\section{Introduction} \label{s:intro}

Let $X$ be a projective variety defined over a field $k_0$.
For every automorphism $f \in \mathrm{Aut}(X)$ and every line bundle $L$ on $X$,
Artin and Van den Bergh introduced in the early 1990s
the notion of \emph{twisted homogeneous coordinate ring} $B(X,f,L)$
associated to the data $(X,f,L)$~\cite{AvB}.
This is an associative $k_0$-algebra which specializes to
the usual homogeneous coordinate ring of a polarized variety
when $f = \mathrm{Id}_X$ and $L$ is ample.
There are natural invariants associated to
the noncommutative algebra $B(X,f,L)$
such as the Gelfand-Kirillov dimension (or GK-dimension for short)
which measures the growth of the $k_0$-algebra when iterating $f$,
generalizing the notion of Krull dimension to the noncommutative setting.
So far, the GK-dimension has been studied extensively,
mostly by specialists in noncommutative algebras~\cite{KL},
and fundamental results of twisted homogeneous coordinate rings have been proven in~\cite{AvB,Ke}
(see also~\cite{ArtinStafford, Smith} among other related work).

One of the aims of this paper is to contribute to the
study of twisted homogeneous coordinate rings
from the dynamical perspective, which is yet to be fully explored.
We hope that the dynamical properties of $(X,f)$ would provide a better understanding of
the algebraic structure of $B(X,f,L)$, and \emph{vice versa}.

As an example,
the main goal of this paper is to present some new upper bounds and lower bounds of
the GK-dimensions of twisted homogeneous coordinate rings
when the latter are finite and
the variety $X$ is defined over an algebraically closed field $k_0$ of characteristic zero.
We obtain these bounds through
the complex dynamical study of automorphisms
of compact K\"ahler manifolds of zero entropy,
especially via the study of polynomial log-volume growth of automorphisms
recently introduced by Cantat and Paris-Romaskevich~\cite{CO}.
It turns out that the GK-dimensions are bounded from the above, in terms of the dimension $d = \dim X$ of the variety only, by an explicit quadratic polynomial in $d$ (Corollaries~\ref{cor-uniform} and \ref{uniformGK}).
When $\dim X = 3$, our upper bound and lower bound
immediately yield a complete determination of
the GK-dimension (Corollaries~\ref{cor-d=3}, \ref{uniformGK}, and Theorem~\ref{main2}),
generalizing a result of Artin--Van den Bergh proven for surfaces~\cite[Theorem 1.7]{AvB}.

The relation between the GK-dimension of a twisted homogeneous coordinate ring
and the polynomial log-volume growth of automorphisms will be clarified in Theorem~\ref{thm-comp}.
In particular, the determination of the GK-dimension
is equivalent to that of the polynomial log-volume growth.
It is the latter problem that we will study in this paper.

In the remainder of the introduction, we will provide further context and state
the main results of the paper.

\subsection{Gelfand-Kirillov dimension}\label{ssec-GK}
\hfill

The reader is referred to Section~\ref{sec-THCR} for the definitions and basic properties of
twisted homogeneous coordinate rings and GK-dimension mentioned in the last paragraph.

Let $X$ be an (irreducible) projective variety.
Throughout the paper, we assume that every
projective variety is
defined over an algebraically closed field $k_0$ of characteristic zero.
Let $f \in \mathrm{Aut}(X)$.
In~\cite{Ke}, Keeler shows that
the GK-dimension of the twisted homogeneous coordinate ring
$B(X,f,L)$ is finite for every ample line bundle $L$ on $X$ if and only if
$f^* : N^1(X) \to N^1(X)$ is quasi-unipotent where $N^1(X) \mathrel{:=} \mathrm{NS}(X)/(\text{torsion})$.
In this case,
${\rm GKdim}\, B(X,f,L)$ is independent of $L$ as long as $L$ is ample (see Theorem~\ref{Keeler}).
We can therefore define the {\it GK-dimension} of $(X,f)$ as
$${\rm GKdim}\, (X,f) \mathrel{:=} {\rm GKdim}\, B(X,f,L)$$
for any ample line bundle $L$.
From the perspective of projective geometry,
we also define the {\it reduced} Gelfand-Kirillov dimension by
$${\rm rGKdim}\, (X, f) \mathrel{:=} {\rm GKdim}\, (X, f) -1,$$
so that ${\rm rGKdim}\, (X, \mathrm{Id}_X) = \dim X$.

In Theorem~\ref{Keeler},
we summarize the fundamental properties of ${\rm GKdim}\, B(X,f,L)$ proven by Keeler and Artin--Van den Bergh.
On top of these results, we also show that ${\rm GKdim}\, (X,f)$ is a birational invariant.

\begin{pro}\label{pro-GKbir}
Let  $\pi : X \dasharrow Y$ be a dominant, generically finite rational map between
projective varieties.
Let $f_X \in \mathrm{Aut}(X)$ and $f_Y \in \mathrm{Aut}(Y)$ be automorphisms such that
$$\pi \circ f_X = f_Y \circ \pi.$$
Then $f^*_X : N^1(X) \circlearrowleft$ is quasi-unipotent
(i.e., a positive power of it is unipotent),
if and only if $f^*_Y : N^1(Y) \circlearrowleft$ is quasi-unipotent.
In this case, we have
$${\rm GKdim}\, (X, f_X) = {\rm GKdim}\,(Y, f_Y).$$
In particular, ${\rm GKdim}\, (X, f_X)$ is a birational invariant.
\end{pro}

We will prove Proposition~\ref{pro-GKbir} in Section~\ref{sec-plov}.
By Proposition~\ref{pro-GKbir},
together with the existence of equivariant projective desingularizations
(see \emph{e.g.} ~\cite[Theorem 3.45]{KollarResSing}),
it is then enough to study the GK-dimension for smooth projective varieties.

\subsection{Automorphisms with zero entropy: polynomial log-volume growth}\label{introPlov}
\hfill

Let $X$ be a compact K\"ahler manifold of dimension $d$ and $f \in \mathrm{Aut} (X)$.
Thanks to Gomov-Yomdin's theorem,
we can compute the topological entropy of $f$
in terms of the cohomological invariants associated to $(X,f)$~\cite{Gromov},~\cite{Yomdin}.
Among the complex dynamical study of $(X,f)$,
there has been a growing interest in
compact K\"ahler manifolds with slow dynamics
 (see \emph{e.g.}  \cite{CantatICM, CO, FanFuOuchi, DLOZ}), and typically, one studies the dynamics of
automorphisms having zero entropy.
In terms of cohomological data, $f$ has zero entropy
if and only if the first dynamical degree of $f$ satisfies
$$d_1(f) = 1,$$
which is further equivalent to the property that
${f^*: H^{1,1}(X) \circlearrowleft}$ (or equivalently ${f^*: N^1(X) \circlearrowleft}$ when $X$ is projective) is quasi-unipotent
(\emph{cf.}  Lemma~\ref{lem:equiv_dyn}).

In the course of their study of the polynomial entropy of slow dynamics, Cantat and Paris-Romaskevich (\cite{CO}) introduced the {\it polynomial logarithmic volume growth} of $(X, f)$, which is defined as follows.
Let $\omega$ be a K\"ahler metric on $X$; by abuse of notation, $\omega \in H^{1,1}(X,{\mathbb R})$
will also denote the underlying K\"ahler class.
For every $n \ge 1$, let $\Gamma_f(n) \subset X^{n+1}$ be the graph of $f \times f^2 \times \cdots \times f^n$ and let $\mathrm{Vol}_\omega(\Gamma_f(n))$ be the volume of $\Gamma_f(n)$ defined by the product metric on $X^{n+1}$ restricted to $\Gamma_f(n)$.
With $d = \dim X $, we have
\begin{equation}\label{eqn-vol}
\mathrm{Vol}_\omega(\Gamma_f(n))  = \int_{\Gamma_f(n)} \frac{1}{d!}\(\sum_{i = 1}^{n+1} \mathrm{pr}_i^*\omega\)^d
= \frac{1}{d!}\Delta_n(f, \omega)^d
\end{equation}
where $\mathrm{pr}_i : X^{n+1} \to X$ is the projection to the $i$-th factor and
$$\Delta_n(f, \omega) \mathrel{:=} \sum_{i = 0}^n (f^i)^*\omega.$$
The {\it polynomial logarithmic volume growth} of $(X,f)$
(or polynomial log-volume growth for short)
is defined to be
$$\mathrm{Plov}(f) \mathrel{:=} \mathrm{Plov}(X, f) \mathrel{:=} \limsup_{n \to \infty} \frac{\log \mathrm{Vol}_\omega(\Gamma_f(n))}{\log n}
	=  \limsup_{n \to \infty} \frac{\log \Delta_n(f, \omega)^d}{\log n}  \in [0,\infty].$$
We will show that $\mathrm{Plov}(f)$ is independent of the choice of a K\"ahler metric $\omega$ (see Lemma~\ref{lem-sminvVol}).

When $X$ is projective, we have defined ${\rm rGKdim}\,(X, f)$ in Section~\ref{ssec-GK}
provided that $f^* : N^1(X) \circlearrowleft$ is quasi-unipotent.
The identification between ${\rm rGKdim}\,(X, f)$ and $\mathrm{Plov} (X, f)$ follows immediately from Keeler's work,
which we will prove in Section~\ref{sec-THCR}.

\begin{thm}[Keeler]\label{thm-comp}
Let $X$ be a complex smooth projective variety.
Assume that $f^* : N^1(X) \circlearrowleft$ is quasi-unipotent,
i.e., a positive power of it is unipotent. Then
$${\rm rGKdim}\, (X,f) = \mathrm{Plov}(X,f).$$
\end{thm}

By the Lefschetz principle,
the study of the GK-dimension of the pair
$(Y, g)$ of
a projective variety $Y$ together with an automorphism $g \in \mathrm{Aut}(Y)$
defined over an algebraically closed field $k_0$ of characteristic zero
is essentially equivalent to the study over ${\mathbb C}$.
Therefore, combining Proposition~\ref{pro-GKbir} and Theorem~\ref{thm-comp},
the study of ${\rm rGKdim}\, (Y, g)$
in noncommutative algebra is essentially
reduced to the study of $\mathrm{Plov} (X, f)$ in complex geometry and complex dynamics,
via an equivariant projective desingularization.
For instance, such an argument works for
the upper bounds and lower bounds of $\mathrm{Plov} (X, f)$ we obtain in this paper,
whence the same bounds are also valid for the GK-dimension (see Corollary~\ref{uniformGK}).

Here are some basic dynamical properties of $\mathrm{Plov}(X, f)$, which will be frequently used in this paper.
We believe that Proposition~\ref{prop1-Kahler} will also be useful in other applications in future.

\begin{pro}\label{prop1-Kahler}
	Let $X$ be a compact K\"ahler manifold.
	\begin{enumerate}
		\item (Independence of the metric; Lemma~\ref{lem-sminvVol})
		The invariant $\mathrm{Plov}(f)$ is independent of $\omega \in H^{1,1}(X,{\mathbb R})$
		whenever $\omega$ is nef and big.
		\item (Finiteness and integrality; Lemmas~\ref{lem:equiv_dyn} and \ref{lem35})
		We have $\mathrm{Plov}(f) < \infty$ if and only if
		$f^* : H^{1,1}(X) \circlearrowleft$ is quasi-unipotent.
		In this case, $\mathrm{Plov}(f)$ is an integer.
		\item (Finite index; Lemma~\ref{lem-invfini})
		We have $\mathrm{Plov}(f) = \mathrm{Plov}(f^N)$
		for any integer $N \not= 0$.
		\item (Product; Lemma~\ref{lem-pruduct}) Let $X_i$ ($i=1$, $2$) be compact K\"ahler manifolds
		and let $f_i \in \mathrm{Aut} (X_i)$.
		Then
		$$\mathrm{Plov} (f_1 \times f_2) = \mathrm{Plov} (f_1) + \mathrm{Plov} (f_2) $$
		for the product automorphism $f_1 \times f_2 \in \mathrm{Aut}(X_1 \times X_2)$.
		\item (Invariance under generically finite maps; Lemma~\ref{lem-invft}) Let $X$ and $Y$ be
	compact K\"ahler manifolds and
	$f_X \in \mathrm{Aut} (X)$ and $f_Y \in \mathrm{Aut} (Y)$.
	Let $\phi : X \dasharrow Y$
	be a generically finite dominant meromorphic map such that $f_Y \circ \phi = \phi \circ f_X$. Then
	$$\mathrm{Plov}(f_X) = \mathrm{Plov}(f_Y).$$
		\item (Restriction; see Lemma~\ref{lem-restr}, also for the precise
		definition of $\mathrm{Plov}(f_{|W})$ when $W$ is not smooth)
		Let $W \subset X$ be a closed subvariety such that $f(W) = W$.
		Then $\mathrm{Plov}(f_{|W}) \le \mathrm{Plov}(f)$ for the automorphism $f_{|W} \in \mathrm{Aut} (W)$ induced from $f$ by restriction.
	\end{enumerate}
\end{pro}

\subsection{Automorphisms with zero entropy: growth of the pullback actions}
\hfill

Another important invariant associated to an automorphism $f$
of a compact K\"ahler manifold $X$ with zero entropy,
is the maximum size of the Jordan block of the Jordan canonical form of $f^*: {H^{1,1}(X)} \circlearrowleft$,
denoted by
$$k(f)+1.$$
We have the following optimal upper bound of $k(f)$ in terms of $\dim X$.

\begin{theorem}[{\cite{DLOZ}, Corollary~\ref{cor-JBboundkd}}]\label{JBbound}
Let $X$ be a compact K\"ahler manifold of dimension $d \geq 1$
and let $\kappa(X)$ be the Kodaira dimension of $X$.
Let $f \in {\rm Aut}\, (X)$ be any automorphism whose topological entropy is zero.
Then $k(f)$ is an even integer and satisfies
$$k(f) \le 2(d -1) \ \text{ and } \ k(f) \le 2(d -\kappa(X)).$$
In other words, for any norm of ${\rm End}_{{\mathbb C}}(H^{1,1}(X))$, we have
$$\|(f^{m})^*: H^{1,1}(X)\circlearrowleft\| = O\big(m^{2(d-1)}\big) \text{ and }  O\big(m^{2(d-\kappa(X))}\big)$$
as $m \to \infty$.
Moreover, these estimates are optimal,
in the sense	that
	for every $d \ge 1$ and $\kappa \ge 1$, there exist examples of $f \in \mathrm{Aut}(X)$ with
	$\dim(X) =  d$, $\kappa(X) = \kappa$, and
	$k(f) = 2(d - \kappa(X))$.
\end{theorem}

Without the Kodaira dimension $\kappa(X)$, Theorem~\ref{JBbound}
was proven together with Tien-Cuong Dinh in our previous work~\cite{DLOZ} for any $d \ge 1$,
and was first
observed and proven by Lo Bianco~\cite{Lo} when $d = 2$ or $3$.
Based on the fundamental properties of
dynamical filtrations of $H^{1,1}(X,{\mathbb R})$
associated to quasi-nef sequences developed in~\cite{DLOZ},
we prove some refinements in Section~\ref{sec-filtr}
which involve fiber space structure, especially the Iitaka fibration
(see Corollaries~\ref{cor25K'} and~\ref{cor-JBboundkd}).
These refinements have their own interest, and we will also use them to prove our main results.

We also prove that $k(f)$ is invariant under bimeromorphic modifications.

\begin{pro}\label{pro-birJB}
	Let  $\pi : X \dashrightarrow Y$ be a dominant, generically finite meromorphic map between
	compact K\"ahler manifolds.
	Let $f_X \in \mathrm{Aut}(X)$ and $f_Y \in \mathrm{Aut}(Y)$ be automorphisms such that
	$$\pi \circ f_X = f_Y \circ \pi.$$
	Assume that $f_X^* : {H^{1,1}(X)} \circlearrowleft$, or equivalently $f_Y^* : {H^{1,1}(Y)} \circlearrowleft$ is quasi-unipotent. Then
	$$k(f_X) = k(f_Y).$$
\end{pro}

\subsection{Upper bounds of polynomial log-volume growth}
\hfill

As we mentioned before,
the main goal of this  paper
is to provide explicit upper bounds and lower bounds of the GK-dimension of
a twisted homogeneous coordinate ring,
which,  by Theorem~\ref{thm-comp}, is now reduced to the study of
$\mathrm{Plov} (X, f)$ where $f$ is an automorphism of a compact K\"ahler manifold
$X$.

The following theorem, essentially due to Keeler (\cite[Lemma 6.13]{Ke}), is the starting point of our study.

\begin{thm}[Keeler]\label{thmkeeler}
Let $X$ be a compact K\"ahler manifold of positive dimension $d$ and let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$. Then
$$d + k(f) \le \mathrm{Plov} (f) \le k(f)(d-1) + d.$$
\end{thm}

A sketch of proof of Theorem~\ref{thmkeeler} is provided at the beginning of Section~\ref{proofuniform}.
Note that Theorem~\ref{thmkeeler} implies that we always have $\mathrm{Plov}(f) \ge d$.
Also, if $k(f) = 0$, then $\mathrm{Plov}(f) = d$.

Combining Theorem~\ref{thmkeeler} with Theorem~\ref{JBbound},
we obtain the following upper bound of $\mathrm{Plov} (f)$,
(or equivalently ${\rm GKdim}\,(X,f)$ for projective $X$).
This is already a non-trivial result from a purely algebraic perspective of ${\rm GKdim}\,(X,f)$,
as there is no {\it a priori} reason why one can estimate $\mathrm{Plov}(X, f)$ and ${\rm rGKdim}\, (X, f)$ uniformly in terms of $\dim X$ only.

\begin{cor}\label{corkeeler}
Let $X$ be a compact K\"ahler manifold of dimension $d \ge 1$ and
let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$.
Then
$\mathrm{Plov} (f) \le 2(d-1)^2 + d = 2d^2 -3d + 2$.
\end{cor}

We call these upper or lower bounds, Keeler's upper or lower bounds.
Keeler's upper bound in Corollary~\ref{corkeeler} is optimal for $d \le 2$,
but it is no longer optimal already for $d = 3$.

