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\datereceived{2024-08-15}
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\begin{document}
\frontmatter

\title{Sharp bounds on the height of K\protect\nobreakdash-semi\-stable\nobreakspace Fano varieties II, the log case}

\author[\initial{R.} \lastname{Andreasson}]{\firstname{Rolf} \lastname{Andreasson}}
\address{Chalmers University of Technology and the University of Gothenburg,\\
Chalmers tvärgata 3, SE-412 96 Göteborg, Sweden}
\email{rolfan@chalmers.se}
\urladdr{}

\author[\initial{R.} \middlename{J.} \lastname{Berman}]{\firstname{Robert} \middlename{J.} \lastname{Berman}}
\address{Chalmers University of Technology and the University of Gothenburg,\\
Chalmers tvärgata 3, SE-412 96 Göteborg, Sweden}
\email{robertb@chalmers.se}
\urladdr{https://www.chalmers.se/en/persons/robertb/}

\thanks{This work was supported by a Wallenberg Scholar grant from the Knut and Alice Wallenberg foundation.}

\begin{abstract}
In our previous work we conjectured---inspired by an algebro-geometric result of Fujita---that the height of an arithmetic Fano variety $\mathcal{X}$ of relative dimension $n$ is maximal when~$\mathcal{X}$ is the projective space $\mathbb{P}_{\mathbb{Z}}^{n}$ over the integers, endowed with the Fubini-Study metric, if~the corresponding complex Fano variety is K-semistable. In~this work the conjecture is settled for diagonal hypersurfaces in $\mathbb{P}_{\mathbb{Z}}^{n+1}$. The proof is based on a logarithmic extension of our previous conjecture, of independent interest, which is established for toric log Fano varieties of relative dimension at most three, hyperplane arrangements on $\mathbb{P}_{\mathbb{Z}}^{n}$, as well as for general arithmetic orbifold Fano surfaces.
\end{abstract}

\subjclass{14G40, 32Q20, 53C25, 11G50, 14J45}

\keywords{Arakelov geometry, heights, Kähler–Einstein metrics, Fano varieties, K-stability}

\altkeywords{Géométrie d'Arakelov, hauteurs, métriques de Kähler–Einstein, variétés de Fano, K-stabilité}

\alttitle{Bornes optimales pour la hauteur des variétés de Fano K-semi-stables II: le cas logarithmique}

\begin{altabstract}
Dans un travail antérieur, nous avons conjecturé --- en nous inspirant d'un résultat algébro-géométrique de Fujita --- que la hauteur d'une variété de Fano arithmétique $\mathcal{X}$ de dimension relative $n$ est maximale lorsque $\mathcal{X}$ est l'espace projectif $\mathbb{P}_{\mathbb{Z}}^{n}$ sur les entiers, muni de la métrique de Fubini-Study, à condition que la variété de Fano complexe correspondante soit K-semi-stable. Dans ce travail, nous démontrons cette conjecture pour les hypersurfaces diagonales dans $\mathbb{P}_{\mathbb{Z}}^{n+1}$. La démonstration repose sur une extension logarithmique de notre conjecture précédente --- d'un intérêt indépendant --- que nous établissons pour les variétés de Fano logarithmiques toriques de dimension relative au plus $3$, les arrangements d'hyperplans dans $\mathbb{P}_{\mathbb{Z}}^{n}$, ainsi que pour les surfaces de Fano arithmétiques orbifoldes générales.
\end{altabstract}

\maketitle

\tableofcontents
\mainmatter

\section{Introduction}

This is a sequel to \cite{a-b}, where a conjectural arithmetic analog
of Fujita's sharp bound for the degree (volume) of K\nobreakdash-semistable Fano
varieties over $\C$ from \cite{fu} was proposed, concerning arithmetic
Fano varieties $\mathcal{X}$. The case when $\mathcal{X}$ is the
canonical integral model of a complex toric Fano variety $X$ was
settled in \cite{a-b}, when the relative dimension $n$ is at most
six (the extension to any $n$ is conditioned on a conjectural gap
hypothesis for the algebro-geometric degree). Here we will, in~particular,
show that the conjecture introduced in \cite{a-b} holds for any diagonal
Fano hypersurface~$\mathcal{X}$ in~$\P_{\Z}^{n+1}$ (see Section
\ref{subsec:Application-to-diagonal} below). The proof is based on
the following extension of the conjecture in \cite{a-b} to the logarithmic
setting, which is the main focus of the present work:
\begin{conjecture}
\label{conj:height log intro}Let $(\mathcal{X},\mathcal{D})$ be
an arithmetic log Fano variety and denote by $(X,\Delta)$ is complexification,
which defines a complex log Fano variety. Then the following inequality
of arithmetic intersection numbers holds for any volume-normalized
continuous metric on $-(K_{X}+\Delta)$ with positive curvature current
if $(X,\Delta)$ is K\nobreakdash-semistable:
\[
(\overline{-\mathcal{K}_{(\mathcal{X},\mathcal{D})}})^{n+1}\leq(\overline{-\mathcal{K}_{\P_{\Z}^{n}}})^{n+1},
\]
where $-K_{\P_{\C}^{n}}$ is endowed with the volume-normalized Fubini-Study
metric. Moreover, if $\mathcal{X}$ is normal equality holds if and
only if $(\mathcal{X},\mathcal{D})=(\P_{\Z}^{n},0)$ and the metric
is Kähler-Einstein, \ie coincides with the Fubini-Study
metric up to the action of an automorphism of $\P_{\C}^{n}$.
\end{conjecture}

By definition, an \emph{arithmetic log Fano variety} $(\mathcal{X},\mathcal{D})$
is a projective flat scheme $\mathcal{X}$ over $\Z$, such that $\mathcal{X}$
is reduced and satisfies Serre's conditions $S_{2}$ (e.g., $\mathcal{X}$
normal), together with an effective $\Q$-divisor $\mathcal{D}$ on
$\mathcal{X}$ such that
\[
-\mathcal{K}_{(\mathcal{X},\mathcal{D})}:=-(\mathcal{K}_{\mathcal{X}}+\mathcal{D})
\]
defines a relatively ample $\Q$-line bundle, where $\mathcal{K}_{\mathcal{X}}$
denotes the relative canonical divisor on $\mathcal{X}$. We also
assume that the corresponding complex variety $X$ is normal. The
complexification $(X,\Delta)$ of $(\mathcal{X},\mathcal{D})$ thus
defines a complex log Fano variety and we shall say that $(\mathcal{X},\mathcal{D})$
is an \emph{integral model} of $(X,\Delta)$ (see Section \ref{subsec:Arithmetic-log-Fano}
for more background on this setup).

Following standard procedure, we denote by $\overline{\mathcal{L}}$
a metrized line bundle, \ie a line bundle $\mathcal{L}$ on $\mathcal{X}$
endowed with an Hermitian metric over the complex points $X$ of $\mathcal{X}$.
Arithmetic intersection numbers of metrized line bundles were introduced
by Gillet-Soulé in the context of Arakelov geometry \cite{b-g-s}.
The top arithmetic intersection number $\overline{\mathcal{L}}^{n+1}$
of $\overline{\mathcal{L}}$ is called the\emph{ height} of $\overline{\mathcal{L}}$.
The height of $\overline{-\mathcal{K}_{\P_{\Z}^{n}}}$ with respect
to the volume-normalized Fubini-Study metric, appearing in the previous
conjecture, is explicitly given by the following formula \cite[Lem.\,3.6]{ab},
which, essentially, goes back to \cite[\S 5.4]{g-s}:
\begin{equation}
(\overline{-\mathcal{K}_{\P_{\Z}^{n}}})^{n+1}=\frac{1}{2}(n+1)^{n+1}\biggl((n+1)\sum_{k=1}^{n}k^{-1}-n+\log(\sfrac{\pi^{n}}{n!})\biggr).\label{eq:expl formul on p n}
\end{equation}
As for the notion of K\nobreakdash-stability it originally appeared in the context
of the Yau-Tian-Donaldson conjecture for Fano manifolds $X$ (see
the survey \cite{x} for recent developments, including connections
to moduli spaces and the minimal model program in birational geometry).
By \cite{li1} and \cite[Th.\,1.6]{l-x-z} a log Fano variety $(X,\Delta)$
over $\C$ is \emph{K\nobreakdash-polystable} (which is a slightly stronger condition
than K\nobreakdash-semistability) if and only if it admits a \emph{log Kähler-Einstein
metric}, \ie a locally bounded metric on $-(K_{X}+\Delta)$, whose
curvature current $\omega$ induces a Kähler metric with
constant Ricci curvature on the complement of $\Delta$ in the regular
locus of $X$. After volume-normalization, any log Kähler-Einstein
for $(X,\Delta)$ maximizes the height $(\overline{-\mathcal{K}_{(\mathcal{X},\mathcal{D})}})^{n+1}$
among all volume-normalized locally bounded metrics on $-(K_{X}+\Delta)$
with positive curvature (as shown precisely as in the case that $\mathcal{D}=0$
considered in \cite[\S 2.3]{a-b}). When $(X,\Delta)$ is log
smooth any log Kähler-Einstein metric has conical singularities
along $\Delta$ (see Section \ref{subsec:Log-Kahler-Einstein-metrics})

The K\nobreakdash-semistability of $(X,\Delta)$ implies that $(X,\Delta)$ is
Kawamata log terminal (klt) in the usual sense of birational geometry
(see Remark \ref{rem:klt}). An important class of klt log Fano varieties
$(X,\Delta)$ is provided by (smooth)\emph{ Fano orbifolds}, where
the coefficients of $\Delta$ are of the form $(1-1/m_{i})$ for positive
integers $m_{i}$. Diophantine aspects of Fano orbifolds have recently
been explored in a number of works, building on Campana's program
\cite{ca} and its developments by Abramovich \cite{ab} (see \cite{tani}
for a very recent survey). In~particular, a logarithmic generalization
of the Manin-Peyre conjecture for the density of rational points of
bounded height on Fano varieties is proposed in~\cite{p-}, which,
for example, is addressed for log Fano hyperplane arrangements and
toric varieties in \cite{b-y} and \cite{p-s}, respectively. See
\cite{ber} for relations between height bounds, K\nobreakdash-stability and the
Manin-Peyre conjecture.

\Subsection{Main results}

\subsubsection{Toric log Fano varieties}

We first consider the case when $(\mathcal{X},\mathcal{D})$ is the
canonical integral model of a complex toric log Fano variety $(X,\Delta)$
(see \cite[\S 2]{ma} and \cite[Def.\,3.5.6]{b-g-p-s}). One advantage
of the logarithmic setup is that on any given toric Fano variety $X$
there exist an infinite number of toric $\Q-$divisors $D$ such that
$-(K_{X}+\Delta)$ is a K\nobreakdash-semistable log Fano variety. Building on
\cite{ab}, where the case when $D=0$ was considered, we show:

\begin{thm}
\label{thm:main log toric intro}Let $(\mathcal{X},\mathcal{D})$
be the canonical integral model of a complex K\nobreakdash-semistable toric log
Fano variety $(X,\Delta)$. Conjecture \ref{conj:height log intro}
holds for $(\mathcal{X},\mathcal{D})$ under anyone of the following
conditions:
\begin{itemize}
\item $n\leq3$ and $X$ is $\Q-$factorial (equivalently, $X$ has at worst
abelian quotient singularities);
\item $X$ is not Gorenstein or has some abelian quotient singularity.
\end{itemize}
\end{thm}

The starting point of the proof is the bound
\begin{equation}
\frac{\overline{(-\mathcal{K}_{(\mathcal{X},\mathcal{D})}})^{n+1}}{(n+1)!}\leq\frac{1}{2}\vol(X,\Delta)\log\Bigl(\frac{(2\pi^{2})^{n}}{\vol(X,\Delta)}\Bigr),\quad \vol(X,\Delta):=\frac{-(K_{X}+\Delta)^{n}}{n!},\label{eq:ineq in pf thm log toric intro}
\end{equation}
shown precisely as in the case when $\Delta=0$, considered in \cite{ab}.
For $X=\P^{n}$ the previous theorem is verified by an explicit calculation. In~the remaining case, $X\neq\P^{n}$, the bound in Conjecture \ref{conj:height log intro}
follows, just as in \cite{ab}, from combining the bound \eqref{eq:ineq in pf thm log toric intro}
with the following logarithmic analog of the ``gap hypothesis''
introduced in \cite{ab}:
\begin{equation}
\vol(X,\Delta)\leq\vol(\P^{n-1}\times\P^{1})\label{eq:log gap intro}
\end{equation}
for any K\nobreakdash-semistable $n$-dimensional log Fano variety $(X,\Delta)$
such that $X\neq\P^{n}$.\footnote{See \cite[\S 3.2.1]{a-b} for a comparison between the gap hypothesis
and the ODP-conjecture in \cite{s-s} (very recently settled in the
toric case \cite{m-s}), which, however, yields a weaker inequality
than the gap hypothesis in our setup.} In~the case that $X$ is singular the logarithmic gap hypothesis
does hold in any dimension, just as in \cite{ab}. In~the non-singular
case there is, for any dimension, only a finite number of toric Fano
varieties $X$. For $n\leq6$ these appear in the database \cite{ob},
which, as observed in \cite{ab}, settles the gap hypothesis for $n\leq6$,
when $\Delta=0$. However, in~the present case there is for any given
toric variety $X$ an infinite number of toric divisors~$\Delta$
on $X$ such that $(X,\Delta)$ is a K\nobreakdash-semistable Fano variety. In~order to establish the logarithmic gap-hypothesis \eqref{eq:log gap intro}
we thus introduce the following invariant of a Fano manifold $X$:
\[
S(X):=\sup_{\Delta}\left\{\vol(X,\Delta):(X,\Delta)\,\text{K\nobreakdash-semistable\,log Fano}\right\}
\]
and show, by~solving the corresponding optimization problem, that
\[
S(X)\leq\vol(\P^{n-1}\times\P^{1})
\]
when $X\neq\P^{n}$ and
$n\leq3$.

The invariant $S(X)$ and the corresponding maximizers $\Delta$ appear
to be of independent interest in Kähler geometry. This
is illustrated by some examples in Section~\ref{subsec:The-logarithmic-gap},
where we make contact with a rigidity property of the corresponding
log Kähler-Einstein metric, first exhibited in \cite{r-z}.

\subsubsection{Hyperplane arrangements}

We next turn to the case when $\mathcal{X}$ is the projective space
over the integers, $\mathcal{X}=\P_{\Z}^{n}$ and $\mathcal{D}$ is
a hyperplane arrangement, \ie its irreducible components are hyperplanes.

\begin{thm}
\label{thm:hyperplane intro}Conjecture \ref{conj:height log intro}
holds when $\mathcal{X}=\P_{\Z}^{n}$ and $\mathcal{D}$ is a hyperplane
arrangement with simple normal crossings.
\end{thm}

The proof employs a convexity argument to reduce the problem to the
case when~$\mathcal{D}$ is toric, which is covered by Theorem \ref{thm:main log toric}.
The argument leverages the explicit characterization of K\nobreakdash-semistable
hyperplane arrangements established in \cite{fu2} and yields the
following explicit bound:
\begin{equation}
\frac{(\overline{-\mathcal{K}_{(\mathcal{X},\mathcal{D})}})^{n+1}}{(n+1)!}\leq\frac{1}{2}\vol(X,\Delta)\log\Bigl(\frac{(n+1)^{n}e^{2a_{n}}}{(n+1)!\vol(X,\Delta)} \Bigr),\quad a_{n}=\frac{(\overline{-\mathcal{K}_{\P_{\Z}^{n}}})^{n+1}}{\left(n+1\right)^{n+1}},\label{eq:estimate by toric intro}
\end{equation}
with equality if and only if $\mathcal{D}$ is toric.

\subsubsection{\label{subsec:Application-to-diagonal}Application to diagonal hypersurfaces}

Given a positive integer $d$ and integers $a_{i}$, consider the
diagonal hypersurface $\mathcal{X}_{a}$ of degree $d$ in $\P_{\Z}^{n+1}$
cut out by the homogeneous polynomial $\sum_{i=0}^{n+1}a_{i}x_{0}^{d}$.
The corresponding complex variety $X_{a}$ is Fano if and only if
$d\leq(n+1)$ and is always K\nobreakdash-polystable (and, in~particular, K\nobreakdash-semistable);
see, for example, \cite{zhu} for an algebraic proof. Using the results
stated in the previous two sections we will establish Conjecture \ref{conj:height log intro}
for $\mathcal{X}_{a}$, endowed with the trivial divisor~$0$:

\begin{thm}
\label{thm:diagonal hypersurface intro}Conjecture \ref{conj:height log intro}
holds for any diagonal hypersurface $\mathcal{X}_{a}$ which is Fano
(\ie $d\leq n+1)$ when the divisor $\mathcal{D}$ is trivial. More
precisely,
\[
(\overline{-\mathcal{K}_{\mathcal{X}_{a}}})^{n+1}\leq(\overline{-\mathcal{K}_{\P_{\Z}^{n}}})^{n+1}+(1-d)(n+2-d)^{n}\sum_{i=0}^{n+1}\log|a_{i}|,
\]
and the inequality is strict if $d\geq2$.
\end{thm}

Note that the schemes $\mathcal{X}_{a}$ are mutually non-isomorphic
over $\Z$, for any given degree~$d$ of at least two. In~fact, in
general, they are not even isomorphic over $\Q$. The proof of the
previous theorem is first reduced to the case of a Fermat hypersurface,
\ie the case when $a_{i}=1$. Expressing $\mathcal{X}_{a}$ as a
Galois cover of $\P_{\Z}^{n}$ the estimate~\eqref{eq:estimate by toric intro}
can then be applied with $\mathcal{X}=\P_{\Z}^{n}$ and $\Delta$
the corresponding branching divisor, which reduces the problem to
a simple toric case.

We recall that the Manin-Peyre conjecture has been settled for Fano
hypersurfaces over $\Q$ of sufficiently small degree \cite{pey,pey2}. In~particular, there is an extensive literature on the diagonal case,
in connection to Waring's classical problem, where the degree bound
has been improved \cite{pi} (see also \cite{bi} and \cite[Th.\,15.6]{woo}
for general number fields).

