# Modified Futaki invariant and equivariant Riemann-Roch formula

Feng Wang Bin $\text{Zhou}^{*}$ Xiaohua $\text{Zhu}^{**}$ Feng Wang,
Department of Mathematics, Zhejiang University, Hangzhou 310027, China  Bin
Zhou, School of Mathematical Sciences, Peking University, Beijing 100871,
China; and Mathematical Sciences Institute, The Australian National
University, Canberra, ACT 2601, Australia.  Xiaohua Zhu, School of
Mathematical Sciences, Peking University, Beijing 100871, China.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we give a new version of the modified Futaki invariant for a
test configuration associated to the soliton action on a Fano manifold. Our
version will naturally come from toric test configurations defined by
Donaldson for toric manifolds. As an application, we show that the modified
$K$-energy is proper for toric invariant Kähler potentials on a toric Fano
manifold.

###### Key words and phrases:

modified Futaki-invariant, Riemann-Roch formula, Kähler-Ricci solitons, toric
manifolds

###### 2000 Mathematics Subject Classification:

Primary: 53C25; Secondary: 53C55, 58J05, 19L10

*Partially supported by NSFC 11101004 and ARC DECRA

** Partially supported by the NSFC Grants 11271022 and 11331001.

##  0\. Introduction

Let $(M,g)$ be a Fano manifold with a Kähler form $\omega_{g}\in 2\pi
c_{1}(M)$ of $g$. Denote $\eta(M)$ to be the linear space of holomorphic
vector fields on $M$. Then by Hodge Theorem, for any $X\in\eta(M)$, there
exists a unique smooth complex-valued function $\theta_{X}(g)$ of $M$ such
that

$\displaystyle\begin{cases}i_{X}\omega_{g}=\sqrt{-1}\bar{\partial}\theta_{X}(g),&\\\\[5.0pt]
\int_{M}e^{\theta_{X}(g)}\omega_{g}^{n}=\int_{M}\omega_{g}^{n}.&\end{cases}$

In [TZ2], Tian and Zhu introduced the modified Futaki invariant on
$\eta(M)\times\eta(M)$,

(0.1)
$F_{X}(v)=\int_{M}v(h_{g}-\theta_{X}(g))e^{\theta_{X}(g)}\frac{\omega_{g}^{n}}{n!},\
\ X,v\in\eta(M),$

where $h_{g}$ is the Ricci potential of $g$ such that

${\rm Ric}(\omega_{g})-\omega_{g}=\sqrt{-1}\partial\bar{\partial}h_{g}.$

It was shown there that $F_{X}(v)$ is a holomorphic invariant independent of
the choice of $g$ with $\omega_{g}\in 2\pi c_{1}(M)$, and so it defines an
obstruction to the existence of Kähler-Ricci solitons with respect to an
element $X\in\eta_{r}(M)$, where $\eta_{r}(M)$ is the reductive part of
$\eta(M)$. In particular, when $X=0$, $F_{X}(v)$ is classical Futaki invariant
[Fut]. It was also proved by Tian and Zhu that there exists a unique $X$ such
that $F_{X}(v)=0$, $\ \forall\ v\in\eta_{r}(M)$. For convenience, we call such
$X$ the soliton vector field on $M$.

Recently, by using Ding-Tian’s idea of generalizing Futaki invariant [DT],
Xiong and Berman, gave a generalization of the modified Futaki invariant
$F_{X}(\cdot)$ for any special degeneration associated to $X$, independently
[Xi, Be2]. As a consequence, they both proved that $F_{X}(\cdot)$ is
nonnegative if $M$ admits a Kähler-Ricci soliton. Berman also gaves an
algebraic formula for $F_{X}(\cdot)$, which depends on weights of the
automorphisms group on holomorphic sections of multi-line bundles on the
center fibre induced by the test configuration. The purpose of present paper
is to define the modified Futaki invariant $F_{X}(\cdot)$ for any test
configuration associated to $X$. Our motivation is inspired by Berman’s
algebraic formula for special degenerations and is to modify his formula for
general test configurations. Then by applying the equivariant Riemann-Roch
formula with a $2$-dimensional torus action we show that our definition
coincides with Xiong-Berman’s for special degenerations. Our definition also
includes Tian and Donaldson’s generalized Futaki invariant as a special case
[T1, D1] when $X=0$.

As examples, we compute the new version of the modified Futaki invariant for
any toric degeneration on toric manifolds introduced by Donaldson [D1]. Then
by using the method in [ZZ1], we are able to prove

######  Theorem 0.1.

Any toric Fano manifold is modified $K$-stable for any toric degeneration.
Furthermore, the modified $K$-energy is proper for toric invariant Kähler
potentials.

Theorem 0.1 gives a new proof of Wang-Zhu’s result for the existence of
Kähler-Ricci solitons on any toric Fano manifold [WZ]. We can also study the
existence of conical Kähler-Ricci solitons on toric Fano manifolds by showing
the properness of modified Log $K$-energy. As a consequence, we give a new
proof of Datar-Guo-Song-Wang Theorem in [DGSW]. In particular, we have

######  Theorem 0.2.

Let $X$ be a soliton vector field on a toric Fano manifold $M$.111$X$ is not
unique in general for the existence of conical Kähler-Ricci solitons while it
is always unique modulo ${\rm Aut}^{0}(M)$ for the existence of Kähler-Ricci
solitons [TZ2]. Then for any $\beta\leq 1$ there exists a unique toric
invariant conical Kähler-Ricci soliton which has conical angle $2\pi\beta$
along each face divisor $D_{i}$ of $M$.

We note that the above energy argument was used by other people, such as in
[JMR, LS, T3, LZ] to study the conical Kähler-Einstein metrics on general Fano
manifolds. Theorem 0.1 and Theorem 0.2 will be proved in Section 2-3 and
Section 4, respectively.

Acknowledgements. The third author would like to thank professor Gang Tian for
his interest to the paper and sharing his insight in Kähler geometry.

##  1\. New version of modified Futaki invariant

According to [D1], a test-configuration on a Fano manifold $M$ is a scheme
$\mathcal{M}$ with a $\mathbb{C}^{*}$\- action which consists of two
integredients:

  1. (1)

an flat $\mathbb{C}^{*}$-equivarant map $\pi:\mathcal{M}\to\mathbb{C}$ such
that $\pi^{-1}(t)$ is biholomorphic to $M$ for any $t\neq 0$;

  2. (2)

an holomorphic line bundle $\mathfrak{L}$ on $\mathcal{M}$ such that
$\mathfrak{L}|_{\pi^{-1}(t)}$ is isomorphic to $K_{M}^{-r}$ for some integer
$r>0$ for any $t\neq 0$.

######  Definition 1.1.

$\mathcal{M}$ is called a test-configuration associated to the soliton action
induced by $X$ if $\sigma_{t}^{v}$ communicate to $\sigma_{t}^{X}$, where
$\sigma_{t}^{X}$ and $\sigma_{t}^{v}$ are two lifting one-parameter subgroups
on $\mathcal{M}$ induced by $X$ and the holomorphic vector field $v$
associated to the $\mathbb{C}^{*}$\- action, respectively. If furthermore the
center fibre $M_{0}=\pi^{-1}(0)$ is a normal variety we call $\mathcal{M}$ is
a special degeneration. In particular, if $\mathcal{M}\cong
M\times\mathbb{C}$, $\mathcal{M}$ is called a trivial test-configuration.

For simplicity, we let $L=\mathfrak{L}|_{M_{0}}$. Let
$\sigma_{t}^{X}(k),\sigma_{t}^{v}(k)$ be two induced one-parameter subgroups
on $H^{0}(M_{0},L^{k})$ by $\sigma_{t}^{X},\sigma_{t}^{v}$, respectively.
Denote by $\\{e^{X_{\alpha}^{k}}\\}$ and $\\{e^{v_{\alpha}^{k}}\\}$ be
eigenvalues of actions $\sigma_{1}^{X}$ and $\sigma_{1}^{v}$. We set

$S_{1}=\sum_{i}e^{\frac{X_{\alpha}^{k}}{k}}v_{\alpha}^{k},\ \
S_{2}=\frac{1}{2}\sum_{i}e^{\frac{X_{\alpha}^{k}}{k}}\frac{X_{\alpha}^{k}}{k}\frac{v_{\alpha}^{k}}{k}.$

Then

$S_{1}=\frac{\partial}{\partial t}{\rm
trace}(e^{sX^{k}+tv^{k}})|_{s=\frac{1}{k},\ t=0},\ \
S_{2}=\frac{1}{2k^{2}}\frac{\partial}{\partial s}\frac{\partial}{\partial
t}{\rm trace}(e^{sX^{k}+tv^{k}})|_{s=\frac{1}{k},t=0},$

where $X^{k}=(X_{\alpha}^{k})$, $v^{k}=(v_{\alpha}^{k})$ are two vectors as
elements of Lie algebra associated to $\sigma_{t}^{X}(k)$,
$\sigma_{t}^{v}(k)$, respectively. Our observation is that both $S_{1}$ and
$S_{2}$ can be computed by the equivariant Riemann-Roch formula with
$G=(S^{1})^{2}$-action. In fact

(1.1) $\displaystyle{\rm trace}(e^{sX^{k}+tv^{k}})=\int_{M_{0}}{\rm
ch}^{G}(-kL){\rm Td}^{G}(X_{0}),$

where ${\rm ch}^{G}(-kL)$ is a $G$-equivariant Chern character of multi-line
bundle $-kL$ and ${\rm Td}^{G}(M_{0})$ is a $G$-equivariant Todd character of
$M_{0}$ [AS]. In particular, for a special degeneration associated to the
soliton action, we can compute both $S_{1}$ and $S_{2}$ precisely in the
following.

According to [DT], for a special degeneration, there exists a hermitian metric
$h$ on $(X_{0},L)$ such that curvature $c(h,L)$ is a r-multiple of an
admissible metric $g$ with property: there exists a $L^{p}$-integrable
function $h_{g}$ (for any $p\geq 0$) with respect to $g$ such that

i) ${\rm Ric}(\omega_{g})-\omega_{g}=\sqrt{-1}\partial\bar{\partial}h_{g}$, on
the smooth part of $M_{0}$;

ii) $v(h_{g})$ is $L^{1}$-integrable with respect to $g$. We note that $v$ is
an admissible holomorphic vector field on $M_{0}$ [DT].