In this work, we make several non-trivial improvements of Keeler's bounds.

\begin{theorem}\label{uniform}
Let $X$ be a compact K\"ahler manifold of dimension $d$ and let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$.
Assume that $d \ge 3$ and $k(f) > 0$. Then
$$ \mathrm{Plov}(f) \le  k(f)(d-1) + d - 2. $$
\end{theorem}

Since $k(f) \le 2d - 2$  by Theorem~\ref{JBbound} and $\mathrm{Plov}(f) = d$ whenever $k(f) = 0$,
Theorem~\ref{uniform} implies that $\mathrm{Plov}(f) \le 2d^2 - 3d$ whenever $d \ge 3$.
When $d = 3$, we obtain $\mathrm{Plov}(f) \le 9$, and this upper bound is optimal (see Theorem~\ref{main3}).
When $d \ge 4$, we can make further improvement in comparison to Corollary~\ref{corkeeler}.

\begin{theorem}\label{uniformdge4}
Let $X$ be a compact K\"ahler manifold of dimension $d \ge 4$ and let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$.
Then
$$
 \mathrm{Plov}(f) \le 2d^2-3d - 2.
$$
\end{theorem}

The following upper bounds of $\mathrm{Plov}(f)$ involving only the dimension of $X$
are consequences of the above two theorems,
which already go beyond
a direct combination of Theorem~\ref{JBbound} and Theorem~\ref{thmkeeler}
proven in~\cite{DLOZ} and~\cite{Ke}.

\begin{cor}\label{cor-uniform}
Let $X$ be a compact K\"ahler manifold of positive dimension $d$ and let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$. Then we have
$$
d \le \mathrm{Plov}(f) \le
\begin{cases}
d^2 \text{ if } d \le 3 \\
2d^2-3d - 2 \text{ if } d \ge 4.
\end{cases}
$$
\end{cor}

\begin{proof}
	The case $d = 1$ or $2$ follows from Theorems~\ref{thmkeeler} and \ref{JBbound}.
	If $k(f) > 0$ (resp. $k(f) = 0$), then Corollary~\ref{cor-uniform}
	follows from Theorem~\ref{uniform} (resp. Theorem~\ref{thmkeeler}).
The statement for $d \ge 4$ is Theorem~\ref{uniformdge4}.
\end{proof}

\begin{remark} The upper bound $\mathrm{Plov} (f) \le 9$ when $d = 3$ is also obtained in the proof of~\cite[Theorem 4.1]{CO}
with a different argument.
As we will see later, we will further determine the exact values
of $\mathrm{Plov}(f)$ when $d = 3$
(see Corollary~\ref{cor-d=3} and Theorem~\ref{main2}).
\end{remark}

The improvements made in Theorems~\ref{uniform} and \ref{uniformdge4} require non-trivial geometric inputs, based on arguments involving the positivity
of certain dynamically relevant cohomology classes initiated in~\cite{DLOZ}.
Just as the proof of Theorem~\ref{JBbound},
the proofs of these new upper bounds make use of the
dynamical filtrations of $H^{1,1}(X,{\mathbb R})$
associated to quasi-nef sequences (\emph{cf.}  Section~\ref{sec-filtr}) in a crucial and efficient way.

Using these filtrations,
not only we can improve Keeler's upper bound, but also refine Keeler's lower bound
$\mathrm{Plov}(f) \ge d + k(f)$ in Theorem~\ref{thmkeeler}.

\begin{thm}\label{thm-lb}
Let $X$ be a compact K\"ahler manifold of dimension $d > 0$ and
let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$.
Then we have
$$\mathrm{Plov}(f) \ge d + 2k(f) - 2.$$
\end{thm}

Recall that $k(f)$ is even (Theorem~\ref{JBbound}) and we always have $\mathrm{Plov}(f) = d$ when $k(f) = 0$.
When $k(f) \ge 2$, Theorem~\ref{thm-lb} improves Keeler's lower bound.

	When $X$ is a projective surface,
	Artin--Van den Bergh proved that $\mathrm{Plov}(f) \in \{2,4\}$~\cite[Theorem 1.7]{AvB}
	and both numbers are realizable.
	When $\dim X = 3$, the upper bound in Theorem~\ref{uniform}
	coincides with the lower bound in Theorem~\ref{thm-lb},
	which immediately implies the analogous result when $\dim X = 3$.

\begin{cor}\label{cor-d=3}
Let $X$ be a compact K\"ahler manifold of
dimension $3$ and let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$.
Then we have
$$\mathrm{Plov}(f) = 2k(f) +1$$
whenever $k(f) > 0$. As a consequence,
$$\mathrm{Plov}(f) \in \{3,5,9\}.$$
\end{cor}

\begin{proof}

	Assume that $k(f) > 0$. It follows from Theorems~\ref{uniform}
	and~\ref{thm-lb} that $\mathrm{Plov}(f) = 2k(f) +1$.
	Recall that $k(f) = 0$ implies $\mathrm{Plov}(f) = d = 3$ by Theorem~\ref{thmkeeler}. Hence
	$$\mathrm{Plov}(f) \in \{3,5,9\}.$$
\end{proof}

In Corollary~\ref{cor-d=3},
the three numbers $3$, $5$, $9$ are all realizable;
we refer to Theorem~\ref{main2} for examples realizing these numbers.

By Theorem~\ref{thm-comp}, together with the existence of equivariant projective desingularization
and the Lefschetz principle, we have the analogous bounds for $ {\rm rGKdim}\, (X, f)$
where $X$ is
a projective variety defined over an algebraically closed field $k_0$ of characteristic zero.
In the following statement, $k(f)$ is defined with $f^* : H^{1,1}(X) \circlearrowleft$ replaced by $f^* : N^1(X) \circlearrowleft$.

\begin{cor}\label{uniformGK}
Let $X$ be
a projective variety of dimension $d > 0$ over $k_0$, not necessarily smooth,
and let $f \in \mathrm{Aut}(X)$ be an automorphism such that $f^* : N^1(X) \circlearrowleft$ is quasi-unipotent.
Then the same inequalities in Theorem~\ref{uniform}, Theorem~\ref{uniformdge4}, Corollary~\ref{cor-uniform},
Theorem~\ref{thm-lb}, and Corollary~\ref{cor-d=3} hold
with $\mathrm{Plov}(f)$ replaced by $ {\rm rGKdim}\, (X, f)$
under the same assumptions on $d$ and $k(f)$.
\end{cor}

\subsection{Polynomial log-volume growth for complex tori and threefolds}
\hfill

We also study $\mathrm{Plov}(f)$ for some particular classes of compact K\"ahler manifolds.
When $X$ is a complex torus, we can completely determine $\mathrm{Plov}(f)$
once we know how $f^*$ acts on $H^{1,0}(X)$.
This is the content of Theorem~\ref{thm51} that we will prove in Section~\ref{sec-MainTheorem3}.
Here we only state the folowing consequence of Theorem~\ref{thm51}.

\begin{theorem}\label{main3}
Let $X$ be a compact complex torus of dimension $d \ge 1$ and let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$. Then we have the upper bound
$$\mathrm{Plov} (X, f) \le d^2$$
and this bound is optimal for each positive integer $d$.
\end{theorem}

Theorem~\ref{main3} also shows that
	the quadratic {\it order} of the upper bounds with respect to $d$ in Corollaries~\ref{cor-uniform} and~\ref{uniformGK} are optimal.

\begin{remark} Besides complex tori (Theorem \ref{thm51}), we can also compute the precise values $\mathrm{Plov}(f)$ and ${\rm rGKdim}\,(X, f)$ for hyper-K\"ahler manifolds $X$ and Mori dream spaces.
See Proposition~\ref{proHK} in Section~\ref{sec-MainTheorem2}.
\end{remark}

Some explicit examples of $(X,f)$ such that $\mathrm{Plov} (X, f) < \infty$ and $\mathrm{Plov} (X, f)$ is as large as possible
are constructed in Section~\ref{sec-MainTheorem2}.
We also have the following result studying $\mathrm{Plov} (f)$ for some explicit threefolds.

\begin{theorem}\label{main2}

Let $X$ be a compact K\"ahler manifold of dimension $3$ and let $f \in \mathrm{Aut}(X)$.
\begin{enumerate}
	\item If $\kappa(X) = 3$, then $\mathrm{Plov} (f) = 3$.
	\item If $\kappa (X) = 2$, then $\mathrm{Plov} (f) \in \{3,5\}$.
	Both numbers are realizable by some smooth
	projective variety $X$.
\item We have $\mathrm{Plov} (f) \in \{3,5,9\}$, all realizable by pairs $(X,f)$,
where $X$ can be taken from any of the following classes of smooth projective varieties:
\begin{itemize}
\item $X$ is rational;
\item $X$ is an abelian threefold;
\item $X$ is a simply connected Calabi-Yau manifold;
\item $X$ has Kodaira dimension $1$.
\end{itemize}
	\item  Without any assumption on $\dim X$, for any integer  $d \ge 1$,
	we can construct $(X,f)$ such that $\dim X = d$, $\mathrm{Plov}(f)  = d^2$, $\kappa(X) = 1$,
	and $X$ is projective.
\end{enumerate}
\end{theorem}

 We prove Theorem~\ref{main2} in Section~\ref{sec-MainTheorem2}.

\subsection{Some questions and a few further observations}
\hfill

As we fail to construct examples of $(X,f)$ such that $d^2 < \mathrm{Plov}(f) < \infty$ where $d = \dim X$,
presumably, the upper bound in Corollary~\ref{cor-uniform} when $d \ge 4$ is still not optimal.
Taking Theorems~\ref{main3} and \ref{main2} into account,
it seems reasonable to ask the following questions.

\begin{question}\label{mainquest2}
Let $X$ be a compact K\"ahler manifold of dimension $d \ge 1$ and of Kodaira dimension $\kappa (X)$. Let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$.
\begin{enumerate}
\item Is $\mathrm{Plov} (f) \le d^2$?
\item More precisely (in view of Theorem~\ref{main3}), are possible values of $\mathrm{Plov}(X,f)$ always
	realizable by $d$-dimensional tori?
\item Is $\mathrm{Plov} (f) < d^2$ when $\kappa (X) \ge 2$?
\end{enumerate}
\end{question}

Question~\ref{mainquest2} is non-trivial only starting from $d \ge 2$.
When $d = 2$,
the determination $\mathrm{Plov}(f) \in \{2,4\}$ by Artin--Van de Bergh~\cite[Theorem 1.7]{AvB}
essentially answers Question~\ref{mainquest2}.
When $\dim X = 3$, all three questions have positive answers
according to Theorem~\ref{main2}.

Question~\ref{mainquest2} (1) is the analogous question of~\cite[Question 4.1]{CO}
asked for polynomial entropy by Cantat--Paris-Romaskevich.
According to~\cite[(2.7)]{CO}, a positive answer to Question~\ref{mainquest2} (1)
also answers~\cite[Question 4.1]{CO} in the affirmative.
Combining the main results proven in this paper,
we have the following partial answers to Question~\ref{mainquest2}
in the positive direction, which also answers Cantat--Paris-Romaskevich's question
when $\kappa(X) \ge \dim X/2$.

\begin{cor}\label{cor-kod}
Let $X$ be a compact K\"ahler manifold of dimension $d \ge 3$
and let $f \in \mathrm{Aut} (X)$ such that $d_1(f) =1$.
If $k(f) \le d$, then $\mathrm{Plov} (f) \le d^2 - 1$.
In particular, $\mathrm{Plov} (f) \le  d^2 - 1$, whenever $\kappa (X) \ge d/2$.

The same results hold when $X$ is
a projective variety
with $\mathrm{Plov}(f)$ replaced by ${\rm rGKdim}\, (X,f)$\footnote{For a singular variety $X$,
the Kodaira dimension of $X$ is defined to be $\kappa(X) : =
\kappa(\widetilde{X})$, where $\widetilde{X}$ is any smooth projective variety birational to $X$.
}.
\end{cor}

\begin{proof}
The main statement of Corollary~\ref{cor-kod} follows from Theorem~\ref{uniform}
(resp. Theorem~\ref{thmkeeler}) when $k(f) > 0$ (resp. $k(f) = 0$).
Together with Theorem~\ref{JBbound}, it follows that $\kappa (X) \ge d/2$ implies $\mathrm{Plov} (f) < d^2$.

Similar statements of ${\rm rGKdim}\, (X,f)$ for projective $X$ follow from
 Theorem~\ref{thm-comp} together with the existence of equivariant projective desingularization
and the Lefschetz principle.
\end{proof}

\subsection{Organization of the paper and a few remarks to the readers}
\hfill

Section~\ref{sec-THCR} is a brief review of twisted homogeneous coordinate rings
and their Gelfand-Kirillov dimensions.
We recall some fundamental results proven in~\cite{Ke,AvB} (Theorem~\ref{Keeler})
and derive Theorem \ref{thm-comp} as a direct consequence.
Section~\ref{sec-plov} is devoted to polynomial log-volume growth,
in which the statements of
Proposition~\ref{prop1-Kahler} are proven separately in several lemmas.
We also prove Proposition~\ref{pro-birJB} in Section~\ref{sec-plov}.
In Sections~\ref{sec-filtr},
we recall the construction of quasi-nef sequences and associated filtrations,
together with their fundamental properties.
We also prove several auxiliary results related to them,
which will be useful in the study of
upper bound and lower bound of $\mathrm{Plov}(f)$.
Section~\ref{sec-filtr} also contains a proof of the statement in Theorem~\ref{JBbound}
involving the Kodaira dimension.
In Section~\ref{proofuniform},
we prove our main results Theorems~\ref{uniform} and \ref{uniformdge4}
about uniform upper bounds of $\mathrm{Plov}(f)$.
Section~\ref{sec-lb} is devoted to lower bound of $\mathrm{Plov}(f)$, in which we prove Theorem~\ref{thm-lb}.
Finally, we study explicit examples in
Sections~\ref{sec-MainTheorem3} and~\ref{sec-MainTheorem2},
which contain
complete descriptions of $\mathrm{Plov}(f)$ for tori (Theorem~\ref{thm51})
and threefolds (Theorem~\ref{main2}) respectively.

Readers who are mainly interested in the polynomial log-volume growth can skip Section \ref{sec-THCR} and the results for ${\rm rGKdim}\, (X, f)$.
Readers who are mainly interested in the case where $X$ is projective
or the results about GK-dimension
can read the statements throughout this paper by replacing $H^{1,1}(X, {\mathbb R})$ by $N^1(X)_{{\mathbb R}}$,
$H^{d-1,d-1}(X, {\mathbb R})$ by $N_1(X)_{{\mathbb R}}$
and a K\"ahler class in $H^{1,1}(X, {\mathbb R})$ by an ample
class in $N^1(X)_{{\mathbb R}}$ and (analytically) nef, big, pseudo-effective classes in $H^{1,1}(X, {\mathbb R})$ by (algebraically) nef, big, pseudo-effective divisor classes in $N^1(X)_{{\mathbb R}}$.
For basics about positive classes and cones of
compact K\"ahler manifolds,
we refer the readers to~\cite{DemaillyMethod}.
\subsection*{ Notations and conventions}
\hfill

Whenever we work on projective varieties, we assume that they are irreducible and defined
over an algebraically closed field $k_0$ of characteristic zero.
All manifolds are assumed to be connected.

For a projective variety $X$, we write
$N^1(X) = \mathrm{NS}(X)/(\mathrm{torsion})$ and
$$N^1(X)_R \mathrel{:=} N^1(X) \otimes_{\mathbb Z} R$$
for any commutative ring $R$. For a compact K\"ahler manifold $X$,
we write
$$H^{i,i}(X,{\mathbb R}) \mathrel{:=} H^{i,i}(X) \cap H^{2i}(X,{\mathbb R}).$$
For every $\alpha \in H^{i,i}(X,{\mathbb R})$, if $\alpha \cdot H^{1,1}(X,{\mathbb R})^{d-i} = 0$ (where $d = \dim X$),
we write
$$\alpha \equiv 0.$$

\medskip\noindent
{\bf Acknowledgments.} We thank Professor Tien-Cuong Dinh for the collaboration of our previous work \cite{DLOZ}, which is crucial in this work, and his warm encouragement. We also thank Professor Yoshinori Gongyo for the invitation to
talk at the Tokyo-Kyoto Algebraic Geometry Zoom Seminar on 11th November 2020, which made our collaboration possible. We thank Professor Serge Cantat for valuable comments,
and Professor S. Paul Smith for his interest
and for bringing his
related work to our attention.
HYL would like to thank the NCTS in Taipei for the hospitality and support during the preparation of this paper.
We are supported by the World Premier International Research centre Initiative (WPI), MEXT, Japan; JSPS grant 20H00111, 20H01809; and an ARF of NUS.

\section{Twisted homogeneous coordinate rings and GK-dimensions: Proof of Theorem~\ref{thm-comp}} \label{sec-THCR}

Following \cite{Ke}, in this section we recall the definition of
twisted homogeneous coordinate rings and related notions,
together with the fundamental properties proven in~\cite{AvB} and~\cite{Ke}.

Let $X$ be a projective variety, defined over an algebraically closed field $k_0$ of characteristic $0$.
Let $f \in \mathrm{Aut}(X)$ be an automorphism.
Given a line bundle $L$ on $X$, we set
$$L^{f} := f^*L.$$
We say that a line bundle $L$ on $X$ is {\it $f$-ample} if for any coherent sheaf $F$ on $X$, there is a positive integer $m_F$ such that
$$H^q(X, F \otimes L \otimes L^f \otimes \ldots \otimes L^{f^{m}}) = 0$$
for any integer $q > 0$ and for any integer $m > m_F$.
A Cartier divisor $D$ is called $f$-ample if $\mathcal{O}(D)$ is $f$-ample.

For any non-negative integer $n$ and any line bundle $L$ on $X$, define\footnote{Our $\Delta_n(f, L)$ is $\Delta_{n+1}(f, L)$ in \cite{Ke}.}
$$\Delta_n(f, L) := \Delta(X, f, L) := L \otimes L^{f} \otimes \ldots \otimes L^{f^n},$$
$$ B_{n+1} := B_{n+1} (X, f, L)  = H^0(X, \Delta_n(f, L))\,\, ,\,\, B_{0} = H^0(X, \mathcal{O}_X) = k_0,$$
$$B := B(X, f, L) := \oplus_{n \in {\mathbb Z}_{\ge 0}} B_n.$$
As $X$ is projective, each $B_n$ is a finite dimensional $k_0$-vector space.
The graded vector space $B$ has a natural (noncommutative) associative graded $k_0$-algebra structure defined by
$$ab := a \otimes (f^{m})^*b$$
for $a \in B_m$ and $b \in B_n$.
We call the graded $k_0$-algebra $B$
the {\it twisted homogeneous coordinate ring} of $X$ associated to $(f, L)$.

\begin{remark}
	By the definition, $L$ is $\mathrm{Id}_X$-ample precisely when $L$ is ample in the usual sense and $B(X, \mathrm{Id}_X, L)$ is nothing but the projective coordinate ring of $X$ associated to the ample line bundle $L$.
\end{remark}

For a (noncommutative) $k_0$-algebra  $A$, its
{\it Gelfand-Kirillov dimension} (or GK-dimension for short) is defined as
$${\rm GKdim}\,\, A := {\rm sup}_{V} \limsup_{N \to \infty} \frac{\log \dim_{k_0} V^{\le N}}{\log N} \in [0, \infty].$$
Here $V$ runs through all finite dimensional vector subspaces of the $k_0$-vector space $A$,
and $V^{\le N}$ is the vector subspace of $A$, generated by the monomials
$$x_1x_2 \cdots x_{\nu}\,\, (0 \le \nu \le N, x_1,x_2 \ldots, x_{\nu} \in V).$$
The supremum is reached by any finite dimensional subspace $V$ generating $A$ ~\cite[p.14, Remark]{KL}.

