#  Kähler-Ricci flow for deformed complex structures

Gang Tian, Liang Zhang and Xiaohua $\text{Zhu}^{*}$ School of Mathematical
Sciences & BICMR, Peking University, Beijing 100871, China.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $(M,J_{0})$ be a Fano manifold which admits a Kähler-Ricci soliton, we
analyze the behavior of the Kähler-Ricci flow near this soliton as we deform
the complex structure $J_{0}$. First, we will establish an inequality of
Lojasiewicz’s type for Perelman’s entropy along the Kähler-Ricci flow. Then we
prove the convergence of Kähler-Ricci flow when the complex structure
associated to the initial value lies in the kernel $Z$ or negative part of the
second variation operator of Perelman’s entropy. As applications, we solve the
Yau-Tian-Donaldson conjecture for the existence of Kähler-Ricci solitons in
the moduli space of complex structures near $J_{0}$, and we show that the
kernel $Z$ corresponds to the local moduli space of Fano manifolds which are
modified $K$-semistable. We also prove an uniqueness theorem for Kähler-Ricci
solitons.

###### Key words and phrases:

Kähler-Ricci flow, Kähler-Ricci solitons, deformation space of complex
structures

###### 2000 Mathematics Subject Classification:

Primary: 53C25; Secondary: 53C55, 58E35 58J35

* Partially supported by NSFC Grants 11771019 and BJSF Grants Z180004.

###### Contents

  1. 0 Introduction
  2. 1  Weighted Hodge-Laplace on $A^{p,q}(M,T^{1,0}M)$
  3. 2  Kuranishi deformation theory for $\Box_{h}$-operator
  4. 3 Restricted entropy $\nu(\cdot)$ and its variations
    1. 3.1 Variation of Kähler metrics
    2. 3.2 The first variation of $\nu(\cdot)$
    3. 3.3 The second variation of $\nu(\cdot)$
  5. 4 A new version of $\delta^{2}\nu$
    1. 4.1  Kernel space $Z$ of $H_{1}$
    2. 4.2 Index of $H_{1}$
    3. 4.3 Maximality of $\lambda(\cdot)$ associated to $Z$
  6. 5 Lojasiewicz inequality on the space of Kähler metrics
  7. 6  Proof of Theorem 0.1
  8. 7 Applications of Theorem 0.1
  9. 8 Generalization of Theorem 0.1
    1. 8.1 Linearly stable case
    2. 8.2 Linearly semistable case

##  0\. Introduction

Let $(M,J_{0})$ be a Fano manifold which admits a Kähler-Ricci, abbreviated as
KR, soliton $\omega_{KS}\in 2\pi c_{1}(M,J_{0})$. 111We always denote a Kähler
metric $g$ by its Kähler form $\omega_{g}$. It is known that for any initial
metric $\omega_{0}\in 2\pi c_{1}(M,J_{0})$, the KR flow will evolve
$\omega_{0}$ to a KR soliton $\omega_{KS}^{\prime}$ smoothly [39, 11].
Moreover, by the uniqueness of KR solitons [35, 36],
$\omega_{KS}^{\prime}=\sigma^{*}\omega_{KS}$ for some $\sigma\in{\rm
Aut}(M,J_{0})$. Thus it is a natural question how to extend the above
convergence result to Kähler manifolds with complex structures near $J_{0}$.
The question is closely related to the existence problem of KR solitons for
the local moduli of complex structures at $J_{0}$ as well as the local moduli
space of KR solitons near $\omega_{KS}$.

When $(M,J_{0})$ admits a Kähler-Einstein (KE) metric, it has been proved that
the KR flow is always convergent to a KE metric in the $C^{\infty}$-topology
for any initial metric $\omega_{0}\in 2\pi c_{1}(M,J)$ so long as $J$ is
sufficiently close to $J_{0}$ [38, 32]. As a consequence, the Mabuchi’s
K-energy on $(M,J)$ is bounded from below [38, Lemma 7.1] (also see [41, 8]),
so all Fano manifolds $(M,J)$ near $(M,J_{0})$ are $K$-semistable. We refer
the reader to [22], [20] [24], [46, Proposition 4.17] etc. for more general
K-semistable Fano manifolds. However, for a KR soliton $(M,J_{0},\omega_{KS})$
which is not KE metric, the flow on $(M,J)$ may not always converge to a KR
soliton smoothly even if $J$ is close to $J_{0}$, as explained in [45, Remark
6.5] by using Pasquier’s example of horospherical variety which is a
degeneration of the Grassmannian manifold ${\rm Gr}_{q}(2,7)$ [26]. Thus we
are led to understanding those deformed complex structures for which the KR
flow is $C^{\infty}$-convergent to a KR soliton.

According to the deformation theory [14, 13], the local moduli of complex
structures at $(M,J_{0})$ can be parameterized by using the
C$\overset{\vee}{e}$ch cohomology class $H^{1}(M,J_{0},\Theta)$, which is the
infinitesimal deformation space of complex structures on $M$. In our case of
Fano manifolds, we introduce the $h$-harmonic space
$\mathcal{H}^{0,1}_{h}(M,T^{1,0}M)$ associated to $H^{1}(M,J_{0},\Theta)$,
where $h$ is a Ricci potential of Kähler metric $\omega$ in $2\pi
c_{1}(M,J_{0})$ (cf. Section 1). In particular, $h$ is same as a potential
function $\theta$ of soliton vector field (VF) if $\omega=\omega_{KS}$. Then
the local deformation of Kähler metrics near $(M,J_{0},\omega)$ can be
parameterized by $\mathcal{U}_{\epsilon}=B(\epsilon)\times C^{\infty}(M)$ (cf.
Section 2), where $B(\epsilon)$ denotes a small $\epsilon$-ball in
$\mathcal{H}^{0,1}_{h}(M,T^{1,0}M)$ centered at the origin. Thus the variation
space $\mathcal{U}$ of Kähler metrics becomes

$\mathcal{U}=\mathcal{H}^{0,1}_{h}(M,T^{1,0}M)\times C^{\infty}(M).$

By computing the second variation of Perelman’s entropy $\lambda(\cdot)$ at
$(M,J_{0},\omega_{KS})$ via the parameter space $\mathcal{U}_{\epsilon}$, we
are able to get the following product formula of the second variation operator
(cf. Proposition 3.9),

$\delta^{2}\lambda(\omega_{KS})=H(\psi,\chi)\,=\,H_{1}(\psi)\oplus
H_{2}(\chi),~{}\forall~{}(\psi,\chi)\in\mathcal{U},$

where $H_{2}(\chi)$ is always non-positive [40]. However, the sign of operator
$H_{1}(\psi)$ is in general not definite according to Pasquier’s example
mentioned above. Thus the convergence problem of KR flow for deformed complex
structure $J_{\psi_{\tau}}$ associated to $\psi_{\tau}\in
B(\epsilon)\subset\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ $(\epsilon<<1)$ via
the Kuranishi map will depend on the sign of $H_{1}(\psi_{\tau})$.

