# Recent advances on Kawaguchi-Silverman conjecture

Yohsuke Matsuzawa  Department of Mathematics, Graduate School of Science,
Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi, Osaka, 558-8585,
Japan<EMAIL_ADDRESS>
###### Abstract.

This is a survey on Kawaguchi-Silverman conjecture.

###### Contents

  1. 1 Introduction
  2. 2 Kawaguchi-Silverman Conjecture
    1. 2.1 Arithmetic degree
    2. 2.2 Kawaguchi-Silverman conjecture
    3. 2.3 Understanding Kawaguchi-Silverman conjecture
  3. 3 A list of known results
  4. 4 First properties
  5. 5 Canonical heights
    1. 5.1 Polarized endomorphisms and nef canonical height
    2. 5.2 Abelian varieties
    3. 5.3 Monomial maps
    4. 5.4 Automorphisms on surfaces and Hyper-Kähler varieties
    5. 5.5 Mori dream spaces and varieties of Fano type
    6. 5.6 Regular affine automorphisms
    7. 5.7 Self-morphisms on ${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$
  6. 6 Minimal Model Program and Projective geometry
    1. 6.1 Varieties admitting an int-amplified endomorphism
    2. 6.2 Projective geometry
    3. 6.3 Some useful facts
  7. 7 Toward rational maps
    1. 7.1 Self-morphisms on affine surfaces
    2. 7.2 Cohomologically hyperbolic maps
  8. 8 Related topics
  9. 9 Future directions

##  1\. Introduction

The Kawaguchi-Silverman conjecture predicts the equivalence of the geometric
complexity of self-maps and the arithmetic complexity of orbits of rational
points. Observations essential to the conjecture were already made in the
1990s, but the conjecture itself was explicitly formulated and stated around
2011 by Kawaguchi and Silverman, building on their earlier work. Following its
formulation, people began to prove many special cases of the conjecture and
study arithmetic degrees, a key concept within the conjecture. In this survey,
we introduce the conjecture, outline some basic observations, and collect
results that affirm special cases of the conjecture. We have done our best to
cover all results directly related to the conjecture, but we might have missed
a few things.

Notation.

  1. (1)

A variety over a field $k$ is (unless otherwise specified) a separated
irreducible reduced scheme of finite type over $k$.

  2. (2)

Let $X$ be a projective variety over an algebraically closed field. For
Cartier divisors (resp. ${\mathbb{Q}}$ or ${\mathbb{R}}$-Cartier divisors)
$D,E$ on $X$, the linear equivalence (resp. ${\mathbb{Q}}$ or
${\mathbb{R}}$-linear equivalence) is denoted by $D\sim E$ (resp.
$D\sim_{{\mathbb{Q}}}E$ or $D\sim_{{\mathbb{R}}}E$). They are said to be
numerically equivalent if $(D\cdot C)=(E\cdot C)$ for all irreducible curves
$C\subset X$. In this case, we write $D\equiv E$. The group of Cartier
divisors modulo numerical equivalence is denoted by $N^{1}(X)$. We write
$N^{1}(X)_{{\mathbb{Q}}}=N^{1}(X){\otimes}_{{\mathbb{Z}}}{\mathbb{Q}}$ and
$N^{1}(X)_{{\mathbb{R}}}=N^{1}(X){\otimes}_{{\mathbb{Z}}}{\mathbb{R}}$.

  3. (3)

The Kodaira dimension of a variety $X$ is denoted by $\kappa(X)$. For a
Cartier divisor $D$ on a projective variety $X$, its Iitaka dimension is
denoted by $\kappa(X,D)$.

  4. (4)

For a dominant rational map $f\colon X\dashrightarrow X$ on a projective
variety $X$ defined over an algebraically closed field of characteristic zero,
the $i$-th dynamical degree of $f$ is denoted by $\delta_{i}(f)$ ($0\leq
i\leq\dim X$).

  5. (5)

For an ${\mathbb{R}}$-Cartier divisor $D$ on a projective variety $X$ defined
over $\overline{\mathbb{Q}}$, $h_{D}$ denotes a logarithmic Weil height
function associated with $D$. We write $h_{D}^{+}=\max\\{1,h_{D}\\}$.

##  2\. Kawaguchi-Silverman Conjecture

###  2.1. Arithmetic degree

Let $X$ be a quasi-projective variety defined over an algebraic closure of
rational numbers $\overline{\mathbb{Q}}$. Let us consider a dominant rational
self-map $f\colon X\dashrightarrow X$ defined over $\overline{\mathbb{Q}}$. As
$f$ is a rational map and not necessarily a morphism, there might be points at
which $f$ is not defined. The smallest closed subset $I_{f}\subset X$ such
that $f$ is a morphism on $X\setminus I_{f}$ is called the indeterminacy locus
of $f$. To discuss the dynamics of $f$, let us introduce the following
notation.

######  Definition 2.1.

The set of $\overline{\mathbb{Q}}$-points whose forward $f$-orbit is well
defined is denoted by $X_{f}(\overline{\mathbb{Q}})$:

(2.1) $\displaystyle X_{f}(\overline{\mathbb{Q}}):=\left\\{x\in
X(\overline{\mathbb{Q}})\ \middle|\ \forall n\geq 0,f^{n}(x)\notin
I_{f}\right\\}.$

For a point $x\in X_{f}(\overline{\mathbb{Q}})$, its $f$-orbit is

(2.2) $\displaystyle O_{f}(x)=\left\\{f^{n}(x)\ \middle|\ n\geq 0\right\\}.$

As $X_{f}(\overline{\mathbb{Q}})$ is defined by countably many conditions, it
is not obvious that this set is non-empty. It is, however, known that this set
contains many points ([2, Corollary 9], [66, Proposition 3.24]). For example,
there is a non-empty adelic open subset $A\subset X(\overline{\mathbb{Q}})$
(in the sense of Xie [66]) such that $A\subset X_{f}(\overline{\mathbb{Q}})$
and the $f$-orbit of every $x\in A$ is infinite.

A point $x\in X(\overline{\mathbb{Q}})$ can be represented using a finite
number of bits (for instance, by taking the coordinates and considering the
minimal polynomial of each coordinate). In typical situations, the number of
bits required to represent $f^{n}(x)$ increases as $n$ goes to infinity. The
Kawaguchi-Silverman conjecture relates this growth rate with the geometric
complexity of $f$, the dynamical degree. Specifically, it predicts that the
arithmetic complexity of any dense orbit equals the geometric complexity of
the map $f$. For more insight into the idea behind the conjecture, see section
2.3.

The arithmetic complexity is measured by using Weil height functions. While
there are many ways to define height function, we present the shortest one
here. For more about the definitions and fundamentals on height functions, see
[16, 6, 29].

######  Definition 2.2 (Weil height functions).

  1. (1)

Let $K$ be a number field. For a point
$x=(a_{0}:\cdots:a_{N})\in{\mathbb{P}}^{N}(\overline{\mathbb{Q}})$ with
$a_{i}\in K$, we define

(2.3) $\displaystyle h_{{\mathbb{P}}^{N}}(x)=\sum_{v\in
M_{K}}\log\max\\{|a_{0}|_{v},\dots,|a_{N}|_{v}\\}$

where $M_{K}$ is the set of absolute values on $K$ normalized as in [6, p11,
(1.6)]. That is, if $v$ is a place of $K$ whose restriction on ${\mathbb{Q}}$
is $p$ ($p$ is a prime number or $\infty$), then

(2.4)
$\displaystyle|a|_{v}=|N_{K_{v}/{\mathbb{Q}}_{p}}(a)|_{p}^{1/[K:{\mathbb{Q}}]}$

where $|\ |_{p}$ is the usual $p$-adic absolute value with $|p|_{p}=p^{-1}$
when $p$ is prime and $|\ |_{\infty}$ is the usual Euclidean absolute value on
${\mathbb{Q}}$. The value $h_{{\mathbb{P}}^{N}}(x)$ is independent of the
choice of $K$ and the homogeneous coordinates. Thus
$h_{{\mathbb{P}}^{N}}\colon{\mathbb{P}}^{N}(\overline{\mathbb{Q}})\longrightarrow{\mathbb{R}}$
is a well-defined function, which is called the naive height function.

  2. (2)

Let $X$ be a projective variety over $\overline{\mathbb{Q}}$ and let $D$ be a
Cartier divisor on $X$. We attach $D$ a real valued function on
$X(\overline{\mathbb{Q}})$ that is well-defined up to difference by bounded
functions. Write $D=H_{1}-H_{2}$ with very ample divisors $H_{1}$ and $H_{2}$
on $X$. Take closed immersions $\varphi_{i}\colon
X\longrightarrow{\mathbb{P}}^{N_{i}}_{\overline{\mathbb{Q}}}\ (i=1,2)$ such
that
$\varphi_{i}^{*}{\mathcal{O}}_{{\mathbb{P}}^{N_{i}}}(1)\simeq{\mathcal{O}}_{X}(H_{i})$.
Then we define

(2.5) $\displaystyle
h_{D}=h_{{\mathbb{P}}^{N_{1}}}\circ\varphi_{1}-h_{{\mathbb{P}}^{N_{2}}}\circ\varphi_{2}.$

For another choice of $H_{i}$’s and $\varphi_{i}$’s, this function changes at
most a bounded function. Moreover, if $D$ and $D^{\prime}$ are linearly
equivalent, then $h_{D}-h_{D^{\prime}}$ is also bounded on
$X(\overline{\mathbb{Q}})$. This correspondence $D\mapsto h_{D}$ defines a
group homomorphism

(2.6) $\displaystyle\\{\text{Cartier divisors on $X$}\\}/\\{\text{principal
divisors}\\}$ (2.7)
$\displaystyle\longrightarrow{\mathrm{Map}}(X(\overline{\mathbb{Q}}),{\mathbb{R}})/\\{\text{bounded
functions}\\}.$

For ${\mathbb{R}}$-Cartier divisors, we extend by linearity (jsut
${\otimes}_{{\mathbb{Z}}}{\mathbb{R}}$ in the left hand side).

Now let us introduce the arithmetic degree.

######  Definition 2.3.

Let $X$ be a quasi-projective variety over $\overline{\mathbb{Q}}$ and let
$f\colon X\dashrightarrow X$ be a dominant rational map. Let us fix an
immersion $i\colon X\longrightarrow P$ to a projective variety $P$ over
$\overline{\mathbb{Q}}$ and fix an ample divisor $H$ on $P$. Let $h_{H}\colon
P(\overline{\mathbb{Q}})\longrightarrow{\mathbb{R}}$ be a logarithmic Weil
height function associated with $H$. We set $h_{H}^{+}:=\max\\{1,h_{H}\\}$.
For a point $x\in X_{f}(\overline{\mathbb{Q}})$, the arithmetic degree of $x$
is

(2.8) $\displaystyle\alpha_{f}(x):=\lim_{n\to\infty}h_{H}^{+}(f^{n}(x))^{1/n}$

provided the limit exists. As we mention below, we do not know if this limit
exists always. We use the following notation and call them upper and lower
arithmetic degrees:

(2.9)
$\displaystyle\overline{\alpha}_{f}(x):=\limsup_{n\to\infty}h_{H}^{+}(f^{n}(x))^{1/n}$
(2.10)
$\displaystyle\underline{\alpha}_{f}(x):=\liminf_{n\to\infty}h_{H}^{+}(f^{n}(x))^{1/n}.$

######  Remark 2.4.

The definition apparently depends on the choice of immersion $i\colon
X\longrightarrow P$, the ample divisor $H$, and the height function $h_{H}$,
but it is actually independent of all of them. The independence of latter two
is immediate from the general fact on height functions. That is, if $H$ and
$H^{\prime}$ are ample divisors on $P$ and $h_{H}$ and $h_{H^{\prime}}$ are
arbitrary choice of height functions associated with $H$ and $H^{\prime}$
respectively, then there are constants $C\geq 1$ and $C^{\prime}\geq 0$ such
that

(2.11) $\displaystyle\frac{1}{C}h_{H}-C^{\prime}\leq h_{H^{\prime}}\leq
Ch_{H}+C^{\prime}$

on $P(\overline{\mathbb{Q}})$. It is easy to see from these inequalities that
the definition is independent of the choice of $H$ and $h_{H}$. The
independence of the immersion $i\colon X\longrightarrow P$ follows from [17,
Lemma 3.8].

We quickly recall the definition of dynamical degrees.