The Gelfand-Kirillov dimension is a natural noncommutative generalization of
Krull dimension in commutative algebra,
in the sense that
when $A$ is a finitely generated commutative $k_0$-algebra,
we have
${\rm GKdim}\, A = \dim \mathrm{Spec \ } A$~\cite[Theorem 4.5]{KL}.
In particular,
$${\rm GKdim}\, B(X, \mathrm{Id}_X, L) = \dim X + 1$$
for any ample line bundle $L$ on a projective scheme $X$.
From the perspective of projective geometry,
it is thus natural to introduce the
\emph{reduced Gelfand-Kirillov dimension} defined as
$${\rm rGKdim}\, B(X, f, L) := {\rm GKdim}\, B(X, f, L) -1.$$
for twisted homogeneous coordinate rings.
While the Krull dimension is an integer (possibly $+\infty$),
any real number $r \ge 2$ can be realized as the
the GK-dimension of a $k_0$-algebra~\cite{IrvingWarfield}.
The reader is referred to~\cite{KL} for further properties about GK-dimension.

The study of twisted homogeneous coordinate rings
was initiated by Artin and Van den Bergh~\cite{AvB}.
Together with the seminal work of Keeler~\cite{Ke},
here is a list of fundamental properties they proved
about twisted homogeneous coordinate rings.

\begin{theorem}[Keeler, Artin--Van den Bergh]\label{Keeler}
Let $X$ be
a projective variety of dimension $d>0$ and let $f \in \mathrm{Aut} (X)$.
Then
\begin{enumerate}
	\item
	$f$-ample line bundles exist
	if and only if $f^* : N^1(X) \circlearrowleft$ is quasi-unipotent.
	\end{enumerate}

Let $L$ be a line bundle and let $B := B(X, f, L)$ be the twisted homogeneous coordinate ring associated to $(f, L)$.
Assume that $f^* : N^1(X) \circlearrowleft$ is quasi-unipotent.
\begin{enumerate}
\setcounter{enumi}{1}
\item
$L$ is $f$-ample if and only if there exists an integer $n > 0$ such that
$\Delta_n(f, L) = L \otimes L^{f} \otimes \ldots \otimes L^{f^{n}}$ is ample.
In particular, any ample line bundle is $f$-ample.
\item
The twisted homogeneous coordinate ring $B$ is a finitely generated $k_0$-algebra,
which is both right and left noetherian.
\item The GK-dimension ${\rm GKdim}\,\, B$
is independent of the choice of an $f$-ample line bundle $L$.
We therefore define the GK-dimension
$${\rm GKdim}\, (X, f) := {\rm GKdim}\, B(X, f, L)$$
and the reduced GK-dimension
$${\rm rGKdim}\, (X, f) \mathrel{:=} {\rm rGKdim}\, B(X, f, L)
\mathrel{:=} {\rm GKdim}\, B(X, f, L) - 1
$$
for any choice of an $f$-ample line bundle $L$.
\item
For any positive integer $N$, we have
$${\rm GKdim}\, (X, f^N) = {\rm GKdim}\, (X, f).$$
\item
The GK-dimension is a positive integer.
More precisely,
after replacing $f$ by its suitable positive power so that $f^* : N^1(X) \circlearrowleft$ is unipotent,
the self intersection number $(\Delta_n(f, L)^d)$  is a polynomial in $n$ and its degree satisfies:
$${\rm rGKdim}\, B = \deg_n (\Delta_n(f, L)^d).$$
\item
Let $k+1$ be the maximum size of the Jordan blocks of the Jordan canonical form of
$f^* : {N^1(X)_{\mathbb C}} \circlearrowleft$. Then
$$k+d \le {\rm rGK\, dim}\, B \le k(d-1) +d.$$
In particular, we have ${\rm rGKdim}\,(X, f) = \dim X$ if and only if $f^*|_{N^1(X)}$ is of finite order.
\end{enumerate}
\end{theorem}

\begin{proof}
	Let us indicate the references where these statements are proven.
	Statements (1) and (3) are contained in~\cite[Theorem 1.2]{Ke}.
	Statements (2), (4), (6), and (7) follow from~\cite[Lemma 4.1]{AvB}, and~\cite[Proposition 6.11, Theorem 6.1.(1), Theorem 6.1.(3)]{Ke}, respectively.
	Statement (5) is also contained in~\cite{AvB, Ke}. Indeed, since
	$\Delta_N(f, L)$ is $f$-ample, we have
	$${\rm GKdim}\, (X, f^N) = {\rm GKdim}\, B(X,f^N, \Delta_{N-1}(f, L) ) = {\rm GKdim}\, B(X,f,L) = {\rm GKdim}\, (X, f) $$
	where the equality in the middle follows from~\cite[(6.3)]{Ke}.
	\end{proof}

\begin{remark}
When $X$ is a complex projective manifold,
Keeler's work implies Theorem~\ref{thm-comp}
as an immediate corollary
that the reduced GK-dimension of $(X,f)$ coincides with
the polynomial log-volume growth of $f$.
Together with Theorem~\ref{Keeler}, this suggests
unexpected relations between noncommutative algebra and
complex dynamics of automorphisms of zero entropy.
\end{remark}

\begin{proof}[Proof of Theorem~\ref{thm-comp}]
		Since $f^* : {N^1(X)} \circlearrowleft$ is quasi-unipotent,
		by Theorem~\ref{Keeler} (6) and the definition of $\mathrm{Plov}(f)$,
		given any ample line bundle $L$ on $X$, we have
		$${\rm rGKdim}\, B = \deg_n (\Delta_n(f, L)^d)
		=  \limsup_{n \to \infty} \frac{\log \Delta_n(f, c_1(L))^d}{\log n} = \mathrm{Plov}(f).$$
\end{proof}

\section{Polynomial log-volume growth: Proofs of Propositions~\ref{pro-GKbir},~\ref{prop1-Kahler}, and~\ref{pro-birJB}
} \label{sec-plov}

Let $X$ be a compact K\"ahler manifold of dimension $d \ge 1$ and
let $f \in \mathrm{Aut}(X)$. Let
$\omega \in H^{1,1}(X,{\mathbb R})$ be a K\"ahler class.
Recall that we have defined
$$\Delta_n(f, \omega) \mathrel{:=} \sum_{i = 0}^n (f^i)^*\omega.$$
for every integer $n \ge 1$ in Section~\ref{introPlov},
and the volume of the graph  $\Gamma_f(n) \subset X^{n+1}$
of $f \times f^2 \times \cdots \times f^n$
with respect to any K\"ahler metric in the class $\omega$
is equal to
$$
\mathrm{Vol}_\omega(\Gamma_f(n)) = \frac{1}{d!}\Delta_n(f, \omega)^d
$$
Note that the class $\Delta_n(f, \omega)$ and the invariant $\mathrm{Vol}_\omega(\Gamma_f(n))$ are defined
more generally for any $\omega \in H^{1,1}(X,{\mathbb R})$.
The class $\Delta_n(f, \omega)$ also makes sense
in $\mathrm{NS}(X)$ for every $\omega = c_1(L) \in \mathrm{NS}(X)$
when $X$ is
a projective variety.
In this case, the self-intersection number $\Delta_n(f, \omega)^d$
coincides with $\Delta_n(f, L)^d$ introduced in Section~\ref{sec-THCR},
as we have already seen in the proof of Theorem~\ref{thm-comp}.

\begin{lem}\label{lem-sminvVol}
For every nef $\omega \in \mathrm{Nef}(X) \subset H^{1,1}(X,{\mathbb R})$, define
	$$\mathrm{Plov}(f,\omega) \mathrel{:=} \limsup_{n \to \infty} \frac{\log \mathrm{Vol}_\omega(\Gamma_f(n))}{\log n}
	=  \limsup_{n \to \infty} \frac{\log \Delta_n(f, \omega)^d}{\log n}  \in [0,\infty].$$
Then $\mathrm{Plov}(f,\omega)$ is a constant in $\omega$ whenever $\omega$ is nef and big.

The same statement also holds when $X$ is
a projective variety.
\end{lem}

	Lemma~\ref{lem-sminvVol} justifies the well-definedness of
	the polynomial logarithmic volume growth of $f$ in the introduction, which
	 is defined to be
	$$\mathrm{Plov}(f) \mathrel{:=} \mathrm{Plov} (X, f) \mathrel{:=} \mathrm{Plov}(f,\omega)$$
	where $\omega$ is any nef and big class.

To prove Lemma \ref{lem-sminvVol}, we need the following easy but useful result.

\begin{lemma}\label{lem:nef_compar}
Let $X$ be a compact K\"ahler manifold (resp. projective variety) of dimension $d$,
and let
$$L_1\,\, ,\,\, \ldots\,\, ,\,\,  L_d\,\, ,\,\, M_1\,\, ,\,\, \ldots\,\, ,\,\, M_d$$
be nef classes in $H^{1,1}(X,{\mathbb R})$ (resp. $N^1(X)_{{\mathbb R}}$) such that $M_i \ge L_i$, i.e., $M_i - L_i$ is pseudo-effective.
Then
$$(M_1 \cdots M_d) \ge (L_1 \cdots L_d) .$$
In particular, $(M_1^d) \ge (L_1^d)$.
\end{lemma}

\begin{proof}
	Inductively, we have
	$$(M_1 \cdots M_d) \ge (L_1 \cdot M_2 \cdots M_d) \ge
	(L_1 \cdots L_j \cdot M_{j+1} \cdots M_d) \ge \cdots
	\ge (L_1 \cdots L_d),$$
which proves Lemma~\ref{lem:nef_compar}.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem-sminvVol}]
	Let $\omega$ and $\omega'$ be two nef and big classes.
	Then there exists some $C > 0$ such that $\omega - C \omega'$ is pseudo-effective.
	Accordingly, $\Delta_n(f,\omega) - C\Delta_n(f,\omega')$ is pseudo-effective,
	so $\Delta_n(f,\omega)^d \ge C^d\Delta_n(f,\omega')^d$ by Lemma~\ref{lem:nef_compar},
	and therefore $\mathrm{Plov}(f,\omega) \ge \mathrm{Plov}(f,\omega')$.
	By symmetry, we have $\mathrm{Plov}(f,\omega) = \mathrm{Plov}(f,\omega')$.
\end{proof}

The same argument as above proves the following.

\begin{lem}\label{lem-nefle}
For every nef class $\alpha \in H^{1,1}(X, {\mathbb R})$,
we have
$$\mathrm{Plov}(f,\alpha) \le \mathrm{Plov}(f).$$
\end{lem}

\begin{proof}
Take a K\"ahler class $\omega$ such that $\omega \ge \alpha$.
By Lemma~\ref{lem:nef_compar}, we have
	$\Delta_n(f,\omega)^d \ge
	\Delta_n(f,\alpha)^d$
	for every integer $n \ge 0$.
Hence $\mathrm{Plov}(f,\alpha) \le \mathrm{Plov}(f)$.
\end{proof}

Next we prove some basic properties about
polynomial logarithmic volume growth.
First we prove that it is invariant under taking finite iterations.

\begin{lem}\label{lem-invfini}
	For every integer $N \ne 0$, we have $\mathrm{Plov}(f^N) = \mathrm{Plov}(f)$.
\end{lem}

\begin{proof}
Since
$$\(\sum_{i = 0}^n (f^{-i})^*\omega\)^d
		= \((f^{-n})^*\sum_{i = 0}^n (f^i)^*\omega\)^d
		= \(\sum_{i = 0}^n (f^i)^*\omega\)^d,$$
	we have $\mathrm{Plov}(f^{-1}) = \mathrm{Plov}(f)$.
	So it suffices to prove Lemma~\ref{lem-invfini} for $N > 0$.

	For every integers $r$ and $j>0$ such that $0 \le r < N$,
	consider the K\"ahler form
	$\omega_{r,j} \mathrel{:=} \sum_{i = r}^{r + j -1} (f^{i})^*\omega$.
	Then
	$$\omega_{r,mN} \le \omega_{0,r} + \omega_{r,mN} = \omega_{0,r + mN} \le \omega_{r-N,(m+1)N}.$$
	So
	$$\mathrm{Vol}_{\omega_{r,mN}}(X) \le \mathrm{Vol}_{\omega_{0,r + mN}}(X) \le \mathrm{Vol}_{\omega_{r-N,(m+1)N}}(X)$$
	by Lemma~\ref{lem:nef_compar}, and thus
	$$\mathrm{Vol}_{\sum_{j= 0}^{N-1}(f^{r+j})^*\omega}(\Gamma_{f^N}(m))
	\le \mathrm{Vol}_{\omega}(\Gamma_{f}(r + mN))
	\le \mathrm{Vol}_{\sum_{j= 1}^{N}(f^{r-j})^*\omega}(\Gamma_{f^N}(m + 1)).$$
	By Lemma~\ref{lem-sminvVol}, we have
	$$\limsup_{m \to \infty}\frac{\log \mathrm{Vol}_{\sum_{j= 0}^{N-1}(f^{r+j})^*\omega}(\Gamma_{f^N}(m))}{\log m}
	= \mathrm{Plov}(f^N) =
	\limsup_{m \to \infty}\frac{\log \mathrm{Vol}_{\sum_{j= 1}^{N}(f^{r-j})^*\omega}(\Gamma_{f^N}(m + 1))}{\log (m+ 1)},$$
	so for every integer $r$ such that $0 \le r < N$, we have
	$$
	\limsup_{m \to \infty}\frac{\log \mathrm{Vol}_{\omega}(\Gamma_{f}(r + mN))}{\log (r+mN)}
	= \limsup_{m \to \infty}\frac{\log \mathrm{Vol}_{\omega}(\Gamma_{f}(r + mN))}{\log (m)}
	= \mathrm{Plov}(f^N).
	$$
	Hence $\mathrm{Plov}(f) = \mathrm{Plov}(f^N)$.
\end{proof}

The polynomial logarithmic volume growth is also compatible with product.

\begin{lem}\label{lem-pruduct}
	Let $X_i$ ($i=1$, $2$) be compact K\"ahler manifolds and let
	 $f_i \in \mathrm{Aut}(X_i)$.
	Then
		$$\mathrm{Plov} (f_1 \times f_2) = \mathrm{Plov} (f_1) + \mathrm{Plov} (f_2).$$
\end{lem}

\begin{proof}
	Let $\omega_i$ be a K\"ahler metric on $X_i$ and
	let $\mathrm{pr}_i : X_1 \times X_2 \to X_i$ be the projection to the $i$-th factor.
	Then
	$$
	\mathrm{Vol}_{\mathrm{pr}_1^*\omega_1 + \mathrm{pr}_2^*\omega_2}(\Gamma_{f_1 \times f_2}(n)) =
	\mathrm{Vol}_{\omega_1}(\Gamma_{f_1}(n))\mathrm{Vol}_{\omega_2}(\Gamma_{f_2}(n)),
	$$
which proves Lemma~\ref{lem-pruduct}.
\end{proof}

For $1 \le i \le d$, recall that
the $i$-th {\it dynamical degree} of $f$, is defined to be:
$$d_i(f) := \lim_{n \to \infty} (\omega^{d - i} \cdot (f^n)^*\omega^i)^{1/n},$$
where $\omega \in H^{1,1}(X)$ is a K\"ahler class~\cite{DS3}. The dynamical degrees of an automorphism $f : X \to X$ of
a projective variety $X$
are defined similarly, with $\omega$ replaced by an ample class in $N^1(X)$.

\begin{lemma}[{\cite{NZ}}]\label{lem:d_1=rho}
Let $X$ be a compact K\"ahler manifold or
a projective variety.
Let $f \in \mathrm{Aut}(X)$. Then
\begin{enumerate}
\item[(1)]
When $X$ is K\"ahler, the dynamical degree
$d_i(f)$ equals the spectral radius of $f^* : {H^{i,i}(X)} \circlearrowleft$.
\item[(2)]
When $X$ is projective, the first dynamical degree
$d_1(f)$ equals the spectral radius of $f^* : N^1(X)_{\mathbb C} \circlearrowleft$.
\item[(3)]
The value $d_i(f)$ does not change if we replace $\omega$
by any other nef and big class in the definition.
\end{enumerate}
\end{lemma}

\begin{proof}
All these statements follow from~\cite[Proposition A.2 and Lemma A.7]{NZ}.
\end{proof}

We can characterize whether a holomorphic automorphism $f \in \mathrm{Aut}(X)$ has
zero entropy based on the finiteness of $\mathrm{Plov}(f)$.

\begin{lemma}\label{lem:equiv_dyn}
Let $X$ be a compact K\"ahler manifold of dimension $d \ge 1$ and let $f \in \mathrm{Aut}(X)$. Then
the following assertions are equivalent.

\begin{enumerate}
\item
$f^* : {H^{\bullet}(X, {\mathbb C})} \circlearrowleft$ is quasi-unipotent, i.e., a positive power of it is unipotent.
\item
$f^* : {H^{1,1}(X)} \circlearrowleft$ is quasi-unipotent.
\item
the first dynamical degree $d_1(f) = 1$.
\item
the topological entropy $h_{\mathrm{top}}(f) = 0$.
\item
$\mathrm{Plov}(f) < \infty$.
\end{enumerate}
If $X$ is projective, then the above assertions are equivalent to the following.
\begin{enumerate}
	\setcounter{enumi}{5}
	\item $f^* : {N^1(X)} \circlearrowleft$ is quasi-unipotent.
\end{enumerate}
\end{lemma}

\begin{proof}
The equivalence of the first four conditions is well-known and is obtained as follows.
By Gromov-Yomdin's theorem (\emph{cf.}  \cite{Gromov}, \cite{Yomdin}; see also~\cite[Theorem 3.6]{OguisoICM}), we have
$$h_{\mathrm{top}}(f) = \mathrm{lov}(f) = \log r(f^*) = \log(\max_{1 \le i \le d} \, \{d_i(f)\})$$
where $r(f^*)$  is the spectral radius of $f^* : H^\bullet(X,{\mathbb C}) \circlearrowleft$, and
$$\mathrm{lov}(f) \mathrel{:=}  \limsup_{n \to \infty} \frac{\log \mathrm{Vol}_\omega(\Gamma_f(n))}{n}.$$
Together with the log concavity of dynamical degrees $d_i(f)$
(which follows from Khovanskii-Teissier's inequality), this
implies that $h_{\mathrm{top}}(f) > 0$ if and only if $d_i(f) > 1$ for some (and hence all)
$i \in \{1, \cdots, d-1\}$.
Thus the equivalence of the first four assertions follows from Kronecker's theorem, and also $(5)$ implies
these assertions.

To see that (2) implies (5),
recall that in order to compute $\mathrm{Plov}(f)$, by Lemma~\ref{lem-invfini}
we can replace $f$ by some iteration of it,
so that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
Hence $\mathrm{Plov}(f) < \infty$ is a consequence of Lemma~\ref{lem35} below.