\subsubsection{\label{subsec:Arithmetic-log-surfaces}Arithmetic log surfaces }\hspace{-0.2cm} Consider a polarized arithmetic log surface $(\mathcal{X},\mathcal{D};\mathcal{L})$,
\ie an arithmetic log surface $(\mathcal{X},\mathcal{D})$ endowed
with a relatively ample line bundle $\mathcal{L}$. Assume that the
complexification $\mathcal{L}\otimes\C$ is isomorphic to $-K_{(X,\Delta)}$
(where $(X,\Delta)$ denotes, as before, the complexification of $(\mathcal{X},\mathcal{D})$).
Given a continuous metric on $\mathcal{L}$ we define the \emph{arithmetic
log Mabuchi functional} of $(\mathcal{X},\mathcal{D};\overline{\mathcal{L}})$
by
\begin{equation}
\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\overline{\mathcal{L}}):=\frac{1}{2}\overline{\mathcal{L}}^{2}+\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}\cdot\overline{\mathcal{L}},\label{eq:def of log Mab surface intro}
\end{equation}
where $\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}$ is endowed
with the volume-normalized metric induced by the curvature current
$\omega$ of $\overline{\mathcal{L}}$, assuming that this metric
on $\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}$ is continuous.
When $\mathcal{D}=0$ the functional $\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\overline{\mathcal{L}})$
coincides with the modular height introduced in \cite{o}, up to normalization
(see \cite[\S 6.4]{a-b} for a comparison between the different
normalizations). For a given integral model $\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\overline{\mathcal{L}})$
is minimized on a log Kähler-Einstein metric, if such
a metric exists, and then
\[
\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\overline{\mathcal{L}})=-(\overline{-\mathcal{K}_{(\mathcal{X},\mathcal{D})}})^{2}/2,
\]
if the metric is volume-normalized and $\mathcal{L}=-\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}$.

\begin{thm}
\label{thm:Mab on log surface}Let $(\mathcal{X},\mathcal{D};\mathcal{L})$
be a polarized arithmetic log surface $(\mathcal{X},\mathcal{D};\mathcal{L})$
with $\mathcal{X}$ normal, such that the complexification $(X,\Delta)$
of $(\mathcal{X},\mathcal{D})$ is a K\nobreakdash-semistable Fano variety and
$\mathcal{L}\otimes\C=-K_{(X,\Delta)}$. If $\Delta$ is supported
on (at most) three points, then
\[
\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\overline{\mathcal{L}})\geq\mathcal{M}_{(\P_{\Z}^{1},0)}(\overline{-\mathcal{K}_{\P_{\Z}^{1}}})\quad (=-1-\log\pi),
\]
where $-\mathcal{K}_{\P_{\Z}^{1}}$ is endowed with the Fubini-Study
metric. Moreover, equality holds if and only if $(\mathcal{X},\mathcal{D})$
is isomorphic to $(\P_{\Z}^{1},0)$ and $\mathcal{L}$ is isomorphic
to $-\mathcal{K}_{\P_{\Z}^{1}}$, endowed with a metric coinciding
with the Fubini-Study metric, up to the application of an automorphism
of~$\P_{\Z}^{1}$ and a scaling of the metric.
\end{thm}

It would be interesting to extend the previous theorem to the case
when $\Delta$ is supported on any number of points. Anyhow, the assumption
that $\Delta$ is supported on three points is always satisfied in
the orbifold case. In~particular, we get:
\begin{cor}
\label{cor:Conjecture holds for log Fano surfaces}Conjecture \ref{conj:height log intro}
holds for arithmetic normal log Fano surfaces, if $\Delta$ is supported
on, at most, three points. In~particular, the conjecture holds for
all normal arithmetic log Fano orbifold surfaces (\ie the case of
coefficients of the form $1-1/m_{i}$ for $m_{i}\in\N$).
\end{cor}

In the setup of the previous theorem the corresponding complex variety
$X$ is always equal to $\P^{1}$ and thus $(X,\Delta)$ is a hyperplane
arrangement. Accordingly, applying Theorem \ref{thm:hyperplane intro},
the proof of Theorem \ref{thm:Mab on log surface} is reduced to showing
that the canonical integral model $(\mathcal{X}_{c},\mathcal{D}_{c};-\mathcal{K}_{(\mathcal{X}_{c},\mathcal{D}_{c})})$
of $(X,\Delta;-K_{(X,\Delta)})$ obtained by setting $\mathcal{X}_{c}=\P_{\Z}^{1}$
and taking $\mathcal{D}_{c}$ to be the Zariski closure of $\{0,1,\infty)\}$
in $\P_{\Z}^{1}$ minimizes $\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\overline{\mathcal{L}})$
over all integral models $(\mathcal{X},\mathcal{D};\mathcal{L})$
of $(X,\Delta;-K_{(X,\Delta)}$), for any fixed metric on $-K_{(X,\Delta)}$.
This minimization property can be viewed as logarithmic version of
Odaka's minimization conjecture (proposed in any dimension in \cite{o}).
Our proof builds on \cite{o}, leveraging log canonical thresholds.
We also show that the minimum is uniquely attained for $(\P_{\Z}^{1},0)$.
See also \cite{h-o} for very recent progress on Odaka's minimization
conjecture in another direction.

\subsubsection*{Acknowledgements}

Thanks to Julia Brandes, Dennis Eriksson, Mattias Jonsson, Gerard
Freixas i Montplet and Yuji Odaka for discussions and the referee
for comments that helped to improve the exposition.

\section{\label{sec:General-setup}General setup}

\Subsection{\label{subsec:Log-Fano-varieties metrics}Log Fano varieties over
$\C$ and volume-normalized metrics on $-(K_{X}+\Delta)$ }

A \emph{log pair }$(X,\Delta)$ over $\C$ is a normal complex projective
variety $X$ together with an effective $\Q$-divisor $\Delta$ on
$X$ such that $K_{X}+\Delta$ is $\Q$-Cartier, \ie defines a $\Q$-line
bundle, where $K_{X}$ denotes the canonical divisor on $X$ \cite{ko0}. In~the logarithmic setting this bundle plays the role of the canonical
line bundle and is thus called the\emph{ log canonical line bundle}
and is denoted by $K_{(X,\Delta)}$. A~log pair $(X,\Delta)$ is said
to be a\emph{ log Fano pair} if~$\Delta$ is effective and $-(K_{X}+\Delta)>0$.
Any continuous metric $\left\Vert \cdot\right\Vert $ on $-(K_{X}+\Delta)$
induces a measure $\mu$ on $X$ in a standard fashion. Indeed, when
$X$ is regular and $\Delta=0$ this follows directly from the definition
of metrics on $-K_{X}$ (see \cite[\S 2.1.2]{a-b}). In~general,
denoting by $X_\mathrm{reg}$ the regular locus of $X$, and using that the
$\Q$-line bundles $-(K_{X}+\Delta)$ and $-K_{X}$ are isomorphic
on the complement $X_\mathrm{reg}\setminus\supp(\Delta)$ in $X_\mathrm{reg}$ of
the support of $\Delta$, the previous construction yields a measure
on $X_\mathrm{reg}\setminus\supp(\Delta)$. Its push-forward to $X$, under
the inclusion map, thus yields a measure on $X$ (see also \cite[\S 3.1]{bbegz}
for a slightly different representation of this measure). This measure
has finite mass if and only if the log pair $(X,\Delta)$ is \emph{klt} in the
standard sense of birational algebraic geometry (see \cite[\S 3.1]{bbegz}
and Remark \ref{rem:klt} below). A~continuous metric on $-(K_{X}+\Delta)$
will be said to be \emph{volume-normalized} if the corresponding measure
is a probability measure.

\subsubsection{\label{subsec:Local-representations-of}Local representations of
metrics and measures}

As in \cite{a-b} we will use additive notation for metrics on holomorphic
line bundles $L\to X$. This means that we identify a continuous
Hermitian metric $\left\Vert \cdot\right\Vert $ on $L$ with a collection
of continuous local functions~$\phi_{U}$ associated to a given covering
of $X$ by open subsets $U$ and trivializing holomorphic sections
$e_{U}$ of $L\to U$:
\begin{equation}
\phi_{U}:=-\log(\left\Vert e_{U}\right\Vert ^{2}).\label{eq:def of phi U}
\end{equation}
The curvature current of the metric may then, locally, be expressed
as
\[
dd^{c}\phi_{U}:=\frac{i}{2\pi}\partial\bar{\partial}\phi_{U}
\]
Accordingly, as is customary, we will symbolically denote by $\phi$
a given continuous Hermitian metric on $L$ and by $dd^{c}\phi$ its
curvature current. We will denote by \hbox{$\mathcal{C}^{0}(L)\cap\mathrm{PSH}(L)$}
the space of all continuous metrics on $L$ whose curvature current
is positive, $dd^{c}\phi\geq0$ (which means that $\phi_{U}$ is plurisubharmonic,
or psh, for short).

Given a log Fano pair $(X,\Delta)$ the measure corresponding to a
given continuous metric $\phi$ on $-K_{(X,\Delta)}$ may be locally
on $X_\mathrm{reg}$ be expressed as
\[
\mu_{\phi}=e^{-\phi_{U}}\left|s_{U}\right|^{-2}(\sfrac{i}{2})^{n^{2}}dz\wedge d\bar{z},\quad dz:=dz_{1}\wedge\cdots\wedge dz_{n}
\]
by taking $e_{U}=\partial/\partial z_{1}\wedge\cdots\wedge\partial/\partial z_{n}\otimes e_{\Delta}$,
where $e_{\Delta}$ is a local trivialization of the $\Q$\nobreakdash-line bundle
over $X_\mathrm{reg}$ corresponding to the divisor $\Delta$ and $s_{U}e_{\Delta}$
is the (multi-valued) holomorphic section cutting out $\Delta$.

\subsubsection{\label{subsec:Log-Kahler-Einstein-metrics}Log Kähler-Einstein
metrics}

Given a log Fano pair $(X,\Delta)$, a metric $\phi$ on $-K_{(X,\Delta)}$
is said to be a\emph{log Kähler-Einstein metric},
if $\phi$ is locally bounded and its curvature current $dd^{c}\phi$
induces a Kähler metric with constant positive Ricci curvature
on the complement of $\Delta$ in $X_\mathrm{reg}$ \cite{bbegz}. When $(X,\Delta)$
is log smooth, \ie $X$ is smooth and $\Delta$ has simple normal
crossings, any log Kähler-Einstein for $(X,\Delta)$ has
conical singularities along $\Delta$. More precisely, $\phi$ is
continuous and if $\Delta$ has coefficient $c_{i}\in{}]0,1[$ along
a smooth prime divisor $\Delta_{i}$, then $dd^{c}\phi$ has cone
angle $2\pi(1-c_{i})$ along $\Delta_{i}$ \cite{j-m-r,g-m,m-r}.

\subsubsection{K-semistability}

We next recall the definition of K\nobreakdash-semistability in terms of intersection
numbers (see the survey \cite{x} for more background). A~\emph{test
configuration} for a log Fano pair $(X,\Delta)$ is a $\C^{*}$-equivariant
normal model $(\mathscr{X},\mathscr{L})$ for $(X,-K_{(X,\Delta)})$
over the complex affine line $\C$. More precisely, $\mathscr{X}$
is a normal complex variety, endowed with a $\C^{*}$-action $\rho$,
a $\C^{*}$-equivariant holomorphic projection $\pi$ to $\C$ and
a relatively ample $\C^{*}$-equivariant $\Q$-line bundle $\mathscr{L}$
(endowed with a lift of $\rho)$:
\begin{equation}
\pi:\mathcal{\mathscr{X}}\to\C,\quad \mathscr{L}\to\mathscr{X},\quad \rho:\mathscr{X}\times\C^{*}\to\mathscr{X}\label{eq:def of pi for test c}
\end{equation}
such that the fiber of $\mathscr{X}$ over $1\in\C$ is equal to $(X,-K_{(X,\Delta)})$. A~log Fano pair $(X,\Delta)$ is said to be \emph{K\nobreakdash-semistable} if
the Donaldson-Futaki invariants $\text{DF}_{\Delta}(\mathscr{X},\mathscr{L})$
are non-negative for any test configuration $(\mathscr{X},\mathscr{L})$
of $(X,\Delta)$:
\[
n!\text{DF}_{\Delta}(\mathscr{X},\mathscr{L})=\frac{n}{(n+1)}\,\overline{\mathscr{L}}^{n+1}+\mathscr{K}_{(\mathcal{\mathscr{\overline{X}}},\mathscr{D})/\P^{1}}\cdot\mathcal{\overline{\mathscr{L}}}^{n},
\]
where $\overline{\mathscr{L}}$ denotes the $\C^{*}$-equivariant
extension of $\mathscr{L}$ to the $\C^{*}$-equivariant compactification
$\mathscr{\overline{X}}$ of $\mathscr{X}$ over $\P^{1}$ and $\mathscr{K}_{(\mathcal{\mathscr{\overline{X}}},\mathscr{D})/\P^{1}}$
denotes the relative log canonical divisor of the pair $(\mathscr{\overline{X}}$$,\mathscr{D}$)
with $\mathscr{D}$ denoting the flat closure in $\mathscr{\overline{X}}$
of the $\C^{*}$-orbit of the divisor~$\Delta$.
\begin{rem}
\label{rem:klt}If a log Fano variety $(X,\Delta)$ is K\nobreakdash-semistable,
then $(X,\Delta)$ is klt (see \cite[Th.\,6.1]{o-s} and \cite[Cor.\,9.6]{b-h-j}).
When $X$ is non-singular and $\Delta$ has simple normal crossings
this means that all the coefficients of $\Delta$ along its irreducible
components are strictly smaller than $1$.
\end{rem}

\subsection{\label{subsec:Arithmetic-log-Fano}Arithmetic log Fano varieties
and integral models}

The notion of log pairs over $\C$ can be extended to schemes over
excellent rings, as explained in the book \cite{ko}. Here we will
consider the case when the ring in question is $\Z$. Henceforth,
$\mathcal{X}$~will denote a projective flat scheme $\mathcal{X}\to\Spec\Z$
of relative dimension $n$ such that $\mathcal{X}$ is reduced and
satisfies Serre's conditions $S_{2}$ (this is, for example, the case
if $\mathcal{X}$ is normal). Such a scheme $\mathcal{X}$ will be
called an \emph{arithmetic variety.} We will denote by $\pi$ the
corresponding structure morphism to $\Spec\Z$,
\[
\pi:\mathcal{X}\to\Spec\Z.
\]
A \emph{log pair} $(\mathcal{X},\mathcal{D})$ over $\Z$ of relative
dimension $n$ is an arithmetic variety $\mathcal{X}$ endowed with
an effective $\Q$-divisor $\mathcal{D}$ on $\mathcal{X}$ such that
$\mathcal{K}_{\mathcal{X}}+\mathcal{D}$ is $\Q$-Cartier, \ie defines
a $\Q$-line bundle, where $\mathcal{K}_{\mathcal{X}}$ denotes the
relative canonical divisor on $\mathcal{X}$ (see \cite[\S 1.1]{ko}
and \hbox{\cite[\S 2.2.1]{a-b}}). We shall call such a pair $(\mathcal{X},\mathcal{D})$
an\emph{ arithmetic log variety.} The complexification of $(\mathcal{X},\mathcal{D})$
will be denoted by $(X,\Delta)$ and $(\mathcal{X},\mathcal{D})$
will be called an \emph{integral model of $(X,\Delta)$.}
\begin{rem}
We are using the notion of an integral model in a generalized sense -- usually a scheme $\mathcal{X}$ over $\Z$ is said to be an integral
model of a scheme $X_{\Q}$ defined over~$\Q$ if $X_{\Q}$ is isomorphic
to $\mathcal{X}\otimes_{\Z}\Q$ over $\Q$. Here $X$ (without any
subscript) will usually denote a variety over $\C$ and a scheme $\mathcal{X}$
over $\Z$ is thus called an integral model of $X$ if $X$ is isomorphic
to $\mathcal{X}\otimes_{\Z}\C$.
\end{rem}

A log pair $(\mathcal{X},\mathcal{D})$ over $\Z$ will be called
an\emph{ arithmetic log Fano variety} if $-(\mathcal{K}_{\mathcal{X}}+\mathcal{D})$
is relatively ample and the corresponding complex variety $X$ is
normal. In~particular, the complexification $(X,\Delta)$ of $(\mathcal{X},\mathcal{D})$
is a log Fano variety over $\C$. More generally, an~arithmetic variety
$\mathcal{X}$ endowed with a relatively ample line bundle $\mathcal{L}$
will be said to be \emph{polarized }and $(\mathcal{X},\mathcal{L})$
will be called an \emph{integral model} of a complex polarized variety
$(X,L)$ if $(X,L)$ is isomorphic to the complexification of $(\mathcal{X},\mathcal{L})$.

\subsection{Arithmetic intersection numbers and heights} \hspace{-0.1cm} We recall some well-known facts about heights (see \cite{a-b} for
more background and references). A\emph{ metrized line bundle} $\overline{\mathcal{L}}$
is a line bundle $\mathcal{L}\to\mathcal{X}$ such that the
corresponding line bundle $L\to X$ is endowed with a metric,
that we shall denote by $\phi$ (as in Section \ref{subsec:Local-representations-of});
$\overline{\mathcal{L}}:=\left(\mathcal{L},\phi\right)$. The \emph{$\chi$-arithmetic
volume} of a polarized arithmetic variety $(\mathcal{X},\mathcal{L})$
is defined by
\begin{equation}
\widehat{\vol}_{\chi}\left(\overline{\mathcal{L}}\right):=\lim_{k\to\infty}k^{-(n+1)}\log\Vol
\bigl\{s_{k}\in H^{0}(\mathcal{X},k\mathcal{L})\otimes\R:\ts\sup_{X}\left\Vert s_{k}\right\Vert _{\phi}\leq1\bigr\},\label{eq:def of xhi vol}
\end{equation}
where $H^{0}(\mathcal{X},k\mathcal{L})\otimes\R$ may be identified
with the subspace of real sections in $H^{0}(X,kL)$. More generally,
$\widehat{\vol}_{\chi}\left(\overline{\mathcal{L}}\right)$
is naturally defined for $\Q-$line bundles, since it is homogeneous
with respect to tensor products of $\overline{\mathcal{L}}$:
\begin{equation}
\widehat{\vol}_{\chi}\left(m\overline{\mathcal{L}}\right)=m^{n+1}\widehat{\vol}_{\chi}\left(\overline{\mathcal{L}}\right),\quad\text{if }m\in\Z_{+}.\label{eq:chi vol for tensor}
\end{equation}
Moreover, $\widehat{\vol}_{\chi}\left(\overline{\mathcal{L}}\right)$
is additively equivariant with respect to scalings of the metric:
\begin{equation}
\widehat{\vol}_{\chi}\left(\mathcal{L},\phi+\lambda\right)=\widehat{\vol}_{\chi}\left(\overline{\mathcal{L}}\right)+\frac{\lambda}{2}\Vol(L),\quad \text{if }\lambda\in\R.\label{eq:scaling of chi vol}
\end{equation}
If the metric on $L$ has positive curvature current (\ie if $\phi$
is psh), then, by~the arithmetic Hilbert-Samuel theorem,
\begin{equation}
\widehat{\vol}_{\chi}(\overline{\mathcal{L}})=\frac{\overline{\mathcal{L}}^{n+1}}{(n+1)!},\label{eq:vol chi as intersection}
\end{equation}
where $\overline{\mathcal{L}}^{n+1}$ denotes the top arithmetic intersection
number in the sense of Gillet-Soulé \cite{g-s}, which,
defines the \emph{height} of $\mathcal{X}$ with respect to $\overline{\mathcal{L}}$
\cite{fa,b-g-s}. For the purpose of the present paper formula \eqref{eq:vol chi as intersection}
may be taken as the definition of $\overline{\mathcal{L}}^{n+1}$
(arithmetic intersections between general $n+1$ metrized line bundles
could then be defined by polarization, \ie using multilinearity).
Following standard practice we will use the shorthand $h_{\phi}(\mathcal{X},\mathcal{L})$
for the height $(\mathcal{L},\phi)^{n+1}$ and $\hat{h}_{\phi}(\mathcal{X},\mathcal{L})$
for the normalized height:
\[
h_{\phi}(\mathcal{X},\mathcal{L}):=(\mathcal{L},\phi)^{n+1},\quad \hat{h}_{\phi}(\mathcal{X},\mathcal{L}):=\frac{(\mathcal{L},\phi)^{n+1}}{(n+1)L^{n}}.
\]
The definition of $\hat{h}_{\phi}(\mathcal{X},\mathcal{L})$ is made
so that
\[
\hat{h}_{\phi+\lambda}(\mathcal{X},\mathcal{L})=\hat{h}_{\phi}(\mathcal{X},\mathcal{L})+\lambda/2\quad \text{if }\lambda\in\R.
\]
We also recall that, given two continuous psh metrics $\phi$ and
$\phi_{0}$ on the complexification $L\to X$ of $\mathcal{L}\to\mathcal{X}$,
we have that
\begin{equation}\label{eq:def of primitive}
\begin{aligned}
2h\left(\mathcal{L},\phi\right)-2h\left(\mathcal{L},\phi_{0}\right)&=\mathcal{E}(\phi,\phi_{0})\\
:\!&=\frac{1}{(n+1)!}\int_{X}(\phi-\phi_{0})\sum_{j=0}^{n}(dd^{c}\phi)^{j}\wedge(dd^{c}\phi_{0})^{n-j}.
\end{aligned}
\end{equation}

\subsection{\label{subsec:The-canonical-height}The canonical height of an arithmetic
log Fano variety}

We define the \emph{canonical height} $h_{\can}(\mathcal{X},\mathcal{D})$
of an arithmetic log Fano variety $(\mathcal{X},\mathcal{D})$ by
\[
h_{\can}(\mathcal{X},\mathcal{D}):=\sup\Bigl\{h_{\phi}\left(-\mathcal{K}_{(\mathcal{X},\mathcal{D})}\right):\phi\,\text{cont.\,psh},\int_{X}\mu_{\phi}=1\Bigr\}
\]
(when $\mathcal{D}=0$ we shall use the short hand $h_{\can}(\mathcal{X})$
for $h_{\can}(\mathcal{X},0)$). As shown precisely as in the
case $\mathcal{D}=0$, considered in \cite{a-b}, $h_{\can}(\mathcal{X},\mathcal{D})<\infty$
if and only if the corresponding log Fano variety $(X,\Delta)$ over $\C$ is
K\nobreakdash-semistable. By \cite[Prop.\,3.3]{a-b2}, the sup defining $h_{\can}(\mathcal{X},\mathcal{D})$
may as well be taken over all locally bounded metrics $\phi$ on $-K_{(X,\Delta)}$
(or, more generally, over all finite energy psh metrics $\phi)$.
Moreover, $(X,\Delta)$ is K\nobreakdash-polystable if and only if the sup defining $h_{\can}(\mathcal{X},\mathcal{D})$
is attained at some locally bounded metric $\phi$, namely a log Kähler-Einstein
metric. Hence, if $(X,\Delta)$ is K\nobreakdash-polystable, then the canonical
height $h_{\can}(\mathcal{X},\mathcal{D})$ is computed by any
volume-normalized log Kähler-Einstein metric.