######  Definition 2.5 (Dynamical degree).

Let $X$ be a projective variety over $\overline{\mathbb{Q}}$ and let $f\colon
X\dashrightarrow X$ be a dominant rational map. For each positive integer $n$,
consider the following diagram

$\textstyle{\Gamma_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcap}$$\scriptstyle{\pi_{n}}$$\scriptstyle{F_{n}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{n}}$$\textstyle{X}$

where $\Gamma_{n}\subset X\times X$ is the graph of $f^{n}$. Let us fix an
ample divisor $H$ on $X$. For an integer $p$ between $0$ and $\dim X$, the
$p$-th dynamical degree of $f$ is

(2.12)
$\displaystyle\delta_{p}(f):=\lim_{n\to\infty}\left(F_{n}^{*}H^{p}\cdot\pi_{n}^{*}H^{\dim
X-p}\right)_{\Gamma_{n}}^{1/n}.$

It is known that this limit exists and independent of the choice of $H$ (cf.
[14, 11, 64]). Moreover, $\delta_{p}(f)$ are invariant under birational
conjugation. We define dynamical degrees of dominant rational self-maps on
quasi-projective varieties by taking a projective model.

Note that if $f\colon X\longrightarrow X$ is a surjective self-morphism on a
projective variety over $\overline{\mathbb{Q}}$, then $\delta_{1}(f)$ is equal
to the spectral radius of $f^{*}\colon N^{1}(X)_{{\mathbb{R}}}\longrightarrow
N^{1}(X)_{{\mathbb{R}}}$, i.e. the maximum of the absolute value of the
eigenvalues of $f^{*}$.

###  2.2. Kawaguchi-Silverman conjecture

Kawaguchi-Silverman conjecture is first stated for dominant rational self-maps
on projective spaces in [60, Conjecture 1] and later formulated for arbitrary
smooth projective varieties in [25, Conjecture 6].

######  Conjecture 2.6 (Kawaguchi-Silverman Conjecture –original version– [25,
Conjecture 6]).

Let $X$ be a smooth projective variety and $f\colon X\dashrightarrow X$ a
dominant rational map both defined over $\overline{\mathbb{Q}}$.

  1. (1)

For all $x\in X_{f}(\overline{\mathbb{Q}})$, the arithmetic degree
$\alpha_{f}(x)$ exists, i.e. the limit

(2.13) $\displaystyle\lim_{n\to\infty}h_{H}^{+}(f^{n}(x))^{1/n}$

exists.

  2. (2)

$\alpha_{f}(x)$ is an algebraic integer.

  3. (3)

The collection of arithmetic degrees

(2.14) $\displaystyle\left\\{\alpha_{f}(x)\ \middle|\ x\in
X_{f}(\overline{\mathbb{Q}})\right\\}$

is a finite set.

  4. (4)

If the orbit $O_{f}(x)$ is Zariski dense in $X$, then
$\alpha_{f}(x)=\delta_{1}(f)$.

Understanding the growth rate of height functions along orbits is a
fundamental subject in arithmetic dynamics. The arithmetic degree is the
simplest measure of this growth rate, and the Kawaguchi-Silverman conjecture
predicts that it is governed by the geometry of $f$. Given the foundational
nature of this conjecture, we believe it has potential applications to other
problems within arithmetic dynamics. Indeed, there are already non-trivial
applications of studying arithmetic degrees to the Zariski dense orbit
conjecture (Zhang and Medvedev-Scanlon’s conjecture) [17, 42]. Additionally,
it’s worth mentioning that the study of the Kawaguchi-Silverman conjecture has
stimulated deeper understanding of the structure of self-maps on varieties.

Let us first remark that part (2) and (3) of the conjecture are already
disproven. Lesieutre and Satriano constructed a counter example to part (3)
using a family of birational self-maps on surfaces whose dynamical degrees
take infinitely many values.

######  Theorem 2.7 ([30, Theorem 2]).

Let
$f\colon{\mathbb{P}}^{4}_{\overline{\mathbb{Q}}}\dashrightarrow{\mathbb{P}}^{4}_{\overline{\mathbb{Q}}}$
be the birational map defined by

(2.15) $\displaystyle[X:Y:Z:A:B]\mapsto[XY+AX:YZ+BX:XZ:AX:BX].$

Then there exists a sequence of points
$P_{n}\in({\mathbb{P}}^{4}_{\overline{\mathbb{Q}}})_{f}(\overline{\mathbb{Q}})$
for which $\alpha_{f}(P_{n})$ exists and $\\{\alpha_{f}(P_{n})\\}_{n}$ is an
infinite set.

Recently, Bell, Diller, and Jonsson constructed an example of dominant
rational self-map whose first dynamical degree is a transcendental number [4]
(cf. [3] as well). Wang and the author gave a counter example to part (2) by
showing that the map admits a point whose arithmetic degree is equal to the
first dynamical degree.

######  Theorem 2.8 ([42, Theorem D]).

There is a dominant rational self-map
$f\colon{\mathbb{P}}^{2}_{\overline{\mathbb{Q}}}\dashrightarrow{\mathbb{P}}^{2}_{\overline{\mathbb{Q}}}$
and
$x\in({\mathbb{P}}^{2}_{\overline{\mathbb{Q}}})_{f}(\overline{\mathbb{Q}})$
such that $\alpha_{f}(x)$ exists, $\alpha_{f}(x)=\delta_{1}(f)$, and
$\alpha_{f}(x)$ is a transcendental number.

The existence of the limit defining arithmetic degree (Conjecture 2.6(1)) has
been known for morphisms (i.e. $I_{f}=\emptyset$) since when the conjecture
was formulated. Moreover, when $f$ is a morphism, part (2)(3) are actually
true.

######  Theorem 2.9 ([24, Theorem 3, Remark 23]).

Let $X$ be a normal projective variety and let $f\colon X\longrightarrow X$ be
a surjective self-morphism both defined over $\overline{\mathbb{Q}}$.

  1. (1)

For any $x\in X(\overline{\mathbb{Q}})$, $\alpha_{f}(x)$ exists and is an
algebraic integer.

  2. (2)

The set

(2.16) $\displaystyle\left\\{\alpha_{f}(x)\ \middle|\ x\in
X(\overline{\mathbb{Q}})\right\\}$

is a finite set.

More precisely, $\alpha_{f}(x)$ is either $1$ or the absolute value of an
eigenvalue of $f^{*}\colon N^{1}(X)_{{\mathbb{Q}}}\longrightarrow
N^{1}(X)_{{\mathbb{Q}}}$.

We already know that parts (2) and (3) of Conjecture 2.6 are not generally
true for rational maps, and parts (1) - (3) have been proven for morphisms.
Therefore, main interest lies in part (4), namely
$\alpha_{f}(x)=\delta_{1}(f)$ for dense orbits. Nonetheless, it is also
intriguing to investigate which maps satisfy parts (2) and (3), given that
these parts still hold true for certain special rational maps like monomial
maps. Further, the existence of the arithmetic degree (part (1) of Conjecture
2.6) for rational maps remains a challenge. But for dense orbits, this part is
typically proven during the process of proving $\alpha_{f}(x)=\delta_{1}(f)$.”

In this note, when we refer to the “Kawaguchi-Silverman conjecture”, it
primarily means this $\alpha_{f}=\delta_{1}(f)$ equality. While the Kawaguchi-
Silverman conjecture is originally stated only for smooth projective
varieties, there is no specific reason to exclude singular varieties.

To simplify the exposition, let us introduce the following conjecture, which
is the essential part of the Kawaguchi-Silverman conjecture.

######  Conjecture 2.10 (Kawaguchi-Silverman $\alpha=\delta$ conjecture).

Let $X$ be a quasi-projective variety defined over $\overline{\mathbb{Q}}$.
Let $f\colon X\dashrightarrow X$ be a dominant rational map. If $x\in
X_{f}(\overline{\mathbb{Q}})$ has Zariski dense orbit, then $\alpha_{f}(x)$
exists and $\alpha_{f}(x)=\delta_{1}(f)$.

######  Remark 2.11 (Smooth vs singular).

At this moment, we do not know if we can reduce the conjecture to the case
where $X$ is smooth. It is always possible to take a resolution of
singularities of $X$, i.e. projective birational morphism $\pi\colon
X^{\prime}\longrightarrow X$ with $X^{\prime}$ is a smooth quasi-projective
variety. Since $\pi$ is birational, $f$ induces a dominant rational map
$f^{\prime}\colon X^{\prime}\dashrightarrow X^{\prime}$ such that
$f\circ\pi=\pi\circ f^{\prime}$. But we do not know if all Zariski dense
orbits of $f$ comes from that of $f^{\prime}$, that is for any point $x\in
X_{f}(\overline{\mathbb{Q}})$, it is not obvious if there is some $l\geq 0$
and $x^{\prime}\in X_{f^{\prime}}(\overline{\mathbb{Q}})$ such that
$f^{l}(x)=\pi(x^{\prime})$. Moreover, even if this is the case, we do not know
if the existence of arithmetic degrees $\alpha_{f}(x)$ and
$\alpha_{f^{\prime}}(x^{\prime})$ are equivalent. See Proposition 4.4 for a
special case of such equivalence.

######  Remark 2.12 (Other ground fields).

The notion of height functions can be generalized to settings beyond number
fields. It is well known that the Weil height machine is available over fields
with proper set of absolute values satisfying product formula. One typical
example of such fields are function fields. But if the coefficient field is
infinite, we do not have the Northcott finiteness property and the exact
analogy of Conjecture 2.10 does not hold. See [41] and [67, section 2] for
more about this direction. In this survey, we focus on number fields as this
is the case that has been most extensively investigated.

Let us introduce the following fundamental inequality between arithmetic
degree and dynamical degree.

######  Theorem 2.13 ([25],[37, Theorem 1.4],[17, Proposition 3.11]).

Let $X$ be a quasi-projective variety defined over $\overline{\mathbb{Q}}$.
Let $f\colon X\dashrightarrow X$ be a dominant rational map. For $x\in
X_{f}(\overline{\mathbb{Q}})$, we have

(2.17) $\displaystyle\overline{\alpha}_{f}(x)\leq\delta_{1}(f).$

This theorem says that the maximal arithmetic complexity that the orbits can
take is at most the first dynamical degree. By this theorem, to prove
Conjecture 2.10, it is enough to show the inequality
$\underline{\alpha}_{f}(x)\geq\delta_{1}(f)$ for $x$ having Zariski dense
orbit.

###  2.3. Understanding Kawaguchi-Silverman conjecture

Roughly speaking, the height of a point is the number of bits you need to
describe the point (cf. [6, Example 1.5.5, Proposition 1.6.6, Lemma 1.6.7]).
Over rational numbers, the number of bits are just the number of digits of
coordinates and Kawaguchi-Silverman conjecture can be interpreted as

(2.18) $\displaystyle\lim_{n\to\infty}\left(\text{number of digits of
coordinates of $f^{n}(x)$}\right)^{1/n}=\delta_{1}(f).$

For a dominant rational self-map
$f\colon{\mathbb{P}}^{N}_{{\mathbb{Q}}}\dashrightarrow{\mathbb{P}}^{N}_{{\mathbb{Q}}}$,
the dynamical degree $\delta_{1}(f)$ has the following equivalent definition:

(2.19) $\displaystyle\delta_{1}(f)=\lim_{n\to\infty}\left(\text{algebraic
degree of $f^{n}$}\right)^{1/n}$

where algebraic degree of $f^{n}$ is the degree of the coprime homogeneous
polynomials defining $f^{n}$. If $f$ is defined by $N+1$ coprime homogeneous
polynomials $P_{0},\dots,P_{N}$ with integer coefficient of degree $d\geq 2$,
then the $n$-times composite of them $P_{0}^{(n)},\dots,P_{N}^{(n)}$ defines
$f^{n}$ and have degree $d^{n}$. But they may have common factors, and
dividing out all the common factors, we get the coprime defining polynomials
and their degree is the algebraic degree of $f^{n}$. Now consider a rational
point $x\in{\mathbb{P}}^{N}({\mathbb{Q}})$. We can write
$x=(a_{0}:\cdots:a_{N})$ with coprime integers $a_{0},\dots,a_{N}$. In this
case, it is easy to see that the height
$h_{{\mathcal{O}}_{{\mathbb{P}}^{N}}(1)}$ can be chosen so that the height of
$x$ is $\log\max\\{|a_{0}|,\dots,|a_{N}|\\}$, which is approximately equal to
the number of digits up to constant multiple. The homogeneous coordinates of
$f^{n}(x)$ are computed by applying the defining homogeneous polynomials of
$f$ repeatedly:
$f^{n}(x)=(P_{0}^{(n)}(a_{0},\dots,a_{N}):\cdots:P_{N}^{(n)}(a_{0},\dots,a_{N}))$.
After $n$-times iterations, we might expect the number of digits to be
$d^{n}$-times larger, but they may have common divisors. Indeed, there will
always be a common factor coming from the common factor of the polynomials
$P_{0}^{(n)},\dots,P_{N}^{(n)}$. Kawaguchi-Silverman conjecture essentially
claims that this is the only common factor:

(2.20)
$\displaystyle\lim_{n\to\infty}\left(\log\max\left\\{\left|\frac{P_{0}^{(n)}(a_{0},\dots,a_{N})}{G_{n}}\right|,\dots,\left|\frac{P_{N}^{(n)}(a_{0},\dots,a_{N})}{G_{n}}\right|\right\\}\right)^{1/n}$
(2.21)
$\displaystyle=\lim_{n\to\infty}\left(\deg\frac{P_{0}^{(n)}}{\gcd(P_{0}^{(n)},\dots,P_{N}^{(n)})}\right)^{1/n}$

where
$G_{n}=\gcd\left(P_{0}^{(n)}(a_{0},\dots,a_{N}),\dots,P_{N}^{(n)}(a_{0},\dots,a_{N})\right)$.