Finally when $X$ is projective,
the equivalence between (6) and (3) follows from Lemma~\ref{lem:d_1=rho}
and Kronecker's theorem.
\end{proof}

Next, we prove the invariance of $\mathrm{Plov}(f)$ under generically finite meromorphic maps.

\begin{lemma}\label{lem-invft}
	Let $X$ and $Y$ be
	compact K\"ahler manifolds of dimension $d \ge 1$ and
	$f_X \in \mathrm{Aut} (X)$ and $f_Y \in \mathrm{Aut} (Y)$.
	Let $\phi : X \dasharrow Y$
	be a generically finite dominant meromorphic map such that $f_Y \circ \phi = \phi \circ f_X$.
	Then $d_p(f_X) = d_p(f_Y)$ for every integer $p \ge 1$, and
	$$\mathrm{Plov}(f_X) = \mathrm{Plov}(f_Y).$$
\end{lemma}

\begin{proof}
	First we reduce to the case where $\phi$ is holomorphic.
	Let $\Gamma$ be the Zariski closure of the graph of $\phi$ in $X \times Y$. Let $p_X : \Gamma \to X$ and $p_Y : \Gamma \to Y$ be the projection. Since $f_X \in \mathrm{Aut}(X)$ and $f_Y \in \mathrm{Aut}(Y)$, it follows that
	$$f_{\Gamma} := (f_X \times f_Y)|_{\Gamma} \in \mathrm{Aut} (\Gamma)$$
	and $f_{\Gamma}$ and $f_X$ (resp. $f_{\Gamma}$ and $f_Y$) are equivariant with respect to a generically finite surjective morphism $p_X$ (resp. $p_Y$).
	By the existence of functorial resolution of singularities~\cite{BM} (see also~\cite[Theorem 3.45]{KollarResSing}),
	there exists a K\"ahler desingularization
	$\nu : \widetilde{\Gamma} \to \Gamma$
	such that
	$f_\Gamma \circ \nu = \nu \circ f_{\widetilde{\Gamma}}$
	for some $f_{\widetilde{\Gamma}} \in \mathrm{Aut}(\widetilde{\Gamma})$.
	If Lemma~\ref{lem-invft} holds whenever $\phi$ is holomorphic,
	then $d_p(f_X) = d_p(f_{\widetilde{\Gamma}}) = d_p(f_Y)$
	and $ \mathrm{Plov}(f_X) = \mathrm{Plov}(f_{\widetilde{\Gamma}}) = \mathrm{Plov}(f_Y)$.

	It remains to prove Lemma~\ref{lem-invft} assuming that $\phi$ is holomorphic.
	The statement $d_p(f_X) = d_p(f_Y)$ is proven in~\cite[Proposition A.8]{NZ}  (see also \cite{DNT}).
	For every K\"ahler class $\omega$ on $Y$, since
	$$
	\(\sum_{i = 0}^n (f_X^i)^*(\phi^*\omega)\)^d
	= \phi^*\(\sum_{i = 0}^n (f_Y^i)^*\omega\)^d
	= \deg(\phi) \cdot \(\sum_{i = 0}^n (f_Y^i)^*\omega\)^d,$$
	we have $\mathrm{Plov}(f_X,\phi^*\omega) = \mathrm{Plov}(f_Y,\omega) = \mathrm{Plov}(f_Y)$.
	As $\phi^*\omega$ is nef and big, it follows from Lemma~\ref{lem-sminvVol}
	that $\mathrm{Plov}(f_X) = \mathrm{Plov}(f_Y)$.
\end{proof}

We will see that the same argument as above implies Proposition~\ref{pro-GKbir}.

\begin{proof}[Proof of Proposition~\ref{pro-GKbir}]
As in Lemma \ref{lem-invft}, up to replacing $\pi$ by its graph, we can assume that
$\pi$ is a morphism. Let $\omega_Y$ be a nef and big class on $X$.
Then $\omega_X \mathrel{:=} \pi^* \omega_Y$ is nef and big as well.
We have
$$ \omega_X^{d - 1} \cdot (f_X^n)^*\omega_X = \pi^*\omega_Y^{d - 1} \cdot  (f_X^n)^*\pi^*\omega_Y
=\pi^*\omega_Y^{d - 1} \cdot  \pi^*(f_Y^n)^*\omega_Y = \deg(\pi) \cdot \(\omega_Y^{d - 1} \cdot (f_Y^n)^*\omega_Y\).$$
By Lemma~\ref{lem:d_1=rho} (3), we can take $\omega_X$ and $\omega_Y$
as nef and big classes in the definitions of
$d_1(f_X)$ and $d_1(f_Y)$ respectively,
so $d_1(f_X) = d_1(f_Y)$.
Hence the first assertion of Proposition~\ref{pro-GKbir} follows from
the equivalence between (3) and (6) in Lemma~\ref{lem:equiv_dyn}.

Since both $\omega_X$ and $\omega_Y$ are nef and big,
by Theorem \ref{Keeler} (6) and Lemma~\ref{lem-sminvVol},
${\rm rGKdim}\, (X, f_X)$ and ${\rm rGKdim}\, (Y, f_Y)$ are
the polynomial degree of $ n \mapsto \Delta_n(f_X, \omega_X)^d$
and $ n \mapsto \Delta_n(f_Y, \omega_Y)^d$ respectively.
The same argument as in Lemma \ref{lem-invft} shows that
$$ \Delta_n(f_X, \omega_X)^d =  \deg(\pi) \cdot \Delta_n(f_Y, \omega_Y)^d,$$
which implies ${\rm rGKdim}\, (X, f) = {\rm rGKdim}\, (Y, f_Y)$.
\end{proof}

\begin{lem}\label{lem-restr}
Let $W \subset X$ be a closed subvariety such that $f(W) = W$.
Then
 $\mathrm{Plov}(f_{|W}) \le \mathrm{Plov}(f)$ for the automorphism $f_{|W} \in \mathrm{Aut} (W)$ induced from $f$ by restriction. Here we define
$$\mathrm{Plov}(f_{|W}) := \mathrm{Plov}(\widetilde{W}, \tilde{f}),$$
where $\tau : \widetilde{W} \to W$
is a K\"ahler desingularization of $W$ such that
 $f_{|W} \circ \tau = \tau \circ \ti{f}$ for some $\tilde{f} \in \mathrm{Aut} (\widetilde{W})$,
which does not depend on the choice of $\tilde{W}$ by Lemma \ref{lem-invft}.
\end{lem}

\begin{proof}
	Let $\nu: \widetilde{W} \to X$ be the composition of $\tau$ with the inclusion $W \hookrightarrow X$.
	Let $d \mathrel{:=} \dim X$ and $e \mathrel{:=} \dim W$.
	Let $\omega$ be a K\"ahler class of $X$.
	Up to replacing $\omega$ by some positive multiple of it,
	we can assume that
	$$(\omega^{d-e} - [W]) \cdot \beta \ge 0$$
	for every $\beta$ in the closed convex cone generated by products of
	$e$ K\"ahler classes.

	Since $\sum_{i = 0}^n(f^i)^*(\omega) - \omega$ is K\"ahler when $n \ge 1$,
	by Lemma~\ref{lem:nef_compar} we have,
	\begin{equation}
	\begin{split}
	\(\sum_{i = 0}^n(\ti{f}^i)^*(\nu^*\omega)\)^e
	& = \(\(\sum_{i = 0}^n(f^i)^*(\omega)\)^e \cdot [W]\) \\
	& \le
	\(\(\sum_{i = 0}^n(f^i)^*(\omega)\)^e \cdot \omega^{d-e}\) \le \(\sum_{i = 0}^n(f^i)^*(\omega)\)^d.
	\end{split}
	\end{equation}
	So $\mathrm{Plov}(\ti{f},\nu^*\omega) \le \mathrm{Plov}(f,\omega) = \mathrm{Plov}(f)$.
	As $\nu^*\omega$ is big and nef, we have $\mathrm{Plov}(\ti{f}) = \mathrm{Plov}(\ti{f},\nu^*\omega)$
	by Lemma~\ref{lem-sminvVol}. Hence $\mathrm{Plov}(f|_{W}) \le \mathrm{Plov}(f)$.
\end{proof}

{\it From now on, we assume that
$f^* : H^{1,1}(X) \circlearrowleft$ is unipotent unless otherwise specified until the end of Section~\ref{sec-plov}.}
The operator $$N \mathrel{:=} f^* - \mathrm{Id} : H^{1,1}(X) \to H^{1,1}(X)$$
is thus nilpotent;
let $k = k(f)$ be the largest integer such that $N^k \ne 0$.
Equivalently, $k(f)+1$ is the maximal size of the Jordan blocks
of the Jordan canonical form of $f^* : H^{1,1}(X) \circlearrowleft$.

Let us first prove Proposition~\ref{pro-birJB},
which implies in particular that $k(f)$ is invariant under bimeromorphic modifications.
We need the following result in linear algebra.

\begin{lem}\label{lem-BPFk}
Let $V$ be a finite dimensional vector space over ${\mathbb R}$
and let $f \in \mathrm{GL}(V)$ be a unipotent operator.
Let $k(f)$ denote the largest integer such that $(f - \mathrm{Id})^{k(f)} \ne 0$.
Assume that $f$ preserves a closed salient convex cone
$\mathcal{C} \subset V$ with nonempty interior.
Then for every linear form $\phi : V \to {\mathbb R}$ such that $\phi(\mathcal{C} \backslash \{0\}) > 0$,
and every $v \in \mathrm{Int}(\mathcal{C})$, we have
$$\phi(f^n(v)) \sim_{n \to \infty} Cn^{k(f)}$$
for some $C > 0$.
\end{lem}

\begin{proof}
Since $\mathcal{C}$ has nonempty interior, there exists $v_0 \in \mathcal{C}$ and $C_0 > 0$ such that
 $$\|f^n(v_0)\| =_{n \to \infty} C_0 n^{k(f)}.$$
 Since $\phi(\mathcal{C} \backslash \{0\}) > 0$ and $\mathcal{C}$ is closed, we have
$$\phi(f^n(v_0)) \sim_{n \to \infty} C_1n^{k(f)}$$
for some $C_1 > 0$.
Let $v \in \mathrm{Int}(\mathcal{C})$. Then there exist $C_2 > 0$ and $w \in \mathcal{C}$ such that $v = C_2v_0 +  w$.
Since
$\phi(f^n(w)) =_{n \to \infty} n^{k(f)} O(1)$ with $O(1) \ge 0$
because $\|f^n(w)\| =_{n \to \infty} O(n^{k(f)})$ by assumption,
we have
$$\phi(f^n(v)) = C_2\phi(f^n(v_0)) + \phi(f^n(w))  \sim_{n \to \infty} \(C_1C_2  + O(1)\)n^{k(f)},$$
which proves the lemma.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{pro-birJB}]
As in Lemma \ref{lem-invft}, up to replacing $X$
by an equivariant desingularization of
the graph of $\pi$, we can assume that $\pi$ is holomorphic.
By assumption together with Lemmas~\ref{lem-invft} and \ref{lem:equiv_dyn},
$f_X^* : {H^{1,1}(X)} \circlearrowleft$ and $f_Y^* : {H^{1,1}(Y)} \circlearrowleft$ are both quasi-unipotent,
so up to replacing $f_X$ and $f_Y$ by some positive iterations,
we can assume that they are both unipotent.

Applying Lemma~\ref{lem-BPFk} to the nef cone
in $H^{1,1}(X,{\mathbb R})$, we see that
for every pair of K\"ahler classes $\omega$, $\eta$ on $X$, we have
\begin{equation}\label{rateX}
 (f_X^*)^n (\omega) \cdot \eta^{\dim\, X -1}   \sim_{n \to \infty} C n^{k(f_X)}
 \end{equation}
for some $C >0$.
Similarly,  for every pair of K\"ahler classes $\omega'$, $\eta'$ on $Y$, we have
\begin{equation}\label{rateY}
 (f_X^*)^n (\pi^*\omega')\cdot \pi^*\eta'^{\dim X -1}
=\deg(\pi) \cdot (f_Y^*)^n (\omega') \cdot \eta'^{\dim Y -1}  \sim_{n \to \infty} C' n^{k(f_Y)}
 \end{equation}
for some $C' > 0$.
Since the classes $\omega,\eta, \pi^*\omega',\pi^*\eta'$ are all nef and big,
with the notation of Lemma~\ref{lem:nef_compar} we have
$$c_1 \pi^*\omega' \le \omega \le c_2 \pi^*\omega' \ \text{ and }  \ c_3 \pi^*\eta' \le \eta \le c_4 \pi^*\eta'$$
for some positive real numbers $c_i$.
It follows from  Lemma~\ref{lem:nef_compar}
that the growth rates of~\eqref{rateX} and~\eqref{rateY} have the same polynomial order.
Hence $k(f_X) = k(f_Y)$.
\end{proof}

By definition of $k \mathrel{:=} k(f)$, for every $\alpha \in H^{1,1}(X,{\mathbb R})$, we have
\begin{equation}\label{eqn-abel}
	\Delta_n(f, \alpha)  =  \sum_{i = 0}^n(f^i)^*(\alpha) = \sum_{i = 0}^n\sum_{j=0}^{k}{i\choose j} N^j\alpha = \sum_{j=0}^{k}\sum_{i = 0}^n{i\choose j} N^j\alpha = \sum_{j=0}^{k}{n\choose j+1} N^j\alpha.
\end{equation}

The following lemma will be useful to prove results
concerning lower bounds of $\mathrm{Plov}(f)$ in this paper.
In the projective setting, this lemma was due to Keeler~\cite[Lemma 6.5(4)]{Ke}
and was applied in his work to prove his lower bound.

\begin{lem}\label{lem-sdP}
	For every integer $i \in [1,d]$, let
	$$P_{f,\omega,i}(n) \mathrel{:=} \Delta_n(f, \omega)^i \omega^{d-i} = \(\sum_{j=0}^{k}{n\choose j+1} N^j\omega\)^i\omega^{d-i},$$
	which is a polynomial in $n$.
	Then we have
	$$\deg P_{f,\omega,i} > \deg P_{f,\omega,i-1}.$$
\end{lem}
\begin{proof}

	For every non-negative integer $m$, define
	$$P_{f,\omega,i,m}(n) \mathrel{:=} \Delta_n(f, \omega)^{i-1}\cdot (f^m)^*\omega \cdot \omega^{d-i} = \(\sum_{j=0}^{k}{n\choose j+1} N^j\omega\)^{i-1} \cdot (\mathrm{Id} + N)^m(\omega) \cdot \omega^{d-i},$$
	which is a polynomial in $n$.
	Note that since both $\omega$ and $(f^m)^*\omega$ are K\"ahler, we have
	$$C_1 \omega \le (f^m)^*\omega \le C_2 \omega$$
	for some $C_1,C_2 > 0$, so
	$$C_1 P_{f,\omega,i,0}(n) \le P_{f,\omega,i,m}(n) \le C_2 P_{f,\omega,i,0}(n)$$
	by Lemma~\ref{lem:nef_compar} and therefore
	$$\deg_n P_{f,\omega,i,m}(n) = \deg_n P_{f,\omega,i,0}(n) = \deg_n P_{f,\omega,i-1}(n).$$
	In particular, $\deg_n P_{f,\omega,i,m}(n)$ is independent of $m$.

	For every $m$,
	since $P_{f,\omega,i,m}(n) > 0$,
	the leading coefficient $C_{f,\omega,i}(m)$ of the polynomial $P_{f,\omega,i,m}$
	satisfies $C_{f,\omega,i}(m) > 0$.
	As $C_{f,\omega,i}(m)$ is polynomial in $m$ (because $N$ is nilpotent),
	the minimum of
	$$	\left\{C_{f,\omega,i}(m) \mid m \in {\mathbb Z}_{\ge 0} \right\}$$
	exists;
	let $\ell \in {\mathbb Z}_{\ge 0}$ such that $C_{f,\omega,i}(\ell)$ is the minimum.

	By construction,  we have
	$$\lim_{n \to \infty} \frac{P_{f,\omega,i,m}(n)}{P_{f,\omega,i,\ell}(n)} = \frac{C_{f,\omega,i}(m)}{C_{f,\omega,i}(\ell)} \ge 1$$
	for every $m \in {\mathbb Z}$, so
	$$\frac{P_{f,\omega,i}(n)}{P_{f,\omega,i,\ell}(n)}
	= \sum_{m = 0}^n\frac{P_{f,\omega,i,m}(n)}{P_{f,\omega,i,\ell}(n)} \succeq_{n \to \infty} \gamma,$$
	for any $\gamma>0$,
	which shows that $\deg_n P_{f,\omega,i,\ell}(n) < \deg_n P_{f,\omega,i}(n)$.
	Hence
	$$\deg_n P_{f,\omega,i-1}(n) = \deg_n P_{f,\omega,i,\ell}(n) < \deg_n P_{f,\omega,i}(n).$$
\end{proof}

\begin{lem} \label{lem35}
	$\mathrm{Plov}(f)$ is equal to the degree of the polynomial
	$$n \mapsto P_{f,\omega}(n) \mathrel{:=} \Delta_n(f, \omega)^d  =  \(\sum_{j=0}^{k(f)}{n\choose j+1} N^j\omega\)^d$$
	for any K\"ahler class $\omega$ on $X$.
	As a consequence, $\mathrm{Plov}(f)$ is a positive integer satisfying
	\begin{equation}\label{eqn-PlovNil}
	k(f) + d \, \le \, \mathrm{Plov}(f) \, \le \, d +
	\max \left\{\sum_{j = 1}^d i_j  \ \bigg{|} \
	i_j \in {\mathbb Z}_{\ge 0}, (N^{i_1}\omega)\cdots(N^{i_d}\omega) \ne 0 \right\}.
	\end{equation}
	where $d = \dim X$.
\end{lem}

\begin{proof}

	The upper bound of $\mathrm{Plov}(f)$ is clear by the equation (\ref{eqn-abel}).