Finally, we note that, by~scaling the metric,
\begin{align}
-\hat{h}_{\can}(\mathcal{X},\mathcal{D})&:=-\frac{h_{\can}(\mathcal{X})}{(n+1)(-K_{(X,\Delta)})^{n}}=\inf_{\phi}\frac{1}{2}\hat{D}_{\Z}(\phi),\notag\\
\hat{D}_{\Z}(\phi)&:=-2\hat{h}_{\phi}\left(-\mathcal{K}_{(\mathcal{X},\mathcal{D})}\right)-\log\int_{X}\mu_{\phi},\label{eq: hcan}
\end{align}
where the inf ranges over all continuous psh metrics $\phi$ on $-K_{(X,\Delta)}$.
Additionally, by~\cite[Prop.\,3.3]{a-b2}, $-h_{\can}(\mathcal{X},\mathcal{D})$
coincides with the infimum of the arithmetic Mabuchi functional attached
to $(\mathcal{X},\mathcal{D};-\mathcal{K}_{(\mathcal{X},\mathcal{D})},\phi)$.
The functional $\hat{D}_{\Z}(\phi)$ in \eqref{eq: hcan} is an arithmetic
analog of the \emph{Ding functional} $\mathcal{D}_{\phi_{0}}$ in
Kähler geometry (that appears in the proof of \cite[Th.\,2.5]{a-b}
in the case when $\mathcal{D}=0$), defined by
\begin{equation}
\mathcal{D}_{\phi_{0}}(\phi):=-\mathcal{E}(\phi,\phi_{0})/\vol(-K_{(X,\Delta)})-\log\int_{X}\mu_{\phi}\label{eq:Ding functional Kahler}
\end{equation}
with respect to a reference metric $\phi_{0}$ on $-K_{(X,\Delta)}$.

\section{\label{sec:Toric-log-Fano}Toric log Fano varieties}

Recall that an $n$-dimensional complex projective variety $X$ is
said to be\emph{ toric }if the complex torus $T_{\C}:=(\mathbb{C}^{*})^{n}$
acts on $X$ with an open dense orbit \cite{c-l-s}. We can thus view
$X$ is a $T_{\C}$-equivariant compactification of $(\mathbb{C}^{*})^{n}$. A~log pair $(X,\Delta)$ over $\C$ is said to be toric if $X$
and $\Delta$ are toric, \ie if $X$ is toric and the $\Q$-divisor
$\Delta$ is invariant under the torus action on $X$. A~line bundle
$L$ over a complex toric variety $X$ is called toric if is endowed
with a $T_{\C}$-action covering the $T_{\C}$-action on $X$. An
ample toric line bundle $L\to X$ corresponds to a convex
rational polytope $P\subset\R^{n}$, called the \emph{moment polytope},
determined by the following property: for any given positive integer~$k$, the complex vector space $H^{0}(X,kL$ decomposes as follows
under the $T_{\C}$-action,
\begin{equation}
H^{0}(X,kL)\cong\bigoplus_{m\in\Z^{n}\cap kP}\C\chi^{m},\quad \chi^{m}(z)=z^{m}:=z_{1}^{m_{1}}\ldots z_{1}^{m_{1}},z\in(\mathbb{C}^{*})^{n},\label{eq:toric decomp}
\end{equation}
where we have identified $L\to X$ with the trivial line bundle
over the dense open orbit of $T_{\C}$ in $X$ (from an invariant
point of view, the real vector space $\R^{n}$ above arises as $M\otimes_{\Z}\R$,
where $M$ is the lattice $\Hom(T_{c},\C^{*})$
of characters of the group $T_{c}$ \cite{c-l-s}).

Any toric ample line bundle $L$ over a complex projective variety
$X$ admits a canonical integral model $\mathcal{L}\to\mathcal{X}$
with $\mathcal{X}$ normal (see \cite[\S 2]{ma} and \cite[Def.\,3.5.6]{b-g-p-s}). It~is determined by the property that the $\Z-$module $H^{0}(\mathcal{X},k\mathcal{L})$
is generated by the sections corresponding to the characters $\chi^{m}$,
appearing in formula \eqref{eq:toric decomp} (see \cite[Prop.\,2.3.10]{ma}).
Likewise, a toric log pair $(X,\Delta)$ admits a canonical integral
model $(\mathcal{X},\mathcal{D})$. Moreover, if $(X,\Delta)$ is
a log Fano variety, then $(\mathcal{X},\mathcal{D})$ is an arithmetic
log Fano variety (as shown precisely as in the case $\mathcal{D}=0$,
considered in \cite[\S 3.1.6]{a-b}).

In this section we will prove the following result, by~building on
the proof of \hbox{\cite[Th.\,1.2]{a-b}}.

\begin{thm}
\label{thm:main log toric}Let $(\mathcal{X},\mathcal{D})$ be the
canonical integral model of a complex K\nobreakdash-semistable toric log Fano
variety $(X,\Delta)$. Conjecture \ref{conj:height log intro} holds
for $(\mathcal{X},\mathcal{D})$ under anyone of the following conditions:
\begin{itemize}
\item $n\leq3$ and $X$ is $\Q$-factorial (equivalently, $X$ has at worst
abelian quotient singularities);
\item $X$ is not Gorenstein or has some abelian quotient singularity.
\end{itemize}
\end{thm}

We start by introducing some notation, following \cite{ber-ber}.
Given a complex toric log Fano variety $(X,\Delta)$ set $L=-(K_{X}+\Delta)$
and denote by $P$ the corresponding moment polytope in $\R^{n}$.
Then\vspace*{-3pt}
\begin{equation}
P=\{p\in\R^{n}:\left\langle l_{F},p\right\rangle \geq-a_{F},\,\,\forall F\},\label{eq:P in log Fano case}
\end{equation}
where $a_{F}\in{}]0,1]$ (generalizing the Fano case when $a_{F}=1$ for all $F$;
see \cite{ber-ber}) and~$l_{F}$ is a primitive integer vector. As
shown in \cite{ber-ber} $(X,\Delta)$ is K\nobreakdash-semistable if and only if $0$ is
the barycenter of $P$ if and only if the log Ding functional $\mathcal{D}_{\phi_{0}}$
is bounded from below. Moreover, the infimum of $\mathcal{D}_{\phi_{0}}$
is attained at a $T$-invariant psh metric~$\phi$ on~$L$. We will
identify the metric~$\phi$ with a continuous convex function on $\R^{n}$,
as~in~\cite{a-b}. More precisely, on $(\mathbb{C}^{*})^{n}\hto X$,
let $x_{i}=\log(|z_{i}|^{2})$. Trivializing $-(K_{X}+\Delta)$ with
$\frac{\mathrm{d}z_{1}}{z_{1}}\wedge\cdots\wedge\frac{\mathrm{d}z_{n}}{z_{n}}\otimes s_{U}e_{\Delta}$
over \hbox{$U=(\mathbb{C}^{*})^{n}$}, and abusing notation slightly, we
let $\phi(x):=\phi_{U}(z)$ in the chosen trivialization over \hbox{$U=(\mathbb{C}^{*})^{n}$}.
Then $\phi$ as a function of $x$ is a continuous convex function
on $\mathbb{R}^{n}$. Via this recipe we also define a reference metric
from the convex function $\psi_{P}(x):=\sup_{p\in P}\langle p,x\rangle$.
As in \cite[Eq.\,(3.8)]{a-b}, we still have that\vspace*{-3pt}
\[
\mathcal{D}_{\psi_{P}}(\phi)=\int_{P}\phi^{*}dy/V-\log\int_{\R^{n}}e^{-\phi(x)}dx-n\log\pi,\quad V:=\vol(P)
\]
(since the support of $\Delta$ is contained in the complement of
$(\C^{*})^{n}$ in $X)$. Thus the inequality in \cite[Prop.\,3.7]{a-b} generalizes to the canonical toric model $\mathcal{L}$
of $L$ (which coincides with $-\mathcal{K}_{(\mathcal{X},\mathcal{D})})$:\vspace*{-3pt}
\begin{equation}
2\widehat{\vol}_{\chi}\left(-\mathcal{K}_{(\mathcal{X},\mathcal{D})},\phi\right)\leq-\vol(X,\Delta)\log\Bigl(\frac{\vol(X,\Delta)}{(2\pi^{2})^{n}}\Bigr),\label{eq:universal log bound}
\end{equation}
where $\vol(X,\Delta):=\vol(-K_{(X,\Delta})$.

We will first prove Theorem \ref{thm:main log toric} in the case
that $X=\P^{n}$, using the following lemma, formulated in terms of
the divisor $D_{0}$ cut out by the $T_{c}$-invariant element of
$H^{0}(X,-K_{X})$ (given by $\frac{\mathrm{d}z_{1}}{z_{1}}\wedge\cdots\wedge\frac{\mathrm{d}z_{n}}{z_{n}}$
over $(\mathbb{C}^{*})^{n}$). In~other words,\vspace*{-3pt}
\[
D_{0}=\sum_{F}D_{F},
\]
where $D_{F}$ is the irreducible divisor corresponding to the facet
$F$ of the moment polytope corresponding to $X$ (see \cite{ber-ber}).
The lemma is a special case of \cite[Prop.\,3.12]{a-b}.

\begin{lem}
\label{lem:scaled polytope}Let $\mathcal{X}$ be the canonical integral
model of an $n$-dimensional complex K\nobreakdash-semistable toric Fano variety
$X$ and denote by $D_{0}$ the standard anti-canonical divisor on
$X$. Then
\begin{align}
\frac{(\overline{-\mathcal{K}_{(\mathcal{X},(1-t)\mathcal{D}_{0})}})^{n+1}/(n+1)!}{\left(-(K_{X}+(1-t)D_{0})\right)^{n}/n!}&=\frac{(\overline{-\mathcal{K}_{\mathcal{X}}})^{n+1}/(n+1)!}{\left(-K_{X}\right)^{n}/n!}-\frac{1}{2}\log(t^{n}),\\
t^{n}&=\Bigl(\frac{\left(-(K_{X}+(1-t)D_{0})\right)^{n}}{\left(-K_{X}\right)^{n}}\Bigr),
\end{align}
with respect to the volume-normalized Kähler-Einstein
metrics.
\end{lem}

We next deduce the following

\begin{lem}
\label{lem: toric case when X is Pn}Let $(\mathcal{X},\mathcal{D})$
be a toric K\nobreakdash-semistable log Fano variety such that $\mathcal{X}=\P_{\Z}^{n}$.
Then $(\overline{-\mathcal{K}_{(\mathcal{X},\mathcal{D})}})^{n+1}\leq(\overline{-\mathcal{K}_{\P_{\Z}^{n}}})^{n+1}$
with equality if and only if $\mathcal{D}=0$.
\end{lem}

\begin{proof}
First observe that there exists $t\in[0,1]$ such that $\mathcal{D}=(1-t)\mathcal{D}_{0}=:\mathcal{D}_{t}$.
This is a special case of \cite[Cor.\,1.6]{fu2}, which applies to $\P^{n}$,
in any dimension $n$, using that toric log Fano varieties are never
uniformly K\nobreakdash-stable. It~will thus be enough to show that $t\mto(\overline{-\mathcal{K}_{(\P_{\Z}^{n},\mathcal{D}_{t})}})^{n+1}$
is increasing on $[0,1]$ (and thus its maximum is attained at $t=1)$.
By the previous lemma\vspace*{-1pt}
\[
\frac{2(\overline{-\mathcal{K}_{(\P_{\Z}^{n},\mathcal{D}_{t})}})^{n+1}/(n+1)!}{(-K_{\P^{n}})^{n}/n!}=t^{n}2\frac{(\overline{-\mathcal{K}_{\P_{\Z}^{n}}})^{n+1}/(n+1)!}{(-K_{\P^{n}})^{n}/n!}-t^{n}\log(t^{n}).
\]
Differentiating with respect to $(t^{n})$ reveals that the right hand side above
is increasing with respect to $t$ if and only if\vspace*{-3pt}
\[
2\,\frac{(\overline{-\mathcal{K}_{\P_{\Z}^{n}}})^{n+1}/(n+1)!}{(-K_{\P^{n}})^{n}/n!}\geq1.
\]
The latter inequality is indeed satisfied, as follows from the explicit
formula \eqref{eq:expl formul on p n}.
\end{proof}
Combining the universal bound \eqref{eq:universal log bound} with \cite[Lem.\,3.8]{a-b}, all that remains to prove Theorem \ref{thm:main log toric}
is to establish the ``logarithmic gap hypothesis''
\begin{equation}
\vol(X,\Delta)\leq\vol(\P^{n-1}\times\P^{1}),\label{eq:log gap}
\end{equation}
assuming that $X\neq\P^{n}$ and that $(X,\Delta)$ satisfies the
assumptions of Theorem \ref{thm:main log toric}.
\begin{rem}
This is a logarithmic generalization of the ``gap hypothesis'',
discussed in \cite[\S 3.2.1]{a-b} and it seems natural to ask
if it holds for any K\nobreakdash-semistable log Fano variety $(X,\Delta)$ such
that $X\neq\P^{n}?$ For example, by~the proof of \cite[Lem.\,3.1]{a-b},
the logarithmic gap hypothesis holds for K\nobreakdash-semistable products. More
generally, as pointed out by the referee, the logarithmic gap hypothesis
holds when $(X,\Delta)$ admits a morphism to $\P^{1}$ with K\nobreakdash-semistable
general fibers, by~\cite[Cor.\,1.17 ]{c-p} (even without assuming that
the total space $(X,\Delta)$ is K\nobreakdash-semistable).
\end{rem}

\begin{prop}
The logarithmic gap hypothesis holds for all toric K\nobreakdash-semistable log
Fano varieties (manifolds) $(X,\Delta)$ such that $X\neq\P^{n}$
if and only if the following bound holds for all Fano varieties (manifolds) $X\neq\P^{n}$
\begin{equation}
S(X)\leq\vol(\P^{n-1}\times\P^{1}),\;\; S(X):=\sup\{\vol(-(K_{X}+\Delta)):(X,\Delta)\,\text{K\nobreakdash-semistable}\}.\label{eq:sup log}
\end{equation}
The ``logarithmic gap hypothesis'' holds for all log Fano varieties
$(X,\Delta)$ such that~$X$ is $\Q$-factorial and of dimension $n\leq3$
and for any dimensions $n$ if $X$ has some abelian quotient singularity
or if $X$ is not Gorenstein.
\end{prop}

\begin{proof}
Since, trivially, $\vol(X,\Delta)\leq S(X)$ the first equivalence
follows directly from the definitions. Next, let us show the last
statement of the proposition, first assuming that $X$ is singular,
which means that the moment polytope $P$ of $(X,\Delta)$ is ``singular''
in the sense that there exists a vertex of $\partial P$ such that
the corresponding primitive vectors $l_{F_{1}},\dots,l_{F_{n}}$ do
not generate $\Z^{n}$. It~follows from the proof of \cite[Lem.\,3.9]{a-b} that
\[
\vol(P)\leq\frac{1}{2}(n+1)^{n}/n!\leq\vol(\P^{n-1}\times\P^{1})
\]
Indeed, since $a_{F}\leq1$ the first inequality follows from the
inequality \cite[(3.13)]{a-b}, using that $\delta\geq2$, according
to the singularity assumption on $P$ (for the second inequality see
formula \cite[(3.14)]{a-b}). All that remains is thus to show
the bound \eqref{eq:sup log} for $S(X)$ when $n\leq3$ and $X$ is
non-singular. First assume that $n=2$. This means, by~classical classification
results, that $X$ is either $\P^{1}\times\P^{1}$ or the blow-up
$X^{(m)}$ of $\P^{2}$ in $m$ points for $m\leq3$. But $(-K_{X^{(m)}})^{2}=(-K_{\P^{2}})^{2}-m$
and thus $\vol(X_{1})\leq4=\vol(\P^{n-1}\times\P^{1})$,
proving the bound \eqref{eq:sup log}. Finally, consider the case when
$n=3$. Starting with the trivial bound $\vol(X,\Delta)\leq\vol(X)$
it follows the classification \cite{ob} of all non-singular toric
Fano varieties of dimension 3 that it is enough to show that the bound
\eqref{eq:sup log} holds when $X$ is $\P^{3}$ blown-up in one point
or $\P(\mathcal{O}(1)\oplus\mathcal{O}(2))$ (whose degrees are~$56$
and~$62$, respectively). According to the following proposition the
corresponding invariants $S(X)n!$ are, approximately, given by $41.8$
and $30.3$, respectively, which are well below the degree $54$ of
$\P^{2}\times\P^{1}$, as desired.
\end{proof}

\subsection{\label{subsec:The-logarithmic-gap}The invariant $S(X)$
for $n\protect\leq3$}