For any admissible holomorphic vector field $w$ on $M_{0}$, we define a
function by

(1.2) $\displaystyle\theta_{w}=-\frac{L_{w}h}{h}.$

Then a direct computation shows

$\displaystyle\sqrt{-1}\overline{\partial}\theta_{w}=i_{w}\omega_{g},$

and consequently

(1.3)
$\displaystyle\Delta\theta_{w}=\frac{L_{w}\omega_{g}^{n}}{\omega_{g}^{n}}.$

######  Lemma 1.1.

$\theta_{w}$ satisfies

(1.4) $\displaystyle\triangle\theta_{w}+w(h_{g})+\theta_{w}=0.$

###### Proof.

It suffices to verify (1.4) on the smooth part of $X_{0}$. Since

${\rm Ric}(\omega_{g})-\omega_{g}={\rm Ric}(\omega_{g})-{\rm
Ric}(h)=\sqrt{-1}\partial\bar{\partial}\log\frac{h}{\omega_{g}^{n}},$

we have

$h_{g}=\log\frac{h}{\omega_{g}^{n}}+const.$

It follows

$w(h_{g})=w(\log\frac{h}{\omega_{g}^{n}}).$

Thus

$\displaystyle\triangle\theta_{w}+w(h_{g})+\theta_{w}=\frac{L_{w}\omega_{g}^{n}}{\omega_{g}^{n}}+w(\log\frac{h}{\omega_{g}^{n}})-\frac{L_{w}h}{h}=0.$

∎

######  Lemma 1.2.

Let $\theta_{X}$ and $\theta_{v}$ be defined by (1.2) for the vectors $X$ and
$v$, respectively. Then, instead of $k$ by $kr$, we have

(1.5) $\displaystyle S_{1}$ $\displaystyle=$ $\displaystyle
k^{n+1}\int_{\mathcal{M}_{0}}\theta_{v}e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}$
$\displaystyle+\frac{k^{n}}{2}\left[n\int_{\mathcal{M}_{0}}\theta_{v}e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}+\int_{\mathcal{M}_{0}}\theta_{X}\theta_{v}e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}-\int_{\mathcal{M}_{0}}v(h_{g}-\theta_{X})e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}\right]+O(k^{n-1}),$
(1.6) $\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{k^{n}}{2}\int_{\mathcal{M}_{0}}\theta_{v}\theta_{X}e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}+O(k^{n-1}).$

###### Proof.

Since

${\rm ch}^{G}(-kK_{M_{0}})=e^{k\omega_{g}+ks\theta_{X}+kt\theta_{v}}$

and

${\rm Td}^{G}(M_{0})=1+\frac{1}{2}c^{G}_{1}+\sum_{i+j\geq
2}a_{ij}s^{i}t^{j}+2l-{\rm forms}~{}(l\geq 2),$

where $c^{G}_{1}$ is the first Chern $G$-equivariant form, we have

$\displaystyle\frac{d}{dt}\Big{|}_{t=0}{\rm trace}(e^{\frac{1}{k}X+tv})$
$\displaystyle=k\int_{M_{0}}\theta_{v}e^{k\omega_{g}+\theta_{X}}{\rm
Td}^{G}(M_{0})+\int_{M_{0}}e^{k\omega_{g}+\theta_{X}}\frac{d}{dt}{\rm
Td}^{G}(M_{0})$
$\displaystyle=k^{n+1}\int_{M_{0}}\theta_{v}e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}\wedge(1+\frac{1}{2}c^{G}_{1})+\frac{1}{2}k^{n}\int_{M_{0}}e^{\theta_{X}}\frac{\omega_{g}^{n-1}}{n!}\wedge
c^{G}_{1}$
$\displaystyle+\int_{M_{0}}e^{k\omega_{g}+\theta_{X}}(\frac{d}{dt}c^{G}_{1})\frac{\omega_{g}^{n}}{n!}+O(k^{n-2}).$

Note that

$c^{G}_{1}={\rm Ric}(\omega_{g})-s\Delta\theta_{V_{0}}-t\Delta\theta_{W_{0}}.$

Thus

$\displaystyle\frac{d}{dt}\Big{|}_{t=0}{\rm trace}(e^{\frac{1}{k}X+tv})$ (1.7)
$\displaystyle=k^{n+1}\int_{M_{0}}e^{\theta_{X}}\theta_{v}\frac{\omega_{g}^{n}}{n!}+\frac{k^{n}}{2}\left(\int_{M_{0}}e^{\theta_{X}}\theta_{v}S\frac{\omega_{g}^{n}}{n!}-\int_{M_{0}}e^{\theta_{X}}\theta_{v}\triangle\theta_{X}\frac{\omega_{g}^{n}}{n!}-\int_{M_{0}}e^{\theta_{X}}\triangle\theta_{v}\frac{\omega_{g}^{n}}{n!}\right)+O(k^{n-1}).$

On the other hand, using the integration by parts, it is easy to see that

(1.8)
$\displaystyle\int_{M_{0}}\theta_{v}e^{\theta_{X}}S\frac{\omega_{g}^{n}}{n!}$
$\displaystyle=$ $\displaystyle
n\int_{M_{0}}\theta_{v}e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}+\int_{M_{0}}\theta_{v}e^{\theta_{X}}\triangle
h_{g}\frac{\omega_{g}^{n}}{n!}$ $\displaystyle=$ $\displaystyle
n\int_{M_{0}}\theta_{v}e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}-\int_{M_{0}}\theta_{v}e^{\theta_{X}}X(h_{g})\frac{\omega_{g}^{n}}{n!}-\int_{M_{0}}v(h_{g})e^{\theta_{X}}\frac{\omega_{g}^{n}}{n!}.$

Hence, by using the relation (1.4) for $\theta_{X}$, we will get (1.5) from
(1) and (1.8) immediately.

The proof of (1.6) is easy. We skip it. ∎

Let $N_{k}={\rm dim}H^{0}(M_{0},L^{k})$. Then by the Riemann-Roch formula, we
have

(1.9)
$N_{k}=\int_{M_{0}}ch(-kK_{M_{0}})Td({M_{0}})=\int_{M_{0}}e^{k\omega_{g}}Td(M_{0}).$

Note that the Todd class is given by

$Td(M_{0})=1+\frac{1}{2}c_{1}(M_{0})+\cdot\cdot\cdot,$

where $c_{1}(M_{0})$ is the first Chern class of $M_{0}$. Thus

(1.10) $N_{k}=A_{0}k^{n}+B_{0}k^{n-1}+O(k^{n-2}),$

where

(1.11) $A_{0}=\int_{M_{0}}\frac{\omega_{g}^{n}}{n!}={\rm Vol}(M_{0}),\ \
B_{0}=\frac{1}{2}\int_{M_{0}}S\,\frac{\omega_{g}^{n}}{n!}=\frac{n}{2}{\rm
Vol}(M_{0}).$

Here $S$ is the scalar curvature of $\omega_{g}$.

By Lemma 1.2 , we can write $\frac{S_{1}-S_{2}}{kN_{k}}$ as an expansion in
$k^{-1}$,

(1.12) $\displaystyle-\frac{S_{1}-S_{2}}{kN_{k}}=F_{0}+F_{1}k^{-1}+o(k^{-2}).$

######  Proposition 1.1.

For a special degeneration on a Fano manifold, we have

$F_{1}=\frac{1}{2}F_{0}=\frac{1}{2{\rm
Vol}(M_{0})}\int_{M_{0}}v(h_{g}-\theta_{X})e^{\theta_{X}}\,\frac{\omega_{g}^{n}}{n!}.$

###### Proof.

By using the relation (1.4) for $\theta_{v}$, we have

$\int_{M_{0}}v(h_{g}-\theta_{X})e^{\theta_{X}}\,\frac{\omega_{g}^{n}}{n!}=-\int_{M_{0}}\theta_{v}e^{\theta_{X}}\,\frac{\omega_{g}^{n}}{n!}.$

Then the Proposition follows from Lemma 1.2 immediately. ∎

For a general test-configuration associated to the soliton action, by the
equivariant Riemann-Roch formula (1.1), we write $S_{1}$ and $S_{2}$ formally
as,

$\displaystyle S_{1}=Ak^{n+1}+Bk^{n}+O(k^{n-1}),$ $\displaystyle
S_{2}=Ck^{n}+Dk^{n-1}+O(k^{n-2}).$

Then the invariant $F_{1}$ in (1.12) is equal to $\frac{2(B-C)-nA}{2A_{0}}$.
We call this quantity the modified Futaki invariant for the test-configuration
$\mathcal{M}$ associated to the soliton vector field $X$.

######  Remark 1.1.

1) In [Be2], Berman defines the modified Futaki invariant by $F_{0}$ for any
special degeneration associated to the soliton vector field $X$. Proposition
1.1 means that our definition coincides Berman’s case. But in general $F_{0}$
will be different to $F_{1}$ as showed in next section for toric degenerations
on a toric Fano manifold. In fact, we will show that the invariant $F_{1}$
comes natrually from the study of modified K-energy on toric manifolds (cf.
Section 3).

2) Proposition 1.1 also shows that the Donaldson invariant $F_{1}$ in [D1]
coincides with the generalized Futaki invariant defined by Tian in [T1] for
special degenerations.

As in [T1, D1], we introduce a notation of modified $K$-stability for any Fano
manifold $M$ via the quantity $F_{1}$.

######  Definition 1.2.

A Fano manifold $M$ is called modified $K$-semi-stable if $F_{1}\geq 0$ for
any test-configuration associated to the soliton action of $M$ and $M$ is
modified $K$-stable if in addition $F_{1}=0$ happens if and only if the test-
configuration is trivial.

Due to the celebrated solving of Yau-Tian-Donaldson’s conjecture for the
existence of Kähler-Einstein metrics [T3, CDS], we propose the following
generalized Yau-Tian-Donaldson’s conjecture for the existence of Kähler-Ricci
solitons.

######  Conjecture 1.1.

A Fano manifold $M$ admits a Kähler-Ricci soliton if and only if $M$ is
modified $K$-stable.

In the remaining sections, we verify Conjecture 1.1 in case of toric Fano
manifolds (also for general conical Kähler-Ricci solitons).