	For the lower bound, by Lemma~\ref{lem-sdP} with the notations therein, we have
	$$\mathrm{Plov}(f) = \deg P_{f,\omega,d} > \deg P_{f,\omega,d-1} > \cdots > \deg P_{f,\omega,1}.$$
	As $N^{k(f)}\omega = k(f)! \cdot \lim_{m \to \infty} \frac{(f^*)^m(\omega)}{m^{k(f)}}$
	is nef and nonzero by definition of $k(f)$, we have $(N^{k(f)}\omega) \cdot \omega^{d-1} \ne 0$.
	So $\deg P_{f,\omega,1} \ge k(f)+1$, which shows that
	$\mathrm{Plov}(f) \ge k(f) + d$.
\end{proof}

The following lemma provides another way to compute $\mathrm{Plov}(f)$, which will turn out to be useful.
Define
$$ \Delta'_n(f, \omega) \mathrel{:=}   \sum_{i = 0}^n \((f^i)^* + (f^{-i})^*\)(\omega).$$

\begin{lem}\label{lem-plovdbend}
	$\mathrm{Plov}(f)$ is also the degree of the polynomial
	$$n \mapsto \Delta'_n(f, \omega)^d = \(\omega + \sum_{i = -n}^n (f^i)^*\omega\)^d.$$
\end{lem}

\begin{proof}
	Recall that $\mathrm{Plov}(f)$ is defined as the polynomial degree of
	$n \mapsto \(\sum_{i = 0}^n (f^i)^*\omega\)^d$, which is also the
	polynomial degree of
	$n \mapsto \(\sum_{i = 0}^{2n} (f^i)^*\omega\)^d$ as well as $n \mapsto \(\sum_{i = 0}^{2n} 2(f^i)^*\omega\)^d$. Hence
	Lemma~\ref{lem-plovdbend} follows from
	$$\(\omega + \sum_{i = -n}^n (f^i)^*\omega\)^d
	= \((f^{-n})^*\((f^{n})^*\omega +\sum_{i = 0}^{2n} (f^i)^*\omega\)\)^d
	=\((f^{n})^*\omega +\sum_{i = 0}^{2n} (f^i)^*\omega\)^d
	$$
	and
	$$\(\sum_{0}^{2n} (f^i)^*\omega\)^d \le \((f^n)^* \omega + \sum_{0}^{2n} (f^i)^*\omega\)^d
	\le \(\sum_{0}
	^{2n} 2(f^i)^*\omega\)^d.$$
\end{proof}

As $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent, $(f^{-1})^* : H^{1,1}(X) \circlearrowleft$ is also unipotent.
Set
$$ N' \mathrel{:=} (f^{-1})^* - \mathrm{Id} :  H^{1,1}(X) \to H^{1,1}(X)$$
and let
$$N_m \mathrel{:=} N^m + N'^m.$$
We have the analogous statement of Lemma~\ref{lem35} with $N^m$ replaced by $N_m$.

\begin{lem}\label{lem-ubN_i}
	We have
	$$\mathrm{Plov}(f) \le
	d + \max \left\{\sum_{j = 1}^d i_j  \ \bigg{|} \
	i_j \in {\mathbb Z}_{\ge 0}, (N_{i_1}\omega)\cdots(N_{i_d}\omega) \ne 0 \right\}.$$
\end{lem}

\begin{proof}
	Lemma~\ref{lem-ubN_i} follows from Lemma~\ref{lem-plovdbend} together with
	$$ \Delta'_n(f, \omega)
	= \sum_{i = 0}^n (f^i)^*\omega + \sum_{i = -n}^0 (f^i)^*\omega
	= \sum_{i = 0}^{k(f)} \binom{n}{i+1}(N^i + N'^i)(\omega)
	= \sum_{i = 0}^{k(f)} \binom{n}{i+1} N_i(\omega).$$
\end{proof}

\begin{lem}\label{lem-sdPbis}
		For every integer $i \in [0,d]$, let
		$$P'_{f,\omega,i}(n) \mathrel{:=} \Delta'_n(f, \omega)^i \omega^{d-i} = \(\sum_{j=0}^{k}{n\choose j+1} N_j\omega\)^i\omega^{d-i},$$
		which is a polynomial in $n$.
		Then we have
		$$\deg P'_{f,\omega,i} > \deg P'_{f,\omega,i-1}.$$
\end{lem}

\begin{proof}
	As in the proof of Lemma~\ref{lem-sdP}, for every non-negative integer $m$, we define
			$$P'_{f,\omega,i,m}(n) \mathrel{:=} \Delta'_n(f, \omega)^{i-1}\cdot( (f^m)^*\omega + (f^{-m})^*\omega) \cdot \omega^{d-i}.$$
		The same argument in Lemma~\ref{lem-sdP} shows that
		$$\deg_n P'_{f,\omega,i,m}(n) = \deg_n P'_{f,\omega,i,0}(n) = \deg_n P'_{f,\omega,i-1}(n)$$
		for every $m$, and there exists $\ell \in {\mathbb Z}_{>0}$ such that
		the leading coefficient $C_{f,\omega,i}(\ell) > 0$ of $P'_{f,\omega,i,\ell}$ is minimum
		among all $\ell \in {\mathbb Z}_{>0}$.
		Since $\Delta'_n(f, \omega)^{i-1}\cdot \omega \cdot \omega^{d-i} > 0$ (because $\omega$ is K\"ahler),
		it follows that
		$$\frac{P'_{f,\omega,i}(n)}{P'_{f,\omega,i,\ell}(n)} =
		 \sum_{m = 0}^n\frac{ P'_{f,\omega,i,m}(n)}{P'_{f,\omega,i,\ell}(n)}
	\succeq_{n \to \infty} \gamma,$$
	for any $\gamma>0$, and we conclude the proof as in Lemma~\ref{lem-sdP}.
\end{proof}

Let $\omega \in H^{1,1}(X)$ and
let $d_{\mathrm{pol}}(f,\omega)$ denote the
polynomial degree of
$$n \mapsto \((f^n)^*\omega + (f^{-n})^*\omega\)^d = \( \sum_{i = 0}^k \binom{n}{i}N_i(\omega) \)^d.$$

\begin{remark}
	The same argument proving Lemma~\ref{lem-sminvVol}
	shows that the polynomial dynamical degree
	$d_{\mathrm{pol}}(f,\omega)$ is independent of $\omega$ whenever $\omega$ is big and nef.
\end{remark}

We will use the next lemma in the proof of Lemma~\ref{lem-vank-2d4}.

\begin{lem}\label{lem-ubdpol}
	We have
	$$d_{\mathrm{pol}}(f,\omega) \le k(f) \left\lfloor \frac{d}{2} \right\rfloor \le \frac{k(f)d}{2} \le d(d-1).$$
\end{lem}
\begin{proof}
	Note that
	$$
	\((f^n)^*\omega + (f^{-n})^*\omega\)^d
	= \sum_{j=1}^d \binom{d}{j} (f^n)^*\omega^{j} \cdot (f^{-n})^*\omega^{d-j}
	= \sum_{j=1}^d \binom{d}{j} \((f^{2n})^*\omega^{j} \)\cdot \omega^{d-j}.
	$$
	It follows from Lemma~\ref{lem-BPFk}
		(applied to the closed convex cone
		 $\mathcal{K}^i(X) \subset H^{i,i}(X,{\mathbb R})$
		 generated by classes of smooth positive $(i,i)$-forms for every $i$) that
	\begin{equation}\label{ineq-dpol}
	d_{\mathrm{pol}}(f,\omega) \le \max_{0 \le i \le d}\,
	\big( \min \left\{c \in {\mathbb Z} \ \ \big{|} \ \ \| (f^n)^* \circlearrowleft H^{i,i}(X) \| = O(n^c) \right\} \big).
	\end{equation}
	Thus the first inequality follows from~\cite[Proposition 5.8]{Dinh-bis}
	and the last inequality from Theorem~\ref{JBbound}.
\end{proof}

\section{Quasi-nef sequences and associated filtrations} \label{sec-filtr}

Let $X$ be a compact K\"ahler manifold of dimension $d \ge 1$.
For every $\alpha \in H^{i,i}(X,{\mathbb R})$, if $\alpha \cdot H^{1,1}(X,{\mathbb R})^{d-i} = 0$, we write
$$\alpha \equiv 0.$$
Let $\mathcal{K}^i(X) \subset H^{i,i}(X,{\mathbb R})$ be
the closed convex cone generated by classes of smooth positive $(i,i)$-forms.
We have $\mathcal{K}^1(X) = \mathrm{Nef}(X)$, which is the nef cone of $X$.
For every $\alpha \in \mathcal{K}^i(X)$, define
$$\mathrm{Nef}(\alpha) \mathrel{:=}  \overline{\alpha \cdot \mathrm{Nef}(X)} \subset H^{i+1,i+1}(X,{\mathbb R}).$$
As $\mathrm{Nef}(X)$ is a convex cone, so is $\mathrm{Nef}(\alpha)$. Since $\mathrm{Nef}(\alpha) \subset \mathcal{K}^{i+1}(X)$
and $\mathcal{K}^{i+1}(X)$ is salient,
$\mathrm{Nef}(\alpha)$ is a closed salient cone.

\begin{construction}[Quasi-nef sequence~\cite{Zhang3}]

Let  $f \in \mathrm{Aut}(X)$ be an automorphism of $X$ such that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
	A \emph{quasi-nef sequence} (with respect to $f$) is a sequence
	$$M_1,\, \ldots, \, M_d \, \in \, H^{1,1}(X,{\mathbb R})$$
	constructed recursively as follows.
	Suppose that $M_1,\ldots,M_i \in H^{1,1}(X,{\mathbb R})$ are constructed,
	then $M_{i+1} \in H^{1,1}(X,{\mathbb R})$ is an element such that
	\begin{itemize}
		\item $f^*(M_1\cdots M_{i+1}) = M_1\cdots M_{i+1} \ne 0$,
		\item $M_1\cdots M_i M_{i+1} \in \mathrm{Nef}(M_1\cdots M_{i})$.
	\end{itemize}
	Since $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent,
	the existence of $M_{i+1}$ follows from
	Birkhoff's Perron--Frobenius theorem~\cite{Bi} applied to $\mathrm{Nef}(M_1\cdots M_{i})$.
	We set $L_0 \mathrel{:=} 1 \in H^0(X,{\mathbb R})$ and define $L_i \mathrel{:=} M_1\cdots M_i \in H^{i,i}(X,{\mathbb R})$.

Note that $M_1,\, \ldots, \, M_d \, \in \, H^{1,1}(X,{\mathbb R})$ is also a quasi-nef sequence with respect to $f^{-1}$.
\qed
\end{construction}

Given a quasi-nef sequence $M_1,\ldots,M_d \in H^{1,1}(X,{\mathbb R})$
with respect to an automorphism $f \in \mathrm{Aut}(X)$ such that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent, define
$$F_i \mathrel{:=} \big\{\alpha \in H^{1,1}(X,{\mathbb R}) \mid L_i\alpha \equiv 0 \big\}$$
and let $F'_i$ be the subspace of $F_i$ spanned by
$$\big\{\alpha \in F_i \mid L_{i-1} \alpha \equiv \beta \text{ for some } \beta \in \mathrm{Nef}(L_{i-1}) \big\}.$$
Recall from~\cite{DLOZ} that these subspaces form an $f^*$-invariant filtration
\begin{equation}\label{filt-F'F}
0 = F_0 \subset F'_1 \subset F_1 \subset \cdots \subset F'_{d-1} \subset F_{d-1} \subset F'_d = H^{1,1}(X,{\mathbb R}).
\end{equation}
We note that the filtration~\eqref{filt-F'F}
\emph{depends on the choice} of
a quasi-nef sequence $M_1,\ldots,M_d \in H^{1,1}(X,{\mathbb R})$.

Let us recall the fundamental properties of these filtrations proven in~\cite{DLOZ}.

\begin{pro}\label{pro-filtr}
	\hfill
	\begin{enumerate}
		\item (\cite[Lemma 3.8]{DLOZ})
		We have $\dim (F'_i/F_{i-1}) \le 1$ and
		$$F'_i = \{ \gamma \in  F_i \mid L_{i-1}\gamma^2 \equiv 0\}.$$
		Moreover the following assertions are equivalent:
		\begin{itemize}
		\item[(i)]
$F_{i-1} \ne F'_i$;
		\item[(ii)]
$F'_i = F_{i-1} \oplus ({\mathbb R} \cdot M_i)$;
		\item[(iii)]
$L_{i-1}M_i^2 \equiv 0$.
		\end{itemize}

		\item (\cite[Proposition 3.9]{DLOZ}) There exists an integer $r \in [1,d-1]$
		and a strictly decreasing sequence of integers
		$$d-1 \ge s_1 > \cdots > s_r \ge 1$$
		such that for every K\"ahler class $\omega \in H^{1,1}(X,{\mathbb R})$ and every integer $j \in [1,r]$,
		$$(f^* - \mathrm{Id})^{2j-1}\omega \in F_{s_j} \backslash F'_{s_j} \ \ \text{ and } \ \
		(f^* - \mathrm{Id})^{2j}\omega \in F'_{s_j} \backslash F_{s_j - 1},$$
		and $(f^* - \mathrm{Id})^{2r+1}\omega = 0$. In particular, $(f^* - \mathrm{Id})^{2r+1} = 0 \in \mathrm{End}(H^{1,1}(X,{\mathbb R}))$.
	\end{enumerate}
\end{pro}
\begin{proof}
	The only statement which is not explicitly stated in~\cite{DLOZ} is that
	$L_{i-1}M_i^2 \equiv 0$ implies $F_{i-1} \ne F_i'$.
	If $L_{i-1}M_i^2 \equiv 0$, then $M_i \in F_i$
	because $L_iM_i = L_{i-1}M_i^2 \equiv 0$, and therefore $M_i \in F_i'$
	by the first statement of Proposition~\ref{pro-filtr}(1).
	As $L_{i-1}M_i = L_i \not\equiv 0$, we have $M_i \not\in F_{i-1}$. Hence $F_{i-1} \ne F_i'$.
\end{proof}

The sequence $s_1 > \cdots > s_r$ in Proposition~\ref{pro-filtr} (2)
	depends on $f$ and is unique for a given
quasi-nef sequence. For the inverse $f^{-1}$, it defines the same sequence
with respect to the same
quasi-nef sequence.
\begin{lem}\label{lem-f-1filtr}
	Let $s_1 > \cdots > s_r$ be the sequence in Proposition~\ref{pro-filtr} (2)
	associated to $f$. Then
	for every K\"ahler class $\omega \in H^{1,1}(X,{\mathbb R})$ and every integer $j \in [1,r]$, we have
	$$((f^{-1})^* - \mathrm{Id})^{2j-1}\omega \in F_{s_j} \backslash F'_{s_j} \ \ \text{ and } \ \
	((f^{-1})^* - \mathrm{Id})^{2j}\omega \in F'_{s_j} \backslash F_{s_j - 1},$$
	and $((f^{-1})^* - \mathrm{Id})^{2r+1}\omega = 0$.
\end{lem}
\begin{proof}
	Since both $F_{s_j}$ and $F_{s_j}'$ are $f^*$-invariant, we have
	$$((f^{-1})^* - \mathrm{Id})^{2j-1}\omega = (-1)^{2j-1}(f^{1-2j})^*(f^* - \mathrm{Id})^{2j-1}\omega \in F_{s_j} \backslash F'_{s_j}.$$
	The same argument shows that $((f^{-1})^* - \mathrm{Id})^{2j}\omega \in F'_{s_j} \backslash F_{s_j - 1}$
	and $((f^{-1})^* - \mathrm{Id})^{2r+1}\omega = 0$.
\end{proof}

The following two lemmas are both consequences of Proposition~\ref{pro-filtr}.

\begin{lem}\label{lem-vanprod}
	For $i \in [1,d] \cap {\mathbb Z}$,
	take $\eta_i \in F'_i$.
	Let $p \in [1,d] \cap {\mathbb Z}$ and $j \in [1,p] \cap {\mathbb Z}$. Then:
	\begin{enumerate}
		\item
		There exists some $C \in {\mathbb R}$
		such that
		$$L_j\eta_{j+1} \cdots \eta_p \equiv CL_p.$$
		\item For any $\eta \in F_p$, we have
		$$L_j\eta_{j+1} \cdots \eta_p\eta \equiv 0.$$
	\end{enumerate}
\end{lem}
\begin{proof}

	Since either $F'_p = F_{p-1}$ or
	$F'_p/F_{p-1}$ is a line spanned by $M_p + F_{p-1}$ by Proposition~\ref{pro-filtr},
	there exists some $C_p \in {\mathbb R}$ such that
	$\eta_p - C_pM_p \in F_{p-1}$.
	As $L_{p-1}F_{p-1} \equiv 0$,
	we have
	$$L_{p-1}\eta_p \equiv C_p L_{p-1}M_p = C_p L_p.$$
	Induction proves that $L_j\eta_{j+1} \cdots \eta_p \equiv CL_p$ for some $C \in {\mathbb R}$.

	Since $L_p F_p \equiv 0$, (2) follows from (1) and the definition of $F_p$.
\end{proof}

\begin{lem}\label{lem-equalM}
	Assume that $M_1 =\cdots = M_i \in H^{1,1}(X,{\mathbb R})$.
	Then
	$$F'_j = F_{j-1}$$
	for every $j \le i-1$.
\end{lem}

\begin{proof}

	Assume to the contrary that $F'_j \ne F_{j-1}$ for some $j \le i-1$.
	By Proposition~\ref{pro-filtr} (1), we would have
	$$L_{j+1} = L_{j-1}M_jM_{j+1} = L_{j-1}M_j^2  \equiv 0,$$
	which is impossible.
	Hence $F'_j = F_{j-1}$ for every $j \le i-1$.
\end{proof}

As a consequence of these results,
we obtain the following refinements of
Theorem~\ref{JBbound}.

\begin{cor}\label{cor25K'}
Let $\phi : X \to B$ be a surjective morphism with connected fibers
	between compact K\"ahler manifolds.
	Let $f \in \mathrm{Aut} (X)$ such that $f^* : H^{1,1}(X, {\mathbb R}) \circlearrowleft$ is unipotent and
	$\phi^*\omega_B$ is $f^*$-invariant for some K\"ahler class $\omega_B \in H^{1,1}(B,{\mathbb R})$. Then
	$$k(f) \le 2(\dim X - \dim B).$$
	Here, we recall that $k(f)+1$ is the maximal size of the Jordan blocks of the Jordan canonical form of the unipotent $f^*|_{H^{1,1}(X, {\mathbb R})}$.
\end{cor}

\begin{proof}
	Let $m \mathrel{:=} \dim B$.
	As $\phi^*\omega_B$ is an $f^*$-invariant nef class and $\phi^*\omega_B^m \not\equiv 0$, we can complete
	$$M_1 = \cdots = M_m = \phi^*\omega_B$$
	to a quasi-nef sequence $M_1,\ldots,M_d$.
	By Lemma~\ref{lem-equalM}, we have
	$F'_j = F_{j-1}$
	for every $j \le m-1$.
	So according to Proposition~\ref{pro-filtr} (2) and the notation therein,
	necessarily $s_r \ge m$, so $r \le \dim X - \dim B$.
	Hence $(f^* - \mathrm{Id})^{2(\dim X - \dim B) +1} (\omega) = 0$ for every $\omega \in H^{1,1}(X,{\mathbb R})$.
\end{proof}

\begin{cor}\label{cor-JBboundkd}
	Let $X$ be a compact K\"ahler manifold of dimension $d \geq 1$ and of Kodaira dimension $\kappa(X)$.
	Let $f \in \mathrm{Aut}(X)$ be any automorphism of zero topological entropy.
	\begin{enumerate}
		\item We have
		$$k(f) \le 2(\dim X - \kappa(X)),$$
		in other words,
		$$\|(f^{m})^*: H^{1,1}(X)\circlearrowleft\| =O\big(m^{2(d-\kappa(X))}\big)$$
		as $m \to \infty$ for any norm of ${\rm End}_{{\mathbb C}}(H^{1,1}(X))$.
		\item The estimate in (1) is optimal, in the sense that
		for every $d \ge 1$ and $\kappa \ge 1$, there exist $X$ and $f \in \mathrm{Aut}(X)$ such that
		$\dim(X) =  d$, $\kappa(X) = \kappa$, and
		$$\|(f^{m})^*: H^{1,1}(X)\circlearrowleft\| \sim_{m \to \infty} m^{2(d-\kappa(X))}.$$
	\end{enumerate}
	\end{cor}

	We prove first Corollary~\ref{cor-JBboundkd} (1).
	We will prove  Corollary~\ref{cor-JBboundkd} (2) in Section~\ref{sec-MainTheorem2}
by constructing explicit examples.

\begin{proof}[Proof of Corollary~\ref{cor-JBboundkd} (1)]
	By an equivariant K\"ahler desingularization, there exists
	a bimeromorphic morphism $\nu : \widetilde{X} \to X$ form a compact K\"ahler manifold $\widetilde{X}$
	such that $f$ lifts to an automorphism $\ti{f} \in \mathrm{Aut}(\widetilde{X})$ and that $\widetilde{X}$
	admits a surjective morphism $\phi : \widetilde{X} \to B$ to a projective manifold
	as a model of its Iitaka fibration.
	As $\phi$ is an Iitaka fibration of $\widetilde{X}$,
	$\ti{f}$ descends to a bimeromorphic self-map of $B$ of finite order
by \cite[Theorem A]{NZ}.
	Up to replacing $f$ by a finite iteration of it,
	we can assume that $\phi$ is $\ti{f}$-invariant.
	In particular, $\phi^*\omega_B$ is
	$\ti{f}^*$-invariant for every K\"ahler class $\omega_B \in H^{1,1}(X,{\mathbb R})$.

