Canonical heights for abelian group actions of maximal dynamical rank In memory of Professor Nessim Sibony

Abstract

Let X be a smooth projective variety of dimension $n\geq 2$ and $G\cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X over $\overline {\mathbf {Q}}$. Suppose that G is of positive entropy. We construct a canonical height function $\widehat {h}_G$ associated with G, corresponding to a nef and big $\mathbf {R}$-divisor, satisfying the Northcott property. By characterizing the zero locus of $\widehat {h}_G$, we prove the Kawaguchi–Silverman conjecture for each element of G. As for other applications, we determine the height counting function for non-periodic points and show that X satisfies potential density.

1 Introduction

Given a surjective holomorphic self-map f of a compact Kähler manifold M of dimension n, the topological entropy $h_{\textrm {top}}(f)$ of f is a key dynamical invariant to measure the divergence of the orbits. A fundamental result due to Gromov [Gro03] and Yomdin [Yom87] establishes its equivalence to the algebraic entropy $h_{\mathrm {alg}}(f)$ , defined as the logarithm of the spectral radius of the linear pullback operation $f^*$ acting on the cohomology group $\bigoplus _{k=0}^{n} H^{k,k}(M,\mathbf {C})$ :

For each k, the quantity $\rho (f^*|_{H^{k,k}(M,\mathbf {C})})$ is called the k-th dynamical degree of f and denoted by $\lambda _k(f)$ ; see Definition 2.12 for an equivalent definition for projective M. We say that f is of positive entropy (resp. null entropy) if $h_{\mathrm {alg}}(f)>0$ (resp. $=0$ ) or, equivalently, if its first dynamical degree satisfies $\lambda _1(f)>1$ (resp. $=1$ ). Denote the full automorphism group of M by $\operatorname {Aut}(M)$ . A subgroup $G\leq \operatorname {Aut}(M)$ is of positive entropy if all the elements of $G{\setminus } \{\operatorname {id}\}$ are of positive entropy.

In their innovative work [DS04], Dinh and Sibony proved that for any abelian subgroup $G\leq \operatorname {Aut}(M)$ , the subset N of G consisting of automorphisms of null entropy is a group; moreover, there exists a free abelian subgroup P of G, of positive entropy and of rank $r\le n-1$ , such that $G\cong P\times N$ . The number r is also called the dynamical rank $\operatorname {dr}(G)$ of the abelian group G. We say that G is of maximal dynamical rank if $\operatorname {dr}(G)=n-1$ . Subsequently, Zhang [Zha09] established a theorem of Tits type for $\operatorname {Aut}(M)$ and extended [DS04] to the solvable group case. Since then, algebraic dynamics on systems with maximal dynamical rank has been intensively studied (see, for example, [Din12, Zha13, CWZ14, DHZ15, Zha16, Les18, Hu20, Zho22, Zho23]).

On the other hand, topological dynamics on higher rank abelian group actions has also been investigated for decades, and various rigidity theorems hold on such systems (see, for example, [KS94, KK01, KKRH14]). However, arithmetic dynamics on these higher rank abelian group actions does not seem to attract as much attention. In this paper, we aim to explore the system of maximal dynamical rank from this perspective. Specifically, we construct a canonical height associated with a $G\cong \mathbf {Z}^{n-1}$ -action of positive entropy on an n-dimensional smooth projective variety X, extending the works in [Sil91, Kaw08] from surfaces to higher dimensions.

Throughout the paper, unless otherwise stated, we will work over the field $\overline {\mathbf {Q}}$ of algebraic numbers. Below is our main result; see Theorem 4.2 for its more precise form and §2.3 for a brief review of Weil’s height theory.

Theorem 1.1 (cf. Theorem 4.2).

Let X be a smooth projective variety of dimension $n\ge 2$ over $\overline {\mathbf {Q}}$ . Let $G\cong \mathbf {Z}^{n-1}$ be a free abelian group of automorphisms of X such that any nontrivial element of G has positive entropy. Then there exist a function $\widehat {h}_G$ on $X(\overline {\mathbf {Q}})$ with respect to a set of generators of G and a G-invariant Zariski closed proper subset Z of X such that

1. (1) $\widehat {h}_G$ is a Weil height function corresponding to a nef and big $\mathbf {R}$ -divisor on X;

2. (2) $\widehat {h}_G$ satisfies the Northcott property on $(X{\setminus } Z)(\overline {\mathbf {Q}})$ ;

3. (3) for any $x\in (X{\setminus } Z)(\overline {\mathbf {Q}})$ , one has $\widehat {h}_G(x) = 0$ if and only if x is g-periodic for any $g\in G$ .

We refer to the function $\widehat {h}_G$ in Theorem 1.1 as a canonical height function associated with the abelian group G (cf. Theorem 2.10 for the one associated with a single endomorphism). Canonical height theory was initially developed by Néron and Tate for abelian varieties in the 1960s, and it has proved to be a crucial concept in arithmetic geometry. In his pioneering work [Sil91], Silverman constructed canonical height functions on the so-called Wehler K3 surfaces that are defined by the smooth complete intersection of two divisors of type $(1,1)$ and $(2,2)$ in $\mathbf {P}^2 \times \mathbf {P}^2$ . Since then, there has been an extensive body of work on canonical height in arithmetic dynamics (see [BIJ⁺19, §16] and the references therein).

Before deriving into the applications of our Theorem 1.1, let us provide a brief sketch of the strategy behind its proof.

Remark 1.2. The idea of our proof of Theorem 1.1 draws significant inspiration from the innovative works of Dinh–Sibony [DS04] and Call–Silverman [CS93]. To be specific, we first follow the approach in [DS04] to construct n distinguished automorphisms $g_i$ in G and n common nef $\mathbf {R}$ -eigendivisors $D_i$ on X. Each $g_i^*$ expands $D_i$ (up to $\mathbf {R}$ -linear equivalence) and shrinks $D_j$ with $j\neq i$ (up to numerical equivalence). We refer to Theorem 3.6 for details.

Furthermore, as established in [CS93], for each pair $(g_i,D_i)$ as described above, there exists a unique nef canonical height function $\widehat {h}_{D_i,g_i}$ . As the sum $\sum _i D_i$ forms a nef and big $\mathbf {R}$ -divisor, the corresponding sum $\sum _{i} \widehat {h}_{D_i,g_i}$ of nef canonical height functions turns out to be a Weil height function (denoted by $\widehat {h}_G$ ) satisfying the Northcott property. See Theorem 4.2 for details.

As the first application of our canonical height, we provide a positive answer to the Kawaguchi–Silverman conjecture under the assumption of maximal dynamical rank. In general, for an arithmetic dynamical system $(X,f)$ over $\overline {\mathbf {Q}}$ and a point $x\in X(\overline {\mathbf {Q}})$ , we use an ample height function to measure the arithmetic complexity of the orbit $\mathcal {O}_f(x)$ . The guiding principle is that ‘geometry governs arithmetic’ in the sense that the height growth rate along the orbit is controlled by the first dynamical degree. The conjecture predicts that if x is sufficiently complicated (e.g., has a Zariski dense orbit), then the height growth achieves the maximum. For precise definitions and the statement of the conjecture, we refer the reader to §2.4; see also [DGH⁺22] for a higher-dimensional analog.

Corollary 1.3. Under the assumption of Theorem 1.1, for each $g\in G$ , the arithmetic degree $\alpha _g(x)$ of g at any $x\in X(\overline {\mathbf {Q}})$ equals the first dynamical degree $\lambda _1(g)$ of g, whenever the forward g-orbit of x is Zariski dense in X.

Remark 1.4. The Kawaguchi–Silverman conjecture (i.e., Conjecture 2.14) has been successfully proved in many cases, leveraging canonical height theory. Notable contributions include works by [Sil91, CS93, Kaw06a, Kaw08, KS14, KS16a, Shi19, LS21], among others. Our approach to proving Corollary 1.3 closely follows this well-established strategy. In comparison, an alternative geometric approach to address Conjecture 2.14 involves the Equivariant Minimal Model Program, as developed by Meng and Zhang. See their survey paper [MZ24a] and the references therein. For the reader interested in this geometric perspective, we recommend exploring [MSS18, Mat20a, MZ22, MY22, MZ23, MZ24b]. For a comprehensive overview of the current state of Conjecture 2.14, we refer to a recent survey [Mat24].

Our second application involves an investigation of the bounded height property for periodic points, as well as an exploration of the height counting function for non-periodic points.

Corollary 1.5 (cf. [Kaw08, Theorem D] and [KS16b, Proposition 3]).

Under the assumption of Theorem 1.1 and with the notation therein, the following assertions hold for any $g\in G{\setminus }\{\operatorname {id}\}$ .

1. (1) The subset below

   $$\begin{align*}\{x\in (X{\setminus} Z)(\overline{\mathbf{Q}}) : x \text{ is } g\text{-periodic}\} \end{align*}$$

   is of bounded height (see Definition 2.9).

2. (2) For any ample divisor $H_X$ on X and any non-g-periodic $x\in (X{\setminus } Z)(\overline {\mathbf {Q}})$ , one has

   $$\begin{align*}\lim_{T\to+\infty}\frac{\#\{ m\in \mathbf{Z}_{\geq 0} : h_{H_X}(g^m(x)) \leq T \}}{\log T} = \frac{1}{\log\lambda_1(g)}. \end{align*}$$

As mentioned in Remark 1.2, one of the crucial ingredients to establish Theorem 1.1 is to construct a special nef and big $\mathbf {R}$ -divisor D, which is the sum of G-common nef eigendivisors (see Theorem 3.6). As a by-product of this construction, we obtain the existence of Zariski dense G-orbits, which also implies that our X satisfies potential density. We refer the reader to [Has03, Cam04, Wit18] and the references therein for the information on potential density.

Corollary 1.6. Under the assumption of Theorem 1.1, there exists a rational point $x\in X(\overline {\mathbf {Q}})$ such that the G-orbit of x is Zariski dense in X. Moreover, if we assume that for some Y defined over a number field K, then there exists a finite field extension $L/K$ such that $Y_L(L)$ is Zariski dense in $Y_L$ .

For the sake of completeness, we recall a few examples satisfying the maximal dynamical rank assumption in Theorem 1.1.

Example 1.7 (cf. [DS04, Exemple 4.5] and [Ogu14, §5.2]).

Let E be an elliptic curve over $\overline {\mathbf {Q}}$ and $E^{n}$ the product variety of E. There is a natural faithful action of $\mathrm {SL}_n(\mathbf {Z})$ on $E^n$ . It is known that $\mathrm {SL}_n(\mathbf {Z})$ admits a free abelian subgroup G of rank $n-1$ which is diagonalizable; any nontrivial element of G has an eigenvalue with modulus greater than $1$ (see, for example, [Din12, Example 1.4]). It is not hard to see that for any $g\in G$ , the first dynamical degree $\lambda _1(g)$ of g as an automorphism of $E^n$ equals the square of the spectral radius of g as a matrix in $\mathrm {SL}_n(\mathbf {Z})$ . So, $G\leq \operatorname {Aut}(E^n)$ is of maximal dynamical rank. In this case, as $E^n$ is an abelian variety, the nef and big $\mathbf {R}$ -divisor D on $E^n$ obtained from Theorem 3.6(2) is indeed ample. Consequently, the G-invariant Zariski closed proper subset Z in Theorem 1.1 is empty.

Further, thanks to Ueno [Uen75] and Campana [Cam11], one can also construct examples of Calabi–Yau varieties and rationally connected varieties. For instance, let E be an elliptic curve over $\overline {\mathbf {Q}}$ such that $\operatorname {Aut}(E)$ is cyclic of order $4$ , and $\sigma $ the generator of $\operatorname {Aut}(E)$ . Let $G \leq \mathrm {SL}_n(\mathbf {Z})$ be as above. Since the diagonal action by $\sigma $ is inside the centralizer of G in $\operatorname {Aut}(E^n)$ , there is an induced faithful G-action on the quotient $E^n/\langle \sigma \rangle $ . Let $X_{n,4}$ be a G-equivariant resolution of singularities of $E^n/\langle \sigma \rangle $ . Then it is well known that the Kodaira dimension of $X_{n,4}$ satisfies

$$ \begin{align*} \kappa(X_{n,4}) = \left\{ \begin{array}{cl} 0 & \text{if } n \geq 4, \\ -\infty & \text{if } n \leq 3. \end{array} \right. \end{align*} $$

Since dynamical degrees are invariant under generically finite rational maps (see Definition 2.12), $G\leq \operatorname {Aut}(X_{n,4})$ is of maximal dynamical rank. In particular, the G-invariant Zariski closed proper subset Z in Theorem 1.1 coincides with the exceptional locus of the equivariant resolution (cf. Lemma 3.16). There is a parallel construction for any elliptic curve E with $\#\operatorname {Aut}(E) = 6$ . See [Ogu14, §5.2] and the references therein for more details.

