S.
 
Filipazzi
,
J.
 
Moraga
, and
R.
 
Svaldi
. “
The Jordan property for local fundamental groups
.”
Geom. Topol.
 
26
, no.
1
(
2022
):
283
–
319
.
10.2140/gt.2022.26.283
.

Google Scholar

Crossref
Search ADS

 
5.

Broustet
,
A.
and
Y.
 
Gongyo
. “
Remarks on log Calabi–Yau structure of varieties admitting polarized endomorphisms
.”
Taiwanese J. Math.
 
21
, no.
3
(
2017
):
569
–
82
.
10.11650/tjm/7968
.

Google Scholar

Crossref
Search ADS

 
6.

Broustet
,
A.
and
A.
 
Höring
. “
Singularities of varieties admitting an endomorphism
.”
Math. Ann.
 
360
, no.
1–2
(
2014
):
439
–
56
.
10.1007/s00208-014-1015-9
.

Google Scholar

Crossref
Search ADS

 
7.

Cox
,
D. A.
,
J. B.
 
Little
, and
H. K.
 
Schenck
. “
Toric varieties
.”
Graduate Studies in Mathematics
, vol. 124.
Providence, RI
:
American Mathematical Society
,
2011
.

Google Scholar

Crossref
Search ADS

 
8.

Dethloff
,
G.
and
H.
 
Grauert
. “
Seminormal complex spaces
.”
Several Complex Variables, VII, Encyclopaedia Math. Sci.
, vol. 74.
183
–
220
.
Berlin
:
Springer
,
1994
.

Google Scholar

Crossref
Search ADS

 
9.

Filipazzi
,
S.
and
J.
 
Moraga
. “
Strong |$\left (\delta , n\right )$|-complements for semi-stable morphisms
.”
Doc. Math.
 
25
(
2020
):
1953
–
96
.
10.4171/DM/790
.

Google Scholar

Crossref
Search ADS

 
10.

Greb
,
D.
,
S.
 
Kebekus
, and
T.
 
Peternell
. “
Étale fundamental groups of kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties
.” (
2013
):
Preprint, arXiv:1307.5718, http://arxiv.org/abs/1307.5718
.

11.

Höring
,
A.
“
Totally invariant divisors of endomorphisms of projective spaces
.”
Manuscripta Math.
 
153
, no.
1–2
,
182
(
2017
):
173
.
10.1007/s00229-016-0881-8
.

Google Scholar

Crossref
Search ADS

 
12.

Hu
,
F.
“
The dimension of automorphism groups of algebraic varieties with pseudo-effective log canonical divisors
.”
Proc. Amer. Math. Soc.
 
146
, no.
5
(
2018
):
1879
–
93
.
10.1090/proc/13893
.

Google Scholar

Crossref
Search ADS

 
13.

Hwang
,
J.-M.
and
N.
 
Nakayama
. “
On endomorphisms of Fano manifolds of Picard number one
.”
Pure Appl. Math. Q.
 
7
, no.
4
(
2011
):
1407
–
26
.
10.4310/PAMQ.2011.v7.n4.a15
.

Google Scholar

Crossref
Search ADS

 
14.

Kawakami
,
T.
and
B.
 
Totaro
. “
Endomorphisms of varieties and Bott vanishing
.”
J. Algebraic Geom.
 
34
, no.
2
(
2025
):
381
–
405
.
10.1090/jag/838
.

Google Scholar

Crossref
Search ADS

 
15.

Kollár
,
J.
“
Singularities of the minimal model program
.”
Cambridge Tracts in Mathematics
, vol. 200.
Cambridge
:
Cambridge University Press
,
2013
.
With a collaboration of Sándor Kovács
.

Google Scholar

OpenURL Placeholder Text

 
16.

Kollár
,
J.
and
C.
 
Xu
. “
Fano varieties with large degree endomorphisms
.” (
2009
):
Preprint,arXiv:0901.1692, http://arxiv.org/abs/0901.1692
.

17.

Laface
,
A.
,
A.
 
Liendo
, and
J.
 
Moraga
. “
The fundamental group of a log terminal |$\mathbb{T}$|-variety
.”
Eur. J. Math.
 
5
, no.
3
(
2019
):
937
–
57
.
10.1007/s40879-018-0296-z
.

Google Scholar

Crossref
Search ADS

 
18.

Lazarsfeld
,
R.
“
Positivity in algebraic geometry. I
.”
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]
, vol. 48.
Berlin
:
Springer
,
2004
.
Classical setting: line bundles and linear series
.