Recall the (normalized) KR flow on a Fano manifold $(M,J)$,

(0.1) $\displaystyle\frac{\partial\widetilde{\omega}(t)}{\partial t}=-{\rm
Ric}\,(\widetilde{\omega}(t))+\widetilde{\omega}(t),~{}\widetilde{\omega}_{0}\in
2\pi c_{1}(M,J).$

By establishing an inequality of Lojasiewicz type for the entropy
$\lambda(\cdot)$ along the flow (0.1), we will prove

######  Theorem 0.1.

Let $(M,J_{0})$ be a Fano manifold which admits a KR soliton $\omega_{KS}\in
2\pi c_{1}(M,J_{0})$ with respect to a holomorphic vector field (HVF) $X$.
Then there exists a small $\epsilon$ such that for any $\tau\in B(\epsilon)$
with $\psi_{\tau}\in{\rm Ker}(H_{1})=Z$ the KR flow (0.1) converges smoothly
to a KR soliton $(M,J_{\infty},\omega_{\infty})$ for any initial metric
$\widetilde{\omega}_{0}$ in $2\pi c_{1}(M,J_{\psi_{\tau}})$. Moreover, $X$ can
be lifted to become a soliton VF of $(M,J_{\infty},\omega_{\infty})$ and

(0.2) $\displaystyle\lambda(\omega_{\infty})=\lambda(\omega_{KS}).$

Also, the convergence is of polynomial rate.

We note that $Z=\mathcal{H}^{0,1}(M,T^{1,0}M)$ in case of
$\omega_{KS}=\omega_{KE}$ [40]. Thus Theorem 0.1 generalizes the results in
[38, 32]. We also note that $\psi_{\tau}\in Z$ if and only if $X$ can be
lifted to a HVF on $(M,J_{\psi_{\tau}})$ (cf. Corollary 4.5). Moreover, as in
the case of $\omega_{KS}=\omega_{KE}$, the set of complex structures
associated to the kernel $Z$ corresponds to the local moduli space of Fano
manifolds which are modified $K$-semistable [48]. In particular, we prove the
following existence theorem of KR solitons in the deformation space of complex
structures.

######  Theorem 0.2.

Let $(M,J_{0})$ be a Fano manifold which admits a KR-soliton $\omega_{KS}\in
2\pi c_{1}(M,J_{0})$. Then there exists a small $\epsilon$ such that
$(M,J_{\psi_{\tau}})$ admits a KR soliton close to $\omega_{KS}\in 2\pi
c_{1}(M)$ in the Cheeger-Gromov topology for $\tau\in B(\epsilon)$ if and only
if the Fano manifold $(M,J_{\psi_{\tau}})$ is modified $K$-polystable.

Theorem 0.2 gives a confirmative answer to the Yau-Tian-Donaldson conjecture
for the existence of KR solitons in the deformation space of complex
structures (cf. [47, 10, 3], etc.). We also mention that Theorem 0.2 was
proved by Inoue for the equivariant deformation space of complex structures by
using the deformation theory [19] (also see Corollary 7.3 and Remark 7.4).
Inoue’s result is a generalization of Székelyhidi’s for KE metrics [33]. Our
theorem gives a more general answer for the existence of KR-solitons and
determines the soliton VFs.

By Theorem 0.1, we can also prove a uniqueness result for KR-solitons in the
closure of the orbit by diffeomorphisms, we refer the reader to Theorem 7.5.

Let us say a few of words about how to prove the Lojasiewicz inequality for
$\lambda(\cdot)$. In fact, we will first derive such an inequality for the
restricted $\nu(\cdot)$ of $\lambda(\cdot)$ on the parameter space
$\mathcal{U}_{\epsilon}$ (cf. Proposition 5.1). The advantage of $\nu(\cdot)$
is that its gradient and the second variation $\delta^{2}\nu(\cdot)$ of
$\nu(\cdot)$ are both of maps to $\mathcal{U}$ (cf. Definition 3.3, 3.8). In
particular, we get an explicit formula for the kernel of
$\delta^{2}\nu(\cdot)$ at a KR soliton (cf. (4.9) and Remark 4.6). Then by the
spectral theorem [31], we prove the Lojasiewicz inequality for the original
$\lambda(\cdot)$ (cf. Lemma 3.6, Corollary 5.4, (6.11)).222 This also answers
a question of Chen-Sun [8, Remark 5.5]. We would like to mention that such an
inequality for the space of Riemannian metrics along the Ricci flow has been
studied by Sun-Wang [32].

Theorem 0.1 will be generalized for Kähler manifolds $(M,J_{\psi_{\tau}})$
with $H_{1}(\psi_{\tau})\leq 0$ as follows.

######  Theorem 0.3.

Let $(M,J_{0})$ be a Fano manifold which admits a KR soliton $\omega_{KS}\in
2\pi c_{1}(M,J_{0})$. Suppose that

  1. (1)

$(M,J_{\psi_{\tau}})$ admits a KR soliton for any $\psi_{\tau}\in
B(\epsilon)\cap Z$;

  2. (2)

$H_{1}(\cdot)\leq 0,~{}{\rm on}~{}\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M).$

Then there exists a small $\epsilon$ such that for any $\tau\in B(\epsilon)$
the KR flow (0.1) converges smoothly to a KR soliton
$(M,J_{\infty},\omega_{\infty})$ for any initial metric
$\widetilde{\omega}_{0}$ in $2\pi c_{1}(M,J_{\psi_{\tau}})$. Moreover,

$\displaystyle\lambda(\omega_{\infty})=\lambda(\omega_{KS}),$

and the convergence is of polynomial rate.

The condition (1) may not be necessary in view of Theorem 0.1. In fact, we
have the following conjecture.

######  Conjecture 0.4.

Let $(M,J_{0})$ be a Fano manifold which admits a KR soliton $\omega_{KS}\in
2\pi c_{1}(M,J_{0})$. Let $A_{-}$ be the linear subspace of
$\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ associated to the negative
eigenvalues of $H_{1}$. Then there exists a small $\epsilon$ such that the
following is true:

  1. (1)

For any $\tau\in B(\epsilon)$ with $\psi_{\tau}\in A_{-}\cup Z$ the KR flow
(0.1) converges smoothly to a KR soliton $(M,J_{\infty},\omega_{\infty})$ for
any initial metric $\omega_{0}$ in $2\pi c_{1}(M,J_{\psi_{\tau}})$. Moreover,
the convergence is of polynomial rate with
$\lambda(\omega_{\infty})=\lambda(\omega_{KS})$.