To end this section, we present an example of numerical computation. Let us
consider the birational self-map

(2.22) $\displaystyle
f\colon{\mathbb{P}}^{2}_{{\mathbb{Q}}}\dashrightarrow{\mathbb{P}}^{2}_{{\mathbb{Q}}},\left(x:y:z\right)\mapsto\left(x(y+z):x(y-z):(y-z)^{2}\right).$

We can easily show that the algebraic degrees $\deg_{a}f^{n}$ of $f^{n}$
satisfies

(2.23) $\displaystyle\deg_{a}f^{0}=1,\deg_{a}f=2$ (2.24)
$\displaystyle\deg_{a}f^{n+2}\leq\deg_{a}f^{n+1}+\deg_{a}f^{n}\quad n\geq 0.$

Thus the degree sequence is bounded above by Fibonacci sequence and the first
dynamical degree of $f$ is at most the golden ratio $(1+\sqrt{5})/2\approx
1.618..$. The following is the first few iterates of the point $(5:3:1)$. Here
we write $f^{n}(5:3:1)=(a(n):b(n):c(n))$ where $a(n),b(n),c(n)$ are coprime
integers.

$n$ | $a(n)$ | $b(n)$ | $c(n)$
---|---|---|---
$0$ | $5$ | $3$ | $1$
$1$ | $10$ | $5$ | $2$
$2$ | $70$ | $30$ | $9$
$3$ | $130$ | $70$ | $21$
$4$ | $1690$ | $910$ | $343$
$5$ | $302510$ | $136890$ | $45927$
$6$ | $682765070$ | $339718730$ | $102151449$
$7$ | $268649620388110$ | $144436902263290$ | $50256645593707$

For this orbit, the sequence defining arithmetic degree

(2.25) $\displaystyle
h_{{\mathcal{O}}_{{\mathbb{P}}^{2}}(1)}(f^{n}(5:3:1))^{1/n}=\left(\log\max\\{|a(n)|,|b(n)|,|c(n)|\\}\right)^{1/n}\quad(n\geq
1)$

is

(2.26) $\displaystyle
2.302..,2.061..,1.694..,1.651..,1.660..,1.652..,1.649..,1.645..,1.643..\cdots.$

##  3\. A list of known results

There are many special cases where the equality $\alpha_{f}(x)=\delta_{1}(f)$
is known. The following is a rough summary of these results. This list may not
be exhaustive and we do not observe the historical order. Additionally, we do
not provide definitions for the terminologies used here. Let us restate
Conjecture 2.10.

######  Conjecture (=Conjecture 2.10. Kawaguchi-Silverman $\alpha=\delta$
conjecture).

Let $X$ be a quasi-projective variety defined over $\overline{\mathbb{Q}}$.
Let $f\colon X\dashrightarrow X$ be a dominant rational map. If $x\in
X_{f}(\overline{\mathbb{Q}})$ has Zariski dense orbit, then $\alpha_{f}(x)$
exists and $\alpha_{f}(x)=\delta_{1}(f)$.

This conjecture is proven in the following cases.

  1. (1)

$X$ is an arbitrary projective variety and $f$ is a polarized surjective self-
morphism. This case includes the case where $f$ is a surjective self-morphism
on a curve, ${\mathbb{P}}^{n}_{\overline{\mathbb{Q}}}$, or more generally a
projective variety with Picard number one. (cf. Theorem 5.4.)

  2. (2)

$X$ is a projective surface and $f$ is a surjective self-morphism. (cf.
Theorems 6.10, 5.15 and 5.14.)

  3. (3)

$X$ is a semi-abelian variety and $f$ is a surjective self-morphism. (cf.
Theorems 5.12, 5.8, 5.7 and 5.5.)

  4. (4)

$X$ is a linear algebraic group and $f$ is a surjective group endomorphism.
(cf. Theorem 5.9.)

  5. (5)

$X$ is a Mori dream space or (not necessarily ${\mathbb{Q}}$-factorial)
variety of Fano type and $f$ is a surjective self-morphism. This case includes
the case where $X$ is a projective toric variety and a smooth projective
rationally connected variety admitting int-amplified endomorphism. (cf.
Theorems 5.19 and 5.20.)

  6. (6)

$X$ is a Hyper-Kähler variety and $f$ is a surjective self-morphism. (cf.
Theorem 5.14.)

  7. (7)

Automorphisms on ${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$ and regular affine
automorphisms on ${\mathbb{A}}^{n}_{\overline{\mathbb{Q}}}$. (cf. Theorems
5.22, 5.23 and 5.24.)

  8. (8)

Self-morphisms on ${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$ with some
conditions on dynamical degrees. (cf. Theorems 5.24, 5.25 and 7.3.)

  9. (9)

There are many special cases proven by using minimal model theory and
projective geometry. These results are mostly on surjective self-morphisms on
projective varieties. Minimal model theory is particularly effective when the
variety admits at least one polarized or more generally int-amplified
endomorphism. See section 6 for details.

  10. (10)

$X$ is a smooth projective variety, $f$ is a $1$-cohomologically hyperbolic
rational map, and the $f$-orbit of $x$ is generic. (cf. Theorem 7.6.)

##  4\. First properties

Let us collect some of basic properties and observations that are frequently
used in the study of Kawaguchi-Silverman conjecture.

######  Proposition 4.1.

Let $X$ be a quasi-projective variety defined over $\overline{\mathbb{Q}}$.
Let $f\colon X\dashrightarrow X$ be a dominant rational map. Let $x\in
X_{f}(\overline{\mathbb{Q}})$. For integers $m\geq 1$ and $l\geq 0$, we have

(4.1)
$\displaystyle\underline{\alpha}_{f^{m}}(f^{l}(x))=\underline{\alpha}_{f}(x)^{m}$
(4.2)
$\displaystyle\overline{\alpha}_{f^{m}}(f^{l}(x))=\overline{\alpha}_{f}(x)^{m}.$

In particular, if one of $\alpha_{f^{m}}(f^{l}(x))$ and $\alpha_{f}(x)$
exists, then the other exists and
$\alpha_{f^{m}}(f^{l}(x))=\alpha_{f}(x)^{m}$.

###### Proof.

Use the same argument as in [37, Corollary 3.4]. The proof of [37, Corollary
3.4] uses [37, Lemma 3.3], which assumes $X$ is smooth projective. Use [17,
Lemma 3.8] instead, then the same calculation works. ∎

The dynamical degree is also compatible with taking iterates:
$\delta_{1}(f^{m})=\delta_{1}(f)^{m}$. Also the orbit $O_{f}(x)$ is Zariski
dense if and only if $O_{f^{m}}(f^{l}(x))$ is Zariski dense, so we can always
replace $f$ with its iterates to prove Conjecture 2.10.

It is easy to see from the definition that the arithmetic degree is compatible
with restriction to closed subvarieties. That is, if $f\colon X\dashrightarrow
X$ is a dominant rational map and $Y\subset X$ is a closed subvariety such
that $Y\not\subset I_{f}$, $f(Y\setminus I_{f})\subset Y$, and $f|_{Y}\colon
Y\dashrightarrow Y$ is dominant, then we can form the following diagram:

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{Y}}$$\scriptstyle{\bigcap}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcap}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X.}$

If $x\in X_{f}(\overline{\mathbb{Q}})\cap Y$, then
$\underline{\alpha}_{f}(x)=\underline{\alpha}_{f|_{Y}}(x)$ and
$\overline{\alpha}_{f}(x)=\overline{\alpha}_{f|_{Y}}(x)$.

When two maps are semi-conjugate by a dominant rational map, we have
inequalities between arithmetic degrees.

######  Proposition 4.2.

Let $X,Y$ be quasi-projective varieties defined over $\overline{\mathbb{Q}}$.
Consider the following commutative diagram of dominant rational maps:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

Let $x\in X_{f}(\overline{\mathbb{Q}})$ be such that

  * •

the orbit $O_{f}(x)$ avoids the indeterminacy locus $I_{\pi}$ of $\pi$:
$O_{f}(x)\cap I_{\pi}=\emptyset$;

  * •

$\pi(x)\in Y_{g}(\overline{\mathbb{Q}})$.

Then we have

(4.3)
$\displaystyle\underline{\alpha}_{g}(\pi(x))\leq\underline{\alpha}_{f}(x)$
(4.4)
$\displaystyle\overline{\alpha}_{g}(\pi(x))\leq\overline{\alpha}_{f}(x).$

###### Proof.

This follows from the definition of arithmetic degree and [17, Lemma 3.8]. ∎

A typical application of this proposition is the following. Suppose $f,g$, and
$\pi$ are morphisms and $\delta_{1}(f)=\delta_{1}(g)$, then for any point
$x\in X(\overline{\mathbb{Q}})$, we have
$\underline{\alpha}_{f}(x)\geq\underline{\alpha}_{g}(\pi(x))$. If $O_{f}(x)$
is Zariski dense, then so is $O_{g}(\pi(x))$ and hence Conjecture 2.10 for
$(X,f)$ follows from that of $(Y,g)$. From this observation, we have for
example:

######  Lemma 4.3.

Let $X,Y$ be projective varieties over $\overline{\mathbb{Q}}$ and $f\colon
X\longrightarrow X$, $g\colon Y\longrightarrow Y$ be surjective self-
morphisms. Then Conjecture 2.10 for $f\times g\colon X\times Y\longrightarrow
X\times Y$ follows from that of $f$ and $g$.

###### Proof.

This follows from $\alpha_{f\times
g}(x,y)=\max\\{\alpha_{f}(x),\alpha_{g}(y)\\}$ and $\delta_{1}(f\times
g)=\max\\{\delta_{1}(f),\delta_{1}(g)\\}$. ∎

In general, there are self-morphisms on a product variety that are not of the
form $f\times g$, but there are several cases where this happens automatically
(at least after some iterates). See [55] for some results to this direction.

When two maps are conjugate by a generically finite morphism and the self-maps
are quasi-finite, we have the invariance of arithmetic degrees for dense
orbits.

######  Proposition 4.4 ([17, Proposition 3.13]).

Let $X,Y$ be quasi-projective varieties over $\overline{\mathbb{Q}}$ and
consider the following commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

where $f,g$ are quasi-finite dominant morphisms and $\pi$ is a generically
finite dominant morphism. Then for $x\in X(\overline{\mathbb{Q}})$, the orbit
$O_{f}(x)$ is Zariski dense in $X$ if and only if $O_{g}(\pi(x))$ is Zariski
dense in $Y$. Moreover, if $O_{f}(x)$ is Zariski dense in $X$, then
$\underline{\alpha}_{f}(x)=\underline{\alpha}_{g}(\pi(x))$ and
$\overline{\alpha}_{f}(x)=\overline{\alpha}_{g}(\pi(x))$. In particular,
Conjecture 2.10 for $f$ is equivalent to that of $g$.

We conclude this section by remarking that Kawaguchi-Silverman conjecture is
vacuous, i.e. there is no Zariski dense orbits, when the Kodaira dimension of
the variety is positive.

######  Proposition 4.5.

Let $X$ be a quasi-projective variety over $\overline{\mathbb{Q}}$ and
$f\colon X\dashrightarrow X$ be a dominant rational map. If the Kodaira
dimension of $X$ is positive, then $f$ does not have any Zariski dense orbits.
Here the Kodaira dimension of $X$ is by definition the Kodaira dimension of
the smooth projective model of $X$. In particular, Conjecture 2.10 is vacuous
in this case.

###### Proof.

This is an application of a variant of the finiteness of pluricanonical
representation. See [49, Theorem A], [40, Remark 1.2]. ∎

As we have this proposition, the study of Kawaguchi-Silverman conjecture
focuses on varieties of non-positive Kodaira dimension.

##  5\. Canonical heights

A typical strategy to prove the Kawaguchi-Silverman conjecture is to construct
a height function that is compatible with the self-map. Such a function is
often referred to as a canonical height function. This function is a dynamical
analogue of the Néron-Tate height function on abelian varieties. Ever since
Call and Silverman established the polarized endomorphism case in [8], many
variants have appeared, reflecting the structure of the dynamical system under
consideration.

Now, let us outline a typical argument used to prove the Kawaguchi-Silverman
conjecture using canonical height functions.