	Since the topological entropy of $f$ is zero,
	we have $d_1(\ti{f}) = 1$
	by Lemmas~\ref{lem-invft} and \ref{lem:equiv_dyn}.
	Replacing $f$ by its finite iteration,
	we can assume that $\ti{f}^*: H^{1,1}(\widetilde{X},{\mathbb C}) \circlearrowleft$ is unipotent by Lemma~\ref{lem:equiv_dyn}.
	Thus by Corollary~\ref{cor25K'}, we have
	$$\|(\ti{f}^{m})^*: H^{1,1}(\widetilde{X},{\mathbb C})\circlearrowleft\|
	=_{m \to \infty} O\big(m^{2(d-\kappa(X))}\big)$$
	As $H^{1,1}({X},{\mathbb C}) \hookrightarrow H^{1,1}(\widetilde{X},{\mathbb C})$ is
	$\ti{f}^*$-stable and the restriction of $\ti{f}^* : H^{1,1}(\widetilde{X},{\mathbb C}) \circlearrowleft$
	to $H^{1,1}({X},{\mathbb C})$ is ${f}^* : H^{1,1}({X},{\mathbb C}) \circlearrowleft$, we have
	$$\|(f^{m})^*: H^{1,1}(X,{\mathbb C})\circlearrowleft\| =_{m \to \infty}
	O\big(m^{2(d-\kappa(X))}\big).$$
\end{proof}

{\it From now on until the end of Section~\ref{sec-filtr},
$f \in \mathrm{Aut}(X)$ is an automorphism such that $f^*: H^{1,1}(X) \circlearrowleft$ is unipotent.}
Then $(f^{-1})^*: H^{1,1}(X) \circlearrowleft$ is also unipotent.
Recall that in Section~\ref{sec-plov}, we have defined
$$N \mathrel{:=} f^* - \mathrm{Id} \in \mathrm{End}(H^{2}(X,{\mathbb R})) \ \ \text{ and } \ \
N' \mathrel{:=} (f^{-1})^* - \mathrm{Id} \in \mathrm{End}(H^{2}(X,{\mathbb R})),$$
and also $N_m \mathrel{:=} N^m + N'^m$.

\begin{lem}\label{lem-N2i-1}
Let $\alpha \in H^{1,1}(X,{\mathbb R})$.
\begin{enumerate}
	\item Let $d-1 \ge s_1 > \cdots > s_r \ge 1$ be the sequence
	associated to $f$ in Proposition~\ref{pro-filtr} (2).
		We have
	$$N_{2i - 1}(\omega), N_{2i}(\omega) \in F'_{s_i} \backslash F_{s_i - 1}$$
	for any K\"ahler class $\omega$.
	In particular,
	$$N_{2i - 1}(\alpha), N_{2i}(\alpha) \in F_{d-i}'.$$
	\item
	Both $N_{k(f) - 1}(\alpha)$ and  $N_{k(f)}(\alpha)$ are $f^*$-invariant.
	\item
	If $\omega$ is nef,
	then both $N_{k(f)}(\omega)$ and $N_{k(f)-1}(\omega)$ are nef.
\end{enumerate}
\end{lem}
\begin{proof}
First we prove (1). Note that $N + N' = -NN'$, so
	\begin{equation*}
	\begin{split}
	N_{2i-1}
	& = N^{2i - 1} +  N'^{2i - 1}
	= (N + N')\sum_{j=0}^{2i-2}(-1)^j N^jN'^{2i-2-j} \\
	& = \sum_{j=0}^{2i-2}(-1)^{j+1} N^{j+1}N'^{2i-1-j} \\
	 & =  \sum_{j=0}^{2i-2} (-1)^{j+1} (f^* - \mathrm{Id})^{j+1}\((f^{-1})^* - \mathrm{Id}\)^{2i-j-1} \\
	 & =  \sum_{j=0}^{2i-2} (f^* - \mathrm{Id})^{2i}(f^{-2i+j+1})^* =
	\sum_{j=0}^{2i-2} N^{2i} \circ (f^{-2i+j+1})^*
	\end{split}
	\end{equation*}
	Since  $\sum_{j=0}^{2i-2}(f^{-2i+j+1})^*\omega$ is K\"ahler, Proposition~\ref{pro-filtr} (2) implies
	$$N_{2i-1}(\omega) = N^{2i} \(\sum_{j=0}^{2i-2}(f^{-2i+j+1})^*\omega\) \in F_{s_i}' \backslash F_{s_i - 1}$$

	By Proposition~\ref{pro-filtr} (2) and Lemma~\ref{lem-f-1filtr},
	we have $N^{2i}(\omega), N'^{2i}(\omega) \in F_{s_i}'$,
	so $N_{2i}(\omega) \in F_{s_i}'$.
	Since $N^{2i}\omega \in F_{s_i}'$,
	we have
	$$L_{s_i - 1}N^{2i}\omega \equiv C L_{s_i}$$
	for some $C \in {\mathbb R}$ by Lemma~\ref{lem-vanprod}. Since $N^{2i}\omega \notin F_{s_i - 1}$, we have $C \ne 0$.
	Moreover, as $L_{s_i - 1}N^{p}\omega \equiv 0$ for every $p > 2i$ by Proposition~\ref{pro-filtr} (2),
	we have
	$$L_{s_i - 1}N^{2i}\omega = (2i)! \cdot \lim_{m \to \infty} L_{s_i - 1}\frac{(f^*)^m(\omega)}{m^{2i}} \in \mathcal{K}^{s_i}(X)/\equiv$$
	where $\mathcal{K}^{s_i}(X)/\equiv$ denotes the image of $\mathcal{K}^{s_i}(X)$ in $H^{s_i,s_i}(X,{\mathbb R})/\equiv$.
	Since $L_{s_i} \in \mathcal{K}^{s_i}(X)$, necessarily $C > 0$.
	Since $s_1 > \cdots > s_r$ is also the sequence
	associated to $f^{-1}$ by Lemma~\ref{lem-f-1filtr},
	the same argument shows that there exists $C' > 0$ such that
	$$L_{s_i - 1}N'^{2i}\omega \equiv C' L_{s_i}.$$
	Hence $$L_{s_i - 1}N_{2i}(\omega) \equiv (C + C') L_{s_i} \not \equiv 0,$$
	namely $N_{2i}(\omega) \notin F_{s_i - 1}$.

For (2), recall that $k(f)$ is an even number (Theorem~\ref{JBbound})
so we can write $k(f) = 2i$.
Since $N^{2i+1} = 0$ and $f^* = \mathrm{Id} + N$, we have
$$N^{2i}/(2i)! = \lim_{m \to \infty} (f^*)^m/m^{2i}$$
whose image is hence $f^*$-invariant.
Since $f^*$ commutes with $N$, and
$N' = -N (f^{-1})^*$, we have $N'^{2i} = N^{2i} (f^{-2i})^*$ whose image is hence $f^*$-invariant too. Thus the images of $N_{2i} = N^{2i} + N'^{2i}$ and $N_{2i-1} = \sum_{j=0}^{2i-2} \, N^{2i} ( (f^{-2i+j+1})^*)$ are also $f^*$-invariant.

For (3),
$$N^{2i}(\omega)= (2i)! \, \lim_{m \to \infty} (f^*)^m(\omega)/m^{2i}, \,\,\, N_{2i-1}(\omega) = \sum_{j=0}^{2i-2} \, N^{2i} ((f^{-2i+j+1})^*(\omega))$$
are clearly all nef.
\end{proof}

\begin{lem}\label{lem-equiv2N}
	Let $m$ be a positive integer and let
	$$\Sigma \mathrel{:=} \left\{ N^{k(f)}(\omega), N_{k(f)}(\omega), N_{k(f)-1}(\omega) \mid \omega \in H^{1,1}(X,{\mathbb R}) \text{ K\"ahler } \right\}.$$
	 The following assertions are equivalent:
	\begin{enumerate}
		\item $M_1\cdots M_m \not\equiv 0$ for some $M_1,\ldots,M_m \in \Sigma$.
		\item $M_1\cdots M_m \not\equiv 0$ for every $M_1,\ldots,M_m \in \Sigma$.
	\end{enumerate}
	\end{lem}
\begin{proof}
	Fix a positive integer $m$.
	It suffices to prove that (1) implies (2).
	To this end, it suffices to prove that given
	$M_1,\ldots,M_m , M_m' \in \Sigma$,
	$$M_1\cdots M_{m-1} M_m \not\equiv 0 \ \ \text{ implies } \ \
	M_1\cdots M_{m-1} M'_m \not\equiv 0.$$
	Then we can replace each factor of $M_1\cdots M_m$
	by any choice of $m$ elements $M'_1,\ldots,M'_m \in \Sigma$ one by one and
	obtain $M'_1\cdots M'_m \not\equiv 0$.

	Since every element of $\Sigma$ is
	nef and $f^*$-invariant by Lemma~\ref{lem-N2i-1}, the sequence
	$M_1, \ldots, M_{m-1}$
	can be completed to a quasi-nef sequence.
	Since $L_{m-1}M_m = M_1\cdots M_{m-1} M_m \not\equiv 0$ and $M_m' \in \Sigma$ by assumption,
	Proposition~\ref{pro-filtr} (2) and Lemmas~\ref{lem-f-1filtr} and~\ref{lem-N2i-1} (1)
	then imply that
	$$M_1\cdots M_{m-1} M'_m = L_{m-1} M'_m
	\not\equiv 0.$$
\end{proof}

\begin{cor}\label{cor-vankk-1}

Let $m$ be a non-negative integer such that $N^{k(f)}(\omega_0)^m \equiv 0$
(or equivalently $N_{k(f)}(\omega_0)^m \equiv 0$ by Lemma~\ref{lem-equiv2N})
for some K\"ahler class $\omega_0$.
Then for every  $\omega \in H^{1,1}(X,{\mathbb R})$, we have
$$N_{k(f)}(\omega)^iN_{k(f) -1}(\omega)^j \equiv 0$$
whenever $i + j \ge m$.

As a consequence, for every  $\omega \in H^{1,1}(X,{\mathbb R})$
and every pair of non-negative integers $i$ and $j$ such that $2i + 2j > 2d - k(f)$, we have
$$N_{k(f)}(\omega)^iN_{k(f) -1}(\omega)^j \equiv 0.$$
\end{cor}

\begin{proof}

Since the vanishing $N_{k(f)}(\omega)^iN_{k(f) -1}(\omega)^j \equiv 0$
is a Zariski closed condition for $\omega \in H^{1,1}(X,{\mathbb R})$
and since the K\"ahler cone is Zariski dense in $H^{1,1}(X,{\mathbb R})$, we can assume that
$\omega \in H^{1,1}(X,{\mathbb R})$ is K\"ahler.
Then the first statement follows from Lemma~\ref{lem-equiv2N}.

Now we prove the second statement. Once again, we can assume that $\omega$ is K\"ahler.
Recall that $ k(f) = 2\ell$ is an even number (Prposition~\ref{pro-filtr}).
By the first statement, it suffices to show that
$$N_{k(f)}(\omega)^{d-\ell + 1} \equiv 0.$$
To this end, we can assume that $N_{k(f)}(\omega)^{d-\ell } \not \equiv 0$
and complete
	$$M_1 = \cdots = M_{d-\ell} \mathrel{:=} N_{k(f)}(\omega)$$
	to a quasi-nef sequence.
	Then Lemma~\ref{lem-N2i-1} implies that
	$$ N_{k(f)}(\omega)= N_{2\ell}(\omega) \in F'_{d-\ell} \subset F_{d-\ell},$$
	hence
	$$  N_{k(f)}(\omega)^{d-\ell+1} = L_{d-\ell } N_{k(f)}(\omega) \equiv 0.$$
\end{proof}

\section{Upper bound of $\mathrm{Plov}(f)$: Proofs of  Theorems~\ref{thmkeeler},~\ref{uniform} and~\ref{uniformdge4}} \label{proofuniform}

Let us first prove Keeler's upper bound in the introduction (Theorem~\ref{thmkeeler}).
We will first provide a sketch of Keeler's original proof,
then an alternative proof using Corollary~\ref{cor-vankk-1}.

\begin{proof}[Keeler's proof of Theorem~\ref{thmkeeler}] Recall that $\mathrm{Plov} (f)$ is the degree of the polynomial $P_{f, \omega}(n)$ which is the same as the polynomial $(\Delta_n(f, L)^d)$ in Theorem \ref{Keeler} (6) if we replace the ample class $L$ by the K\"ahler class $\omega$.
Therefore, by setting $D = \omega$ and $P = f^*$ in the proof of \cite[Lemma 6.13]{Ke}, the purely cohomological proof of \cite[Lemma 6.13]{Ke} works without any further change, which proves the result.
\end{proof}

\begin{proof}[Second proof of Theorem~\ref{thmkeeler}]
	The lower bound is proven in Lemma~\ref{lem35}.

	It remains to prove the upper bound.
	By Lemmas~\ref{lem-invfini} and~\ref{lem:equiv_dyn},
	we can assume that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
	Let $\alpha \in H^{1,1}(X,{\mathbb R})$ and let
	$$k(f) \ge i_1 \ge \cdots i_d \ge 0$$
	be $d$ integers such that
	$$\sum_{j = 1}^d i_j > k(f)(d-1).$$
	Elementary computation shows that
	the product $N_{i_1}(\alpha) \cdots N_{i_d}(\alpha)$
	is of the form
	$$\Pi \mathrel{:=} N_{k(f)}(\alpha)^a N_{k(f) - 1}(\alpha)^b N_{i_{a+b+1}}(\alpha) \cdots N_{i_{d}}(\alpha)$$
	with $2a + 2b > 2d - k(f)$.
	It follows from Corollary~\ref{cor-vankk-1} that $\Pi = 0$.
Thus $\mathrm{Plov}(f) \le k(f)(d-1) + d$ by Lemma~\ref{lem-ubN_i}.
\end{proof}

In the remainder of this section, we prove Theorems~\ref{uniform} and~\ref{uniformdge4}.
Let us first prove Theorem~\ref{uniform} for $k(f) = 2$. The proof relies on the following lemma.

\begin{lem}\label{lem-N2d-1}
Let $X$ be a compact K\"ahler manifold of dimension $d \ge 3$ and let $f \in \mathrm{Aut} (X)$ such that
$f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
Assume that $k(f) = 2$. Then for every $\omega \in H^{1,1}(X,{\mathbb R})$, we have
$$(N^2\omega)^{d-1} = 0.$$
\end{lem}

\begin{proof}
We can assume that $\omega$ is K\"ahler.
Assume to the contrary that $(N^2\omega)^{d-1} \ne 0$.
Consider $\gamma \mathrel{:=} (N^2\omega)^{d-3}\omega^2 \in H^{d-1,d-1}(X,{\mathbb R})$.
Since $k(f) = 2$, $N^2\omega$ is $f^*$-invariant, so
$$(f^* - \mathrm{Id})^j(\gamma) =  (N^2\omega)^{d-3} \cdot (f^* - \mathrm{Id})^j(\omega^2).$$
Note that for every pair of integers $a_1$ and $a_2$, $(f^* - \mathrm{Id})(N^{a_1}\omega N^{a_2}\omega)$ is a sum of terms of the form $N^{b_1}\omega N^{b_2}\omega$ with $b_1 + b_2 > a_1 + a_2$, so induction (together with the vanishing $N^p\omega = 0$ for $p \ge 3$) shows that
$$(f^* - \mathrm{Id})^4(\gamma) = C (N^2\omega)^{d-1}$$
for some $C>0$.
As $k(f) = 2$, we have $(f^* - \mathrm{Id})^4(\gamma) = 0$
by Poincar\'e duality, which contradicts the initial assumption.
Hence $(N^2\omega)^{d-1} = 0$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{uniform} when $k(f) = 2$]
By Lemmas~\ref{lem-invfini} and~\ref{lem:equiv_dyn}, we can assume that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
Let $\omega$ be a K\"ahler class and let $i$ be the largest integer such that $(N^2\omega)^i \ne 0$.
By Lemma~\ref{lem-N2d-1}, we have $i \le d - 2$.
Since $(N^2\omega)^{i+1} = 0$, it follows from Corollary~\ref{cor-vankk-1} that
$$(N_2\omega)^a(N_1\omega)^b \equiv 0$$
whenever $a + b > i $. Hence by Lemma~\ref{lem-ubN_i},
$$\mathrm{Plov}(f) \le d + 2i \le 3d - 4 = k(f)(d-1) + d - 2.$$
\end{proof}

The proof of Theorem~\ref{uniform} for $k(f) > 2$ follows a different argument.
In Lemmas~\ref{lem-vank-2d} and \ref{lem-vank-2d4} below, let $X$ be a compact K\"ahler manifold of dimension $d \ge 1$ and $f \in \mathrm{Aut}(X)$ an automorphism such that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
As before, $k(f)+1$ denotes the maximal size of the Jordan blocks
of the Jordan canonical form of $f^* : H^{1,1}(X) \circlearrowleft$.

\begin{lem}\label{lem-vank-2d}
Assume $k(f) > 0$.
	Let $(a,b)$ be a pair of non negative integers such that $2a + 2b = 2d-k(f)$. Then $d' \mathrel{:=} d - a - b  = k(f)/2 \ge 1$. Further,
	let $i_1 \ge \cdots \ge i_{d'} \ge 0$ be integers such that
	$$\sum_{j = 1}^{d'} i_j > (k(f)-4)d' + 2.$$
	Then
	$$N_{k(f)}(\alpha)^a N_{k(f) - 1}(\alpha)^b N_{i_1}(\alpha) \cdots N_{i_{d'}}(\alpha) = 0$$
	for every $\alpha \in H^{1,1}(X,{\mathbb R})$.
\end{lem}

\begin{proof}
We can assume that $N_{k(f)}(\alpha)^a N_{k(f) - 1}(\alpha)^b  \not\equiv 0$ and $\alpha \in H^{1,1}(X)$ is K\"ahler.
If  $i_1 \ge k(f) - 1$,
then since $2(a + b + 1) > 2d - k(f)$ and by Corollary~\ref{cor-vankk-1}, we already have
$$N_{k(f)}(\alpha)^a N_{k(f) - 1}(\alpha)^b N_{i_1}(\alpha) \equiv 0.$$
Thus we can assume that $i_1 \le k(f) - 2$.