  2. (2)

For any $\tau\in B(\epsilon)$ with
$\psi_{\tau}\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\setminus(A_{-}\cup Z)$,
the KR flow (0.1) converges to a singular KR soliton $\omega_{\infty}$ for any
initial metric $\omega_{0}$ in $2\pi c_{1}(M,J_{\psi_{\tau}})$ with
$\lambda(\omega_{\infty})$ strictly bigger than $\lambda(\omega_{KS})$.

The above conjecture means that the set of complex structures, for which the
KR flow (0.1) is stable, corresponds to the linear semistable subspace
$A_{-}\cup Z$ of $\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ in the deformation
space of Fano manifolds which admit a KR-soliton. Otherwise, the flow on
$(M,J_{\psi_{\tau}})$ is unstable, but will in general converge to a singular
KR soliton $\omega_{\infty}$ with $\lambda(\omega_{\infty})$ strictly bigger
than $\lambda(\omega_{KS})$ by the Hamilton-Tian conjecture [34, 2, 7, 46].

The organization of the paper is as follows. In Section 1, we introduce the
weighted Hodge-Laplace operator $\Box_{h}$ and the $h$-harmonic space of
$\mathcal{H}^{0,1}_{h}(M,T^{1,0}M)$ on a Fano manifold. In Section 2, we study
the Kuranishi deformation theory of complex structures associated to
$\Box_{h}$ and prove that the solution of modified Kuranishi’s equation is
divergent-free (cf. Proposition2.1)). In Section 3, we compute the first and
second variations of $\nu(\cdot)$ on $\mathcal{U}_{\epsilon}$. In Section 4,
we give a new version of the second variations of $\nu(\cdot)$ and prove the
local maximality of $\lambda(\cdot)$ (cf. Theorem 4.4 and Proposition 4.11).
Section 5 is devoted to prove the Lojasiewicz inequality (cf. Proposition
5.1). Theorem 0.1 is proved in Section 6 while Both of Theorem 0.2 and Theorem
7.5 are proved in Section 7. In Section 8, we prove Theorem 0.3.

##  1\.  Weighted Hodge-Laplace on $A^{p,q}(M,T^{1,0}M)$

In this section, we introduce a weighted Hodge-Laplace operator on the space
$A^{p,q+1}(M,T^{1,0}M)$ on a Fano manifold $(M,J_{0})$ with a Kähler form
$\omega\in 2\pi c_{1}(M)$. This is very similar with the weighted Laplace
operator associated to the Bakry-Émery Ricci curvature studied in [44].

Let $h$ be a Ricci potential of $\omega$, which satisfies that

(1.1) $\displaystyle\rm{Ric}(\omega)-\omega=\sqrt{-1}\partial\bar{\partial}h.$

We denote $\delta_{h}=\overline{\partial}_{h}^{*}$ the dual operator of
$\overline{\partial}$ with respect to the inner product

$\int_{M}<,>e^{h}\omega^{n},$

where $<,>=<,>_{\omega}$ is the inner product on $\Lambda^{p,q}T^{*}M\otimes
T^{1,0}M$ induced by $\omega$. Namely, for any $\varphi\in
A^{p,q}(M,T^{1,0}M)$, $\varphi^{\prime}\in A^{p,q+1}(M,T^{1,0}M)$, we have

$\int_{M}<\bar{\partial}\varphi,\varphi^{\prime}>e^{h}\omega^{n}=\int_{M}<\varphi,\delta_{h}\varphi^{\prime}>e^{h}\omega^{n}.$

Then weighted Hodge-Laplace operator $\Box_{h}$ on $A^{p,q+1}(M,T^{1,0}M)$ is
defined by

$\Box_{h}=\overline{\partial}\cdot\delta_{h}+\delta_{h}\cdot\overline{\partial}.$

Thus, similarly with the Hodge-Laplace operator, we see that

$\Box_{h}\varphi=0$

if only if

$\bar{\partial}\varphi=0~{}{\rm and}\ \delta_{h}\varphi=0.$

For simplicity, we denote

$\mathcal{H}^{p,q}_{h}(M,T^{1,0}M)=\\{\varphi\in
A^{p,q}(M,T^{1,0}M)|~{}\Box_{h}\varphi=0\\}$

the harmonic space of $\Box_{h}$ on $A^{p,q+1}(M,T^{1,0}M)$.

In this paper, we are interested in the space $A^{0,1}(M,T^{1,0}M)$. Then
under local holomorphic coordinates, for any

$\varphi=\varphi^{i}_{\overline{j}}d\overline{z}^{j}\otimes\frac{\partial}{\partial
z^{i}}\in A^{0,1}(M,T^{1,0}M),$

we have

(1.2)
$\displaystyle{}\delta_{h}\varphi=-(\varphi^{i}_{\overline{j},k}+\varphi^{i}_{\overline{j}}h_{k})g^{k\overline{j}}\frac{\partial}{\partial
z^{i}}.$

Moreover,

(1.3) $\displaystyle\mathcal{H}^{0,1}_{h}(M,T^{1,0}M)\cong H^{1}(M,J,\Theta).$

The latter is the C$\overset{\vee}{e}$ch cohomology group associated to the
infinitesimal deformation of complex structures on $(M,J_{0})$ [14].

We define a $h$-divergence operator on $A^{p,q+1}(M,T^{1,0}M)$ by

${\rm div}_{h}\varphi=e^{-h}{\rm div}(e^{h}\varphi).$

In particular, if

$\varphi=\varphi^{i}_{\overline{j}}d\bar{z}^{j}\otimes\frac{\partial}{\partial
z^{i}}\in A^{0,1}(M,T^{1,0}M),$

then

(1.4) $\displaystyle{}{\rm
div}_{h}\varphi=(\varphi^{i}_{\overline{j},i}+\varphi^{i}_{\overline{j}}h_{i})d\bar{z}^{j}.$

Also we introduce an inner product $\varphi\lrcorner\mu$ with respect to

$\mu=\sqrt{-1}\mu_{k\bar{l}}dz^{k}\wedge d\bar{z}^{l}\in A^{1,1}(M)$

by

$\varphi\lrcorner\mu=\sqrt{-1}\varphi^{k}_{\overline{j}}\mu_{k\overline{l}}d\bar{z}^{j}\wedge
d\bar{z}^{l}\in A^{0,2}(M).$

We list some identities for the above two operators in the following lemma.