Let $f\colon X\dashrightarrow X$ be a dominant rational self-map and $h_{H}$
be a height function associated with an ample divisor $H$ on $X$. Let $K$ be a
number field over which $X$ and $f$ are both defined. Suppose we have
constructed a function $\hat{h}\colon
X_{f}(\overline{\mathbb{Q}})\longrightarrow{\mathbb{R}}$ with the following
properties:

(5.1) $\displaystyle\hat{h}\leq C_{1}h_{H}+C_{2}\quad\text{for some constants
$C_{1},C_{2}>0$;}$ (5.2) $\displaystyle\hat{h}\circ f=\delta_{1}(f)\hat{h};$
(5.5) $\displaystyle(*)\quad$ The set $\left\\{x\in X(L)\ \middle|\
\hat{h}(x)\leq 0\right\\}$ is not Zariski dense in $X$ for any field extension
$K\subset L\subset\overline{\mathbb{Q}}$ with $[L:K]<\infty$.

Note that if $\hat{h}$ is itself equal to, for example, an ample height
function, then the first and the third conditions are automatically satisfied.
Now, for $x\in X_{f}(\overline{\mathbb{Q}})$ we have

(5.6) $\displaystyle
C_{1}h_{H}(f^{n}(x))+C_{2}\geq\hat{h}(f^{n}(x))=\delta_{1}(f)^{n}\hat{h}(x).$

If $\hat{h}(x)\leq 0$, then $\hat{h}(f^{n}(x))=\delta_{1}(f)^{n}\hat{h}(x)\leq
0$ as well. If the point $x$ is defined over $L$, where $L$ is a finite
extension of $K$, then all $x,f(x),f^{2}(x),\dots$ are defined over $L$. Thus
if the orbit $O_{f}(x)$ is Zariski dense, we must have $\hat{h}(x)>0$. Thus we
get

(5.7)
$\displaystyle\underline{\alpha}_{f}(x)=\liminf_{n\to\infty}h_{H}^{+}(f^{n}(x))^{1/n}\geq\liminf_{n\to\infty}h_{H}(f^{n}(x))^{1/n}$
(5.8)
$\displaystyle=\liminf_{n\to\infty}(C_{1}h_{H}(f^{n}(x))+C_{2})^{1/n}\geq\liminf_{n\to\infty}(\delta_{1}(f)^{n}\hat{h}(x))^{1/n}=\delta_{1}(f).$

Since we know $\overline{\alpha}_{f}(x)\leq\delta_{1}(f)$ (Theorem 2.13), we
get $\alpha_{f}(x)$ exists and is equal to $\delta_{1}(f)$.

Let us add some remark on the condition $(*)$. It is a weakened version of
Northcott finiteness property for height functions associated with ample
divisors (see, for example, [6, Theorem 2.4.9] for the Northcott property of
ample height functions). For $(*)$, we do not need ampleness and we have the
following.

######  Proposition 5.1 (cf. [36, Proposition 3.5],[33, Proposition 5.1]).

Let $X$ be a geometrically irreducible normal projective variety over a number
field $K$. Let $D$ be an ${\mathbb{R}}$-Cartier divisor on $X_{\overline{K}}$
and $h_{D}$ be a height function associated with $D$. If one of the following
conditions is satisfied, then the set

(5.9) $\displaystyle\left\\{x\in X(L)\ \middle|\ K\subset
L\subset\overline{K},[L:K]\leq d,h_{D}(x)\leq B\right\\}$

is not Zariski dense in $X_{\overline{K}}$ for all $B,d\geq 1$.

  1. (1)

$D$ is Cartier and $\kappa(X,D)>0$, i.e. $\dim
H^{0}(X,{\mathcal{O}}_{X}(mD))\geq 2$ for some $m\in{\mathbb{Z}}_{>0}$.

  2. (2)

$X_{\overline{K}}$ is ${\mathbb{Q}}$-factorial and $D$ is
${\mathbb{R}}$-linearly equivalent to two effective ${\mathbb{R}}$-divisors,
i.e. $\exists D^{\prime},D^{\prime\prime}$ such that
$D^{\prime},D^{\prime\prime}$ are effective ${\mathbb{R}}$-divisors on
$X_{\overline{K}}$, $D^{\prime}\neq D^{\prime\prime}$, and
$D\sim_{{\mathbb{R}}}D^{\prime}\sim_{{\mathbb{R}}}D^{\prime\prime}$.

If $\hat{h}$ can be understood as a height function associated with an
${\mathbb{R}}$-Cartier divisor $D$ and one of the conditions in Proposition
5.1 is satisfied, then we can confirm $(*)$.

Now, let us introduce several cases where we can construct a nice canonical
height functions and prove Kawaguchi-Silverman conjecture by using them.

###  5.1. Polarized endomorphisms and nef canonical height

Polarized endomorphism is probably the first and easiest case that Kawaguchi-
Silverman conjecture was proven. This case is a model case in the study of the
conjecture.

######  Definition 5.2.

Let $X$ be a projective variety over $\overline{\mathbb{Q}}$ and let $f\colon
X\longrightarrow X$ be a surjective self-morphism. Then $f$ is said to be
polarized if there is an ample Cartier divisor $H$ such that $f^{*}H\sim dH$
for some integer $d>1$.

Note that in this case we have $d=\delta_{1}(f)$.

######  Remark 5.3.

It is known that the linear equivalence in the definition can be replaced with
numerical equivalence. More precisely, $f$ is polarized if and only if there
is a rational number $d>1$ and a big ${\mathbb{R}}$-Cartier divisor $D$ such
that $f^{*}D\equiv dD$ [45, Proposition 1.1].

######  Theorem 5.4 ([8],[25, Theorem 5]).

Notation as in Definition 5.2. For any $x\in X(\overline{\mathbb{Q}})$, the
limit

(5.10)
$\displaystyle\hat{h}_{H,f}(x):=\lim_{n\to\infty}\frac{h_{H}(f^{n}(x))}{d^{n}}$

exists. Moreover the function $\hat{h}_{H,f}$ satisfies

(5.11) $\displaystyle\hat{h}_{H,f}=h_{H}+O(1)$ (5.12)
$\displaystyle\hat{h}_{H,f}\circ f=d\hat{h}_{H,f}.$

In particular, Conjecture 2.10 is true for $f$. Moreover,
$\alpha_{f}(x)=d=d_{1}(f)$ if and only if $x$ has infinite $f$-orbit.

Note that for any surjective self-morphism $f\colon X\longrightarrow X$ on a
projective variety $X$, there is always a numerically non-zero nef
${\mathbb{R}}$-divisor $D$ such that $f^{*}D\equiv\delta_{1}(f)D$. (This is an
application of Perron-Frobenius type theorem [5]. Note that the nef cone
$\operatorname{Nef}(X)$ is strictly convex and $f^{*}$ preserves the nef
cone.) If $\delta_{1}(f)>1$, we can construct the similar canonical height
function

(5.13)
$\displaystyle\hat{h}_{D,f}(x)=\lim_{n\to\infty}\frac{\hat{h}_{D}(f^{n}(x))}{\delta_{1}(f)^{n}}$

called nef canonical height function. This function also satisfies similar
properties

(5.14) $\displaystyle\hat{h}_{D,f}=h_{D}+O\left(\sqrt{h_{H}^{+}}\right)$
(5.15) $\displaystyle\hat{h}_{D,f}\circ f=\delta_{1}(f)\hat{h}_{D,f}$

where $h_{H}$ is any ample height function. Here the $O(1)$ is replaced with
$O\left(\sqrt{h_{H}^{+}}\right)$ due to the fact that
$f^{*}D\equiv\delta_{1}(f)D$ is numerical equivalence and not the linear
equivalence. Note that if $\delta_{1}(f)>1$ and at least $X$ is smooth, it is
actually possible to choose nef divisor $D$ so that
$f^{*}D\sim_{{\mathbb{R}}}\delta_{1}(f)D$ (cf. [53, Remark 5.11]) and
$\hat{h}_{D,f}=h_{D}+O\left(1\right)$. But the problem is that we do not have
the Northcott property in general for $h_{D}$ when $D$ is nef and not ample.
Let us also remark that a function $\hat{h}_{D,f}$ satisfying eq. 5.14 is
unique.

###  5.2. Abelian varieties

For group endomorphisms on abelian varieties, Kawaguchi and Silverman gave a
description of the vanishing locus of nef canonical height function and as a
consequence proved Conjecture 2.10.

######  Theorem 5.5 ([24]).

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and let $f\colon
X\longrightarrow X$ be a surjective group endomorphism. Suppose
$\delta_{1}(f)>1$. Let $D$ be a non-zero symmetric nef ${\mathbb{R}}$-divisor
on $A$ such that $f^{*}D\equiv\delta_{1}(f)D$. (We can always make $D$
symmetric by replacing $D$ with $D+[-1]^{*}D$. Note that $f$ commutes with
$[-1]$.) Then for $x\in A(\overline{\mathbb{Q}})$, we can define the canonical
height function

(5.16)
$\displaystyle\hat{h}_{D,f}(x):=\lim_{n\to\infty}\frac{h_{D}(f^{n}(x))}{\delta_{1}(f)^{n}}.$

Moreover, there is a proper abelian subvariety $B\subset A$ such that

(5.17) $\displaystyle\left\\{x\in A(\overline{\mathbb{Q}})\ \middle|\
\hat{h}_{D,f}(x)=0\right\\}=B(\overline{\mathbb{Q}})+A(\overline{\mathbb{Q}})_{\rm
tors}$

where $A(\overline{\mathbb{Q}})_{\rm tors}$ is the set of torsion points.

Since torsion points defined over a fixed number field are finite, the
vanishing locus of $\hat{h}_{D,f}$ is Zariski non-dense over a fixed number
field. In particular, Conjecture 2.10 is true for $f$.

######  Remark 5.6.

In [24], instead of $\hat{h}_{D,f}$, they use

(5.18)
$\displaystyle\hat{q}_{A,D}(x)=\lim_{n\to\infty}\frac{h_{D}(nx)}{n^{2}},$

which is usual quadratic part of canonical height function on abelian
varieties. As we choose $D$ symmetric, we can check that this function
satisfies eq. 5.14 and hence $\hat{h}_{D,f}=\hat{q}_{A,D}$.

######  Remark 5.7.

By a following work by Silverman [61], Conjecture 2.10 is proven for
surjective self-morphism on abelian varieties. Note that every dominant
rational self-map on an abelian variety is automatically a morphism. This
follows from more general fact that every rational map from a smooth variety
to abelian variety is a morphism because there is no rational curves on an
abelian variety (cf. [28, Corollary 1.5]). Therefore, Conjecture 2.10 is
completely solved for abelian varieties.

Based on this abelian variety case and a prior work by Silverman on the
monomial maps on algebraic torus (see section 5.3), Sano and the author proved
Conjecture 2.10 for all surjective self-morphisms on semi-abelian varieties.

######  Theorem 5.8 ([39, Theorem 1.1]).

Let $X$ be a semi-abelian variety over $\overline{\mathbb{Q}}$ and let
$f\colon X\longrightarrow X$ be a surjective self-morphism. Then Conjecture
2.10 holds for $f$.

In this case, other parts of Conjecture 2.6 are actually true, i.e. the
arithmetic degrees are algebraic integers and they take only finitely many
values. See [39] for details.

Having seen that the conjecture has been solved for self-morphisms on semi-
abelian varieties, it is natural to ask about its validity for an arbitrary
algebraic groups (they are automatically quasi-projective cf. [63, Tag 0BF7]).
As for linear algebraic groups and their group endomorphisms, the answer is
positive.

######  Theorem 5.9 ([36, Theorem 6.1]).

Let $X$ be a connected linear algebraic group over $\overline{\mathbb{Q}}$ and
let $f\colon X\longrightarrow X$ be a surjective group endomorphism. Then
Conjecture 2.10 holds for $f$.

Note that it is very difficult to remove the assumption that $f$ is a group
endomorphism, because for example the unipotent group consisting of upper
triangular matrices with diagonal entries $1$ is an affine space as varieties.
More generally, every linear algebraic group is birational to a projective
space as varieties (cf. [7, 14.14]).

###  5.3. Monomial maps

A monomial map is a group endomorphism of an algebraic torus
${\mathbb{G}}_{{\mathrm{m}}}^{N}$. It is determined by an $N\times N$ matrix
$A=(a_{ij})$ with integer coefficients. The map corresponds to $A$ is

(5.19) $\displaystyle\varphi_{A}\colon$
$\displaystyle{\mathbb{G}}_{{\mathrm{m}}}^{N}\longrightarrow{\mathbb{G}}_{{\mathrm{m}}}^{N}$
(5.20)
$\displaystyle(X_{1},\dots,X_{N})\mapsto(X_{1}^{a_{11}}X_{2}^{a_{12}}\cdots
X_{N}^{a_{1N}},\dots,X_{1}^{a_{N1}}X_{2}^{a_{N2}}\cdots X_{N}^{a_{NN}}).$

This map is dominant if and only if $\det A\neq 0$. Note that the first
dynamical degree of $\varphi_{A}$ is equal to the spectral radius of $A$.
Silverman introduced the following canonical height function:

######  Definition 5.10.

Let
$h\colon{\mathbb{G}}_{{\mathrm{m}}}^{N}(\overline{\mathbb{Q}})\longrightarrow{\mathbb{R}}$
be the naive logarithmic height function (defined by embedding
${\mathbb{G}}_{{\mathrm{m}}}^{N}\subset{\mathbb{P}}^{N}$). Let

(5.23) $\displaystyle l(A)=\left(\hbox{\hbox{\kern
0.0pt\raise-7.91667pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
the size of the maximal Jordan blocks of $A$}\\\\\hbox{{\ignorespaces
corresponding to eigenvalues of maximal absolute value}\crcr}}}}\right)-1.}}$

Then define

(5.24)
$\displaystyle\hat{h}_{\varphi_{A}}(x)=\limsup_{n\to\infty}\frac{h(\varphi_{A}^{n}(x))}{n^{l(A)}\delta_{1}(f)^{n}}$

for $x\in{\mathbb{G}}_{{\mathrm{m}}}^{N}(\overline{\mathbb{Q}})$.

