# A uniform version of the Yau-Tian-Donaldson correspondence for extremal
Kähler metrics on polarized toric manifolds

Yasufumi Nitta  and  Shunsuke Saito  Department of Mathematics, Faculty of
Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo
162-8601, Japan<EMAIL_ADDRESS>Department of Mathematics, Faculty of
Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo
162-8601, Japan<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to solve a uniform version of the Yau-Tian-Donaldson
conjecture for polarized toric manifolds. Also, we show a combinatorial
sufficient condition for uniform relative K-polystability.

###### Key words and phrases:

Relative K-stability, extremal Kähler metrics, the Yau-Tian-Donaldson
correspondence, toric varieties

###### 2010 Mathematics Subject Classification:

Primary 53C25,Secondary 32Q26,14M25

###### Contents

  1. 1 Introduction
  2. 2 Background materials
    1. 2.1 Notation and convensions
    2. 2.2 Extremal Kähler metrics
    3. 2.3 The Futaki invariant and the extremal Kähler vector field
    4. 2.4 Riemannian geometry of the space of Kähler potentials
    5. 2.5 Energy functionals
    6. 2.6 Vector field energies and the relative K-energy
    7. 2.7 $d_{1}$-distances, reduced $d_{1}$-distances and the reduced J-functionals
  3. 3 Test configurations and relative K-stability
    1. 3.1 Test configurations
    2. 3.2 Duistermaat-Heckman measures
    3. 3.3 Non-Archimedean functionals
    4. 3.4 Filtrations
    5. 3.5 Limit measures
    6. 3.6 Reduced non-Archimedean $J$-functional
    7. 3.7 Uniform relative K-stability
  4. 4 Convex Preliminaries
    1. 4.1 Polarized toric manifolds
    2. 4.2 Symplectic potentials
    3. 4.3 Toric test configurations
    4. 4.4 Energy functionals on polarized toric manifolds
    5. 4.5 Non-Archimedean functionals on polarized toric manifolds
  5. 5 Proof of Theorem 1.0.1
    1. 5.1 Spaces of convex functions
    2. 5.2 Approximations and compactness of convex functions
    3. 5.3 The non-Archimedean K-energy
    4. 5.4 The J-norm
    5. 5.5 Proof of Theorem 1.0.1
  6. 6 Proof of Theorem 1.0.5

##  1\. Introduction

Let $(X,L)$ be an $n$-dimensional polarized algebraic manifold. We denote
$\mathcal{H}(X,L)$ by the set of all Kähler metrics of $X$ in $c_{1}(L)$. A
Kähler metric $\omega\in\mathcal{H}(X,L)$ is called a _Calabi’s extremal
Kähler metric_ if it is a critical point of the Calabi functional

$\displaystyle\mathcal{H}(X,L)\ni\alpha\mapsto\int_{X}(s(\alpha)-\overline{s})^{2}\,\frac{\alpha^{n}}{n!}\in\mathbf{R},$

where $s(\alpha)$ is a scalar curvature of $\alpha$ and
$\overline{s}=-n(K_{X}\cdot L^{n-1})/(L^{n})$ is its average. Calabi showed in
[14] that this condition is equivalent to that
$\operatorname{grad}_{\omega}^{(1,0)}(s(\omega)-\overline{s})$ is a
holomorphic vector field on $X$. Extremal Kähler metrics have been widely
studied as canonical Kähler metrics containing constant scalar curvature
Kähler metrics and in particular Kähler-Einstein metrics. Especially, the
existence problem of extremal Kähler metrics is known as the _Yau-Tian-
Donaldson conjecture_ , and is one of the central problem in Kähler geometry.

The Yau-Tian-Donaldson conjecture predicts that the existence of an extremal
Kähler metric is equivalent to the stability of $(X,L)$ in some sense of
geometric invariant theory. Székelyhidi introduced in [73] the notion of
_relative K-polystability_ as a candidate of such a stability condition.
Roughly speaking, relative K-polystability requires the positivity of the
_relative Donaldson-Futaki invariant_ $DF_{V}>0$ for any nontrivial test
object. Here $V$ is the extremal Kähler vector field of $(X,L)$ ([40]), and
test object is a special kind of degeneration of $(X,L)$, which is called a
_test configuration_.

On the other hand, calculations by Apostolov-Calderbank-Gauduchon-Tønnesen-
Friedman [4] suggest that relative K-polystability might not be sufficient to
ensure the existence of a extremal Kähler metric, and leads to an expectation
that we need to strengthen the notion of relative K-polystability. For
strengthenings, the following two approaches are known:

  1. (a)

Requiring that $DF_{V}$ is bounded from below by a multiple of some ‘norm’ of
test configurations ([72], [32], [12, 13]).

  2. (b)

Testing $DF_{V}>0$ for more general test objects than test configurations
([76], [63]).

In this paper, we will show the equivalence of both approaches for polarized
toric manifolds. As a consequence, we will solve a uniform version of the Yau-
Tian-Donaldson conjecture for polarized toric manifolds.

An $n$-dimensional polarized toric manifold $(X,L)$ corresponds to an
$n$-dimensional integral Delzant polytope $P\subset\mathbf{R}^{n}$ which is
written by an intersection of half spaces

(1.1) $\displaystyle
P=\\{x\in\mathbf{R}^{n}\mid\langle\lambda_{j},x\rangle+d_{j}\geq 0\
(j=1,\ldots,r)\\},$

where $\langle\cdot,\cdot\rangle$ is the standard inner product on
$\mathbf{R}^{n}$, $r$ is the number of facets of $P$,
$\lambda_{j}\in\mathbf{Z}^{n}$, $d_{j}\in\mathbf{Z}$ and each $\lambda_{j}$ is
a primitive vector. Then, any toric Kähler metric representing $c_{1}(L)$
corresponds to a smooth strictly convex function on $P^{\circ}$ which
satisfies the _Guillemin’s boundary condition_ (see Section 4.2). Also, any
toric test configuration $(\mathcal{X},\mathcal{L})$ for $(X,L)$ corresponds
to a rational piecewise affine function $f$ on $P$, and the relative
Donaldson-Futaki invariant $DF_{V}(\mathcal{X},\mathcal{L})$ of
$(\mathcal{X},\mathcal{L})$ can be written as

$\displaystyle L_{V}(f)=\int_{\partial
P}f\,d\sigma-\int_{P}(\overline{s}+V)f\,dx,$

where $\sigma$ is the Borel measure on $\partial P$ which is the multiple of
the $(n-1)$-dimensional Lebesgue measure by $|\lambda_{j}|^{-1}$ on each facet
$F_{j}=\\{x\in P\mid\langle\lambda_{j},x\rangle+d_{j}=0\\}$ (see [80]).

Let $\mathcal{S}$ be the set of all strictly convex functions on $P^{\circ}$
which satisfy the Guillemin’s boundary condition. Let
$\mathcal{C}_{PL}^{\mathbf{Q}}$ be the set of all rational piecewise affine
functions on $P$. As mentioned above, we can regard
$\mathcal{C}_{PL}^{\mathbf{Q}}$ as the set of all toric test configurations
for $(X,L)$. Also, we define $\mathcal{C}_{PL}$ to be the set of all piecewise
affine functions on $P$. Let $\mathcal{C}_{\infty}$ be the set of all
continuous convex functions on $P$ which is smooth in the interior. Further,
let $P^{\ast}$ be the union of relative interiors of faces of $P$ up to
codimension one, and $\mathcal{C}_{\ast}$ be the set of all continuous convex
functions on $P^{\ast}$ which are integrable on $\partial P$ with respect to
the measure $\sigma$. We fix a point $x_{0}\in P$ in the interior, and denote
by $\widetilde{\mathcal{C}}_{\ast}$ the set of all $f\in\mathcal{C}_{\ast}$
satisfying the _normalized condition_ $\inf_{P^{\ast}}f=f(x_{0})=0$. Also, set
$\widetilde{\mathcal{F}}=\mathcal{F}\cap\widetilde{\mathcal{C}}_{\ast}$ for
any $\mathcal{F}\subset\mathcal{C}_{*}$. Finally, for each
$f\in\mathcal{C}_{\ast}$ we denote the _$J$ -norm_ by $\|f\|_{J}$ (see Section
5.4).

Under the notations above, our first main result is the following equivalence.

######  Theorem 1.0.1.

Let $\mathcal{F}$ be one of the spaces $\mathcal{E}_{1}$,
$\mathcal{C}_{\ast}$, $\mathcal{C}_{PL}$, $\mathcal{C}_{PL}^{\mathbf{Q}}$,
$\mathcal{C}_{\infty}$ or $\mathcal{S}$. Then the followings are equivalent.

$(b)_{\mathcal{F}}$

There exists a $\delta>0$ such that $L_{V}(f)\geq\delta\int_{\partial
P}f\,d\sigma$ for any $f\in\widetilde{\mathcal{F}}$.

$(J)_{\mathcal{F}}$

There exists a $\delta>0$ such that $L_{V}(f)\geq\delta\|f\|_{J}$ for any
$f\in\mathcal{F}$.

$(\mathrm{K})_{\mathcal{C}_{\ast}}$

$L_{V}(f)\geq 0$ for any $f\in\mathcal{C}_{\ast}$ and $L_{V}(f)=0$ if and only
if $f$ is affine.

$(\mathrm{K})_{\mathcal{E}_{1}}$

$L_{V}(f)\geq 0$ for any $f\in\mathcal{E}_{1}$ and $L_{V}(f)=0$ if and only if
$f$ is affine.

Note that the proof of Theorem 1.0.1 does not require the Delzant condition
and that this equivalence also holds for polarized toric _varieties_.

Let $(X,L)$ be a polarized toric manifold associated to an integral Delzant
polytope $P\subset\mathbf{R}^{n}$. We say that $(X,L)$ is _uniformly
relatively K-polystable_ if $P$ satisfies the condition
$(J)_{\mathcal{C}_{PL}^{\mathbf{Q}}}$. $(X,L)$ is _uniformly K-polystable_ if
in addition that $V=0$. By combining Theorem 1.0.1 with the works of [16],
[80], [47] and [52], we can prove a uniform version of the Yau-Tian-Donaldson
correspondence for polarized toric manifolds.

######  Theorem 1.0.2 (The toric Yau-Tian-Donaldson correspondence).

Let $(X,L)$ be an $n$-dimensional polarized toric manifold. Let $T$ be a
maximal algebraic torus of $\mathrm{Aut}^{0}(X,L)$, and $S$ be the maximal
compact subgroup. Then the followings are equivalent.

  1. (1)

$(X,L)$ admits an $S$-invariant extremal Kähler metric.

  2. (2)

$(X,L)$ is uniformly relatively K-polystable.

  3. (3)

The relative K-energy of $(X,L)$ is $T$-coercive.

The implications (1) $\Rightarrow$ $(b)_{\mathcal{C}_{\infty}}$ and
$(b)_{\mathcal{C}_{\infty}}$ $\Rightarrow$ (3) are proved in [16] and [80],
respectively. The implication (3) $\Rightarrow$ (1) can be obtained by the
work of He [47] and its toric reduction by Legendre (see Section 3.2 of [52]).
Finally, the equivalence
$(J)_{\mathcal{C}_{PL}^{\mathbf{Q}}}\Leftrightarrow(b)_{\mathcal{C}_{\infty}}$
is proved in Theorem 1.0.1.

If we consider the case that the extremal Kähler vector field of $(X,L)$ is
zero, we obtain the following:

######  Corollary 1.0.3.

Let $(X,L)$, $T$, $S$ be as above. Then the followings are equivalent.

  1. (1)

$(X,L)$ admits an $S$-invariant Kähler metric of constant scalar curvature.

  2. (2)

$(X,L)$ is uniformly K-polystable.

  3. (3)

The K-energy of $(X,L)$ is $T$-coercive.

We mention the very recent papers [50] and [56], which have some overlaps with
the present paper.

As a simple application of Theorem 1.0.2, we obtain the following product
theorem for uniform relative K-polystability.

######  Corollary 1.0.4.

Let $(X_{1},L_{1})$ and $(X_{2},L_{2})$ be polarized toric manifolds, and set
$X=X_{1}\times X_{2}$ and $L=L_{1}\boxtimes L_{2}$. If $(X_{1},L_{1})$ and
$(X_{2},L_{2})$ are uniformly relatively K-polystable, then so is $(X,L)$.

In fact, if $(X_{1},L_{1})$ and $(X_{2},L_{2})$ are uniformly relatively
K-polystable, then they admit torus invariant extremal Kähler metrics by
Theorem 1.0.2. Since the product metric of extremal Kähler metrics is also an
extremal Kähler metric, $(X,L)$ is uniformly relatively K-polystable by
Theorem 1.0.2 again.

Next, we give a combinatorial sufficient condition for unirofm relative
K-polystability. Let $(X,L)$ be a polarized toric manifold associated an
integral Delzant polytope $P$ defined by (1.1). We fix a point $x_{0}\in P$ in
the interior, and put $d_{x_{0},j}=\langle\lambda_{j},x_{0}\rangle+d_{j}$ for
each $j=1,\ldots,r$. Note that $d_{x_{0},j}>0$ for any $j$. Then the following
theorem strengthens [80, Theorem 0.1], where relative K-polystability was
deduced under the same assumptions.

######  Theorem 1.0.5.

Let $d_{x_{0}}=\max\\{d_{x_{0},1},\ldots,d_{x_{0},r}\\}$. Suppose that $P$
satisfies either

(1.2) $\displaystyle
V=0\quad\text{and}\quad\overline{s}<\frac{n+1}{d_{x_{0}}},$

or

(1.3) $\displaystyle V\neq
0\quad\text{and}\quad\overline{s}+\max_{P}V\leq\frac{n+1}{d_{x_{0}}}.$

Then $(X,L)$ is uniformly relatively K-polystable.

In case that $(X,L)=(X,-K_{X})$ is a toric Fano manifold with anticanonical
polarization, then $\overline{s}=n$ and $d_{j}=1$ for all $j$. Moreover, we
can choose $x_{0}$ to be the origin of $\mathbf{R}^{n}$. Hence the conditions
(1.2) and (1.3) reduce

(1.4) $\displaystyle V=0$

and

(1.5) $\displaystyle V\neq 0\quad\text{and}\quad\max_{P}V\leq 1,$

respectively. Combining this with the computations in [80] and [66] yields the
following:

######  Corollary 1.0.6.

  1. (1)

All toric del Pezzo surfaces are uniformly relatively K-polystable.

  2. (2)

In 3-dimensional case, at least 13 toric Fano manifolds out of 18 are
uniformly relatively K-polystable.

It is known that all toric del Pezzo surfaces admit an extremal Kähler metric
in its anticanonical class by the works of [14] (degree 8 case), [71] and [77]
(degree 6 case) and [25] (degree 7 case). Corollary 1.0.6 gives the stability
counterpart of this fact.

We finally concern uniform K-polystability of toric Fano manifolds. Note that
the condition (1.4) is equivalent to that the Futaki invariant of $X$ vanishes
([40]). Also, it is known that the Futaki invariant vanishes if and only if
the barycenter of $P$ coincides with the origin of $\mathbf{R}^{n}$ ([60]).
Thus,

######  Corollary 1.0.7.

Let $(X,-K_{X})$ be an $n$-dimensional toric Fano manifold with anticanonical
polarization associated to an integral Delzant polytope $P$. Then the
followings are equivalent:

  1. (1)

$(X,-K_{X})$ is uniformly K-polystable.

  2. (2)

$(X,-K_{X})$ is K-polystable.

  3. (3)

$(X,-K_{X})$ is K-semistable.

  4. (4)

The Futaki invariant of $X$ vanishes.

  5. (5)

The barycenter of $P$ coincides with the origin of $\mathbf{R}^{n}$.

  6. (6)

$V=0$.

##  2\. Background materials

###  2.1. Notation and convensions

Throughout this paper, we use the following notation and convensions: In this
paper, we mean an $n$-dimensional polarized manifold a pair $(X,L)$ of an
$n$-dimensional compact complex manifold $X$ and an ample holomorphic line
bundle $L$ over $X$.

  * •

$S^{1}=\mathbf{R}/\mathbf{Z}$, $\mathrm{Lie}(S^{1})=2\pi\sqrt{-1}\mathbf{R}$,
$\mathrm{vol}(S^{1})=1$.

  * •

$\mathbf{C}=(-1/2)\mathbf{R}\oplus
2\pi\sqrt{-1}\mathbf{R}=(-\sqrt{-1}/4\pi)\mathrm{Lie}(S^{1})\oplus\mathrm{Lie}(S^{1})$.

  * •

$\mathbf{C}^{\times}=\mathbf{C}\setminus\\{0\\}=\exp((-1/2)\mathbf{R}\oplus
2\pi\sqrt{-1}\mathbf{R})$.

  * •

$\omega(\cdot,\cdot)=(1/2\pi)g(J\cdot,\cdot)$ for any Kähler metric $g$. Here
$J$ is the integrable almost complex structure of the complex manifold $X$.

  * •

$\Delta=\Delta_{d}=2\Delta_{\bar{\partial}}=\nabla^{i}\nabla_{i}=\mathrm{tr}_{g}\mathrm{Hess}_{g}$
for any Kähler metric $g$.

Let $T$ be a maximal algebraic torus of $\mathrm{Aut}^{0}(X)$, with
$\mathfrak{t}=\mathrm{Lie}(T)$.

  * •

$\mathbf{A}^{1}=\operatorname{Spec}\mathbf{C}[t]$,
$\mathbf{G}_{m}=\operatorname{Spec}\mathbf{C}[t,t^{-1}]$.

  * •

$N=\mathrm{Hom}(\mathbf{G}_{m},T)$, which is a lattice of rank
$\dim_{\mathbf{C}}T$.

  * •

$N_{k}=N\otimes_{\mathbf{Z}}k$ for $k=\mathbf{Q}$ or $\mathbf{R}$.

  * •

$M=\mathrm{Hom}(N,\mathbf{Z})$, $M_{k}=M\otimes_{\mathbf{Z}}k$ for
$k=\mathbf{Q}$ or $\mathbf{R}$.

  * •

$S=N\otimes_{\mathbf{Z}}S^{1}$, which is the maximal compact subgroup of $T$,
with $\mathfrak{s}=\mathrm{Lie}(S)=2\pi\sqrt{-1}N_{\mathbf{R}}$, which
satisfies
$\mathfrak{t}=(-\sqrt{-1}/4\pi)\mathfrak{s}\oplus\mathfrak{s}=(-1/2)N_{\mathbf{R}}\oplus
2\pi\sqrt{-1}N_{\mathbf{R}}$. Note that $M_{\mathbf{R}}=\mathfrak{s}^{\vee}$.

###  2.2. Extremal Kähler metrics

In this section, we recall the definition of extremal Kähler metrics. Let
$(X,L)$ be an $n$-dimensional polarized manifold, and fix a maximal algebraic
torus $T$ of $\mathrm{Aut}^{0}(X)$. We denote by $\mathcal{H}(X,L)^{S}$ the
set of all $S$-invariant Kähler metrics $\omega$ such that $\omega\in
c_{1}(L)$. By setting $\overline{s}=-n(K_{X}\cdot L^{n-1})/(L^{n})$, we define
$\Phi\colon\mathcal{H}(X,L)^{S}\to\mathbf{R}$ by

$\displaystyle\Phi(\omega)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}(S(\omega)-\overline{s})^{2}\omega^{n}=\frac{1}{\mathrm{vol}_{\omega}(X)}\int_{X}(s(\omega)-\overline{s})^{2}\omega^{n},$

where $\mathrm{vol}_{\omega}(X)=\int_{X}\omega^{n}$ and
$\overline{s}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}s(\omega)\omega^{n}$.
Note that $\mathrm{vol}_{\omega}(X)$ and $\overline{s}$ is independent of
choice of $\omega\in\mathcal{H}(X,L)^{S}$ since
$\mathrm{vol}_{\omega}(X)=(L^{n})=\langle c_{1}(L)^{n},[X]\rangle$ and
$\overline{s}=-n(K_{X}\cdot L^{n-1})/(L^{n})$. We call $\Phi$ the Calabi
functional.