This paper is organized as follows. In Section 2, we provide a review of essential concepts and results, including weak numerical equivalence, stable base loci, Weil’s height theory and dynamical/arithmetic degrees. In Section 3, we proceed to construct n distinguished automorphisms $g_i$ in G and a nef and big $\mathbf {R}$ -divisor as a sum of common nef eigendivisors $D_i$ , possessing favorable properties from an arithmetic perspective (see Theorem 3.6 and Remark 3.7); then we prove Corollary 1.6 as a direct application. Following this, in Section 4, we define a canonical height function associated with the abelian group G of maximal dynamical rank. Finally, we present the proofs of Theorem 1.1 and Corollaries 1.3 and 1.5.

We conclude the introduction with the following remark.

Remark 1.8 (About the generalization of Theorem 1.1).

It is noteworthy that extending our main result to normal projective varieties over $\overline {\mathbf {Q}}$ poses no essential difficulties, using intersection theory and Weil’s height theory on $\mathbf {R}$ -Cartier divisors. Furthermore, the extension of our main Theorem 1.1 to global fields in positive characteristic is also possible using the language in [Tru20, Dan20, Hu20], along with the result from [Hu24]. However, given that the primary focus of this paper is to present a distinctive perspective for studying abelian group actions of maximal dynamical rank, we choose to concentrate on smooth projective varieties over $\overline {\mathbf {Q}}$ . This decision is made in the interest of maintaining clarity and simplicity in our exposition. We anticipate that this alternative viewpoint may provide valuable insights into the classification problem of abelian group actions with lower dynamical rank (see [Din12, Problem 1.5]).

2 Preliminaries

We start with notation and terminology. Let X be a smooth projective variety of dimension n over $\overline {\mathbf {Q}}$ . The symbols $\sim $ , $\approx $ and $\equiv $ stand for rational equivalence, algebraic equivalence and numerical equivalence for algebraic cycles, respectively. Abusing the notation, we also denote $\mathbf {Z}$ -, $\mathbf {Q}$ -, $\mathbf {R}$ -linear equivalence for divisors by $\sim $ , $\sim _{\mathbf {Q}}$ , $\sim _{\mathbf {R}}$ .

For any $0\leq i\leq n$ , denote by $\mathsf {CH}^i(X)$ the Chow group of algebraic cycles of codimension i on X modulo rational equivalence. It is well known in intersection theory that is a graded commutative ring with respect to the intersection product, called the Chow ring. When working with a coefficient field $\mathbf {K} = \mathbf {Q}$ or $\mathbf {R}$ , we write

In particular, when $i=1$ , the Chow group $\mathsf {CH}^1(X)$ coincides with the Picard group $\mathsf {Pic}(X)$ of X. Denote by $\mathsf {Pic}^0(X)$ the subgroup of $\mathsf {Pic}(X)$ consisting of all integral divisors on X algebraically equivalent to zero (modulo linear equivalence); it has a structure of an abelian variety. The quotient group

is called the Néron–Severi group of X, which is a finitely generated abelian group.

Let $\mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}$ denote the Picard variety of X over $\overline {\mathbf {Q}}$ , i.e., the neutral connected component of the Picard group scheme $\mathbf {Pic}_{X/\overline {\mathbf {Q}}}$ of X over $\overline {\mathbf {Q}}$ ; it is also the dual abelian variety of the Albanese variety $\operatorname {Alb}(X)$ of X (see [Kle05, Theorem 9.5.4 and Remark 9.5.25]). Note that the group of the $\overline {\mathbf {Q}}$ -points of the Picard variety $\mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}$ is exactly $\mathsf {Pic}^0(X)$ .

For any $0\le i\le n$ , denote by $\mathsf {N}^i(X)$ the finitely generated free abelian group of algebraic cycles of codimension i on X modulo numerical equivalence (cf. [Ful98, Definition 19.1]), i.e.,

For $\mathbf {K}=\mathbf {Q}$ or $\mathbf {R}$ , denote by the associated finite-dimensional $\mathbf {K}$ -vector space. It is also well known that when $i=1$ ,

see, for example, [Kle05, Theorem 9.6.3]. The Picard number $\rho (X)$ of X is defined as the rank of $\mathsf {N}^1(X)$ or $\dim _{\mathbf {R}} \mathsf {N}^1(X)_{\mathbf {R}}$ . We henceforth endow $\mathsf {N}^1(X)_{\mathbf {R}}$ with the standard Euclidean topology and fix a norm $\|\cdot \|$ on it.

A divisor D on X is nef if the intersection number $D\cdot C$ is nonnegative for any curve C on X. The cone of all nef $\mathbf {R}$ -divisors in $\mathsf {N}^1(X)_{\mathbf {R}}$ is called the nef cone $\mathsf {Nef}(X)$ of X, which is a salient closed convex cone of full dimension. Its interior is called the ample cone $\mathsf {Amp}(X)$ of X. An integral divisor D on X is big if the linear system $|mD|$ of some multiple of D induces a birational map $\Phi _{|mD|}$ from X onto its image. An $\mathbf {R}$ -divisor D is big if it is a positive combination of integral big divisors. It is well known that for any nef $\mathbf {R}$ -divisor D on X, it is big if and only if $D^n>0$ .

2.1 Weak numerical equivalence

It turns out to be convenient to consider the following notion (implicitly) introduced by Dinh–Sibony [DS04] and Zhang [Zha09].

Definition 2.1 (Weak numerical equivalence).

Let X be a smooth projective variety of dimension n over $\overline {\mathbf {Q}}$ . An algebraic cycle Z of codimension i on X is called weakly numerically trivial and denoted by $Z\equiv _{\mathsf {w}} 0$ if

$$\begin{align*}Z\cdot H_1\cdots H_{n-i} = 0 \end{align*}$$

for all ample (and hence for all) $\mathbf {R}$ -divisors $H_1,\ldots ,H_{n-i}$ on X.

Clearly, weak numerical equivalence is coarser than numerical equivalence. An important property of weak numerical equivalence is the following result due to Dinh and Sibony [DS04]. It essentially comes from the Hodge–Riemann bilinear relations and is crucial for deducing the nonvanishing of intersection numbers of divisors.

Lemma 2.2 (cf. [DS04, Lemme 4.4]).

Let $D_1,\dots ,D_j,D^{\prime }_j$ be nef $\mathbf {R}$ -divisors on X with $1\leq j\leq n-1$ such that $D_1 \cdots D_{j-1}\cdot D_j \not \equiv _{\mathsf {w}} 0$ and $D_1 \cdots D_{j-1}\cdot D^{\prime }_j \not \equiv _{\mathsf {w}} 0$ . Let f be an automorphism of X such that

$$ \begin{align*} f^*(D_1 \cdots D_{j-1}\cdot D_j) & \equiv_{\mathsf{w}} \lambda D_1 \cdots D_{j-1}\cdot D_j \ \text{ and} \\ f^*(D_1 \cdots D_{j-1}\cdot D^{\prime}_j) & \equiv_{\mathsf{w}} \lambda' D_1 \cdots D_{j-1}\cdot D^{\prime}_j \end{align*} $$

with positive real numbers $\lambda \neq \lambda '$ . Then $D_1 \cdots D_{j-1}\cdot D_j\cdot D^{\prime }_j \not \equiv _{\mathsf {w}} 0$ .

2.2 Stable and augmented base loci

In the course of our construction of a canonical height function associated with an abelian group G of automorphisms of X, we actually first construct a nef and big $\mathbf {R}$ -divisor on X. In dealing with the height functions of $\mathbf {R}$ -divisors, it becomes necessary to consider the so-called augmented base loci of $\mathbf {R}$ -divisors.

Definition 2.3 (Augmented base loci).

Let X be a smooth projective variety over $\overline {\mathbf {Q}}$ . The stable base locus $\mathbf {B}(D)$ of a $\mathbf {Q}$ -divisor D is the Zariski closed subset of X defined by

where $\operatorname {Bs}(mD)$ denotes the base locus of the linear system $|mD|$ . It is an elementary fact that there is an $M\geq 1$ such that $\mathbf {B}(D) = \operatorname {Bs}(MD)$ .

The augmented base locus $\mathbf {B}_+(D)$ of an $\mathbf {R}$ -divisor D is the Zariski closed subset of X defined by

where the intersection is taken over all ample $\mathbf {R}$ -divisors A such that $D-A$ is a $\mathbf {Q}$ -divisor.

For a detailed study of augmented base loci, we direct the reader to [ELM⁺06] and references therein. Here, we only state a few of them which will be utilized in the proofs of Theorem 1.1 and its corollaries.

Proposition 2.4 (cf. [ELM⁺06, Propositions 1.4 and 1.5, Example 1.7]).

Let X be a smooth projective variety over $\overline {\mathbf {Q}}$ . Then the following assertions hold.

1. (1) For any $\mathbf {R}$ -divisor D on X, it is big if and only if $\mathbf {B}_+(D)\neq X$ .

2. (2) If $D_1$ and $D_2$ are numerically equivalent $\mathbf {R}$ -divisors on X, then $\mathbf {B}_+(D_1) = \mathbf {B}_+(D_2)$ .

3. (3) For any $\mathbf {R}$ -divisor D on X, there is a positive number $\varepsilon $ such that for any ample $\mathbf {R}$ -divisor A with $\|A\|\leq \varepsilon $ and such that $D-A$ is a $\mathbf {Q}$ -divisor, $\mathbf {B}_+(D) = \mathbf {B}(D-A)$ .

Lesieutre and Satriano [LS21] observed that for two nef $\mathbf {R}$ -divisors $D_1,D_2$ on X, the augmented base locus of $\mathbf {B}_+(a_1D_1 + a_2D_2)$ is independent of the positive coefficients $a_1$ and $a_2$ . By induction, we can easily deduce the following.

Lemma 2.5 (cf. [LS21, Lemma 2.16]).

Let $D_1,\dots ,D_m$ be nef $\mathbf {R}$ -divisors on X. Then for any $a_1,\dots ,a_m>0$ , one has

$$\begin{align*}\mathbf{B}_+(a_1D_1 + \dots + a_mD_m) = \mathbf{B}_+(D_1 + \dots + D_m). \end{align*}$$

2.3 Weil height and canonical height

We refer to [HS00, Part B] for an introduction to Weil’s height theory. Among others, we collect some important facts from there.

Theorem 2.6 (cf. [HS00, Theorems B.3.2, B.3.6, and B.5.9] and [Kaw06b, Theorem 1.1.1]).

Let X be a smooth projective variety over $\overline {\mathbf {Q}}$ . Then there exists a unique homomorphism

satisfying the following properties.

1. (i) (Normalization) Let D be a very ample divisor on X and $\phi _D\colon X\hookrightarrow \mathbf {P}^N$ the associated embedding. Then we have

   $$\begin{align*}h_{X,D} = h\circ \phi_D + O(1), \end{align*}$$

   where h is the absolute logarithmic height on $\mathbf {P}^N$ (see [HS00, Definition, Page 176]).

2. (ii) (Functoriality) Let $\pi \colon X\to Y$ be a morphism of smooth projective varieties and $D_Y\in \mathsf {Pic}(Y)_{\mathbf {R}}$ . Then we have

   $$\begin{align*}h_{X,\pi^*\!{D_Y}} = h_{Y,D_Y}\circ \pi + O(1). \end{align*}$$

3. (iii) (Additivity) Let $D_1,D_2$ be $\mathbf {R}$ -divisors on X. Then we have

   $$\begin{align*}h_{X,D_1+D_2} = h_{X,D_1} + h_{X,D_2} + O(1). \end{align*}$$

4. (iv) (Positivity) Let D be an effective integral divisor on X. Then $h_{X,D} \geq O(1)$ outside the base locus $\operatorname {Bs}(D)$ of D.