##  2\. Modified Futaki invariant for toric degenerations

In this section we compute the modified Futaki invariant $F_{1}$ for a toric
degeneration on a toric Fano manifold $M$. Let
$T=T^{n}_{\mathbb{C}}=(\mathbb{C}^{*})^{n}=(S^{1})^{n}\times\mathbb{R}^{n}$ be
torus action on $M$ and denote $G_{0}=(S^{1})^{n}$. Choose an
$G_{0}$-invariant Kähler metric $g$ with $\omega_{g}\in 2\pi c_{1}(M)$. The
open dense orbit of $T_{\mathbb{C}}^{n}$ in $M$ induces an global coordinates
$(w_{1},...,w_{n})\in(\mathbb{C}^{*})^{n}$. To do the reduction we use the
affine logarithmic coordinates $z_{i}=\log w_{i}=\xi_{i}+\sqrt{-1}\eta_{i}$.
Then $\omega_{g}$ is determined by a strictly convex function $\varphi_{0}$
which depends only on $\xi_{1},...,\xi_{n}\in{\mathbb{R}^{n}}$ in the
coordinates $(z_{1},......,z_{n})$, namely
$\omega_{g}=\sqrt{-1}\partial\bar{\partial}\varphi_{0}$ on
$(\mathbb{C}^{*})^{n}$. Since the torus action $T$ is Hamiltonian, there
exists a moment map $m:M\rightarrow\mathfrak{t}^{*}$, where $\mathfrak{t}^{*}$
is the dual of the Lie algebra of $T$ which can be identified with
$\mathbb{R}^{n}$. By the convexity theorem the image is a convex polytope in
$\mathbb{R}^{n}$. Moreover, the moment map can be given by

$(m_{1},......,m_{n})=\nabla\varphi_{0}=\left(\frac{\partial\varphi_{0}}{\partial\xi_{1}},......,\frac{\partial\varphi_{0}}{\partial\xi_{n}}\right).$

Denote the image by $P=D\varphi_{0}(\mathbb{R}^{n})$. Then $P$ is a convex
polytope represented by a set of inequalities of the form (up to translation
of coordinates)

(2.1) $P=\\{x\in\mathbb{R}^{n}:\ \langle x,\ell_{i}\rangle\leq 1,\
i=1,2,\cdots,d\\},$

where $\ell_{i}$ is the outer normal vector to a face of $P$ and $d$ is the
number of faces of $P$. This polytope is independent of the choice of the
metric $g$ in $2\pi c_{1}(M)$. See [Ab1, Ab2, Gu] for more details.

On the other hand, the soliton vector field $X$ can be written as
$X=\sum_{i=1}^{n}\theta_{i}w_{i}\frac{\partial}{\partial
w_{i}}=\sum_{i=1}^{n}\theta_{i}\frac{\partial}{\partial z_{i}}$. Let
$\theta_{X}(\omega_{g})$ be the potential function determined by

$i_{X}\omega_{g}=\sqrt{-1}\bar{\partial}\theta_{X}(\omega_{g}),$

then $\theta_{X}(\omega_{g})=X(\varphi_{0})+c$ for some $c\in\mathbb{R}^{n}$.
By (1.4), it is easy to see

(2.2) $\int_{M}\theta_{X}(\omega_{g})e^{h_{g}}\frac{\omega_{g}^{n}}{n!}=0.$

Note that

$\int_{M}\theta_{\frac{\partial}{\partial
z_{i}}}(\omega_{g})e^{h_{g}}\frac{\omega_{g}^{n}}{n!}=C\int_{\mathbb{R}^{n}}\frac{\partial\varphi}{\partial\xi_{i}}e^{-\varphi},\
i=1,2,\cdot\cdot\cdot,$

for a constant $C$. Then $c=0$ under the above normalization, and we have

(2.3)
$\theta_{X}(\omega_{g})=\sum_{i=1}^{n}\theta_{i}\frac{\partial\varphi_{0}}{\partial
z_{i}}=\frac{1}{2}\sum_{i=1}^{n}\theta_{i}\frac{\partial\varphi_{0}}{\partial\xi_{i}}=\sum_{i=1}^{n}{\theta_{i}x_{i}}:=\theta(x)$

in the symplectic coordinates. One can see that $\theta(x)$ is also
independent of the choice of metric $g$.

Accordimg to [D1], a toric degeneration is induced by positive rational,
piecewise linear functions on $P$. Note that a piecewise linear(PL) convex
function $u$ on $P$ is of the form

$u=\text{max}\\{u^{1},...,u^{r}\\},$

where $u^{\lambda}=\sum a_{i}^{\lambda}x_{i}+c^{\lambda},\lambda=1,...,r,$ for
some vectors $(a_{1}^{\lambda},...,a^{\lambda}_{n})\in\mathbb{R}^{n}$ and some
numbers $c^{\lambda}\in\mathbb{R}$. $u$ is called a rational piecewise linear
convex function if the coefficients $a^{\lambda}_{i}$ and numbers
$c^{\lambda}$ are all rational.

For a positive rational PL convex function $u$ on $P$, we choose a positive
integer $R$ so that

$Q=\\{(x,t)\ |\ x\in P,\ 0<t<R-u(x)\\}$

is a convex polytope in $\mathbb{R}^{n+1}$. Without loss of generality, we may
assume that the coefficients $a^{\lambda}_{i}$ are integers and $Q$ is an
integral polytope. Otherwise we replace $u$ by $lu$ and $Q$ by $lQ$ for some
integer $l$, respectively. Then the $n+1$-dimensional polytope $Q$ determines
an $(n+1)$-dimensional toric variety $M_{Q}$ with a holomorphic line bundle
$\mathfrak{L}\rightarrow M_{Q}$. Note that the face
$\bar{Q}\cap\\{\mathbb{R}^{n}\times\\{0\\}\\}$ of $Q$ is a copy of the
$n$-dimensional polytope $P$, so we have a natural embedding $i:M\rightarrow
M_{Q}$ such that $\mathfrak{L}|_{M}=-K_{M}$. Decomposing the torus action
$T_{\mathbb{C}}^{n+1}$ on $M_{Q}$ as $T_{\mathbb{C}}^{n}\times\mathbb{C}^{*}$
so that $T_{\mathbb{C}}^{n}\times\\{\text{Id}\\}$ is isomorphic to the torus
action on $M$, we get $\mathbb{C}^{*}$-action $\sigma^{u}$ by
$\\{\text{Id}\\}\times{\mathbb{C}^{*}}$. Hence, we define an equivariant map

$\pi:M_{Q}\rightarrow\mathbb{C}\mathbb{P}^{1}$

satisfying $\pi^{-1}(\infty)=i(M)$. One can check that
$\mathcal{W}=M_{Q}\backslash i(M)$ is a test configuration for the pair $M$,
called a toric degeneration.

Let $kP$ be the polytope which corresponds to the bundle $-kK_{M}$. Let
$B_{k,P}=\mathbb{Z}^{n}\cap k\overline{P}$ be the lattices set of $kP$. Let
$d\sigma=\langle\vec{n},x\rangle\,d\sigma_{0}$, where $\vec{n}$ is the unit
outer normal vector field, and $d\sigma_{0}$ is the Lebesgue measure on on
$\partial P$. We need the following lemma.

######  Lemma 2.1.

Let $\phi$ be a continous function on $\overline{P}$, then

(2.4)
$\sum_{I\in{B_{k,P}}}\phi(I/k)=k^{n}\int_{P}\phi\,dx+\frac{k^{n-1}}{2}\int_{\partial
P}\phi\,d\sigma+O(k^{n-2}).$

In particular,

$N_{k}=k^{n}|P|+\frac{k^{n-1}}{2}|\partial P|+O(k^{n-2}).$

Note that $\frac{|\partial P|}{|P|}=n$ if $P$ corresponds to $2\pi c_{1}(M)$.

This lemma was proved in [D1] for convex rational PL functions. It is easy to
see that the formula can be extended to continuous function by approximation
arguments.

######  Proposition 2.1.

Let

(2.5) $\displaystyle\mathcal{L}(u)=\int_{\partial
P}ue^{\theta(x)}\,d\sigma-\int_{P}(n+\theta(x))e^{\theta(x)}u\,dx.$

Then for a toric degeneration on $M$ induced by a positive rational PL-convex
function $u$, we have

(2.6) $F_{1}=\frac{1}{2Vol(P)}\mathcal{L}(u).$

###### Proof.

We consider the space $H^{0}(\mathcal{W},\mathfrak{L}^{k})$ of holomorphic
sections over $\mathcal{W}$. It is well-known that
$H^{0}(\mathcal{W},\mathfrak{L}^{k})$ has a basis $\\{S_{I,i}\\}$, where $I$
is a lattice in $B_{k,P}$ and $0\leq i\leq k(R-u)(I/k)$. By using the exact
sequence for large $k$,

$0\longrightarrow
H^{0}(\mathcal{W},\mathfrak{L}^{k}\otimes\pi^{*}(\vartheta(-1)))\longrightarrow
H^{0}(\mathcal{W},\mathfrak{L}^{k})\longrightarrow
H^{0}(M_{0},\mathfrak{L}^{k})\longrightarrow 0,$

$H^{0}(M_{0},\mathfrak{L}^{k})$ has a basis
$\\{S_{I,k(R-u)(I/k)}|_{M_{0}}\\}_{I\in{B_{k,P}}}$. According to [ZZ1], the
action $\sigma^{X}$ induced by $X$ acts on $S_{I,k(R-u)(\frac{I}{k})}$ with
weight $k\theta(I/k)$. The action $\sigma^{u}$ induced by $u$ acts on
$S_{I,k(R-u)(\frac{I}{k})}$ with weight $k(R-u)(I/k)$. Then by Lemma 2.1, it
is easy to see that

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\sum_{I\in{B_{k,P}}}e^{\theta(I/k)}k(R-u)(I/k)$ $\displaystyle=$
$\displaystyle\int_{P}e^{\theta(x)}(R-u)\,dx-\frac{k^{n}}{2}\int_{\partial
P}e^{\theta(x)}(R-u)\,d\sigma+O(k^{n-1})+\frac{1}{2}\left[k^{n}\int_{P}e^{\theta(x)}(R-u)\cdot\theta(x)\,dx\right],$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\sum_{I\in{B_{k,P}}}e^{\theta(I/k)}\theta(I/k)\cdot(R-u)(I/k)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[k^{n}\int_{P}e^{\theta(x)}(R-u)\cdot\theta(x)\,dx-\frac{k^{n-1}}{2}\int_{\partial
P}e^{\theta(x)}(R-u)\cdot\theta(x)\,d\sigma+O(k^{n-2})\right].$

Then

$\displaystyle-(S_{1}-S_{2})$
$\displaystyle=-k^{n+1}\int_{P}e^{\theta(x)}(R-u)\,dx-\frac{k^{n}}{2}\left[\int_{\partial
P}e^{\theta(x)}(R-u)\,d\sigma-\int_{P}e^{\langle
v,x\rangle}(R-u)\cdot\theta(x)\,dx\right]+O(k^{n-1}).$