In the proof above we used the following result.
\begin{prop}
After rounding to the nearest decimal place the invariant $n!S(X)$
(formula \eqref{eq:sup log}) is given by $41.8$ and $30.3$ when $X$
equals $\mathbb{P}^{3}$ blown up in one point and $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(2))$,
respectively.
\end{prop}

\begin{proof}
Given a convex subset $P$ of $\R^{n}$ let
\[
s(P):=\sup\{\vol(P_{0}):P_{0}\subset P,\,b_{P_{0}}=0\},
\]
where $P_{0}$ is a closed subset of $P$ with barycenter $b_{P_{0}}$
at the origin. We will compute $s(P)$ when $P$ is the moment polytope
of the manifolds $X$ appearing in the proposition, showing at the
same time that $s(P)=S(X)$. The moment polytopes $P$ of both $\mathbb{P}^{3}$
blown up in one point and $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(2))$
are of the form a simplex, with a simplex subset removed, by~chopping
off a vertex (see ID 20 and ID7 in the database \cite{ob})). After
a general linear transformation, they are of the form $(a\Delta_{3}-\mathbf{1})\setminus(b\Delta_{3}-\mathbf{1})$,
where~$\Delta_{3}$ is the standard unit simplex in dimension three,
$\mathbf{1}$ is the vector with all entries equal to~$1$, and $a$ and $b$ are positive
real numbers. For $\mathbb{P}^{3}$ blown up in one point we can transform
the moment polytope to $(4\Delta_{3}-\mathbf{1})\setminus(2\Delta_{3}-\mathbf{1})$
and for $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(2))$ we get $(5\Delta_{3}-\mathbf{1})\setminus(\Delta_{3}-\mathbf{1})$. In~the first case, the linear transformation is unimodular, but in
the second case the transformation has determinant $2$. This will
not matter when computing $s(P)$ as long as we correct for the non-unit
determinant. Next we compute the barycenter $b_{P}$ of these polytopes,
a simple task using the explicit barycenter of the standard unit simplex,
$b_{\Delta_{n}}=\mathbf{1}/(n+1)$, and then scaling and linearity
properties of the volume times the barycenter. The barycenter of $(a\Delta_{3}-\mathbf{1})\setminus(b\Delta_{3}-\mathbf{1})$
is given~by
\[
\frac{a^{3}/3!(a/4-1)-b^{3}/3!(b/4-1)}{a^{3}/3!-b^{3}/3!}\,\mathbf{1}.
\]
Next we use a general fact, to be proved in the lemma below, stating
that the closed subset $P'$ of $P$ which maximizes volume, with
the relaxed constraint
\begin{equation}
b_{P'}\cdot\mathbf{1}=0\label{eq:constraint}
\end{equation}
is the one given by $P\cap H$ where $H$ is a half-space with normal
$\mathbf{1}$. In~our case, by~symmetry, this $P'$ automatically
satisfies the stronger constraint $b_{P'}=0$. Moreover, since the
boundary of $P\cap H$ is parallel to a facet of $P$ it corresponds
to a divisor $\Delta$ on $X$ defining a log Fano pair $(X,\Delta)$.
Thus $(X,\Delta)$ is also the K\nobreakdash-semistable log Fano pair realizing
the sup in the definition of $S(X)$, showing that $s(P)=S(X)$. We
can find $H$ by imposing the constraint. We introduce the weight
$w$ such that
\[
P\cap H=((a-w)\Delta_{3}-\mathbf{1})\setminus(b\Delta_{3}-\mathbf{1}).
\]
From here it is clear that if $b_{P'}\cdot\mathbf{1}=0$, then, in
fact, the entire barycenter will vanish and the condition $b_{P'}\cdot\mathbf{1}=0$
turns into the following fourth order polynomial equation for $w$:
\[
(a-w)^{3}/3!((a-w)/4-1)-b^{3}/3!(b/4-1)=0.
\]
The solution $w$ and the corresponding value $s(P)$ for $\mathbb{P}^{3}$
blown up in one point, is given by
\begin{align*}
w&=\frac{2}{3}\biggl(5-\frac{4}{\sqrt[3]{19-3\sqrt{33}}}-\sqrt[3]{19-3\sqrt{33}}\biggr)\\
\tag*{and}
n!s(P)&=n!\vol(P')=((4-w)^{3}-2^{3})\approx41.8
\end{align*}
and for $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(2))$,
\begin{align*}
w&=\biggl(4-\sqrt[3]{\frac{4}{2-\sqrt{2}}}-\sqrt[3]{2(2-\sqrt{2})}\biggr)\\
\tag*{and}
n!S(P)&=\frac{1}{2}n!\vol(P')=\frac{1}{2}((5-w)^{3}-1^{3})\approx30.3,
\end{align*}
where we have corrected for the non-unimodular transformation used
in the second case.
\end{proof}
In the above proof we used the following

\begin{lem}
Let $P$ be a closed subset of $\mathbb{R}^{n}$ with the origin as
an interior point. Given $v\in\mathbb{R}^{n}$ assume that $\int_{P}x\cdot v>0$.
Then the maximum
\[
\max_{Q\subset P:\int_{Q}v\cdot x\mathrm{d\lambda(x)=0}}\int_{Q}\mathrm{d}\lambda
\]
is attained at $Q=P\cap H$ with $H$ a closed half-space with outward
pointing normal $v$. Here $\mathrm{d}\lambda$ is Lebesgue measure.
\end{lem}

\begin{proof}
Without loss of generality we can assume that $v=(0,\dots,0,1)$. Denote
by $(x_{1},x_{2},\dots,x_{n-1},y)$ the coordinates on $\mathbb{R}^{n}$.
Since the origin is an interior point of $P$ and $\int_{P}x\cdot v>0$
there is a closed half-space $H$ as in the lemma satisfying $\int_{P\cap H}y\mathrm{d\lambda=0}$.
Hence, any candidate $Q$ for the maximum in question satisfies $\int_{P\cap H}y\mathrm{d}\lambda=\int_{Q}y\mathrm{d}\lambda$.
Subtracting the left hand side from the right hand side and vice versa
yields $\int_{P\cap H\setminus Q}y\mathrm{d}\lambda=\int_{Q\setminus P\cap H}y\mathrm{d}\lambda$.
Since $\sup_{P\cap H\setminus Q}y\leq\inf_{Q\setminus(P\cap H)}y$
it follows that $\vol(P\cap H\setminus Q)\geq\vol(Q\setminus P\cap H)$,
so that $\vol(P\cap H)\geq\vol(Q),$
as desired.
\end{proof}
In fact, with just a slight variation of the argument above, any maximizer
must be of the special form above and, in~addition, assuming connectedness
of $P$, the maximizer is unique. The proof of the previous proposition
thus reveals that the unique toric divisor $\Delta$ on $X$ realizing
the sup defining the invariant $S(X)$ is a multiple of the prime
divisor $D_{F}$ defined by the zero-section of $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(2))\to\P^{2}$
and hyperplane ``at infinity'' in $\P^{3}$ blown up at the origin
in $\C^{3}\subset\P^{3}$, respectively (\ie the zero-section of $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1))\to\P^{2})$. A~similar argument also applies when $X$ is the blow-up of $\P^{2}$
at the origin in $\C^{2}$ (\ie the first Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1))\to\P^{2})$.
The unique maximizer for the invariant $S(X)$ is then a log Fano
pair $(X,\Delta)$ for a multiple of the hyperplane $D$ ``at infinity''
(\ie the zero-section of $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1))\to\P^{2})$.
Interestingly, this K\nobreakdash-polystable log pair $(X,\Delta)$ was also singled
out in \cite[Cor.\,1.5]{r-z} by the following rigidity property (answering
a question of Cheltsov): it admits a rigid Kähler-Einstein
metric in the sense that for any other multiple $cD$ the log pair
$(X,cD)$ does not admit a Kähler-Einstein metric. The
same rigidity property holds for the two three-dimensional log pairs
discussed above (since there is a unique half-space $H$ satisfying
the constraint in formula \eqref{eq:constraint}).

\subsection{Estimates on the canonical height}

Theorem 1.3 from \cite{a-b} (and its corollary) generalizes directly
to the case of log Fano pairs and their Kähler-Einstein
metrics in any relative dimension $n$ (with the same proof, by~letting
$P$ be the moment polytope corresponding to $(X,\Delta))$:
\[
\frac{1}{2}\vol(X,\Delta)\log\Bigl(\frac{n!m_{n}\pi^{n}}{\vol(X,\Delta)}\Bigr)\leq\frac{h_{\can}(\mathcal{X},\mathcal{D})}{(n+1)!}\leq\frac{1}{2}\vol(X,\Delta)\log\Bigl(\frac{(2\pi)^{n}\pi^{n}}{\vol(X,\Delta)}\Bigr).
\]
Interestingly, Lemma \ref{lem:scaled polytope} reveals that the family
of log Fano pairs $(\mathcal{X},\mathcal{D})$ appearing in the lemma
may be explicitly expressed in terms of the algebro-geometric volume
$\vol(X,\Delta)$ in the same functional form as the one appearing
in the previous upper and lower bounds:
\begin{gather*}
\frac{(\overline{-\mathcal{K}_{(\mathcal{X},\mathcal{D})}})^{n+1}}{(n+1)!}=\frac{1}{2}\vol(X,\Delta)\log\Bigl(\frac{be^{2a}}{\vol(X,\Delta)}\Bigr)\\
\tag*{with}a:=\frac{(\overline{-\mathcal{K}_{\mathcal{X}}})^{n+1}/(n+1)!}{\left(-K_{X}\right)^{n}/n!}\quad\text{and}\quad b=\vol(X).
\end{gather*}

\section{\label{sec:Hyperplane-arrangements}Hyperplane arrangements}

In this section we prove Theorem \ref{thm:hyperplane intro} concerning
hyperplane arrangements. Recall that a log Fano pair $(X,\Delta)$
is called a \emph{log Fano hyperplane arrangement }if $X=\mathbb{P}^{n}$
and $\Delta=\sum_{i=1}^{m}w_{i}H_{i}$ where $w_{i}\in\Q_{>0}$ and
the $H_{i}$ are distinct hyperplanes. Furthermore we will call $(X,\Delta)$
\emph{simple normal crossing}, abbreviated snc, if the support of
$\Delta$ has simple normal crossings.

For an snc log Fano hyperplane arrangement, if $m=n+1$ and all the
weights $w_{i}$ are equal, then $(X,\Delta)$ is a toric log-pair
(see Lemma \ref{lem:scaled polytope}). The following lemma shows
that for given hyperplanes $H_{1},\dots,H_{m}$ and a fixed volume $\vol(X,\Delta)$,
the ``toric'' weights form the vertices of the convex polytope of
all weights $w_{i}$ corresponding to K\nobreakdash-semistable $(X,\Delta)$.

\begin{lem}
Fix $m\geq1$ and a real number $0<D\leq(-K_{\mathbb{P}^{n}})^{n}=(n+1)^{n}$.
Let as before for a real $m$-tuple $w$, $\Delta=\sum_{i=1}^{m}w_{i}H_{i}$
for distinct hyperplanes $H_{i}$. Then the set of weights
\[
S=\{w\in\R^{n}:(-(K_{\mathbb{P}^{n}}+\Delta))^{n}=D\mathrm{\ and\ }(\mathbb{P}^{n},\Delta)\mathrm{\ is\ K}\text{-}\mathrm{semistable}\}
\]
is either empty or $m\geq n+1$ and $S$ is a polytope with $\binom{n}{m}$
vertices given by any reordering of the tuple $w_{1}=w_{2}=\cdots=w_{n+1}=\frac{1}{m}(n+1-D^{1/n})$, $w_{\ell}=0$ $\forall \ell>n+1$.\label{lem: stability polytope}
\end{lem}

\begin{proof}
By \cite{fu2}, for $w\in\mathbb{R}^{n}$ and $\Delta=\sum_{i=1}^{m}w_{i}H_{i}$, $(\mathbb{P}^{n},\Delta)$ is a K\nobreakdash-semistable log Fano pair if and
only if $w$ is in the convex set $C$ defined by the following inequalities:\vspace*{-3pt}
\begin{align}
0 & \leq w_{i}<1\quad \forall i=1,\dots,m\nonumber \\
k & \sum_{i=1}^{m}w_{i}\geq(n+1)\sum_{j=1}^{k}w_{i_{j}}\quad \begin{cases}
\forall k=1,\dots,n,\\
\forall i_{1},\cdots,i_{k}\text{ with }1\leq i_{1}<\cdots<i_{k}\leq m.
\end{cases}
\label{eq:def of C}
\end{align}
Here it should be noted that in fact, it suffices to consider the
second inequality for index combinations $i_{j}$ of length $1$. The
other inequalities for larger index combinations follows. Hence, K\nobreakdash-semistability
of $(\mathbb{P}^{n},\Delta)$ is equivalent to\vspace*{-3pt}
\begin{align}
0 & \leq w_{i}<1\quad \forall i=1,\dots,m,\nonumber \\
w_{i} & \leq\frac{1}{n+1}\sum_{j=1}^{m}w_{j}\quad \forall i=1,\dots,m.\label{eq:weight cond}
\end{align}
Fix $m$ and $D$ as in the statement of the theorem. The goal is
to understand the intersection of the above set with the set $\{w:-(K_{\P^{n}}+\Delta)=D\}$.
Note first that
\[\ts
\bigl(-(K_{\P^{n}}+\sum_{i=1}^{m}w_{i}H_{i})\bigr)^{n}=\bigl(n+1-\sum_{i=1}^{m}w_{i}\bigr)^{n}.
\]
Let $C:=n+1-D^{1/n}$, so that $\{w:-(K_{\P^{n}}+\Delta)=D\}=\{w:\sum_{i=1}^{m}w_{i}=C\}$.
Thus with S defined as in Lemma \ref{lem: stability polytope}\vspace*{-3pt}
\[\ts
S=\biggl\{w:\sum_{i=1}^{m}w_{i}=C,\ \Bigl\{\begin{array}{ll}
0\leq w_{i}<1&\forall i=1,\dots,m\\
w_{i}\leq\spfrac{C}{n+1}&\forall i=1,\dots,m
\end{array}
\biggr\}.
\]
Observe that since $0\leq C<n+1$, the inequality $w_{i}<1$ is superfluous.
After a convenient rescaling we get
\[\ts
\dfrac{n+1}{C}\,S=\{w:\sum_{i=1}^{m}w_{i}=n+1,\
0\leq w_{i}\leq1\ \forall i=1,\dots,m\}.
\]
Clearly if $m<n+1$, $\Psfrac{n+1}{C}S$ is empty. For $m\geq n+1$,
any vertex of $\Psfrac{n+1}{C}S$ is given by the intersection of $\Psfrac{n+1}{C}S$
with some collection of the inequalities put to equality. But clearly
all such points must be of the form of having $n+1$ ones and $m-(n+1)$
zeros. And on the other hand any such point is a vertex.
\end{proof}
Fixing the volume $\vol(X,\Delta)$ is, when $X=\P^{n}$, tantamount
to fixing the isomorphism class of the $\Q$-line bundle $-(K_{X}+\Delta)$
(since the rank of the Picard group of $\P^{n}$ is one). The following
lemma shows that, in~this case, the maximal height is convex with
respect to the weights of $\Delta$.

\begin{lem}
\label{lemma:convexity of height}Consider an arithmetic Fano variety
$\mathcal{X}$ and a curve $t\mto(\mathcal{X},\mathcal{D}_{t})$
of arithmetic log Fano varieties where $\mathcal{D}_{t}=\sum_{i=1}^{m}w_{i}(t)\mathcal{D}_{i}$
for some $m\geq1$, irreducible divisors $\mathcal{D}_{i}$ over $\Z$
and $w:[0,1]\to\R^{m}$ an affine function. Additionally assume
that all the $\mathcal{D}_{t}$ are linearly equivalent, which equivalently
means that $-(\mathcal{K}+\mathcal{D}_{t})$ isomorphic to $\mathcal{L}$
for a line bundle $\mathcal{L}\to\mathcal{X}$ independent
of $t$. Then the function $h:[0,1]\to{}]-\infty,\infty]$ defined
as
\begin{equation}
t\mto h_{\can}(\mathcal{X},\mathcal{D}_{t})\label{eq: convex height equation}
\end{equation}
is strictly convex. Equivalently the function $t\mto\hat{h}_{\can}(\mathcal{X},\mathcal{D}_{t})$
is strictly convex.
\end{lem}

\begin{proof}
By assumption we can identify $-(\mathcal{K}+\mathcal{D}_{t})$ with
$\mathcal{L}$ for a line bundle $\mathcal{L}$ independent of $\phi$.
Thus the height $h_{\phi}(\mathcal{X},\mathcal{D}_{t})$
for a fixed metric on $\mathcal{L}$ is independent of~$t$. Likewise,
$h_{\can}(\mathcal{X},\mathcal{D}_{t})$ coincides with $\hat{h}_{\can}(\mathcal{X},\mathcal{D}_{t})$
up to multiplication by a constant independent of~$t$. Next, express
\[
\hat{h}_{\can}(\mathcal{X},\mathcal{D}_{t})=\sup_{\phi}\hat{h}(\mathcal{X},\mathcal{D}_{t})+\frac{1}{2}\log\int_{X}\mu_{(\phi,\mathcal{D}_{t})},
\]
where the sup ranges over all continuous psh metrics on $\mathcal{L}$.
Introducing an arbitrary volume form $\mathrm{d}V$ on $X$ we can
rewrite
\[
\int_{X}\mu_{(\phi,\mathcal{D}_{t})}=\int_{X}\exp(-\phi-\sum_{i=1}^{m}w_{i}(t)\psi_{D_{i}}-\log\mathrm{d}V)dV.
\]
By Hölder's inequality this expression is convex in $t$, since $w_{i}(t)$
is affine. It~is even strictly convex since the $D_{i}$ are distinct.
This means that $\hat{h}_{\can}(\mathcal{X},\mathcal{D}_{t})$
is the supremum over a set independent of $t$, of a collection of
strictly convex functions and thus is itself, strictly convex.
\end{proof}
The above lemmas will reduce the proof of Theorem \ref{thm:hyperplane intro}
to the case of when the support of $\Delta$ consists of $n+1$ distinct
hyperplanes. The following lemma shows that we can further reduce
to the case when these hyperplanes are the standard toric ones.