######  Remark 5.11.

The factor $n^{l(A)}$ in the definition of canonical height corresponds to the
following fact on the growth rate of degree sequence [60, Theorem 24]. Let
$\deg_{a}(\varphi_{A}^{n})$ be the algebraic degree, i.e. degree of defining
(coprime) homogeneous polynomials of $\varphi_{A}$ considered as a rational
self-map of ${\mathbb{P}}^{N}$. Then $\deg_{a}(\varphi_{A}^{n})\gg\ll
n^{l(A)}\delta_{1}(\varphi_{A})^{n}$ for $n\geq 1$.

######  Theorem 5.12 ([60, Theorem 4]).

Notation as above. Suppose $\delta_{1}(\varphi_{A})>1$.

  1. (1)

The arithmetic degrees of $\varphi_{A}$ satisfy

$\\{\alpha_{\varphi_{A}}(x)\mid
x\in{\mathbb{G}}_{{\mathrm{m}}}^{N}(\overline{\mathbb{Q}})\\}\subset\\{\text{eigenvalues
of }A\\}\cup\\{1\\}.$

In particular, $\alpha_{\varphi_{A}}(x)$ is an algebraic integer for all
$x\in{\mathbb{G}}_{{\mathrm{m}}}^{N}(\overline{\mathbb{Q}})$.

  2. (2)

Let $x\in{\mathbb{G}}_{{\mathrm{m}}}^{N}(\overline{\mathbb{Q}})$ be a point
with $\hat{h}_{\varphi_{A}}(x)=0$. Then the orbit $O_{\phi}(x)$ is contained
in a proper (possibly disconnected) algebraic subgroup of
${\mathbb{G}}_{{\mathrm{m}}}^{N}$. In particular, Conjecture 2.10 holds for
$\varphi_{A}$.

  3. (3)

If the matrix $A$ is diagonalizable over $\mathbb{C}$, then

$\hat{h}_{\varphi_{A}}(x)=0\Longleftrightarrow\alpha_{\varphi_{A}}(x)<\delta_{1}(\varphi_{A}).$

  4. (4)

If the characteristic polynomial of the matrix $A$ is irreducible over
$\mathbb{Q}$, then

$\hat{h}_{\varphi_{A}}(x)=0\Longleftrightarrow O_{\varphi_{A}}(x)\text{ is
finite.}$

######  Remark 5.13.

Based on this result, Lin proved the existence of arithmetic degrees for
monomial maps on projective toric varieties. (Note that Conjecture 2.10 for
this case follows directly from Theorem 5.12.) Moreover, Lin constructed an
example of monomial maps for which the sequence in eq. 5.24 has infinitely
many accumulation points [34, Example 3.1].

###  5.4. Automorphisms on surfaces and Hyper-Kähler varieties

When you have more than one eigendivisors and can construct canonical height
functions associated with them, it is reasonable to expect that the sum of
them might have desired positivity even if each individual function does not.
This idea already appeared in [59], and now has numerous variants.

Let us consider an automorphism $f\colon X\longrightarrow X$ with
$\delta_{1}(f)>1$. Then we have two nef canonical height functions
$\hat{h}_{D_{+},f}$ and $\hat{h}_{D_{-},f^{-1}}$ with respect to $f$ and its
inverse $f^{-1}$. Here $D_{+}$ and $D_{-}$ are non-zero nef
${\mathbb{R}}$-divisors such that $f^{*}D_{+}\equiv\delta_{1}(f)D_{+}$ and
$(f^{-1})^{*}D_{-}\equiv\delta_{1}(f^{-1})D_{-}$. (Note that we have
$\delta_{1}(f^{-1})=\delta_{\dim X-1}(f)$ and log concavity implies
$\delta_{\dim X-1}(f)>1$.) Then

(5.25)
$\displaystyle\hat{h}_{D_{+},f}+\hat{h}_{D_{-},f^{-1}}=h_{D_{+}+D_{-}}+O\left(\sqrt{h_{H}^{+}}\right)$

would have positivity property if so does the divisor $D_{+}+D_{-}$. Kawaguchi
originally used this strategy for surfaces and later Lesieutre and Satriano
extended this idea to Hyper-Kähler varieties.

######  Theorem 5.14 ([21], [23, Theorem 2], [31, Theorem 1.2]).

Conjecture 2.10 is true for automorphisms on

  1. (1)

smooth projective surfaces;

  2. (2)

Hyper-Kähler varieties.

In particular, Conjecture 2.10 holds for all surjective self-morphisms on
Hyper-Kähler varieties because a surjective self-morphism on a Hyper-Kähler
variety is automatically an automorphism.

######  Remark 5.15.

Conjecture 2.10 is now completely proven for surjective self-morphisms on
projective surfaces (cf. Theorem 6.10). Non-isomorphic surjective self-
morphisms on smooth projective surfaces were proven by Sano, Shibata, and the
author [40], and later Meng and Zhang proved the singular case [47]. Except
the Kawaguchi’s automorphism case, the proof heavily relies on classification
(or minimal model theory) of surfaces.

Theorem 5.14 follows from the following.

######  Proposition 5.16 ([21],[31]).

Let $X$ be one of the classes of varieties in Theorem 5.14. Let $f\colon
X\longrightarrow X$ be an automorphism and suppose $\delta_{1}(f)>1$. Then we
have $\delta_{1}(f^{-1})=\delta_{1}(f)$ and there are nef
${\mathbb{R}}$-divisors $D_{+}$ and $D_{-}$ on $X$ such that

(5.26) $\displaystyle f^{*}D_{+}\sim_{{\mathbb{R}}}\delta_{1}(f)D_{+}$ (5.27)
$\displaystyle(f^{-1})^{*}D_{-}\sim_{{\mathbb{R}}}\delta_{1}(f)D_{-}$ (5.28)
$\displaystyle\text{$D_{+}+D_{-}$ is nef and big}.$

In particular, the function $\hat{h}_{D_{+},f}+\hat{h}_{D_{-},f^{-1}}$ is not
bounded above on any Zariski dense set of $K$-rational points with bounded
extension degree $[K:{\mathbb{Q}}]$.

Using the same method, Lesieutre and Satriano proved Conjecture 2.10 for
automorphisms on smooth projective varieties of Picard number two [31, Theorem
2.30].

######  Remark 5.17.

Once you get the bigness of $D_{+}+D_{-}$, it is easy to deduce Conjecture
2.10 from eq. 5.25. Indeed, bigness of $D_{+}+D_{-}$ implies
$h_{D_{+}+D_{-}}\geq Ch_{H}^{+}$ on a non-empty open subset of $X$ where $C$
is a positive constant. Therefore, for a point $x\in X(\overline{\mathbb{Q}})$
with Zariski dense orbit, there is a sequence $n_{1},n_{2},\dots$ such that

(5.29) $\displaystyle
h_{D_{+}+D_{-}}(f^{n_{j}}(x))+O\left(\sqrt{h_{H}^{+}(f^{n_{j}}(x))}\right)$
(5.30) $\displaystyle\geq
h_{H}^{+}(f^{n_{j}}(x))\left(C+O\left(\frac{1}{\sqrt{h_{H}^{+}(f^{n_{j}}(x))}}\right)\right)$

and the right hand side goes to $\infty$ as $j\to\infty$. Thus by eq. 5.25,

(5.31)
$\displaystyle\hat{h}_{D_{+},f}(f^{n_{j}}(x))+\hat{h}_{D_{-},f^{-1}}(f^{n_{j}}(x))$
(5.32)
$\displaystyle=\delta_{1}(f)^{n_{j}}\hat{h}_{D_{+},f}(x)+\delta_{1}(f^{-1})^{-n_{j}}\hat{h}_{D_{-},f^{-1}}(x)$

goes to infinity as $j\to\infty$. Hence we must have $\hat{h}_{D_{+},f}(x)>0$
and this implies $\alpha_{f}(x)\geq\delta_{1}(f)$. In particular, it is not
necessary to find $D_{+}$ and $D_{-}$ so that they are eigenvectors up to
linear equivalence (numerical equivalence is enough).

######  Remark 5.18.

Jonsson and Reschke extended the construction of $\hat{h}_{D_{+}}$ and
$\hat{h}_{D_{-}}$ to birational self-maps on smooth projective surfaces [18,
Theorem D]. But in the paper they say they were not able to prove that their
canonical height is not constant zero. By Theorems 7.6 and 7.7, Kawaguchi-
Silverman conjecture is now almost proven in this case and we know that there
are points with arithmetic degree equal to dynamical degree, but we still do
not know wether Jonsson-Reschke’s canonical height is not identically zero or
not (cf. [42, Remark 4.11]).

###  5.5. Mori dream spaces and varieties of Fano type

A surjective self-morphism $f\colon X\longrightarrow X$ on a normal projective
variety over $\overline{\mathbb{Q}}$ is automatically a finite morphism and it
induces a linear automorphism $f^{*}$ on $N^{1}(X)_{{\mathbb{R}}}$ by pull-
back. Moreover, $f^{*}$ induces a bijection on the nef cone
$\operatorname{Nef}(X)$. Therefore, if $\operatorname{Nef}(X)$ is generated by
finitely many vectors, it is generated by finitely many extremal rays and
these extremal rays are permuted by $f^{*}$. By replacing $f$ with $f^{n}$ for
some $n\geq 1$, we may assume all the extremal rays are eigenvectors of
$f^{*}$. In particular, one of them is the leading eigenvector, i.e.
eigenvector with eigenvalue $\delta_{1}(f)$. If we further assume all the
extremal rays of $\operatorname{Nef}(X)$ are generated by base point free
Cartier divisors, then we can find a nef canonical height function with nice
properties.

######  Theorem 5.19 ([36]).

Let $f\colon X\longrightarrow X$ be a surjective self-morphism on a normal
projective variety over a number field $K$. Suppose

  1. (1)

$N^{1}(X_{\overline{\mathbb{Q}}})_{{\mathbb{Q}}}\simeq\operatorname{Pic}(X_{\overline{\mathbb{Q}}})_{{\mathbb{Q}}}$

  2. (2)

the nef cone $\operatorname{Nef}(X_{\overline{\mathbb{Q}}})$ is generated by
finitely many base point free Cartier divisors

Then after replacing $f$ with its iterates, there is a non-zero nef and base
point free divisor $D$ such that $f^{*}D\sim\delta_{1}(f)D$ and the
corresponding nef canonical height function $\hat{h}_{D,f}$ satisfies the
following property. For any $B\in{\mathbb{R}}$ and $d\geq 1$,

(5.33) $\displaystyle\left\\{x\in X(L)\ \middle|\ [L:K]\leq
d,\hat{h}_{D,f}(x)\leq B\right\\}$

is not Zariski dense. In particular, Conjecture 2.10 is true for $f$.

The assumptions in the theorem are well-studied in algebraic geometry and
there are many varieties that satisfy them.

######  Corollary 5.20.

Conjecture 2.10 is true for surjective self-morphism on Mori dream spaces,
varieties of Fano type, and projective toric varieties.

###### Proof.

The conditions in Theorem 5.19 are actually parts of the definition of Mori
dream spaces. We can check that varieties of Fano type satisfy the conditions
by basic theorems in minimal model theory (base point free theorem and cone
theorem). For projective toric varieties, the conditions follow from [10,
Theorem 6.3.12, Theorem 6.3.15, Theorem 6.3.20]. ∎

The key point in Theorem 5.19, as well as polarized case, is that we can find
a good eigendivisor for $f^{*}$. In [50], Nasserden explored what conditions
on eigenspaces of $f^{*}$ would be sufficient for proving Kawaguchi-Silverman
conjecture.

###  5.6. Regular affine automorphisms

When our self-map is a rational map and not a morphism, it is usually very
difficult to construct canonical height function with nice properties. There
is an early work by Kawaguchi on regular affine automorphisms that can be
understood as a generalization of the strategy in section 5.4.