We have

$\frac{-(S_{1}-S_{2})}{kN_{k}}=F_{0}+F_{1}k^{-1}+\cdot\cdot\cdot,$

where

$\displaystyle F_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{|P|}\int_{P}e^{\theta(x)}(R-u)\,dx,$ (2.7)
$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2|P|}\left[\int_{\partial
P}e^{\theta(x)}(R-u)\,d\sigma-\int_{P}e^{\theta(x)}(R-u)\cdot\theta(x)\,dx-\frac{|\partial
P|}{|P|}\int_{P}e^{\theta(x)}(R-u)\,dx\right]$ $\displaystyle=$
$\displaystyle\frac{R}{2|P|}\left[\int_{\partial
P}e^{\theta(x)}\,d\sigma-\int_{P}(n+\theta(x))e^{\theta(x)}\,dx\right]-\frac{1}{2|P|}\left[\int_{\partial
P}e^{\theta(x)}u\,d\sigma-\int_{P}(n+\theta(x))e^{\theta(x)}u\,dx\right].$

Integrating by parts, we have

$\displaystyle\int_{P}e^{\theta(x)}\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{n}\int_{\partial
P}e^{\theta(x)}\,d\sigma-\frac{1}{n}\int_{P}\theta(x)e^{\theta(x)}\,dx.$

Therefore, the coefficient of $R$ in (2.7) vanishes, and we have

(2.8) $F_{1}=\frac{1}{2|P|}\left[\int_{\partial
P}e^{\theta(x)}u\,d\sigma-\int_{P}(n+\theta(x))e^{\theta(x)}u\,dx\right].$

∎

######  Remark 2.1.

As can be seen in the above lemma, the weights of the action depend on $R$ and
$F_{0}$ also depends on the integer $R$. But $F_{1}$ is independent of $R$. In
particular, $F_{0}$ is different to $F_{1}$.

Since $X$ is the soliton vector field, $\mathcal{L}(u)=0$ for any linear
function $u$. This implies that $\mathcal{L}(u)$ is invariant when adding $u$
by a linear function. We call a convex function is normalized at $0\in P$ if
$\inf_{P}u=u(0)$. Let $\mathcal{C}_{\infty}$ be the set of smooth convex
functions on $\overline{P}$ and $\tilde{\mathcal{C}}_{\infty}$ be the set of
smooth convex functions normalized at $0\in P$. It is clear that the PL
functions can be approximated uniformly by functions in
$\mathcal{C}_{\infty}$.

######  Lemma 2.2.

There exists a $\lambda>0$ such that

(2.9) $\mathcal{L}(u)\geq\lambda\int_{\partial P}ue^{\theta(x)}\,d\sigma,\ \
u\in\tilde{\mathcal{C}}_{\infty}.$

###### Proof.

We note that $\mathcal{L}(u)$ can be rewritten as

(2.10) $\mathcal{L}(u)=\int_{P}[(\sum
x_{i}u_{i}-u)+u]e^{\theta(x)}\,dx\geq\int_{P}ue^{\theta(x)}\,dx.$

By the contradiction, we suppose that (2.9) is not true. Then there is a
sequence of functions $\\{u_{k}\\}$ in $\tilde{\mathcal{C}}_{\infty}$ such
that

(2.11) $\int_{\partial P}u_{k}e^{\theta(x)}\,d\sigma=1$

and

(2.12) $\mathcal{L}(u_{k})\longrightarrow 0,\text{as}\ \
k\longrightarrow\infty.$

By (2.11), there exists a subsequence (still denoted by $\\{u_{k}\\}$) of
$\\{u_{k}\\}$, which converges locally uniformly to a convex function
$u_{\infty}\geq 0$ on $P$. By (2.10) and (2.12), we have

$\int_{P}u_{k}e^{\theta(x)}\,dx\leq\mathcal{L}(u_{k})\longrightarrow 0.$

It follows

$\int_{P}u_{\infty}e^{\theta(x)}\,dx=0.$

Hence, we obtain $u_{\infty}\equiv 0$ in $P$. On the other hand,

$\displaystyle\mathcal{L}(u_{k})$ $\displaystyle=$
$\displaystyle\int_{\partial
P}u^{k}e^{\theta(x)}d\sigma-\int_{P}(n+\sum{x_{i}\theta_{i}})u_{k}e^{\theta(x)}\,dx$
$\displaystyle\longrightarrow$ $\displaystyle
1-\int_{P}(n+\sum{x_{i}\theta_{i}})u_{\infty}e^{\theta(x)}\,dx=1>0.$

This contradicts with (2.12). The lemma is proved. ∎

By Lemma 2.2, we immediately get

######  Theorem 2.1.

Any toric Fano manifold is modified $K$-stable for toric degenerations.

##  3\. Modified $K$-energy on a toric Fano manifold

Let $K_{X}$ be a one parameter compact subgroup generated by the image part
$\text{Im}(X)$ and denote by $\mathcal{H}_{X}(\omega_{g})$ a set of
$K_{X}$-invariant Kähler potentials. In the study of Kähler-Ricci solitons,
the modified Mabuchi’s $K$-energy $\mu_{\omega_{g}}(\phi)$ plays an important
role [TZ2, CTZ], where

$\displaystyle\mu_{\omega_{g}}(\phi)=-\frac{1}{V}\int_{0}^{1}\int_{M}\dot{\phi}_{t}[S(\phi_{t})-n-tr_{\omega_{\phi_{t}}}(\nabla_{\omega_{\phi_{t}}}X)-X(h_{\omega_{\phi_{t}}}-\theta_{X}(\omega_{\phi_{t}}))]e^{\theta_{X}(\omega_{\phi_{t}})}\frac{\omega_{\phi_{t}}^{n}}{n!}\wedge
dt.$

Here $\phi\in\mathcal{H}_{X}(\omega_{g})$ and $g$ is chosen to be
$K_{X}$-invariant. Recall two Aubin typed functionals introduced in [Z1],

$\displaystyle I_{\omega_{g}}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{V}\int_{M}\phi(e^{\theta_{X}(\omega_{g})}\frac{\omega_{g}^{n}}{n!}-e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}),$
$\displaystyle J_{\omega_{g}}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{V}\int_{0}^{1}\int_{M}\dot{\phi}_{s}(e^{\theta_{X}(\omega_{g})}\frac{\omega_{g}^{n}}{n!}-e^{\theta_{X}(\omega_{\phi_{s}})}\frac{\omega_{\phi_{s}}^{n}}{n!})\wedge
ds,$

where $\omega_{\phi}=\omega_{g}+\sqrt{-1}\partial\bar{\partial}\phi$ and
$\phi_{s}$ is a path in $\mathcal{H}_{X}(\omega_{g})$. It is known that

(3.1) $c_{1}I_{\omega_{g}}(\phi)\geq J_{\omega_{g}}(\phi)\geq
c_{2}I_{\omega_{g}}(\phi),\ \forall\ \phi\in\mathcal{H}_{X}(\omega_{g}),$

for two positive constants $c_{1}$, $c_{2}$. Then $\mu_{\omega_{g}}(\cdot)$
can also be written as

(3.2) $\displaystyle\mu_{\omega_{g}}(\phi)$ $\displaystyle=$
$\displaystyle-\frac{1}{V}\int_{M}\log\left(\frac{e^{\theta_{X}(\omega_{\phi})}\omega_{\phi}^{n}}{e^{\theta_{X}(\omega_{g})}\omega_{g}^{n}}\right)e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}-(I_{\omega_{g}}(\phi)-J_{\omega_{g}}(\phi))$
$\displaystyle+\frac{1}{V}\int_{M}(h_{g}-\theta_{X}(\omega_{g}))(e^{\theta_{X}(\omega_{g})}\frac{\omega_{g}^{n}}{n!}-e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}).$

######  Definition 3.1.

Let $(M,g)$ be a Fano manifold $M$. Let $G$ be a reductive subgroup of
automorphisms group ${\rm Aut}(M)$ which contains $K_{X}$. We call
$\mu_{\omega_{g}}(\phi)$ proper modulo $G$ if there is a continuous function
$p(t)$ in $\mathbb{R}$ with the property

$\lim_{t\to+\infty}p(t)=+\infty,$

such that

(3.3) $\mu_{\omega_{g}}(\phi)\geq\inf_{\sigma\in
G}p(I_{\omega_{g}}(\phi_{\sigma})),$

where $\phi_{\sigma}$ is defined by

$\omega_{g}+\sqrt{-1}\partial\bar{\partial}\phi_{\sigma}=\sigma^{*}(\omega_{g}+\sqrt{-1}\partial\bar{\partial}\phi).$

The properness of $\mu_{\omega_{g}}(\phi)$ is a sufficient condition for the
existence of Kähler-Ricci solitons due to the following lemma.

######  Lemma 3.1.

Suppose that $\mu_{\omega_{g}}(\phi)$ is proper modulo a reductive subgroup
$G$ of automorphisms group ${\rm Aut}(M)$ which contains $K_{X}$. Then $M$
admits a Kähler-Ricci soliton.

Lemma 3.1 was proved by using Kähler-Ricci flow as in [TZ3, Z2, BB] and can be
also proved by uisng the continuity method as in [CTZ, T2].

Let $P_{g}=\triangle_{g}+X(\cdot)$ be a linear elliptic operator defined on
the space

$\mathcal{N}_{X}=\\{u\in C^{\infty}(M)|~{}{\rm Im}(X(u))=0\\},$

associated to $\omega_{g}$ and a holomorphic on $M$. $P_{g}$ is a self-adjoint
elliptic operator on $\mathcal{N}_{X}$ with respect to the inner product,

$(\phi,\psi)=\int_{M}\phi\psi
e^{\theta_{X}(\omega_{g})}\frac{\omega_{g}^{n}}{n!}.$

The following lemma shows that the properness given in Definition 3.1
coincides with one as defined in [T1, CTZ], when $g$ is a Kähler-Ricci
soliton.