######  Lemma 1.1.

  1. (1)

$\overline{\partial}(\varphi\lrcorner\mu)=\overline{\partial}\varphi\lrcorner\mu+\varphi\lrcorner\overline{\partial}\mu.$

  2. (2)

$\delta_{h}({\rm div}_{h}\varphi)={\rm div}_{h}(\delta_{h}\varphi).$

  3. (3)

$\delta_{h}(\varphi\lrcorner\omega)=-\delta_{h}\varphi\lrcorner\omega-\sqrt{-1}{\rm
div}_{h}\varphi.$

  4. (4)

$\overline{\partial}{\rm div}_{h}\varphi-{\rm
div}_{h}(\overline{\partial}\varphi)=-\sqrt{-1}\varphi\lrcorner\omega.$

  5. (5)

${\rm div}_{h}[\varphi,\psi]=\psi\lrcorner\partial{\rm
div}_{h}\varphi+\varphi\lrcorner\partial{\rm div}_{h}\psi.$

  6. (6)

$[\varphi,\psi]\lrcorner\omega=\varphi\lrcorner\partial(\psi\lrcorner\omega)+\psi\lrcorner\partial(\varphi\lrcorner\omega).$

###### Proof.

All identities can be verified directly. $(1)$ is simple. For $(2)$, we note
that

$\delta_{h}({\rm
div}_{h}\varphi)=-(\varphi^{i}_{\overline{j},is}+\varphi^{i}_{\overline{j}s}h_{i}+\varphi^{i}_{\overline{j}}h_{is}+(\varphi^{i}_{\overline{j},i}+\varphi^{i}_{\overline{j}}h_{i})h_{s})g^{s\overline{j}}$

and

${\rm
div}_{h}(\delta_{h}\varphi)=-(\varphi^{i}_{\overline{j},si}+\varphi^{i}_{\overline{j}i}h_{s}+\varphi^{i}_{\overline{j}}h_{si}+(\varphi^{i}_{\overline{j},s}+\varphi^{i}_{\overline{j}}h_{s})h_{i})g^{s\overline{j}}.$

Thus $(2)$ is true.

Since
$\varphi\lrcorner\omega=\sqrt{-1}\varphi^{p}_{\overline{k}}g_{p\overline{j}}d\overline{z}^{k}\wedge
d\overline{z}^{j}$, we have

$\displaystyle\delta_{h}(\varphi\lrcorner\omega)$
$\displaystyle=-\sqrt{-1}(\varphi^{p}_{\overline{k},s}g_{p\overline{j}}+\varphi^{p}_{\overline{k}}g_{p\overline{j}}h_{s})g^{s\overline{j}}d\overline{z}^{k}+\sqrt{-1}(\varphi^{p}_{\overline{k},s}g_{p\overline{j}}+\varphi^{p}_{\overline{k}}g_{p\overline{j}}h_{s})g^{s\overline{k}}d\overline{z}^{j}$
$\displaystyle=-\sqrt{-1}(\varphi^{s}_{\overline{k},s}+\varphi^{s}_{\overline{k}}h_{s})d\overline{z}^{k}-\sqrt{-1}(\delta_{h}\varphi)^{p}g_{p\overline{j}}d\overline{z}^{j}$
$\displaystyle=-\sqrt{-1}{\rm
div}_{h}\varphi-(\delta_{h}\varphi\lrcorner\omega).$

This is $(3)$.

In order to prove $(4)$, we choose a normal coordinate $(z^{1},\ldots,z^{n})$
around each $p\in M$. By (1.4), we have

$\overline{\partial}{\rm
div}_{h}\varphi=(\varphi^{s}_{\overline{k},s\overline{l}}+\varphi^{s}_{\overline{k},\overline{l}}h_{s}+\varphi^{s}_{\overline{k}}h_{s\overline{l}})d\overline{z}^{l}\wedge
d\overline{z}^{k}.$

Since

$\overline{\partial}\varphi=(\varphi^{s}_{\overline{k},\overline{l}}+\Gamma^{\overline{h}}_{\overline{k}\overline{l}}\varphi^{s}_{\overline{h}})d\overline{z}^{l}\wedge
d\overline{z}^{k}\otimes\frac{\partial}{\partial z^{s}},$

we see that at $p$,

${\rm
div}_{h}\overline{\partial}\varphi=(\varphi^{s}_{\overline{k},\overline{l}s}+\partial_{s}\Gamma^{\overline{h}}_{\overline{k}\overline{l}}\varphi^{s}_{\overline{h}}+\varphi^{s}_{\overline{k},\overline{l}}h_{s})d\overline{z}^{l}\wedge
d\overline{z}^{k}.$

Thus we get

(1.5) $\displaystyle\overline{\partial}{\rm div}_{h}\varphi-{\rm
div}_{h}(\overline{\partial}\varphi)=(\varphi^{s}_{\overline{k},s\overline{l}}-\varphi^{s}_{\overline{k},\overline{l}s}+\varphi^{s}_{\overline{k}}h_{s\overline{l}}-\partial_{s}\Gamma^{\overline{h}}_{\overline{k}\overline{l}}\varphi^{s}_{\overline{h}})d\overline{z}^{l}\wedge
d\overline{z}^{k}.$

On the other hand,

$\displaystyle\varphi^{s}_{\overline{k},s\overline{l}}-\varphi^{s}_{\overline{k},\overline{l}s}$
$\displaystyle=-R_{h\overline{t}s\overline{l}}g^{s\overline{t}}\varphi^{h}_{\overline{k}}-R_{t\overline{k}s\overline{l}}g^{t\overline{h}}\varphi^{s}_{\overline{h}}$
$\displaystyle=-R_{h\overline{l}}\varphi^{h}_{\overline{k}}+\partial_{s}\Gamma^{\overline{h}}_{\overline{k}\overline{l}}\varphi^{s}_{\overline{h}}.$

By (1.5) it follows that

$\overline{\partial}{\rm div}_{h}\varphi-{\rm
div}_{h}(\overline{\partial}\varphi)=(-R_{s\overline{l}}+h_{s\overline{l}})\varphi^{s}_{\overline{k}}d\overline{z}^{l}\wedge
d\overline{z}^{k}.$

We note that

$(-R_{s\overline{l}}+h_{s\overline{l}})\varphi^{s}_{\overline{k}}d\overline{z}^{l}\wedge
d\overline{z}^{k}=\varphi\lrcorner\frac{{\rm
Ric}(\omega)-\sqrt{-1}\partial\overline{\partial}h}{\sqrt{-1}}.$

Together with the assumption
${\rm{Ric}}(\omega)-\omega=\sqrt{-1}\partial\bar{\partial}h$, we conclude that

$\overline{\partial}{\rm div}_{h}\varphi-{\rm
div}_{h}(\overline{\partial}\varphi)=\varphi\lrcorner\frac{{\rm
Ric}(\omega)-\sqrt{-1}\partial\overline{\partial}h}{\sqrt{-1}}=-\sqrt{-1}\varphi\lrcorner\omega,$

which proves $(4)$.