######  Definition 5.21.

An automorphism $f\colon{\mathbb{A}}^{N}\longrightarrow{\mathbb{A}}^{N}$ over
$\overline{\mathbb{Q}}$ is called regular affine automorphism (regular
polynomial automorphism) if the indeterminacy loci of $f$ and $f^{-1}$
considered as birational maps on ${\mathbb{P}}^{N}$ are disjoint.

Regular affine automorphism is algebraically stable when it is considered as a
rational self-map on the projective space ${\mathbb{P}}^{N}$ via the natural
embedding ${\mathbb{A}}^{N}\subset{\mathbb{P}}^{N}$ (i.e.
$(f^{n})^{*}=(f^{*})^{n}\colon\operatorname{Pic}{\mathbb{P}}^{N}\longrightarrow\operatorname{Pic}{\mathbb{P}}^{N}$,
cf. for example [23, Lemma 19]).

######  Theorem 5.22 ([22]).

Let
$f\colon{\mathbb{A}}^{N}_{\overline{\mathbb{Q}}}\longrightarrow{\mathbb{A}}^{N}_{\overline{\mathbb{Q}}}$
be a regular affine automorphism with degree $d\geq 2$ (i.e. $d$ is the max of
the degrees of the coordinate polynomials). Then $d=\delta_{1}(f)$ since $f$
is algebraically stable as a self-map on
${\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}$. Let $d_{-}$ be the degree of
$f^{-1}$ (note that $d_{-}=\delta_{1}(f^{-1})=\delta_{N-1}(f)\geq 2$). Let
$h\colon{\mathbb{A}}^{N}(\overline{\mathbb{Q}})\longrightarrow{\mathbb{R}}$ be
the naive logarithmic height function (defined by embedding
${\mathbb{A}}^{N}_{\overline{\mathbb{Q}}}\subset{\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}$).
Then the limits

(5.34)
$\displaystyle\hat{h}_{f}^{+}(x)=\lim_{{\mathfrak{n}}\to\infty}\frac{h(f^{n}(x))}{d^{n}}\
\ x\in{\mathbb{A}}^{N}(\overline{\mathbb{Q}})$ (5.35)
$\displaystyle\hat{h}_{f}^{-}(x)=\lim_{{\mathfrak{n}}\to\infty}\frac{h(f^{-n}(x))}{d_{-}^{n}}\
\ x\in{\mathbb{A}}^{N}(\overline{\mathbb{Q}})$

exists and $\hat{h}_{f}^{+}+\hat{h}_{f}^{-}\gg\ll h+O(1)$ (implicit constants
depend at most on $f$). In particular, Conjecture 2.10 is true for $f$.

######  Remark 5.23.

In [20], Kawaguchi proved two dimensional version of Theorem 5.22 for all
automorphisms on ${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$ with dynamical
degree $>1$. We remark that, every automorphism
$f\colon{\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}\longrightarrow{\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$
with $\delta_{1}(f)>1$ is conjugate to a regular affine automorphism (more
precisely, composite of Hénon maps) cf. [20, Theorem 3.1, Proposition 3.2].
Here conjugate means conjugate by an automorphism on
${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$. Therefore, Theorem 5.22 implies
Conjecture 2.10 for all automorphisms on
${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$.

###  5.7. Self-morphisms on ${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$

Jonsson and Wulcan studied the case for self-morphisms on affine place
${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$. They use some results about
equivariant compactification of self-morphisms on
${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$, which is based on prior works by
Favre and Jonsson on valuative trees.

######  Theorem 5.24 ([19, Main Theorem]).

Let
$f\colon{\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}\longrightarrow{\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$
be a dominant self-morphism with $\delta_{1}(f)>\delta_{2}(f)$ ($f$ is said to
have small topological degree). Let
$h\colon{\mathbb{A}}^{2}(\overline{\mathbb{Q}})\longrightarrow{\mathbb{R}}$ be
the naive logarithmic height function (defined by embedding
${\mathbb{A}}^{2}\subset{\mathbb{P}}^{2}$). Then the limit

(5.36)
$\displaystyle\hat{h}(x)=\lim_{n\to\infty}\frac{h(f^{n}(x))}{\delta_{1}(f)^{n}}$

exists for all $x\in{\mathbb{A}}^{2}(\overline{\mathbb{Q}})$. Further,
$\hat{h}$ is not constant $0$ and $\hat{h}(x)=0$ implies

(5.37)
$\displaystyle\overline{\alpha}_{f}(x)=\limsup_{n\to\infty}h(f^{n}(x))^{1/n}\leq\delta_{2}(f)<\delta_{1}(f).$

Moreover, if $f$ is an automorphism, then $\hat{h}(x)=0$ if and only if $x$ is
$f$-periodic. In particular, Conjecture 2.10 holds for automorphisms on
${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$.

We remark that Conjecture 2.10 for automorphisms on
${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$ was already proven by Kawaguchi in
[20] (cf. Remark 5.23). But Jonsson and Wulcan’s proof is different and has
the following extension.

######  Remark 5.25.

There is an unpublished work by Jonsson-Wulcan-Xie on Kawaguchi-Silverman
conjecture for self-morphisms on ${\mathbb{A}}^{2}$. For a dominant self-
morphism
$f\colon{\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}\longrightarrow{\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$,
if $f$ satisfies one of the following conditions, then Conjecture 2.10 holds
for $f$:

  1. (1)

$\delta_{1}(f)\geq\delta_{2}(f)$;

  2. (2)

$\delta_{1}(f)\notin{\mathbb{Z}}$;

  3. (3)

$\delta_{1}(f)\in{\mathbb{Z}}$ and $\gcd(\delta_{1}(f),\delta_{2}(f))\neq 1$.

Also, they proved that Vojta’s conjecture implies Kawaguchi-Silverman
conjecture for $f$. I appreciate them sharing this result with me and allowing
me to include it here.

Let us mention that there is another relevant result Theorems 7.3 and 7.5,
which can be understood as a generalization of Kawaguchi’s approach.

##  6\. Minimal Model Program and Projective geometry

Recently, there have been many advancements in the study of the Kawaguchi-
Silverman conjecture that use techniques from projective geometry and the
minimal model program. The fundamental idea is as follows: Suppose we have a
surjective self-morphism $f\colon X\longrightarrow X$ on a certain projective
variety $X$. If we can find another projective variety $Y$ that is birational
to $X$ and is geometrically well understood (e.g. ${\mathbb{P}}^{N}$, an
abelian variety, or a product of them), then we could deduce the Kawaguchi-
Silverman conjecture for $f$ from that of the induced map $f_{Y}$ on $Y$.
However, the challenge lies in the fact that even if $Y$ is a well-understood
variety, the Kawaguchi-Silverman conjecture for $Y$ is very difficult if
$f_{Y}$ is a rational map. Therefore, the crucial aspect of this strategy is
to construct $Y$ in such a way that $f_{Y}$ is a morphism.

###  6.1. Varieties admitting an int-amplified endomorphism

For a projective variety, the presence of certain special types of surjective
self-morphisms, such as polarized self-morphisms, generally imposes a very
strong condition. There have been efforts to classify all such varieties and
to understand the structure of these self-morphisms. In this context, Meng
introduced the following class of surjective self-morphisms on projective
varieties:

######  Definition 6.1 ([44]).

Let $f\colon X\longrightarrow X$ be a surjective self-morphism on a projective
variety defined over an algebraically closed field. Then $f$ is said to be
int-amplified if there is an ample Cartier divisor $H$ such that $f^{*}H-H$ is
ample.

######  Remark 6.2.

Polarized endomorphisms are int-amplified. Int-amplifiedness is characterized
by the following property: the modulus of all the eigenvalues of $f^{*}\colon
N^{1}(X)\longrightarrow N^{1}(X)$ is greater than $1$ [44, Theorem 3.3].
Contrast to polarized case, being int-amplified is preserved under taking
products. For additional properties of int-amplified endomorphisms, see
Section 3 in [44].

Int-amplified endomorphism was first introduced in the study of equivariant
minimal model program as described below, but the notion seems to be useful in
other context, e.g. [26]. Also int-amplifiedness can be understood as
cohomological hyperbolicity (cf. Definition 7.2): $f\colon X\longrightarrow X$
is int-amplified if and only if $\delta_{\dim X-1}(f)<\delta_{\dim X}(f)=\deg
f$ (at least when $X$ is smooth), i.e. $f$ is $\dim X$-cohomologically
hyperbolic.

Minimal model theory provides us strong tools to study the structure of
algebraic varieties. The basic idea it to perform a series of special type of
transformations that simplify the variety. Meng and Zhang proved that if the
original variety admits an int-amplified endomorphism, then all the process in
MMP can be made equivariant under any surjective self-morphisms up to
iterates. The following is a part of the consequence of their results ([46,
Theorem 1.2], [45, Theorem 1.8],[44, Theorem 1.10]).

######  Theorem 6.3 (Equivariant MMP (Meng-Zhang)).

Let $X$ be a ${\mathbb{Q}}$-factorial klt projective variety over
$\overline{\mathbb{Q}}$ admitting an int-amplified endomorphism. Then for any
surjective self-morphism $f\colon X\longrightarrow X$, there exists a positive
integer $n>0$ and the following commutative diagram of rational maps

$\textstyle{X=X_{0}}$$\scriptstyle{\pi_{0}}$$\scriptstyle{g_{0}=f^{n}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{\pi_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{r-1}}$$\scriptstyle{g_{r-1}}$$\textstyle{X_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{r}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{h}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{r-1}}$$\textstyle{X_{r}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
A}$$\scriptstyle{\nu}$

such that

  1. (1)

$\pi_{i}\colon X_{i}\dashrightarrow X_{i+1}$ is either the divisorial
contraction, flip, or Fano contraction of a $K_{X_{i}}$-negative extremal ray;

  2. (2)

$A$ is an abelian variety and $\nu$ is a finite morphism that is étale in
codimension one, i.e. quasi-étale morphism (Note that $A$ could be a point);

  3. (3)

$g_{i}\colon X_{i}\longrightarrow X_{i}$ for $i=0,\dots,r$ and $h\colon
A\longrightarrow A$ are surjective self-morphisms.

In [47], Meng and Zhang studied a strategy of proving Kawaguchi-Silverman
conjecture and extracted the following special case called Case TIR as the
obstruction to their strategy.

######  Definition 6.4 (Case ${\rm TIR}_{n}$ –Totally Invariant Ramification
case–).

Let $X$ be a normal projective variety of dimension $n\geq 1$, which has only
${\mathbb{Q}}$-factorial klt singularities and admits an int-amplified
endomorphism. Let $f:X\longrightarrow X$ be an arbitrary surjective self-
morphism. Moreover, we impose the following conditions.

  1. (A1)

The anti-Kodaira dimension is zero: $\kappa(X,-K_{X})=0$, and $-K_{X}$ is nef,
whose class is extremal in both the nef cone $\operatorname{Nef}(X)$ and the
pseudo-effective divisors cone $\overline{\operatorname{Eff}}(X)$.

  2. (A2)

There is a prime divisor $D$ such that $f^{*}D=\delta_{1}(f)D$ and
$D\sim_{{\mathbb{Q}}}-K_{X}$. (Here the equality is an equality as divisors.
In particular, $\delta_{1}(f)$ is an integer.)

  3. (A3)

The ramification divisor of $f$ satisfies $\operatorname{Supp}R_{f}=D$.

  4. (A4)

There is an $f$-equivariant Fano contraction $\pi:X\longrightarrow Y$ with
$\delta_{1}(f)>\delta_{1}(f_{Y})(\geq 1)$, where $f_{Y}$ is the induced
surjective self-morphism on $Y$.

  5. (A5)

$\dim(X)\geq\dim(Y)+2\geq 3$.

######  Theorem 6.5 ([47, Theorem 1.7]).

Let $X$ be a ${\mathbb{Q}}$-factorial klt projective variety over
$\overline{\mathbb{Q}}$ admitting an int-amplified endomorphism.

  1. (1)

If $K_{X}$ is pseudo-effective, then Conjecture 2.10 is true for all
surjective self-morphisms on $X$. Note that in this case $X$ is an Q-abelian
variety (i.e. there is a finite surjective morphism from an abelian variety
that is étale in codimension one).

  2. (2)

Assume Conjecture 2.10 is true for Case ${\rm TIR}_{n}$ for $n\leq\dim X$ (it
is enough to assume for the fibrations appearing in an equivariant MMP
starting from X). Then Conjecture 2.10 is true for all surjective self-
morphisms on $X$.

Note that when $X$ is a Q-abelian variety, then every surjective self-morphism
on $X$ lifts to a surjective self-morphism on the abelian variety covering it
and hence Conjecture 2.10 follows from the abelian variety case [47, Theorem
2.8].

The following theorem gives us some insight about the conditions in Case TIR.