######  Lemma 3.2.

Suppose that $M$ admits a Kähler-Ricci soliton $g_{KS}$. Then the modified
$K$-energy $\mu_{\omega_{KS}}(\phi)$ is proper with respect to $X$ modulo
${\rm Aut}^{0}(M)$ iff there is a continuous function $\bar{p}(t)$ in
$\mathbb{R}$ with the property

$\displaystyle\lim_{t\to+\infty}\bar{p}(t)=+\infty$

such that

(3.4) $\mu_{\omega_{KS}}(\phi)\geq\bar{p}(I_{\omega_{KS}}(\phi)),\ \ \forall\
\phi\in\Lambda_{1}^{\bot}(M,g_{KS}),$

where $\Lambda_{1}(M,g_{KS})$ denotes the first non-zero eigenfunctions space
for the operator $P_{g_{KS}}$ associated to the metric $g_{KS}$, i.e.,
$\Lambda_{1}(M,g_{KS})={\rm Ker}(P_{g_{KS}}+I)$.

###### Proof.

First we prove the necessary part of the lemma. We choose the Kähler-Ricci
soliton metric $g_{KS}$ as an initial metric. Then we induce a functional on
$\text{Aut}^{0}(M)$ for any $\phi\in\mathcal{H}_{X}(\omega_{KS})$ by

$\Phi(\sigma)=I_{\sigma^{*}\omega_{\phi}}(-\phi_{\sigma})-J_{\sigma^{*}\omega_{\phi}}(-\phi_{\sigma}),$

where $\phi_{\sigma}$ is an induced Kähler potential defined by

$\omega_{KS}+\sqrt{-1}\partial\bar{\partial}\phi_{\sigma}.$

$I_{\sigma^{*}\omega_{\phi}}(\psi)$ and $J_{\sigma^{*}\omega_{\phi}}(\psi)$
are functionals $I(\psi)$ and $J(\psi)$ respectively while the initial metric
$\omega_{KS}$ is replaced by $\sigma^{*}\omega_{\phi}$. According to [TZ1],
one can show that there exists a $\tau\in\text{Aut}^{0}(M)$ such that

$\Phi(\tau)=\inf_{\sigma\in\text{Aut}^{0}(M)}\Phi(\sigma)$

and consequently $\phi_{\tau}\in\Lambda_{1}^{\bot}(M,g_{KS})$. In fact, from
the proof of uniqueness of Kähler-Ricci solitons in [TZ1] it can be proved
that $\phi\in\Lambda_{1}^{\bot}(M,g_{KS})$ iff

$I_{\omega_{\phi}}(-\phi)-J_{\omega_{\phi}}(-\phi)=\inf_{\sigma\in\text{Aut}^{0}(M)}\Phi(\sigma).$

Thus by the assumption (3.3) and relation (3.1), for any
$\phi\in\Lambda_{1}^{\bot}(M,g_{KS})$, we have

$\displaystyle\mu_{\omega_{KS}}(\phi)$ $\displaystyle\geq$
$\displaystyle\inf_{\sigma\in\text{Aut}^{0}(M)}p(I_{\omega_{KS}}(\phi_{\sigma}))$
$\displaystyle=$
$\displaystyle\inf_{\sigma\in\text{Aut}^{0}(M)}p(I_{\sigma^{*}\omega_{\phi}}(-\phi_{\sigma}))$
$\displaystyle\geq$
$\displaystyle\inf_{\sigma\in\text{Aut}^{0}(M)}\tilde{p}(\Phi(\sigma))$
$\displaystyle=$
$\displaystyle\tilde{p}(I_{\omega_{\phi}}(-\phi)-J_{\omega_{\phi}}(-\phi))$
$\displaystyle\geq$ $\displaystyle\tilde{p}(I_{\omega_{\phi}}(-\phi))$
$\displaystyle=$ $\displaystyle\tilde{p}(I_{\omega_{KS}}(\phi)),$

where $\tilde{p}(t)$ is another continuous function in $\mathbb{R}$ which
satisfying (2.9).

Next we prove the sufficient part. We note that the modified $K$-energy is
invariant under ${\rm Aut}^{0}(M)$ [TZ2]. Then by the discussion at last
paragraph, for any $\phi\in\mathcal{H}_{X}(\omega_{KS})$, we can choose a
$\tau\in\text{Aut}^{0}(M)$ such that
$\phi_{\tau}\in\Lambda_{1}^{\bot}(M,g_{KS})$ and

$\mu(\phi)=\mu(\phi_{\tau}).$

Thus by (3.4), we get

$\mu(\phi)\geq\bar{p}(I(\phi_{\tau}))\geq\inf_{\sigma\in\text{Aut}^{0}(M)}\bar{p}(I(\phi_{\sigma})).$

∎

The converse of Lemma 3.1 was conjectured by Tian in sense of (3.4) in the
case of Kähler-Einstein metrics [T1] and was proved by him under the
assumption that there is no any holomorphic vector field on $M$. Thus one may
believe that the converse of Lemma 3.1 is also true as a generalization of
Tian’s conjecture for the case of Kähler-Ricci solitons [CTZ]. In this section
we give a positive answer in the case of toric Fano manifolds. Namely, we
shall prove

######  Theorem 3.1.

On a toric Fano manifold, the modified $K$-energy is proper for toric
invariant Kähler potentials modula toric action.

Theorem 3.1 has been proved under the assumption that the Futaki invariant
vanishes [ZZ1]. In the following, we alway assume that $M$ is a toric Fano
manifold.

###  3.1. The reduction of modified $K$-energy

Denote $\mathcal{H}_{G_{0}}(\omega_{g})\subset\mathcal{H}_{X}(\omega_{g})$ to
be the set of $G_{0}$-invariant Kähler potentials. Then
$\mathcal{H}_{G_{0}}(\omega_{g})$ is equal to the set

$\\{\phi\in C^{\infty}({\mathbb{R}^{n}})|~{}|\phi|<\infty~{}{\rm
and}~{}\varphi_{0}+\phi~{}{\rm is~{}strictly~{}convex}\\}.$

By using the Legendre transformation $\xi=(D\varphi_{0})^{-1}(x)$, one sees
that the function (Legendre dual function) defined by

$u_{0}(x)=\langle\xi,D\varphi_{0}(\xi)\rangle-\varphi_{0}(\xi)=\langle\xi(x),x\rangle-\varphi_{0}(\xi(x)),\
\forall\ x\in P$

is strictly convex. Set the space of symplectic functions by

$\mathcal{C}=\\{u=u_{0}+f\ |\ u\ \text{is a strictly convex function in }\ P,\
f\in C^{\infty}(\overline{P})\\}.$

It was shown in [Ab1] that there is a bijection between $\mathcal{C}$ and
$\mathcal{H}_{G_{0}}(\omega_{g})$.

######  Proposition 3.1.

Let $\phi\in\mathcal{H}_{G_{0}}(\omega_{g})$ and $u$ be the Legendre dual
function of $\varphi_{0}+\phi$. Then the modified $K$-energy is given by

(3.5)
$\displaystyle\mu_{\omega_{g}}(\omega_{\phi})=\frac{(2\pi)^{n}}{V}\mathcal{F}(u)+C,$

where

(3.6)
$\mathcal{F}(u)=-\int_{P}\log\mbox{det}(u_{ij})e^{\theta(x)}\,dx+\mathcal{L}(u),$

and $C$ is a constant.

###### Proof.

By (3.2), a direct computation shows

$\displaystyle\mu_{\omega_{g}}(\phi)$
$\displaystyle=\frac{1}{V}\int_{M}\log\left(\frac{e^{\theta_{X}(\phi)}\omega_{\phi}^{n}}{e^{\theta_{X}(\omega_{g})}\omega_{g}^{n}}\right)e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}-\left[\frac{1}{V}\int_{0}^{1}\int_{M}\dot{\phi}_{t}e^{\theta_{X}(\omega_{\phi_{t}})}\frac{\omega_{\phi_{t}}^{n}}{n!}\wedge
dt-\frac{1}{V}\int_{M}\phi
e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}\right]$
$\displaystyle\ \ \
-\frac{1}{V}\int_{M}(h_{g}-\theta_{X}(g))e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}+\frac{1}{V}\int_{M}(h_{g}-\theta_{X}(\omega_{g}))e^{\theta_{X}(\omega_{g})}\frac{\omega_{g}^{n}}{n!}$
$\displaystyle=\frac{1}{V}\int_{M}\log\left(\frac{\omega_{\phi}^{n}}{\omega_{g}^{n}}e^{\phi-
h_{g}}\right)e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}-\frac{1}{V}\int_{0}^{1}\int_{M}\dot{\phi}_{t}e^{\theta_{X}(\omega_{\phi_{t}})}\frac{\omega_{\phi_{t}}^{n}}{n!}\wedge
dt+\frac{1}{V}\int_{M}\theta_{X}(\omega_{\phi})e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}$
$\displaystyle\ \ \
+\frac{1}{V}\int_{M}(h_{g}-\theta_{X}(\omega_{g}))e^{\theta_{X}(\omega_{g})}\frac{\omega_{g}^{n}}{n!}$
(3.7)
$\displaystyle=\frac{1}{V}\int_{M}\log\left(\frac{\omega_{\phi}^{n}}{\omega_{g}^{n}}e^{\phi-
h_{g}}\right)e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}-\frac{1}{V}\int_{0}^{1}\int_{M}\dot{\phi}_{t}e^{\theta_{X}(\omega_{\phi_{t}})}\frac{\omega_{\phi_{t}}^{n}}{n!}\wedge
dt+{\rm const.}$

On the other hand,

$h_{g}=-\varphi_{0}-\log\mbox{det}(\varphi_{0ij})+C.$

Then

$\frac{\omega_{\phi}^{n}}{\omega_{g}^{n}}e^{\phi-
h_{g}}=C\mbox{det}(\varphi_{ij})e^{\varphi}.$

It follows

$\displaystyle\int_{M}\log\left(\frac{\omega_{\phi}^{n}}{\omega_{g}^{n}}e^{\phi-
h_{g}}\right)e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}$ (3.8)
$\displaystyle=(2\pi)^{n}\left[\int_{\mathbb{R}^{n}}\log\mbox{det}(\varphi_{ij})e^{X(\varphi)}\mbox{det}(\varphi_{ij})\,d\xi+\int_{\mathbb{R}^{n}}\varphi
e^{X(\varphi)}\mbox{det}(\varphi_{ij})\,d\xi\right].$

By using the relations

$\displaystyle\varphi=\sum_{i=1}^{n}{x_{i}u_{i}}-u,\
\mbox{det}(\varphi_{ij})\,d\xi=dx,\ \ \dot{\phi}_{t}=-\dot{u}_{t},$

where $\phi_{t}$ is a path in $\mathcal{H}_{G_{0}}(\omega_{g})$ and $u_{t}$ is
the symplectic potential of $\varphi_{t}=\varphi_{0}+\phi_{t}$, we also get