For $(5)$, we recall that for any
$\varphi=\varphi^{l}_{\overline{k}}d\overline{z}^{k}\otimes\frac{\partial}{\partial
z^{l}}$ and
$\psi=\psi^{j}_{\overline{i}}d\overline{z}^{i}\otimes\frac{\partial}{\partial
z^{j}}$ it holds

$\displaystyle[\varphi,\psi]$
$\displaystyle=\varphi^{l}_{\overline{k}}\frac{\partial\psi^{j}_{\overline{i}}}{\partial
z^{l}}d\overline{z}^{k}\wedge d\overline{z}^{i}\otimes\frac{\partial}{\partial
z^{j}}+\psi^{j}_{\overline{i}}\frac{\partial\varphi^{l}_{\overline{k}}}{\partial
z^{j}}d\overline{z}^{i}\wedge d\overline{z}^{k}\otimes\frac{\partial}{\partial
z^{l}}$
$\displaystyle=\varphi^{l}_{\overline{k}}\frac{\partial\psi^{j}_{\overline{i}}}{\partial
z^{l}}d\overline{z}^{k}\wedge d\overline{z}^{i}\otimes\frac{\partial}{\partial
z^{j}}-\psi^{l}_{\overline{i}}\frac{\partial\varphi^{j}_{\overline{k}}}{\partial
z^{l}}d\overline{z}^{k}\wedge d\overline{z}^{i}\otimes\frac{\partial}{\partial
z^{j}}$
$\displaystyle=(\varphi^{l}_{\overline{k}}\frac{\partial\psi^{j}_{\overline{i}}}{\partial
z^{l}}-\psi^{l}_{\overline{i}}\frac{\partial\varphi^{j}_{\overline{k}}}{\partial
z^{l}})d\overline{z}^{k}\wedge
d\overline{z}^{i}\otimes\frac{\partial}{\partial z^{j}}$
$\displaystyle=(\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},l}-\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},l}-(\varphi^{l}_{\overline{k}}\Gamma^{j}_{lh}\psi^{h}_{\overline{i}}-\psi^{l}_{\overline{i}}\Gamma^{j}_{hl}\varphi^{h}_{\overline{k}}))d\overline{z}^{k}\wedge
d\overline{z}^{i}\otimes\frac{\partial}{\partial z^{j}}$
$\displaystyle=(\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},l}-\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},l})d\overline{z}^{k}\wedge
d\overline{z}^{i}\otimes\frac{\partial}{\partial z^{j}}.$

Then,

$\displaystyle{\rm div}[\varphi,\psi]$
$\displaystyle=(\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},l})_{j}d\overline{z}^{k}\wedge
d\overline{z}^{i}-(\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},l})_{j}d\overline{z}^{k}\wedge
d\overline{z}^{i}$
$\displaystyle=(\varphi^{l}_{\overline{k},j}\psi^{j}_{\overline{i},l}+\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},lj}-\psi^{l}_{\overline{i},j}\varphi^{j}_{\overline{k},l}-\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},lj})d\overline{z}^{k}\wedge
d\overline{z}^{i}$
$\displaystyle=(\varphi^{j}_{\overline{k},l}\psi^{l}_{\overline{i},j}+\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},lj}-\psi^{l}_{\overline{i},j}\varphi^{j}_{\overline{k},l}-\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},lj})d\overline{z}^{k}\wedge
d\overline{z}^{i}$
$\displaystyle=(\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},lj}-\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},lj})d\overline{z}^{k}\wedge
d\overline{z}^{i}.$

Hence by (1.4), we get

(1.6) $\displaystyle{\rm div}_{h}[\varphi,\psi]$
$\displaystyle=(\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},lj}-\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},lj}+\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},l}h_{j}-\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},l}h_{j})d\overline{z}^{k}\wedge
d\overline{z}^{i}.$

On the other hand, by (1.4) we also have

$\partial{\rm
div}_{h}\varphi=(\varphi^{l}_{\overline{k},lj}+\varphi^{l}_{\overline{k},j}h_{l}+\varphi^{l}_{\overline{k}}h_{lj})dz^{j}\wedge
d\overline{z}^{k}.$

It follows that

(1.7) $\displaystyle\psi\lrcorner\partial{\rm
div}_{h}\varphi=(\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},lj}+\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},l}h_{j}+\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k}}h_{jl})d\overline{z}^{i}\wedge
d\overline{z}^{k}.$

Similarly we have

(1.8) $\displaystyle\varphi\lrcorner\partial{\rm
div}_{h}\psi=(\varphi^{l}_{\overline{i}}\psi^{j}_{\overline{k},lj}+\varphi^{l}_{\overline{i}}\psi^{j}_{\overline{k},l}h_{j}+\varphi^{l}_{\overline{i}}\psi^{j}_{\overline{k}}h_{jl})d\overline{z}^{i}\wedge
d\overline{z}^{k}.$

We note that $h_{ij}=h_{ji}$. Hence, combing the above (1.6)-(1.8) we will
derive $(5)$ immediately.

Finally we prove $(6)$. From the proof of $(5)$ we see that

$[\varphi,\psi]=(\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},l}-\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},l})d\overline{z}^{k}\wedge
d\overline{z}^{i}\otimes\frac{\partial}{\partial z^{j}}.$

Thus we have

(1.9)
$\displaystyle[\varphi,\psi]\lrcorner\omega=\sqrt{-1}\varphi^{l}_{\overline{k}}\psi^{j}_{\overline{i},l}g_{j\overline{t}}d\overline{z}^{k}\wedge
d\overline{z}^{i}\wedge
d\overline{z}^{t}-\sqrt{-1}\psi^{l}_{\overline{i}}\varphi^{j}_{\overline{k},l}g_{j\overline{t}}d\overline{z}^{k}\wedge
d\overline{z}^{i}\wedge d\overline{z}^{t}.$

On the other hand, we have

$\displaystyle\partial(\psi\lrcorner\omega)=\sqrt{-1}\psi_{\overline{i},t}^{j}g_{j\overline{k}}dz^{t}\wedge
d\overline{z}^{i}\wedge d\overline{z}^{k}.$

It follows that

(1.10)
$\displaystyle\varphi\lrcorner\partial(\psi\lrcorner\omega)=\sqrt{-1}\varphi^{l}_{\overline{k}}\psi_{\overline{i},l}^{j}g_{j\overline{t}}d\overline{z}^{i}\wedge
d\overline{z}^{t}\wedge d\overline{z}^{k}.$

Similarly we have

(1.11)
$\displaystyle\psi\lrcorner\partial(\varphi\lrcorner\omega)=\sqrt{-1}\psi^{l}_{\overline{i}}\varphi_{\overline{k},l}^{j}g_{j\overline{t}}d\overline{z}^{k}\wedge
d\overline{z}^{t}\wedge d\overline{z}^{i}.$

Hence, combining the above (1.9)-(1.11) we get $(6)$. ∎

######  Lemma 1.2.

Suppose that

$\delta_{h}\varphi=0~{}{\rm
and}~{}\overline{\partial}(\varphi\lrcorner\omega)=0.$