######  Definition 2.2.1 ([14]).

A Kähler metric $\omega\in\mathcal{H}(X,L)^{S}$ is called an extremal Kähler
metric if $\omega$ is a critical point of $\Phi$.

Kähler-Einstein metrics are extremal metrics, and (more generally) constant
scalar curvature Kähler metrics in $\mathcal{H}(X,L)^{S}$ are typical examples
of extremal Kähler metrics. The following fact is well-known.

######  Proposition 2.2.2 ([14, Theorem 2.1]).

A Kähler metric $\omega\in\mathcal{H}(X,L)^{S}$ is an extremal Kähler metric
if and only if $\mathrm{grad}_{g}(s(\omega)-\overline{s})$ is a real
holomorphic vector field on $X$.

We fix $\omega\in\mathcal{H}(X,L)^{S}$. Set

$\displaystyle C^{\infty}(X,\omega,\mathbf{R})_{0}^{S}=\\{f\in
C^{\infty}(X,\mathbf{R})\mid\int_{X}f\omega^{n}=0\\},$
$\displaystyle\mathfrak{s}(X,\omega)_{0}=\\{u\in
C^{\infty}(X,\omega,\mathbf{R})_{0}^{S}\mid 4\pi
J\mathrm{grad}_{g}u\in\mathfrak{s}\\}.$

Conversely, for each $W\in\mathfrak{s}$ there exists unique $u_{\omega}^{W}\in
C^{\infty}(X,\mathbf{R})^{S}$ such that

$\displaystyle
i_{W}\omega=-du_{\omega}^{W},\quad\int_{X}u_{\omega}^{W}\omega^{n}=0.$

Then we have

$\displaystyle-
du_{\omega}^{W}=g(JJ\mathrm{grad}_{g}u_{\omega}^{W},\cdot)=i_{2\pi
J\mathrm{grad}_{g}u_{\omega}^{W}}\omega$

and $2\pi J\mathrm{grad}_{g}u_{\omega}^{W}=W\in\mathfrak{s}$, which implies
$u_{\omega}^{W}\in\mathfrak{s}(X,\omega)_{0}$. Hence the correspondence
$\mathfrak{s}\ni W\mapsto u_{\omega}^{W}\in\mathfrak{s}(X,\omega)_{0}$ gives a
Lie algebra isomorphism. The following is also ovbious.

######  Proposition 2.2.3.

A Kähler metric $\omega\in\mathcal{H}(X,L)^{S}$ is extremal if and only if
$s(\omega)-\overline{s}\in\mathfrak{s}(X,\omega)_{0}$.

###  2.3. The Futaki invariant and the extremal Kähler vector field

Let $(X,L)$ be a polarized manifold and $\omega\in\mathcal{H}(X,L)^{S}$. Since
$\mathfrak{t}=(-1/4\pi)J\mathfrak{s}\oplus\mathfrak{s}$, every
$W\in\mathfrak{t}$ can be written as

$\displaystyle W=-\frac{1}{4\pi}JW_{1}+W_{2}$

for some $W_{1},W_{2}\in\mathfrak{s}$. Then we define $u_{\omega}^{W}\in
C^{\infty}(X,\omega,\mathbf{C})_{0}^{S}$ by

$\displaystyle
u_{\omega}^{W}=-\frac{1}{4\pi}\sqrt{-1}u_{\omega}^{W_{1}}+u_{\omega}^{W_{2}}.$

######  Definition 2.3.1 ([38], [39], [15], [7]).

We call the Lie algebra character $F\colon\mathfrak{t}\to\mathbf{C}$ defined
by

$\displaystyle
F(W)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}u_{\omega}^{W}(s(\omega)-\overline{s})\omega^{n}$

the Futaki invariant of $(X,L)$.

We remark that (i) the definition of $F$ is independent of choice of
$\omega\in\mathcal{H}(X,L)^{S}$, (ii) if $\mathcal{H}(X,L)^{S}$ contains a
Kähler metric of constant scalar curvature, then $F$ vanishues identically,
and (iii) an extremal Kähler metric $\omega\in\mathcal{H}(X,L)^{S}$ has
constant scalar curvature if and inly if $F=0$ (see [38], [39], [15], [7]).

Let $\mathrm{pr}_{\omega}\colon
C^{\infty}(X,\omega,\mathbf{R})_{0}^{S}\to\mathfrak{s}(X,\omega)_{0}$ be the
$L^{2}$-orthogonal projection with respect to the volume form $\omega^{n}$.

######  Definition 2.3.2 ([40]).

Let
$\theta_{\omega}=\mathrm{pr}_{\omega}(s(\omega)-\overline{s})\in\mathfrak{s}(X,\omega)_{0}$.
We call the real holomorphic vector field defined by

$\displaystyle V\coloneqq\mathrm{grad}_{g}\theta_{\omega}\in N_{\mathbf{R}}$

the extremal Kähler vector field of $(X,L)$ with respect to $T$.

It is known that (i) the definition of $V$ is independent of choice of
$\omega\in\mathcal{H}(X,L)^{S}$, (ii) a Kähler metric
$\omega\in\mathcal{H}(X,L)^{S}$ is extremal if and only if
$s(\omega)-\overline{s}=\theta_{\omega}$, and (iii)
$\mathrm{grad}_{g}\theta_{\omega}\in N_{\mathbf{Q}}$, that is,
$\exp(mV)=\mathrm{id}_{X}$ for some $m\in\mathbf{Z}_{>0}$ (see [40] and [64]).

######  Remark 2.3.3.

We remark that the definition of $V$ depends only on the choice of $T$ and
independent of neither the choice of $K$ nor $K$-invariant Kähler metric in
$c_{1}(L)$. Indeed, by [40] $V$ is independent of the choice of $K$-invariant
Kähler metric in $c_{1}(L)$ for fixed $K$. If $K^{\prime}$ is another choice,
then $K$ and $K^{\prime}$ have a common maximal torus $S$. Hence there exists
$\tau\in T$ so that $K^{\prime}=\tau K\tau^{-1}$ by [55, Proposition A.2].
Finally, by [40, Corollary E] we obtain $V^{\prime}=\mathrm{Ad}(\tau)V=V$.

######  Definition 2.3.4 ([40]).

We define $B\colon\mathfrak{t}\times\mathfrak{t}\to\mathbf{C}$ by

$\displaystyle
B(W,W^{\prime})=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}u_{\omega}^{W}u_{\omega}^{W^{\prime}}\omega^{n}$

and call the Futaki-Mabuchi bilinear form.

It is known that (i) the definition of $B$ is independent of choice of
$\omega\in\mathcal{H}(X,L)^{S}$, (ii) $B$ is a nondegenerate and symmetric
bilinear form on $\mathfrak{t}$, and (iii) for any $W\in\mathfrak{t}$

$\displaystyle B(W,V)=-F(W),$

namely, $F$ is dual to $-V$ (see [40]). In paricular, the Futaki invariant
vanishes identically if and only if $V=0$.

###  2.4. Riemannian geometry of the space of Kähler potentials

In [59], Mabuchi studied a natural Riemannian structure on the space of Kähler
potentials. Let $(X,L)$ be an $n$-dimensional polarized manifold and fix
$\omega_{0}\in\mathcal{H}(X,L)^{S}$. By the well-known $dd^{c}$-lemma,
$\mathcal{H}(X,L)^{S}$ can be identified the space

$\displaystyle\mathcal{H}^{S}=\mathcal{H}(X,\omega_{0})^{S}\coloneqq\\{\varphi\in
C^{\infty}(X,\mathbf{R})^{S}\mid\omega_{\varphi}\coloneqq\omega_{0}+dd^{c}\varphi>0\\}.$

If $\varphi\in\mathcal{H}^{S}$, the tangent space $T_{\varphi}\mathcal{H}^{S}$
at $\varphi$ can be identified with $C^{\infty}(X,\mathbf{R})^{S}$. Then the
natural Riemannian metric on $T_{\varphi}\mathcal{H}^{S}$ is given by

$\displaystyle\langle
u,v\rangle_{\varphi}\coloneqq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}uv\omega_{\varphi}^{n}.$

Let $\\{\varphi^{t}\\}$ be a smooth path in $\mathcal{H}^{S}$, and
$\\{u^{t}\\}$ be a smooth vector filed of $\mathcal{H}^{S}$ along the path
$\\{\varphi^{t}\\}$, namely, $u^{t}\in
T_{\varphi^{t}}\mathcal{H}^{S}=C^{\infty}(X,\mathbf{R})^{S}$ for each $t$.
Then the Levi-Civita connection is given by

$\displaystyle\nabla_{\dot{\varphi}^{t}}u^{t}=\dot{u}^{t}-\frac{1}{2}\langle
d\dot{\varphi}^{t},du^{t}\rangle_{g_{\varphi^{t}}}.$

This immidiately implies that $t\mapsto\varphi^{t}$ is a geodesic if and only
if $\nabla_{\dot{\varphi}^{t}}\dot{\varphi}^{t}=0$, or equivalently

(2.1)
$\displaystyle\ddot{\varphi}^{t}-\frac{1}{2}|d\dot{\varphi}^{t}|_{g_{\varphi^{t}}}^{2}=0.$

As discovered by Semmes [69] and Donaldson [33], the above equation can be
understood as a complex Monge-Ampère equation as follows. For each
$T\in(0,\infty]$, let $A=\\{w\in\mathbf{C}\mid e^{-T/2}\leq|w|\leq 1\\}$. Here
we propose $A=\\{w\in\mathbf{C}\mid 0<|w|\leq 1\\}$ if $T=\infty$. By setting
$w=\exp(-t/2+2\pi\sqrt{-1}\theta)$, we have $t=-\log|w|^{2}$ and

$\displaystyle e^{-T/2}\leq|w|\leq 1\iff 0\leq t\leq T.$

Let $\overline{X}=X\times A$. We regard $\overline{X}$ as a complex manifold
with boundary. Let $\pi\colon\overline{X}\to X$ be the natural projection. For
each smooth path $\\{\varphi^{t}\\}$ in $\mathcal{H}^{S}$, we define
$\Phi\colon\overline{X}\to\mathbf{R}$ by

$\displaystyle\Phi(x,w)\coloneqq\varphi^{-\log|w|^{2}}(x).$

######  Theorem 2.4.1 ([69]).

We denote the $d$ and $d^{c}$-operator on $\overline{X}$ by $\overline{d}$ and
$\overline{d}^{c}$, respectively. Then

$\displaystyle(\pi^{*}\omega_{0}+\overline{d}\,\overline{d}^{c}\Phi)^{n+1}=(n+1)\left(\ddot{\varphi}^{t}-\frac{1}{2}|d\dot{\varphi}^{t}|_{g_{\varphi^{t}}}^{2}\right)\pi^{*}\omega_{\varphi^{t}}^{n}+\frac{\sqrt{-1}}{2\pi}\frac{dw\wedge
d\overline{w}}{|w|^{2}}.$

In particular, $\\{\varphi^{t}\\}$ is a geodesic if and only if $\Phi$
satisfies

(2.2)
$\displaystyle(\pi^{*}\omega_{0}+\overline{d}\,\overline{d}^{c}\Phi)^{n+1}=0.$

Unfortunately, the Dirichlet problem of the equation (2.2) does not usually
have a smooth solution (see [53]). This means that in general one cannot find
a smooth geodesic segment connecting two points in $\mathcal{H}_{0}^{S}$. On
the other hand, Chen [18] proved that the Dirichlet problem of (2.2) always
has a unique solution in a weak sense ([8]). After successive refinement in
[11] and [26], it is proved that the weak solution has $C^{1,1}$ regularity.

###  2.5. Energy functionals

In this subsection, we recall functionals on the space of Kähler potentials.

######  Definition 2.5.1.

  1. (1)

We define the Monge-Ampère energy of $\varphi\in\mathcal{H}^{S}$ by

$\displaystyle
E(\varphi)=\frac{1}{n+1}\sum_{i=0}^{n}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\varphi\omega_{\varphi}^{i}\wedge\omega_{0}^{n-i}.$

  2. (2)

We define the $J$-functional of $\varphi\in\mathcal{H}^{S}$ by

$\displaystyle
J(\varphi)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\varphi\omega_{0}^{n}-E(\varphi).$

######  Fact 2.5.2.

  1. (1)

$E(\varphi+c)=E(\varphi)+c$ for any $\varphi\in\mathcal{H}^{S}$ and
$c\in\mathbf{R}$.

  2. (2)

$J(\varphi)\geq 0$ and $J(\varphi+c)=J(\varphi)$ for any
$\varphi\in\mathcal{H}^{S}$ and $c\in\mathbf{R}$.

  3. (3)

For any smooth path $\\{\varphi^{t}\\}$ in $\mathcal{H}^{S}$ we have

$\displaystyle\frac{d}{dt}E(\varphi^{t})$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}\omega_{\varphi^{t}}^{n},$
$\displaystyle\frac{d}{dt}J(\varphi^{t})$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}(\omega_{0}^{n}-\omega_{\varphi^{t}}^{n}).$

Here we mean $\dot{\varphi}^{t}$ the derivative of $\varphi^{t}$ with
respective to $t$.

By using $E$, we define

$\displaystyle\mathcal{H}_{0}^{S}=\mathcal{H}(X,\omega)_{0}^{S}\coloneqq\\{\varphi\in\mathcal{H}^{S}\mid
E(\varphi)=0\\}.$

Then we have the identification
$\mathcal{H}^{S}\ni\varphi\mapsto(\varphi-E(\varphi),E(\varphi))\in\mathcal{H}_{0}^{S}\mathbf{R}$.

######  Definition 2.5.3.

We define the functionals on $\mathcal{H}_{0}^{S}$ as follows:

  1. (1)

We define th Ricci energy of $\varphi\in\mathcal{H}^{S}$ by

$\displaystyle
R(\varphi)=-\sum_{i=0}^{n-1}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\varphi\mathrm{Ric}(\omega_{0})\wedge\omega_{\varphi}^{i}\wedge\omega_{0}^{n-i-1}.$

  2. (2)

We define the entropy of $\varphi\in\mathcal{H}^{S}$ by

$\displaystyle
H(\varphi)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\log\left(\frac{\omega_{\varphi}^{n}}{\omega_{0}^{n}}\right)\omega_{\varphi}^{n}.$

  3. (3)

We define the K-energy of $\varphi\in\mathcal{H}^{S}$ by

$\displaystyle
M(\varphi)=-\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}(s(\omega_{\varphi^{t}})-\overline{s})\omega_{\varphi^{t}}^{n},$

where $\\{\varphi^{t}\\}_{t\in[0,1]}$ is a smooth path in $\mathcal{H}^{S}$ so
that $\varphi^{0}=0$ and $\varphi^{1}=\varphi$.

######  Fact 2.5.4.

  1. (1)

For any smooth path $\\{\varphi^{t}\\}$ in $\mathcal{H}^{S}$ we have

$\displaystyle\frac{d}{dt}R(\varphi^{t})$
$\displaystyle=-n\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}\mathrm{Ric}(\varphi_{0})\wedge\omega_{\varphi^{t}}^{n-1}.$

  2. (2)

$M(\varphi)=H(\varphi)+R(\varphi)+\overline{s}E(\varphi)$ for any
$\varphi\in\mathcal{H}_{0}^{S}$, and in particular the definition of
$M(\varphi)$ is independent of choice of the path
$\\{\varphi^{t}\\}_{t\in[0,1]}$ in $\varphi\in\mathcal{H}_{0}^{S}$ ([19], [9,
Proposition 3.2]).

  3. (3)

For any smooth path $\\{\varphi^{t}\\}$ in $\mathcal{H}^{S}$ we have

$\displaystyle\frac{d}{dt}M(\varphi^{t})$
$\displaystyle=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}(s(\omega_{\varphi^{t}})-\overline{s})\omega_{\varphi^{t}}^{n}.$

In particlar, $\varphi\in\mathcal{H}_{0}^{S}$ is a critical point of $M$ if
and only if the scalar curvature of $\omega_{\varphi}$ is constant.

  4. (4)

For each $\varphi\in\mathcal{H}_{0}^{S}$ and $W\in\mathfrak{s}$, put

$\displaystyle\varphi^{t}=\exp\left(-\frac{t}{4\pi}JW\right)^{*}\varphi.$

Then $\dot{\varphi}^{t}=u_{\omega_{\varphi^{t}}}^{W}$ and

$\displaystyle\frac{d}{dt}M(\varphi^{t})=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}u_{\omega_{\varphi^{t}}}^{W}(s(\omega_{\varphi^{t}})-\overline{s})\omega_{\varphi^{t}}^{n}=F(W).$

In other word, the K-energy functional is the integration of the Futaki
invariant.

  5. (5)

For a smooth geodesic $\\{\varphi^{t}\\}$ in $\mathcal{H}_{0}^{S}$ we have

$\displaystyle\frac{d^{2}}{dt^{2}}M(\varphi^{t})=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}|\overline{\partial}\mathrm{grad}_{g_{\varphi^{t}}}\dot{\varphi^{t}}|_{g_{\varphi^{t}}}^{2}\omega_{\varphi^{t}}^{n}\geq
0.$

Hence $M$ is convex along the geodesic $\\{\varphi^{t}\\}$.

###  2.6. Vector field energies and the relative K-energy

In this subsection, we introduce the vector field energy functionals defined
by Mabuchi [61] and the relative K-energy functional defined by [61], [23],
[41], and [70] independently.

Let $(X,L)$ be an $n$-dimensional polarized manifold. Fix
$\omega_{0}\in\mathcal{H}(X,L)^{S}$, and define

$\displaystyle
H_{V}(\varphi)=\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}\theta_{\varphi^{t}}\omega_{\varphi^{t}}^{n}$

for any $\varphi\in\mathcal{H}^{S}$, where $\\{\varphi^{t}\\}_{t\in[0,1]}$ is
a smooth path in $\mathcal{H}^{S}$ so that $\varphi^{0}=0$ and
$\varphi^{1}=\varphi$.

######  Fact 2.6.1 ([61]).

  1. (1)

For any $\varphi\in\mathcal{H}_{0}^{S}$

$\displaystyle H_{V}(\varphi)$
$\displaystyle=\frac{1}{n+1}\sum_{i=0}^{n}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\varphi\theta_{\omega_{\varphi}}\omega_{\varphi}^{i}\wedge\omega_{0}^{n-i}$
$\displaystyle\quad-\frac{1}{(n+1)(n+2)}\sum_{i=0}^{n}(n-i+1)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\varphi
V\varphi\omega_{\varphi}^{i}\wedge\omega_{0}^{n-i}.$

In particular, the definition of $H_{V}(\varphi)$ is independent of choice of
the path $\\{\varphi^{t}\\}_{t\in[0,1]}$ in $\varphi\in\mathcal{H}_{0}^{S}$.