(3.9)
$\displaystyle\int_{0}^{1}\int_{M}\dot{\phi}_{t}e^{\theta_{X}(\omega_{\phi_{t}})}\frac{\omega_{\phi_{t}}^{n}}{n!}\wedge
dt$
$\displaystyle=(2\pi)^{n}\int_{0}^{1}\int_{\mathbb{R}^{n}}\dot{\phi}_{t}e^{X(\varphi_{t})}\mbox{det}({\varphi_{t}}_{ij})\,d\xi\,\wedge
dt$ $\displaystyle=-(2\pi)^{n}\int_{P}ue^{\theta(x)}dx+{\rm const.},$
$\displaystyle\int_{\mathbb{R}^{n}}\log\mbox{det}(\varphi_{ij})e^{X(\varphi)}\mbox{det}(\varphi_{ij})\,d\xi+\int_{\mathbb{R}^{n}}\varphi
e^{X(\varphi)}\mbox{det}(\varphi_{ij})\,d\xi$ (3.10)
$\displaystyle=-\int_{P}\log\mbox{det}(u_{ij})e^{\theta(x)}\,dx+\int_{P}(\sum_{i=1}^{n}{x_{i}u_{i}}-u)e^{\theta(x)}\,dx.$

Hence inserting (3.1)-(3.1) into (3.1), we obatin

$\displaystyle\mu_{\omega_{g}}(\phi)=\frac{(2\pi)^{n}}{V}\left[-\int_{P}\log\mbox{det}(u_{ij})e^{\theta(x)}\,dx+\int_{P}\sum_{i=1}^{n}{x_{i}u_{i}}e^{\theta(x)}\,dx\right]+C.$

Integrating by parts, we deduce (3.5) immediately. ∎

###  3.2. Properness of $\mathcal{F}(u)$

In this subsection, we show the properness of $\mathcal{F}(u)$ by similar
arguments as in [D1, ZZ1]. First, we have

######  Lemma 3.3.

There exists a constant $C>0$ such that for any $u\in\mathcal{C}_{\infty}$, it
holds

(3.11)
$\int_{P}\log\mbox{det}(u_{ij})e^{\theta(x)}\,dx\leq\mathcal{L}_{B}(u)+C$

where
$B=(u_{0})_{ij}^{ij}+2(u_{0})_{i}^{ij}\theta_{j}+(u_{0})^{ij}\theta_{i}\theta_{j}$
is a bouned function, and

(3.12) $\mathcal{L}_{B}(u)=\int_{\partial
P}ue^{\theta(x)}\,d\sigma+\int_{P}Bue^{\theta(x)}\,dx.$

###### Proof.

Let $f=u-u_{0}$. By the convexity of $-\log\mbox{det}$, we have

$\log\mbox{det}(u_{ij})\leq\log\mbox{det}((u_{0})_{ij})+(u_{0})^{ij}f_{ij}.$

For any $\delta>0$, let $P_{\delta}$ be the interior polygon with faces
parallel to those of $P$ separated by distance $\delta$, then $f$ is smooth
over the closure of $P_{\delta}$.

Integrating by parts,

$\int_{P_{\delta}}(u_{0})^{ij}f_{ij}e^{\theta(x)}\,dx=\int_{\partial
P_{\delta}}(u_{0})^{ij}f_{i}n_{j}e^{\theta(x)}\,d\sigma_{0}-\int_{P_{\delta}}(u_{0})^{ij}_{j}f_{i}e^{\theta(x)}\,dx-\int_{P_{\delta}}(u_{0})^{ij}f_{i}\theta_{j}e^{\theta(x)}\,dx.$

Integrating by parts for the last two terms again, we have

$\displaystyle\int_{P_{\delta}}(u_{0})^{ij}f_{ij}e^{\theta(x)}\,dx$
$\displaystyle=$ $\displaystyle\int_{\partial
P_{\delta}}(u_{0})^{ij}f_{i}n_{j}e^{\theta(x)}\,d\sigma_{0}-\int_{\partial
P_{\delta}}(u_{0})^{ij}_{j}n_{i}fe^{\theta(x)}d\sigma_{0}-\int_{\partial
P_{\delta}}(u_{0})^{ij}n_{i}\theta_{j}fe^{\theta(x)}\,d\sigma_{0}$
$\displaystyle+\int_{P_{\delta}}((u_{0})^{ij}_{ij}+(u_{0})^{ij}_{j}\theta_{i}+(u_{0})^{ij}_{i}\theta_{j}+(u_{0})^{ij}\theta_{i}\theta_{j})fe^{\theta(x)}\,dx.$

Note that

$(u_{0})^{ij}n_{j}\,d\sigma_{0}\to 0,\ \
-(u_{0})^{ij}_{j}n_{i}d\sigma_{0}\to\,d\sigma$

as $\delta\rightarrow 0$ [D1, D2]. Then

$\displaystyle\int_{\partial
P_{\delta}}(u_{0})^{ij}f_{i}n_{j}e^{\theta(x)}\,d\sigma_{0},\ \int_{\partial
P_{\delta}}(u_{0})^{ij}n_{i}\theta_{j}fe^{\theta(x)}\,d\sigma_{0}\longrightarrow
0,$

and

$\displaystyle\int_{\partial
P_{\delta}}(u_{0})^{ij}_{j}n_{i}fe^{\theta(x)}\,d\sigma_{0}\longrightarrow\int_{\partial
P}fe^{\theta(x)}\,d\sigma$

as $\delta\rightarrow 0$. In conclusion,

$\displaystyle\int_{P}(u_{0})^{ij}f_{ij}e^{\theta(x)}\,dx$ $\displaystyle=$
$\displaystyle\int_{\partial
P}fe^{\theta(x)}\,d\sigma+\int_{P}Bfe^{\theta(x)}\,dx.$

Hence,

$\displaystyle\int_{P}\log\mbox{det}(u_{ij})e^{\theta(x)}\,dx$
$\displaystyle\leq$ $\displaystyle\int_{\partial
P}ue^{\theta(x)}\,d\sigma+\int_{P}Bue^{\theta(x)}\,dx+\int_{\partial
P}u_{0}e^{\theta(x)}\,d\sigma-\int_{P}Bu_{0}e^{\theta(x)}\,dx+\int_{P}\log\mbox{det}((u_{0})_{ij})e^{\theta(x)}\,dx$
$\displaystyle=$ $\displaystyle\int_{\partial
P}ue^{\theta(x)}\,d\sigma+\int_{P}Bue^{\theta(x)}\,dx+const.$

∎

######  Remark 3.1.

As in [ZZ2], Lemma 3.3 can be extended for any $u\in\mathcal{C}_{\star}$,
where

$\mathcal{C}_{\star}=\\{u|~{}\text{$u$ is convex and satisfies $\int_{\partial
P}u\,d\sigma<\infty$}\\}.$

Denote

(3.13) $H(u)=\int_{P}ue^{\theta(x)}\,dx.$

######  Proposition 3.2.

For any $0<\delta<1$, there exsits $C_{\delta}>0$ such that

(3.14) $\mathcal{F}(u)\geq\delta H(u)-C_{\delta},\ \ \forall
u\in\tilde{\mathcal{C}}_{\infty}.$

###### Proof.

First, we compute the difference of $\mathcal{L}$(u) and $\mathcal{L}_{B}(u)$

$\displaystyle|\mathcal{L}(u)-\mathcal{L}_{B}(u)|$ $\displaystyle=$
$\displaystyle\left|\int_{P}(n+\sum\theta_{i}x_{i}+B)ue^{\theta(x)}\,dx\right|$
$\displaystyle\leq$ $\displaystyle C^{\prime}\int_{P}ue^{\theta(x)}\,dx$
$\displaystyle\leq$ $\displaystyle(1+\delta)C_{0}C^{\prime}\int_{\partial
P}ue^{\theta(x)}\,d\sigma-\delta C^{\prime}\int_{P}ue^{\theta(x)}\,dx$

where $C^{\prime}=\|n+\sum\theta_{i}x_{i}+B\|_{L^{\infty}}$. Note

$\displaystyle\int_{P}ue^{\theta(x)}\,dx\leq C_{0}\int_{\partial
P}ue^{\theta(x)}\,d\sigma,~{}\forall~{}u\in\tilde{\mathcal{C}}_{\infty}.$

Then by (2.9), it follows

$\displaystyle|\mathcal{L}(u)-\mathcal{L}_{B}(u)|\leq$
$\displaystyle\frac{(1+\delta)C_{0}C^{\prime}}{\lambda}\mathcal{L}(u)-\delta
C^{\prime}\int_{P}ue^{\theta(x)}dx.$

Thus

$\left(1+\frac{(1+\delta)C_{0}C^{\prime}}{\lambda}\right)\mathcal{L}(u)\geq\mathcal{L}_{B}(u)+\delta
C^{\prime}\int_{P}ue^{\theta(x)}\,dx.$

Now let $r=\left(1+\frac{(1+\delta)C_{0}C^{\prime}}{\lambda}\right)^{-1}$, we
get

$\mathcal{L}(u)\geq\mathcal{L}_{B}(ru)+r\delta
C^{\prime}\int_{P}ue^{\theta(x)}\,dx.$

Applying the inequality (3.11) to $ru$, we obtain

$-\int_{P}\log\mbox{det}(u_{ij})e^{\theta(x)}\,dx\geq-\mathcal{L}_{B}(ru)-C+n\log
r.$

Hence,

$\displaystyle\mathcal{F}(u)\geq r\delta
C^{\prime}\int_{P}ue^{\theta(x)}\,dx-C+n\log r.$

∎

###  3.3. Properness of $\mu_{\omega_{g}}(\cdot)$

In this subsection, we show that the properness of $\mathcal{F}(u)$ in the
above subsection is equivalent to the properness of $\mu_{\omega_{g}}(\phi)$.
We need a lemma as follows.

######  Lemma 3.4.

There exists $C>0$ such that

$|J_{\omega_{g}}(\tilde{\phi})-H(u_{\tilde{\phi}})|\leq
C,~{}\forall~{}\phi\in\mathcal{H}_{G_{0}}(\omega_{g}),$

where $\tilde{\phi}=\phi_{\sigma}$ is a normalization of $\phi$ after a
transformation $\sigma\in T$ so that

$(\psi_{0}+\tilde{\phi})(0)=0,~{}D(\psi_{0}+\tilde{\phi})(0)=0.$

###### Proof.

By the relation $\dot{\phi_{t}}=-\dot{u_{t}}$, it is easy to see

$\displaystyle J_{\omega_{g}}(\phi)=\frac{1}{V}\int_{M}\phi
e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}+H(u_{\phi})-H(u_{0}),~{}\forall~{}\phi\in\mathcal{H}^{G_{0}}_{X}(\omega_{g}).$

In particular,

$J_{\omega_{g}}(\tilde{\phi})-H(u_{\tilde{\phi}})=\frac{1}{V}\int_{M}\tilde{\phi}e^{\theta_{X}(g)}\omega^{n}_{g}-H(u_{0}).$

We claim that

$\left|\frac{1}{V}\int_{M}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}\right|\leq
C$

for some uniform constant $C$.