Then

(1.12) $\displaystyle\Box_{h}(\varphi\lrcorner\omega)=-\sqrt{-1}({\rm
div}_{h}\overline{\partial}\varphi)-\varphi\lrcorner\omega.$

###### Proof.

Using the assumption $\overline{\partial}(\varphi\lrcorner\omega)=0$, we have

$\displaystyle\Box_{h}(\varphi\lrcorner\omega)$
$\displaystyle=\overline{\partial}\delta_{h}(\varphi\lrcorner\omega)+\delta_{h}\overline{\partial}(\varphi\lrcorner\omega)$
$\displaystyle=\overline{\partial}\delta_{h}(\varphi\lrcorner\omega).$

Then by Lemma 1.1-$(3)$ and the assumption $\delta_{h}\varphi=0$, we get

$\displaystyle\Box_{h}(\varphi\lrcorner\omega)$
$\displaystyle=\overline{\partial}(-\delta_{h}\varphi\lrcorner\omega-\sqrt{-1}{\rm
div}_{h}\varphi)$ $\displaystyle=\overline{\partial}(-\sqrt{-1}{\rm
div}_{h}\varphi).$

Hence, by Lemma 1.1-$(4)$, we obtain (1.12). ∎

######  Corollary 1.3.

Let $\varphi\in\mathcal{H}^{0,1}_{h}(M,T^{1,0}M).$ Then

$\Box_{h}(\varphi\lrcorner\omega)=-\varphi\lrcorner\omega.$

As a consequence,

(1.13) $\displaystyle\varphi\lrcorner\omega=0.$

###### Proof.

Note that $\overline{\partial}\varphi=0$. Then by Lemma 1.1-$(1)$ we have
$\overline{\partial}(\varphi\lrcorner\omega)=0$. On the other hand, since
$\delta_{h}\varphi=0$, by Lemma 1.2 together with
$\overline{\partial}\varphi=0$, we see that

$\displaystyle\Box_{h}(\varphi\lrcorner\omega)$ $\displaystyle=-\sqrt{-1}{\rm
div}_{h}(\overline{\partial}\varphi)-\varphi\lrcorner\omega$
$\displaystyle=-\varphi\lrcorner\omega.$

Thus

$\displaystyle 0\leq(\varphi\lrcorner\omega,\varphi\lrcorner\omega)_{h}$
$\displaystyle=-(\Box_{h}(\varphi\lrcorner\omega),\varphi\lrcorner\omega)_{h}$
$\displaystyle=-(\delta_{h}(\varphi\lrcorner\omega),\delta_{h}(\varphi\lrcorner\omega))_{h}-(\overline{\partial}(\varphi\lrcorner\omega),\overline{\partial}(\varphi\lrcorner\omega))_{h}$
$\displaystyle\leq 0.$

It follows that

$(\varphi\lrcorner\omega,\varphi\lrcorner\omega)_{h}=0,$

which implies (1.13).

∎

##  2\.  Kuranishi deformation theory for $\Box_{h}$-operator

We choose a basis $\\{e_{i}\\}_{i=1,...,l}$ of
$\mathcal{H}^{0,1}_{h}(M,T^{1,0}M)$ and denote an $\epsilon$-ball by

$B(\epsilon)=\\{\tau=(\tau_{1},...,\tau_{l})|~{}\psi_{\tau}=\sum\tau_{i}e_{i}\in\mathcal{H}^{0,1}_{h}(M,T^{1,0}M)~{}{\rm
with}~{}\sum_{i}\tau_{i}^{2}<\epsilon^{2}\\}.$

Then by the Kodaira deformation theory for complex structures on $(M,J_{0})$,
there is a map

$\Psi:B(\epsilon)\mapsto A^{0,1}(M,T^{1,0}M)$

as long as $\epsilon$ is small enough such that $\Psi(t)$ is a solution of
Cartan-Maurer equation,

(2.1)
$\displaystyle\overline{\partial}\Psi(\tau)=\frac{1}{2}[\Psi(\tau),\Psi(\tau)].$

More precisely, analogous to the Kodaira theory [14], we can reduce (2.1) to
solving the following equation with gauge fixed,

(2.4) $\displaystyle\left\\{\begin{array}[]{ll}\delta_{h}\varphi(\tau)=0\\\
\overline{\partial}\varphi(\tau)=\frac{1}{2}[\varphi(\tau),\varphi(\tau)].\\\
\end{array}\right.$

The above equation is equivalent to

(2.5)
$\displaystyle\varphi(\tau)=\sum^{m}_{i=1}\tau_{i}e_{i}+\frac{1}{2}\delta_{h}G_{h}[\varphi(\tau),\varphi(\tau)],$

where $G_{h}$ the Green operator associated to $\Box_{h}$. Following the
Kuranshi’s method by the implicity function theorem [13], there is a unique
solution of (2.5) as long as $\sum|\tau_{i}|^{2}<\epsilon^{2}$. Thus there is
a map

(2.6) $\displaystyle\Phi:B(\epsilon)\mapsto A^{0,1}(M,T^{1,0}M)$

such that $\Phi(\tau)=\varphi(\tau)=\varphi_{\tau}$. For simplicity, we call
$\Phi$ the Kuranishi map associated to $\Box_{h}$-operator.

On the other hand, according to [14], we may write $\varphi(\tau)$ as a
convergent expansion to solve (2.1),

(2.7) $\displaystyle\varphi(\tau)=\sum\tau_{i}e_{i}+\sum_{|I|\leq
2}\tau^{I}\varphi_{I}.$

By (2.7), we prove

######  Proposition 2.1.

Let $\varphi(\tau)$ be a Kuranishi solution of (2.5) as the form of (2.7).
Then

(2.8) $\displaystyle\varphi(\tau)\lrcorner\omega=0~{}{\rm and}~{}{\rm
div}_{h}\varphi(\tau)=0.$

###### Proof.

Let

(2.9) $\displaystyle\psi_{k}=\sum_{|I|\leq
k}\tau^{I}\varphi_{I},k=1,2,\ldots.$

Then

$\delta_{h}\psi_{k}=0,~{}{\rm for~{}each}~{}k\geq 1.$

We need to prove (2.8) for each $\psi_{k}$ by induction. Since

$\delta_{h}\psi_{1}=0,\overline{\partial}\psi_{1}=0,$

by Corollary 1.3, we have

$\psi_{1}\lrcorner\omega=0.$

By Lemma 1.1-$(3)$, it follows that

(2.10) $\displaystyle{\rm div}_{h}\psi_{1}=0.$

Thus we may assume that

(2.11) $\displaystyle\psi_{k}\lrcorner\omega=0~{}{\rm and}~{}{\rm
div}_{h}\psi_{k}=0,~{}{\rm for~{}any}~{}k<l.$

By (2.4), we see that

(2.12)
$\displaystyle\overline{\partial}\psi_{l}=\frac{1}{2}\sum_{k=1}^{l-1}[\varphi_{k},\varphi_{l-k}],\delta_{h}\psi_{l}=0.$