  2. (2)

For any smooth path $\\{\varphi^{t}\\}$ in $\mathcal{H}_{0}^{S}$ we have

$\displaystyle\frac{d}{dt}H_{V}(\varphi^{t})=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}\theta_{\varphi^{t}}\omega_{\varphi^{t}}^{n}.$

  3. (3)

For each $\varphi\in\mathcal{H}_{0}^{S}$ and $W\in\mathfrak{s}$, put
$\varphi^{t}=\exp(-tJW/4\pi)^{*}\varphi$. Then
$\dot{\varphi}^{t}=u_{\omega_{\varphi^{t}}}^{W}$ and

$\displaystyle\frac{d}{dt}H_{V}(\varphi^{t})=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}u_{\omega_{\varphi^{t}}}^{W}\theta_{\omega_{\varphi}}\omega_{\varphi^{t}}^{n}=B(W,V).$

  4. (4)

For a smooth geodesic $\\{\varphi^{t}\\}$ in $\mathcal{H}_{0}^{S}$, we have

$\displaystyle\frac{d^{2}}{dt^{2}}H_{V}(\varphi^{t})=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\theta_{\omega_{\varphi^{t}}}\left(\ddot{\varphi}^{t}-\frac{1}{2}|d\dot{\varphi}^{t}|_{g_{\varphi^{t}}}^{2}\right)\omega_{\varphi^{t}}^{n}=0.$

Hence $H_{V}$ is affine along the geodesic $\\{\varphi^{t}\\}$.

######  Definition 2.6.2 ([61], [23], [41], [70]).

We define the relative K-energy functional
$M_{V}\colon\mathcal{H}^{S}\to\mathbf{R}$ by

$\displaystyle M_{V}(\varphi)\coloneqq
M(\varphi)+H_{V}(\varphi)=-\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}(s(\omega_{\varphi^{t}})-\overline{s}-\theta_{\varphi^{t}})\omega_{\varphi^{t}}^{n}.$

######  Fact 2.6.3.

  1. (1)

For any smooth path $\\{\varphi^{t}\\}$ in $\mathcal{H}_{0}^{S}$ we have

$\displaystyle\frac{d}{dt}M_{V}(\varphi^{t})=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}\dot{\varphi}^{t}(s(\omega_{\varphi^{t}})-\overline{s}-\theta_{\varphi^{t}})\omega_{\varphi^{t}}^{n}.$

Hence $\omega_{\varphi}$ is an extremal Kähler metric if and only if
$\varphi\in\mathcal{H}_{0}^{S}$ is a critical point of $M_{V}$.

  2. (2)

For each $\varphi\in\mathcal{H}_{0}^{S}$ and $W\in\mathfrak{s}$, put
$\varphi^{t}=\exp(-tJW/4\pi)^{*}\varphi$. Then
$\dot{\varphi}^{t}=u_{\omega_{\varphi^{t}}}^{W}$ and

$\displaystyle\frac{d}{dt}M_{V}(\varphi^{t})=F(W)+B(W,V)=0.$

  3. (3)

For a smooth geodesic $\\{\varphi^{t}\\}$ in $\mathcal{H}_{0}^{S}$ we have

$\displaystyle\frac{d^{2}}{dt^{2}}M_{V}(\varphi^{t})=\frac{d^{2}}{dt^{2}}M(\varphi^{t})\geq
0.$

Hence $M_{V}$ is also convex along the geodesic $\\{\varphi^{t}\\}$.

###  2.7. $d_{1}$-distances, reduced $d_{1}$-distances and the reduced
J-functionals

Let $(X,L)$ be an $n$-dimensional polarized manifold. For any smooth path
$\\{\varphi^{t}\\}_{t\in[0,1]}\in\mathcal{H}_{0}^{S}$, we define the
$L^{1}$-length of $\\{\varphi^{t}\\}_{t\in[0,1]}$ by

$\displaystyle\ell_{1}(\\{\varphi^{t}\\}_{t\in[0,1]})\coloneqq\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}|\dot{\varphi^{t}}|\omega_{\varphi^{t}}^{n}.$

######  Definition 2.7.1.

For each $\varphi_{0},\varphi_{1}\in\mathcal{H}_{0}^{S}$, we define the
$L^{1}$-distance of $\varphi_{0}$ and $\varphi_{1}$ by

$\displaystyle
d_{1}(\varphi_{0},\varphi_{1})\coloneqq\inf_{\\{\varphi^{t}\\}_{t\in[0,1]}}\ell_{1}(\\{\varphi^{t}\\}_{t\in[0,1]}),$

where $\\{\varphi^{t}\\}_{t\in[0,1]}$ runs through all smooth paths in
$\mathcal{H}_{0}^{S}$ so that $\varphi^{0}=\varphi_{0}$ and
$\varphi^{1}=\varphi_{1}$.

######  Fact 2.7.2 ([28]).

The $L^{1}$-distance $d_{1}$ is in fact a distance function on
$\mathcal{H}_{0}^{S}$. However, the metric space $(\mathcal{H}_{0}^{S},d_{1})$
is not complete.

Next we consider the reduced $L^{1}$-distance. Let us consider the action of
$T$ on $\mathcal{H}_{0}^{S}$. For each $\tau\in T$ and
$\varphi\in\mathcal{H}_{0}^{S}$, define $\varphi_{\tau}\in\mathcal{H}_{0}^{S}$
by

$\displaystyle\tau^{*}\omega_{\varphi}=\omega_{0}+dd^{c}\varphi_{\tau}=\omega_{\varphi_{\tau}}.$

Since

$\displaystyle\tau^{*}\omega_{\varphi}=\tau^{*}(\omega_{0}+dd^{c}\varphi)=\omega_{0}+dd^{c}(0_{\tau}+\tau^{*}\varphi)$

and

$\displaystyle E(0_{\tau}+\tau^{*}\varphi)$
$\displaystyle=E(0_{\tau}+\tau^{*}\varphi)-E(0_{\tau})$
$\displaystyle=\frac{1}{n+1}\sum_{i=0}^{n}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}(\tau^{*}\varphi)(\tau^{*}\omega_{\varphi})^{i}\wedge(\tau^{*}\omega_{0})^{n-i}$
$\displaystyle=E(\varphi)=0,$

we have $0_{\tau}+\tau^{*}\varphi\in\mathcal{H}_{0}^{S}$ and
$\varphi_{\tau}=0_{\tau}+\tau^{*}\varphi$.

######  Proposition 2.7.3.

The $T$-action on $(\mathcal{H}_{0}^{S},d_{1})$ is isometric, that is, for
each $\tau\in T$ and $\varphi_{0},\varphi_{1}\in\mathcal{H}_{0}^{S}$, we have

$\displaystyle
d_{1}((\varphi_{0})_{\tau},(\varphi_{1})_{\tau})=d_{1}(\varphi_{0},\varphi_{1}).$

###### Proof.

Let $\\{\varphi^{t}\\}_{t\in[0,1]}$ be a smooth path in $\mathcal{H}_{0}^{S}$
connecting $\varphi_{0}$ and $\varphi_{1}$. Then
$\\{\varphi_{\tau}^{t}\\}_{t\in[0,1]}$ is a smooth path in
$\mathcal{H}_{0}^{S}$ connecting $(\varphi_{0})_{\tau}$ and
$(\varphi_{1})_{\tau}$, and

$\displaystyle\dot{\varphi}_{\tau}^{t}=\frac{d}{dt}(0_{\tau}+\tau^{*}\varphi^{t})=\tau^{*}\dot{\varphi}^{t}.$

Hence we have

$\displaystyle\ell_{1}(\\{\varphi_{\tau}^{t}\\}_{t\in[0,1]})$
$\displaystyle=\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}|\dot{\varphi}_{\tau}^{t}|\omega_{\varphi_{\tau}^{t}}^{n}$
$\displaystyle=\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}|\tau^{*}\dot{\varphi}^{t}|\tau^{*}\omega_{\varphi^{t}}^{n}=\ell_{1}(\\{\varphi^{t}\\}_{t\in[0,1]}),$

as required. ∎

######  Definition 2.7.4.

For each $\varphi_{0},\varphi_{1}\in\mathcal{H}_{0}^{S}$, we define the
reduced $L^{1}$-distance of $\varphi_{0}$ and $\varphi_{1}$ by

$\displaystyle
d_{1,T}(\varphi_{0},\varphi_{1})\coloneqq\inf_{\tau_{0},\tau_{1}\in
T}d_{1}((\varphi_{0})_{\tau_{0}},(\varphi_{1})_{\tau_{1}})=\inf_{\tau\in
T}d_{1}((\varphi_{0})_{\tau},\varphi_{1}).$

Similarly, we introduce the reduced $J$-functional.

######  Definition 2.7.5.

We define the reduced $J$-functional by

$\displaystyle J_{T}(\varphi)\coloneqq\inf_{\tau\in T}J(\varphi_{\tau}),$

for any $\varphi\in\mathcal{H}_{0}^{S}$.

######  Proposition 2.7.6.

The $J$-functional is equivalent to the $L^{1}$-distance in the following
sense: there exists a $C>0$ such that for any $\varphi\in\mathcal{H}_{0}^{S}$

$\displaystyle\frac{1}{C}J(\varphi)-C\leq d_{1}(\varphi,0)\leq CJ(\varphi)+C.$

######  Proposition 2.7.7.

The reduced $J$-functional is equivalent to the reduced $L^{1}$-distance in
the following sense: there exists a $C>0$ such that for any
$\varphi\in\mathcal{H}_{0}^{S}$

$\displaystyle\frac{1}{C}J_{T}(\varphi)-C\leq d_{1,T}(\varphi,0)\leq
CJ_{T}(\varphi)+C.$

By using the theory of Chen-Cheng and He [20], [21], [22], [47], and the
arguments in [55], we shall show the following theorem.

######  Theorem 2.7.8.

For a polarized manifold $(X,L)$, the followings are equivalent.

  1. (1)

$(X,L)$ admits an $S$-invariant extremal Kähler metric.

  2. (2)

There exists $\delta,C>0$ such that

$\displaystyle M_{V}(\varphi)\geq\delta d_{1,T}(\varphi,0)-C$

for any $\varphi\in\mathcal{H}_{0}^{S}$.

  3. (3)

There exists $\delta,C>0$ such that

$\displaystyle M_{V}(\varphi)\geq\delta J_{T}(\varphi)-C$

for any $\varphi\in\mathcal{H}_{0}^{S}$.

We say that the relative K-energy is $T$-coercive if $M_{V}$ satisfies the
condition (2) and (or) (3) in Theorem 2.7.8.

###### Proof of Theorem 2.7.8.

The equivalence of (2) and (3) is ovbious from Proposition 2.7.7. We shall
show the equivalence of (1) and (2). For this, we apply Darvas-Rubinstein’s
existence/properness principle [30, Theorem 3.4]. In their notation, we
consider the following data

$\displaystyle\mathcal{R}=\mathcal{H}_{0}^{S},\quad d=d_{1},\quad
F=M_{V},\quad G=T.$

Note that the lower semicontinuity of $M_{V}$ is obtained in [47, Corollary
2.2]. For (P1), existence of $d_{1}$-geodesic can be found in [18, Theorem 3]
and [28, Theorem 2]. Continuity and convexity of $M_{V}$ can be found in [9,
Theorem 3.4] and [47, Proposition 2.2]. The property (P2) is obtained by the
compactness theorem in [10, Theorem 2.17] and the lower semicontinuity of
$M_{V}$. The property (P3) is due to [47, Theorem 3.6]. The property (P4) is
already showed in Proposition 2.7.3. The property (P6) can be omitted (see
Remark 4.8 of [29]). The property (P7) is easily proved.

We need to check the property (P5). Let
$\mathfrak{aut}(X,V)=\\{W\in\mathfrak{aut}(X)\mid L_{V}W=0\\}$. Then
$\mathfrak{aut}(X,V)$ is a Lie subalgebra of $\mathfrak{aut}(X)$. We denote by
$H\coloneqq\mathrm{Aut}(X,V)_{0}$ the connected Lie subgroup of
$\mathrm{Aut}(X)_{0}$ corresponding to $\mathfrak{aut}(X,V)$. Let
$\omega_{i}$, $i=1,2$ be two $S$-invariant extremal Kähler metrics in
$\mathcal{H}(X,L)^{S}$, and

$\displaystyle K_{i}=\mathrm{Isom}(X,\omega_{i})_{0}.$

By a theorem of Calabi [15], $\mathrm{Aut}(X,V)_{0}$ is a reductive and each
$K_{i}$ is a maximal compact subgroup of $\mathrm{Aut}(X,V)_{0}$. Since
$\omega_{i}$ is $S$-invariant, we have $S\subset K_{1}\cap K_{2}$ and that $S$
is a maximal compact torus of both $K_{1}$ and $K_{2}$. Hence by [55,
Proposition A.2] $K_{2}=\tau K_{1}\tau^{-1}$ for some $\tau\in T$.

By Berman-Berndttson’s uniqueness theorem of extremal Kähler metrics, we can
find $f\in H$ satisfying $\omega_{1}=f^{*}\omega_{2}$ (see the proof of [9,
Theorem 4.16]). Then we have $K_{1}=fK_{2}f^{-1}=(f\tau)K_{1}(f\tau)^{-1}$ and
hence $f\tau\in N_{H}(K_{1})$, where $N_{H}(K_{1})$ is the normalizer of
$K_{1}$ in $H$. Furthermore, by [55, Proposition A.1] we have $f\tau\in
C(H)_{0}K_{1}$, where $C(H)_{0}$ is the identity component of the center of
$H$. Hence there exists $\tau^{\prime}\in C(H)_{0}\subset T$ and $k_{1}\in
K_{1}$ such that $f\tau=\tau^{\prime}k_{1}$. By setting
$\tau_{1}=\tau^{\prime}\tau^{-1}$, we have $\tau_{1}\in T$, $f=k_{1}\tau_{1}$
and

$\displaystyle\omega_{1}=f^{*}\omega_{2}=(k_{1}\tau_{1})^{*}\omega_{2}=\tau_{1}^{*}k_{1}^{*}\omega_{2}=\tau_{1}^{*}\omega_{2}.$

∎

##  3\. Test configurations and relative K-stability

###  3.1. Test configurations

Let $(X,L)$ be an $n$-dimensional polarized manifold.

######  Definition 3.1.1.

A test configuration for $(X,L)$ of exponent $r$ consists of

  1. (1)

a scheme $\mathcal{X}$ with a $\mathbf{G}_{m}$-action,

  2. (2)

a $\mathbf{G}_{m}$-equivariant flat and proper morphism of schemes
$\pi\colon\mathcal{X}\to\mathbf{A}^{1}$, where $\mathbf{G}_{m}$ acts on
$\mathbf{A}^{1}$ by multiplication,

  3. (3)

a $\mathbf{G}_{m}$-linearlized $\pi$-very ample line bundle $\mathcal{L}$ over
$\mathcal{X}$, and

  4. (4)

an isomorphism $(\mathcal{X}_{1},\mathcal{L}_{1})\cong(X,L^{r})$.

Here $\mathcal{X}_{t}$ is the fiber of $\pi$ over $t\in\mathbf{A}^{1}$, and
$\mathcal{L}_{t}=\mathcal{L}|_{\mathcal{X}_{t}}$. A test configuration
$(\mathcal{X},\mathcal{L})$ is called product if $\mathcal{X}$ is isomorphic
to the fiber product of $X$ and $\mathbf{A}^{1}$ over $\mathbf{A}^{1}$, and
trivial if in addition $\mathbf{G}_{m}$ acts only on the second factor. A test
configuration $(\mathcal{X},\mathcal{L})$ is called normal if $\mathcal{X}$ is
a normal variety. For an algebraic subgroup $G$ of $\mathrm{Aut}(X,L)$, a
$G$-equivariant test configuration is a test configuration
$(\mathcal{X},\mathcal{L})$ with a lifted $G$-action on
$(\mathcal{X},\mathcal{L})$ which preserves each fiber, commutes with the
fiber $\mathbf{G}_{m}$-action of $(\mathcal{X},\mathcal{L})$, and coincides
with $G$-action when actiong on
$(\mathcal{X}_{1},\mathcal{L}_{1})\cong(X,L^{r})$.

For our purpose, it is convenient to compactify test configurations. Let
$(\mathcal{X},\mathcal{L})$ be a test configuration for $(X,L)$ of exponent
$r$. The $\mathbf{G}_{m}$-action on $(\mathcal{X},\mathcal{L})$ and the
isomorphism $(\mathcal{X}_{1},\mathcal{L}_{1})\cong(X,L^{r})$ induces an
equivariant trivialization
$(\mathcal{X},\mathcal{L})|_{\mathbf{G}_{m}}\cong(X\times\mathbf{G}_{m},p_{1}^{\ast}L^{r})$,
where $p_{1}\colon X\times\mathbf{G}_{m}\to X$ is the projection to the first
factor, and $\mathbf{G}_{m}$ acts only on the second factor. Then we can
compactify $(\mathcal{X},\mathcal{L})$ by gluing it with the product
$(X\times(\mathbf{P}^{1}\setminus\\{0\\}),p_{1}^{\ast}L^{r})$ along their
respective open sets $\mathcal{X}\setminus\mathcal{X}_{0}$ and
$X\times(\mathbf{A}^{1}\setminus\\{0\\})$. The resulting
$(\overline{\mathcal{X}},\overline{\mathcal{L}})$ with
$\overline{\pi}\colon\overline{\mathcal{X}}\to\mathbf{P}^{1}$ is a
$\mathbf{G}_{m}$-equivariant flat family over $\mathbf{P}^{1}$ with fibers
isomorphic to $(X,L^{r})$ except over 0. We call
$(\overline{\mathcal{X}},\overline{\mathcal{L}})$ _the canonical
compactification_ of the test configuration $(\mathcal{X},\mathcal{L})$. Note
that the action of $\mathbf{G}_{m}$ on the $\infty$-fiber
$(\overline{\mathcal{X}}_{\infty},\overline{\mathcal{L}}_{\infty})$ is
trivial.

######  Example 3.1.2.

Since $L$ is ample, there exists an embeddings

$\displaystyle\Phi_{m}\colon X\to\mathbf{P}(H^{0}(X,L^{m}))^{\vee}$

for any sufficiently divisible $m\in\mathbf{Z}_{>0}$. By replacing $m$ if
necessary, we obtain a faithful representation

$\displaystyle\theta_{m}\colon T\to\mathrm{GL}(H^{0}(X,L^{m}))$

satisfying

$\displaystyle\theta_{m}(\tau)\circ\Phi_{m}=\Phi_{m}\circ\tau$

for any $\tau\in T$. Then the infinitesimal representaion
$\theta_{m*}\colon\mathfrak{t}\to\mathfrak{gl}(H^{0}(X,L^{m}))$ is given by

$\displaystyle-\frac{1}{4\pi}\theta_{m*}(JW)=\nabla_{W}+mu_{\omega}^{W}\mathrm{id}_{H^{0}(X,L^{m})}.$

Let $C_{m}(T)$ be the centralizer of a subset $\theta_{m}(T)$ in
$\mathrm{GL}(H^{0}(X,L^{m}))$. For any morphism
$\lambda\colon\mathbf{G}_{m}\to C_{m}(T)$, we define $\mathcal{X}$ as the
Zariski closure of the set

$\displaystyle\bigcup_{\tau\in\mathbf{G}_{m}}\lambda^{\vee}(\tau)(\Phi_{m}(X))\times\\{\tau\\}$

in $\mathbf{P}(H^{0}(X,L^{m}))^{\vee}\times\mathbf{A}^{1}$, and set
$\mathcal{L}=\mathcal{O}_{\mathcal{X}}(1)$. Then $(\mathcal{X},\mathcal{L})$
is a test confiduration for $(X,L)$ of exponent $m$. The central fiber
$\mathcal{X}_{0}$ is obtained as the flat limit of the image of $\Phi_{m}(X)$
under $\lambda^{\vee}$ as $\tau\to 0$.