5. (v) (Algebraic equivalence) Let $H_X, D\in \mathsf {Pic}(X)_{\mathbf {R}}$ be $\mathbf {R}$ -divisors with $H_X$ ample and D algebraically equivalent to zero. Then there is a constant $C>0$ such that

   $$\begin{align*}h_{X,D} \leq C \sqrt{h_{X,H_X}^+}, \end{align*}$$
   where .

Remark 2.7. (1) It is worth mentioning that the terms $O(1)$ only depend on varieties, divisors and morphisms but are independent of rational points of varieties. This is why we omit the points $x\in X(\overline {\mathbf {Q}})$ in various height equations. See [HS00, Remarks B.3.2.1(ii)].

(2) When the ambient variety X is clear, we often use $h_D$ to stand for $h_{X,D}$ for simplicity.

The following finiteness property, originally established by Northcott for integral ample divisors, becomes a fundamental tool in Weil’s height theory (see [HS00, Theorem B.3.2(g)] and [Kaw06b, Theorem 1.1.2]). Lesieutre and Satriano proved a version for big $\mathbf {R}$ -divisors (see [LS21, Lemma 2.26]). For the reader’s convenience, we restate it here.

Theorem 2.8 (Northcott finiteness property).

Let X be a smooth projective variety over a number field K and D a big $\mathbf {R}$ -divisor on X. Then for any $d\in \mathbf {Z}_{>0}$ and $T\in \mathbf {R}$ , the set

$$\begin{align*}\big\{ x\in (X{\setminus}\mathbf{B}_+(D))(\overline{\mathbf{Q}}) : [K(x):K] \leq d, \, h_D(x)\leq T \big\} \end{align*}$$

is finite, where $\mathbf {B}_+(D)$ denotes the augmented base locus of D. In particular, if D is an ample $\mathbf {R}$ -divisor, then $\mathbf {B}_+(D)=\emptyset $ , and this is the usual Northcott finiteness property.

Definition 2.9 (Bounded height).

Fix an ample divisor $H_X$ on a smooth projective variety X over $\overline {\mathbf {Q}}$ . A subset $S\subseteq X(\overline {\mathbf {Q}})$ is called a set of bounded height if there is a constant C such that $h_{H_X}(s)\leq C$ for all $s\in S$ . This property is independent of the choice of the ample divisor $H_X$ .

The Northcott finiteness property Theorem 2.8 implies that if X is a smooth projective variety over a number field K and $S\subseteq X(\overline {\mathbf {Q}})$ is a set of bounded height, then $\{s\in S : [K(s) : K]\leq d\}$ is finite for any positive integer d.

In what follows, we recall a classical result on the canonical height associated with a single endomorphism due to Call and Silverman [CS93, Theorem 1.1], though in the book [HS00] the authors still attributed it to Néron and Tate.

Theorem 2.10 (cf. [HS00, Theorem B.4.1]).

Let $f\colon X \to X$ be an endomorphism of a smooth projective variety X over $\overline {\mathbf {Q}}$ . Let D be an $\mathbf {R}$ -divisor on X such that $f^*D \sim _{\mathbf {R}} \alpha D$ for some $\alpha>1$ . Then there is a unique function $\widehat {h}_{D,f}\colon X(\overline {\mathbf {Q}})\to \mathbf {R}$ , called the canonical height on X with respect to f and D, satisfying the following properties:

1. (1) $\widehat {h}_{D,f} = h_{D} + O(1)$ ;

2. (2) $\widehat {h}_{D,f}\circ f = \alpha \, \widehat {h}_{D,f}$ .

Moreover, the function $\widehat {h}_{D,f}$ is constructed by the limit

Remark 2.11. In the above Theorem 2.10, if we replace $\mathbf {R}$ -linear equivalence with algebraic or numerical equivalence – often more practical to verify – Kawaguchi and Silverman introduced a ‘canonical height’ in a similar manner. Specifically, they showed that given $f^*D^{\prime } \equiv \beta D^{\prime }$ for some $\mathbf {R}$ -divisor $D^{\prime }$ and some $\beta> \sqrt {\lambda _1(f)}$ , the limit

exists and satisfies the following properties:

1. (1) $\widehat {h}_{D^{\prime },f} = h_{D^{\prime }} + O\big (\sqrt {h_{H}^+}\big )$ ;

2. (2) $\widehat {h}_{D^{\prime },f}\circ f = \beta \, \widehat {h}_{D^{\prime },f}$ .

See [KS16b, Theorem 5]. Its proof relies on their height estimate [KS16b, Theorem 24]. Yet, the correct proof of this height estimate is due to Matsuzawa (see [Mat20b, Theorem 1.4]). Further, we remark that if one needs to control the term $O(\sqrt {h_{H}^+})$ , this height estimate is inevitable (see, for example, Lemmas 3.13 and 4.1).

2.4 Dynamical degrees and arithmetic degrees

In this subsection, we let $f\colon X\to X$ be a surjective endomorphism of a smooth projective variety X of dimension n over $\overline {\mathbf {Q}}$ and $H_X$ an ample divisor on X. We recall the definitions of dynamical degrees and arithmetic degrees of this algebraic dynamical system $(X,f)$ , as well as the Kawaguchi–Silverman conjecture which reveals the relationship between these two dynamical invariants.

Definition 2.12 (Dynamical degrees).

For each $0\leq k\leq n$ , the k-th dynamical degree of f is defined by

It is well known that this definition is equivalent to the one given in the introduction. Dynamical degrees are invariant under generically finite rational maps and independent of the choice of the ample $H_X$ . They also satisfy the log-concavity property as follows:

$$\begin{align*}\lambda_{k}(f)^2\geq \lambda_{k-1}(f)\lambda_{k+1}(f) \text{ for } 1\leq k\leq n-1. \end{align*}$$

We refer to [DS05, Tru20, Dan20, HT21] for more detailed discussions on dynamical degrees. The algebraic entropy of f is defined by

Definition 2.13 (Arithmetic degrees).

Let $h_{H_X}$ be an absolute logarithmic Weil height function associated with $H_X$ . Set $h_{H_X}^+ = \max (1,h_{H_X})$ . For each $x\in X(\overline {\mathbf {Q}})$ , we define the arithmetic degree of f at x by

It is known that the limit always exists and is also independent of the choice of the ample divisor $H_X$ (see [KS16a, Theorem 3] and [KS16b, Proposition 12], respectively).

The following conjecture due to Kawaguchi and Silverman asserts the properties of arithmetic degrees. Throughout this paper, we shall only consider this conjecture for automorphisms. We refer to [KS16b, Conjecture 6] for a general version of dominant rational self-maps.

Conjecture 2.14 (cf. [KS16b, Conjecture 6]).

Let $f\colon X\to X$ be a surjective endomorphism of a smooth projective variety X over $\overline {\mathbf {Q}}$ . Then for any point $x\in X(\overline {\mathbf {Q}})$ , if the forward f-orbit of x is Zariski dense in X, then

$$\begin{align*}\alpha_f(x)=\lambda_1(f). \end{align*}$$

3 Construction of distinguished automorphisms and divisors

This section is devoted to the construction of n distinguished automorphisms in the rank $n-1$ free abelian subgroup $G\leq \operatorname {Aut}(X)$ of positive entropy and n common nef $\mathbf {R}$ -divisors whose sum is a big $\mathbf {R}$ -divisor (see Theorem 3.6). Hinging on them, we shall define a canonical height function in Section 4. As a by-product, we prove Corollary 1.6 at the end of this section.

3.1 Commuting families of linear maps preserving cones

Throughout this subsection, V is a finite-dimensional topological $\mathbf {R}$ -vector space, and C is a salient closed convex cone in V of full dimension (i.e., $C\cap (-C)=\{0\}$ and C spans V, respectively). We recall a few facts on them from linear algebra which are crucial to the construction in the next §3.2. The following is Garrett Birkhoff’s generalization of the classical Perron–Frobenius theorem.

Theorem 3.1 (cf. [Bir67]).

Let $\varphi \in \operatorname {End}(V)$ be an $\mathbf {R}$ -linear endomorphism of V such that C is $\varphi $ -invariant (i.e., $\varphi (C) \subseteq C$ ). Then the spectral radius $\rho (\varphi )$ is an eigenvalue of $\varphi $ , and there is an eigenvector $v_\varphi \in C$ of $\varphi $ associated with $\rho (\varphi )$ .

It is well known that a commuting family $\mathcal {F} \subseteq \operatorname {M}_n(\mathbf {C})$ of complex matrices possesses a nonzero common eigenvector $v \in \mathbf {C}^n$ . Below is its analog when $\mathcal {F}$ is a family of real matrices preserving a salient closed convex cone of full dimension. It is essentially due to Dinh and Sibony (see [DS04, Proposition 4.1]).

Proposition 3.2. Let $\mathcal {F} \subseteq \operatorname {End}(V)$ be a commuting family of $\mathbf {R}$ -linear endomorphisms of V such that C is $\mathcal {F}$ -invariant (i.e., $\psi (C) \subseteq C$ for any $\psi \in \mathcal {F}$ ). Then for any $\varphi \in \mathcal {F}$ , there exists a nonzero vector $v_\varphi \in C$ such that

1. (1) for any $\psi \in \mathcal {F}$ , $\psi (v_\varphi ) \in \mathbf {R}_{\geq 0} \cdot v_\varphi $ (i.e., $v_\varphi $ is a common eigenvector for all $\psi \in \mathcal {F}$ associated with some nonnegative eigenvalues); and moreover,

2. (2) $\varphi (v_\varphi ) = \rho (\varphi )v_\varphi $ .

Proof. Let $\varphi \in \mathcal {F}$ be fixed. By Birkhoff’s Theorem 3.1, the spectral radius $\rho (\varphi )$ of $\varphi $ is an eigenvalue of $\varphi $ , and the corresponding eigenvector can be chosen to lie in C. In particular,

is a nonzero salient closed convex cone in V. It is easy to see that $C_\varphi $ is $\mathcal {F}$ -invariant. Indeed, for any $\psi \in \mathcal {F}$ and any $v \in C_\varphi \subseteq C$ , by assumption that C is $\mathcal {F}$ -invariant, we have $\psi (v) \in C$ . Then it follows from the commutativity of $\mathcal {F}$ that

$$\begin{align*}\varphi(\psi(v)) = \psi(\varphi(v)) = \psi(\rho(\varphi)v) = \rho(\varphi) \psi(v). \end{align*}$$

Hence $\psi (v) \in C_\varphi $ by the definition of $C_\varphi $ .

Denote by $V_\varphi $ the $\mathbf {R}$ -vector subspace of V spanned by $C_\varphi $ . Clearly, $V_\varphi $ is nonzero, contained in the eigenspace of $\varphi $ associated with the eigenvalue $\rho (\varphi )$ , and $\mathcal {F}$ -invariant, since so is $C_\varphi $ . It suffices to show that there exists a common eigenvector $v_\varphi \in C_\varphi $ for any $\psi \in \mathcal {F}$ . In other words, let $\psi $ be arbitrary in $\mathcal {F}$ and denote

which is a nonzero salient closed (possibly nonconvex) cone in $V_\varphi $ . Then it remains to show that

(3.1) $$ \begin{align} \bigcap_{\psi \in \mathcal{F}} \widetilde{C}_\psi \neq \{0\}, \end{align} $$

or equivalently, in the quotient space (think of $\mathbf {R}_{>0}$ as the multiplicative subgroup of ),

(3.2) $$ \begin{align} \bigcap_{\psi \in \mathcal{F}} \mathbf{P}^+(\widetilde{C}_\psi) \neq \emptyset, \end{align} $$

where each $\mathbf {P}^+(\widetilde {C}_\psi )$ denotes the image of $\widetilde {C}_\psi {\setminus }\{0\}$ under the natural quotient map

$$\begin{align*}\pi\colon V_\varphi{\setminus}\{0\} \to \mathbf{P}^+(V_\varphi). \end{align*}$$

Note that $\mathbf {P}^+(V_\varphi )$ endowed with the quotient topology is homeomorphic with the $(\dim V_\varphi -1)$ -sphere and hence compact. Moreover, $\mathbf {P}^+(\widetilde {C}_\psi )$ is closed in $\mathbf {P}^+(V_\varphi )$ since so is $\widetilde {C}_{\psi }{\setminus } \{0\}$ in $V_\varphi {\setminus }\{0\}$ . We are thus reduced to show Equation (3.2) or Equation (3.1) for any finite $\mathcal {F}'\subseteq \mathcal {F}$ .