\begin{lem}
\label{lem: reduction to the standard toric case}Assume that the
log pair $(\mathbb{P}_{\Z}^{n},\mathcal{D})$ is isomorphic to the
standard toric one $(\mathbb{P}_{\Z}^{n},\mathcal{D}_{0})$ over $\C$.
Then, for any $t\in[0,1]$,
\[
h_{\can}(\P_{\Z}^{n},(1-t)\mathcal{D})\leq h_{\can}(\P_{\Z}^{n},(1-t)\mathcal{D}_{0}).
\]
\end{lem}

\begin{proof}
Denote by $s_{0},\dots,s_{n}$ the integral sections of $\mathcal{O}(1)$
cutting out the irreducible components of $\mathcal{D}$. We can express
$s_{i}:=\sum_{j}A_{ij}x_{j}$ with $A_{ij}\in\Z$, where $(x_{0},\dots,x_{n})$
are the standard affine coordinates on $\C^{n+1}$. It~will be enough
to show that
\begin{equation}
\hat{h}_{\can}(\P_{\Z}^{n},(1-t)\mathcal{D})=\hat{h}_{\can}(\P_{\Z}^{n},(1-t)\mathcal{D}_{0})+(t-1)\log\left|\det A\right|\label{eq:desired id}
\end{equation}
(since $t\leq1$ and $\left|\det A\right|\geq1)$. To this end, denote
by $F$ the invertible $\C$-linear map from $\C_{x}^{n+1}\to\C_{y}^{n+1}$
satisfying
\[\ts
F^{*}(\sum_{j}A_{ij}y_{j})=x_{i}
\]
(the existence of $F$ is equivalent to the invertibility of the matrix
$A$ which, in~turn, is equivalent to the assumption about an isomorphism
over $\C)$. We will use the same symbol $F$ for the induced map
$\P^{n}\to\P^{n}$ and its standard lift to $\mathcal{O}(1)$
(as well as its tensor powers). By basic linear algebra
\[
F^{*}dy_{0}\wedge\cdots\wedge dy_{n}=(\det A)^{-1}dx_{0}\wedge\cdots\wedge dx_{n}.
\]
Now, observe that $-K_{(\P_{\Z}^{n},(1-t)\mathcal{D})}\simeq t\mathcal{O}(n+1)$,
using standard isomorphisms over $\Z$. In~particular, any given metric
$\phi$ on $\mathcal{O}(n+1)$ induces a metric $t\phi$ on $-K_{(\P_{\Z}^{n},(1-t)\mathcal{D})}$
and thus, using the log pair $(\P_{\Z}^{n},(1-t)\mathcal{D})$, a
measure $F^{*}\mu_{t\phi}$ on $\P_{\Z}^{n}$. Note that
\begin{equation}
\int_{\P^{n}}\mu_{t\phi}=\left|\det A\right|^{-2}\int_{\P^{n}}\mu_{tF^{*}\phi},\label{eq:integrals same}
\end{equation}
where $\mu_{tF^{*}\phi}$ is the measure associated to the metric
$tF^{*}\phi$ on $-K_{(\P_{\Z}^{n},(1-t)\mathcal{D}_{0})}$ (now using
the toric log pair $(\P_{\Z}^{n},(1-t)\mathcal{D}_{0}))$. Indeed,
consider the (singular) metric $\psi$ on $(n+1)\mathcal{O}(1)$ defined
by
\[
\psi:=t\phi+(1-t)\sum_{i}\log|s_{i}|^{2}.
\]
The measure $\mu_{t\phi}$ coincides with the measure $\mu_{\psi}$
attached to the metric on $-K_{\P^{n}}$ induced by $\psi$. Hence,
denoting by $G$ the canonical lift of the map $F$ to $-K_{\P^{n}}$,
we~have, by~definition, that
\[
G^{*}\mu_{\psi}=\mu_{G^{*}\psi},
\quad
G^{*}\psi=F^{*}\psi+\log(\left|\det A\right|^{2}),\quad F^{*}\psi:=tF^{*}\phi+(1-t)\sum_{i}\log|F^{*}s_{i}|^{2}.
\]
This proves formula \eqref{eq:integrals same}. Next, note that, by~Lemma \ref{lem:append} in the appendix,
\[
\hat{h}(\mathcal{O}(n+1),G^{*}\phi)=\hat{h}(\mathcal{O}(n+1),\phi).
\]
Hence, we get, as above, that
\begin{align*}
\hat{h}(\mathcal{O}(n+1),F^{*}\phi)&=\hat{h}(\mathcal{O}(n+1),G^{*}\phi-\log(\left|\det A\right|))
\\
&=\hat{h}(\mathcal{O}(n+1),G^{*}\phi)-\log(\left|\det A\right|^{2})\\
&=\hat{h}(\mathcal{O}(n+1),\phi)-\log(\left|\det A\right|)),
\end{align*}
giving
\[
\hat{h}(\mathcal{O}(t(n+1)),t\phi)=\hat{h}(\mathcal{O}(t(n+1)),F^{*})(t\phi))+t\log(\left|\det A\right|).
\]
All in all, this means that
\[
\hat{h}(\mathcal{O}(t(n+1)),t\phi)+\log\int\mu_{t\phi}
=\hat{h}(\mathcal{O}(t(n+1)),F^{*}t\phi)+\log\int\mu_{F^{*}t\phi}+(t-1)\log(\left|\det A\right|).
\]
Taking the sup over all continuous metrics $\phi$ on $\mathcal{O}(n+1)$
with positive curvature thus concludes the proof of the desired identity
\eqref{eq:desired id}.
\end{proof}

\subsection{Conclusion of the proof of Theorem \ref{thm:hyperplane intro}}

In the following we will use the notation $\mathcal{D}_{w}=\sum_{i=1}^{m}w_{i}H_{i}$
for $w\in\mathbb{R}^{m}$ and for fixed hyperplanes $H_{i}$ defined
over~$\Z$. Let $(\P_{\Z}^{n},\mathcal{D}_{w'})$ be a K\nobreakdash-semistable
snc log Fano hyperplane arrangement. Define for brevity $d:=(-(K_{\P^{n}}+\Delta(w')))^{n}$.
Set, as in Lemma \ref{lem: stability polytope},
\[
S=\{w\in\R^{n}:(-(K_{\mathbb{P}^{n}}+\Delta_{w}))^{n}=d\mathrm{\ and\ }(\mathbb{P}_{\Z}^{n},\mathcal{D}_{w})\text{\ is\ K-semistable}\}.
\]
Consider the function $h(w)$ defined by
\[
h(w)=h_{\can}(\P_{\Z}^{n},\mathcal{D}_{w}).
\]
Restricted to the convex set $S$, $h|_{S}$ is convex by Lemma \ref{lemma:convexity of height}.
Next by Lemma \ref{lem: stability polytope}, $S$~is the convex hull
of weight vectors $(w^{k})_{k=1,\dots,\binom{m}{n+1}}$, each corresponding
to toric log Fano pairs, equivalent to $(\mathbb{P}_{\Z}^{n},(1-t)\mathcal{D}_{0})$
over $\C$, where $\mathcal{D}_{0}$ is the toric standard anti-canonical
divisor and $t$ is the unique number such that $(-(K_{\P^{n}}+(1-t)\mathcal{D}_{0})^{n}=d$.
By the convexity of $h$,
\[
h(w')\leq\max_{k}h(w^{k})\leq h_{\can}(\P_{\Z}^{n},(1-t)\mathcal{D}_{0}),
\]
where the second inequality is the content of Lemma \ref{lem: reduction to the standard toric case}.
We have thus reduced to the standard toric case, which we have already
handled. Specifically, the bound \eqref{eq:estimate by toric intro}
follows directly from Lemma \ref{lem:scaled polytope}. For Theorem
\ref{thm:hyperplane intro}, recall that it was observed in the proof
of Lemma \ref{lem: toric case when X is Pn} that the volume dependent
bound in \eqref{eq:estimate by toric intro} is strictly increasing
with volume, so that a universal bound is uniquely given for maximal
volume, \ie when $\Delta=0$, yielding the result.

\section{Diagonal hypersurfaces}

In this section we will deduce Theorem \ref{thm:diagonal hypersurface intro}
from the results in the previous sections. The starting point of the
proof is the following analytic representation of the height:

\begin{lem}[Restriction formula]
\label{lem:(Restriction-formula)-Let}
Let $\mathcal{X}$
be the subscheme of $\P_{\Z}^{n+1}$ cut out by a homogeneous polynomial
$s$ of degree $d$ with integer coefficients and $\phi$ a continuous
psh metric on $\mathcal{O}(d)\to\P_{\C}^{n+1}$. Then the
height $h_{\phi}(\mathcal{X}_{d},\mathcal{O}(d))$ of the restriction
of $(\mathcal{O}(d),\phi)$ to $\mathcal{X}$ may be expressed as
\[
\frac{2h_{\phi}(\mathcal{X}_{d},\mathcal{O}(d))}{(n+1)!}=(n+2)\mathcal{E}(\phi,d\phi_{0})+\int_{\P^{n+1}}\log\bigl(\left\Vert s\right\Vert _{\phi}^{2}\bigr)\frac{(dd^{c}\phi)^{n+1}}{(n+1)!},
\]
where $\phi_{0}$ is the Weil metric on $\mathcal{O}(1)$ and $\mathcal{E}$
is the functional defined by formula \eqref{eq:def of primitive}, corresponding
to $\mathcal{O}(d)\to\P^{n+1}$.
\end{lem}

\begin{proof}
This is well-known, but for completeness we provide a proof. Consider
first the general situation where $\mathcal{X}$ is a subscheme (of
relative dimension $n)$ of a regular projective flat scheme $\mathcal{Y}$
cut out by a section $s$ of a relatively ample line bundle $\mathcal{L}\to\mathcal{Y}$.
Then, given a metric $\phi$ on the complexification $L$ of $\mathcal{L}\to\mathcal{Y}$,
the restriction formula for arithmetic intersection numbers \cite[Prop.\,2.3.1]{b-g-s}
gives
\begin{equation}
(\mathcal{L},\phi)^{n+2}\cdot\mathcal{Y}=(\mathcal{L},\phi)^{n+1}\cdot\mathcal{X}-\int_{Y}\log||s||_{\phi}(dd^{c}\phi)^{n+1}.\label{eq:restriction formula}
\end{equation}
In particular, setting $\mathcal{Y}=\P_{\Z}^{n+1}$ and $\mathcal{L}=\mathcal{O}(d)$
gives
\[
\frac{2h_{\phi}(\mathcal{X}_{d},\mathcal{O}(d))}{(n+1)!}=(n+2)\,\frac{h_{\phi}(\P_{\Z}^{n+1},\mathcal{O}(d))}{(n+2)!}+\int\log||s||_{\phi}^{2}\frac{(dd^{c}\phi)^{n+1}}{(n+1)!}.
\]
The proof is thus concluded by invoking the well-known fact that
\[ h_{\phi}(\P_{\Z}^{n+1},\mathcal{O}(d))/(n+2)!=\mathcal{E}_{\P^{n+1}}(\phi,\phi_{0}).
\]
For example, this is a special case of the toric formula \cite[Eq.\,3.7]{ab}.
\end{proof}
In general, if $\mathcal{X}$ is subscheme of $\P_{\Z}^{n+1}$ of
codimension one, then $\mathcal{K}_{\mathcal{X}}$ is well-defined
as line bundle over $\mathcal{X}$. More precisely, by~the adjunction
formula, there is an isomorphism of line bundles over $\Z$,
\[
\mathcal{K}_{\mathcal{X}}\simeq\bigl(\mathcal{K}_{\P_{\Z}^{n+1}}-\mathcal{O}(\mathcal{I}/\mathcal{I}^{2})\bigr)_{\mathcal{X}},
\]
where $\mathcal{I}$ is the ideal sheaf cutting out $\mathcal{X}$
\cite[Eq.\,1.6.2, p.\,8]{ko}. In~particular, if $\mathcal{X}$
is cut out by a homogeneous polynomial $s$ of degree $d$, then
\begin{equation}
-\mathcal{K}_{\mathcal{X}}\simeq-\mathcal{K}_{\P_{\Z}^{n+1}}-\mathcal{O}(d)\simeq\mathcal{O}(n+2-d).\label{eq:adjunction isomo}
\end{equation}
Hence, $-\mathcal{K}_{\mathcal{X}}$ is relatively ample if and only if $d\leq n+1$.
Now assume that the complex variety $X$ defined by the complex points
of $\mathcal{X}$ is non-singular. Then, by~the adjunction isomorphism
\eqref{eq:adjunction isomo}, a metric $\phi_{0}$ on $\mathcal{O}(n+2-d)|_{X}$
may be identified with a metric on~$-K_{X}$.

\subsection{Reduction to Fermat hypersurfaces}

Given integers $a_{i}$ consider the subscheme $\mathcal{X}_{a}$
of $\P_{\Z}^{n+1}$ cut out by the homogeneous polynomial
\[
s_{a}:=\sum_{i=0}^{n+1}a_{i}x_{i}^{d}.
\]
Denote by $X_{a}$ the corresponding complex variety, which is non-singular
and consider the map
\begin{equation}
F_{a}:\C^{n+2}\to\C^{n+2},\quad x\mto F_{a}(x):=(a_{0}^{-1/d}x_{0},\dots,a_{n+1}^{-1/d}x_{n+1}).\label{eq:def of F a}
\end{equation}
It descends to an automorphism of $\P^{n+1}$ with the property that
\[
X_{a}=F_{a}(X_{1}).
\]
By identifying $\C^{n+2}\setminus\{0\}$ with the complement of the zero-section
of $\mathcal{O}(1)\to\P^{n+1}$, the map $F_{a}$ induces,
for any given integer $m$, an automorphism of $\mathcal{O}(m)\to\P^{n+1}$
(that we also denote by $F_{a})$. In~the statement of the following
proposition we will use that, by~the adjunction isomorphism recalled
above, a metric $\phi$ on $\mathcal{O}(n+2-d)|_{X_{a}}$ may be identified
with a metric on $-K_{X_{a}}$ and thus induces a measure $\mu_{\phi}$
on $X$ (Section~\ref{subsec:Local-representations-of}).
\begin{prop}
\label{prop:red to fermat}Let $k$ be a positive integer and $\phi$
a metric on $\mathcal{O}(n+2-d)|_{X_{a}}$. Then
\begin{multline*}
2\hat{h}_{\phi}(\mathcal{X}_{a},\mathcal{O}(n+2-d))+\log\int_{X_{a}}\mu_{\phi}
\\
=2\hat{h}_{F_{a}^{*}\phi}(\mathcal{X}_{1},\mathcal{O}(n+2-d))+\log\int_{X_{1}}\mu_{F_{a}^{*}\phi}+\Bigl(\frac{n+2-d}{(n+1)}-1\Bigr)d^{-1}\sum_{i}\log(|a_{i}|^{2}).
\end{multline*}
\end{prop}

The proof of the proposition follows from combining the following
two lemmas:

\begin{lem}
\label{lem:transf of height}Let $k$ be a positive integer and $\phi$
a metric on $\mathcal{O}(k)$. Then
\[
2\widehat{h}_{\phi}(\mathcal{X}_{a},\mathcal{O}(k))=2\widehat{h}_{F_{a}^{*}\phi}(\mathcal{X}_{1},\mathcal{O}(k))+\frac{1}{d}\sum_{i}\log(|a_{i}|^{2})\,\frac{k}{(n+1)}.
\]
\end{lem}

\begin{proof}
We will use that any continuous psh metric $\phi$ on $\mathcal{O}(k)|_{X}$
is the restriction of a continuous psh metric on $\mathcal{O}(k)\to\P^{n+1}$,
that we shall denote by the same symbol~$\phi$ \cite{c-g-z}. First
consider the case when $k=d$ and denote by $s_{a}$ the section of
$\mathcal{O}(d)$ cutting out the scheme $\mathcal{X}_{a}$. By the
restriction formula (Lemma \ref{lem:(Restriction-formula)-Let}),
setting $\text{MA}(\psi):=\sfrac{(dd^{c}\psi)^{n+1}}{(n+1)!}$,
\[
\frac{2}{(n+1)!}\,h_{\phi}(\mathcal{X}_{a},\mathcal{O}(k))=(n+2)\mathcal{E}_{\P^{n+1}}(\phi,d\phi_{0})+\int_{\P^{n+1}}\log|s_{a}|_{\phi}^{2}\text{MA}(\phi).
\]
Rewriting
\[
\int_{\P^{n+1}}\log|s_{a}|_{\phi}^{2}MA(\phi)=\int_{\P^{n+1}}\log|(F_{a}^{-1})^{*}s_{1}|_{\phi}^{2}\text{MA}(\phi)=\int_{\P^{n+1}}\log|s_{1}|_{F_{a}^{*}\phi}^{2}\text{MA}(F_{a}^{*}\phi),
\]
thus reveals that
\begin{multline*}
\frac{2}{(n+1)!}\,h_{F_{a}^{*}\phi}(X_{1},\mathcal{O}(k))-\frac{2}{(n+1)!}\,h_{\phi}(X_{a},\mathcal{O}(k))\\
=(n+2)\bigl(\mathcal{E}_{\P^{n+1}}(F_{a}^{*}\phi,d\phi_{0})-\mathcal{E}_{\P^{n+1}}(\phi,d\phi_{0})\bigr).
\end{multline*}
Now, denote by $G_{a}$ the standard lift of $F_{a}$ from $X_{1}$
to $-K_{X_{1}}$ and its tensor powers, which has the property that
$F_{a}^{*}\mu_{\phi}=\mu_{G_{a}^{*}\phi}$ for any metric $\phi$
on $-K_{X_{a}}$. We can then express
\[
G_{a}^{*}\phi=F_{a}^{*}\phi+c_{a},\quad c_{a}:=\frac{k}{(n+2)}\frac{1}{d}\sum_{i}\log(|a_{i}|^{2}).
\]
Indeed,
\begin{equation}
G_{a}^{*}(e^{-\frac{n+2}{k}\phi}dzd\bar{z})=(e^{-\frac{n+2}{k}F_{a}^{*}\phi})F_{a}^{*}(dzd\bar{z})=(e^{-\frac{n+2}{k}F_{a}^{*}\phi})\prod_{i}|a_{i}|^{-2/d}(dzd\bar{z}).\label{eq:pull-back volume form}
\end{equation}
Hence,
\[
2h_{F_{a}^{*}\phi}(\P^{n+1},\mathcal{O}(k))=2h_{G_{a}^{*}\phi-c_{a}}(\P^{n+1},\mathcal{O}(k))=2h_{G_{a}^{*}\phi}(\P^{n+1},\mathcal{O}(k))-c_{a}\,\frac{k^{n+1}}{(n+1)!}.
\]
But, by~Lemma \ref{lem:append} in the appendix,
\[
\mathcal{E}_{\P^{n+1}}(G_{a}^{*}\phi,d\phi_{0})=\mathcal{E}_{\P^{n+1}}(\phi,d\phi_{0}).
\]
Hence,
\begin{multline*}
\frac{2}{(n+1)!}\,h_{F_{a}^{*}\phi}(X_{1},\mathcal{O}(k))-\frac{2}{(n+1)!}\,h_{\phi}(X_{a},\mathcal{O}(k))\\
=(n+2)\left(\mathcal{E}_{\P^{n+1}}(F_{a}^{*}\phi,d\phi_{0})-\mathcal{E}_{\P^{n+1}}(\phi,d\phi_{0})\right)=-(n+2)c_{a}\,\frac{k^{(n+1)}}{(n+1)!}.
\end{multline*}
As a consequence,
\begin{align*}
2\hat{h}_{F_{a}^{*}\phi}(X_{1},\mathcal{O}(k))-2\hat{h}_{\phi}(X_{a},\mathcal{O}(k))&=-(n+2)c_{a}\,\frac{k^{n+1}}{(n+1)!}\cdot\frac{1}{dk^{n}/n!}
\\
=-(n+2)c_{a}\,\frac{k}{d(n+1)}&=-\frac{1}{d}\sum_{i}\log(|a_{i}|^{2})(n+2)\frac{k}{(n+2)}\,\frac{k}{d(n+1)}
\\
&=-\frac{1}{d}\sum_{i}\log(|a_{i}|^{2})\,\frac{k^{2}}{d(n+1)}.
\end{align*}
Since we have assumed that $k=d$ this means that
\[
2\widehat{h}_{\phi}(X_{a},\mathcal{O}(k))-2\widehat{h}_{F_{a}^{*}\phi}(X_{1},\mathcal{O}(k))=\frac{1}{d}\sum_{i}\log(|a_{i}|^{2})\,\frac{d}{(n+1)}.
\]
Finally, for any given integer $k$ we can express $k=d\lambda$ for
$\lambda=k/d$ and use the basic scaling property
\[
\widehat{h}_{\lambda\phi}(\mathcal{X},\lambda\mathcal{L})=\lambda\widehat{h}_{\phi}(\mathcal{X},\mathcal{L}),
\]
to get
\[
2\widehat{h}_{\phi}(X_{a},\mathcal{O}(k))-2\widehat{h}_{F_{a}^{*}\phi}(X_{1},\mathcal{O}(k))=\frac{1}{d}\sum_{i}\log(|a_{i}|^{2})\,\frac{k}{(n+1)},
\]
which concludes the proof.
\end{proof}