######  Example 3.1.3.

The set of test configurations for $(X,L)$ admits two natural operations:
translations and scalings. $(\mathcal{X},\mathcal{L})$ be a test configuration
for $(X,L)$.

  1. (1)

Let $c\in\mathbf{Q}$. A translation of $(\mathcal{X},\mathcal{L})$ is defined
to be
$(\mathcal{X},\mathcal{L}+c\mathcal{X}_{0})\coloneqq(\mathcal{X},\mathcal{L}\otimes_{\mathcal{X}}\mathcal{O}_{\mathcal{X}}(c\mathcal{X}_{0}))$.

  2. (2)

Let $d\in\mathbf{Z}_{>0}$. A scaling of $(\mathcal{X},\mathcal{L})$ is defined
to be the normalization of the base change of $(\mathcal{X},\mathcal{L})$ by
$\tau\mapsto\tau^{d}$. We denote such a test configuration by
$(\mathcal{X}_{d},\mathcal{L}_{d})$.

Associated to a test configuration, the Donaldson-Futaki invariant is defined
as follows. Let $(\mathcal{X},\mathcal{L})$ be a test configuration for
$(X,L)$ of exponent $r$. By the flatness of
$\pi\colon\mathcal{X}\to\mathbf{A}^{1}$ we have

$\displaystyle
N_{rm}\coloneqq\dim_{\mathbf{C}}H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})=\dim_{\mathbf{C}}H^{0}(X,L^{rm})$

for sufficiently large $m\in\mathbf{Z}_{>0}$. Moreover, the asymptotic
Riemann-Roch theorem tells us that

$\displaystyle N_{m}=a_{0}m^{n}+a_{1}m^{n-1}+O(m^{n-2}),$

where

$\displaystyle a_{0}=\frac{(L^{n})}{n!},\quad
a_{1}=-\frac{1}{2}\frac{(K_{X}\cdot L^{n-1})}{(n-1)!}.$

Note that

$\displaystyle\overline{s}=-n\frac{(K_{X}\cdot
L^{n-1})}{(L^{n})}=-2\frac{a_{1}}{a_{0}}.$

Since $0\in\mathbf{A}^{1}$ is fixed by the $\mathbf{G}_{m}$-action, the
$\mathbf{G}_{m}$-action on $(\mathcal{X},\mathcal{L})$ induces a coaction
$\lambda_{m}\colon H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})\to
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})\otimes\mathbf{C}[t,t^{-1}]$. Let

$\displaystyle
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})=\bigoplus_{i=1}^{N_{rm}}H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})_{\lambda_{i}^{(rm)}}$

be the weight decomposition. Here
$H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})_{\lambda}\coloneqq\\{s\in
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})\mid\lambda_{m}(s)=s\otimes\tau^{\lambda}\\}$
is the $\lambda$-weight space for this action. By [12, Theorem 3.1], we have

$\displaystyle
w_{m}\coloneqq\sum_{i=1}^{N_{m}}\lambda_{i}^{(m)}=b_{0}m^{n+1}+b_{1}m^{n}+O(m^{n-1})$

for sufficiently large $m$ divisible by $r$. We then consider the coeffients

$\displaystyle\frac{w_{m}}{mN_{m}}=F_{0}(\mathcal{X},\mathcal{L})+F_{1}(\mathcal{X},\mathcal{L})m^{-1}+O(m^{-2}),$
$\displaystyle F_{0}(\mathcal{X},\mathcal{L})=\frac{b_{0}}{a_{0}},\quad
F_{1}(\mathcal{X},\mathcal{L})=\frac{a_{0}b_{1}-a_{1}b_{0}}{a_{0}^{2}}.$

######  Definition 3.1.4.

We call

$\displaystyle
DF(\mathcal{X},\mathcal{L})\coloneqq-2F_{1}(\mathcal{X},\mathcal{L})=-2\frac{a_{0}b_{1}-a_{1}b_{0}}{a_{0}^{2}}$

the Donaldson-Futaki invariant of a test configuration
$(\mathcal{X},\mathcal{L})$.

When the test configuration $(\mathcal{X},\mathcal{L})$ is normal, we can
describe the Donaldson-Futaki invariant via intersection numbers ([78,
Proposition 17], [57, Proposition 6], [12, Proposition 3.12]). Indeed, by
using the canonical compactification and asymptotic Riemann-Roch theorem, we
have

$\displaystyle w_{rm}$
$\displaystyle=\chi(\overline{\mathcal{X}},\overline{\mathcal{L}}^{m})-\chi(X,L^{rm})$
$\displaystyle=\frac{(\overline{\mathcal{L}}^{n+1})}{r^{n+1}(n+1)!}(rm)^{n+1}-\frac{(K_{\overline{\mathcal{X}}}\cdot\overline{\mathcal{L}}^{n})}{2r^{n}n!}(rm)^{n}+O(m^{n-1})$
$\displaystyle\quad-\frac{(L^{n})}{n!}(rm)^{n}+O(m^{n-1})$
$\displaystyle=\frac{(\overline{\mathcal{L}}^{n+1})}{r^{n+1}(n+1)!}(rm)^{n+1}-\frac{1}{2r^{n}n!}\left((K_{\overline{\mathcal{X}}}\cdot\overline{\mathcal{L}}^{n})+2r^{n}(L^{n})\right)(rm)^{n}$
$\displaystyle\quad+O(m^{n-1})$

for any sufficiently large $m\in\mathbf{Z}_{>0}$ (see the proof of [12,
Proposition 3.12]). By setting $K_{\mathcal{X}/\mathbf{P}^{1}}\coloneqq
K_{\overline{\mathcal{X}}}-\overline{\pi}^{*}K_{\mathbf{P}^{1}}$, we obtain

$\displaystyle 2r^{n}(L^{n})$
$\displaystyle=2((L^{r})^{n})=2(\overline{\mathcal{L}}|_{\mathcal{\overline{X}_{1}}})^{n}=2(\overline{\mathcal{X}}_{1}\cdot\overline{\mathcal{L}}_{1}^{n})$
$\displaystyle=(\overline{\pi}^{*}(2[1])\cdot\overline{\mathcal{L}}_{1}^{n})=-(\overline{\pi}^{*}K_{\mathbf{P}^{1}}\cdot\overline{\mathcal{L}}^{n})$

and

$\displaystyle
b_{0}=\frac{(\overline{\mathcal{L}}^{n+1})}{r^{n+1}(n+1)!},\quad
b_{1}=-\frac{1}{2r^{n}n!}(K_{\overline{\mathcal{X}}/\mathbf{P}^{1}}\cdot\overline{\mathcal{L}}^{n}).$

Therefore, we obtain the intersection number formula

(3.1) $\displaystyle
DF(\mathcal{X},\mathcal{L})=\overline{s}\frac{(\overline{\mathcal{L}}^{n+1})}{r(n+1)(L^{n})}+\frac{1}{(L^{n})}(K_{\overline{\mathcal{X}}/\mathbf{P}^{1}}\cdot\overline{\mathcal{L}}^{n}).$

###  3.2. Duistermaat-Heckman measures

Let $(\mathcal{X},\mathcal{L})$ be a test configuration for $(X,L)$ of
exponent $r$. Then the central fiber $(\mathcal{X}_{0},\mathcal{L}_{0})$ is a
polarized $\mathbf{G}_{m}$-scheme. For sufficiently divisible
$m\in\mathbf{Z}_{>0}$, we can consider the induced $\mathbf{G}_{m}$-action on
$H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})$. Then there is a weak limit of
the measure

$\displaystyle\mathrm{DH}(\mathcal{X},\mathcal{L})=\lim_{m\to\infty}\frac{1}{N_{rm}}\sum_{\lambda\in\mathbf{Z}}\dim
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})_{\lambda}\delta_{\lambda/rm}$

on $\mathbf{R}$, where $\delta_{\lambda/rm}$ is the Dirac delta measure
concentrated at $\lambda/rm$ (see [12]). This is called _the Duistermaat-
Heckman measure_ of the test configuratin $(\mathcal{X},\mathcal{L})$.

###  3.3. Non-Archimedean functionals

In this subsection we introduce non-Archimedean functionals. Let $(X,L)$ be an
$n$-dimensional polarized manifold and $(\mathcal{X},\mathcal{L})$ a test
configuration for $(X,L)$ of exponent $r$.

######  Definition 3.3.1 ([12]).

  1. (1)

We define the non-Archimedean Monge-Ampère energy of $\varphi$ by

$\displaystyle
E^{NA}(\mathcal{X},\mathcal{L})\coloneqq\frac{(\overline{\mathcal{L}}^{n+1})}{(n+1)(L^{n})}=\int_{\mathbf{R}}\lambda\,d\mathrm{DH}(\mathcal{X},\mathcal{L}).$

  2. (2)

We define the non-Archimedean $J$-functional of $\varphi$ by

$\displaystyle
J^{NA}(\mathcal{X},\mathcal{L})\coloneqq\sup\mathrm{supp}(\mathrm{DH}(\mathcal{X},\mathcal{L}))-E^{NA}(\mathcal{X},\mathcal{L}).$

When $\mathcal{X}$ dominates the product $X\times\mathbf{P}^{1}$, namely,
there exists $\rho\colon\mathcal{X}\to X\times\mathbf{P}^{1}$ which is
equivariant with respect to the trivial action on the target, $J^{NA}$ can be
written as

$\displaystyle
J^{NA}(\mathcal{X},\mathcal{L})=\frac{1}{(L^{n})}(\overline{\mathcal{L}}\cdot\rho^{*}p_{1}^{*}L^{n})-\frac{(\overline{\mathcal{L}}^{n+1})}{r(n+1)(L^{n})}.$

Let

$\displaystyle K_{\overline{\mathcal{X}}}^{\mathrm{log}}\coloneqq
K_{\overline{\mathcal{X}}}+\overline{\mathcal{X}}_{0,\mathrm{red}}+\overline{\mathcal{X}}_{\infty,\mathrm{red}}=K_{\overline{\mathcal{X}}}+\overline{\mathcal{X}}_{0,\mathrm{red}}+\overline{\mathcal{X}}_{\infty},$
$\displaystyle K_{\mathbf{P}^{1}}^{\mathrm{log}}\coloneqq
K_{\mathbf{P}^{1}}+[0]+[\infty],$ $\displaystyle
K_{\overline{\mathcal{X}}/\mathbf{P}^{1}}^{\mathrm{log}}\coloneqq
K_{\overline{\mathcal{X}}}^{\mathrm{log}}-\overline{\pi}^{*}K_{\mathbf{P}^{1}}^{\mathrm{log}}=K_{\overline{\mathcal{X}}/\mathbf{P}^{1}}+(\overline{\mathcal{X}}_{0,\mathrm{red}}-\overline{\mathcal{X}}_{0}).$

######  Definition 3.3.2 ([12]).

Let $(\mathcal{X},\mathcal{L})$ be a test configuration for $(X,L)$.

  1. (1)

The non-Archimedean Ricci energy is defined by

$\displaystyle
R^{NA}(\mathcal{X},\mathcal{L})\coloneqq\frac{1}{(L^{n})}(\rho^{*}K_{X\times\mathbf{P}^{1}/\mathbf{P}^{1}}^{\mathrm{log}}\cdot\overline{\mathcal{L}}^{n}).$

  2. (2)

The non-Archimedean entropy is defined by

$\displaystyle
H^{NA}(\mathcal{X},\mathcal{L})\coloneqq\frac{1}{(L^{n})}(K_{\overline{\mathcal{X}}/\mathbf{P}^{1}}^{\mathrm{log}}\cdot\overline{\mathcal{L}}^{n})-\frac{1}{(L^{n})}(\rho^{*}K_{X\times\mathbf{P}^{1}/\mathbf{P}^{1}}^{\mathrm{log}}\cdot\overline{\mathcal{L}}^{n}).$

  3. (3)

The non-Archimedean K-energy is defined by

$\displaystyle M^{NA}(\varphi)$ $\displaystyle\coloneqq
H^{NA}(\mathcal{X},\mathcal{L})+R^{NA}(\mathcal{X},\mathcal{L})+\overline{s}E^{NA}(\mathcal{X},\mathcal{L})$
$\displaystyle=\overline{s}\frac{(\overline{\mathcal{L}}^{n+1})}{(n+1)(L^{n})}+\frac{1}{(L^{n})}(K_{\overline{\mathcal{X}}/\mathbf{P}^{1}}^{\mathrm{log}}\cdot\overline{\mathcal{L}}^{n}).$

The non-Archimedean K-energy is closely related to the Donaldson-Futaki
invariant. In fact, by comparing (3.1) we have

$\displaystyle M^{NA}(\mathcal{X},\mathcal{L})$
$\displaystyle=DF(\mathcal{X},\mathcal{L})+\frac{1}{(L^{n})}(K_{\overline{\mathcal{X}}/\mathbf{P}^{1}}^{\mathrm{log}}\cdot\overline{\mathcal{L}}^{n})-\frac{1}{(L^{n})}(K_{\overline{\mathcal{X}}/\mathbf{P}^{1}}\cdot\overline{\mathcal{L}}^{n})$
$\displaystyle=DF(\mathcal{X},\mathcal{L})+\frac{1}{(L^{n})}((\overline{\mathcal{X}}_{0,\mathrm{red}}-\overline{\mathcal{X}}_{0})\cdot\overline{\mathcal{L}}^{n}).$

Hence $M^{NA}(\mathcal{X},\mathcal{L})\leq DF(\mathcal{X},\mathcal{L})$ for
any test configuration $(\mathcal{X},\mathcal{L})$ and equality holds if and
only if $\mathcal{X}$ is regular in codimension one and $\mathcal{X}_{0}$ is
generically reduced ([12, Proposition 7.15], [13, Proposition 2.8]). An
advantage of $M^{NA}$ is the following homogeneity with respect to base
change, which fails for $DF$ when the central fiber is non-reduced.

######  Proposition 3.3.3 ([12, Proposition 7.14]).

Let $(\mathcal{X},\mathcal{L})$ be a test configuration for $(X,L)$. For each
$d\in\mathbf{Z}_{>0}$, let $(\mathcal{X}_{d},\mathcal{L}_{d})$ be the test
configuration obtained by the base change $z\mapsto z^{d}$. Then
$M^{NA}(\mathcal{X}_{d},\mathcal{L}_{d})=dM^{NA}(\mathcal{X},\mathcal{L})$.

On the other hand, Mumford’s semistable reduction theorem ([51, p.53,
pp.100–101]. See [5, Theorem 3.8] and [57, Lemma 5] for equivariant version)
gives us the following proposition.

######  Proposition 3.3.4 ([12, Proposition 7.16]).

For each test configuration $(\mathcal{X},\mathcal{L})$ for $(X,L)$, there
exists $d_{0}\in\mathbf{Z}_{>0}$ such that
$DF(\mathcal{X}_{d},\mathcal{L}_{d})=M^{NA}(\mathcal{X}_{d},\mathcal{L}_{d})=dM^{NA}(\mathcal{X},\mathcal{L})$
for all $d\in\mathbf{Z}_{>0}$ divisible by $d_{0}$.

Now we consider the $T$-action on $(X,L)$. Let $(\mathcal{X},\mathcal{L})$ be
a $T$-equivariant test configuration for $(X,L)$. Let $V$ be the extremal
Kähler vector field for $T$, and $T_{V}$ the one-dimensional algebraic torus
generated by $V$. Then $T_{V}\subset T$ and induces a coaction

$\displaystyle\theta_{m}\colon H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})\to
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})\otimes\mathbf{C}[t,t^{-1}]$

for each $m\in\mathbf{Z}_{>0}$. We denote by
$(\theta_{1}^{(rm)},\ldots,\theta_{N_{rm}}^{(rm)})$ the weight of
$\theta_{m}^{\vee}$. Let $\Theta_{rm}\coloneqq\theta_{m*}(1)$ be the generator
of $\theta_{m}$ and $\Theta_{rm}^{\circ}$ the its traceless part.

######  Definition 3.3.5.

Let $(\mathcal{X},\mathcal{L})$ a $T$-equivariant test configuration for
$(X,L)$.

  1. (1)

${\displaystyle
H_{V}^{NA}(\mathcal{X},\mathcal{L})\coloneqq\lim_{m\to\infty}\frac{1}{(rm)^{2}N_{rm}}\mathrm{tr}(\Lambda_{rm}^{\circ}\Theta_{rm}^{\circ})}$.

  2. (2)

We define the non-Archimedean relatice K-energy of $\varphi$ by

$\displaystyle M_{V}^{NA}(\mathcal{X},\mathcal{L})\coloneqq
M^{NA}(\mathcal{X},\mathcal{L})+H_{V}^{NA}(\mathcal{X},\mathcal{L}).$

  3. (3)

We define the relative Donaldson-Futaki invariant of $\varphi$ by

$\displaystyle DF_{V}(\mathcal{X},\mathcal{L})\coloneqq
DF(\mathcal{X},\mathcal{L})+H_{V}^{NA}(\mathcal{X},\mathcal{L}).$

The following slope formula relates the Archimedean and non-Archimedean
functionals. Let $(\mathcal{X},\mathcal{L})$ be a $T$-equivariant test
configuration for $(X,L)$ and
$\lambda\colon\mathbf{G}_{m}\to\mathrm{Aut}^{0}(\mathcal{X},\mathcal{L})$ the
$\mathbf{G}_{m}$-action. We denote by $h_{\mathrm{FS}}$ the pull back of the
Fubini-Study metric on
$\mathbf{P}H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})^{\vee}$ by the
composition

$\displaystyle\mathcal{X}\hookrightarrow\mathbf{P}H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})^{\vee}\times\mathbf{A}^{1}\to\mathbf{P}H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})^{\vee},$

and $h_{\mathrm{FS}}|_{\tau}$ the restriction of $h_{\mathrm{FS}}$ to
$(\mathcal{X}_{\tau},\mathcal{L}_{\tau})$ for any $\tau\in\mathbf{G}_{m}$. By
setting

$\displaystyle
h_{t}\coloneqq\lambda(e^{-t/2})^{*}h_{\mathrm{FS}}|_{e^{-t/2}},\quad\omega_{0}\coloneqq
c_{1}(L,h_{0})$

for each $t\in[0,\infty)$, we have
$\varphi^{t}\in\mathcal{H}(X,\omega_{0})_{0}^{S}$ by

$\displaystyle c_{1}(L,h_{t})-\omega_{0}=dd^{c}\varphi^{t}.$

######  Theorem 3.3.6 ([13, Theorem 3.6], [79, Theorem 12]).

Let $(\mathcal{X},\mathcal{L})$ and $\\{\varphi^{t}\\}_{t\in[0,\infty)}$ be as
above, and $F$ be one of the following functionals: $E$, $J$, $M$, $H_{V}$,
and $M_{V}$. Then

$\displaystyle\lim_{t\to\infty}\frac{F(\varphi^{t})}{t}=F^{NA}(\mathcal{X},\mathcal{L}).$

###  3.4. Filtrations

To introduce the non-Archimedean version of the reduced $J$-functional, it is
convenient to use the notion of the _filtration_ of the section ring

$\displaystyle R=R(X,L)\coloneqq\bigoplus_{m=0}^{\infty}R_{m},\quad
R_{m}=H^{0}(X,L^{m})$

of a polarized manifold $(X,L)$. Following [12], we give a brief review of
filtrations.