Suppose now that $\mathcal {F}' = \{\psi _0,\psi _1,\ldots ,\psi _m\}$ is any fixed finite subset of $\mathcal {F}$ . By adding the fixed endomorphism $\varphi $ to $\mathcal {F}'$ , if necessary, we may assume that $\psi _0=\varphi $ . Let and . We shall inductively construct pairs $(V_k, C_k)$ , $0 \leq k \leq m$ , satisfying the following properties:

1. (i) $V_0 \supseteq V_1 \supseteq \dots \supseteq V_m \neq \{0\}$ is a decreasing sequence of nonzero $\mathcal {F}$ -invariant $\mathbf {R}$ -vector subspaces of V;

2. (ii) $C_0 \supseteq C_1 \supseteq \dots \supseteq C_m \neq \{0\}$ is a decreasing sequence of nonzero $\mathcal {F}$ -invariant salient closed convex cones in V;

3. (iii) for each $0 \leq k \leq m$ , $C_k$ spans $V_k$ and

   $$\begin{align*}C_k \subseteq \bigcap_{i=0}^k \widetilde{C}_{\psi_i}. \end{align*}$$

As an immediate consequence of this construction, one gets $\bigcap _{i=0}^m \widetilde {C}_{\psi _i} \neq \{0\}$ since it contains the nonzero cone $C_m$ , which completes the proof of Proposition 3.2.

By the definition of $\widetilde {C}_\varphi $ , one has $C_\varphi = \widetilde {C}_{\varphi } = \widetilde {C}_{\psi _0}$ . The assertion for $(V_0,C_0)$ is hence true. By the inductive hypothesis, suppose that we have constructed pairs $(V_i, C_i)$ for all $0 \leq i \leq k-1$ with $1 \leq k \leq m$ . We then construct $(V_k, C_k)$ satisfying all properties i to iii. Note that $V_{k-1}$ and the spanning cone $C_{k-1}$ are both $\mathcal {F}$ -invariant and hence $\psi _k$ -invariant. It follows from Theorem 3.1, applied to the triplet $(V_{k-1}, \psi _k|_{V_{k-1}}, C_{k-1})$ , that

is a nonzero salient closed convex cone in $V_{k-1}$ . Again by the commutativity of $\mathcal {F}$ and the $\mathcal {F}$ -invariance of $C_{k-1}$ , we see that $C_k$ is $\mathcal {F}$ -invariant. Let $V_k$ denote the $\mathbf {R}$ -vector subspace of $V_{k-1}$ spanned by $C_k$ . Then $V_k$ is also $\mathcal {F}$ -invariant. Moreover, as $C_k \subseteq C_{k-1} \cap \widetilde {C}_{\psi _k}$ by construction, the property iii for $C_k$ follows by inductive hypothesis.

Remark 3.3. Note that in the above Proposition 3.2, if we replace $\mathcal {F} \subseteq \operatorname {End}(V)$ with an abelian subgroup $\mathcal {G}$ of the $\mathbf {R}$ -linear automorphism group $\mathrm {GL}(V)$ of V, then for any $\varphi \in \mathcal {G}$ , there is a common eigenvector $v_\varphi \in C$ such that for any $\psi \in \mathcal {G}$ , $\psi (v_\varphi ) \in \mathbf {R}_{>0} \cdot v_\varphi $ and $\varphi (v_\varphi ) = \rho (\varphi )v_\varphi $ . It gives rise to a multiplicative group character $\chi _\varphi \colon \mathcal {G} \to (\mathbf {R}_{>0}, \times )$ defined by $\psi (v_\varphi ) = \chi _\varphi (\psi )v_\varphi $ for any $\psi \in \mathcal {G}$ ; the character $\chi _\varphi $ also satisfies that $\chi _\varphi (\varphi )=\rho (\varphi )$ .

A priori, it is still unknown that for different $\varphi $ and $\varphi '$ , the above characters $\chi _{\varphi }$ and $\chi _{\varphi '}$ are distinct, neither the uniqueness of $\chi _\varphi $ for each $\varphi $ . Nonetheless, the following lemma shows that there are at most $\dim _{\mathbf {R}} V$ distinct characters of $\mathcal {G}$ constructed via common eigenvectors in C.

Lemma 3.4. Let $\mathcal {G}$ be an abelian subgroup of $\mathrm {GL}(V)$ such that C is $\mathcal {G}$ -invariant. As in Remark 3.3, let $\{\chi _1, \dots , \chi _m\}$ denote the set of all distinct multiplicative group characters of $\mathcal {G}$ , where each $\chi _i$ is associated with some (mutually noncolinear) common eigenvectors $v_i \in C$ ; that is, for any $\psi \in \mathcal {G}$ and any $1\leq i\leq m$ ,

$$\begin{align*}\psi(v_i) = \chi_i(\psi)v_i. \end{align*}$$

Then the above $v_1,\dots ,v_m$ are linearly independent; in particular, $m\leq \dim _{\mathbf {R}} V$ holds. Moreover, for any $\varphi \in \mathcal {G}$ , there exists some $1 \leq i \leq m$ such that $\chi _i(\varphi ) = \rho (\varphi )$ , i.e., $\varphi (v_i) = \rho (\varphi )v_i$ .

Proof. We prove the first half by induction. Suppose by inductive hypothesis that $v_1,\dots ,v_{k}$ are linearly independent for $1\leq k\leq m-1$ . Suppose that we have

(3.3) $$ \begin{align} \sum_{i=1}^{k+1} a_i v_i = 0 \end{align} $$

for some $a_i \in \mathbf {R}$ . Fix an arbitrary index j with $1\leq j\leq k$ . Since $\chi _{k+1} \neq \chi _j$ , there is some $\psi _\circ \in \mathcal {G}$ such that $\chi _{k+1}(\psi _\circ ) \neq \chi _{j}(\psi _\circ )$ . Note that for any $1\leq i\leq k+1$ , one has $\psi _\circ (v_i) = \chi _i(\psi _\circ )v_i$ by definition. Hence, applying the above $\psi _\circ $ to Equation (3.3) yields that

(3.4) $$ \begin{align} \sum_{i=1}^{k+1} a_i \chi_i(\psi_\circ)v_i = 0. \end{align} $$

Using the above two Equations (3.3) and (3.4) to cancel the coefficient of $v_{k+1}$ , we obtain that

$$\begin{align*}\sum_{i=1}^{k} a_i(\chi_{k+1}(\psi_\circ)-\chi_i(\psi_\circ))v_i = 0. \end{align*}$$

Since $v_1,\dots ,v_k$ are linearly independent, $a_i(\chi _{k+1}(\psi _\circ )-\chi _i(\psi _\circ )) = 0$ for all $1\leq i\leq k$ . In particular, $a_j(\chi _{k+1}(\psi _\circ )-\chi _j(\psi _\circ )) = 0$ , and hence, $a_j=0$ . We thus prove the linear independence of $v_1,\dots ,v_m$ by induction.

For the second half, suppose to the contrary that there exists some $\varphi _\circ \in \mathcal {G}$ such that $\chi _i(\varphi _\circ ) \neq \rho (\varphi _\circ )$ for all $1\leq i\leq m$ . Then by Proposition 3.2, there is a group character $\chi _{\varphi _\circ }$ of $\mathcal {G}$ associated with some common eigenvector $v_{\varphi _\circ } \in C$ such that $\chi _{\varphi _\circ }(\varphi _\circ ) = \rho (\varphi _\circ )$ , i.e., $\varphi _\circ (v_{\varphi _\circ }) = \rho (\varphi _\circ )v_{\varphi _\circ }$ . Clearly, this new character $\chi _{\varphi _\circ }$ is different from any other $\chi _i$ , a contradiction.

At the end of this subsection, we provide an auxiliary module-theoretic result that will be used in the proof of Lemma 3.13 to strengthen eigenequations of $\mathbf {R}$ -divisors modulo numerical equivalence to equations modulo $\mathbf {R}$ -linear equivalence.

Lemma 3.5. Let $R\subseteq S$ be integral domains, M an (unnecessarily finitely generated) R-module and $\varphi \colon M\to M$ an R-linear map such that $P(\varphi )=0$ for some polynomial $P(t)\in R[t]$ . Denote the field of fractions of S by K. Let be the vector space over K and the induced K-linear map on $M_K$ . Let $s\in S$ such that $P(s)\neq 0$ in S. Then $\varphi _K-s\operatorname {id}_{M_K}$ is an isomorphism of $M_K$ .

Proof. We first prove that $\varphi _K-s\operatorname {id}_{M_K}$ is injective. Let $v\in M_K$ such that $(\varphi _K-s\operatorname {id}_{M_K})(v)=0$ , i.e., $\varphi _K(v) = s v$ . It is easy to verify that $P(\varphi _K)=(P(\varphi ))_K=0$ . Then we have

$$\begin{align*}0 = P(\varphi_{K})(v) = P(s) v. \end{align*}$$

As $P(s)\neq 0$ is invertible in K, it follows that $v=0$ .

We next show that $\varphi _K-s\operatorname {id}_{M_K}$ is also surjective. Denote the degree of the polynomial $P(t)$ by n. Let $w\in M_K$ be arbitrary. As $P(\varphi _K)(w)=0$ , there exist $c_0,\dots ,c_{n-1}\in K$ (or rather, in the field of fractions of R), such that

$$\begin{align*}\varphi_K^n(w) = c_{n-1}\varphi_K^{n-1}(w) + \cdots + c_1\varphi_K(w) + c_0 w. \end{align*}$$

In other words, the vector space $B_w$ generated by $\{w,\varphi _K(w),\varphi _K^2(w),\dots \}$ over K is a finite-dimensional $\varphi _K$ -invariant subspace of $M_K$ . It follows that the restriction map $(\varphi _K-s\operatorname {id}_{M_K})|_{B_w}$ of $\varphi _K-s\operatorname {id}_{M_K}$ on $B_w$ is also injective. As $B_w$ is of finite dimension, $(\varphi _K-s\operatorname {id}_{M_K})|_{B_w}$ is surjective. Since $w\in M_K$ is arbitrary, we see that $\varphi _K-s\operatorname {id}_{M_K}$ itself is surjective.

3.2 Automorphisms and divisors associated with abelian groups of maximal dynamical rank

Below is the main result of this section. Given an abelian subgroup $\mathbf {Z}^{n-1}\cong G\leq \operatorname {Aut}(X)$ of positive entropy, we construct n distinguished automorphisms in G and a nef and big $\mathbf {R}$ -divisor associated with G. This construction forms a crucial ingredient for defining a canonical height in Section 4 (see Theorem 4.2).

Theorem 3.6. Let X be a smooth projective variety of dimension $n\geq 2$ defined over $\overline {\mathbf {Q}}$ and $G \cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X of positive entropy. Then the following assertions hold.

1. (1) The set

   $$\begin{align*}\bigg\{\chi\colon G\to (\mathbf{R}_{>0},\times)\,\bigg|\, \begin{array}{ll} \chi\text{ is a group homomorphism and there is a nef }\mathbf{R}\text{-divisor}\\ D\not\equiv 0 \text{ such that }g^*D\equiv \chi(g)D \text{ for any }g\in G \end{array} \bigg\} \end{align*}$$

   has n elements $\chi _i$ with $1\leq i \leq n$ , where each $\chi _i$ is associated with some common nef eigendivisor $D_i$ (i.e., $g^*D_i \equiv \chi _i(g)D_i$ for any $g\in G$ ).

2. (2) $D_1\cdots D_n \in \mathbf {R}_{>0}$ ; in particular, is a nef and big $\mathbf {R}$ -divisor on X.

3. (3) For any $g\in G$ , there is some $1\leq i\leq n$ such that $\chi _{i}(g) = \lambda _1(g)$ .