\begin{lem}
\label{lem:scaling of integral}Given a metric $\phi$ on $\mathcal{O}(n+2-d)$
we have that
\[
\log\int_{X_{a}}\mu_{\phi}=\log\int_{X_{1}}\mu_{F_{a}^{*}\phi}-d^{-1}\sum_{i}\log(|a_{i}|^{2}).
\]
\end{lem}

\begin{proof}
Let $X$ be the non-singular hypersurface of $\P^{n+1}$ cut out by
a given homogeneous polynomial $s$ in $\C^{n+2}$. Let $A_{X}$ be
the zero-locus of $s$ in $\C^{n+2}\setminus\{0\}$ and assume that $ds\neq0$
on $A_{X}$. Let $\Omega_{s}$ be the holomorphic top form on $A_{X}$
defined by the relation $\Omega_{s}\wedge ds=dz$ on $A_{X}$, where
$dz:=dz_{0}\wedge\cdots\wedge dz_{n+1}$. We can identify $X$ with
$A_{X}/\C^{*}$ using the standard $\C^{*}$-action on $\C^{n+2}$.
Denote by $\delta$ interior multiplication with the holomorphic vector
field generating the $\C^{*}$-action. Assume now that $X$ is Fano. A~given metric $\phi$ on $-K_{X}$ then corresponds to a one-homogeneous
function $r$ on $A_{X}$ (using the adjunction isomorphism \eqref{eq:adjunction isomo}
and by identifying $\C^{n+2}\setminus\{0\}$ with the complement of the zero-section
in $\mathcal{O}(1)^{*}\to X)$. Moreover, lifting the adjunction
isomorphism \eqref{eq:adjunction isomo} to $A_{X}$ yields the following
well-known formula (which applies in the general setup of Fano varieties
over local fields; cf. \cite[Lem.\,4.2.2]{pey}):
\[
\int_{X}\mu_{\phi}=c\int_{\{s=0\}/\C^{*}}r^{d-(n+2)}(\delta\Omega_{s}\wedge\overline{\delta\Omega_{s}})
\]
for a non-zero constant $c$ only depending on $n$ and $d$ (since
$\Omega_{s}$ has degree $(n+2-d)$ with respect to the $\C^{*}$-action, the
$(n,n)$-form $r^{d-(n+2)}(\delta\Omega_{s}\wedge\overline{\delta\Omega_{s}})$
is $\C^{*}$-invariant and thus descends to a real top form on $X)$.
Hence, setting $F:=F_{a}$,
\[
\int_{X}\mu_{\phi}=c\int_{\{F^{*}s=0\}/\C^{*}}(F^{*}r)^{d-(n+2)}(\delta F^{*}\Omega_{s}\wedge\overline{\delta F^{*}\Omega_{s}}).
\]
To conclude the proof it will thus be enough to verify that
\[
F^{*}\Omega_{s}=\Omega_{F^{*}s}a_{0}^{-1/d}\cdots a_{n+1}^{-1/d}.
\]
To this end, note that applying $F^{*}$ to the defining relation
for $\Omega_{s}$ yields $F^{*}\Omega_{s}\wedge\nobreak d(F^{*}f)\allowbreak=F^{*}dz$.
Since $F^{*}dz=a_{0}^{-1/d}\cdots a_{n+1}^{-1/d}dz$ this concludes
the proof.
\end{proof}
It follows directly from Proposition \ref{prop:red to fermat} and
the definition of $h_{\can}(\mathcal{X})$ that
\begin{equation}
h_{\can}(\mathcal{X}_{a})-h_{\can}(\mathcal{X}_{1})=(n+1)(n+2-d)^{n}d\Bigl(\frac{n+2-d}{(n+1)}-1\Bigr)d^{-1}\sum_{i}\log(|a_{i}|^{2}).\label{eq:canonical height of X a}
\end{equation}
In particular, since $n+2-d\leq n+1$, this means that
\begin{equation}
h_{\can}(\mathcal{X}_{a})\leq h_{\can}(\mathcal{X}_{1})\label{eq:ineq X a},
\end{equation}
and thus the proof of Theorem \ref{thm:diagonal hypersurface intro}
is reduced to the case of the Fermat hypersurface~$\mathcal{X}_{1}$.
\begin{rem}
For any sufficiently large finite field extension $\F$ of $\Q$,
the schemes $\mathcal{X}_{a}\otimes_{\Z}\F$ and $\mathcal{X}_{1}\otimes_{\Z}\F$
are isomorphic over $\F$ and, in~particular, over $\C$ -- the
isomorphism is provided by the map $F_{a}$. Moreover, since $X_{1}$
admits a Kähler-Einstein metric \hbox{\cite[\S 6.3]{ti}},
$h_{\can}(\mathcal{X}_{1})=h_{\phi_{1}}(\mathcal{X}_{1})$,
for any volume-normalized Kähler-Einstein metric $\phi_{\text{1}}$
on $-K_{X_{1}}$. Denoting by $\phi_{a}$ the corresponding metric
on $-K_{X_{a}}$, induced by the natural lift $G_{a}$ of $F_{a}$
from $-K_{X_{1}}$ to $-K_{X_{a}}$, the identity \eqref{eq:canonical height of X a}
thus holds for $h_{\phi_{a}}(\mathcal{X}_{a})-h_{\phi_{1}}(\mathcal{X}_{1})$
(this also follows directly from Proposition \ref{prop:red to fermat}).
Such a difference~is, by~general principles \cite[Lem.\,2.4]{a-b2},
a purely arithmetic quantity (independent of the choice of metric~$\phi_{1})$, since $G_{a}$ induces an isomorphism $\mathcal{K}_{\mathcal{X}_{a}}\otimes_{\Z}\F\simeq\mathcal{K}_{\mathcal{X}_{a}}\otimes_{\Z}\F$. It~seems thus natural to ask if there is a direct scheme-theoretic
proof of the identity \eqref{eq:canonical height of X a}? Interestingly,
the corresponding inequality \eqref{eq:ineq X a} is reminiscent of
Odaka's minimization conjecture \cite{o,h-o}. But an important difference
is that $h_{\can}(\mathcal{X}_{1})$ does not, in~general,
maximize $h_{\can}(\mathcal{X})$ over \emph{all} integral
models $\mathcal{X}$ of $\mathcal{X}_{1}\otimes_{\Z}\Q$ (and likewise
when~$\Q$ is replaced by a number field $\F)$. For example, when
$d=2$, $\mathcal{X}_{1}\otimes_{\mathcal{O}_{\F}}\F$ is isomorphic
to $\P_{\F}^{1}$ for $\F=\Q(\sqrt{-1})$, but $h_{\can}(\mathcal{X}_{1}\otimes_{\Z}\mathcal{O}_{\F})<h_{\can}(\P_{\mathcal{O}_{\F}}^{1})$,
as follows from the conclusion of Theorem \ref{thm:hyperplane intro}
(after a base change). Alternatively, the strict inequality follows
from Corollary \ref{cor:Conjecture holds for log Fano surfaces}, applied
to $\Delta=0$, since $\mathcal{X}_{1}\otimes_{\Z}\mathcal{O}_{\F}$
is not isomorphic to $\P_{\mathcal{O}_{\F}}^{1}$ over~$\mathcal{O}_{\F}$.
\end{rem}

Before continuing we also make a final remark.
\begin{rem}
\label{rem:formula with T}Let $\mathcal{X}$ be a hypersurface in
$\P_{\Z}^{n+1}$ cut-out by a homogeneous polynomial~$s$ of degree
$d$ of the form $T^{*}s_{1}$ where $T\in \mathrm{GL}(n+2,\C)$. Then formula
\eqref{eq:canonical height of X a} can be generalized as follows (shown
in essentially the same manner as before):
\[
h_{\can}(\mathcal{X})=h_{\can}(\mathcal{X}_{1})+(n+1)(n+2-d)^{n}d\Bigl(\frac{n+2-d}{(n+1)}-1\Bigr)\log(|\det T|^{2}).
\]
\end{rem}

\subsection{Reduction to log hyperplane arrangements }

Fix a degree $d(\leq n+1)$ and denote by $\mathcal{X}$ the corresponding
Fermat hypersurface. The Fermat hypersurface of degree one will be
denoted by $\mathcal{Y}$. We will next express the canonical height
$h_{\can}(\mathcal{X})$ in terms of the canonical height $h_{\can}(\mathcal{Y},\mathcal{D})$
where $\mathcal{D}$ is the divisor on $\mathcal{Y}$ defined by
\[
\mathcal{D}=(1-1/d)\left[x_{0}=0\right]+\cdots+(1-1/d)\left[x_{n}=0\right]+(1-1/d)\bigl[\ts\sum_{i=0}^{n}x_{i}=0\bigr],
\]
where $x_{i}$ denotes the homogeneous coordinates on $\P_{\Z}^{n+1}$
restricted to $\mathcal{Y}$.
\begin{prop}
\label{prop:red to log hyper}Denote by $\mathcal{X}$ the Fermat
hypersurface of a given degree $m(\leq n+1)$ and by $\mathcal{Y}$
the Fermat hypersurface of degree one, endowed with the divisor $\mathcal{D}$.
Then
\begin{equation}
\hat{h}_{\can}(\mathcal{X})=\hat{h}(\mathcal{Y},\mathcal{D})-\frac{1}{2}\log\frac{V(X)}{V(Y,\Delta)}.\label{eq: height hypersurface to plane arrangement}
\end{equation}
\end{prop}

\begin{proof}
By the adjunction formula we have isomorphisms $-K_{\mathcal{X}}\simeq(n+2-m)\mathcal{O}(1)|_{\mathcal{X}}$
and $-\mathcal{K}_{(\mathcal{Y},\Delta)}\simeq(n+2-m)(1/m)\mathcal{O}(1)|_{\mathcal{Y}}$.
Consider the following morphism:
\[
F:\C^{n+2}\to\C^{n\text{+2}},\quad (x_{0},\dots,x_{n+1})\mto(x_{0}^{m},\dots,x_{n+1}^{m}),
\]
which induces a map $\P^{n+1}\to\P^{n\text{+1}}$ and a lift
to $\mathcal{O}(1)$, which is naturally defined over~$\Z$ and satisfies
$F^{*}\mathcal{O}(1)\simeq m\mathcal{O}(1)$. In~particular, it induces
a morphism
\[
F:\mathcal{X}\to\mathcal{Y},\quad F^{*}(-\mathcal{K}_{(\mathcal{Y},\Delta)})\simeq-K_{\mathcal{X}},
\]
under the adjunction isomorphisms. We recall that, by~basic functorial
properties of heights,
\begin{equation}
\hat{h}(\mathcal{X},F^{*}\mathcal{L})=\hat{h}(\mathcal{Y},\mathcal{L}).\label{eq:normalized height under morphism}
\end{equation}
In fact, in~this case this formula follows directly from the analytic
representation of the height in Lemma \ref{lem:(Restriction-formula)-Let},
using that $F$ preserves the Weil metric $\phi_{0}$. In~particular,
setting $\mathcal{L}:=(n+2-m)(1/m)\mathcal{O}(1)_{|\mathcal{Y}}$
and using the adjunction isomorphisms yields $\hat{h}(\mathcal{X},-K_{\mathcal{X}},F^{*}\phi)=\hat{h}(\mathcal{Y},-\mathcal{K}_{(\mathcal{Y},\Delta)},\phi)$.
Thus, all that remains is to show that
\[
\int_{X}\mu_{F^{*}\phi}=m^{-(n+1)}\int_{Y}\mu_{\phi},
\]
where we have used that $F^{*}\phi$ induces a metric on $-K_{X}$
and $\phi$ induces a metric on $-K_{(Y,\Delta)}$. Since $F$ has
topological degree $m^{(n+1)}$ we have $F_{*}[X_{m}]=m^{(n+1)}[Y]$
as homology classes and thus it will be enough to show that
\[
F^{*}\mu_{\phi}=m^{2(n+1)}\mu_{F^{*}\phi}.
\]
To this end consider the affine piece $\C^{n+1}$ of $\P^{n+1}$ where
$x_{0}\neq0$. Setting $z_{i}=x_{i}/x_{0}$ for $i=1,\dots,n+1)$ we
can, locally, parametrize $X$ by the coordinates $z_{1},\dots,z_{n}$. In~these coordinates a metric $\psi$ on the restriction of $\mathcal{O}(1)$
to $X$ induces, by~the adjunction isomorphism, a metric on $-K_{X}$
and thus a measure on $X$ locally expressed as
\begin{equation}
\mu_{\psi}=\frac{e^{-(n+2-m)\psi}}{\left(m|z_{n+1}|^{m-1}\right)^{2}}\,\frac{i}{2}\,\mathrm{d}z_{1}\wedge\mathrm{d}\bar{z}_{1}\cdots\frac{i}{2}\,\mathrm{d}z_{n}\wedge\mathrm{d}\bar{z}_{n}.\label{eq:mu phi on quadric-1}
\end{equation}
To see this, recall that, by~definition,
\[
\mu_{\psi}:=\left\Vert \mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\right\Vert _{\psi}^{-2}\frac{i}{2}\,\mathrm{d}z_{1}\wedge\mathrm{d}\bar{z}_{1}\cdots\frac{i}{2}\,\mathrm{d}z_{n}\wedge\mathrm{d}\bar{z}_{n}.
\]
In the affine piece $\C^{n+1}$ of $\P^{n+1}$ we can express
\[
s=fx_{0}^{\otimes2},\quad f=1+\sum_{i=1}^{n}z_{i}^{m}+z_{n+1}^{m}.
\]
By the adjunction isomorphism \eqref{eq:adjunction isomo} we have
\[
\left\Vert \mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\right\Vert :=\left\Vert \mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\wedge\mathrm{d}s\right\Vert =\left\Vert \mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\wedge\mathrm{d}f\right\Vert\Vert x_{0}^{\otimes m}\Vert.
\]
Since $\mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\wedge\mathrm{d}f=\mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\wedge\mathrm{d}z_{n+1}\partial f/\partial z_{n+1}$
this means that
\[
\left\Vert \mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\right\Vert ^{2}:=\Bigl|\frac{\partial f}{\partial z_{n+1}}\Bigr|^{2}\left\Vert \mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\wedge\mathrm{d}z_{n+1}\right\Vert\Vert x_{0}^{\otimes m}\Vert =e^{(n+2)\psi}e^{-m\psi},
\]
giving
\begin{equation}
\mu_{\phi}=\frac{e^{-(n+2-m)\psi}}{|\partial f/\partial z_{n+1}|^{2}}\,\frac{i}{2}\,\mathrm{d}z_{1}\wedge\mathrm{d}\bar{z}_{1}\cdots\frac{i}{2}\,\mathrm{d}z_{n}\wedge\mathrm{d}\bar{z}_{n},\label{eq: representation of muphi}
\end{equation}
which proves \eqref{eq:mu phi on quadric-1}. Likewise, we can parametrize
the affine piece of $Y$ by the coordinates $z_{1},\dots,z_{n}$. A~given metric $\phi$ on the restriction of $\mathcal{O}(1)$ to $Y$
induces, by~the adjunction isomorphism a measure (defined with respect
to the divisor $\mathcal{D})$
\begin{multline*}
\mu_{\phi}=e^{-(n+2-m)m^{-1}\phi}\left|z_{n+1}\right|^{-2(1-1/m)}\left|z_{1}\right|^{-2(1-1/m)}\cdots\left|z_{n}\right|^{-2(1-1/m)}\\
{} \cdot\frac{i}{2}\,\mathrm{d}z_{1}\wedge\mathrm{d}\bar{z}_{1}\cdots\frac{i}{2}\,\mathrm{d}z_{n}\wedge\mathrm{d}\bar{z}_{n}.
\end{multline*}
Since $F^{*}z_{i}=z_{i}^{m}$ this means that
\begin{multline*}
F^{*}\mu_{\phi}=e^{-(n+2-m)f^{*}(m^{-1}\phi)}\left|z_{n+1}\right|^{-2(m-1)}\left|z_{1}\right|^{-2(m-1)}\cdots\left|z_{n}\right|^{-2(m-1)}\\
{}\cdot\frac{i}{2}\,\mathrm{d}(z_{1}^{m})\wedge\mathrm{d}(\bar{z}_{1}^{m})\cdots\frac{i}{2}\,\mathrm{d}(z_{n}^{m})\wedge\mathrm{d}(\bar{z}_{n}^{m}).
\end{multline*}
Finally, since $\mathrm{d}(z^{m})=mz^{m-1}$ this proves the desired
identity \eqref{eq: height hypersurface to plane arrangement}, using
the representation \eqref{eq: representation of muphi} with $\psi=F^{*}(m^{-1}\phi)$.
\end{proof}

\subsection{Conclusion of the proof of Theorem \ref{thm:diagonal hypersurface intro}} The affine projection
\[
(x_{0},\dots,x_{n+1})\mto(x_{0},\dots,x_{n})
\]
induces an isomorphism from $\mathcal{Y}$ to $\P_{\Z}^{n}$, identifying
$(\mathcal{Y},\mathcal{D})$ with a hyperplane arrangement $(\P_{\Z}^{n},\mathcal{D})$
with simple normal crossings. It~follows readily from the definition
of~$\mathcal{D}$ and the criterion \eqref{eq:def of C} that $(\P_{\Z}^{n},\mathcal{D})$
is K\nobreakdash-semistable. Hence, combining Proposition \ref{prop:red to log hyper}
with refined bound following the statement of Theorem \ref{thm:hyperplane intro}
yields
\[
\hat{h}_{\can}(\mathcal{X})\leq\hat{h}_{\can}(\P_{\Z}^{n},\mathcal{D}_{t})-\frac{1}{2}\log\frac{V(X)}{V(\P^{n},\Delta_{t})},
\]
where $\mathcal{D}_{t}$ is the toric divisor on $\P_{\Z}^{n}$ such
that $(\P_{\Z}^{n},\mathcal{D}_{t})$ is K\nobreakdash-semistable and $V(\P^{n},\Delta_{t})=V(\P^{n},\Delta)$.
The explicit formula for $\hat{h}_{\can}(\P_{\Z}^{n},\mathcal{D}_{t})$
thus yields
\[
\hat{h}_{\can}(\mathcal{X})\leq\hat{h}_{\can}(\P_{\Z}^{n})-\frac{1}{2}\log\frac{V(X)}{V(\P^{n})}.
\]
Multiplying both sides with $V(X)$ reveals that
\[
h_{\can}(\mathcal{X})\leq\lambda h_{\can}(\P_{\Z}^{n})-\frac{1}{2}V(X)\log\lambda,\qquad \lambda:=V(X)/V(\P^{n}).
\]
Since $\lambda\in{}]0,1[$ it thus follows from Lemma \ref{lem:scaled polytope}
that the right hand side above is increasing with respect to $\lambda$
and thus maximal when $\lambda=1$, giving $h_{\can}(\mathcal{X},\mathcal{D})\leq h_{\can}(\P_{\Z}^{n})$.
Moreover, the equality is strict if $d\geq2$ since then $\lambda<1$.