Let $G(p,p^{\prime})$ be the Green function of $P_{\omega_{g}}$ so that

$\int_{M}G(p,\cdot)e^{\theta_{X}(g)}\frac{\omega_{g}^{n}}{n!}=0.$

It is proved in [CTZ] that there exists a $C>0$ depending only on $g$ such
that

$G(p,p^{\prime})\geq-C.$

Then applying the Green’s formula to potential $\tilde{\phi}$, we have

(3.15) $\displaystyle\tilde{\phi}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{V}\int_{M}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}-\int_{M}G(x,\cdot)(\triangle\tilde{\phi}(\cdot)+X(\tilde{\phi}))e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}$
$\displaystyle\leq$
$\displaystyle\frac{1}{V}\int_{M}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}+C_{0},$

where $C_{0}$ is a uniform constant. The second inequality follows from
$\triangle\tilde{\phi}\geq-n$ and that $X(\tilde{\phi})$ is uniformly bounded
[Z1]. Thus

(3.16)
$\frac{1}{V}\int_{M}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}\geq\sup_{M}{\\{\tilde{\phi}\\}}-C_{0}=\sup_{\mathbb{R}^{n}}{\\{\tilde{\phi}\\}}-C_{0}.$

Set

$\Omega_{N}=\\{\xi\in
M|~{}\tilde{\phi}(\xi)\leq\sup_{M}{\\{\tilde{\phi}\\}}-N\\}.$

Note that

$\displaystyle\frac{1}{V}\int_{M}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}$
$\displaystyle=$
$\displaystyle\frac{1}{V}\int_{M\cap\Omega_{N}}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}+\frac{1}{V}\int_{M\setminus\Omega_{N}}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}$
$\displaystyle\leq$
$\displaystyle\frac{1}{V}[(\sup_{M}{\\{\tilde{\phi}\\}}-N)\widetilde{\rm
Vol}(M\cap\Omega_{N})+\sup_{M}{\\{\tilde{\phi}\\}}\widetilde{\rm
Vol}(M\setminus\Omega_{N})]$ $\displaystyle=$
$\displaystyle\sup_{M}{\\{\tilde{\phi}\\}}-\frac{N\cdot\widetilde{\rm
Vol}(M\cap\Omega_{N})}{\rm Vol(M)}.$

Here $\widetilde{\rm Vol}(\cdot)$ represents the volume with form
$e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}$. Hence by (3.16), we derive

(3.17) $\widetilde{\rm Vol}(M\cap\Omega_{N})\leq\frac{C_{0}{\rm
Vol}(M)}{N}=\frac{C_{0}V}{N}\to 0,$

as $N\to\infty$.

On the other hand, by the normalization, we have
$\tilde{\phi}(0)=-\psi_{0}(0)$. Note $D(\tilde{\phi}+\psi_{0})\in P$. Then we
have

$|D\tilde{\phi}|\leq 2\sup\\{|p|:~{}p\in P\\}$

and

$\tilde{\phi}(x)\leq\tilde{\phi}(0)+2r\sup\\{|p|:~{}p\in P\\}\leq
C(r),~{}\forall~{}x\in B_{r}(0),$

where $C(r)$ depends only on the radius $r$ of ball $B_{r}(0)$ centered at the
original. Since the volume of domain $B_{1}(0)\times(0,2\pi)^{n}\subset M$
associated the metric
$\omega_{g}=\sqrt{-1}\partial\overline{\partial}\psi_{0}$ is bigger than some
number $\epsilon>0$, by (3.17), it is easy to see that there is at least a
point $x_{0}\in B_{1}(0)$ such that

$\tilde{\phi}(x_{0})\geq\sup_{M}\tilde{\phi}-N$

if $N$ is sufficiently large. Hence

$\sup_{M}\tilde{\phi}\leq N+C,$

and consequently

$\frac{1}{V}\int_{M}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}\leq
N+C.$

By (3.16), we also get

$\frac{1}{V}\int_{M}\tilde{\phi}e^{\theta_{X}(g)}\frac{\omega^{n}_{g}}{n!}\geq\tilde{\phi}(0)-C_{0}=-\psi(0)-C_{0}.$

Therefore the claim is true and the lemma is proved. ∎

######  Theorem 3.2.

There exist numbers $\delta>0$ and $C$ such that

(3.18)
$\displaystyle\mu_{\omega_{g}}(\phi)\geq\delta\inf_{\tau\in{T}}{I_{\omega_{g}}(\phi_{\tau})}-C,~{}\forall~{}\phi\in\mathcal{H}_{G_{0}}(\omega_{g}).$

In particular, $\mu(\phi)$ is proper for any
$\phi\in\mathcal{H}_{G_{0}}(\omega_{g})$ modulo $G_{0}$.

###### Proof.

Let $\phi\in\mathcal{H}_{G_{0}}(\omega_{g})$. Then there exists a $\sigma\in
T$ such that the Legendre function $u_{\phi_{\sigma}}$ associated to
$\phi_{\sigma}$ is belonged to $\tilde{\mathcal{C}}_{\infty}$. By Proposition
3.2, we see that

$\mu_{\omega_{g}}(\phi_{\sigma})\geq\delta H(\phi_{\sigma})-C_{\delta}.$

Note that $\mu_{\omega_{g}}(\phi)=\mu_{\omega_{g}}(\phi_{\sigma})$. Thus by
Lemma 3.4, we get

$\displaystyle\mu_{\omega_{g}}(\phi)=\mu_{\omega_{g}}(\phi_{\sigma})$
$\displaystyle\geq$ $\displaystyle\delta
J_{\omega_{g}}(\phi_{\sigma})-C_{\delta}^{\prime}$ $\displaystyle\geq$
$\displaystyle\delta\inf_{\tau\in{T}}J_{\omega_{g}}(\phi_{\tau})-C_{\delta}^{\prime}$
$\displaystyle\geq$
$\displaystyle\frac{\delta}{n+1}\inf_{\tau\in{T}}I_{\omega_{g}}(\phi_{\tau})-C_{\delta}^{\prime}.$

Here at the last inequality we used the relation (3.1). ∎

######  Remark 3.2.

Theorem 3.2 seems to overlap Theorem 4.5 in [BB] where the Aubin-Ding typed
functional is studied instead of modified $K$-energy by using the geodesic
theory for Kähler potentials. 222We are indebted to Chi Li for telling us
Berman-Berndtsson’s results in [BB].

As an application of Theorem 3.2 together with Lemma 3.1, we give a new proof
of the following Wang-Zhu Theorem [WZ].

######  Theorem 3.3.

There exists a Kähler-Ricci soliton on any toric Fano manifold.

##  4\. Generalization to conic metrics case

Singular Kähler-Ricci solitons on toric manifolds have been extensivedly
studied by [SZ, Le, BB, DGSW], etc. In this section, we generalize the
discussion in former sections to give a new approach by showing the properness
of modified Log $K$-energy.

Let $M$ be a toric Fano manifold and $K_{M}$ is the canonical line bundle. Let
$\\{D_{i}\\}_{i=1}^{d}$ be the toric divisors corresponding to the faces of
the moment polytope. Suppose $\beta>0$ and
$D=\sum_{i=1}^{d}(1-\beta_{i})D_{i}\in|-(1-\beta)K_{M}|$ be an effective
$\mathbb{R}$-divisor with strictly normal crossing support and
$0<\beta_{i}\leq 1$ for each $i$. A conical Kähler metric $g$ on $M$ with
angle $2\pi\beta_{i}$ along $D_{i}$ is a closed positive $(1,1)$ current in
$2\pi c_{1}(M)$, which is a smooth Kähler metric $\omega_{D}$ in $M\setminus
D$ and satisfies: for any $p\in D$, there is a coordinates neighborhood $U$
with local holomorphic coordinates $(z^{1},\cdot\cdot\cdot,z^{n})$ of $p$ such
that $D\cap U=\\{z^{i}=0,1\leq i\leq r\\}$ and the metric is asymptotically
equivalent along the model conic metric

$\sqrt{-1}\sum_{j=1}^{r}|z^{j}|^{2\beta_{j}-2}dz^{j}\wedge
d\bar{z}^{j}+\sqrt{-1}\sum_{j=r+1}^{n}dz^{j}\wedge d\bar{z}^{j}.$

One can check that the Guillemin metric
$\omega_{g}=\omega_{D}=\sqrt{-1}\partial\overline{\partial}\varphi_{0}$
induced by symplectic potential

$u_{0}=\sum_{i}\beta_{i}^{-1}l_{i}\log l_{i}$

is a conical Kähler metric (cf. [Ab2, Le, DGSW]), where
$l_{i}=1-\langle\ell_{i},x\rangle$. In fact, if we let a set of symplectic
potentials

$\mathcal{C}_{\beta}=\\{u=u_{0}+f\ |\ u\ \text{is a strictly convex function
in }\ P,\ f\in C^{\infty}(\overline{P})\\},$

then there is a one-to-one correspondence between $\mathcal{C}_{\beta}$ and
$\mathcal{H}_{D}^{G_{0}}(\omega_{g}),$ where
$\mathcal{H}^{G_{0}}_{D}(\omega_{g})$ consists all $G_{0}$-invariant Kähler
potentials which are asymptotically equilvalent to $\omega_{D}$.