Then by Lemma 1.1-(6) together with the first relation in (2.11), we get

$\overline{\partial}\psi_{l}\lrcorner\omega=0.$

By Lemma 1.1-(1), it follows that

(2.13) $\displaystyle\overline{\partial}(\psi_{l}\lrcorner\omega)=0.$

Thus by Lemma 1.2 together with the second relation in (2.12), we derive

(2.14) $\displaystyle\Box_{h}(\psi_{l}\lrcorner\omega)=-\sqrt{-1}{\rm
div}_{h}\overline{\partial}\psi_{l}-\psi_{l}\lrcorner\omega.$

On the other hand,

$\displaystyle{\rm div}_{h}\overline{\partial}\psi_{l}$ $\displaystyle={\rm
div}_{h}(\frac{1}{2}\sum_{k=1}^{l-1}[\varphi_{k},\varphi_{l-k}])$
$\displaystyle=\frac{1}{2}\sum_{k=1}^{l-1}{\rm
div}_{h}[\varphi_{k},\varphi_{l-k}].$

By Lemma 1.1-(5), we have

${\rm div}_{h}\overline{\partial}\psi_{l}=0.$

Thus by (2.14), we get

$\Box_{h}(\psi_{l}\lrcorner\omega)=-\psi_{l}\lrcorner\omega.$

Hence, as in the proof of (1.13), we obtain

(2.15) $\displaystyle\psi_{l}\lrcorner\omega=0.$

This proves the first relation of (2.11) for $k=l$.

By Lemma 1.1-(3), we also have

$\delta_{h}(\psi_{l}\lrcorner\omega)=-\delta_{h}\psi_{l}\lrcorner\omega-\sqrt{-1}{\rm
div}_{h}\psi_{l}.$

Thus (2.15) implies the second relation of (2.11) for $k=l$. The proposition
is proved. ∎

By (2.8), for $v,w\in T^{0,1}M$ we have

$\omega(v+\varphi_{\tau}(v),w+\varphi_{\tau}(w))=0.$

Since $T^{0,1}_{\varphi_{\tau}}=(id+\varphi_{\tau})(T^{0,1}M)$ and $\omega$ is
real, $\omega$ is still a $(1,1)$-form with respect to the complex structure
$J_{\psi_{\tau}}$ defined by $\varphi_{\tau}$. Namely, $\varphi_{\tau}$ is
compatible with the Kähler form $\omega$. As a consequence,
$(\omega,J_{\psi_{\tau}})$ defines a family of Kähler metrics by

(2.16) $\displaystyle g_{\tau}={\omega}(\cdot,J_{\psi_{\tau}}\cdot).$

Hence, we get

######  Corollary 2.2.

For any $\tau\in B(\epsilon)$, $g_{\tau}$ defined in (2.16) is a family of
Kähler metrics with the same Kähler form $\omega$.

The following proposition gives a relationship between the first and second
relations in (2.8), which means that the Kuranishi equation is equivalent to
the Cartan-Maurer equation with the divergence free gauge.

######  Proposition 2.3.

Suppose that $\varphi\in A^{0,1}(M,T^{1,0}M)$ satisfies

$\overline{\partial}\varphi=\frac{1}{2}[\varphi,\varphi]~{}{\rm with}~{}{\rm
div}_{h}\varphi=0.$

Then

$\displaystyle\varphi\lrcorner\omega=0~{}{\rm and}~{}\delta_{h}\varphi=0.$

###### Proof.

By Lemma 1.1-$(5)$ and the assumption that ${\rm div}_{h}\varphi=0$, we have

$\displaystyle{\rm div}_{h}(\overline{\partial}\varphi)$
$\displaystyle=\frac{1}{2}{\rm div}_{h}[\varphi,\varphi]$
$\displaystyle=\varphi\lrcorner\partial{\rm div}_{h}\varphi$
$\displaystyle=0.$

By Lemma 1.1-$(4)$ together with the assumption ${\rm div}_{h}\varphi=0$, it
follows that

(2.17) $\displaystyle\varphi\lrcorner\omega=0.$

On the other hand, by Lemma 1.1-$(3)$ and (2.17), we see that

$\delta_{h}\varphi\lrcorner\omega=0,$

which means that $\delta_{h}\varphi=0$. The proposition is proved. ∎

##  3\. Restricted entropy $\nu(\cdot)$ and its variations

From this section, we will always assume that $(M,J_{0})$ is a Fano Kähler
manifold which admits a KR soliton $(\omega_{KS},X)$. Here the soliton VF $X$
can be regarded as an element in the center of Lie algebra $\eta_{r}(M,J_{0})$
of a maximal reductive subgroup ${\rm Aut}_{r}(M,J_{0})$ of ${\rm
Aut}(M,J_{0})$ [35]. Namely $(\omega_{KS},X)$ satisfies the soliton equation,

$\rm{Ric}(\omega_{KS})-\omega_{KS}=\mathcal{L}_{X}\omega_{KS},$

where $\mathcal{L}_{X}\omega_{KS}$ is the Lie derivative of $\omega_{KS}$
along $X$. Then there is a real smooth function
$\theta=\theta_{X}(\omega_{KS})$ which satisfies

(3.1) $\displaystyle
i_{X}\omega_{KS}=\sqrt{-1}\overline{\partial}\theta,~{}\int_{M}e^{\theta}\omega_{KS}^{n}=\int_{M}\omega_{KS}^{n}.$

Thus by Corollary 2.2, there is a small ball

$B(\epsilon)=\\{\tau|~{}\psi_{\tau}=\sum\tau_{i}e_{i}\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M),\sum_{i}\tau_{i}^{2}<\epsilon^{2}\\},$

such that the Kuranishi map:

$\Phi:B(\epsilon)\mapsto A^{0,1}(M,T^{1,0}M)$

induces a family of Kähler metrics $g_{\tau}$ with the same Kähler form
$\omega_{KS}$. As a consequence, any Kähler metric $g$ in $2\pi c_{1}(M)$ with
small perturbed integral complex structure of $J_{0}$ can be parameterized by
the following map:

$L:~{}B(\epsilon)\times C^{\infty}(M)\mapsto{\rm Sym}^{2}(T^{*}M)$

with $g_{\tau,\chi}(\cdot,\cdot)=L(\psi,\chi)(\cdot,\cdot)$ satisfying

(3.2) $\displaystyle
g_{\tau,\chi}=(\omega_{KS}+\sqrt{-1}\partial_{J_{\psi_{\tau}}}\overline{\partial}_{J_{\psi_{\tau}}}\chi)(\cdot,J_{\psi_{\tau}}\cdot).$