\section{\label{sec:Arithmetic-surfaces}Arithmetic log surfaces }

In this section we will prove Theorem \ref{thm:Mab on log surface}.
Throughout the section $\mathcal{X}$ will be assumed normal. Given
an effective divisor $\Delta_{\Q}$ on $\P_{\Q}^{1}$ supported at
three points, such that $-K_{(\P^{1},\Delta)}$ is ample, we define
the \emph{canonical model} of $(\P_{\Q}^{1},\Delta_{\Q};-K_{(\P_{\Q}^{1},\Delta_{\Q})})$
over $\Z$ as $(\P_{\Z}^{1},\mathcal{D}_{c};-K_{(\P_{\Z}^{1},\mathcal{D}_{c})})$
where $\mathcal{D}_{c}$ is the Zariski closure of the divisor on
$\P_{\Q}^{1}$ supported on $\{0,1,\infty\}$, having the same coefficients
as $\Delta_{\Q}$. That this is a model over $\Z$ follows from the
basic fact that any three points on $\P_{\Q}^{1}$ can be mapped to
$\{0,1,\infty\}$, by~an automorphism of $\P_{\Q}^{1}$,

As explained in Section \ref{subsec:Arithmetic-log-surfaces}, by~Theorem \ref{thm:hyperplane intro}, the proof is reduced to showing
that for any fixed metric on $-K_{(\P^{1},\Delta)}$:
\begin{itemize}
\item The canonical integral model $(\P_{\Z}^{1},\mathcal{D}_{c};-\mathcal{K}_{(\P_{\Z}^{1},\mathcal{D}_{c})})$
of $(\P_{\Q}^{1},\Delta_{\Q};-K_{(\P_{\Q}^{1},\Delta_{\Q})})$ minimizes
$\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\overline{\mathcal{L}})$
over all integral models $(\mathcal{X},\mathcal{D};\mathcal{L})$
of $(\P_{\Q}^{1},\Delta_{\Q};-K_{(\P_{\Q}^{1},\Delta_{\Q})})$.
\item When $\Delta_{\Q}=0$, the minimum is uniquely attained for $\mathcal{X}=\P_{\Z}^{1}$,
up to isomorphisms over $\Z$.
\end{itemize}

\subsection{Preliminaries on log canonical thresholds}

Following \cite{ko,ta} a log pair $(\mathcal{X},\mathcal{D})$ is
said to be \emph{log canonical (lc)} if for any normal blow-up morphism
$p:\mathcal{Y}\to\mathcal{X}$
\[
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-p^{*}\mathcal{D}=\sum_{i}a_{i}E_{i},\qquad a_{i}\geq-1,\quad \mathcal{K}_{\mathcal{Y}/\mathcal{X}}:=\mathcal{K}_{\mathcal{Y}}-p^{*}\mathcal{K}_{\mathcal{X}},
\]
where the prime divisor $E_{i}$ is either an exceptional divisor
of $p$ or the proper transform of a component of $\mathcal{D}$.
The\emph{ log canonical threshold} of a $\Q-$divisor $F$ on $\mathcal{X}$
with respect to the log pair $(\mathcal{X},\mathcal{D})$ is defined
by
\[
\lct(\mathcal{X},\mathcal{D};F):=\sup_{t>0}\{t:(\mathcal{X},tF+\mathcal{D})\,\text{is lc}\}.
\]
The following lemma follows readily from the definition:

\begin{lem}
\label{Lemma:lct as inf}For any normal blow-up morphism $p:\mathcal{Y}\to\mathcal{X}$
\[
\lct(\mathcal{X},\mathcal{D};F)\leq\inf_{i}\frac{a_{i}-b_{i}+1}{c_{i}},
\]
where $a_{i},b_{i}$ and $c_{i}$ denote the order of vanishing along
the $p$-exceptional prime divisor~$E_{i}$ of $\mathcal{K}_{\mathcal{Y}/\mathcal{X}}$,
$p^{*}\mathcal{D}$ and $p^{*}F$, respectively and $i$ ranges over
all $p$-exceptional prime divisors.
\end{lem}

\subsection{Preparations for the proof of Theorem \ref{thm:Mab on log surface}}

The following result is a logarithmic generalization of \cite[Th.\,2.14\,(3)]{o}
(in the case of arithmetic surfaces).

\begin{lem}
\label{lem:Odaka log alpha }Let $(X,D)$ be a log Fano curve over
$\C$ and $(\mathcal{X},\mathcal{D})$ an arithmetic log Fano model
for $(X,D)$ such the fibers $\mathcal{X}_{b}$ of $\mathcal{X}$
are reduced and irreducible and the divisor $\mathcal{D}$ is horizontal
(\ie $\mathcal{D}$ is the Zariski closure of $D)$. Assume that
\[
\alpha(\mathcal{X},\mathcal{D}):=\inf_{b,F}\lct(\mathcal{X},\mathcal{D}+\mathcal{X}_{b};\,F)\geq1/2,
\]
where the inf runs over all effective $\Q-$divisors $F$ on $\mathcal{X}$
linearly equivalent to $-\mathcal{K}_{(\mathcal{X},\mathcal{D})}$
and closed points $b$ in the base $\mathcal{B}:=\Spec\Z$
such that $F$ does not contain the support of $\mathcal{X}_{b}$.
Then
\[
\frac{1}{2}\overline{\mathcal{L}}^{2}+\overline{\mathcal{K}}_{(\mathcal{X}',\mathcal{D})}\cdot\overline{\mathcal{L}}\geq\frac{1}{2}\overline{\mathcal{L}}^{2}+\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}\cdot\overline{\mathcal{L}}
\]
for any relatively ample model $(\mathcal{X}',\mathcal{D}';\mathcal{L}')$
of $(X,D;-K_{(X,D)})$ and given metrics on $-K_{(X,D)}$ and $L$.
\end{lem}

In the proof we fix once and for all metrics on $-K_{(X,D)}$ and
$L$ and set
\begin{equation}
\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\mathcal{L}):=\frac{1}{2}\overline{\mathcal{L}}^{2}+\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}\cdot\overline{\mathcal{L}}\label{eq:more general log Mab}
\end{equation}
for the corresponding metrized lines bundles $\overline{\mathcal{L}}$
and $\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}$. Thus $\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\mathcal{L})$
specializes to the arithmetic log Mabuchi functional $\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\overline{\mathcal{L}})$
(formula \eqref{eq:def of log Mab surface intro}) precisely when the
metric on $\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}$ is
the one induced from the curvature form $\omega$ of $\overline{\mathcal{L}}$.
But here it will be convenient to consider the present more general
setup.

\subsubsection*{Proof when $D$ is trivial}

To fix ideas we first consider the case when $D$ is trivial. Set
$\mathcal{B}:=\Spec\Z$ and $\mathcal{L}:=-\mathcal{K}_{\mathcal{X}}$.
To simplify the notation we will remove the bar indicating the metric
in the notation for the arithmetic intersection numbers. Anyhow, all
the arithmetic intersections will be computed over the closed points
$b$ in the base~$\mathcal{B}$ and are thus independent of the choice
of metric (since they are proportional to the algebraic intersections
on the scheme $\pi^{-1}(b)$ over the residue field of $b$). If $F_{1}$
and $F_{2}$ are $\Q$-divisors we will write $F_{1}\geq F_{2}$ if
$F_{1}-F_{2}$ is effective.

\subsubsection*{Step 1} \emph{It is enough to consider the case of a relatively
semi-ample model of the form $(\mathcal{X}',\mathcal{L}')=(\mathcal{Y},p^{*}\mathcal{L}-E)$
where $p:\mathcal{Y}\to\mathcal{X}$ is the blow-up along
a closed subscheme~$\mathcal{Z}$ of $\mathcal{X}$ and $E$ is an
effective $p$-exceptional divisor on $\mathcal{Y}$ whose support
contains all the $p$-exceptional prime divisors and such that for
any $b\in\pi(\mathcal{Z})$ $p^{*}\mathcal{L}-E$ admits a global
section $s_{b}$ not vanishing identically along $\mathcal{Y}_{b}$.}

\smallskip
This is shown precisely as in the proof of \cite[Prop.\,3.10]{od0} -- for completeness, a proof is given in Step 1 in Section \ref{subsec:Conclusion-of-the proof log surface}
below.

\subsubsection*{Step 2: The inequality holds in the case of Step 1}

First observe that
\begin{equation}
\mathcal{M}_{\mathcal{Y}}(\mathcal{L}')-\mathcal{M}_{\mathcal{X}}(\mathcal{L})=\mathcal{L}'\cdot\Bigl(\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-\frac{1}{2}E\Bigr).\label{eq:difference of Mab}
\end{equation}
Indeed, rewriting
\[
\mathcal{M}_{\mathcal{X}}(\mathcal{L})=-\frac{\mathcal{L}^{2}}{2}+\cdot\mathcal{L}\cdot(\mathcal{L}+\mathcal{K}_{\mathcal{X}})
\]
(and likewise for $(\mathcal{Y},\mathcal{L}')$) the left hand side
in formula \eqref{eq:difference of Mab} may be expressed as
\[
\frac{p^{*}\mathcal{L}^{2}-\mathcal{L}'^{2}}{2}+\mathcal{L}'\cdot(\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-E)=\frac{E\cdot(p^{*}\mathcal{L}+\mathcal{L}')}{2}+\mathcal{L}'\cdot(\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-E).
\]
Since $E\cdot p^{*}\mathcal{L}=0$ this proves formula \eqref{eq:difference of Mab}.

Since $\mathcal{L}'$ is relatively semi-ample it will thus be enough
to show that the vertical exceptional divisor $\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-\frac{1}{2}E$
is effective. This means, by~the assumption on \hbox{$\alpha(\mathcal{X})\,(:=\alpha(\mathcal{X},0))$},
that it is enough to show that
\begin{equation}
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-\alpha(\mathcal{X})E\geq0.\label{eq:K-alphaE is nonneg}
\end{equation}
To fix ideas first assume that $\pi(\mathcal{Z})$ is supported on
a single point that we denote by $b$. By Step 1, we can express $s_{b}=p^{*}s$
for a global section $s$ of $\mathcal{L}\to\mathcal{X}$
whose zero-divisor~$F$ does not vanish identically on $\mathcal{X}_{b}$
and such that $p^{*}F-E$ is effective. Since $F$ is a contender
for the inf defining $\alpha(\mathcal{X})$ we have
\[
\alpha(\mathcal{X})\leq\lct(\mathcal{X},\mathcal{X}_{b};\,F),
\]
Next, since $p^{*}F\geq E$ it follows from Lemma \ref{Lemma:lct as inf}
that
\[
\lct(\mathcal{X},\mathcal{X}_{b};\,F)\leq\inf_{i}\frac{a_{i}+1-b_{i}}{c_{i}},
\]
where $i$ runs over the $p$-exceptional irreducible prime divisors
$E_{i}$ of $\mathcal{Y}$ and $a_{i}$,~$b_{i}$ and~$c_{i}$ denote
the order of vanishing along $E_{i}$ of $\mathcal{K}_{\mathcal{Y}/\mathcal{X}}$,
$\mathcal{Y}_{b}$ and $E$ respectively. Note that $c_{i}>0$ (since
the support of $E$ contains the support of all $p$-exceptional divisors)
and $b_{i}\geq1$ (since $\mathcal{Z}$ is assumed to be supported
in $\mathcal{X}_{b})$. Thus
\[
\alpha(\mathcal{X})\leq\inf_{i}\frac{a_{i}+1-b_{i}}{c_{i}}\leq\inf_{i}\frac{a_{i}}{c_{i}},
\]
giving
\[
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-\alpha(\mathcal{X})E\geq\sum_{i}a_{i}E_{i}-\left(\ts\min_{j}\sfrac{a_{j}}{c_{j}}\right)E_{i}=\sum_{i}\left(\ts\sfrac{a_{i}}{c_{i}}-\left(\min_{j}\sfrac{a_{j}}{c_{j}}\right)\right)c_{i}E_{i}\geq0,
\]
which proves \eqref{eq:K-alphaE is nonneg}. Finally, consider the general
case when the support of $\pi(\mathcal{Z})$ consists of a finite
number of points $b_{m}$ in $\mathcal{B}$. We then split the vertical
divisors $\mathcal{K}_{\mathcal{Y}/\mathcal{X}}$ and $E$ into the
components $\mathcal{K}_{\mathcal{Y}/\mathcal{X}}^{(m)}$ and $E^{(m)}$
over $b_{m}$:
\[
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-\alpha(\mathcal{X})E=\sum_{m}\mathcal{K}_{\mathcal{Y}/\mathcal{X}}^{(m)}-\alpha(\mathcal{X})E^{(m)},
\]
and apply the previous bound for each fixed $m$ (with $b$ replaced
by $b_{m})$ to get that $\mathcal{K}_{\mathcal{Y}/\mathcal{X}}^{(m)}-\alpha(\mathcal{X})E^{(m)}\geq0$
and thus $\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-\alpha(\mathcal{X})E\geq0$,
as desired.

\subsubsection*{Proof for log pairs }

Just as in the previous case it is enough to consider the special
case of Step 2. In~this case formula \eqref{eq:difference of Mab} readily
generalizes to
\begin{equation}
\mathcal{M}_{(\mathcal{Y},q^{*}\mathcal{D}')}(\mathcal{L}')-\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\mathcal{L})=\mathcal{L}'\cdot\Bigl(\mathcal{D}'-p^{*}\mathcal{D}+\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-\frac{1}{2}E\Bigr).\label{eq:difference Mab}
\end{equation}
As before it will thus be enough to show that
\begin{equation}
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}+q^{*}\mathcal{D}'-p^{*}\mathcal{D}-\alpha(\mathcal{X},\mathcal{D})E\geq0.\label{eq:log K minus alpha E}
\end{equation}
To simplify the exposition we will assume that $\pi(\mathcal{Z})$
is a single closed point in $\mathcal{B}$, denoted by $b$ (the general
case is shown in a similar way by decomposing with respect to the components of
$\pi(\mathcal{Z})$ as above). By the definition of $\alpha(\mathcal{X},\mathcal{D})$,
\begin{equation}
\alpha(\mathcal{X},\mathcal{D})\leq\lct(\mathcal{X},\mathcal{D}+\mathcal{X}_{b};\,F),\label{eq:upper bound on log alpha}
\end{equation}
Next, since $p^{*}F-E$ is effective, \ie $p^{*}F\geq E$, Lemma \ref{Lemma:lct as inf}
yields
\[
\lct(\mathcal{X},\mathcal{D}+\mathcal{X}_{b};\,F)\leq\inf_{i}\frac{a_{i}+1-d_{i}-b_{i}}{c_{i}},
\]
where $a_{i}$, $b_{i}$, $c_{i}$ and $d_{i}$ are the order of vanishing
along $E_{i}$ of $\mathcal{K}_{\mathcal{Y}/\mathcal{X}}$, $\mathcal{Y}_{b}$,
$E$ and $p^{*}\mathcal{D}$ respectively. In~particular, $b_{i}\geq1$
since $\mathcal{Z}$ is supported in $\mathcal{X}_{b}$, Hence,
\[
\alpha(\mathcal{X},\mathcal{D})\leq\inf_{i}\frac{a_{i}+1-d_{i}-b_{i}}{c_{i}}.
\]
Next, we may decompose
\[
p^{*}\mathcal{D}=\left(p^{*}\mathcal{D}\right)_{\hor}+(p^{*}\mathcal{D})_{\text{ex}},
\]
where $\left(p^{*}\mathcal{D}\right)_{\hor}$ is the horizontal
divisor obtained as the proper transform of the horizontal divisor
$\mathcal{D}$ and $(p^{*}\mathcal{D})_{\text{ex}}$ is $p$-exceptional.
By \eqref{eq:upper bound on log alpha}
\begin{align*}
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-(p^{*}\mathcal{D})_{\text{ex}}-\alpha E_{\text{ex}}&\geq\sum_{i}(a_{i}-d_{i})E_{i}-\Bigl(\min_{j}\frac{a_{j}-b_{j}+1-d_{j}}{c_{j}}\Bigr)c_{i}E_{i}
\\
&\geq\sum_{i}\Bigl(\frac{(a_{i}-d_{i})}{c_{i}}E_{i}-\Bigl(\min_{j}\frac{a_{j}-d_{j}}{c_{j}}\Bigr)\Bigr)c_{i}E_{i}\geq0,
\end{align*}
using that $b_{j}\geq1$. Hence,
\[
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}+\mathcal{D}'-p^{*}\mathcal{D}-\alpha(\mathcal{X},\mathcal{D})E_{\text{ex}}\geq\mathcal{D}'-(p^{*}\mathcal{D})_{\hor}.
\]
But, since both $(\mathcal{X}',\mathcal{D}')$ and $(\mathcal{X},\mathcal{D})$
are models for $(X,\Delta)$ and $\mathcal{D}$ is assumed horizontal
it follows that $q^{*}\mathcal{D}'-\left(p^{*}\mathcal{D}\right)_{\hor}$
is an effective vertical divisor and hence
\[
q^{*}\mathcal{D}'-\left(p^{*}\mathcal{D}\right)_{\hor}\geq0,
\]
which concludes the proof of the inequality \eqref{eq:log K minus alpha E}.

\begin{lem}
\label{lem:alpha geq one half}If $(X,\Delta)$ is a K\nobreakdash-semistable
log Fano curve over $\C$, then the canonical model $(\P_{\Z}^{1},\mathcal{D}_{c})$
of $(X,\Delta)$ satisfies
\[
\alpha(\mathcal{X},\mathcal{D})\geq1/2,
\]
and the inequality is strict if $(X,\Delta)$ is K\nobreakdash-stable.
\end{lem}

\begin{proof}
By inversion of adjunction on surfaces over excellent schemes \cite{ta}
\[
\lct(\mathcal{X},\mathcal{D}+\mathcal{X}_{b};\,F)=\lct(\mathcal{X},\mathcal{D}_{|\mathcal{X}_{b}};F_{|\mathcal{X}_{b}}),
\]
if $F$ does not contain the support of the divisor $\mathcal{X}_{b}$. In~the present case $\mathcal{X}_{b}=\P_{\F_{b}}^{1}$, where $b$
has been identified with a prime number and $\F_{b}$ denotes the
field with $b$ elements. Decomposing
\[
\mathcal{D}_{c}=\sum w_{i}\mathcal{D}_{i},
\]
the K\nobreakdash-semistability assumption is, by~\eqref{eq:weight cond}, equivalent
to the condition
\begin{equation}
w_{j}\leq\frac{1}{2}\sum_{i}w_{i},\quad \forall j.\label{eq:weight cond in pf log alpha geq half}
\end{equation}
We recall that for any curve $C$ over a perfect field (here taken
to be $\P_{\F_{b}}^{1}$) an effective $\Q$-divisor $F$ on $C$
is lc if and only if all its coefficients are less then are equal to one \cite{ko,ta}.
Since $-K_{\P_{\F_{b}}^{1}}$ is linearly equivalent to $\mathcal{O}(2)$
and $D_{i|\mathcal{X}_{b}}$ is linearly equivalent to $\mathcal{O}(1)$
it thus follows from the weight condition \eqref{eq:weight cond in pf log alpha geq half}
that $\lct(\mathcal{X},\mathcal{D}_{|\mathcal{X}_{b}};F_{|\mathcal{X}_{b}})\geq1/2$.
Indeed, it is enough to consider the case when $F=(2-\sum_{i}w_{i})[x]$,
where $[x]$ denotes the prime divisor on $\P_{\F_{b}}^{1}$ corresponding to a closed point $x$ in $\P_{\F_{b}}^{1}$. Then\vspace*{-3pt}
\[
\frac{1}{2}F+\mathcal{D}_{|\mathcal{X}_{b}}=\Bigl(1-\frac{1}{2}\sum_{i}w_{i}\Bigr)[x]+\sum w_{i}[x_{i}].
\]
The definition of $\mathcal{D}_{c}$ ensures that $[x_{i}]=[x_{j}]$
if and only if $i=j$. In~the case that $x\neq x_{i}$ for any $i$ the coefficients
of $F/2+\mathcal{D}_{|\mathcal{X}_{b}}$ are indeed less than or equal
to $1$, since $w_{i}\in[0,1]$. Moreover, if $x=x_{j}$ then the
coefficient of index $j$ equals $(1-\frac{1}{2}\sum_{i}w_{i})+w_{j}$
which is less than are equal to $1$, by~the weight condition \eqref{eq:weight cond in pf log alpha geq half}.