Let $s_{i}$ be the defining section of $D_{i}$ and $h_{i}$ a Hermitian metric
on $D_{i}$. Denote $\|s_{i}\|^{2}$ the norm of $s_{i}$. Then
$h=\otimes_{i=1}^{d}h_{i}^{1-\beta_{i}}$ defines a Hermitian metric on $D$ and
gives a norm $\|s_{D}\|$ for the defining section
$s=\otimes_{i=1}^{d}s^{1-\beta_{i}}$ of $D$. By the Poincare-Lelong identity,

$\sqrt{-1}\partial\bar{\partial}\log\|s_{i}\|^{2}=-c_{1}([D_{i}],h_{i})+\\{D_{i}\\},$

where $\\{D_{i}\\}$ denotes the current of integration along $D_{i}$, we have

(4.1)
$\sqrt{-1}\partial\bar{\partial}\log\|s_{D}\|^{2}=-c_{1}([D],h)+\\{D\\}.$

(4.1) implies that there exists $\tau\in\mathbb{R}^{n}$ such that

(4.2)
$\displaystyle\log\|s_{D}\|^{2}=-(1-\beta)\varphi_{0}-\beta\langle\tau,\xi\rangle+const.$

A conical Kähler metric $\omega$ with $2\pi\beta_{i}$ angle along each $D_{i}$
is called a conical Kähler-Ricci soliton if there is a holomorphic vector
field $X$ on $M$ for some $\beta\in(0,1]$ such that

(4.3) $\displaystyle{\rm Ric}(\omega)-\beta\omega-\\{D\\}=L_{X}\omega.$

We will investigate a solution of (4.3) in
$\mathcal{H}^{G_{0}}_{D}(\omega_{g})$. Let
$X=\sum\theta_{\alpha}\zeta_{\alpha}$, where $\zeta_{\alpha}$ is a basis of
Lie algebra $\eta_{T}$ of $T$. Then a lemma in [DGSW] shows that
$\tau=(\tau_{1},...,\tau_{n})$ in (4.2) is uniquely determined by relation,

(4.4)
$\tau_{\alpha}=\frac{\int_{P}x_{\alpha}e^{\theta(x)}dx}{\int_{P}e^{\theta(x)}dx},\
\ \alpha=1,...,n.$

Moreover, $\beta_{i}=1-\beta l_{i}(\tau),$ $i=1,...,d$.

Let $\mathcal{H}_{X,D}(\omega_{g})$ be a class of $K_{X}$-invariant functions
$\phi\in C^{2,\alpha}(M)$ such that
$\omega_{g}+\sqrt{-1}\partial\overline{\partial}\phi$ are conical metrics with
$2\pi\beta_{i}$ angle along each $D_{i}$. Following [LS] (also see [Be1, Li]),
we consider the following modified Log $K$-energy functional on
$\mathcal{H}_{X,D}(\omega_{g})$,

(4.5) $\displaystyle\mu_{\omega_{g},D}(\phi)$ $\displaystyle=$
$\displaystyle\mu_{\omega_{g}}(\phi)+(1-\beta)(I_{\omega_{g}}(\phi)-J_{\omega_{g}}(\phi))+\int_{M}\log\|s_{D}\|^{2}e^{\theta_{X}(\omega_{\phi})}\frac{\omega^{n}_{\phi}}{n!}.$

It is easy to check that a soultion of (4.3) is a critical point of
$\mu_{\omega_{g},D}(\phi)$. We need to study the properness of
$\mu_{\omega_{g},D}(\phi)$.

Define

(4.6)
$\mathcal{L}_{\beta,\tau}(u)=\beta\left(\mathcal{L}(u)-\int_{P}\langle\tau,\nabla
u\rangle e^{\theta(x)}\,dx\right),~{}\forall~{}u\in\mathcal{C}_{\infty}.$

By (4.4), it is clear

(4.7) $\mathcal{L}_{\beta,\tau}(u)=0,~{}\forall~{}u=\sum
a_{\alpha}x_{\alpha},~{}(a_{1},...,a_{n})\in\mathbb{R}^{n}.$

######  Lemma 4.1.

Let $\phi\in\mathcal{H}^{G_{0}}_{D}(\omega_{g})$ and $u$ be the Legendre dual
function of $\varphi_{0}+\phi$. Then

(4.8)
$\displaystyle\mu_{\omega_{g},D}(\phi)=\frac{(2\pi)^{n}}{V}\mathcal{F}_{\beta,\tau}(u)+C,$

where

(4.9)
$\mathcal{F}_{\beta,\tau}(u)=-\int_{P}\log\mbox{det}(u_{ij})e^{\theta(x)}\,dx+\mathcal{L}_{\beta,\tau}(u),$

and $C$ is a constant.

###### Proof.

It suffices to transform the latter two terms in (4.5) under symplectic
potentials. Note that

$\displaystyle
I_{\omega_{g}}(\phi)-J_{\omega_{g}}(\phi)=-\frac{1}{V}\int_{M}\phi
e^{\theta_{X}(\omega_{\phi})}\frac{\omega_{\phi}^{n}}{n!}+\frac{1}{V}\int_{0}^{1}\int_{M}\dot{\phi}_{s}e^{\theta_{X}(\omega_{\phi_{s}})}\frac{\omega_{\phi_{s}}^{n}}{n!}ds$

Hence by similar expression as in Propositon 3.5 we have

$\displaystyle(1-\beta)(I_{\omega_{g}}(\phi)-J_{\omega_{g}}(\phi))+\int_{M}\log\|s_{D}\|^{2}e^{\theta_{X}(\omega_{\phi})}\frac{\omega^{n}_{\phi}}{n!}$
$\displaystyle=$
$\displaystyle\frac{(2\pi)^{n}}{V}\left[-(1-\beta)\int_{\mathbb{R}^{n}}\varphi
e^{X(\varphi)}\mbox{det}D^{2}\varphi\,d\xi+\beta\int_{\mathbb{R}^{n}}\langle\tau,\xi\rangle
e^{X(\varphi)}\mbox{det}D^{2}\varphi\,d\xi\right]$
$\displaystyle+\frac{(2\pi)^{n}}{V}(1-\beta)\int_{0}^{1}\int_{M}\dot{\phi}_{s}e^{\theta_{X}(\omega_{\phi_{s}})}\frac{\omega_{\phi_{s}}^{n}}{n!}ds$
$\displaystyle=$
$\displaystyle\frac{(2\pi)^{n}}{V}\left[-(1-\beta)\mathcal{L}(u)-\beta\int_{P}\langle\tau,\nabla
u\rangle e^{\theta(x)}\,dx\right]$

Combing with (3.6), we obtain (4.9). ∎

To prove the properness of $\mathcal{F}_{\beta,\tau}(u)$, by (4.7), it
suffices to consider the function space $\tilde{\mathcal{C}}_{\infty,\tau}$
which contains all the functions in $\mathcal{C}_{\infty}$ normalized at
$\tau$.

######  Proposition 4.1.

If $\tau\in P$, then for any $0<\delta<1$, there exists $C_{\delta,\beta}>0$
such that

$\mathcal{F}_{\beta,\tau}(u)\geq\delta\int_{P}ue^{\theta(x)}\,dx-
C_{\delta,\beta},\ \ \forall u\in\tilde{\mathcal{C}}_{\infty,\tau}.$

###### Proof.

The proof is similar to Propositon 3.2. Note that by the definition of
Legendre transform, we have $(x-\tau)\cdot\nabla u-u\geq 0$ for
$u\in\tilde{\mathcal{C}}_{\infty,\tau}$. 333For any $x_{0}\in P$, let
$l_{x_{0}}(x)$ be the tangent function at $x_{0}$, then $l_{x_{0}}(x)=\nabla
u(x_{0})(x-x_{0})+u(x_{0})$. Sine $u$ is normalized at $\tau$, we always have
$l_{x_{0}}(\tau)\leq 0$. It is easy to see that $\nabla
u(x_{0})(x_{0}-\tau)-u(x_{0})=-l_{x_{0}}(\tau)\geq 0$. Then one can replace
(2.10) in Lemma 2.2 by

$\mathcal{L}_{\beta,\tau}(x)=\beta\left(\int_{P}[(x-\tau)\cdot\nabla
u-u]e^{\theta(x)}\,dx+\int_{P}ue^{\theta(x)}\,dx\right)\geq\beta\int_{P}ue^{\theta(x)}\,dx$

and check the arguments line by line in Section 3.2. ∎

Applying Lemma 4.1 and Proposition 4.1, we can give a new proof of following
Datar-Guo-Song-Wang Theorem [DGSW].444 The statement here is a bit different
to one in [DGSW].

######  Theorem 4.1.

Let $X=\sum_{i=1}^{n}\theta_{i}\zeta_{i}\in\eta_{T}$. Let
$\overline{\beta}=\sup\\{\beta|~{}\beta l_{i}(\tau)<1,~{}i=1,...,d\\}$.
Suppose that $\tau\in P$. Then for any $\beta\leq\overline{\beta}$ there
exists a unique toric invariant conical Kähler-Ricci soliton $\omega$ which
solves (4.3) with $D=\sum(1-\beta_{i})D_{i}$ and $\beta_{i}=\beta
l_{i}(\tau).$ Moreover, the conical angles of $\omega$ are $2\pi\beta_{i}$
along $D_{i}$.

###### Proof.

Since higher order estimates depend on $C^{0}$-estimate for a family of Kähler
potentials $\phi_{t}$ as in [TZ1, JMR] (also see [DGSW] for toric manifolds),
it suffices to get the $C^{0}$-estimate when we use the continuity method to
solve (4.3). Since $\mu_{\omega_{g},D}(\phi)$ is monotonic for $\phi_{t}$ as
in the smooth metrics case [TZ1, CTZ], the properness of
$\mu_{\omega_{g},D}(\phi)$ implies that $I_{\omega_{g}}(\phi_{t})$ is
uniformly bonuded. As a consequence, we get an upper bound of $\phi_{t}$. The
lower bound can be also obtained by establishing a uniform lower bonud for the
Green functions of $\omega_{\phi_{t}}$ as in [Ma, CTZ] for smooth metrics.
There is another way to get a Hölder estimate for Legendre functions $u_{t}$
of $\phi_{t}$ by using an observation in [D3] for toric invariant metrics, if
one knows the upper bound of $u_{t}$, which is equal to one of $\phi_{t}$.
Then $u_{t}$ is uniformly bounded, and so is $\phi_{t}$ ( also see [SZ]). In
fact, the latter argument was presented by Datar-Guo-Song-Wang [DGSW] while
they got an upper bound of $u_{t}$ by studying a class of real Monge- Ampére
equations as done in [WZ]. The uniqueness of $\omega$ follows from the
convexity of $\mathcal{F}_{\beta,\tau}(u)$.

∎

In case that $X$ is chosen as a soliton vector field on $M$, $\tau=0$ by
(4.4). Then in Theorem 4.1, $\beta_{i}=\beta\leq 1,~{}i=1,...,d,$ and
$D=\sum(1-\beta)D_{i}$. Thus Theorem 0.2 is a corollary of Theorem 4.1.

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