######  Definition 3.4.1.

A filtration $\mathcal{F}$ of the graded $\mathbf{C}$-algebra $R$ consists of
a family of subspaces $\\{F^{\lambda}R_{m}\\}$ of $R_{m}$ for each
$m\in\mathbf{Z}_{\geq 0}$ satisfying:

  1. (1)

decreasing: $F^{\lambda}R_{m}\subset F^{\lambda^{\prime}}R_{m}$ if
$\lambda\geq\lambda^{\prime}$,

  2. (2)

left-continuous:
$F^{\lambda}R_{m}=\bigcap_{\lambda^{\prime}<\lambda}F^{\lambda^{\prime}}R_{m}$,

  3. (3)

multiplicative: $F^{\lambda}R_{m}\cdot
F^{\lambda^{\prime}}R_{m^{\prime}}\subset
F^{\lambda+\lambda^{\prime}}R_{m+m^{\prime}}$, and

  4. (4)

linearly bounded: there exist $e_{-},e_{+}\in\mathbf{Z}$ such that
$F^{me_{-}}R_{m}=R_{m}$ and $F^{me_{+}}R_{m}=\\{0\\}$ for all
$m\in\mathbf{Z}_{\geq 0}$.

We say that $\mathcal{F}$ is a $\mathbf{Z}$-filtration if
$F^{\lambda}R_{m}=F^{\lceil\lambda\rceil}R_{m}$ for any $\lambda\in\mathbf{R}$
and $m\in\mathbf{Z}_{\geq 0}$. A filtration $FR$ is $T$-equivariant if
$F^{\lambda}R_{m}$ is $T$-invariant subspace of $R_{m}$ for all
$\lambda\in\mathbf{R}$.

######  Definition 3.4.2.

Let $\mathcal{F}=\\{F^{\lambda}R_{m}\\}$ be a $\mathbf{Z}$-filtration of $R$.

  1. (1)

The Rees algebra of $\mathcal{F}$ is a $\mathbf{C}[t]$-algebra defined by

$\displaystyle\mathrm{Rees}(\mathcal{F})\coloneqq\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}t^{-\lambda}F^{\lambda}R_{m}.$

We say that $\mathcal{F}$ is finitely generated if its Rees algebra is a
finitely generated graded $\mathbf{C}[t]$-algebra.

  2. (2)

The associated graded algebra of $\mathcal{F}$ is a graded
$\mathbf{C}$-algebra defined by

$\displaystyle\mathrm{Gr}^{\mathcal{F}}\mathcal{R}=\bigoplus_{m=0}^{\infty}\mathrm{Gr}_{m}^{\mathcal{F}}\mathcal{R}\coloneqq\mathcal{R}/t\mathcal{R}=\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}F^{\lambda}R_{m}/F^{\lambda+1}R_{m}.$

Given a finitely generated $\mathbf{Z}$-filtration
$\mathcal{F}=\\{F^{\lambda}R_{m}\\}$, one can construct a test configuration
for $(X,L)$ as follows. Let $\mathcal{R}\coloneqq\mathrm{Rees}(\mathcal{F})$
be the Rees algebra of the filtration $FR$. Then $\mathcal{R}$ is a torsion
free $\mathbf{C}[t]$-algebra. Since $\mathbf{C}[t]$ is a principal ideal
domain, $\mathcal{R}$ is flat over $\mathbf{C}[t]$. Let

$\displaystyle\mathcal{X}\coloneqq\operatorname{Proj}\mathcal{R},\quad\mathcal{L}\coloneqq\mathcal{O}_{\mathcal{X}}(1),$

and $\pi\colon\mathcal{X}\to\mathbf{A}^{1}$ be the structure morphism. Then
$(\mathcal{X},\mathcal{L})$ is a flat family over $\mathbf{A}^{1}$. Further,
for any $a\in\mathbf{A}^{1}(\mathbf{C})$ we have

$\displaystyle\mathcal{X}_{a}$
$\displaystyle=\mathcal{X}\times_{\mathbf{A}^{1}}\operatorname{Spec}(\mathbf{C}[t]/(t-a)\mathbf{C}[t])$
$\displaystyle=\operatorname{Proj}(\mathcal{R}/(t-a)\mathcal{R})$
$\displaystyle=\begin{cases}\operatorname{Proj}R&a\neq 0,\\\
\operatorname{Proj}\mathrm{Gr}^{\mathcal{F}}\mathcal{R}&a=0.\end{cases}$

Hence $(\mathcal{X}_{a},\mathcal{L}_{a})=(X,L)$ for any
$a\in\mathbf{A}^{1}\setminus\\{0\\}$ and $(\mathcal{X},\mathcal{L})$ is a test
configuration for $(X,L)$.

######  Example 3.4.3 (Test configurations).

Let $(\mathcal{X},\mathcal{L})$ be a test configuration for $(X,L)$ of
exponent $1$. For each $\lambda\in\mathbf{Z}$, define $F^{\lambda}R_{m}$ to be
the image of the restriction map

$\displaystyle H^{0}(\mathcal{X},\mathcal{L}^{m})_{\lambda}\to
H^{0}(\mathcal{X}_{1},\mathcal{L}_{1}^{m})=H^{0}(X,L^{m}).$

By regarding $H^{0}(\mathcal{X}\setminus\mathcal{X}_{0},\mathcal{L}^{m})$ as a
$\mathbf{C}[t,t^{-1}]$ module, $F^{\lambda}R_{m}$ can be written as

$\displaystyle F^{\lambda}R_{m}=\\{\sigma\in H^{0}(X,L^{m})\mid
t^{-\lambda}\overline{\sigma}\in H^{0}(\mathcal{X},\mathcal{L}^{m})\\},$

where $\overline{\sigma}\in
H^{0}(\mathcal{X}\setminus\mathcal{X}_{0},\mathcal{L}^{m})$ is the
$\mathbf{G}_{m}$-equivariant section defined by $\sigma\in H^{0}(X,L^{m})$. By
setting $F^{\lambda}R_{m}=F^{\lceil\lambda\rceil}R_{m}$ for each
$\lambda\in\mathbf{R}$, we obtain a $\mathbf{Z}$-filtration of $R$. We denote
the filtration by $\mathcal{F}(\mathcal{X},\mathcal{L})$.

######  Example 3.4.4 (Translations and scalings).

Let $\mathcal{F}=\\{F^{\lambda}R_{m}\\}$ be a filtration of $R$, and set
$\mathcal{R}\coloneqq\mathrm{Rees}(\mathcal{F})$.

  1. (1)

Let $c\in\mathbf{R}$. A translation $\mathcal{F}(c)=\\{F(c)^{\lambda}R_{m}\\}$
of $\mathcal{F}$ is defined by

$\displaystyle F(c)^{\lambda}R_{m}\coloneqq F^{\lambda-mc}R_{m}.$

If $c\in\mathbf{Z}$ and $\mathcal{F}$ is a $\mathbf{Z}$-filtraion, then
$\mathcal{F}(c)$ is also a $\mathbf{Z}$-filtraion. Suppose $c\in\mathbf{Z}$,
and $\mathcal{F}$ is a finitely generated $\mathbf{Z}$-filtration. Let
$(\mathcal{X},\mathcal{L})$ be a test configuration associated to
$\mathcal{F}$. Then $\mathcal{X}=\operatorname{Proj}\mathcal{R}$ and
$\mathcal{L}=\mathcal{O}_{\mathcal{X}}(1)=\widetilde{\mathcal{R}(1)}$. By
setting $\mathcal{I}_{\mathcal{X}_{0}}$ to be the ideal sheaf of
$\mathcal{X}_{0}$ in $\mathcal{X}$, we have

$\displaystyle\mathcal{O}_{\mathcal{X}}(c\mathcal{X}_{0})=\mathcal{I}_{\mathcal{X}_{0}}^{-c}=\widetilde{t^{-c}\mathcal{R}}.$

Hence we obtain

$\displaystyle(\mathcal{L}+c\mathcal{X}_{0})^{m}$
$\displaystyle=\mathcal{L}^{m}\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{I}_{\mathcal{X}_{0}}^{-mc}$
$\displaystyle=(\mathcal{R}(1)^{m}\otimes_{\mathcal{R}}t^{-mc}\mathcal{R})^{\widetilde{}}$
$\displaystyle=(t^{-mc}\mathcal{R}(m))^{\widetilde{}}$

and

$\displaystyle H^{0}(\mathcal{X},(\mathcal{L}+c\mathcal{X}_{0})^{m})$
$\displaystyle=(t^{-mc}\mathcal{R}(m))_{0}=t^{-mc}\mathcal{R}_{m}$
$\displaystyle=t^{-mc}\bigoplus_{\lambda\in\mathbf{Z}}t^{-\lambda}F^{\lambda}R_{m}$
$\displaystyle=\bigoplus_{\lambda\in\mathbf{Z}}t^{-(\lambda+mc)}F^{\lambda}R_{m}$
$\displaystyle=\bigoplus_{\lambda\in\mathbf{Z}}t^{-\lambda}F^{\lambda-
mc}R_{m}$
$\displaystyle=\bigoplus_{\lambda\in\mathbf{Z}}t^{-\lambda}F(c)^{\lambda}R_{m}.$

This shows that

$\displaystyle\bigoplus_{m=0}^{\infty}H^{0}(\mathcal{X},(\mathcal{L}+c\mathcal{X}_{0})^{m})=\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}t^{-\lambda}F(c)^{\lambda}R_{m}=\mathrm{Rees}(\mathcal{F}(c))$

and that the translation $(\mathcal{X},\mathcal{L}+c\mathcal{X}_{0})$ is the
test configuration associated to $\mathcal{F}(c)$. Note that

$\displaystyle\mathcal{R}(c)\coloneqq\mathrm{Rees}(\mathcal{F}(c))\cong\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}t^{-\lambda}F^{\lambda}R_{m}=\mathcal{R}$

as a graded ring. Hence we have
$\operatorname{Proj}\mathcal{R}(c)=\operatorname{Proj}\mathcal{R}=\mathcal{X}$.

  2. (2)

Let $d\in\mathbf{Z}_{>0}$. A scaling
$\mathcal{F}_{d}=\\{F_{d}^{\lambda}R_{m}\\}$ of $\mathcal{F}$ is defined by

$\displaystyle F_{d}^{\lambda}R_{m}\coloneqq F^{\lceil\lambda/d\rceil}R_{m}.$

If $\mathcal{F}$ is a $\mathbf{Z}$-filtraion, then $\mathcal{F}_{d}$ is also a
$\mathbf{Z}$-filtraion. Suppose $\mathcal{F}$ is a finitely generated
$\mathbf{Z}$-filtration. Let $(\mathcal{X},\mathcal{L})$ be a test
configuration associated to $\mathcal{F}$. Then the base change
$\mathcal{X}_{d}$ of $\mathcal{X}$ by the homomorphism $\mathbf{C}[t]\ni
t\mapsto s^{d}\in\mathbf{C}[s]$ is given by

$\displaystyle\mathcal{X}_{d}=\operatorname{Proj}\mathcal{R}\times_{\operatorname{Spec}\mathbf{C}[t]}\operatorname{Spec}\mathbf{C}[s]=\operatorname{Proj}(\mathcal{R}\otimes_{\mathbf{C}[t]}\mathbf{C}[s]),$
$\displaystyle\mathcal{L}_{d}=p^{\ast}\mathcal{L}=p^{\ast}\mathcal{O}_{\mathcal{X}}(1)=p^{\ast}\widetilde{\mathcal{R}(1)}=((\mathcal{R}\otimes_{\mathbf{C}[t]}\mathbf{C}[s])(1))\,\tilde{}=\mathcal{O}_{\mathcal{X}_{d}}(1),$

where $p\colon\mathcal{X}_{d}\to\operatorname{Proj}\mathcal{R}=\mathcal{X}$ is
the natural projection. As a $\mathbf{C}[t]$-algebra, we have

$\displaystyle\mathcal{R}\otimes_{\mathbf{C}[t]}\mathbf{C}[s]$
$\displaystyle=\left(\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}t^{-\lambda}F^{\lambda}R_{m}\right)\otimes_{\mathbf{C}[t]}\mathbf{C}[s]$
$\displaystyle\cong\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}s^{-\lambda}F^{\lceil\frac{\lambda}{d}\rceil}R_{m}$
$\displaystyle=\mathrm{Rees}(\mathcal{F}_{d}),$

which is also an isomprphism as $\mathbf{C}[s]$-algebras. Hence the scaling
$(\mathcal{X}_{d},\mathcal{L}_{d})$ is the test configuration associated to
$\mathcal{F}_{d}$.

###  3.5. Limit measures

Let $(X,L)$ be a polarized manifold and $\mathcal{F}$ a filtration of
$R=R(X,L)$. For each $m\in\mathbf{Z}_{>0}$, we define a Borel probability
measure $\nu_{m}$ on $\mathbf{R}$ by

$\displaystyle\nu_{m}\coloneqq-\frac{d}{d\lambda}\left(\frac{1}{N_{m}}\dim
F^{m\lambda}R_{m}\right),$

where the derivative is taken in the sense of distributions. We call it _the
normalized weight measure_. By the linearly boundedness of $\mathcal{F}$,
$\nu_{m}$ has uniformly bounded support. To describe its limit, to each
$\lambda\in\mathbf{R}$, we define a graded subalgebra $R^{(\lambda)}$ of
$R(X,L)$ by

$\displaystyle
R^{(\lambda)}\coloneqq\bigoplus_{m=0}^{\infty}F^{m\lambda}R_{m}.$

_The volume_ of $R^{(\lambda)}$ is defined by

$\displaystyle\mathrm{vol}(R^{(\lambda)})\coloneqq\limsup_{m\to\infty}\frac{1}{N_{m}}\dim
F^{m\lambda}R_{m}.$

######  Theorem 3.5.1 ([12, Theorem 5.3]).

The sequence of normalized weight measures $\\{\nu_{m}\\}_{m=0}^{\infty}$
converges weakly to the probability measure

$\displaystyle\mathrm{DH}(\mathcal{F})\coloneqq-\frac{d}{d\lambda}\mathrm{vol}(R^{(\lambda)}).$

We call $\mathrm{DH}(\mathcal{F})$ _the Duistermaat-Heckman measure_ of the
filtration $\mathcal{F}$.

######  Example 3.5.2 (Test configurations).

Let $(\mathcal{X},\mathcal{L})$ be a test configuration for $(X,L)$ of
exponent $1$ and $\mathcal{F}$ the filtration of $R$ as in Example 3.4.4. By
[68, Lemma 8.5], the normalized weight measure is given by

$\displaystyle\nu_{m}$
$\displaystyle=-\frac{d}{d\lambda}\left(\frac{1}{N_{m}}\dim
F^{m\lambda}R_{m}\right)$
$\displaystyle=\frac{1}{N_{m}}\sum_{\lambda\in\mathbf{Z}}\dim
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{m})_{\lambda}\delta_{\lambda/m}$

for any $m\in\mathbf{Z}_{\geq 0}$. Hence the limit coincides with the
Duistermaat-Heckman measure:

$\displaystyle\mathrm{DH}(\mathcal{F})=\mathrm{DH}(\mathcal{X},\mathcal{L}).$

###  3.6. Reduced non-Archimedean $J$-functional

Let $(X,L)$ be a polarized manifold, and set $R=R(X,L)$.

######  Definition 3.6.1.

Let $\mathcal{F}=\\{F^{\lambda}R_{m}\\}$ be a filtration of $R$.

  1. (1)

The non-Archimedean Monge-Ampère energy of $\mathcal{F}$ is defined by

$\displaystyle
E^{NA}(\mathcal{F})\coloneqq\int_{\mathbf{R}}\lambda\,d\mathrm{DH}(\mathcal{F}).$

  2. (2)

The non-Archimedean J-functional of $\mathcal{F}$ is defined by

$\displaystyle
J^{NA}(\mathcal{F})\coloneqq\sup\mathrm{supp}(\mathrm{DH}(\mathcal{F}))-E^{NA}(\mathcal{F}).$

######  Definition 3.6.2.

Let $\mathcal{F}$ be a $T$-equivariant filtration of $R$. We denote the weight
decomposition of $T$-action on each $F^{\lambda}R_{m}$ by

$\displaystyle F^{\lambda}R_{m}=\bigoplus_{\alpha\in
M}(F^{\lambda}R_{m})_{\alpha}.$

Here $(F^{\lambda}R_{m})_{\alpha}$ is the $\alpha$-weight subspace of
$F^{\lambda}R_{m}$. For each $\xi\in N_{\mathbf{R}}$, we define the
$\xi$-twist $\mathcal{F}_{\xi}=\\{F_{\xi}^{\lambda}R_{m}\\}$ of $\mathcal{F}$
by

$\displaystyle F_{\xi}^{\lambda}R_{m}\coloneqq\bigoplus_{\alpha\in
M}(F_{\xi}^{\lambda}R_{m})_{\alpha},\quad(F_{\xi}^{\lambda}R_{m})_{\alpha}=(F^{\lambda-\langle\alpha,\xi\rangle}R_{m})_{\alpha}$

for each $\lambda\in\mathbf{R}$ and $m\in\mathbf{Z}_{\geq 0}$.

Note that the $\xi$-twist of a filtration $\mathcal{F}$ may be an
$\mathbf{R}$-filtration even if $\mathcal{F}$ is a $\mathbf{Z}$-filtration.

######  Definition 3.6.3.

Let $\mathcal{F}$ be a $T$-equivariant filtration of $R$. The reduced non-
Archimedean $J$-functional of $\mathcal{F}$ is defined by

$\displaystyle J_{T}^{NA}(\mathcal{F})\coloneqq\inf_{\xi\in
N_{\mathbf{R}}}J^{NA}(\mathcal{F}_{\xi}).$

If $(\mathcal{X},\mathcal{L})$ is a test configuration for $(X,L)$, the
reduced non-Archimedean $J$-functional of $(\mathcal{X},\mathcal{L})$ is
defined by $J_{T}^{NA}(\mathcal{X},\mathcal{L})\coloneqq
J_{T}^{NA}(\mathcal{F}(\mathcal{X},\mathcal{L}))$.

######  Theorem 3.6.4 ([48, Theorem B], [55, Theorem 3.14]).

Let $(\mathcal{X},\mathcal{L})$ be a $T$-equivariant test configuration for
$(X,L)$. Then

$\displaystyle\lim_{t\to\infty}\frac{J_{T}(\varphi^{t})}{t}=J_{T}^{NA}(\mathcal{X},\mathcal{L}).$

######  Definition 3.6.5 (see also [12]).

Let $p\in[1,\infty]$.

  1. (1)

Let $\mathcal{F}$ be a filtration of $R$. The $L^{p}$-norm
$\|\mathcal{F}\|_{p}$ of $\mathcal{F}$ is defined as the $L^{p}$-norm of
$\lambda-\overline{\lambda}$ with respect to $\mathrm{DH}(\mathcal{F})$, where

$\displaystyle\overline{\lambda}\coloneqq\int_{\mathbf{R}}\lambda\,d\mathrm{DH}(\mathcal{F})$

is the barycenter of $\mathrm{DH}(\mathcal{F})$. If
$(\mathcal{X},\mathcal{L})$ is a test configuration for $(X,L)$, the
$L^{p}$-norm of $(\mathcal{X},\mathcal{L})$ is defined by
$\|(\mathcal{X},\mathcal{L})\|_{p}\coloneqq\|\mathcal{F}(\mathcal{X},\mathcal{L})\|_{p}$.