We will also need the following lemma, shown precisely as in the case
when $\mathcal{D}=0$ considered in \cite[Prop.\,5.3]{ab}.
\end{proof}

\begin{lem}
\label{lem:inf is inf}Let $(\mathcal{X},\mathcal{D};\mathcal{L})$
be a polarized arithmetic log surfaces $(\mathcal{X},\mathcal{D};\mathcal{L})$
such that the complexification $(X,\Delta)$ of $(\mathcal{X},\mathcal{D})$
is a log Fano variety and $\mathcal{L}\otimes\C=-K_{(X,\Delta)}$. A~metric realizes the infimum\vspace*{-3pt}
\[
\inf_{\left\Vert \cdot\right\Vert }\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\mathcal{L},\left\Vert \cdot\right\Vert)
\]
over all locally bounded metrics on $-(K_{(X,D)})$ with positive
curvature current if and only if the metric is a log Kähler-Einstein
metric. In~particular, in~the case when $D=0$ any minimizer coincides
with the Fubini-Study metric up to the application of an automorphism
of $X$ and a scaling of the metric. Moreover,\vspace*{-3pt}
\[
\inf_{\left\Vert \cdot\right\Vert }\mathcal{M}_{(\P_{\Z}^{1},\mathcal{D})}\bigl(-\mathcal{K}_{(\P_{\Z}^{1},\mathcal{D})},\left\Vert \cdot\right\Vert \bigr)=-\sup_{\left\Vert \cdot\right\Vert }\frac{1}{2}\bigl(-\mathcal{K}_{(\P_{\Z}^{1},\mathcal{D})},\left\Vert \cdot\right\Vert \bigr)^{2},
\]
where the sup in the right-hand side is restricted to volume-normalized
metrics.
\end{lem}

\subsection{\label{subsec:Conclusion-of-the proof log surface}Conclusion of
the proof of Theorem \ref{thm:main log toric intro} and Corollary
\ref{cor:Conjecture holds for log Fano surfaces}}

Combining the previous first two lemmas immediately yields\vspace*{-3pt}
\begin{equation}
\mathcal{M}_{(\mathcal{X}',\mathcal{D}')}(\overline{\mathcal{L}'})\geq\mathcal{M}_{(\P_{\Z}^{1},\mathcal{D})}(\overline{-\mathcal{K}_{(\P_{\Z}^{1},\mathcal{D})}}).\label{eq:ineq log Mab in pf optimal}
\end{equation}
Applying the third lemma above thus gives\vspace*{-3pt}
\[
\mathcal{M}_{(\mathcal{X}',\mathcal{D}')}(\overline{\mathcal{L}'})\geq-\sup_{\left\Vert \cdot\right\Vert }\Bigl(\frac{1}{2}\bigl(-\mathcal{K}_{(\P_{\Z}^{1},\mathcal{D})},\left\Vert \cdot\right\Vert \bigr)^{2}\Bigr),
\]
where the infimum in the left hand side is restricted to volume-normalized
metrics. Invoking Theorem \ref{thm:hyperplane intro} and using that
the Fubini-Study metric is a minimizer when $\mathcal{D}=0$ (by Lemma
\ref{lem:inf is inf}) thus proves the inequality in Theorem \ref{thm:Mab on log surface}
and it corollary. Moreover, Theorem \ref{thm:hyperplane intro} implies
that the inequality is strict, as soon as $D$ is non-trivial.

\subsubsection{\label{subsec:The-equality-case}The equality case}

Consider now the case of equality in Theorem \ref{thm:Mab on log surface} (and, as~a consequence, $D=0)$:\vspace*{-3pt}
\begin{equation}
\mathcal{M}_{(\mathcal{X}',\mathcal{D}')}(\overline{\mathcal{L}'})=\mathcal{M}_{(\P_{\Z}^{1},0)}(\overline{-\mathcal{K}_{\P_{\Z}^{1}}}),\label{eq:equality Mab in pf of equality case}
\end{equation}
where, in~the right-hand side, $-\mathcal{K}_{\P_{\Z}^{1}}$ is endowed with the
Fubini-Study metric. By the minimizing property in Lemma \ref{Lemma:lct as inf},
when $D=0$, the metric on $\mathcal{L}$ coincides with the Fubini-Study
metric up to the application of an automorphism of $X$ and scaling
of the metric. All that remains is to show is thus that $(\mathcal{X}',\mathcal{D}')$
is isomorphic to $(\P_{\Z}^{1},0)$. To this end first note that since
$\mathcal{X}$ $(=\P_{\Z}^{1})$ and $\mathcal{X}'$ have the same generic
fiber they are birationally equivalent. Thus, there exists a normal
variety $\mathcal{Y}$, which is flat and projective over $\mathcal{B}$,
dominating both $\mathcal{X}$ and $\mathcal{X}'$, with birational
morphisms\vspace*{-3pt}
\begin{equation}
p:\mathcal{Y}\to\mathcal{X},\quad q:\mathcal{Y}\to\mathcal{X}',\label{eq:morphisms p and q}
\end{equation}
which are the identity over the generic point in $\mathcal{B}$ (a
concrete construction is given in Step 1 below). It~will thus be enough
to show that the equality \eqref{eq:equality Mab in pf of equality case}
implies that $p$ can be taken to be an isomorphism. Indeed, if $p$
is an isomorphism we get a birational morphism $q$ from $\P_{\Z}^{1}$
to $\mathcal{X}'$ and any such morphism is an isomorphism (since
the fibers of $\P_{\Z}^{1}$ over $\mathcal{B}$ are all reduced and
irreducible). Moreover, when $\mathcal{X}'$ is equal to $\P_{\Z}^{1}$
any~$\mathcal{L}$ whose complexification equals $-K_{\P^{1}}$ is
isomorphic to $-\mathcal{K}_{\P_{\Z}^{1}}$ and the components of
any divisor $\mathcal{D}'$ on $\mathcal{X}'$ whose complexification
is trivial are fibers $\mathcal{X}_{b_{i}}$ of $\P_{\Z}^{1}$ (using
again that the fibers of $\P_{\Z}^{1}$ over $\mathcal{B}$ are all
reduced and irreducible). Hence, the assumed equality \eqref{eq:equality Mab in pf of equality case}
implies, since $\mathcal{X}_{b_{i}}^{2}=0$ and $-\mathcal{K}_{\P_{\Z}^{1}}$
is relatively ample that $\mathcal{D}'$ is trivial, \ie $\mathcal{D}'=0$.

Thus all that remains is to show that the assumed equality in formula
\eqref{eq:equality Mab in pf of equality case} implies that the morphism
$p:\mathcal{Y}\to\mathcal{X}$ (in formula \eqref{eq:morphisms p and q})
can be taken to be an isomorphism.

\subsubsection*{Step 1}
{\itshape In the case of arithmetic surfaces $p:\mathcal{Y}\to\mathcal{X}$
can be taken as the successive blow-ups of $\mathcal{X}$ along a
finite number of closed points $x_{i}$ in regular surfaces $\mathcal{X}_{i}$
and there exists a $p$-exceptional and $p$-ample effective divisor
$E$ on $\mathcal{Y}$ and a morphism $q$ from }$\mathcal{Y}$ to
$\mathcal{X}$ \emph{such that $q^{*}\mathcal{L}'=p^{*}\mathcal{L}-E$. In~particular, $\mathcal{M}_{\mathcal{X}'}(\mathcal{L}')=\mathcal{M}_{\mathcal{Y}}(p^{*}\mathcal{L}-E)$.}

\smallskip
This is shown as in the beginning of the proof of \cite[Prop.\,3.10]{od0},
as next explained. First note that, since $\mathcal{X}$ and $\mathcal{X}'$
have the same generic fiber, they are birationally equivalent. Since
$\mathcal{X}$ is normal this means that there exists a morphism $h:U\to\mathcal{X}'$
from a Zariski open subset $U$ in $\mathcal{X}$ of codimension two.
As a consequence, $h^{*}\mathcal{L}'$ extends to a $\Q$-line bundle
$\mathcal{L}''$ on $\mathcal{X}$ coinciding with $-\mathcal{K}_{\mathcal{X}}$
on the generic fiber. Since $\mathcal{X}=\P_{\Z}^{1}$ this implies
that $\mathcal{L}''$ is isomorphic to $-\mathcal{K}_{\mathcal{X}}$
(using that $\pi:\mathcal{X}\to\Spec\Z$
has reduced irreducible fibers). Now fix a positive integer $k$ such
that $k\mathcal{L}'$ is a relatively very ample line bundle and take
a basis $s'_{i}$ in the free $\Z-$module $H^{0}(\mathcal{X}',k\mathcal{L}'$).
Then $s_{i}:=h^{*}s_{i}$ extends, since $\mathcal{X}$ is normal,
to a unique element in $H^{0}(\mathcal{X},k\mathcal{L})$. Denote
by $\mathcal{J}$ the ideal sheaf on $\mathcal{X}$ generated by the
sections $s_{i}$. Since $\mathcal{X}$ is a regular surface we get
after successive blow-ups (as stated in Step 1) a morphism $p:\mathcal{Y}\to\mathcal{X}$
from a regular surface $\mathcal{Y}$ to $\mathcal{X}$ with the property
that $p^{*}\mathcal{J}$ defines an effective $p$-exceptional divisor~$E_{k}$ on $\mathcal{Y}$ (using that $\Z$ is an excellent ring)
\cite{st}. Set\vspace*{-3pt}
\[
E:=k^{-1}E_{k}\quad (E_{k}:=p^{*}\mathcal{J}).
\]
By construction, $E_{k}$ is $p$-ample,\vspace*{-3pt}
\begin{equation}
H^{0}(\mathcal{Y},kp^{*}\mathcal{L}-E_{k})\cong H^{0}(\mathcal{X},k\mathcal{L}\otimes\mathcal{J})\cong H^{0}(\mathcal{X}',k\mathcal{L}'),\label{eq:isom}
\end{equation}
and the global sections of $kp^{*}\mathcal{L}-E_{k}$ induce a morphism
$q$ to $\mathcal{X}'$ such that $q^{*}\mathcal{L}'=p^{*}\mathcal{L}-E$.
Finally, note that\vspace*{-3pt}
\[
\mathcal{M}_{\mathcal{X}'}(\mathcal{L}')=\mathcal{M}_{\mathcal{Y}}(q^{*}\mathcal{L}'),
\]
as follows directly from the fact that $p$ is an isomorphism between
Zariski open subsets of $\mathcal{X}'$ and $\mathcal{Y}$ and, as
a consequence, the $\Q-$line bundle $q^{*}\mathcal{L}'$ is trivial
on the support of the divisor $q^{*}\mathcal{K}_{\mathcal{X}}-\mathcal{K}_{\mathcal{Y}}$.

\subsubsection*{Step 2}
{\itshape $\mathcal{M}_{\mathcal{Y}}(p^{*}\mathcal{L}-E)=\mathcal{M}_{\mathcal{X}}(\mathcal{L})\implies p\text{ is an isomorphism, when }\mathcal{X}=\P_{\Z}^{1}$.}

\smallskip
Replacing $q^{*}\mathcal{L}'$ with $p^{*}\mathcal{L}-E$ in formula
\eqref{eq:difference of Mab} yields, since $\mathcal{M}_{\mathcal{Y}}(p^{*}\mathcal{L}-E)=\mathcal{M}_{\mathcal{X}}(\mathcal{L})$,\vspace*{-3pt}\enlargethispage{\baselineskip}
\[
(p^{*}\mathcal{L}-E)\cdot\Bigl(\mathcal{K}_{\mathcal{Y}/\mathcal{X}}-\frac{1}{2}E\Bigr)=0.
\]
It follows, since, by~construction, $p^{*}\mathcal{L}-E$ is $p$-ample,\vspace*{-3pt}
that
\begin{equation}
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}=\frac{1}{2}E.\label{eq:pf equality case identity}
\end{equation}
Now, since $p:\mathcal{Y}\to\mathcal{X}$ is the blow-up
along a finite number of closed points $x_{i}$ in regular surfaces
$\mathcal{X}_{i}$,\vspace*{-3pt}
\begin{equation}
\mathcal{K}_{\mathcal{Y}/\mathcal{X}}=\sum c_{i}E_{i},\quad c_{i}\geq1,\label{eq:pf equality case ineq}
\end{equation}
where the sum runs over all prime $p$-exceptional divisors $E_{i}$.
Hence,\vspace*{-3pt}
\[
E=\sum_{i}2c_{i}E_{i}\geq\sum_{i}2E_{i}.
\]
But this contradicts the isomorphisms \eqref{eq:isom}, if the number
of points $x_{i}$ is non-zero. Indeed, denote by $E_{1}$ the strict
transform of the $p$-exceptional divisor on $\mathcal{Y}$ induced
from the exceptional divisor on the first point $x_{1}$ blown-up
on $\mathcal{X}(=\P_{\Z}^{1})$. Then it follows from the previous
inequality and the construction of $E$ that the restriction of the
ideal sheaf $\mathcal{J}$ on $\mathcal{X}$ to a neighbourhood of
$x_{1}$ in the fiber $\mathcal{X}_{\pi(x_{1})}$ is contained in
the $2k$th power $\mathfrak{m}_{x_{1}}^{2k}$ of the
maximal ideal $\mathfrak{m}_{x_{1}}$ on $\mathcal{X}_{\pi(x_{1})}$
defined by the point $x_{1}$. But, in~general, for $\mathcal{X}=\P_{\Z}^{1}$,
the line bundle $k\mathcal{L}_{|\mathcal{X}_{\pi(x)}}\otimes\mathfrak{m}_{x}^{2k}$
on $\mathcal{X}_{\pi(x)}$ is trivial for any closed point $x$ on
$\mathcal{X}$ (since $\mathcal{L}_{|\mathcal{X}_{b}}:=-K_{\mathcal{X}_{b}}=\mathcal{O}_{\P_{\F_{b}}^{1}}(2))$.
But this contradicts the isomorphism~\eqref{eq:isom}, since $\mathcal{L}'$
is relatively ample. Hence, the number of points $x_{i}$ must be
zero, as desired.

Combing these two steps thus concludes, as discussed above, the proof
of Theorem~\ref{thm:Mab on log surface}. Finally, Corollary \ref{cor:Conjecture holds for log Fano surfaces}
can be deduced from Theorem \ref{thm:Mab on log surface} using a
generalization of Lemma \ref{lem:inf is inf}. But here we instead
proceeds as follows. Given an arithmetic log Fano surface $(\mathcal{X},\mathcal{D})$
set $\mathcal{L}:=-\mathcal{K}_{\mathcal{X}}$ and endow $\mathcal{L}$
and $-\mathcal{K}_{\mathcal{X}}$ with the same metric induced from
a volume-normalized metric on $-K_{X}$ with positive curvature current.
Then, by~definition \eqref{eq:more general log Mab},\vspace*{-3pt}
\[
-\frac{1}{2}\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}^{2}=\mathcal{M}_{(\mathcal{X},\mathcal{D})}(\mathcal{L}).
\]
Hence, combining Step one and Step two above yields\vspace*{-3pt}
\[
-\frac{1}{2}\overline{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}^{2}\geq\mathcal{M}_{(\P_{\Z}^{1},0)}(\overline{-\mathcal{K}_{\P_{\Z}^{1}}})=-\frac{1}{2}(\overline{-\mathcal{K}_{\P_{\Z}^{1}}})^{2},
\]
and the equality case is deduced precisely as before.
\begin{rem}
When $(\P^{1},\Delta)$ is K\nobreakdash-stable equality holds in the inequality
\eqref{eq:ineq log Mab in pf optimal} if and only if $(\mathcal{X}',\mathcal{D}')=(\P^{1},\mathcal{D}^{0})$.
Indeed, by~Lemma \ref{lem:alpha geq one half}, the K\nobreakdash-stability of
$(\P^{1},\Delta)$ implies that $\alpha(\P^{1},\mathcal{D}^{0})>1/2$.
Hence, if equality holds in \eqref{eq:ineq log Mab in pf optimal},
then formula \eqref{eq:difference Mab} forces $E=0$, showing that
$p$ is an isomorphism. We can then conclude precisely as in the beginning
of Section \ref{subsec:The-equality-case}.
\end{rem}

\section{Appendix}

In the proof of Lemma \ref{lem:transf of height} we used the following
result (applied to $X=\P_{\C}^{n}$).

\begin{lem}
\label{lem:append}Let $X$ be a Fano manifold and $V$ a holomorphic
vector field on $X$. Denote by $G_{t}$ the flow of the real part
of $V$ on $X$ at time $t$ and by $(G_{t}^{V})^{*}\phi$ its action
on a given continuous metric $\phi$ on $-K_{X}$ with positive curvature
current. If $X$ admits a Kähler-Einstein metric, then
\[
\frac{d}{dt}\mathcal{E}(G_{t}^{*}\phi,\psi_{0})=0
\]
for any fixed metric $\psi_{0}$ on $-K_{X}$.
\end{lem}

\begin{proof}
This is well-known and essentially goes back to \cite{fut}, but for
the convenience of the reader we provide a proof in the spirit of
the present paper and its precursor~\cite{a-b}. Consider the \emph{Ding
functional} $\mathcal{D}_{\phi_{0}}$ on the space of all continuous
metrics on $-K_{X}$ with positive curvature current, defined by formula
\eqref{eq:Ding functional Kahler}. Since, $\mu_{G^{*}\phi}=G^{*}\mu_{\phi}$
for any biholomorphism $G$ of $X$ it follows that $\mathcal{E}(G_{t}^{*}\phi)$
and $\mathcal{D}_{\phi_{0}}(G_{t}{}^{*}\phi)$ have the same derivative.
Moreover, in~general, the function $t\mto\mathcal{D}_{\phi_{0}}(G_{t}^{*}\phi)$
is linear. Indeed, its derivative is the Futaki invariant of $V$
(see the claim in \cite[p.\,73]{ti}, where $\mathcal{D}_{\phi_{0}}$
is denoted by $F_{\omega}$). Hence, all that remains is to verify
that $\mathcal{D}_{\phi_{0}}$ is bounded from below (since then $t\mto\mathcal{D}_{\phi_{0}}(G_{t}^{*}\phi)$
must be constant). But this follows from the existence of a Kähler-Einstein
metric, since such a metric minimizes $\mathcal{D}_{\phi_{0}}$, as
recalled in \cite[\S 2.3]{a-b}.
\end{proof}

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