By Lemma \ref{lem-N2i-1}, both $N_{k(f)}(\alpha)$ and $ N_{k(f) - 1}(\alpha)$ are $f^*$-invariant nef,
so we can complete
	$$M_1 = \cdots = M_a \mathrel{:=} N_{k(f)}(\alpha)  , \ \ M_{a+1} = \cdots = M_{a+b} \mathrel{:=} N_{k(f)-1}(\alpha)$$
	to a quasi-nef sequence.

	Since $\sum_{j = 1}^{d'} i_j > (k(f)-4)d' + 2,$
	we have
	$$i_1,i_2 \in [k(f)-3, k(f)-2].$$
	Indeed,
	otherwise we would have $i_2 \le k(f) - 4$ and since $k(f) = 2d'$, we would have
	$$\sum_{j = 1}^{d'} i_j \le (k(f) - 2) + (d'-1)(k(f) - 4)
	= 2d'^2 - 4d' + 2 = (k(f)-4)d' + 2,$$
	which contradicts the assumption.
		It follows from Lemma~\ref{lem-N2i-1} that
	$$N_{i_1}(\alpha), N_{i_{2}}(\alpha) \in  F'_{d-\frac{k(f)}{2} + 1 } = F'_{a + b + 1 }.$$
	So
	$$N_{k(f)}(\alpha)^a N_{k(f) - 1}(\alpha)^bN_{i_1}(\alpha) N_{i_{2}}(\alpha)
	= L_{a+b}N_{i_1}(\alpha) N_{i_{2}}(\alpha)  \equiv 0,$$
	by Lemma~\ref{lem-vanprod}, which proves Lemma~\ref{lem-vank-2d}.
\end{proof}

\begin{proof}[End of proof of Theorem~\ref{uniform}]

Recall that $k(f)$ is an even number (Theorem~\ref{JBbound}), and we already
proved Theorem~\ref{uniform} for $k(f) =2$.
It remains to prove Theorem~\ref{uniform} for $k(f) > 2 $.

By Lemmas~\ref{lem-invfini} and~\ref{lem:equiv_dyn},
we can assume that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
Let $\alpha \in H^{1,1}(X,{\mathbb R})$ and let
$$k(f) \ge i_1 \ge \cdots i_d \ge 0$$
be $d$ integers such that
$$\sum_{j = 1}^d i_j > k(f)(d-1) - 2.$$
Then the product $N_{i_1}(\alpha) \cdots N_{i_d}(\alpha)$
is of the form
$$\Pi \mathrel{:=} N_{k(f)}(\alpha)^a N_{k(f) - 1}(\alpha)^b N_{i_{a+b+1}}(\alpha) \cdots N_{i_{d}}(\alpha) .$$
We now show that
$\Pi = 0$.
For this, if $a +b > d - \frac{k(f)}{2}$, then $\Pi = 0$ by Corollary~\ref{cor-vankk-1}.
Suppose $a +b \le d - \frac{k(f)}{2}$, i.e., $d' \mathrel{:=} d - a - b \ge \frac{k(f)}{2}$.
Then
$$(a + b)k(f) + \sum_{j = a + b + 1}^di_j \ge  \sum_{j = 1}^{a + b}i_j + \sum_{j = a + b + 1}^di_j
= \sum_{j = 1}^d i_j > k(f)(d-1) - 2.$$
Since $2d' \ge k(f) \ge 4$ by assumption, we have
$$\sum_{j = a + b + 1}^di_j  > k(f)(d-a-b-1) - 2 = k(f)d' - k(f) - 2 \ge (k(f)-4)d' + 2.$$
So we also have $\Pi = 0$ by Lemma~\ref{lem-vank-2d}.
It follows from Lemma~\ref{lem-ubN_i} that
$$\mathrm{Plov}(f) \le  d + k(f)(d-1) - 2 .$$
\end{proof}

\begin{lem}\label{lem-vank-2d4}
Assume that $k \mathrel{:=} k(f) = 2d-2$ and $d \ge 4$. Take $d$ non negative integers $i_1 \ge \cdots \ge i_d$ such that $k  = 2d-2 \ge i_1$.
	Assume that
	$$\sum_{j = 1}^d i_j \ge  (k-2)d-1.$$
Then for every $\alpha \in H^{1,1}(X)$, we have
$$N_{i_1}(\alpha) \cdots N_{i_{d}}(\alpha) = 0 .$$
\end{lem}

\begin{proof}
	First we assume that $i_1 \ge k-1$. We have
	\begin{equation}
	\begin{split}
	\sum_{j = 2}^d i_j & = \(\sum_{j = 1}^d i_j\)  - i_1
	> (k-2)d-1 - k = 2d^2 -6d + 1 \\
	& \ge 2d^2 - 8d + 8 = (k - 4)(d-1) + 2,\end{split}
	\end{equation}
	where the second inequality follows from $d \ge 4$.
	So $N_{i_1}(\alpha) \cdots N_{i_{d}}(\alpha) = 0$ by Lemma~\ref{lem-vank-2d}.

	Assume that $i_1 \le k-2$.
	Since $\sum_{j = 1}^d i_j \ge (k-2)d - 1$ and the sequence $i_j$ is decreasing,
	necessarily
	$$i_1 = \cdots = i_{d-1} = k-2 \ \ \ \text{ and } \ \ \  i_d = k-2 \text{ or } k-3.$$
	Since we have already proven that $N_{j_1}(\alpha) \cdots N_{j_{d}}(\alpha) = 0$
	whenever $j_1 \ge k - 1$, in particular whenever
	$$\sum_{l = 1}^d j_l > (k-2)d,$$
	we have
	\begin{equation}\label{eqn-Pganmod}
	\begin{split}
	P_\alpha(n)  & \mathrel{:=} \(\sum_{i = 0}^k \binom{n}{i}N_i(\alpha) \)^d \\
	& =_{n \to \infty} \binom{n}{k-2}^d N_{k-2}(\alpha)^d + d\binom{n}{k-2}^{d-1}\binom{n}{k-3} N_{k-2}(\alpha)^{d-1}N_{k-3}(\alpha) \\
	& + O(n^{(k-2)d - 2}).
	\end{split}
	\end{equation}
	Recall that  $\deg(P_\alpha(n)) \le d(d-1)$ by Lemma~\ref{lem-ubdpol}.
	Since
	$d(d-1) \le (k-2)d - 2$ (because $d \ge 4$),
	it follows from~\eqref{eqn-Pganmod} that
	$$N_{k-2}(\alpha)^d = 0$$
	and then
	$$N_{k-2}(\alpha)^{d-1}N_{k-3}(\alpha) = 0,$$
	which finishes the proof of Lemma~\ref{lem-vank-2d4}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{uniformdge4}]
	By Theorem~\ref{JBbound}, we have $k(f) \le 2d-2$ and $k(f)$ is even.
	Since $d \ge 4$, Theorem~\ref{uniformdge4}
	in the case $k(f) < 2d-2$ (resp. $k(f) = 2d-2$) follows from Theorem~\ref{thmkeeler}
	(resp. Lemma~\ref{lem-ubN_i} and Lemma~\ref{lem-vank-2d4}).
\end{proof}

\section{A refined lower bound: Proof of Theorem~\ref{thm-lb}}\label{sec-lb}

In this section we prove Theorem~\ref{thm-lb}, which concerns lower bound of $\mathrm{Plov}(f)$.

\begin{proof}[Proof of Theorem~\ref{thm-lb}]

	We can assume that $\dim X \ge 2$, otherwise $k(f) = 0$, and Theorem~\ref{thm-lb} holds trivially.

	By Lemmas~\ref{lem-invfini} and~\ref{lem:equiv_dyn},
	we can assume that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
	Let $\omega$ be a K\"ahler class. Recall that we have
	$$\Delta_n'(f,\omega) \mathrel{:=} \sum_{i= 0}^n \((f^i)^*\omega + (f^{-i})^*\omega\)= \sum_{i = 0}^{k(f)} \binom{n}{i+1} N_i(\omega)$$
	from the computation in the proof of Lemma~\ref{lem-ubN_i}.
	By Lemma~\ref{lem-sdPbis}, and using the notations therein, we have
	$$\mathrm{Plov}(f) = \deg P'_{f,\omega,d} > \deg P'_{f,\omega,d-1} > \cdots > \deg P'_{f,\omega,2}.$$
	Therefore it suffices to show that
	$$\deg P'_{f,\omega,2} \ge 2k(f).$$

	Recall that
	\begin{equation}\label{eqn-P'2}
	P'_{f,\omega,2} = \Delta'_n(f, \omega)^2 \omega^{d-2} = \(\sum_{j=0}^{k(f)}{n\choose j+1} N_j\omega\)^2\omega^{d-2}.
	\end{equation}

	Assume that $(N_{k(f)}\omega)^2 \ne 0$. Since $N_{k(f)}\omega$
	is nef, we have $(N_{k(f)}\omega)^2 \cdot \omega^{d-2} \ne 0$.
	Hence $\deg P'_{f,\omega,2} \ge 2k(f) + 2$ by~\eqref{eqn-P'2}.

	Now assume that $(N_{k(f)}\omega)^2 = 0$.
	Then
	\begin{equation}\label{van-k-1k}
	(N_{{k(f)}-1}\omega)^2 \equiv 0, \ \ (N_{k(f)}\omega)(N_{{k(f)}-1}\omega) \equiv 0
	\end{equation}
	by Lemma~\ref{lem-equiv2N}.
	Since $N_{k(f)}\omega$ is nef and $f^*$-invariant by Lemma~\ref{lem-N2i-1},
	we can construct a quasi-nef sequence $M_1,\ldots,M_d$ with $M_1 = N_{k(f)}\omega$.
	Suppose that $(N_{k(f)}\omega)(N_{{k(f)}-2}\omega) \equiv 0$. Then $(N_{{k(f)}-2}\omega) \in F_1$,
	and we would have
	$(N_{{k(f)}}\omega) \in F_0 = 0$ by Lemma~\ref{lem-N2i-1}, which contradicts the assumption that $N_{k(f)}\omega \ne 0$. Hence $(N_{k(f)}\omega)(N_{{k(f)}-2}\omega) \not\equiv 0$.
	Together with the vanishings~\eqref{van-k-1k} and~\eqref{eqn-P'2},
	we obtain $\deg P'_{f,\omega,2} \ge 2k(f)$.
\end{proof}

\section{Complex tori: Proof of Theorem \ref{main3} and a few remarks} \label{sec-MainTheorem3}

In this section, we prove the following theorem,
a more precise version of Theorem~\ref{main3};
see Remark~\ref{rem51} for further discussion.

\begin{theorem}\label{thm51}
	Let $X$ be a complex torus of dimension $d$ and $f \in \mathrm{Aut} (X)$ an automorphism
	such that $d_1(f) = 1$.
	Assume that the Jordan canonical form of $f^* : H^{1,0}(X)\circlearrowleft$
	consists of Jordan blocks of sizes $k_1,\ldots,k_p$, counted with multiplicities. Then
	$$\mathrm{Plov}(f) = \sum_{i=1}^{p} k_i^2.$$
	In particular, we have $\mathrm{Plov}(f) \le d^2$, and this upper bound is optimal.
\end{theorem}

\begin{proof}

	First we perform some reduction.
	By Lemmas~\ref{lem-invfini} and~\ref{lem:equiv_dyn}, up to replacing $f$
	by some finite iteration of it, we can assume that $f^* : H^{1,0}(X)\circlearrowleft$ is unipotent.
	Fix a basis
	$$dz_{1,1},dz_{1,2},\ldots,dz_{1,k_1},\ldots,dz_{p,1},\ldots,dz_{p,k_p}$$
	of $H^{1,0}(X)$ such that for every $i= 1,\ldots,p$,
	$$
		 f^*dz_{i,j} =
	\begin{cases}
		dz_{i,1} \text{ if } j = 1 \\
		dz_{i,j} +  dz_{i,j-1} \text{ if } 2 \le j \le k_i.
	\end{cases}
	$$
 As $\mathrm{Plov}(f)$ is a cohomological invariant,
 and the $f^*$-action on $H^{1,0}(X)$
 	determines the $f^*$-action on $H^\bullet(X,{\mathbb C})$ when $X$ is a torus,
 we can assume that $X = E^d$ with $E$ being an elliptic curve, or even $E = {\mathbb C}/{\mathbb Z}[\sqrt{-1}]$, and $(z_{i,j})_{1 \le i \le p,  2 \le j \le k_i}$ are the global coordinates of $E^d$, so that
	$$f : E^d = \prod_{i = 1}^p E^{k_i} \to  \prod_{i = 1}^p E^{k_i} = E^d$$
	is the product of $E^{k_i} \to E^{k_i}$ defined by the unipotent Jordan matrix of size $k_i$.

By the product formula (Proposition~\ref{prop1-Kahler} (4)), it suffices to prove Theorem~\ref{thm51} for the case $p = 1$. So, from now on until the end of proof, we assume that $p=1$. Namely $f^* : H^{1,0}(X)\circlearrowleft$ has only one Jordan block.

	  Set $e_i = dz_{i}$ and $\bar{e}_i = d\bar{z}_{i}$.
	  For every $\sigma = \sum_{i,j} a_{ij}e_i \wedge \bar{e}_j
\in H^{1,1}(X,{\mathbb C}) \setminus \{0\}$, define
	$$w(\sigma) \mathrel{:=} \max\{i+j \mid a_{ij} \ne 0\},$$
	and for every $p = 2,\ldots, 2d$, define
	$$\sigma(p) \mathrel{:=} \sum_{i + j = p} a_{ij}e_i \wedge \bar{e}_j.$$

We need the following.
\begin{lem}\label{lem-van}
		Let $\sigma_1,\ldots,\sigma_d \in H^{1,1}(X,{\mathbb C}) \setminus \{0\}$ and let $w_i \mathrel{:=} w(\sigma_i)$.
		\begin{enumerate}
			\item If $\sum_{i = 1}^d w_i < d(d+1)$, then $\sigma_1 \wedge \cdots \wedge \sigma_d = 0$.
			\item If $\sum_{i = 1}^d w_i = d(d+1)$, then $\sigma_1 \wedge \cdots \wedge \sigma_d = \sigma_1({w_1}) \wedge \cdots \wedge \sigma_d({w_d})$.
		\end{enumerate}
\end{lem}

\begin{proof}
	By multi-linearity of $\sigma_1 \wedge \cdots \wedge \sigma_d$, it is clear that (1) implies (2), and that it suffices to prove (1) for $\sigma_1,\ldots,\sigma_d$ of the form $\sigma_{i} = e_{i_1} \wedge \bar{e}_{j_1},\ldots, \sigma_d = e_{i_d} \wedge \bar{e}_{j_d}$. If $\sigma_1 \wedge \cdots \wedge \sigma_d \ne 0$, then necessarily
	$$\{i_1,\ldots,i_d\} = \{1,\ldots,d\} = \{j_1,\ldots,j_d\},$$
	so $\sum_{i = 1}^d w_i = d(d+1)$.
\end{proof}

We return to the proof of Theorem \ref{thm51}.
Let $N \mathrel{:=} f^* - \mathrm{Id}$  and let
	$$\omega \mathrel{:=} \sqrt{-1} \sum_{i=1}^d e_i \wedge \bar{e}_i,$$
	which is a K\"ahler class on $X$.
	For every $q = 0,\ldots, 2d-2$, by induction on $q$ we have
	$$(N^q\omega)(p) = 0$$
	for every $p > 2d-q$ and
	$$(N^q\omega)(2d-q) = \sqrt{-1}(N^q(e_d \wedge \bar{e}_d))(2d-q) =
		\sqrt{-1}\sum_{i + j = q}\binom{q}{i}e_{d-q+i} \wedge \bar{e}_{d-q+j} \ne 0.$$
	Therefore,
	$$w(N^q\omega) = 2d-q.$$
	Let $q_1,\ldots,q_d \ge 0$ be non-negative integers. If $\sum_{i=1}^d q_i > d^2 - d$, then by Lemma~\ref{lem-van}
	$$(N^{q_1}\omega) \cdots (N^{q_d}\omega) = 0,$$
	so $\mathrm{Plov}(f) \le d^2$ by~\eqref{eqn-PlovNil}.