  2. (2)

Let $\mathcal{F}$ be a $T$-equivariant filtration of $R$. Let $\mathcal{F}$ be
a filtration of $R$. The reduced $L^{p}$-norm $\|\mathcal{F}\|_{p}$ of
$\mathcal{F}$ is defined by

$\displaystyle\|\mathcal{F}\|_{p,T}\coloneqq\inf_{\xi\in
N_{\mathbf{R}}}\|\mathcal{F}_{\xi}\|_{p}.$

If $(\mathcal{X},\mathcal{L})$ is a $T$-equivariant test configuration for
$(X,L)$, the reduced $L^{p}$-norm of $(\mathcal{X},\mathcal{L})$ is defined by
$\|(\mathcal{X},\mathcal{L})\|_{p,T}\coloneqq\|\mathcal{F}(\mathcal{X},\mathcal{L})\|_{p,T}$.

According to [12, Lemma 7.10], we have the following equivalence between the
(reduced) $L^{1}$-norm and the (reduced) non-Archimedean $J$-functional.

######  Theorem 3.6.6 (see also [12, Theorem 7.9]).

Let $c_{n}\coloneqq 2n^{2}/(n+1)^{n+1}$.

  1. (1)

For each filtration $\mathcal{F}$, we have

$\displaystyle c_{n}J^{NA}(\mathcal{F})\leq\|\mathcal{F}\|_{1}\leq
2J^{NA}(\mathcal{F}).$

  2. (2)

For each $T$-equivariant filtration $\mathcal{F}$, we have

$\displaystyle c_{n}J_{T}^{NA}(\mathcal{F})\leq\|\mathcal{F}\|_{1,T}\leq
2J_{T}^{NA}(\mathcal{F}).$

######  Corollary 3.6.7.

Let $(\mathcal{X},\mathcal{L})$ be a normal test configuration for $(X,L)$.

  1. (1)

([12, Theorem 6.8 and Theorem 7.9]) The following conditions are equivalent:

    1. (i)

$(\mathcal{X},\mathcal{L})$ is trivial.

    2. (ii)

$\|(\mathcal{X},\mathcal{L})\|_{p}=0$ for some $p\in[1,\infty]$.

    3. (iii)

$J^{NA}(\mathcal{X},\mathcal{L})=0$.

  2. (2)

([49, Theorem B]) Suppose $(\mathcal{X},\mathcal{L})$ is $T$-equivariant. Then
the following conditions are equivalent:

    1. (i)

$(\mathcal{X},\mathcal{L})$ is product.

    2. (ii)

$\|(\mathcal{X},\mathcal{L})\|_{p,T}=0$ for some $p\in[1,\infty]$.

    3. (iii)

$J_{T}^{NA}(\mathcal{X},\mathcal{L})=0$.

###  3.7. Uniform relative K-stability

######  Definition 3.7.1.

Let $(X,L)$ be a polarized algebraic manifold and $T$ a maximal algebraic
torus of $\mathrm{Aut}^{0}(X)$.

  1. (1)

$(X,L)$ is relatively K-semistable if $M_{V}^{NA}(\mathcal{X},\mathcal{L})\geq
0$ for any $T$-equivariant normal test configuration
$(\mathcal{X},\mathcal{L})$ for $(X,L)$.

  2. (2)

$(X,L)$ is relatively K-polystable if $(X,L)$ is relatively K-semistable and
$M_{V}^{NA}(\mathcal{X},\mathcal{L})=0$ if and only if
$(\mathcal{X},\mathcal{L})$ is product.

  3. (3)

$(X,L)$ is relatively K-unstable if $(X,L)$ is not relatively K-semistable.

  4. (4)

$(X,L)$ is uniformly relatively K-polystable if there exists a $\delta>0$ such
that

$\displaystyle M_{V}^{NA}(\mathcal{X},\mathcal{L})\geq\delta
J_{T}^{NA}(\mathcal{X},\mathcal{L})$

for any $T$-equivariant normal test configuration $(\mathcal{X},\mathcal{L})$
for $(X,L)$.

By the slope formula, we have the following

######  Theorem 3.7.2.

A polarized manifold $(X,L)$ is uniformly relatively K-polystable if the
relative K-energy is $T$-coercive.

###### Proof.

Assume that the relative K-energy is $T$-coercive. Then there exists
$\delta,C>0$ such that

$\displaystyle M_{V}(\varphi)\geq\delta J_{T}(\varphi)-C$

for any $\varphi\in\mathcal{H}_{0}^{S}$. Let
$\varphi=(\mathcal{X},\mathcal{L})$ be a test configuration for $(X,L)$ of
exponent $r$, and $\\{\varphi^{t}\\}_{t\in[0,+\infty)}$ the subgeodesic ray
corresponding to $\varphi$. Then, Theorem 3.3.6 and Theorem 3.6.4 allow us to
conclude

$\displaystyle M_{V}^{NA}(\mathcal{X},\mathcal{L})\geq\delta
J_{T}^{NA}(\mathcal{X},\mathcal{L}),$

as required. ∎

Combining Theorems 2.7.8 and 3.7.2, we further obtain the following corollary.

######  Corollary 3.7.3.

A polarized manifold $(X,L)$ is uniformly relatively K-polystable if $(X,L)$
admits an $S$-invariant extremal Kähler metric.

##  4\. Convex Preliminaries

###  4.1. Polarized toric manifolds

In this subsection, we briefly review the definition and construction of
polarized toric manifolds. See [42] for details.

A $2n$-dimensional symplectic toric manifold is a $2n$-dimensional compact
connected symplectic manifold $(X,\omega)$ endowed with an effective
Hamiltonian action of an $n$-dimensional compact torus $S=(S^{1})^{n}$ and the
moment map $\mu\colon X\to M_{\mathbf{R}}$. Let $(X,\omega)$ be a
$2n$-dimensional symplectic toric manifold. According to the convexity theorem
of Atiyah [6] and Guillemin-Sternberg [44], the image of the moment map
$P\coloneqq\mu(X)\subset M_{\mathbf{R}}$ is a convex polytope obtained as the
convex hull of the image of the fixed points of the $S$-action. We call $P$
the _moment polytope_. Further, since $(X,\omega)$ is toric, $P$ satisfies the
following special conditions, usually called the _Delzant condition_ :

  1. (1)

There are exactly $n$ edges meeting at each vertex.

  2. (2)

Each edge meeting at the vertex $v\in P$ is of the form $v+tu_{i}$, $t\geq 0$,
where $u_{i}\in N$ is primitive.

  3. (3)

For each vertex, the corresponding $u_{1},\ldots,u_{n}$ generate the lattice
of $N_{\mathbf{R}}$ over $\mathbf{Z}$.

A convex polytope satisfiying the conditions above is called a _Delzant
polytope_. In [31], Delzant showed that there is a one to one correspondence
between isomorphism classes of symplectic toric manifolds and Delzant
polytopes. More precisely, for each Delzant polytope $P$, one can construct in
a canonical way a symplectic toric manifold in such a way that $P$ may be
identified with the moment polytope. This construction is known as the
_Delzant construction_. Let $P$ be a Delzant polytope and $(X_{P},\omega_{P})$
the corresponding symplectic toric manifold with the moment map $\mu_{P}\colon
X_{P}\to M_{\mathbf{R}}$. By the Delzant construction, there is an
$S$-invariant complex structure $J_{P}$ compatible with $\omega_{P}$. In
particular, $(X_{P},\omega_{P},J_{P})$ is a Kähler manifold. Since the
$S$-action preserves the complex structure $J_{P}$, it can be canonically
extended to an action of $T\coloneqq S^{\mathbf{C}}=(\mathbf{G}_{m})^{n}$
preserving $J_{P}$ ([45, Theorem 4.4]). Then $X_{P}$ contains $T$ as an open
dense orbit, and hence $X_{P}$ has a structure of a nonsingular toric variety
with the fan associated to the polytope $P$ ([54, Lemma 9.2]).

Let

(4.1) $\displaystyle P=\\{x\in
M_{\mathbf{R}}\mid\langle\lambda_{j},x\rangle+d_{j}\geq 0\ (j=1,\ldots,r)\\}$

be the facet representaion of $P$. Here $\langle\cdot,\cdot\rangle$ is the
natural pairing between $N_{\mathbf{R}}$ and $M_{\mathbf{R}}$, $r$ is the
number of facets of $P$, $\lambda_{j}\in N$, $d_{j}\in\mathbf{R}$, and each
$\lambda_{j}$ is primitive. Let $P^{\circ}$ denote the interior of $P$. Then
$X_{P}^{\circ}\coloneqq\mu_{P}^{-1}(P^{\circ})$ is a dense open subset of
$X_{P}$ where the $S$-action is free. This coincides with the open dense orbit
of the $T$-action described above. Also, for any $k$-dimensional face $F$ of
$P$, $\mu_{P}^{-1}(F)$ is an $T$-invariant $k$-dimensional connected complex
submanifold of $X_{P}$. In particular, if $F$ is a facet of $P$ then
$\mu_{P}^{-1}(F)$ is a $T$-invariant prime divisor of $X_{P}$. For each
$j=1,\ldots,r$, let $F_{j}$ denote the facet of $P$ defined by

$\displaystyle F_{j}=\\{x\in P\mid\langle\lambda_{j},x\rangle+d_{j}=0\\},$

and $D_{j}\coloneqq\mu_{P}^{-1}(F_{j})$. By setting $c_{j}$ the Poincare dual
of $D_{j}$, we have

$\displaystyle[\omega_{P}]=\sum_{j=1}^{r}d_{j}c_{j}$

as a de Rham cohomology class. If the Delzant polytope $P$ is _integral_ ,
that is, each vertex of $P$ belongs to $\mathbf{Z}^{n}$, then
$d_{j}\in\mathbf{Z}$ ($j=1,\ldots,r$) and hence $[\omega_{P}]\in
H^{2}(X,\mathbf{Z})$. Hence, in this case there exists a $T$-equivariant
holomorphic line bundle $L_{P}$ over $X_{P}$ which satisfies
$c_{1}(L_{P})=[\omega_{P}]$. By the Kodaira embedding theorem, $L_{P}$ is an
ample line bundle over $X_{P}$. We call $(X_{P},L_{P})$ a _polarized toric
manifold associated to $P$_.

###  4.2. Symplectic potentials

In this subsection, we quickly review differential-geometric aspects of toric
Kähler manifolds. For details, see [1] and [2], [42], [43].

Let $P\subset M_{\mathbf{R}}$ be an $n$-dimensional integral Delzant polytope,
and $(X_{P},L_{P})$ the polarized toric manifold corresponding to $P$. We
choose an $S$-invariant Kähler metric $\omega\in c_{1}(L_{P})$. Since
$X_{P}^{\circ}$ is holomorphically isomorphic to $T$, there is an
$S$-invariant smooth function $\phi\colon T\to\mathbf{R}$ so that

$\displaystyle\omega=dd^{c}\phi$

on $X_{P}^{\circ}$ ([43, Theorem 4.3]). Through the identification
$X_{P}^{\circ}\cong(\mathbf{C}^{\times})^{n}\cong\mathbf{R}^{n}\times S$, we
can regard $\phi$ as a smooth function defined on $\mathbf{R}^{n}$.

######  Remark 4.2.1.

Let $\tau=(\tau^{1},\ldots,\tau^{n})$,
$\tau^{i}=\exp((-1/2)y^{i}+2\pi\sqrt{-1}\theta^{i})$ ($i=1,\ldots,n$) denote
the coordinate of $T$. Then, for each $i=1,\ldots,n$ we have

$\displaystyle\frac{d\tau^{i}}{\tau^{i}}=-\frac{1}{2}dy^{i}+2\pi\sqrt{-1}d\theta^{i},\quad\frac{d\bar{\tau}^{i}}{\bar{\tau}^{i}}=-\frac{1}{2}dy^{i}-2\pi\sqrt{-1}d\theta^{i}$

and

$\displaystyle\frac{\sqrt{-1}}{2\pi}\frac{d\tau^{i}\wedge
d\bar{\tau}^{i}}{|\tau^{i}|^{2}}=dy^{i}\wedge d\theta^{i}.$

Further, since

$\displaystyle\tau^{i}\frac{\partial}{\partial\tau^{i}}=-\frac{\partial}{\partial
y^{i}}-\frac{\sqrt{-1}}{4\pi}\frac{\partial}{\partial\theta^{i}},\quad\bar{\tau}^{i}\frac{\partial}{\partial\bar{\tau}^{i}}=-\frac{\partial}{\partial
y^{i}}+\frac{\sqrt{-1}}{4\pi}\frac{\partial}{\partial\theta^{i}},$

we obtain

$\displaystyle
dd^{c}\phi=\frac{\sqrt{-1}}{2\pi}\frac{\partial^{2}\phi}{\partial\tau^{i}\partial\bar{\tau}^{j}}d\tau^{i}\wedge
d\bar{\tau}^{i}=\frac{\sqrt{-1}}{2\pi}\frac{\partial^{2}\phi}{\partial
y^{i}\partial
y^{j}}\frac{d\tau^{i}}{\tau^{i}}\wedge\frac{d\bar{\tau}^{i}}{\bar{\tau}^{j}}$

for any $\phi\in C^{\infty}(T,\mathbf{R})^{S}$. In particular, $dd^{c}\phi$ is
a Kähler form on $T$ if and only if $\phi$ is a strictly convex function on
$\mathbf{R}^{n}$.

By using the moment map, one can describe $\phi$ in terms of a convex function
on $P$, which is called a symplectic potential. Indeed, up to an additive
constant, the derivative $\nabla\phi$ coincides with the moment map for the
Kähler metric $\omega$. Hence we may assume that $\nabla\phi$ precisely
coincides with the moment map by adding a suitable linear function. Since
$\phi$ is strictly convex, the moment map $\nabla\phi$ is a diffeomorphism
from $\mathbf{R}^{n}$ onto $P^{\circ}$. Now we introduce the coordinate

$\displaystyle x=(\nabla\phi)(y),$

and define the function $u\colon P^{\circ}\to\mathbf{R}$ as the Legendre dual
of $\phi$, i.e.,

$\displaystyle u(x)+\phi(y)=\langle x,y\rangle.$

$u$ is called the _symplectic potential_ of the Kähler metric $\omega$.

######  Example 4.2.2 ([43]).

Let $\ell_{j}(x)\coloneqq\langle\lambda_{j},x\rangle+d_{j}$ ($j=1,\ldots,r$)
and $\ell_{\infty}\coloneqq\sum_{j=1}^{r}\ell_{j}$. Then, by setting
$\phi_{P}\coloneqq\mu_{P}^{\ast}(\ell_{\infty}-\sum_{j=1}^{r}d_{j}\ell_{j})$,
we have

$\displaystyle\omega_{P}=dd^{c}\phi_{P}$

on $X_{P}^{\circ}$. The symplectic potential of the Kähler metric $\omega_{P}$
is given by

$\displaystyle u_{P}\coloneqq\sum_{j=1}^{r}\ell_{j}\log\ell_{j},$

usually called the Guillemin potential.

By the theory of Guillemin [43] and Abreu [1], there is the following
correspondence for $S$-invariant Kähler metrics in $c_{1}(L_{P})$ and
symplectic potentials. Let $u$ be a strictly convex function on $P^{\circ}$.
The function $u$ is said to satisfy the Guillemin’s boundary condition if it
satisfies the following conditions:

  1. (1)

$u-u_{P}\in C^{\infty}(P)$;

  2. (2)

For any face $F$ of $P$, $u|_{F^{\circ}}$ is a strictly convex function on
$F^{\circ}$.

Let $\mathcal{S}$ denote the set of all strictly convex functions on
$P^{\circ}$ which satisfy the Guillemin’s boundary condition.

######  Theorem 4.2.3 ([1], [3], [35], [43]).

Every $S$-invariant Kähler metric in the class $c_{1}(L_{P})$ has a symplectic
potential $u$ in $\mathcal{S}$. Conversely, every function belongs to
$\mathcal{S}$ is a symplectic potential for some $S$-invariant Kähler metric
in $c_{1}(L_{P})$.

For later use, for each $u\in\mathcal{S}$ we denote
$\varphi_{u}\in\mathcal{H}(X_{P},\omega_{P})^{S}$ by the Kähler potential
corresponding to $u$, which is determined by
$\varphi_{u}|_{X_{P}^{\circ}}=\phi-\phi_{P}$, where $\phi$ is the Legendre
dual of $u$. By using the correspodence in Theorem 4.2.3, the scalar curvature
of an $S$-invariant Kähler metric $\omega\in c_{1}(L_{P})$ is given by
derivatives of the corresponding symplectic potential $u$ with respect to the
symplectic coordinates $(x_{1},\ldots,x_{n})$.

######  Theorem 4.2.4 (Abreu [2]).

Let $\omega$ be an $S$-invariant Kähler metric in the class $c_{1}(L_{P})$ and
$u$ the symplectic potential of $\omega$. Then the scalar curvature
$s(\omega)$ of $\omega$ is given by

$\displaystyle
s(\omega)=s(u)\coloneqq-\sum_{i,j=1}^{n}\frac{\partial^{2}u^{ij}}{\partial
x_{i}\partial x_{j}},$

where $u^{ij}$ is the inverse of the Hessian $(u_{ij})$ of $u$:

$\displaystyle(u_{ij})=\left(\frac{\partial^{2}u}{\partial x_{i}\partial
x_{j}}\right).$

Furthermore, by using Donaldson’s integration by parts formula ([34]), the
average of the scalar curvature $\bar{s}$ is obtained as follows. Let us
recall the Borel measure $\sigma$ on $\partial P$ defined by

$\displaystyle\sigma(A)=\sum_{j=1}^{r}\frac{|A\cap
F_{j}|_{n-1}}{|\lambda_{j}|},$

where $|\cdot|_{n-1}$ be the $(n-1)$-dimensional Lebesgue measure. This
measure arises naturally as the subleading coefficient of the Ehrhart
polynomial of an integral polytope

$\displaystyle E_{P}(m)\coloneqq\\#(mP\cap
M)=\mathrm{vol}(P)m^{n}+\frac{\sigma(\partial P)}{2}m^{n-1}+O(m^{n-2}).$

Let $\mathcal{C}_{\infty}$ denote the set of all continuous convex functions
on $P$ which are smooth in the interior.