4. (4) There exist n automorphisms $g_1,\dots ,g_{n} \in G$ such that for any $1\leq i\neq j\leq n$ , we have $\chi _j(g_i)<1$ , and moreover,

   $$\begin{align*}g_i^*D_i\sim_{\mathbf{R}} \chi_i(g_i)D_i = \lambda_1(g_i)D_i. \end{align*}$$

Remark 3.7. The construction of the common nef eigendivisors $D_i$ is essentially attributed to Dinh and Sibony, who initially constructed $n-1$ of them and then separately constructed the last one (see the last paragraph of the proof of [DS04, Théorème 4.4]). This separateness is a crucial aspect, preventing a direct use of their construction to define a canonical height in an appropriate way. Upon revisiting [DS04] and examining each character $\chi _i\colon G\to (\mathbf {R}_{>0},\times )$ , we demonstrate that there exist n distinguished automorphisms $g_1,\dots ,g_n$ in the rank $n-1$ abelian group G such that $\chi _i(g_i)=\lambda _1(g_i)$ and $\chi _j(g_i)<1$ for all $j\neq i$ . Consequently, all n nef eigendivisors $D_i$ share the same status as in Theorem 3.6. It is worth mentioning that the automorphisms $g_1,\dots ,g_n$ we constructed may not necessarily be the generators of G. Instead, any $n-1$ of them generate a finite index subgroup of G (see Remark 3.15).

Before proving the above Theorem 3.6 at the end of this subsection, we prepare all necessary ingredients. Recall that $\mathsf {N}^1(X)_{\mathbf {R}}$ is the real Néron–Severi space of $\mathbf {R}$ -divisors on X modulo numerical equivalence $\equiv $ ; its real dimension $\dim _{\mathbf {R}} \mathsf {N}^1(X)_{\mathbf {R}}$ is called the Picard number of X, denoted by $\rho (X)$ . The nef cone $\mathsf {Nef}(X)$ , consisting of the classes of all nef $\mathbf {R}$ -divisors on X, is a salient closed convex cone in $\mathsf {N}^1(X)_{\mathbf {R}}$ of full dimension. The pullback action of automorphisms on $\mathsf {N}^1(X)_{\mathbf {R}}$ induces a natural representation of $\operatorname {Aut}(X)$ :

$$ \begin{align*} \operatorname{Aut}(X) \to \mathrm{GL}(\mathsf{N}^1(X)_{\mathbf{R}}), \quad g \mapsto g^*|_{\mathsf{N}^1(X)_{\mathbf{R}}}. \end{align*} $$

Note that any automorphism preserves the nef cone $\mathsf {Nef}(X) \subseteq \mathsf {N}^1(X)_{\mathbf {R}}$ . For any subgroup G of $\operatorname {Aut}(X)$ , denote by $G|_{\mathsf {N}^1(X)_{\mathbf {R}}} \leq \mathrm {GL}(\mathsf {N}^1(X)_{\mathbf {R}})$ the image of the above representation.

First of all, as a straightforward application of the previous discussion in §3.1 to the triplet $(\mathsf {N}^1(X)_{\mathbf {R}}, G|_{\mathsf {N}^1(X)_{\mathbf {R}}}, \mathsf {Nef}(X))$ , we get some nonzero common nef eigendivisors $D_i$ that naturally define multiplicative group characters $\chi _i$ of $G|_{\mathsf {N}^1(X)_{\mathbf {R}}}$ . Composing them with the group homomorphism $G \to G|_{\mathsf {N}^1(X)_{\mathbf {R}}}$ yields group characters of G itself, still denoted by $\chi _i$ . In summary, we obtain the following.

Proposition 3.8. Let X be a smooth projective variety of dimension $n\geq 2$ defined over $\overline {\mathbf {Q}}$ and G an abelian subgroup of $\operatorname {Aut}(X)$ . Let m be the number of all distinct multiplicative group characters $\chi _i\colon G\to (\mathbf {R}_{>0},\times ) \, (1\leq i\leq m)$ , where $\chi _i$ is associated with some common nef eigendivisor $D_i$ (i.e., $g^*D_i\equiv \chi _i(g)D_i$ for any $g\in G$ ). Then $1\leq m\leq \rho (X)$ holds. Further, for any $g\in G$ , there is some $1\leq i\leq m$ such that $\chi _i(g) = \rho (g^*|_{\mathsf {N}^1(X)_{\mathbf {R}}}) = \lambda _1(g)$ .

Proof. It follows readily from Proposition 3.2, Remark 3.3 and Lemma 3.4.

By the above Proposition 3.8, following [DS04], we define a group homomorphism

$$ \begin{align*} \pi \colon G & \to (\mathbf{R}^{m}, +) \\ g & \mapsto (\log\chi_1(g),\dots,\log\chi_m(g)). \end{align*} $$

Lemma 3.9 (cf. [DS04, Corollaire 2.2]).

Let X be a smooth projective variety of dimension $n\geq 2$ defined over $\overline {\mathbf {Q}}$ . Then the set of the first dynamical degrees of surjective endomorphisms of X is discrete in $[1,+\infty )$ .

Proof. It suffices to show that for any $M>1$ , the following set

is finite. We note that the first dynamical degree $\lambda _1(f)$ of a surjective endomorphism $f\colon X\to X$ is the spectral radius of $f^*|_{\mathsf {N}^1(X)_{\mathbf {R}}}$ , which is induced from $f^*|_{\mathsf {N}^1(X)}$ . Since $\mathsf {N}^1(X)$ is a free abelian group of rank , all the eigenvalues of $f^*|_{\mathsf {N}^1(X)_{\mathbf {R}}}$ are algebraic integers. That is, every $\lambda _1(f)$ is the maximal modulus of the roots of a monic polynomial of degree $\rho $ with integer coefficients. Let $P(t) = t^\rho + c_1t^{\rho -1} + \cdots + c_\rho \in \mathbf {Z}[t]$ be such a polynomial that the maximal modulus of all roots $\alpha _1,\dots ,\alpha _\rho $ , counting with multiplicities, is no more than M. It is well known that each $c_i=(-1)^i s_i(\alpha _1,\dots ,\alpha _\rho )$ , where $s_i$ is the i-th elementary symmetric polynomial. Since the $|\alpha _i|$ are bounded and $\rho $ is fixed, the $|c_i|$ are also bounded. In particular, $\#S_M$ is finite. We finish the proof of the lemma.

From now on, we consider abelian groups G of automorphisms of positive entropy. Recall that a lattice $\Gamma $ in an $\mathbf {R}$ -vector space V is a (possibly non-cocompact) discrete free subgroup. It is called complete if its rank equals the dimension of V.

Proposition 3.10 (cf. [DS04, Proposition 4.2]).

Let $G\leq \operatorname {Aut}(X)$ be an abelian group of automorphisms of positive entropy. Then $\pi $ is injective, and its image $\pi (G)$ is discrete in $\mathbf {R}^m$ . In particular, G is free abelian, and $\pi (G)$ is a lattice in $\mathbf {R}^m$ of rank $r \leq m$ .

Proof. For any $g\in G{\setminus } \{\operatorname {id}\}$ , it follows from Proposition 3.8 that one of the coordinates of $\pi (g)$ coincides with $\log \lambda _1(g)$ which is positive; hence, $\pi $ is injective. As $\pi $ is a group homomorphism, to show that the image $\pi (G)$ is discrete, it is sufficient to show that $\pi (\operatorname {id}) = 0$ is an isolated point in the image $\pi (G)$ . Applying Lemma 3.9, we see for each $g\in G{\setminus } \{\operatorname {id}\}$ that $\log \lambda _1(g)$ has a uniform lower bound (which is independent of g) and hence $\{0\}$ is an isolated point in $\pi (G)$ . Note that the image $\pi (G)$ is an additive subgroup which is also discrete in $\mathbf {R}^m$ . Hence, $\pi (G)$ is a lattice in $\mathbf {R}^m$ . Because $\pi $ is injective, G is free abelian of rank $r\leq m$ .

The lemma below provides a more accurate range of the number m of all distinct multiplicative group characters $\chi _i$ of G (cf. Proposition 3.8).

Lemma 3.11. Let $G\leq \operatorname {Aut}(X)$ be a free abelian group of rank r of positive entropy. Let $\chi _1,\dots ,\chi _m$ and $D_1,\dots ,D_m$ be in Proposition 3.8. Then we have

$$\begin{align*}D_1\cdots D_m\not\equiv_{\mathsf{w}} 0 \text{ and } r+1 \leq m \leq \min(n, \rho(X)). \end{align*}$$

Proof. We first prove that $D_{i_1}\cdots D_{i_k} \not \equiv _{\mathsf {w}} 0$ for any multi-index $1\leq i_1 < \dots < i_k \leq \min (m,n)$ by induction. By the higher-dimensional Hodge index theorem (see, for example, [Zha16, Lemma 3.2]), we have $D_i \not \equiv _{\mathsf {w}} 0$ for each $1\leq i\leq m$ . Suppose that the intersection product of any $j\leq k-1$ different divisors choosing from $D_1,\dots ,D_m$ is not weakly numerically trivial. Fix a multi-index $1\leq i_1 < \dots < i_j < i_{j+1} \leq \min (m,n)$ . By inductive hypothesis, we have

$$\begin{align*}D_{i_1}\cdots D_{i_{j-1}} \cdot D_{i_j} \not\equiv_{\mathsf{w}} 0 \text{ and } D_{i_1}\cdots D_{i_{j-1}} \cdot D_{i_{j+1}} \not\equiv_{\mathsf{w}} 0. \end{align*}$$

Since the $\chi _i$ are distinct, one has $\chi _{i_j}\neq \chi _{i_{j+1}}$ , i.e., there is some $g_\circ \in G$ such that $\chi _{i_j}(g_\circ )\neq \chi _{i_{j+1}}(g_\circ )$ . It follows from G-invariance of the $D_i$ that

$$ \begin{align*} g_\circ^*(D_{i_1}\cdots D_{i_{j-1}} \cdot D_{i_j}) & \equiv_{\mathsf{w}} \chi_{i_1}(g_\circ)\cdots \chi_{i_{j-1}}(g_\circ) \, \chi_{i_j}(g_\circ) (D_{i_1}\cdots D_{i_{j-1}} \cdot D_{i_j}) \ \text{ and} \\ g_\circ^*(D_{i_1}\cdots D_{i_{j-1}}\cdot D_{i_{j+1}}) & \equiv_{\mathsf{w}} \chi_{i_1}(g_\circ)\cdots \chi_{i_{j-1}}(g_\circ) \, \chi_{i_{j+1}}(g_\circ) (D_{i_1}\cdots D_{i_{j-1}}\cdot D_{i_{j+1}}). \end{align*} $$

Noting that $j\leq k-1\leq n-1$ by assumption, one has $D_{i_1}\cdots D_{i_j} \cdot D_{i_{j+1}} \not \equiv _{\mathsf {w}} 0$ , thanks to Lemma 2.2. In other words, we have proved by induction that the product of any $k\leq \min (m,n)$ different divisors from $D_1,\dots ,D_m$ is not weakly numerically trivial.

By Proposition 3.10 and Lemma 3.4, we already know that $r \leq m \leq \rho (X)$ . Suppose to the contrary that there are $m\geq n+1$ distinct group characters $\chi _1,\dots ,\chi _{n+1},\dots ,\chi _m$ of G associated with some common nef eigendivisors $D_1,\dots ,D_m$ . By what we just proved, one has $D_1\cdots D_n>0$ and $D_2\cdots D_{n+1}>0$ . Then by the projection formula, we have for any $g\in G$ ,

$$\begin{align*}0\neq D_1\cdots D_n = \deg(g)(D_1\cdots D_n) = g^*D_1\cdots g^*D_n=\chi_1(g)\cdots\chi_n(g)(D_1\cdots D_n), \end{align*}$$

and hence, $\chi _1(g)\cdots \chi _n(g) = 1$ . Similarly, we also have $\chi _2(g)\cdots \chi _{n+1}(g) = 1$ , so that $\chi _1(g) = \chi _{n+1}(g)$ , a contradiction. So we get $m\leq n$ , and hence, $D_1\cdots D_m\not \equiv _{\mathsf {w}} 0$ , as desired.

It remains to show that $m\geq r+1$ . Suppose to the contrary that $m=r$ . Then $\pi (G)$ is a complete lattice in $\mathbf {R}^m$ (see Proposition 3.10). Namely, $\pi (G)$ spans $\mathbf {R}^m$ . Therefore, there is some $g_\diamond \in G{\setminus } \{\operatorname {id}\}$ such that all m coordinates of $\pi (g_\diamond )$ are negative (i.e., $\chi _i(g_\diamond )<1$ for all $1\leq i\leq m$ ). On the other hand, Proposition 3.8 asserts that for such $g_\diamond $ , there is some $1\leq i_\diamond \leq m$ such that $\chi _{i_\diamond }(g_\diamond ) = \lambda _1(g_\diamond )> 1$ , since $g_\diamond $ is of positive entropy. This is a contradiction.