######  Theorem 6.6 ([47, Proposition 1.6]).

Let $f:X\longrightarrow X$ be a surjective self-morphism of a
${\mathbb{Q}}$-Gorenstein normal projective variety $X$ over
$\overline{\mathbb{Q}}$ with positive anti-Kodaira dimension:
$\kappa(X,-K_{X})>0$. Suppose $f^{*}K_{X}\equiv\delta_{1}(f)K_{X}$. Then
Conjecture 2.10 holds for $f$.

######  Remark 6.7.

Using equivariant MMP, we may reduce the problem to the case where
$f^{*}K_{X}\equiv\delta_{1}(f)K_{X}$ holds. This theorem ensures once you have
$\kappa(X,-K_{X})>0$, then we get the desired result. Further investigation of
the case $\kappa(X,-K_{X})=0$ leads to the Case TIR (Definition 6.4). See [47]
for more details.

Yoshikawa and the author proved that if the original variety $X$ is smooth
rationally connected, then Case TIR does not happen.

######  Theorem 6.8 ([43]).

Let $X$ be a smooth projective rationally connected variety defined over
$\overline{\mathbb{Q}}$ such that $X$ admits at least one int-amplified
endomorphism. Then for any surjective self-morphism $f\colon X\longrightarrow
X$, Conjecture 2.10 is true.

######  Remark 6.9.

This theorem is first proven in dimension three by Meng and Zhang [47, Theorem
1.11]. After we proved this theorem, Yoshikawa proved that such a variety is
actually a variety of Fano type [68]. Therefore Kawaguchi-Silverman conjecture
follows from Theorem 5.19. But let us remark that Yoshikawa’s proof in [68] is
an extension of our arguments in [43], so it does not give an easier proof of
Kawaguchi-Silverman conjecture.

As another application of minimal model program, Meng and Zhang proved
Kawaguchi-Silverman conjecture for surjective self-morphisms on projective
surfaces (not necessarily smooth).

######  Theorem 6.10 ([47, Theorem 1.3]).

Conjecture 2.10 is true for all surjective self-morphisms on projective
surfaces.

######  Remark 6.11.

For automorphisms, this theorem follows from smooth case, which was proven by
Kawaguchi [21], by taking equivariant resolution of singularities. For
surjective self-morphisms on smooth projective surfaces, this was proven in
[40].

###  6.2. Projective geometry

There are many works that utilize techniques from projective and birational
geometry to prove special cases of Kawaguchi-Silverman conjecture. A common
strategy involves reducing the problem to cases that have already been proven,
or demonstrating that there are no Zariski dense orbits, thereby confirming
Conjecture 2.10 in a vacuous sense. This approach often requires exploiting
the algebro-geometric structures of the varieties in question. In this
subsection, we collect such results.

A smooth projective variety $X$ over $\overline{\mathbb{Q}}$ is said to be
Calabi–Yau if $\dim X\geq 3$,
${\mathcal{O}}_{X}(K_{X})\simeq{\mathcal{O}}_{X}$, and $\dim
H^{0}(\Omega_{X}^{p})=0$ for $0<p<\dim X$. According to the Beauville-
Bogomolov decomposition, Calabi-Yau varieties together with abelian varieties
and Hyper-Kähler varieties are building blocks of smooth projective varieties
with numerically trivial canonical divisors. From this point of view,
Lesieutre and Satriano proved:

######  Theorem 6.12 ([31, Corollary 1.5]).

Let $n$ be a positive integer. Then Conjecture 2.10 is true for all
automorphisms of smooth projective varieties $X$ over $\overline{\mathbb{Q}}$
with dimension at most $n$ and $K_{X}$ numerically trivial if and only if
Conjecture 2.10 is true for all automorphisms of smooth Calabi–Yau varieties
over $\overline{\mathbb{Q}}$ with dimension at most $n$.

See [33, Proposition 3.1] for a generalization of this theorem to singular
varieties. See [32] as well for a related result.

For a smooth projective variety $X$, let us write the irregularity of $X$ by
$q(X)$, i.e. $q(X)=\dim H^{1}(X,{\mathcal{O}}_{X})$. Chen, Lin, and Oguiso
studied Kawaguchi-Silverman conjecture and existence of Zariski dense orbits
for varieties with positive irregularities $q(X)>0$ in [9]. In any dimension,
using albanese morphism and geometry on abelian varieties, they proved:

######  Theorem 6.13 ([9, Theorem 1.5]).

Let $X$ be a smooth projective variety over $\overline{\mathbb{Q}}$ with
Kodaira dimension zero.

  1. (1)

If $q(X)=\dim X$, then Conjecture 2.10 holds for all dominant rational self-
maps $f\colon X\dashrightarrow X$.

  2. (2)

If $q(X)=\dim X-1$, then any birational map $f\colon X\dashrightarrow X$ has
no Zariski dense orbits and hence Conjecture 2.10 is vacuously true.

See [32] for a related result.

######  Remark 6.14.

For a smooth projective variety $X$ with Kodaira dimension zero, the
irregularity is at most $\dim X$: $0\leq q(X)\leq\dim X$. Moreover, if
$q(X)=\dim X$, then the albanese morphism is birational and hence $X$ is
birational to an abelian variety. See [27].

Constructing a fibration is the most common way to reduce Kawaguchi-Silverman
conjecture to easier cases. Let $f\colon X\longrightarrow X$ be a surjective
self-morphism. Suppose we have constructed a fibration (i.e. proper surjective
morphism with connected fibers) $\pi\colon X\longrightarrow Y$ and a self-
morphism $g$ on $Y$ such that $g\circ\pi=\pi\circ f$. Such an equivariant
fibration structure is helpful to understand the dynamics of $f$. Indeed, if
$\delta_{1}(f)=\delta_{1}(g)$, then Kawaguchi-Silverman conjecture for $f$
follows from that of $g$ (cf. Proposition 4.2 and comments after that). But
when $\delta_{1}(f)>\delta_{1}(g)$, there is no way to deduce the conjecture
to $g$. (Case TIR (Definition 6.4) can be understood as an appearance of this
difficulty.) Projective bundles are probably the easiest fibration and this
case is already non-trivial in general.

######  Theorem 6.15 ([31, Theorem 1.10, Corollary 6.8.]).

  1. (1)

Conjecture 2.10 holds for all surjective self-morphisms on $n$-fold rational
normal scrolls over $\overline{\mathbb{Q}}$.

  2. (2)

Let $C$ be a smooth projective curve over $\overline{\mathbb{Q}}$ and
${\mathcal{E}}$ be a vector bundle on $C$. Then Conjecture 2.10 holds for all
surjective self-morphisms on the projective bundle
${\mathbb{P}}_{C}({\mathcal{E}})$ if it holds for all surjective self-
morphisms on ${\mathbb{P}}_{C}({\mathcal{E}})$ with ${\mathcal{E}}$ semistable
of degree $0$.

######  Theorem 6.16 ([33, Proposition 4.1]).

Let $Y$ be a smooth Fano variety over $\overline{\mathbb{Q}}$ of Picard number
one. Let $X={\mathbb{P}}_{Y}({\mathcal{E}})$ be a projective bundle over $Y$.
Then Conjecture 2.10 holds for all surjective self-morphisms
$f:X\longrightarrow X$.

We end this subsection with a list of results on $3$-folds.

######  Theorem 6.17.

  1. (1)

[31, Proposition 1.7] Let $X$ be a smooth projective variety over
$\overline{\mathbb{Q}}$ with $\dim X=3$ and Kodaira dimension zero. Then
Conjecture 2.10 is true for surjective self-morphisms $f\colon
X\longrightarrow X$ with $\deg f\geq 2$.

  2. (2)

[31, Theorem 1.8] Let $X$ be a smooth Calabi-Yau three fold and $f\colon
X\longrightarrow X$ an automorphism. Suppose either

     * •

the second Chern class $c_{2}(X)$ is strictly positive on
$\operatorname{Nef}(X)$, or

     * •

there is a non-zero semi-ample nef divisor $D$ such that $(c_{2}(X)\cdot
D)=0$. 111Oguiso pointed out that the proof contains an error. The proof
employs Oguiso’s theorem on the finiteness of $c_{2}$-contractions, but it is
only up to automorphisms and therefore one cannot deduce that $D$ is fixed by
some iterates of $f$. Although not yet published, Oguiso has provided an
alternative proof for this case and has also proven some generalizations [52].

Then Conjecture 2.10 holds for $f$.

  3. (3)

[31, Theorem 1.10 (1)] Let $X$ be a projective variety over
$\overline{\mathbb{Q}}$ with $\dim X=3$. Suppose $X$ has a Mori fiber space
structure (see [31, Definition 6.1] for the definition). Then Conjecture 2.10
holds for automorphisms on $X$.

  4. (4)

[9, Theorem 1.6] Let $X$ be a smooth projective variety over
$\overline{\mathbb{Q}}$ with $\dim X=3$ and $q(X)>0$.

    1. (a)

Conjecture 2.10 is true for automorphisms on $X$.

    2. (b)

Suppose $X$ is not covered by rational surfaces. Then Conjecture 2.10 is true
for birational self-maps on $X$.

###  6.3. Some useful facts

In this subsection, we introduce some basic facts that can be used to
construct fibrations equivariant under self-morphisms. For more about this
topic, you may refer to, for instance, [48, Theorem 4.19], [45, section 4,
Lemma 6.2], [47, Theorem 7.2], [17, section 4]. Let us start with the
following fundamental fact.

######  Proposition 6.18.

Let $f\colon X\longrightarrow X$ be a surjective self-morphism on a normal
projective variety $X$ defined over $\overline{\mathbb{Q}}$. Let $D$ be a
Cartier divisor on $X$ such that

  * •

$f^{*}D\equiv dD$ for some $d\in{\mathbb{Z}}_{>0}$;

  * •

$D$ is base point free and (its sublinear system) defines a morphism
$\varphi\colon X\longrightarrow{\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}$.

Let $X\xrightarrow{\pi}Y\to{\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}$ be the
Stein factorization of $\varphi$. Then $f$ induces a surjective self-morphism
$g\colon Y\longrightarrow Y$ such that

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

commutes.

###### Proof.

An irreducible curve $C\subset X$ is mapped to a point by $\pi$ if and only if
$(D\cdot C)=0$. Therefore $f$ preserves the fibers of $\pi$. Use the Zariski
main theorem to conclude the proof. ∎

Every surjective self-morphism on a projective bundle descends to a surjective
self-morphism on the base after replacing with some iterate.

######  Proposition 6.19.

Let $Y$ be a normal projective variety over $\overline{\mathbb{Q}}$ and let
${\mathcal{E}}$ be a locally free ${\mathcal{O}}_{Y}$-module of finite rank.
Let $X={\mathbb{P}}({\mathcal{E}}):=\operatorname{Proj}\oplus_{n\geq
0}\operatorname{Sym}^{n}{\mathcal{E}}$ be the projective bundle over $Y$ with
structure morphism $\pi\colon X\longrightarrow Y$. Let $f\colon
X\longrightarrow X$ be a surjective self-morphism. Then there is a positive
integer $m$ and a surjective self-morphism $g\colon Y\longrightarrow Y$ such
that

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{m}}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

commutes.

###### Proof.

Use the same argument before [1, Theorem 2]. ∎

Albanese morphism is compatible with self-morphisms because of the universal
property.

######  Theorem 6.20 ([33, Proposition 3.7],[9, Proposition 5.1]).

Let $X$ be a normal projective variety over $\overline{\mathbb{Q}}$. Let
$\alpha\colon X\longrightarrow A$ be the albanese morphism.

  1. (1)

Let $f\colon X\longrightarrow X$ be a self-morphism. Then $f$ induces a self-
morphism $g\colon A\longrightarrow A$ such that $\alpha\circ f=g\circ\alpha$.
Moreover, if $f$ has Zariski dense orbit, then $\alpha$ is surjective.

  2. (2)

Suppose $X$ is smooth. Let $f\colon X\dashrightarrow X$ be a rational map.
Then $f$ induces a self-morphism $g\colon A\longrightarrow A$ such that
$\alpha\circ f=g\circ\alpha$. Moreover, if $f$ has Zariski dense orbit, then
$\alpha$ is surjective.

######  Remark 6.21.

To deduce the last part of Theorem 6.20(2) from [9, Proposition 5.1], we have
to confirm that a Zariski dense set of $\overline{\mathbb{Q}}$-points is
Zariski dense in $X_{{\mathbb{C}}}$. We can prove this by using, for example,
the openness of the morphism $X_{{\mathbb{C}}}\longrightarrow X$, which is
because the morphism
$\operatorname{Spec}{\mathbb{C}}\longrightarrow\operatorname{Spec}\overline{\mathbb{Q}}$
is universally open (cf. [63, Tag 0383]).