######  Theorem 4.2.5 ([34, Lemma 3.3.5]).

Let $u\in\mathcal{S}$ and $f\in\mathcal{C}_{\infty}$. Then $u^{ij}f_{ij}$ is
integrable on $P$ and

$\displaystyle\int_{P}u^{ij}f_{ij}\,dx$ $\displaystyle=\int_{\partial
P}fd\sigma+\int_{P}f(u^{ij})_{ij}\,dx$ $\displaystyle=\int_{\partial
P}fd\sigma-\int_{P}fs(u)\,dx.$

######  Corollary 4.2.6.

The average of the scalar curvature $\overline{s}$ is given by

$\displaystyle\overline{s}=\frac{\sigma(\partial P)}{\mathrm{vol}(P)}.$

###  4.3. Toric test configurations

In this subsection we explain the construction of toric test configurations.
For this, we first recall the algebraic construction of toric varieties
following [27, Chapter 7] (see also [81]). Let $P\subset\mathbf{R}^{n}$ be an
$n$-dimensional integral (possibly unbounded) polyhedron. Then $P$ has a
unique facet representaion

$\displaystyle P=\\{x\in
M_{\mathbf{R}}\mid\langle\lambda_{j},x\rangle+d_{j}\geq 0\ (j=1,\ldots,r)\\},$

where $\lambda_{j}\in N$, $d_{j}\in\mathbf{Z}$ and $\lambda_{j}$ is primitive.
Also, it can be written as a Minkowski sum

$\displaystyle P=Q+C,$

where $Q$ is an integral polytope and $C$ is a strongly convex rational
polyhedral cone ([81, Theorem 1.2]). The cone part of $P$ is determined by

$\displaystyle C=\\{x\in M_{\mathbf{R}}\mid\langle\lambda_{j},x\rangle\geq 0\
(j=1,\ldots,r)\\}.$

([81, Proposition 1.12]). Following [81], we call $C$ the _recession cone_ of
$P$. Let $C(P)$ denote the _cone of_ $P$ defined by

$\displaystyle C(P)=\\{(x,\rho)\in M_{\mathbf{R}}\times\mathbf{R}\mid\rho\geq
0,\langle\lambda_{j},x\rangle+\rho d_{j}\geq 0\ (j=1,\ldots,r)\\}.$

Then it is easy to check that

$\displaystyle C(P)\cap(M_{\mathbf{R}}\times\\{0\\})=C\times\\{0\\},$

and $\rho P$ is the slice of $C(P)$ at height $\rho$ for any $\rho>0$. Let
$S_{P}\coloneqq C(P)\cap(M\times\mathbf{Z})$. Also, we let $\sigma\coloneqq
C\times\\{0\\}$ and $S_{\sigma}\coloneqq\sigma\cap(M\times\mathbf{Z})$. From
Gordan’s lemma, $S_{P}$ and $S_{\sigma}$ are affine semigroups. Let
$R\coloneqq\mathbf{C}[S_{P}]$ be the semigroup ring of $S_{P}$. Then $R$ is a
finitely generated $\mathbf{C}$-algebra. The character associated to
$(\alpha,m)\in M\times\mathbf{Z}$ is written $\chi^{\alpha}s^{m}$, and $R$ is
graded by height, i.e. $\deg(\chi^{\alpha}s^{m})=m$. Consequently, we obtain a
graded $\mathbf{C}$-algebra

$\displaystyle R=\bigoplus_{m=0}^{\infty}R_{m},\quad
R_{0}=\mathbf{C}[S_{\sigma}].$

The inclusion $R_{0}\hookrightarrow R$ gives a structure of finitely generated
$R_{0}$-algebra on $R$. Let $U_{P}\coloneqq\operatorname{Spec}R_{0}$,
$X_{P}\coloneqq\operatorname{Proj}R$, and $\pi_{P}\colon X_{P}\to U_{P}$ be
the structure morphism. Then $U_{P}$ is an affine toric variety, $X_{P}$ is a
projective toric variety, and $\pi_{P}$ is a projective toric morphism. For
each $j\in\\{1,\ldots,r\\}$, let $F_{j}$ denote the facet of $P$ defined by
$F_{j}=\\{x\in P\mid\langle\lambda_{j},x\rangle+d_{j}=0\\}$. Associated to
$F_{j}$, we have a $T$-invariant prime divisor $D_{j}$ of $X_{P}$. Then,

$\displaystyle D_{P}\coloneqq\sum_{j=1}^{r}d_{j}D_{j}$

is a $\pi_{P}$-ample Cartier divisor of $X_{P}$ ([27, Example 7.2.8]). An
important fact is that if $P$ is an integral Delzant polytope then
$(X_{P},L_{P})\cong(X_{P},\mathcal{O}_{X_{P}}(D_{P}))$.

Now let $P\subset M_{\mathbf{R}}$ be an integral Delzant polytope, and
identify $(X_{P},L_{P})$ and $(X_{P},\mathcal{O}_{X_{P}}(D_{P}))$ constructed
above. Recall that a function $f\colon P\to\mathbf{R}$ is called _rational
piecewise affine convex_ if $f$ is a convex function of the form

(4.2) $\displaystyle f=\max\\{\ell_{1},\ldots,\ell_{m}\\}$

with each $\ell_{j}$ an affine function having rational coefficients. Given a
rational piecewise affine convex function $f$, we can construct a
$T$-equivariant test configuration $(\mathcal{X}_{f},\mathcal{L}_{f})$ for
$(X_{P},L_{P})$ as follows. Choose an integer $L$ so that $L>\max_{P}f$, we
define a rational convex polyhedron $\mathcal{P}\subset
M_{\mathbf{R}}\times\mathbf{R}$ by

$\displaystyle\mathcal{P}=\\{(x,y)\in M_{\mathbf{R}}\times\mathbf{R}\mid x\in
P,\ f(x)-L\leq y\\}.$

By replacing $f$, $L$, $\mathcal{P}$ by $kf$, $kL$, $k\mathcal{P}$ for
suitable $k\in\mathbf{Z}_{>0}$ if necessary, we may assume that each
$\ell_{j}$ has integral coefficients and $\mathcal{P}$ is an integral
polyhedron. Then the recession cone of $\mathcal{P}$ is given by
$\mathcal{C}\coloneqq\\{0\\}\times\mathbf{R}_{\geq 0}\subset
M_{\mathbf{R}}\times\mathbf{R}$, and the cone of $\mathcal{P}$ is given by

$\displaystyle
C(\mathcal{P})=(\mathcal{C}\times\\{0\\})\cup\left(\bigcup_{\rho>0}\rho\mathcal{P}\times\\{\rho\\}\right).$

Let $S_{\mathcal{P}}\coloneqq C(\mathcal{P})\cap(M\times\mathbf{Z}^{2})$,
$\sigma\coloneqq\mathcal{C}\times\\{0\\}$, and
$S_{\sigma}\coloneqq(\mathcal{C}\times\\{0\\})\cap(M\times\mathbf{Z}^{2})$.
Then $S_{\mathcal{P}}$ and $S_{\sigma}$ are affine semigroups. Let
$\mathcal{R}\coloneqq\mathbf{C}[S_{\mathcal{P}}]$ be the semigroup ring of
$S_{\mathcal{P}}$. Then $\mathcal{R}$ is a finitely generated
$\mathbf{C}$-algebra. The character associated to $(\alpha,\lambda,m)\in
M\times\mathbf{Z}^{2}$ is written $\chi^{\alpha}t^{\lambda}s^{m}$, and the
grading of $R_{\mathcal{P}}$ is given by
$\deg(\chi^{\alpha}t^{\lambda}s^{m})=m$. Consequently, we have the graded
$\mathbf{C}$-algebra

$\displaystyle\mathcal{R}=\bigoplus_{m=0}^{\infty}\mathcal{R}_{m},$
$\displaystyle\mathcal{R}_{m}=\begin{cases}\mathbf{C}[S_{\sigma}]=\mathbf{C}[t]&m=0,\\\
\mathrm{Vect}_{\mathbf{C}}\left\\{\chi^{\alpha}t^{\lambda}s^{m}\left|\begin{array}[]{l}\alpha\in
mP\cap M,\lambda\in\mathbf{Z},\\\
m(f(\alpha/m)-L)\leq\lambda\end{array}\right.\right\\}&m>0.\end{cases}$

The inclusion $\mathbf{C}[t]=\mathcal{R}_{0}\hookrightarrow\mathcal{R}$ gives
a structure of $\mathbf{C}[t]$-algebra on $\mathcal{R}$. Note also that
$\mathcal{R}$ is finitely generated as a $\mathbf{C}[t]$-algebra. By setting
$\mathcal{D}_{\mathcal{P}}$ the divisor of $\operatorname{Proj}\mathcal{R}$
defined above, we have a polarized scheme

$\displaystyle(\mathcal{X}_{f},\mathcal{L}_{f})=(\operatorname{Proj}\mathcal{R},\mathcal{O}_{\operatorname{Proj}\mathcal{R}}(\mathcal{D}_{\mathcal{P}})).$

Then one can check that
$\mathcal{O}_{\operatorname{Proj}\mathcal{R}}(\mathcal{D}_{\mathcal{P}})=\mathcal{O}_{\operatorname{Proj}\mathcal{R}}(1)$.
Let $\pi_{f}\colon\mathcal{X}_{f}\to\mathbf{A}^{1}$ be the structure morphism.
Then $\pi_{f}$ is a projective morphism and hence proper. By construction,
$\mathcal{X}_{f}$ is a normal toric variety and $\mathcal{L}_{f}$ is
$\pi_{f}$-ample line bundle over $\mathcal{X}_{f}$. There is a
$\mathbf{G}^{m+1}$-action on $\mathcal{R}$, which has weight
$(\alpha,\lambda)\in M\times\mathbf{Z}$ for any $\chi^{\alpha}t^{\lambda}\in
S_{\mathcal{P}}$. This induces a $T_{f}\coloneqq T\times\mathbf{G}_{m}$-action
on $\mathcal{X}_{f}$ over $\mathbf{A}^{1}$. We claim
$(\mathcal{X}_{f},\mathcal{L}_{f})$ is a test configuration for
$(X_{P},L_{P})$. To see this, it is sufficient to show that $\mathcal{R}$ is
the Rees algebra of a finitely generated $\mathbf{Z}$-filtration of $R$ (see
Section 3.4). Let $\mathcal{F}=\\{F^{\lambda}R_{m}\\}$ be a
$\mathbf{Z}$-filtration of $R$ defined by

$\displaystyle F^{\lambda}R_{0}=\begin{cases}\mathbf{C}&\lambda\leq 0,\\\
\\{0\\}&\lambda>0\end{cases}$

and

$\displaystyle
F^{\lambda}R_{m}=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}s^{m}\mid\alpha\in
mP\cap M,\ \lambda\leq m(L-f(\alpha/m))\\}$

for any $\lambda\in\mathbf{R}$ and $m\in\mathbf{Z}_{>0}$. Then the Rees
algebra of $\mathcal{F}$ is given by

$\displaystyle\mathrm{Rees}(\mathcal{F})$
$\displaystyle=\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}t^{-\lambda}F^{\lambda}R_{m}$
$\displaystyle=\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}\mathrm{Vect}_{\mathbf{C}}\left\\{\chi^{\alpha}t^{-\lambda}s^{m}\left|\begin{array}[]{c}\alpha\in
mP\cap M,\\\ \lambda\leq m(L-f(\alpha/m))\end{array}\right.\right\\}$
$\displaystyle=\bigoplus_{m=0}^{\infty}\bigoplus_{\lambda\in\mathbf{Z}}\mathrm{Vect}_{\mathbf{C}}\left\\{\chi^{\alpha}t^{\lambda}s^{m}\left|\begin{array}[]{c}\alpha\in
mP\cap M,\\\ m(f(\alpha/m)-L)\leq\lambda\end{array}\right.\right\\}$
$\displaystyle=\bigoplus_{m=0}^{\infty}\mathcal{R}_{m}=\mathcal{R}.$

Furthermore, $\mathcal{F}$ is finitely generated since $\mathcal{R}$ is a
finitely generated $\mathbf{C}[t]$-algebra. This shows the claim. We call
$(\mathcal{X}_{f},\mathcal{L}_{f})$ the _toric test configuration_ associated
to $f$. The induced $\mathbf{G}_{m}$-coaction $\mu_{m}$ on the central fiber
$H^{0}((\mathcal{X}_{f})_{0},(\mathcal{L}_{f}^{m})_{0})=\mathrm{Gr}_{m}^{\mathcal{F}}\mathcal{R}$
is given by

(4.3)
$\displaystyle\mu_{m}(\chi^{\alpha}s^{m})=t^{m(L-f(\alpha/m))}\otimes\chi^{\alpha}s^{m}.$

Easy computation shows that the canonical compactification of
$(\mathcal{X}_{f},\mathcal{L}_{f})$ is given as a polarized toric variety
$(\mathcal{X}_{\mathcal{Q}},\mathcal{L}_{\mathcal{Q}})$ corresponding to a
rational convex polytope

(4.4) $\displaystyle\mathcal{Q}=\\{(x,y)\in M_{\mathbf{R}}\times\mathbf{R}\mid
x\in P,\ f(x)-L\leq y\leq 0\\}.$

######  Example 4.3.1 (Translations and scalings: toric cases).

Let $f$ be a rational piecewise affine convex function on an integral Delzant
polytope $P$. Let us describe translations and scalings of
$(\mathcal{X}_{f},\mathcal{L}_{f})$ in terms of the convex function $f$. for
simplicity, we assume $f$ is integral, i.e. $f$ is of the form (4.2) with each
$\ell_{j}$ an affine function having integral coefficients.

  1. (1)

Let $c\in\mathbf{Z}$, and choose $L\in\mathbf{Z}_{>0}$ so that
$\max\\{f,f-c\\}<L$ on $P$. Then the filtration $\mathcal{F}_{c}$
corresponding to the translation $(\mathcal{X}_{f},(\mathcal{L}_{f})_{c})$ is
given by

$\displaystyle F_{c}^{\lambda}R_{m}=F^{\lambda-mc}R_{m}.$

Claerly we have $F_{c}^{\lambda}R_{0}=F^{\lambda}R_{0}$. If $m>0$, then we
have

$\displaystyle F_{c}^{\lambda}R_{m}$
$\displaystyle=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}s^{m}\mid\alpha\in
mP\cap M,\ \lambda-mc\leq m(L-f(\alpha/m))\\}$
$\displaystyle=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}s^{m}\mid\alpha\in
mP\cap M,\ \lambda\leq m(L-f(\alpha/m)+c)\\}.$

Hence $\mathcal{F}_{c}$ coincides with the filtration corresponding to the
toric test configuration $(\mathcal{X}_{f-c},\mathcal{L}_{f-c})$.

  2. (2)

Let $d\in\mathbf{Z}_{>0}$. Then the filtration $\mathcal{F}_{d}$ corresponding
to the scaling $((\mathcal{X}_{f})_{d},(\mathcal{L}_{f})_{d})$ is given by

$\displaystyle F_{d}^{\lambda}R_{m}=F^{\lceil\frac{\lambda}{d}\rceil}R_{m}.$

Claerly we have $F_{d}^{\lambda}R_{0}=F^{\lambda}R_{0}$. If $m>0$, then we
have

$\displaystyle F_{d}^{\lambda}R_{m}$
$\displaystyle=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}s^{m}\mid\alpha\in
mP\cap M,\ \lceil\lambda/d\rceil\leq m(L-f(\alpha/m))\\}$
$\displaystyle=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}s^{m}\mid\alpha\in
mP\cap M,\ \lambda/d\leq m(L-f(\alpha/m))\\}$
$\displaystyle=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}s^{m}\mid\alpha\in
mP\cap M,\ \lambda\leq m(dL-df(\alpha/m))\\}.$

Hence $\mathcal{F}_{d}$ coincides with the filtration corresponding to the
toric test configuration $(\mathcal{X}_{df},\mathcal{L}_{df})$.

###  4.4. Energy functionals on polarized toric manifolds

By using Duistermaat-Heckman theorem and Legendre dual functions, we can write
down energy functionals in terms of convex functions.

######  Lemma 4.4.1.

Let $\\{u^{t}\\}\subset\mathcal{S}$ be a smooth path, and $\\{\phi^{t}\\}$ be
their Legendre duals. Then

$\displaystyle\dot{u}^{t}(x)=-\dot{\phi}^{t}(\nabla u^{t}(x))$

for any $x\in P^{\circ}$.

###### Proof.

Since $\phi^{s}$ is a strictly convex function, $\nabla u^{s}(x)$ is a unique
minimizer of $y\mapsto\phi^{s}(y)-\langle y,x\rangle$. By setting
$y^{s}=\nabla u^{s}(x)$, we have

$\displaystyle 0=\left.\frac{d}{ds}\right|_{s=t}(\phi^{t}(y^{s})-\langle
y^{s},x\rangle)=\left.\frac{d}{ds}\right|_{s=t}\nabla u^{s}(x).$

Hence we obtain

$\displaystyle\dot{u}^{t}(x)$
$\displaystyle=\left.\frac{d}{ds}\right|_{s=t}u^{s}(x)$
$\displaystyle=\left.\frac{d}{ds}\right|_{s=t}(\langle
y^{s},x\rangle-\phi^{s}(y^{s}))$
$\displaystyle=\left.\frac{d}{ds}\right|_{s=t}(\langle
y^{s},x\rangle-\phi^{t}(y^{s}))+\left.\frac{d}{ds}\right|_{s=t}(\phi^{t}(y^{s})-\phi^{s}(y^{s}))$
$\displaystyle=\langle\nabla\phi^{t},\dot{y}^{t}\rangle-\dot{\phi}^{t}(y^{t})-\langle\nabla\phi^{t},\dot{y}^{t}\rangle$
$\displaystyle=-\dot{\phi}^{t}(y^{t}),$

as required. ∎

######  Proposition 4.4.2.

Let $\varphi\in\mathcal{H}(X_{P},\omega_{P})^{S}$, and $u\in\mathcal{S}$ be
the symplectic potential of $\omega_{\varphi}$. Then

$\displaystyle
E(\varphi)=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(u-u_{P})\,dx.$

###### Proof.

Let $\\{u^{t}\\}_{t\in[0,1]}\subset\mathcal{S}$ be a smooth path so that
$u^{0}=u_{P}$ and $u^{1}=u$. Let
$\varphi^{t}\coloneqq\varphi_{u^{t}}\in\mathcal{H}(X_{P},\omega_{P})^{S}$ for
each $t\in[0,1]$. Note that $\varphi^{0}=0$ and $\varphi^{1}=\varphi$. Then we
have

$\displaystyle E(\varphi)$
$\displaystyle=\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}\dot{\varphi}^{t}\omega_{\varphi^{t}}^{n}$
$\displaystyle=\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}^{\circ}}\dot{\phi}^{t}(dd^{c}\phi^{t})^{n}$
$\displaystyle=-\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\dot{u}^{t}\,dx$
$\displaystyle=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(u-u_{P})\,dx.$

∎

We define a subset $\mathcal{S}_{0}$ of $\mathcal{S}$ by

$\displaystyle\mathcal{S}_{0}\coloneqq\left\\{u\in\mathcal{S}\left|-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(u-u_{P})\,dx=0\right.\right\\}$

######  Proposition 4.4.3.

Let $\varphi\in\mathcal{H}(X_{P},\omega_{P})^{S}$ and $u\in\mathcal{S}$ be the
symplectic potential of $\omega_{\varphi}$.

  1. (1)

Let $\phi$ be the Legendere dual of $u$. Then

$\displaystyle
J(\varphi)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}^{\circ}}(\phi-\phi_{P})\omega_{P}^{n}+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(u-u_{P})\,dx.$

  2. (2)

Let ${\displaystyle
F(u)\coloneqq-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\log\det(u_{ij})\,dx+\frac{1}{\mathrm{vol}(P)}\int_{\partial
P}u\,d\sigma-\overline{s}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}u\,dx}$.
Then

$\displaystyle M(\varphi)=F(u)-F(u_{P}).$

###### Proof.

For (1), since $\varphi=\phi-\phi_{P}$ on $X_{P}^{\circ}$ we have

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}\varphi\omega_{P}^{n}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}^{\circ}}(\phi-\phi_{P})\,\omega_{P}^{n}.$

Hence we have

$\displaystyle J(\varphi)$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}\varphi\omega_{P}^{n}-E(\varphi)$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}^{\circ}}(\phi-\phi_{P})\omega_{P}^{n}+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(u-u_{P})\,dx$

by Proposition 4.4.2.