Remark 3.12. Note that all these common nef eigendivisors $D_i$ in Proposition 3.8 are constructed numerically. Namely, they only satisfy eigenequations modulo numerical equivalence. Though it is enough to define nef canonical height functions in the sense of [KS16b, Theorem 5(a)], the difference may not be bounded (see [KS16b, Theorem 5(b)]). In dimension two, Kawaguchi [Kaw08, Lemma 3.8] managed to improve them to eigenequations modulo $\mathbf {R}$ -linear equivalence so that the difference is indeed bounded. Such an eigenequation modulo linear equivalence also appears in [Zha06, Proposition 1.1.3], [NZ10, Lemma 2.3] and [MMS+22, Theorem 6.4(1)].

Below is a higher-dimensional analog of [Kaw08, Lemma 3.8], which will be used in the proof of Theorem 3.6(4) shortly.

Lemma 3.13. Let X be a smooth projective variety of dimension n over $\overline {\mathbf {Q}}$ and f a surjective endomorphism of X of positive entropy (i.e., $\lambda _1(f)>1$ ). Then the following assertions hold:

1. (1) there is a nef $\mathbf {R}$ -divisor $D_f$ on X such that $f^*D_f \equiv \lambda _1(f)D_f$ ; further,

2. (2) for any $D_f$ in the assertion (1), there is a unique nef $\mathbf {R}$ -divisor $D^{\prime }_f$ on X, up to $\mathbf {R}$ -linear equivalence, such that $D^{\prime }_f \equiv D_f$ and $f^*D^{\prime }_f \sim _{\mathbf {R}} \lambda _1(f)D^{\prime }_f$ .

Proof. It is well known that the assertion (1) follows from Birkhoff’s Theorem 3.1. Fix such a nef $\mathbf {R}$ -divisor $D_f\in \mathsf {Nef}(X)$ such that $f^*D_f \equiv \lambda _1(f)D_f$ . Let $\mathsf {Pic}^0(X)$ denote the subgroup of the Picard group $\mathsf {Pic}(X)$ consisting of integral divisors on X algebraically equivalent to zero (modulo linear equivalence), which has the structure of an abelian variety. Consider the exact sequence of $\mathbf {R}$ -vector spaces:

$$\begin{align*}0\to \mathsf{Pic}^0(X)_{\mathbf{R}} \to \mathsf{Pic}(X)_{\mathbf{R}} \to \mathsf{NS}(X)_{\mathbf{R}} \cong \mathsf{N}^1(X)_{\mathbf{R}} \to 0. \end{align*}$$

If the irregularity , then $\mathsf {N}^1(X)_{\mathbf {R}} \cong \mathsf {Pic}(X)_{\mathbf {R}}$ , and hence, the assertion (2) follows. So let us consider the case $q(X)>0$ . Note that $f^*D_f - \lambda _1(f)D_f \in \mathsf {Pic}^0(X)_{\mathbf {R}}$ .

Claim 3.14. The $\mathbf {R}$ -linear map

$$\begin{align*}f^* \otimes_{\mathbf{Z}} 1_{\mathbf{R}} - \lambda_1(f) \operatorname{id} \colon \mathsf{Pic}^0(X)_{\mathbf{R}} \to \mathsf{Pic}^0(X)_{\mathbf{R}} \end{align*}$$

on the (possibly infinite-dimensional) $\mathbf {R}$ -vector space $\mathsf {Pic}^0(X)_{\mathbf {R}}$ is bijective.

Assuming Claim 3.14 for the time being, up to $\mathbf {R}$ -linear equivalence, there is a unique $\mathbf {R}$ -divisor $E\in \mathsf {Pic}^0(X)_{\mathbf {R}}$ such that $f^*E-\lambda _1(f)E \sim _{\mathbf {R}} f^*D_f - \lambda _1(f)D_f$ , which yields that $f^*(D_f-E) \sim _{\mathbf {R}} \lambda _1(f)(D_f-E)$ . Hence, suffices to conclude the assertion (2).

Proof of Claim 3.14.

Let $\mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}$ denote the Picard variety of X. The pullback $f^*$ of divisors on X induces an isogeny g of $\mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}$ . Denote by $P_g(t) \in \mathbf {Z}[t]$ the characteristic polynomial of g, which has degree $2q(X)$ and satisfies that

where $H^{2n-1}(X, \mathbf {Q})$ is canonically isomorphic to $H^1(\mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}, \mathbf {Q})$ via the Poincaré divisor on $X\times \mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}$ and the last equality follows from Poincaré duality. Thanks to [Din05, Proposition 5.8], all roots of $P_g(t)$ have moduli at most $\sqrt {\lambda _1(f)}$ . In particular, $P_g(\lambda _1(f)) \neq 0$ since $\lambda _1(f)>1$ by assumption. Besides, since the rational representation $\operatorname {End}(\mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}) \to \operatorname {End}_{\mathbf {Q}}(H^1(\mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}, \mathbf {Q}))$ is an injective homomorphism, $P_g(g) = 0$ as an endomorphism of $\mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}$ . It follows that $P_g(f^*) = 0$ as a group homomorphism of $\mathsf {Pic}^0(X) = \mathbf {Pic}^0_{X/\overline {\mathbf {Q}}}(\overline {\mathbf {Q}})$ . Now by applying Lemma 3.5 to the $\mathbf {Z}$ -module $\mathsf {Pic}^0(X)$ , Claim 3.14 follows.

Now we are ready to prove Theorem 3.6.

Proof of Theorem 3.6.

Assertions 1 to 3 follow readily from Proposition 3.8, Lemma 3.11 and the assumption that $\operatorname {rank} G = n-1$ . We are left to construct n automorphisms $g_1,\dots ,g_n\in G$ satisfying Assertion (4).

Let $1\leq i \leq n$ be fixed. Let $p_i$ denote the natural projection from $\mathbf {R}^n$ to $\mathbf {R}^{n-1}$ by omitting the i-th coordinate $x_i$ . Recall that the group homomorphism $\pi \colon G \to (\mathbf {R}^n, +)$ , defined by sending $g\in G$ to $(\log \chi _1(g),\dots ,\log \chi _n(g))$ , is injective and $\pi (G)$ is a lattice in $\mathbf {R}^n$ of rank $n-1$ (see Proposition 3.10). Besides, by the projection formula, we have for any $g\in G$ ,

$$\begin{align*}D_1\cdots D_n=\deg(g)(D_1\cdots D_n)=g^*D_1\cdots g^*D_n=\chi_1(g)\cdots\chi_n(g)(D_1\cdots D_n). \end{align*}$$

Since $D_1\cdots D_n>0$ by Assertion (2), one has $\chi _1(g) \cdots \chi _n(g)=1$ . Hence, the image $\pi (G)$ of G is actually contained in the hyperplane .

Consider the following commutative diagram:

Clearly, $p_i \circ \iota \colon H \to \mathbf {R}^{n-1}$ is an isomorphism of $\mathbf {R}$ -vector spaces. By the open mapping theorem, it is also an isomorphism of topological vector spaces. Denote $p_i \circ \pi $ by $\pi _i$ . Since $\tau (G)$ is a lattice in H of rank $n-1$ and $p_i\circ \iota $ is a topological isomorphism, $\pi _i(G) = (p_i \circ \iota )(\tau (G))$ is a lattice in $\mathbf {R}^{n-1}$ of rank $n-1$ . Therefore, there is some $g_i \in G{\setminus }\{\operatorname {id}\}$ such that all $n-1$ coordinates of $\pi _i(g_i)$ are negative (i.e., $\chi _j(g_i)<1$ for all $j\neq i$ ). Further, by Proposition 3.8, for such $g_i$ , there is some $1\leq t_i \leq n$ such that $\chi _{t_i}(g_i) = \lambda _1(g_i)>1$ . Clearly, $t_i$ has to be i. We thus prove that $g_i^*D_i \equiv \chi _{i}(g_i)D_i = \lambda _1(g_i)D_i$ . By Lemma 3.13(2), there is a nef $\mathbf {R}$ -divisor $D^{\prime }_i \equiv D_i$ such that $g_i^*D^{\prime }_i \sim _{\mathbf {R}} \lambda _1(g_i)D^{\prime }_i$ .

In the end, we replace $D_i$ with $D^{\prime }_i$ for all i. Since $D^{\prime }_i \equiv D_i$ , this does not affect Assertions (1) and (2). We thus complete the proof of Theorem 3.6.

Remark 3.15. In the above Theorem 3.6, denote by M the matrix $(\log \chi _i(g_j))_{1\leq i,j\leq n}$ . Clearly, the rank of M is at most $n-1$ since $\sum _{i=1}^n \log \chi _i = 0$ . On the other hand, as $\log \chi _i(g_j)<0$ for all $i\neq j$ , one can see that any submatrix $M_i$ of M obtained by deleting the i-th row and the i-th column is strictly diagonally dominant and hence nonsingular (see [HJ13, Theorem 6.1.10]). It follows that $\operatorname {rank} M=n-1$ . Hence, $g_1,\dots ,g_n$ generate a free abelian subgroup of G of full rank $n-1$ ; moreover, by the same argument, so do any $n-1$ automorphisms from $g_1,\dots ,g_n$ .

At the end of this section, we quote a lemma of Zhang [Zha16], who essentially proved the existence of Zariski dense G-orbits (though he did not write it down explicitly). We will also use it later in the proof of Theorem 1.1 (or rather, proofs of Theorem 4.2 and its corollaries). For the sake of completeness, we present a proof. Recall that for a nef $\mathbf {R}$ -divisor D on X, the null locus $\operatorname {Null}(D)$ of D is the union of all subvarieties V of X of positive dimension such that the restriction $D|_V$ is not big, i.e., $D^{\dim V}\cdot V=0$ .

Lemma 3.16 (cf. [Zha16, Lemma 3.9]).

With the notation as in Theorem 3.6, the augmented base locus $\mathbf {B}_+(D)$ of the nef and big $\mathbf {R}$ -divisor D is a G-invariant Zariski closed proper subset of X equal to

(3.5) $$ \begin{align} \bigcup_{G\text{-periodic }V \subsetneq X} V, \end{align} $$

where V runs over all G-periodic proper subvarieties of X of positive dimension. In particular, $\mathbf {B}_+(D)$ is g-invariant for any $g\in G$ , and hence, for any $x\in (X{\setminus } \mathbf {B}_+(D))(\overline {\mathbf {Q}})$ , one has $\mathcal {O}_g(x)\cap \mathbf {B}_+(D) = \emptyset $ for any $g\in G$ .

Proof. It is well known that $\mathbf {B}_+(D)$ is a Zariski closed proper subset of X, as D is big; see Proposition 2.4(1). We shall first show that $\mathbf {B}_+(D)$ is G-invariant. Let $g\in G$ be an arbitrary fixed automorphism of X. Thanks to an observation by Lesieutre and Satriano (see Lemma 2.5), for any positive numbers $a_1,\dots ,a_n$ , we have

$$\begin{align*}\mathbf{B}_+(a_1D_1+\dots+a_nD_n)=\mathbf{B}_+(D_1+\dots+D_n). \end{align*}$$

It thus follows from Proposition 2.4(2) that

$$\begin{align*}g(\mathbf{B}_+(D)) = \mathbf{B}_+((g^{-1})^*(D_1+\cdots+D_n)) = \mathbf{B}_+\bigg(\sum_{i=1}^n\chi_i(g)^{-1}D_i\bigg) = \mathbf{B}_+(D). \end{align*}$$

Therefore, $\mathbf {B}_+(D)$ is g-invariant and hence G-invariant.