It remains to prove that $\mathrm{Plov}(f) \ge d^2$. Note that since $\omega({2d}) = \sqrt{-1} e_d \wedge \bar{e}_d$ is nef,
by Lemma~\ref{lem-nefle} we have
$\mathrm{Plov}(f) \ge \mathrm{Plov}(f,\omega({2d}))$.
Until the end of the proof, we formally define $e_i \wedge \bar{e}_j = 0$
whenever $i$ and $j$ are integers such that $i \notin [1,d]$ or $j \notin [1,d]$.

\begin{claim}\label{claim-Pascalpyr}
	We have
	$$N^q(e_d \wedge \bar{e}_d) =
	\sum_{i + j \le q}\binom{q}{i,j,q-i-j}e_{d-q+i} \wedge \bar{e}_{d-q+j},$$
\end{claim}
\begin{proof}
	Let $V \mathrel{:=} {\mathbb C}[X,Y]/(X^d,Y^d)$.
	We have an isomorphism of ${\mathbb C}$-vector spaces
	$V \simeq H^{1,1}(X)$ sending each
	$X^iY^j$ to $e_{d-i} \wedge {\bar e}_{d-j}$. Under this isomorphism, $N: H^{1,1}(X) \to H^{1,1}(X)$ becomes
	$$
	\begin{aligned}
	N : V & \to V \\
	P & \mapsto (XY + X + Y)P  \mod (X^d , Y^d),
	\end{aligned}
	$$
	so
	\begin{equation}\label{eqn-Nq(1)}
	N^q(1) = (XY + X + Y)^q = \sum_{i + j \le q}\binom{q}{i,j,q-i-j}X^{q-i}Y^{q-j} \mod (X^d , Y^d).
	\end{equation}
	Translating~\eqref{eqn-Nq(1)} back to $N: H^{1,1}(X) \to H^{1,1}(X)$
 	proves the claim.
\end{proof}

For every integer $n > 0$,
by Claim~\ref{claim-Pascalpyr} we have
 $$\Omega \mathrel{:=}
 \sum_{q=0}^{2d-2}{n\choose q+1} N^q(\omega({2d}))
 = \sqrt{-1} \sum_{q=0}^{2d-2}  \sum_{i + j \le q}
 {n\choose q+1}\binom{q}{i,j,q-i-j}e_{d-q+i} \wedge \bar{e}_{d-q+j}.$$
 For each pair of integers $1 \le i,j \le d$,
 define the polynomial $P_{i,j}(n)$ in $n$ by
 \begin{equation}\label{eqn-defPij}
 \Omega =
 \sqrt{-1} \sum_{1 \le i , j \le d}
P_{i,j}(n) e_{i} \wedge \bar{e}_{j}.
\end{equation}

\begin{claim}\label{claim-deglc}
The polynomial $P_{d-i,d-j}(n)$ in $n$ has degree
$i + j + 1$ and leading coefficient
	$$\frac{1}{(i+j+1)!}
	 \binom{i+j}{i}.$$
\end{claim}

\begin{proof}
	As $e_{d-i} \wedge \bar{e}_{d-j} = e_{d-q + (q-i)} \wedge \bar{e}_{d-q + (q-j)}$,
	by construction we have (with $q$ varying in the sum)
$$P_{d-i,d-j}(n) =
\sum_{(q-i)+(q-j) \le q} {n\choose q+1}\binom{q}{q-i,q-j,i+j-q}.$$
So the degree and the leading coefficient of $P_{d-i,d-j}$, are equal to
those of the polynomial ${n\choose q+1}\binom{q}{q-i,q-j,i+j-q}$ in $n$ when $q$ is maximal and satisfying $(q-i)+(q-j) \le q$ (that is, when $q = i+j$). This proves the claim.
\end{proof}

By~\eqref{eqn-defPij}, we have
 $$\Omega^d = (\sqrt{-1})^d d! P(n)  (e_1 \wedge \bar{e}_1)  \wedge \cdots  \wedge (e_d \wedge \bar{e}_d)$$
 where $P(n)$ is the determinant of the matrix $(P_{i,j}(n))_{1 \le i,j\le d}$.
 By Claim~\ref{claim-deglc},
 we have $\deg_n(P(n)) \le d^2$ and the coefficient in front of $n^{d^2}$ is $\det M$,
  where $M = (M_{i+1, j+1})_{0 \le i,j \le d-1}$ is the $(d \times d)$ matrix defined by
 $$M_{i+1,j+1} = \frac{1}{(i+j+1)!} \binom{i+j}{i} = \frac{1}{i!j!} \cdot \frac{1}{(i+j+1)} , \,\,\, 0 \le i,j \le d-1.$$
 We have
$$\det M = \frac{1}{\(\prod_{p=0}^{d-1}p!\)^2} \det\(\frac{1}{(i+j+1)}\)_{0 \le i,j\le d-1} =  \frac{\prod_{p=0}^{d-1}p!}{\prod_{p=d}^{2d-1}p!} \ne 0,$$
where the second equality follows from
the determinant of the Cauchy matrix (see \emph{e.g.} ~\cite[(1.1)]{CauchyId}).
Since $\mathrm{Plov}(f,\omega({2d})) = \deg_n(P(n))$ by~\eqref{eqn-abel} and the definition of $\Omega$,
it thus follows that
$$\mathrm{Plov}(f) \ge \mathrm{Plov}(f,\omega({2d})) = \deg_n(P(n)) = d^2 .$$

This completes the proof of the main statement of Theorem~\ref{thm51}.
The optimality of the upper bound is provided by Example~\ref{ex-prod_E} below.
\end{proof}

\begin{example}\label{ex-prod_E}
Let $E$ be a complex elliptic curve and let
$X = E^d$. Let
\begin{equation}
\begin{split}
f:  X & \to X \\
  (x_1, \dots, x_d) & \mapsto (x_1, x_2+x_1, \dots, x_d+x_{d-1})
\end{split}
\end{equation}
Then $f^* : H^{1,0}(X)\circlearrowleft$ is the $(d \times d)$-Jordan matrix, and we have
$$\mathrm{Plov}(X, f) = d^2$$
as a consequence of the main statement of Theorem~\ref{thm51}.
\end{example}

\begin{remark}\label{rem51}
\hfill
\begin{enumerate}
	\item Consider $f \in \mathrm{Aut} (E^3)$ with $f(x_1,x_2, x_3) = (x_1, x_1+x_2, x_3)$.
	Then $f^* : H^{1,0}(E^3)\circlearrowleft$ has two Jordan blocks, of sizes $2$ and $1$ respectively.
	By Theorem \ref{thm51}, we have
	$$\mathrm{Plov} (f) =  2^2 + 1^2 = 5,$$
	which is also consistent with \cite[Example 6.14]{Ke}.
	\item The upper bound in Theorem~\ref{thm51}
	is also asserted in the proof of~\cite[Proposition 4.3]{CO} (without optimality).
	However, the estimates (4.6)-(4.7) using $\ell_1$ in their proof have to be suitably modified,
	 otherwise
	as we can see that if $f$ is the identity, the estimate
	$\mathrm{Vol}(\Gamma(n)) \le C n^{\ell_1^2}$ in~\cite[(4.6)-(4.7)]{CO}
	would imply that $\mathrm{Vol}(\Gamma(n))$ grows at most linearly in $n$,
	which contradicts the equality $\mathrm{Plov}(X) = d$.
\end{enumerate}
\end{remark}

Finally, note that in this paper, whenever
we prove that $\mathrm{Plov}(f)$ is bounded from above by some constant $C$
for an automorphism $f \in \mathrm{Aut}(X)$ such that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent,
we actually prove that the right hand side of the inequality in
Lemma~\ref{lem35} or \ref{lem-ubN_i} is bounded by $C$.
In view of Question~\ref{mainquest2}, we ask the following.

\begin{question}\label{que-mainquest2St}
	Let $X$ be a compact K\"ahler manifold of dimension $d \ge 1$ and
	let $f \in \mathrm{Aut} (X)$ such that $f^* : H^{1,1}(X) \circlearrowleft$ is unipotent.
	For every $\omega \in H^{1,1}(X)$, do we have
	$$(N^{i_1}\omega)\cdots(N^{i_d}\omega) = 0$$
	and
	$$(N_{i_1}\omega)\cdots(N_{i_d}\omega) = 0$$
	whenever the $i_j$ are non-negative integers satisfying
	$\sum_{j = 1}^d i_j > d(d-1)$?
\end{question}

So long one of the vanishings in Question~\ref{que-mainquest2St} holds for every such $d$-tuple of integers $i_1,\ldots,i_d$,
Question~\ref{mainquest2} can be answered readily in the affirmative
by Lemma~\ref{lem35} or \ref{lem-ubN_i}.
\section{Three dimensional case and explicit examples: Proof of Theorem~\ref{main2}} \label{sec-MainTheorem2}

We know that $\mathrm{Plov} (f) = 1$ for a compact Riemann surface, $\mathrm{Plov} (X, f) = 2$ or $4$ for a compact K\"ahler surface (\cite[Theorem 1.7]{AvB}; or Theorem~\ref{thmkeeler} together with Theorem~\ref{JBbound})
and $\mathrm{Plov} (X, f) = d$ for a projective variety $X$ of dimension $d$ whose desingularization $\tilde{X}$ is of general type (as $|{\rm Bir}\, (\tilde{X})| < \infty$ by \cite[Corollary 14.3]{Ueno}).
Besides complex tori and these three cases,
we can also determine the value $\mathrm{Plov} (f)$ or
${\rm rGKdim}\, (X, f)$ for some other classes of compact K\"ahler manifolds $X$ or projective varieties.

\begin{pro}\label{proHK}
	\hfill
	\begin{enumerate}
		\item Let $X$ be a compact hyper-K\"ahler manifold of dimension $2d$ and let $f \in \mathrm{Aut} (X)$ such that $d_1(f) = 1$. Then
		$\mathrm{Plov} (f) = 2d$ if $f$ is of finite order, and $\mathrm{Plov} (f) = 4d$ if $f$ is of infinite order; both cases are realizable, with $X$ projective. (Note that $\mathrm{Plov} (f) = {\rm rGKdim}\, (X, f)$ when $X$ is projective.)
		\item Let $X$ be a projective variety whose nef cone is a finite rational polyhedral cone. Let $\dim X = d$ and $f \in \mathrm{Aut} (X)$. Then $f$ is quasi-unipotent and ${\rm rGKdim}\, (X, f) = d$. In particular, this is a case when $X$ is a Mori dream space, especially when $X$ is a projective toric variety or a Fano manifold.
	\end{enumerate}
\end{pro}

\begin{proof}
	(1) The reader is referred to \cite{Hu} for basics about compact hyper-K\"ahler manifolds. Note that a compact hyper-K\"ahler manifold has no global vector field other than $0$. Hence $\mathrm{Aut} (X)$ is discrete. Thus $f$ is of finite order if and only if $f^*|_{H^{1,1}(X, {\mathbb R})}$ is of finite order.

	So, replacing $f$ by its power and using Proposition~\ref{prop1-Kahler} (2), we can assume that $f = {\rm id\hspace{.1ex}}_X$ or $f^*|_{H^{1,1}(X, {\mathbb R})}$ is unipotent of infinite order. The result is clear when $f = {\rm id\hspace{.1ex}}_X$. In the rest, we will assume that $f^*|_{H^{1,1}(X, {\mathbb R})}$ is unipotent of infinite order.

	Let $q_X(x)$ be Beauville-Bogomolov-Fujiki's quadratic form on $H^{1,1}(X, {\mathbb R})$. The signature of $q_X(x)$ is $(1, h^{1,1}(X) -1)$.

	Let $\omega$ be a K\"ahler class on $X$. Then the degree of the polynomial
	$q_X(P_{f,\omega}(n))$ in Lemma \ref{lem35},
	with respect to $n$ is $2^2 = 4$ by~\cite[Lemma 5.4]{AvB}. The first part of (1)
	then follows from Fujiki's relation below (with positive constant $c_X > 0$):
	$$(x^{2d})_X = c_X(q_X(x))^d .$$

	For the realization part of (1), let $\varphi : S \to \P^1$ be a projective elliptic K3 surface whose Mordell-Weil group ${\rm MW}\,(\varphi)$ has an element of infinite order, say $f$. There are plenty of such K3 surfaces. Then $f \in \mathrm{Aut} (S)$ and it induces an automorphism $f^{[d]} \in \mathrm{Aut} ({\rm Hilb}^d(S)/\P^d)$ of infinite order. Here the Hilbert scheme $X := {\rm Hilb}^d(S)$ is a projective hyper-K\"ahler manifold of dimension $2d$ with the Lagrangian fibration ${\rm Hilb}^d(S) \to \P^d$ induced by $\varphi$. Hence $d_1(f^{[d]}) = 1$ as it preserves the pullback $h$ of the hyperplane class of $\P^d$, which is non-zero nef class on $X$ such that $q_X(h) = 0$.
	Thus $(X, f^{[d]})$ provides an example such that ${\rm rGKdim}\, (X, f^{[d]}) = \mathrm{Plov} (f^{[d]}) = 2 \dim X = 4d$. This completes the proof of (1).

	(2) By the assumption, $f^*|_{N^1(X)}$ is always of finite order (even though the order of $f$ itself can be often infinite). Thus we have ${\rm rGKdim}\, (X,f) = d$ by Theorem~\ref{Keeler}(7),
	as claimed.
\end{proof}

We finish this section with proofs of
 Corollary~\ref{cor-JBboundkd} (2) and Theorem~\ref{main2}
by constructing explicit examples.
The examples that we will construct
also appear in other complex dynamical contexts~\cite{OT, DLOZ}.

\begin{proof}[Proof of Corollary~\ref{cor-JBboundkd} (2)]

	Let $X_d = E^d$ ($d \ge 2$) be the $d$-fold self-product of an elliptic curve $E$
	and $f_d$ the automorphism of $X_d$ defined by
	$$f_d(x_1, x_2, \ldots, x_d) = (x_1, x_2 + x_1, \ldots, x_{d} + x_{d-1}),$$
	as in Example \ref{ex-prod_E}. We verify that $k(f_d) = 2d-2$~\cite[Remark 3.10]{DLOZ}.

	Let $C$ be a smooth projective curve of genus $g(C) \ge 2$ with a surjective morphism
	$$\pi : C \to E.$$
	For instance, the double cover of $E$ branched at two points is such an example.
	Let
	$$Y_d := C \times E^{d-1}.$$
	Then
	$Y_d$ is a smooth projective variety with $\dim Y_d = d$
	and Kodaira dimension $\kappa(Y_d) = 1$.
	We define $g_d \in \mathrm{Aut} (Y_d)$ by
	$$g_d(P, x_2, x_3, \ldots, x_{d}) = (P, x_2+\pi(P), x_3+x_2 \ldots, x_{d}+x_{d-1})$$
	We also define
	$$p : Y_d \to X_d\,\, ;\,\, (P, x_2, x_3, \ldots, x_{d}) \mapsto (\pi(P), x_2, x_3, \ldots, x_{d}).$$
	Then $p$ is a finite surjective morphism such that
	$$f_d \circ p = p \circ g_d,$$
	so $k(g_d) = k(f_d) = 2d-2$ by Proposition~\ref{pro-birJB}.
	Finally, for every smooth projective variety $V$ with $\kappa(V) = \dim V$,
	let $V_d \mathrel{:=} Y_d \times V$
	and consider $\phi_d \mathrel{:=} g_d \times \mathrm{Id}_V \in \mathrm{Aut}(V_d)$.
	We have
	$$2(d-1) = k(g_d) \le k(g_d \times \mathrm{Id}_V) \le 2(\dim V_d - \kappa(V_d)) = 2(d-1),$$
	where the second inequality follows from the first statement of Corollary~\ref{cor-JBboundkd}.
	So
	$$ k(\phi_d) = 2(\dim V_d - \kappa(V_d)).$$
	When $d$ and $V$ vary, any pair of positive integers
	$\dim V_d \ge 1$ and $\kappa(V_d) \ge 1$ is realizable, which finishes the proof.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{main2}]

	Recall that by Corollary~\ref{cor-d=3}, we have
	$\mathrm{Plov}(f) \in \{3,5,9\}$ when $\dim X = 3$.
	If $\kappa(X) = 2$ (resp. $\kappa(X) = 3$),
	then $k(f) \in \{0,2\}$ (resp. $k(f) = 0$) by Theorem~\ref{JBbound},
	so $\mathrm{Plov}(f) \in \{5,9\}$ (resp. $\mathrm{Plov}(f) = 9$) by Theorems~\ref{uniform} and~\ref{thm-lb}.

Now we construct explicit examples for Theorem~\ref{main2}, starting with Theorem \ref{main2} (4).
We continue to use the automorphisms constructed
in the previous proof of Corollary~\ref{cor-JBboundkd},
	together with the same notations.
Since
$$f_d \circ p = p \circ g_d,$$
we have
$$\mathrm{Plov} (Y_d, g_d) = \mathrm{Plov} (X_d, f_d) = d^2,$$
by Proposition~\ref{prop1-Kahler} (5) and Example \ref{ex-prod_E}.
Hence $(Y_d, g_d)$ provides an example satisfying description (4) in Theorem~\ref{main2}.

Note that since $\mathrm{Plov}(\mathrm{Id}_X) = 3$ for every threefold $X$,
all we need to construct for Theorem \ref{main2} (1), (2), (3)
are examples with $\mathrm{Plov}(f) = 5$ or $9$.
The automorphism  $(\mathrm{Id}_C \times g_2) : C \times Y_2 \circlearrowleft$ satisfies $\kappa(C \times Y_2) = 2$
and $\mathrm{Plov}(\mathrm{Id}_C \times g_2) = 5$
by Proposition~\ref{prop1-Kahler} (4), which provides an example for Theorem \ref{main2} (2).
We also have $\mathrm{Plov}(f_3) = 9$ and $\mathrm{Plov}(f'_3) = 5$;
here  $f_d' \mathrel{:=} f_{d-1} \times \mathrm{Id}_E \in \mathrm{Aut}(E^d)$.
These automorphisms provide examples when $X$ is an abelian threefold for Theorem \ref{main2} (3).

Next, let $E$ be an elliptic curve admitting an automorphism $\alpha : E \to E$ which lifts to
the multiplication ${\mathbb C} \to {\mathbb C}$ by a primitive $m$-th root of unity (so  $m = 2,3, 4, 6$).
Then the diagonal automorphism $\tau \mathrel{:=} \alpha \times \cdots \times \alpha \in \mathrm{Aut}(E^d) = \mathrm{Aut}(X_d)$
has finite order.
Let
$$\nu : Z_{d, \alpha} \to \overline{X_{d, \alpha}} := X_d/\langle \tau \rangle$$
be the blowup at the maximal ideals of the image of the fixed points of $\langle \tau \rangle$, which consists of finitely many isolated fixed points. Then $Z_{d, \alpha}$ is a smooth projective variety of dimension $d$ and Kodaira dimension $\kappa (Z_{d, \alpha}) \le 0$.

Let
$$\varphi : X_d \dasharrow Z_{d, \alpha}$$
denote the dominant rational map induced by the quotient morphism and $\nu$. Clearly, $\varphi$ is generically finite.
As $f_d \circ \tau = \tau \circ f_d$, by the universal property of blowup,
$f_d$ induces $h_d \in \mathrm{Aut} (Z_{d, \alpha})$ such that $\varphi \circ f_d = h_d \circ \varphi$.
Once again, we have
$$\mathrm{Plov} (Z_{d, \alpha}, h_d) = \mathrm{Plov} (X_d, f_d) = d^2$$
by Proposition~\ref{prop1-Kahler} (5) and Example~\ref{ex-prod_E}.

As is well known (see e.g.~\cite[Example 16.12]{Ueno}), $Z_{d, \alpha}$ is a (simply connetced) Calabi-Yau threefold when $d = 3$, $m = 3$, and $E$ is an elliptic curve of period $\zeta_3$.
By~\cite{OT}, $Z_{d, \alpha}$ is a rational threefold when $d = 3$, $m = 6$, and $E$ is an elliptic curve of period $\zeta_3$. These two cases provide examples of $\mathrm{Plov} = 9$ satisfying description (3) in Theorem~\ref{main2}.

Following the same argument,
replacing $f_d$ with $f_d' = f_{d-1} \times \mathrm{Id}_E \in \mathrm{Aut}(E^d)$ with $d=3$
in the above constructions
gives examples of $\mathrm{Plov} = 5$ satisfying Theorem~\ref{main2} (3).
\end{proof}