Next consider (2). Let $\\{\varphi^{t}\\}_{t\in[0,1]}$ be a smooth path in
$\mathcal{H}_{0}^{S}$ satisfying $\varphi^{0}=0$ and $\varphi^{1}=\varphi$,
and $\\{u^{t}\\}_{t\in[0,1]}$ be the coressponding symplectic potentials. Note
that $u^{0}=u_{P}$ and $u^{1}=u$. Then

$\displaystyle\frac{d}{dt}M(\varphi^{t})$
$\displaystyle=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}\dot{\varphi}^{t}(s(\varphi^{t})-\overline{s})\omega_{\varphi^{t}}^{n}$
$\displaystyle=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}(-\dot{u}^{t})(-{(u^{t})^{ij}}_{ij}-\overline{s})\,dx$
$\displaystyle=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}(\dot{u}^{t}){(u^{t})^{ij}}_{ij}\,dx-\overline{s}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\dot{u}^{t}\,dx.$

By Theorem 4.2.5, we obtain

$\displaystyle-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}(\dot{u}^{t}){(u^{t})^{ij}}_{ij}\,dx$
$\displaystyle=\frac{1}{\mathrm{vol}(P)}\int_{\partial
P}\dot{u}^{t}\,d\sigma-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(u^{t})^{ij}(\dot{u}^{t})_{ij}\,dx$
$\displaystyle=\frac{1}{\mathrm{vol}(P)}\int_{\partial
P}\dot{u}^{t}\,d\sigma-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\frac{d}{dt}\log\det((u^{t})_{ij})\,dx$

and hence

$\displaystyle\frac{d}{dt}M(\varphi^{t})$
$\displaystyle=\frac{1}{\mathrm{vol}(P)}\int_{\partial
P}\dot{u}^{t}\,d\sigma-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\frac{d}{dt}\log\det((u^{t})_{ij})\,dx-\overline{s}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\dot{u}^{t}\,dx$
$\displaystyle=\frac{d}{dt}\left(-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\log\det((u^{t})_{ij})\,dx+\frac{1}{\mathrm{vol}(P)}\int_{\partial
P}u^{t}\,d\sigma-\overline{s}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}u^{t}\,dx\right),$

as required. ∎

######  Proposition 4.4.4.

Let $\varphi\in\mathcal{H}(X_{P},\omega_{P})^{S}$, and $u\in\mathcal{S}$ be
the symplectic potential of $\omega_{\varphi}$.

  1. (1)

${\displaystyle
H_{V}(\varphi)=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(u-u_{P})V\,dx}$.

  2. (2)

Let ${\displaystyle F_{V}(u)\coloneqq F(u)+H_{V}(\varphi)}$. Then
$M_{V}(\varphi)=F_{V}(u)-F_{V}(u_{P})$.

###### Proof.

For (1), let $\\{\varphi^{t}\\}_{t\in[0,1]}$ be a smooth path in
$\mathcal{H}_{0}^{S}$ satisfying $\varphi^{0}=0$ and $\varphi^{1}=\varphi$,
and $\\{u^{t}\\}_{t\in[0,1]}$ be the corresponding symplectic potentials. Note
that $u^{0}=u_{P}$ and $u^{1}=u$. Then

$\displaystyle\frac{d}{dt}H_{V}(\varphi^{t})$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}\dot{\varphi}^{t}\theta_{\varphi^{t}}\omega_{\varphi^{t}}^{n}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X_{P}}\dot{\varphi}^{t}(-\left\langle\mu_{\varphi^{t}},-\frac{1}{2}\theta\right\rangle)\omega_{\varphi^{t}}^{n}$
$\displaystyle=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\dot{u}^{t}\frac{1}{2}\theta\,dx$
$\displaystyle=\frac{d}{dt}\left(-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}u^{t}\frac{1}{2}\theta\,dx\right),$

as required.

The claim (2) is easily proved from (1) and Proposition 4.4.3. ∎

As shown by Guan in [41], any pair of Kähler potentials in
$\mathcal{H}_{0}^{S}$ can be joined by a unique geodesic segment, which are
obtained as a line segment of symplectic potentials. Thus, we have the
following;

######  Proposition 4.4.5.

Let $\varphi\in\mathcal{H}_{0}^{S}$, and $u\in\mathcal{S}$ be the symplectic
potential of $\omega_{\varphi}$. Then

$\displaystyle d_{1}(\varphi,0)$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}|u-u_{P}|\,dx.$

###### Proof.

Let $u^{t}=(1-t)u_{P}+tu$ ($t\in[0,1]$), and
$\varphi^{t}\coloneqq\varphi_{u^{t}}$. Then $\\{\varphi^{t}\\}_{t\in[0,1]}$ is
a unique geodesic segment joining $0$ and $\varphi$. Hence we have

$\displaystyle d_{1}(\varphi,0)$
$\displaystyle=\ell_{1}(\\{\varphi^{t}\\}_{t\in[0,1]})=\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{X}|\dot{\varphi}^{t}|\omega_{\varphi^{t}}^{n}$
$\displaystyle=\int_{0}^{1}dt\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}|-\dot{u}^{t}|\,dx=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}|u-u_{P}|\,dx.$

∎

Consider the action of $T$ on $\mathcal{S}_{0}$. Let $u\in\mathcal{S}_{0}$ and
$\tau\in T$. Then $(\varphi_{u})_{\tau}\in\mathcal{H}_{0}^{S}$ can be written
as

$\displaystyle\tau^{*}\omega_{\varphi_{u}}=\omega_{P}+dd^{c}(\varphi_{u})_{\tau}.$

On $X_{P}^{\circ}\cong T$, we can further express

$\displaystyle\tau^{*}\omega_{\varphi_{u}}=\tau^{*}dd^{c}u^{\vee}=dd^{c}(\tau^{*}u^{\vee}).$

If we write $\tau=\exp(-\xi/2)$ for some $\xi\in N_{\mathbf{R}}$, we have

$\displaystyle\tau\cdot\exp(-y/2+2\pi\sqrt{-1}\theta)=\exp(-(y+\xi)/2+2\pi\sqrt{-1}\theta)$

and

$\displaystyle\tau^{*}u^{\vee}(y)$ $\displaystyle=u^{\vee}(y+\xi)$
$\displaystyle=\sup_{x\in P^{\circ}}(\langle x,y+\xi\rangle-u)$
$\displaystyle=(u-\xi)^{\vee}(y).$

Therefore we obtain

$\displaystyle\tau^{*}\omega_{u}=dd^{c}(\tau^{*}u^{\vee})=dd^{c}(u-\xi)=\omega_{u-\xi}$

and hence

$\displaystyle
dd^{c}(\varphi_{u})_{\tau}=\tau^{*}\omega_{u}-\omega_{P}=\omega_{u-\xi}-\omega_{P}=dd^{c}\varphi_{u-\xi}$

on $X_{P}^{\circ}$. By replacing the coordinate $(x_{1},\ldots,x_{n})$ on
$M_{\mathbf{R}}$ so that

$\displaystyle\int_{P}x_{i}\,dx=0,\quad i=1,\ldots,n,$

we obtain $(\varphi_{u})_{\tau}=\varphi_{u-\xi}$. This shows the following

######  Proposition 4.4.6.

Let $\varphi\in\mathcal{H}_{0}^{S}$, and $u\in\mathcal{S}$ be the symplectic
potential of $\omega_{\varphi}$. Then

$\displaystyle d_{1,T}(\varphi,0)$ $\displaystyle=\inf_{\xi\in
N_{\mathbf{R}}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}|(u-\xi)-u_{P}|\,dx.$

We define the _reduced $L^{1}$-norm_ by

$\displaystyle\|u\|_{1,T}\coloneqq\inf_{\xi\in
N_{\mathbf{R}}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}|(u-\xi)-(\overline{u-\xi})|\,dx,$

where

$\displaystyle(\overline{u-\xi})\coloneqq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(u-\xi)\,dx.$

Then

######  Corollary 4.4.7.

Let $\varphi\in\mathcal{H}_{0}^{S}$, and $u\in\mathcal{S}$ be the symplectic
potential of $\omega_{\varphi}$. Then

$\displaystyle\|u\|_{1,T}-\|u_{P}\|_{1}\leq
d_{1,T}(\varphi,0)\leq\|u\|_{1,T}+\|u_{P}\|_{1}.$

######  Corollary 4.4.8.

For a polarized toric manifold $(X_{P},L_{P})$, the followings are equivalent.

  1. (1)

The relative K-energy is $T$-coercive.

  2. (2)

There exists $\delta,C>0$ such that

$\displaystyle F_{V}(u)\geq\delta\|u\|_{1,T}-C$

for any $u\in\mathcal{S}_{0}$.

  3. (3)

There exists $\delta,C>0$ such that

$\displaystyle F_{V}(u)\geq\delta J_{T}(u)-C$

for any $u\in\mathcal{S}_{0}$.

###  4.5. Non-Archimedean functionals on polarized toric manifolds

In this subsection, we give convex analytic expression of non-Archimedean
functionals for toric test configurations. Let
$f\in\mathcal{C}_{PL}^{\mathbf{Q}}$. For simplicity we assume $f$ is integral.
Choose $L\in\mathbf{Z}_{>0}$ so that $L-f>0$ on $P$. Then the corresponding
$(\mathcal{X}_{f},\mathcal{L}_{f})$ is a toric test configuration of exponent
1.

######  Proposition 4.5.1.

The Duistermaat-Heckmann measureof $(\mathcal{X}_{f},\mathcal{L}_{f})$ s given
by

$\displaystyle\mathrm{DH}(\mathcal{X}_{f},\mathcal{L}_{f})=(L-f)_{\\#}\frac{dx}{\mathrm{vol}(P)}.$

In particular,
$\mathrm{supp}(\mathrm{DH}(\mathcal{X}_{f},\mathcal{L}_{f}))=[L-\max_{P}f,L-\min_{P}f]$.

###### Proof.

Let $\rho$ be a bounded continuous function on $\mathbf{R}$. Then we have

$\displaystyle\int_{\mathbf{R}}\rho\,d\mathrm{DH}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=\lim_{m\to\infty}\frac{1}{E_{P}(m)}\sum_{\alpha\in mP\cap
M}\rho\left(L-f\left(\frac{\alpha}{m}\right)\right)$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\rho(L-f)\,dx.$

∎

######  Proposition 4.5.2.

For each $f\in\mathcal{C}_{PL}^{\mathbf{Q}}$ we have

$\displaystyle
E^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})=L-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}f\,dx.$

###### Proof.

Let $f\in\mathcal{C}_{PL}^{\mathbf{Q}}$. Then we have

$\displaystyle E^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=\int_{\mathbf{R}}\lambda\,d\mathrm{DH}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=\lim_{m\to\infty}\frac{1}{E_{P}(m)}\sum_{\alpha\in mP\cap
M}\left(L-f\left(\frac{\alpha}{m}\right)\right)$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(L-f)\,dx$
$\displaystyle=L-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}f\,dx,$

as required. ∎

######  Proposition 4.5.3.

For each $f\in\mathcal{C}_{PL}^{\mathbf{Q}}$ we have

$\displaystyle
J^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})=\|f\|_{J}\coloneqq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}f\,dx-\min_{P}f,$
$\displaystyle
J_{T}^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})=\|f\|_{J}\coloneqq\inf_{\text{$\xi\colon$affine}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(f+\xi)\,dx-\min_{P}(f+\xi)\right).$

###### Proof.

The formula for $J^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$ is a direct
consequence of Propositions 4.5.1 and 4.5.2. Let us consider
$J_{T}^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$. For the graded ring
$R(X_{P},L_{P})$, $R_{m}$ can be identified with

$\displaystyle
R_{m}=H^{0}(X_{P},L_{P}^{m})\cong\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}t^{m}\mid\alpha\in
mP\cap M\\}.$

The induced $T$-equivariant filtration
$\mathcal{F}(\mathcal{X}_{f},\mathcal{L}_{f})$ is then given by

$\displaystyle
F^{\lambda}R_{m}=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}t^{m}\mid\alpha\in
mP\cap M,m(f(\alpha/m)-L)\leq\lambda\\},$
$\displaystyle(F^{\lambda}R_{m})_{\alpha}=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}t^{m}\mid
m(f(\alpha/m)-L)\leq\lambda\\}.$

Let $\xi\in N_{\mathbf{R}}$. If necessary, replace $L\in\mathbf{Z}_{>0}$ so
that $L-f-\xi>0$. Then the $\xi$-twist $\mathcal{F}_{\xi}$ of
$\mathcal{F}(\mathcal{X}_{f},\mathcal{L}_{f})$ is given by

$\displaystyle(F_{\xi}^{\lambda}R_{m})_{\alpha}$
$\displaystyle=(F^{\lambda-\langle\alpha,\xi\rangle}R)_{\alpha}$
$\displaystyle=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}t^{m}\mid
m(f(\alpha/m)-L)\leq\lambda-\langle\alpha,\xi\rangle\\}$
$\displaystyle=\mathrm{Vect}_{\mathbf{C}}\\{\chi^{\alpha}t^{m}\mid
m(f(\alpha/m)+\xi(\alpha/m)-L)\leq\lambda\\},$ $\displaystyle
F_{\xi}^{\lambda}R_{m}$ $\displaystyle=\bigoplus_{\alpha\in
M}(F_{\xi}^{\lambda}R_{m})_{\alpha}.$

Hence we have

$\displaystyle\nu_{m}$
$\displaystyle=-\frac{d}{d\lambda}\frac{1}{E_{P}(m)}\dim
F_{\xi}^{m\lambda}H^{0}(X,L^{m})$
$\displaystyle=\frac{1}{E_{P}(m)}\sum_{\alpha\in mP\cap
M}\delta_{L-f(\alpha/m)-\xi(\alpha/m)}$

and

$\displaystyle\int_{\mathbf{R}}\rho\,d\nu$
$\displaystyle=\lim_{m\to\infty}\frac{1}{E_{P}(m)}\sum_{\alpha\in mP\cap
M}\rho(L-f(\alpha/m)-\xi(\alpha/m))$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\rho(L-f-\xi)\,dx$

for any bounded continuous function $\rho$ on $\mathbf{R}$. Therefore, for
each Borel measurable set $A$ in $P$ we have

$\displaystyle\nu(A)=\frac{1}{\mathrm{vol}(P)}\mathrm{vol}(P)\\{x\in P\mid
L-f-\xi\in A\\}$

and

$\displaystyle\int_{\mathbf{R}}\lambda\,d\nu=L-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(f+\xi)\,dx,$
$\displaystyle\mathrm{supp}(\nu)=[L-\max_{P}(f+\xi),L-\min_{P}(f+\xi)].$

Hence, we obtain

$\displaystyle J^{NA}(\mathcal{F}_{\xi})$
$\displaystyle=\sup\mathrm{supp}(\nu)-\int_{\mathbf{R}}\lambda\,\nu=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(f+\xi)\,dx-\min_{P}(f+\xi),$
$\displaystyle J_{T}^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=\inf_{\text{$\xi\colon$affine}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(f+\xi)\,dx-\min_{P}(f+\xi)\right).$

∎

######  Proposition 4.5.4.

For each $f\in\mathcal{C}_{PL}^{\mathbf{Q}}$ we have

$\displaystyle M^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=\frac{1}{\mathrm{vol}(P)}L(f)$
$\displaystyle\coloneqq\frac{1}{\mathrm{vol}(P)}\left(\int_{\partial
P}f\,d\sigma-\overline{s}\int_{P}f\,dx\right).$

###### Proof.

Let $f\in\mathcal{C}_{PL}^{\mathbf{Q}}$. First we assume the central fiber of
$(\mathcal{X}_{f},\mathcal{L}_{f})$ is generically reduced. Then
$M^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$ coincides with
$DF(\mathcal{X}_{f},\mathcal{L}_{f})$. Further, by (4.3) and [80, Lemma 3.3]
we have

$\displaystyle w_{m}$ $\displaystyle=\sum_{\alpha\in mP\cap
M}m\left(L-f\left(\frac{\alpha}{m}\right)\right)$
$\displaystyle=m\left(m^{n}\int_{P}(L-f)\,dx+\frac{m^{n-1}}{2}\int_{\partial
P}(L-f)\,d\sigma+O(m^{n-2})\right)$
$\displaystyle=m^{n+1}\int_{P}(L-f)\,dx+\frac{m^{n}}{2}\int_{\partial
P}(L-f)\,d\sigma+O(m^{n-1})$

for any $m\in\mathbf{Z}_{>0}$. By noting $\mathrm{vol}(P)=(L^{n})/n!$ and
$\overline{s}=\sigma(\partial P)/\mathrm{vol}(P)$, we obtain

$\displaystyle M^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=DF(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=\frac{n!}{(L^{n})}\left(-\int_{\partial
P}(L-f)\,d\sigma+\overline{s}\int_{P}(L-f)\,dx\right)$
$\displaystyle=\frac{1}{\mathrm{vol}(P)}\left(\int_{\partial
P}f\,d\sigma-\overline{s}\int_{P}f\,dx\right).$

For general cases, we consider $df$ for $d\in\mathbf{Z}_{>0}$. This
corresponds to replacing $(\mathcal{X}_{f},\mathcal{L}_{f})$ with its base
change $(\mathcal{X}_{df},\mathcal{L}_{df})$ (see Example 4.3.1 (2)). Then, by
Proposition 3.3.4 there exists $d_{0}\in\mathbf{Z}_{>0}$ such that
$DF(\mathcal{X}_{df},\mathcal{L}_{df})=M^{NA}(\mathcal{X}_{df},\mathcal{L}_{df})=dM^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
for all $d\in\mathbf{Z}_{>0}$ divisible by $d_{0}$. Further, by Proposition
3.3.3 we have

$\displaystyle M^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=\frac{1}{d}M^{NA}(\mathcal{X}_{df},\mathcal{L}_{df})$
$\displaystyle=\frac{1}{d}\left[\frac{1}{\mathrm{vol}(P)}\left(\int_{\partial
P}df\,d\sigma-\overline{s}\int_{P}df\,dx\right)\right]$
$\displaystyle=\frac{1}{\mathrm{vol}(P)}\left(\int_{\partial
P}f\,d\sigma-\overline{s}\int_{P}f\,dx\right),$

as required. ∎

######  Proposition 4.5.5.

For each $f\in\mathcal{C}_{PL}^{\mathbf{Q}}$ we have

$\displaystyle H_{V}^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}fV\,dx,$
$\displaystyle M_{V}^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle\coloneqq\frac{1}{\mathrm{vol}(P)}\left(\int_{\partial
P}f\,d\sigma-\int_{P}(\overline{s}+V)f\,dx\right).$

###### Proof.

It is enough to show the formula for $H_{V}^{NA}$. Since $\int_{P}V\,dx=0$, we
have

$\displaystyle H_{V}^{NA}(\mathcal{X}_{f},\mathcal{L}_{f})$
$\displaystyle=\lim_{m\to\infty}\frac{1}{m^{2}}\left\\{\frac{1}{E_{P}(m)}\sum_{\alpha\in
mP\cap M}m\left(L-f\left(\frac{\alpha}{m}\right)\right)V(\alpha)\right.$
$\displaystyle\quad\left.-\frac{1}{(E_{P}(m))^{2}}\sum_{\alpha\in mP\cap
M}m\left(L-f\left(\frac{\alpha}{m}\right)\right)\sum_{\alpha\in mP\cap
M}V(\alpha)\right\\}$
$\displaystyle=\lim_{m\to\infty}\frac{1}{E_{P}(m)}\sum_{\alpha\in mP\cap
M}\left(L-f\left(\frac{\alpha}{m}\right)\right)V\left(\frac{\alpha}{m}\right)$
$\displaystyle=\frac{1}{\mathrm{vol}(P)}\int_{P}(L-f)V\,dx$
$\displaystyle=-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}fV\,dx,$

as required. ∎

By Proposition 4.5.5, we have the following

######  Corollary 4.5.6.

Let $(X_{P},L_{P})$ be a polarized toric manifold. If $(X_{P},L_{P})$ is
uniformly relatively K-polystable, then there exists $\delta>0$ such that

(4.5) $\displaystyle L_{V}(f)\geq\delta\|f\|_{J}$

for any $f\in\mathcal{C}_{PL}^{\mathbf{Q}}$.

Our main theorem guarantees that the converse is also true, that is, a