We then prove that $\mathbf {B}_+(D)$ equals the union (3.5). First, by G-invariance of $\mathbf {B}_+(D)$ , every irreducible component of the closed $\mathbf {B}_+(D)$ is G-periodic. It is also known that $\mathbf {B}_+(D)$ has no isolated points (see [ELM⁺09, Proposition 1.1]). Hence, $\mathbf {B}_+(D)$ is contained in the union (3.5). On the other hand, thanks to [ELM⁺09, Corollary 5.6], $\mathbf {B}_+(D)$ coincides with the null locus $\operatorname {Null}(D)$ of D. It remains to show that $\operatorname {Null}(D)$ contains the union (3.5). Suppose that $V\subsetneq X$ is a G-periodic proper subvariety of dimension k with $1\leq k < n$ . We shall prove that $D|_V$ is not big or, equivalently, $D^k \cdot V=0$ . Since D is the sum of nef divisors $D_i$ , it suffices to show that $D_{i_1}\cdots D_{i_k}\cdot V=0$ for any multi-index $1\leq i_1\leq \dots \leq i_{k}\leq n$ . Fix such a multi-index $1\leq i_1\leq \dots \leq i_{k}\leq n$ . Then there always exists an $i_\circ $ with $1\leq i_\circ \leq n$ different from all $i_j$ with $1\leq j\leq k$ . Hence, by Theorem 3.6(4), one has $\chi _{i_j}(g_{i_\circ })<1$ for any $1\leq j\leq k$ . Suppose that $g_{i_\circ }^e(V)=V$ for some $e\geq 1$ (depending on $g_{i_\circ }$ ). Then it follows from Theorem 3.6(1) and the projection formula that

$$\begin{align*}D_{i_1}\cdots D_{i_k}\cdot V = (g_{i_\circ}^e)^*D_{i_1}\cdots (g_{i_\circ}^e)^*D_{i_k}\cdot (g_{i_\circ}^e)^*V = \prod_{j=1}^{k} \chi_{i_j}(g_{i_\circ})^e (D_{i_1}\cdots D_{i_k}\cdot V). \end{align*}$$

This implies that $D_{i_1}\cdots D_{i_k}\cdot V=0$ and hence concludes the proof of Lemma 3.16.

We are now in the position to prove the existence of Zariski dense G-orbits.

Proof of Corollary 1.6.

According to Theorem 3.6 and Lemma 3.16, the augmented base locus $\mathbf {B}_+(D)$ of the nef and big $\mathbf {R}$ -divisor D is a G-invariant Zariski closed proper subset of X. Thanks to [Ame11, Corollary 9 and the paragraph after it], there is a closed point $x\in X(\overline {\mathbf {Q}})$ away from $\mathbf {B}_+(D)$ such that the G-orbit $\mathcal {O}_G(x)$ of x is infinite. Let Z be the Zariski closure of $\mathcal {O}_G(x)$ in X. Then Z is a positive-dimensional G-invariant Zariski closed subset of X. Suppose that $Z\neq X$ . Then by Lemma 3.16, Z is contained in $\mathbf {B}_+(D)$ , contradicting the choice of x. Therefore, the G-orbit $\mathcal {O}_G(x)$ of x is Zariski dense in X. The first assertion is thus verified.

Let us prove the second assertion on the potential density of Y. By assumption, Y is defined over a number field K, and $X=Y\times _{\operatorname {Spec} K}\operatorname {Spec}\overline {\mathbf {Q}}$ . By the first assertion, there is a closed point $x\in X(\overline {\mathbf {Q}})$ such that its G-orbit $\mathcal {O}_{G}(x)$ is Zariski dense in X. On the other hand, such a closed point $x\in X(\overline {\mathbf {Q}})$ is in fact defined over a number field L, which is a finite extension of the defining field K. Similarly, we denote by $Y_L$ the base extension of Y and let $\varphi \colon X\to Y_L$ be the natural projection. Then $\varphi (x)$ is an L-rational point of $Y_L$ , and the $\varphi $ -image $\varphi (\mathcal {O}_G(x))$ of the G-orbit $\mathcal {O}_G(x)$ is Zariski dense in $Y_L$ by noting that $\varphi $ is a finite surjective morphism. It follows that the set $Y_L(L)$ of all L-points of $Y_L$ is also Zariski dense in $Y_L$ , as we have a natural inclusion $\varphi (\mathcal {O}_G(x))\subseteq Y_L(L)$ . We thus finish the proof of our Corollary 1.6.

4 Canonical heights for abelian group actions

Throughout this section, X is a smooth projective variety of dimension $n\geq 2$ defined over $\overline {\mathbf {Q}}$ , and $G \cong \mathbf {Z}^{n-1}$ is a free abelian group of automorphisms of X of positive entropy. Thanks to Theorem 3.6, we can choose n automorphisms $g_1,\dots ,g_{n} \in G$ and n common nef $\mathbf {R}$ -eigendivisors $D_1,\dots ,D_n$ on X such that

1. (1) for any $1\leq i\leq n$ and any $g\in G$ , one has $g^*D_i \equiv \chi _i(g)D_i$ , which defines the group characters $\chi _i$ of G;

2. (2) is a nef and big $\mathbf {R}$ -divisor on X;

3. (3) for any $g\in G$ , there is some $1\leq i\leq n$ such that $\chi _i(g) = \lambda _1(g)$ ;

4. (4) for any $1\leq i\neq j\leq n$ , one has $\chi _j(g_i)<1$ and $g_i^*D_i\sim _{\mathbf {R}} \lambda _1(g_i)D_i$ .

We shall stick to the above notations throughout and construct a height function $\widehat {h}_G$ associated with G as the sum of the following individual canonical heights.

Lemma 4.1. For any $1\leq i\leq n$ and any $x\in X(\overline {\mathbf {Q}})$ , the limit

exists and satisfies the following properties:

1. (1) $\widehat {h}_{D_i,g_i} = h_{D_i} + O(1)$ .

2. (2) $\widehat {h}_{D_i,g_i}\circ g = \chi _i(g) \, \widehat {h}_{D_i,g_i}$ for any $g\in G$ ; in particular, $\widehat {h}_{D_i,g_i}\circ g_i = \lambda _1(g_i) \, \widehat {h}_{D_i,g_i}$ .

Proof. Note that $g_i^*D_i \sim _{\mathbf {R}} \lambda _1(g_i)D_i$ by Theorem 3.6(4). Hence, the existence of each $\widehat {h}_{D_i,g_i}$ and the property (1) follow immediately from Theorem 2.10. For the property (2), fix an integer i with $1\leq i\leq n$ , an automorphism $g\in G$ of positive entropy, an ample divisor $H_X$ on X, and a height function $h_{H_X}$ associated with $H_X$ . Then thanks to Matsuzawa [Mat20b, Theorem 1.7(2)], there is a constant $C_1>0$ such that for any rational point $x\in X(\overline {\mathbf {Q}})$ and any $m\geq 1$ ,

(4.1) $$ \begin{align} h_{H_X}^+(g_i^m(x)) \leq C_1 m^{\rho(X)-1}\lambda_1(g_i)^m h_{H_X}^+(x). \end{align} $$

Observe that by Theorem 3.6(1), we have $g^*D_i \equiv \chi _i(g)D_i$ . Then according to Theorem 2.6 ii (Functoriality) and v (Algebraic equivalence), there is a constant $C_2>0$ such that

$$ \begin{align*} |h_{g^*D_i} - h_{D_i}\circ g| & \leq C_2, \\ |h_{g^*D_i} - \chi_i(g) \, h_{D_i}| & \leq C_2 \sqrt{h_{H_X}^+}. \end{align*} $$

Combining them together yields that

$$\begin{align*}|h_{D_i}\circ g - \chi_i(g) \, h_{D_i}| \leq 2C_2 \sqrt{h_{H_X}^+}. \end{align*}$$

In particular, for any $m\geq 1$ , one has

(4.2) $$ \begin{align} |h_{D_i}(g(g_i^m(x))) - \chi_i(g) \, h_{D_i}(g_i^m(x))| \leq 2C_2 \sqrt{h_{H_X}^+(g_i^m(x))}. \end{align} $$

As G is abelian, it follows from Equations (4.1) and (4.2) that

$$ \begin{align*} \left|\frac{h_{D_i}(g_i^m(g(x)))}{\lambda_1(g_i)^m}-\frac{\chi_i(g)h_{D_i}(g_i^m(x))}{\lambda_1(g_i)^m}\right| \leq \frac{2C_2 \sqrt{h_{H_X}^+(g_i^m(x))}}{\lambda_1(g_i)^m} \leq C_3 \sqrt{\frac{m^{\rho(X)-1}}{\lambda_1(g_i)^{m}}}, \end{align*} $$

where $C_3>0$ is a constant independent of $m\geq 1$ . Note that both limits of the left-hand side exist (see Theorem 2.10). So taking $m\to \infty $ yields that $\widehat {h}_{D_i,g_i}(g(x)) = \chi _i(g) \, \widehat {h}_{D_i,g_i}(x)$ .

The theorem below is a precise version of our main result Theorem 1.1. Indeed, one just takes the Zariski closed proper subset Z to be the augmented base locus $\mathbf {B}_+(D)$ of the nef and big $\mathbf {R}$ -divisor D constructed in Theorem 3.6(2). It extends [Sil91, Theorem 1.1] and [Kaw08, Theorem 5.2] to higher dimensions (under the maximal dynamical rank assumption).

Theorem 4.2. Let X be a smooth projective variety of dimension $n\geq 2$ defined over $\overline {\mathbf {Q}}$ and $G \cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X of positive entropy. Define a function $\widehat {h}_G \colon X(\overline {\mathbf {Q}})\to \mathbf {R}$ by

where the $\widehat {h}_{D_i,g_i}$ are from Lemma 4.1. Then the following assertions hold.

1. (1) The function $\widehat {h}_{G}$ is a Weil height corresponding to the nef and big $\mathbf {R}$ -divisor D; that is,

   $$\begin{align*}\widehat{h}_{G} = h_{D} + O(1). \end{align*}$$

2. (2) For any $g\in G$ , one has

   $$\begin{align*}\widehat{h}_{G} \circ g = \sum_{i=1}^n \chi_i(g) \, \widehat{h}_{D_i,g_i}. \end{align*}$$

3. (3) For any $1\leq i\leq n$ and any $x\in (X{\setminus }\mathbf {B}_+(D))(\overline {\mathbf {Q}})$ , one has $\widehat {h}_{D_i,g_i}(x)\geq 0$ , and hence, $\widehat {h}_{G}(x)\geq 0$ , where $\mathbf {B}_+(D)$ is the augmented base locus of D.

4. (4) The Weil height function $\widehat {h}_{G}$ satisfies the Northcott finiteness property on $X{\setminus }\mathbf {B}_+(D)$ ; in other words, for any positive integer d and real number T, the set

   $$\begin{align*}\big\{x \in (X{\setminus}\mathbf{B}_+(D))(\overline{\mathbf{Q}}) : [K(x):K]\leq d,~\widehat{h}_{G}(x)\leq T \big\} \end{align*}$$

   is finite, where K is a defining number field of X.

5. (5) For any $x\in (X{\setminus }\mathbf {B}_+(D))(\overline {\mathbf {Q}})$ , the following statements are equivalent.

   1. (i) $\widehat {h}_{G}(x)=0$ .

   2. (ii) $\widehat {h}_{D_i,g_i}(x)=0$ for all $1\leq i\leq n$ .

   3. (iii) x is g-periodic for any $g\in G$ .

   4. (iv) x is g-periodic for some $g\in G{\setminus }\{\operatorname {id}\}$ .

   5. (v) $\widehat {h}_{D_i,g_i}(x)=0$ for some $1\leq i\leq n$ .

   6. (vi) x is G-periodic (i.e., the G-orbit of x is finite).

We call $\widehat {h}_G$ a canonical height function associated with the abelian group G of maximal dynamical rank.

Proof. Assertion (1) follows immediately from Lemma 4.1(1) and Theorem 2.6 iii (Additivity). Assertion (2) follows from Lemma 4.1(2). Since D is a big $\mathbf {R}$ -divisor, the height function $h_D$ satisfies the Northcott finiteness property (see Theorem 2.8), and so does $\widehat {h}_G$ by Assertion (1). We have thus proved Assertion (4).

Next, we shall show Assertion (3). Fix an index i with $1\leq i\leq n$ . According to Lemma 3.16, the augmented base locus $\mathbf {B}_+(D)$ of D is a G-invariant Zariski closed proper subset of X. Fix a point $x\in (X{\setminus }\mathbf {B}_+(D))(\overline {\mathbf {Q}})$ ; in particular, $\mathcal {O}_{g_i}(x)\cap \mathbf {B}_+(D) = \emptyset $ . We notice by [LS21, Lemma 2.26] that $h_D \geq O(1)$ outside $\mathbf {B}_+(D)$ . On the other hand, by Assertion (1), we have

$$\begin{align*}h_D = \widehat{h}_G + O(1) = \sum_{j=1}^n \widehat{h}_{D_j,g_j} + O(1). \end{align*}$$

It follows that for any $m\geq 1$ ,

(4.3) $$ \begin{align} O(1) \leq \sum_{j=1}^n \widehat{h}_{D_j,g_j}(g_i^m(x)) = \sum_{j=1}^n \chi_j(g_i)^m \, \widehat{h}_{D_j,g_j}(x), \end{align} $$

where the equality is from Assertion (2). Furthermore, by Theorem 3.6(4), one has $\chi _j(g_i) < 1$ for all $j\neq i$ and $\chi _i(g_i) = \lambda _1(g_i)>1$ . Dividing (4.3) by $\chi _i(g_i)^m$ from both sides and letting m tend to infinity, it is easy to see that $\widehat {h}_{D_i,g_i}(x)$ has to be nonnegative. This thus shows Assertion (3).