##  7\. Toward rational maps

We have seen that there are many partial results and potential strategies to
prove Kawaguchi-Silverman conjecture for surjective self-morphisms on
projective varieties. However, the situation for rational maps is markedly
different. In certain special cases, there are nice canonical height functions
as discussed in section 5. Yet, it is unclear if we can expect the existence
of such nice functions for general rational maps. The effectiveness of
algebro-geometric methods arises from the fact that height functions have
functorial property with respect to morphisms. When our self-map is merely a
rational map, and not a morphism defined everywhere on a projective variety,
we usually cannot interpret geometric relationships between divisors directly
as equations of height functions. There are several highly non-trivial results
for rational maps that confirm the conjecture e.g. Theorems 6.13, 5.24, 5.22
and 5.12. In this section, we introduce some recent works concerning rational
maps.

###  7.1. Self-morphisms on affine surfaces

Among rational maps, those which can be regarded as self-morphisms on quasi-
projective varieties are relatively more tractable. In [17], Jia, Shibata,
Xie, and Zhang studied Kawaguchi-Silverman conjecture and Zariski dense orbit
conjecture for surjective self-morphisms on quasi-projective varieties.
Studying the structure of self-maps and the varieties, they proved:

######  Theorem 7.1 ([17, Theorem 1.8, Theorem 1.9]).

Let $X$ be a smooth affine surface over $\overline{\mathbb{Q}}$ and let
$f\colon X\longrightarrow X$ be a finite surjective self-morphism. Let
$\overline{\kappa}(X)$ be the log Kodaira dimension.

  1. (1)

If $\overline{\kappa}(X)\geq 0$ and $\deg f\geq 2$, then Conjecture 2.10 holds
for $f$.

  2. (2)

If $\overline{\kappa}(X)=-\infty$ and $\deg f=1$, then Conjecture 2.10 holds
for $f$.

  3. (3)

If the étale fundamental group $\pi_{1}^{et}(X)$ is infinite, $\deg f\geq 2$,
and $X\not\simeq{\mathbb{A}}^{1}\times{\mathbb{G}}_{{\rm m}}$, then Conjecture
2.10 holds for $f$.

###  7.2. Cohomologically hyperbolic maps

Recently, Wang extended the idea of the proof of positivity of canonical
height functions for automorphisms on surfaces and adapt it to a class of
rational maps called cohomologically hyperbolic maps [35]. In this
generalization, he did not take the approach to construct a canonical height
function, and only proved that the height sequence ”$h_{H}(f^{n}(x))$”
satisfies a certain inequality.

######  Definition 7.2.

Let $X$ be a projective variety over $\overline{\mathbb{Q}}$. Let
$p\in{\mathbb{Z}}$ with $1\leq p\leq\dim X$. A dominant rational map $f\colon
X\dashrightarrow X$ is said to be $p$-cohomologically hyperbolic if
$\delta_{p}(f)>\delta_{i}(f)$ for all $i\in\\{1,\dots,\dim
X\\}\setminus\\{p\\}$.

This class of rational maps were first introduced by Guedj [15] as a
cohomological analogue of hyperbolicity. Wang proved:

######  Theorem 7.3 ([35, Theorem 1.4]).

Let $X$ be a smooth projective surface over $\overline{\mathbb{Q}}$. Let
$f\colon X\dashrightarrow X$ be a $1$-cohomologically hyperbolic dominant
rational map (i.e. $\delta_{1}(f)>\delta_{2}(f)$). Assume there is an open
dense subset $W\subset X\setminus I_{f}$ with $f(W)\subset W$ such that either

  1. (1)

$f|_{W}\colon W\longrightarrow W$ is étale, or

  2. (2)

$W={\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$.

Then Conjecture 2.10 holds for $f$.

######  Remark 7.4.

In [35, Theorem 1.4], $f|_{W}\colon W\longrightarrow W$ is assumed to be
surjective, but it is not necessary.

######  Remark 7.5.

The second case (i.e. affine space case) also follows from Remark 5.25.
Jonsson-Wulcan-Xie’s proof uses valuative trees as a key ingredient to prove
the positivity of canonical height functions. Wang’s approach is completely
different, but it actually uses valuative tree indirectly as the proof
requires dynamical Mordell-Lang conjecture for
${\mathbb{A}}^{2}_{\overline{\mathbb{Q}}}$, which is proven by Xie using
valuative tree [65].

Wang needs the assumptions (1)(2) in Theorem 7.3 entirely because he uses
dynamical Mordell-Lang conjecture. If you assume that the orbit is generic and
not merely Zariski dense, then you can avoid this difficulty. Recall that an
orbit $O_{f}(x)$ is generic if it is infinite and its intersection with any
proper closed subset is finite. We remark that dynamical Mordell-Lang
conjecture implies every Zariski dense orbit is generic. In our joint work,
Wang and the author proved the following higher dimensional generalization of
Theorem 7.3.

######  Theorem 7.6 ([42, Theorem A, Corollary 1.8]).

Let $X$ be a smooth projective variety over $\overline{\mathbb{Q}}$. Let
$f\colon X\dashrightarrow X$ be a $p$-cohomologically hyperbolic map. Let
$x\in X_{f}(\overline{\mathbb{Q}})$ be a point with generic $f$-orbit. Then we
have

(7.1)
$\displaystyle\underline{\alpha}_{f}(x)\geq\frac{\delta_{p}(f)}{\delta_{p-1}(f)}.$

If $p=1$, then $\alpha_{f}(x)$ exists and $\alpha_{f}(x)=\delta_{1}(f)$.

Note that if $f$ is a birational self-map on a smooth projective surface, then
$1$-cohomological hyperbolicity is equivalent to $\delta_{1}(f)>1$. For such a
case, Conjecture 2.10 with the stronger assumption that $O_{f}(x)$ is generic
follows from Theorem 7.6.

######  Remark 7.7.

As an application of some properties of arithmetic degrees and dynamical
degrees, we proved that $1$-cohomologically hyperbolic map has points with
generic orbits [42, Theorem B].

##  8\. Related topics

There are several works concerning arithmetic degrees.

Kawaguchi-Silverman conjecture asks if a point with dense orbit has arithmetic
degree equal to dynamical degree, but it is natural to ask if there exists at
least one point $x$ that satisfies the equality $\alpha_{f}(x)=\delta_{1}(f)$.
This is proven for surjective self-morphisms on projective varieties [40,
Lemma 9.1,Corollary 9.3] (in this paper, the variety is assumed smooth, but
the same proof works for singular varieties). By the following works by Sano
and Shibata, it is proven that such points are Zariski dense over
$\overline{\mathbb{Q}}$ and over number fields under some additional
assumptions [57, 56]. For rational maps, it is not known in general that there
is at least one point $x$ with $\alpha_{f}(x)=\delta_{1}(f)$. There are some
results to this direction [23, Theorem 3], [42, Theorem B, Proposition 4.6].

There are several works studying which values arithmetic degrees can take. In
general, the set $\\{\alpha_{f}(x)\mid x\in X_{f}(\overline{\mathbb{Q}})\\}$
of arithmetic degrees can be infinite (Theorem 2.7), but for surjective self-
morphisms on normal projective varieties, it is finite and described by the
eigenvalues of $f^{*}\colon N^{1}(X)\longrightarrow N^{1}(X)$ (Theorem 2.9).
Lin determined the set of arithmetic degrees of monomial maps on projective
toric varieties [34, Theorem A]. In [39], we determined the set of arithmetic
degrees of surjective self-morphisms on semi-abelian varieties. For a
surjective self-morphism on a normal projective variety $f\colon
X\longrightarrow X$, members of the set $\\{|\mu|\mid\text{$\mu$ is an
eigenvalue of $f^{*}$ with $|\mu|>1$}\\}$ are candidates of arithmetic
degrees. In [51], Nasserden studies the difference of this set and the set of
arithmetic degrees ($\neq 1$). For surjective self-morphisms on
${\mathbb{Q}}$-factorial toric varieties, he proved that all such values can
be realized as arithmetic degrees. Also he constructed examples of self-
morphisms on abelian varieties such that some of such candidate values are not
realized as arithmetic degrees.

For surjective self-morphism on a projective variety, we can observe that the
set of points $x$ with $\alpha_{f}(x)<\delta_{1}(f)$ is small. More precisely,
it seems that the set of points $x$ with $\alpha_{f}(x)<\delta_{1}(f)$ and
defined over number fields of bounded degree is not Zariski dense. This is
stated as “small Arithmetic Non-density conjecture (sAND)” in [38, Conjecture
1.3] and confirmed for several cases. See [58] and [51, Theorem 1.7] for
related results. Note that sAND implies Kawaguchi-Silverman conjecture. It is
also worth mentioning that we cannot expect the same statement for rational
maps [38, Remark 1.4(6)].

Kawaguchi-Silverman conjecture compares arithmetic degree $\alpha_{f}(x)$ with
the first dynamical degree $\delta_{1}(f)$ of $f$. It is natural to ask what
the arithmetic counter part of higher dynamical degrees $\delta_{p}(f)$ is.
Dang, Ghioca, Hu, Lesieutre, and Satriano defined higher arithmetic degrees
using Arakelov theory and extend Kawaguchi-Silverman conjecture to that
setting [12]. In their formulation, higher arithmetic degrees are defined for
subvarieties (not points) and measures arithmetic complexity of orbits of
subvarieties. Very recently, Jiarui Song established basic properties of
higher arithmetic degrees and gave a conceptual proof of the inequality
$\overline{\alpha}_{f}(x)\leq\delta_{1}(f)$ for arbitrary dominant rational
maps on normal projective variety over a number field [62]. Moreover, he gave
a counter example to DGHLS’s higher dimensional version of Kawaguchi-Silverman
conjecture using the morphism that produces a counter example to the original
dynamical Manin-Mumford conjecture.

##  9\. Future directions

Let us conclude this survey by discussing potential future directions.

The most challenging and crucial case of the Kawaguchi-Silverman conjecture
might be the dominant rational maps on the projective space
${\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}$. The projective space has densely
many rational points and a rich structure of self-maps. Another demanding case
is self-maps on Calabi-Yau varieties. In broad terms, Calabi-Yau varieties are
one of the building blocks of varieties of Kodaira dimension zero, alongside
Abelian varieties and Hyper-Kähler varieties. The Kawaguchi-Silverman
conjecture is fully resolved for Abelian varieties, and for Hyper-Kähler
varieties, it is solved for all surjective self-morphisms. However, we lack
such general results for Calabi-Yau varieties and even the solution for
automorphism case would be very interesting (See Theorem 6.12). It’s worth
mentioning that Calabi-Yau varieties are also expected to have many rational
points and exhibit a rich dynamical structure.

Solving the Kawaguchi-Silverman conjecture in full generality may be quite
difficult, but exploring, for instance, the cohomologically hyperbolic case
could be intriguing. It would also be interesting to examine how prevalent
cohomologically hyperbolic maps are. Given that they are defined by the non-
coincidence of successive dynamical degrees, it is reasonable to anticipate
the existence of many such maps, particularly on specific varieties such as
projective spaces.

The Case TIR (Definition 6.4), which emerged during the study of the
Kawaguchi-Silverman conjecture, is interesting from the viewpoint of the
structure theory of surjective self-morphisms on projective varieties as well.
It is predicted that the Case TIR does not occur, but this remains an open
question in general to the best of my knowledge.

In addition to proving the equality $\alpha_{f}(x)=\delta_{1}(f)$,
establishing the existence of the arithmetic degree for arbitrary dominant
rational self-maps presents a significant challenge. This is tied to a deeper
understanding of the sequence $h_{H}(f^{n}(x))$. For surjective self-morphisms
on projective varieties, this sequence is well-understood, at least when
$\alpha_{f}(x)>1$ (See [54, Theorem 1.1]). For rational maps, this sequence
behaves wildly (See [34, Example 3.1]), making it intriguing to discern some
patterns.

Investigating the function field analogue of the Kawaguchi-Silverman
conjecture could also be of interest. When the coefficient field of the
function field is infinite, some additional assumptions are necessary to avoid
“constant points”, but it seems reasonable to expect that the same essentially
holds. Refer to Remark 2.12 and [41, 13, 67] for more on this direction.

As demonstrated in [17, 42], the study of arithmetic degrees has applications
to the Zariski dense orbit conjecture. The arithmetic degree serves as a
measure of the complexity of the orbits, meaning that, under certain
additional assumptions on the geometric structure of the self-maps, points
with maximal arithmetic degrees have dense orbits. Pursuing this direction and
discovering further applications to the Zariski dense orbit conjecture would
be highly fascinating.

######  Acknowledgements.

The author would like to thank Mattias Jonsson, John Lesieutre, Sheng Meng,
Keiji Oguiso, Kaoru Sano, Matt Satriano, Takahiro Shibata, Joe Silverman,
Elizabeth Wulcan, Junyi Xie, Shou Yoshikawa and D.Q. Zhang for valuable
comments on the manuscript of this survey and answering his questions. Some of
the works related to the equivariant MMP referred in this survey started
during the Simons symposium “Algebraic, Complex and Arithmetic Dynamics
(2019)”. The author would like to thank Simons foundation and the organizers,
Laura DeMarco and Mattias Jonsson, for providing a stimulating and
collaborative environment.

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