Google Scholar

OpenURL Placeholder Text

 
19.

Meng
,
S.
“
Building blocks of amplified endomorphisms of normal projective varieties
.”
Math. Z.
 
294
, no.
3–4
(
2020
):
1727
–
47
.
10.1007/s00209-019-02316-7
.

Google Scholar

Crossref
Search ADS

 
20.

Meng
,
S.
and
D.-Q.
 
Zhang
. “
Characterizations of toric varieties via polarized endomorphisms
.”
Math. Z.
 
292
, no.
3—4
(
2019
):
1223
–
31
.
10.1007/s00209-018-2160-8
.

Google Scholar

Crossref
Search ADS

 
21.

Meng
,
S.
and
D.-Q.
 
Zhang
. “
Normal projective varieties admitting polarized or int-amplified endomorphisms
.”
Acta Math. Vietnam.
 
45
, no.
1
(
2020
):
11
–
26
.
10.1007/s40306-019-00333-6
.

Google Scholar

Crossref
Search ADS

 
22.

Meng
,
S.
,
D.-Q.
 
Zhang
, and
G.
 
Zhong
. “
Non-isomorphic endomorphisms of Fano threefolds
.”
Math. Ann.
 
383
, no.
3–4
(
2022
):
1567
–
96
.
10.1007/s00208-021-02274-8
.

Google Scholar

Crossref
Search ADS

 
23.

Meng
,
S.
and
G.
 
Zhong
. “
Rigidity of rationally connected smooth projective varieties from dynamical viewpoints
.”
Math. Res. Lett.
 
30
, no.
2
(
2023
):
589
–
610
.
10.4310/MRL.2023.v30.n2.a10
.

Google Scholar

Crossref
Search ADS

 
24.

Moraga
,
J.
“
On a toroidalization for klt singularities
.” (
2021
):
Preprint, arXiv:2106.15019, http://arxiv.org/abs/2106.15019
.

25.

Moraga
,
J.
“
Coregularity of Fano varieties
.”
Geom. Dedicata
 
218
, no.
2
(
2024
):
55
.
10.1007/s10711-023-00882-z
.
Paper No. 40
.

Google Scholar

Crossref
Search ADS

 
26.

Nakayama
,
N.
“
Ruled surfaces with non-trivial surjective endomorphisms
.”
Kyushu J. Math.
 
56
, no.
2
(
2002
):
433
–
46
.
10.2206/kyushujm.56.433
.

Google Scholar

Crossref
Search ADS

 
27.

Nakayama
,
N.
and
D.-Q.
 
Zhang
. “
Polarized endomorphisms of complex normal varieties
.”
Math. Ann.
 
346
, no.
4
(
2010
):
991
–
1018
.
10.1007/s00208-009-0420-y
.

Google Scholar

Crossref
Search ADS

 
28.

Paranjape
,
K. H.
and
V.
 
Srinivas
. “
Self-maps of homogeneous spaces
.”
Invent. Math.
 
98
, no.
2
(
1989
):
425
–
44
.
10.1007/BF01388861
.

Google Scholar

Crossref
Search ADS

 
29.

Saltman
,
D. J.
“
Noether’s problem over an algebraically closed field
.”
Invent. Math.
 
77
, no.
1
(
1984
):
71
–
84
.
10.1007/BF01389135
.

Google Scholar

Crossref
Search ADS

 
30.

Watanabe
,
K.
“
Some remarks concerning Demazure’s construction of normal graded rings
.”
Nagoya Math. J.
 
83
(
1981
):
203
–
11
.
10.1017/S0027763000019498
.

Google Scholar

Crossref
Search ADS

 
31.

Yoshikawa
,
S.
“
Structure of Fano fibrations of varieties admitting an int-amplified endomorphism
.”
Adv. Math.
 
391
(
2021
):
32
,
Paper No. 107964
.
10.1016/j.aim.2021.107964
.

Google Scholar

Crossref
Search ADS

 
32.

Yoshikawa
,
S.
“
Global |$F$|-splitting of surfaces admitting an int-amplified endomorphism
.”
Manuscripta Math.
 
169
, no.
1–2
(
2022
):
271
–
96
.
10.1007/s00229-021-01331-5
.

Google Scholar

Crossref
Search ADS

 
© The Author(s) 2025. Published by Oxford University Press.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact [email protected] for reprints and translation rights for reprints. All other permissions can be obtained through our RightsLink service via the Permissions link on the article page on our site-for further information please contact [email protected].
Issue Section:
Article
Download all slides