At last, we prove Assertion (5). Fix a rational point $x\in (X{\setminus }\mathbf {B}_+(D))(\overline {\mathbf {Q}})$ ; in particular, $\mathcal {O}_{g}(x)\cap \mathbf {B}_+(D) = \emptyset $ for any $g\in G$ . We shall prove the equivalence in the following order:

By definition, 5ii $\Rightarrow $ 5i is trivial, while 5i $\Rightarrow $ 5ii follows from Assertion (3). The implications 5vi $\Rightarrow $ 5iii $\Rightarrow $ 5iv are also trivial.

We first show 5iii $\Rightarrow $ 5vi. Let $\{f_1,\dots ,f_{n-1}\}$ be a generating set of G and let s be the common period of x under the $f_i$ . It follows from the commutativity of G that any point in the G-orbit $\mathcal {O}_G(x)$ of x is of the form $f_1^{a_1} \circ \dots \circ f_{n-1}^{a_{n-1}}(x)$ such that $1-s\leq a_i\leq s-1$ for each $1\leq i\leq n-1$ . In particular, the set $\mathcal {O}_G(x)$ is finite.

We next prove 5ii $\Rightarrow $ 5iii. Let $g\in G$ be fixed. By Assertion (2), for any $m\geq 1$ ,

$$\begin{align*}\widehat{h}_{G}(g^m(x)) = \sum_{i=1}^n \chi_i(g)^m \, \widehat{h}_{D_i,g_i}(x) = 0. \end{align*}$$

Since these rational points $g^m(x)$ are of bounded degree over $\mathbf {Q}$ , it follows from Assertion 4 that the forward g-orbit $\mathcal {O}_g(x)$ of x is finite, that is, x is g-periodic, noting that g is an automorphism.

We now show 5iv $\Rightarrow $ 5v. Suppose that x is g-periodic for some $g \in G{\setminus }\{\operatorname {id}\}$ . Note that by Theorem 3.6(3), there is some $1\leq i\leq n$ such that $\chi _i(g) = \lambda _1(g)$ . Consider the growth of the function $\widehat {h}_G$ along the forward g-orbit $\mathcal {O}_g(x)$ of x, which is finite by assumption. In other words, we have

$$\begin{align*}O(1) = \widehat{h}_{G}(g^m(x)) \geq \lambda_1(g)^m \, \widehat{h}_{D_i,g_i}(x), \end{align*}$$

where the last inequality is due to Assertion (3). As $\lambda _1(g)>1$ , we see that $\widehat {h}_{D_i,g_i}(x)$ has to be zero by letting m tend to infinity.

It remains to prove 5v $\Rightarrow $ 5ii. Without loss of generality, we may assume that $\widehat {h}_{D_1,g_1}(x) = 0$ . Via a similar argument in the proof of 5ii $\Rightarrow $ 5iii, we claim that x is $g_1$ -periodic. Indeed, by Assertion (2) and Theorem 3.6(4), we have for any $m\geq 1$ ,

$$\begin{align*}\widehat{h}_{G}(g_1^m(x)) = \sum_{j=2}^n \chi_j(g_1)^m \, \widehat{h}_{D_j}(x) < \sum_{j=2}^n \widehat{h}_{D_j}(x). \end{align*}$$

Note that the $g_1^m(x)$ are of bounded degree over $\mathbf {Q}$ . By Assertion 4, x is $g_1$ -periodic. Denote the finite period by $e_1\in \mathbf {Z}_{>0}$ , i.e., $g_1^{e_1}(x)= x$ . Consider the growth of the function $\widehat {h}_{G}$ along the orbit $\mathcal {O}_{g_1^{-e_1}}(x)$ of x under the automorphism $g_1^{-e_1}$ . Precisely, for any $m\geq 1$ , we have

$$\begin{align*}\widehat{h}_{G}(x) = \widehat{h}_{G}(g_1^{-e_1}(x)) = \widehat{h}_{G}(g_1^{-e_1m}(x)) = \sum_{j=2}^n \chi_j(g_1)^{-e_1m} \, \widehat{h}_{D_j}(x), \end{align*}$$

which forces $\widehat {h}_{D_j}(x) = 0$ for all $j>1$ by noting that $\chi _j(g_1)<1$ . This verifies 5v $\Rightarrow $ 5ii.

We thus complete the proof of Theorem 4.2.

As a direct consequence of Theorem 4.2, we also obtain the following.

Corollary 4.3 (cf. [KS14, Proposition 7 and Proof of Theorem 2(c)]).

Under the assumption of Theorem 4.2, for any $g\in G$ and any point $x\in (X{\setminus }\mathbf {B}_+(D))(\overline {\mathbf {Q}})$ , we have

$$\begin{align*}\alpha_g(x) = \left\{ \begin{array}{cl} 1 & \text{if } x \text{ is } g\text{-periodic}, \\ \lambda_1(g) & \text{if } x \text{ is not } g\text{-periodic}. \end{array} \right. \end{align*}$$

Proof. Let $g\in G$ be fixed. First, we assume that $x\in (X{\setminus }\mathbf {B}_+(D))(\overline {\mathbf {Q}})$ is a non-g-periodic point. As we mentioned before, the limit defining $\alpha _g(x)$ exists and is independent of the choice of the ample divisor (see [KS16a, Theorem 3] and [KS16b, Proposition 12], respectively). Choose an ample divisor $H_X$ such that $H_X - D$ is ample (noting that the ample cone is open). It follows from Theorem 2.6 iii (Additivity) and iv (Positivity) that for any $m\geq 1$ ,

$$\begin{align*}h_{H_X}(g^m(x)) = h_{D}(g^m(x)) + h_{H_X-D}(g^m(x)) + O(1) \geq h_{D}(g^m(x)) + O(1). \end{align*}$$

On the other hand, Theorem 4.2(1) asserts that

$$\begin{align*}h_{D}(g^m(x)) = \widehat{h}_G(g^m(x)) + O(1). \end{align*}$$

Putting them together yields that

$$\begin{align*}h_{H_X}(g^m(x)) \geq \widehat{h}_G(g^m(x)) + O(1) = \sum_{j=1}^n \chi_j(g)^m \, \widehat{h}_{D_j,g_j}(x) + O(1), \end{align*}$$

where the equality is from Theorem 4.2(2). Furthermore, by Theorem 3.6(3), there is some $1\leq i\leq n$ such that $\chi _i(g) = \lambda _1(g)$ . We thus obtain that

$$\begin{align*}h_{H_X}(g^m(x)) \geq \lambda_1(g)^m \, \widehat{h}_{D_i,g_i}(x) + O(1). \end{align*}$$

Note that the term $O(1)$ does not depend on x nor m. Also, according to Theorem 4.2 5, one has $\widehat {h}_{D_i,g_i}(x)> 0$ for all $1\leq i\leq n$ . Now by taking m-th roots and letting $m\to \infty $ , one easily has $\alpha _g(x) \geq \lambda _1(g)$ . The reverse inequality is due to [KS16b, Theorem 4] or [Mat20b, Theorem 1.4].

Secondly, we assume that x is g-periodic. Then with the ample divisor $H_X$ chosen as above, it is clear that $(h_{H_X}(g^m(x)))_{m\in \mathbf {N}}$ is a finite set. Therefore,

$$\begin{align*}1\leq \alpha_g(x)=\lim_{m\to\infty}(h_{H_X}^+(g^m(x)))^{1/m}\leq 1. \end{align*}$$

We finish the proof of Corollary 4.3.

Remark 4.4. Following [Sil91], we can also define a function

$$\begin{align*}\widehat{H}_G\colon X(\overline{\mathbf{Q}}) \to \mathbf{R} \quad \text{by} \quad \widehat{H}_G(x) = \prod_{i=1}^n \widehat{h}_{D_i,g_i}(x). \end{align*}$$

Note that by the projection formula, for any $g\in G$ , one has

$$\begin{align*}0\neq D_1\cdots D_n=\deg(g)(D_1\cdots D_n)=g^*D_1\cdots g^*D_n=\chi_1(g)\cdots\chi_n(g)(D_1\cdots D_n); \end{align*}$$

in particular, $\prod _{i=1}^n \chi _i(g) = 1$ . It follows that

$$\begin{align*}\widehat{H}_G \circ g = \prod_{i=1}^n \widehat{h}_{D_i,g_i} \circ g = \prod_{i=1}^n \chi_i(g) \, \widehat{h}_{D_i,g_i} = \prod_{i=1}^n \widehat{h}_{D_i,g_i} = \widehat{H}_G. \end{align*}$$

In other words, the function $\widehat {H}_G$ is G-invariant. Let $x\in (X{\setminus }\mathbf {B}_+(D))(\overline {\mathbf {Q}})$ be arbitrary. Then according to Theorem 4.2(3), each $\widehat {h}_{D_i,g_i}(x)\geq 0$ . This yields that

$$\begin{align*}\sqrt[\leftroot{-1}\uproot{6} n]{\widehat{H}_G(x)} \leq \frac{\widehat{h}_G(x)}{n}. \end{align*}$$

Moreover, by Theorem 4.2 5, $\widehat {H}_G(x)=0$ if and only if x is g-periodic for any $g\in G$ .

Inspired by [Sil91, Theorem 1.3(a)], we ask the following.

Question 4.5. For any point $x\in (X{\setminus }\mathbf {B}_+(D))(\overline {\mathbf {Q}})$ with infinite G-orbit, is there any lower bound of $\widehat {H}_G(x)$ in terms of $\widehat {h}_G(x)$ ?

Note that if the above question has an affirmative answer, then one could prove a similar result as [Sil91, Theorem 1.2(b)] using the Northcott property for $\widehat {h}_G$ (i.e., Theorem 4.2 4). Namely, there are only finitely many infinite G-orbits in $(X{\setminus }\mathbf {B}_+(D))(K)$ , where K is any number field.

4.1 Proofs of Theorem 1.1 and Corollaries 1.3 and 1.5

Proof of Theorem 1.1.

By Theorem 3.6, we take Z to be the augmented base locus $\mathbf {B}_+(D)$ of D, which is a G-invariant Zariski closed proper subset of X (see Lemma 3.16). Theorem 1.1 then follows easily from Theorem 4.2.

Proof of Corollary 1.3.

It follows readily from Lemma 3.16 and Corollary 4.3.

Proof of Corollary 1.5.

Take Z to be the augmented base locus $\mathbf {B}_+(D)$ of D as in the proof of Theorem 1.1. Fix a g-periodic point $x\in (X{\setminus } Z)(\overline {\mathbf {Q}})$ and an ample divisor $H_X$ on X. By Theorem 4.2(1) and (5), we have $h_D(x) = \widehat {h}_G(x) + O(1) = O(1)$ . On the other hand, according to Proposition 2.4(3), there is $\varepsilon>0$ such that $D-\varepsilon H_X$ is an effective $\mathbf {Q}$ -divisor and $\mathbf {B}_+(D) = \mathbf {B}(D-\varepsilon H_X) = \operatorname {Bs}(M(D-\varepsilon H_X))$ for some $M\geq 1$ . Since $x\not \in \mathbf {B}_+(D)$ , by applying Theorem 2.6 iii (Additivity) and iv (Positivity) to $M(D-\varepsilon H_X)$ , we obtain that $h_{D-\varepsilon H_X}(x)\geq O(1)$ . It thus follows that

$$ \begin{align*} O(1) = h_D(x) = h_{D-\varepsilon H_X}(x)+h_{\varepsilon H_X}(x)+O(1) \geq h_{\varepsilon H_X}(x) + O(1). \end{align*} $$

Therefore, $h_{\varepsilon H_X}(x)$ and $h_{H_X}(x)$ are both bounded. Assertion (1) is thus proved.

Assertion (2) follows easily from [KS16b, Proposition 3] and Corollary 4.3.