The purpose of this section is to compute the variation of Perelman’s entropy
$\lambda(\cdot)$ for Kähler metrics $g_{\tau,\chi}$. Recall that the
Perelman’s W-functional for Kähler metrics $g$ in $2\pi c_{1}(M)$ is defined
for a pair $(g,f)$ by (cf. [40]),

(3.3) $\displaystyle W(g,f)=(2\pi)^{-n}\int_{M}[R(g)+|\nabla
f|^{2}+f]e^{-f}\omega_{g}^{n},$

where $f$ is a real smooth function normalized by

(3.4) $\displaystyle\int_{M}e^{-f}\omega_{g}^{n}=\int_{M}\omega_{g}^{n}.$

Then $\lambda(g)$ is defined by

$\lambda(g)=\inf_{f}\\{~{}W(g,f)|~{}\text{$(g,f)$ satisfies (\ref{nor})}\
\\}.$

The number $\lambda(g)$ can be attained by some $f$ (cf. [30]). In fact, such
a $f$ is a solution of the equation,

(3.5) $\displaystyle 2\triangle f+f-|Df|^{2}+R=\lambda(g).$

In particular, $f=\theta$ if $\omega_{g}=\omega_{KS}$, so the minimizer of
$W(g,\cdot)$ is unique near a KR soliton [38, 32]. Thus by the relation (3.2),
we get the restricted entropy $\nu(\cdot)$ of $\lambda(g)$ on
$\mathcal{U}_{\epsilon}=B(\epsilon)\times C^{\infty}(M)$ by

$\nu(\psi,\chi)=\lambda(L(\psi,\chi)),$

which is a smooth functional near $\omega_{KS}$. In fact, $\nu(\cdot)$ is
analytic (cf. [32]).

###  3.1. Variation of Kähler metrics

We calculate the variation of Kähler metrics $g_{\tau,\chi}$ at $g=g_{0,0}$
with its Kähler form $\omega=\sqrt{-1}g_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}.$
Let $\varphi\in A^{0,1}(M,T^{1,0}M)$ with the almost complex structure
$J_{\varphi}$ associated with $\varphi$. Under local coordinates
$(z^{1},\ldots z^{n})$ on $(M,J_{0})$, $\varphi$ is written as a Beltrami
differential by

$\varphi=\varphi_{\overline{i}}^{j}d\overline{z}^{i}\otimes\frac{\partial}{\partial
z^{j}}.$

Decompose $T_{\mathbb{C}}M$ with respect to $J_{\varphi}$ by

$T_{\mathbb{C}}M=T^{1,0}_{J_{\varphi}}M\oplus T_{J_{\varphi}}^{0,1}M.$

Then

$T_{J_{\varphi}}^{0,1}M=(id+\varphi)T^{0,1}M,$

where $\varphi$ is viewed as a map from $T^{1,0}M\mapsto T^{1,0}M$.

Let

$\left\\{\begin{array}[]{ll}J_{\varphi}\frac{\partial}{\partial
z^{i}}=J^{j}_{i}\frac{\partial}{\partial
z^{j}}+J^{\overline{j}}_{i}\frac{\partial}{\partial\overline{z}^{j}}\\\
J_{\varphi}\frac{\partial}{\partial\overline{z}^{i}}=J^{j}_{\overline{i}}\frac{\partial}{\partial
z^{j}}+J^{\overline{j}}_{\overline{i}}\frac{\partial}{\partial\overline{z}^{j}}.\\\
\end{array}\right.$

Since $J_{\varphi}$ is real,

$\overline{J^{j}_{i}}=J^{\overline{j}}_{\overline{i}},\overline{J^{\overline{j}}_{i}}=J^{j}_{\overline{i}}.$

Thus for a family of $\varphi=\varphi(\tau)\in A^{0,1}(T^{1,0}M)$ with
$\phi(0)=0$, we get the derivative of almost complex structure $J_{\varphi}$
as follows,

$\displaystyle\frac{dJ^{j}_{\overline{i}}}{d\tau}|_{\tau=0}$
$\displaystyle=-2\sqrt{-1}\frac{d\varphi^{j}_{\overline{i}}}{d\tau},$
$\displaystyle\frac{dJ^{\overline{j}}_{\overline{i}}}{ds}|_{\tau=0}$
$\displaystyle=0.$

By Corollary 2.2, $g_{\tau}$ given in (2.16) is a family of Kähler metrics
with the fixed Kähler form ${\omega}$. Locally, as a Riemannian tensor,
$g_{\tau}$ is of form,

$\displaystyle g_{\tau}=2{\rm
Re}(-\sqrt{-1}{g_{k\overline{i}}}J^{k}_{\overline{j}}d\overline{z}^{i}\otimes
d\overline{z}^{j}-\sqrt{-1}{g_{k\overline{i}}}J^{k}_{j}d\overline{z}^{i}\otimes
dz^{j}).$

Then the derivative of $L$ at $(0,0)$ is given by

$\eta=DL_{(0,0)}(\psi,0)=-4{\rm
Re}(g_{k\overline{i}}\psi^{k}_{\overline{j}}d\overline{z}^{i}\otimes
d\overline{z}^{j})$

and

$DL_{(0,0)}(0,\chi)=2{\rm Re}(\chi_{j\overline{i}}d\overline{z}^{i}\otimes
dz^{j}),$

where $\psi\in\mathcal{H}_{h}^{0,1}(M,T^{1,0}M)$ and $\chi\in C^{\infty}(M).$
Clearly, $\eta$ is anti-hermitian symmetric and $DL_{(0,0)}(0,\chi)$ is
hermitian symmetric.

We define the divergence for a $(0,2)$-type tensor $\eta$ by

${\rm div}_{h}\eta=e^{-h}{\rm div}(e^{h}\eta)=2{\rm
Re}[\eta_{\overline{i}\overline{j},k}+\eta_{\overline{i}\overline{j}}h_{k})g^{k\overline{j}}d\bar{z}^{i}].$

The following lemma shows that the tensor $\eta$ is also divergence-free as
$\psi$.

######  Lemma 3.1.

(3.6) $\displaystyle{\rm div}_{h}\eta=0~{}{\rm
and}~{}\eta_{\overline{i}\overline{j},\overline{k}}=\eta_{\overline{i}\overline{k},\overline{j}}.$

###### Proof.

By (1.4) we have

$\displaystyle({\rm div}_{h}\eta)_{\overline{i}}$
$\displaystyle=(\eta_{\overline{i}\overline{j},k}+\eta_{\overline{i}\overline{j}}h_{k})g^{k\overline{j}}$
$\displaystyle=-4(g_{l\overline{i}}(\psi^{l}_{\overline{j},k}+\psi^{l}_{\overline{j}}h_{k})g^{k\overline{j}}$
$\displaystyle=-4g_{l\overline{i}}({\rm div}_{h}\psi)^{l}$ $\displaystyle=0.$

The last equality comes from (2.8). Thus ${\rm div}_{h}\eta=0.$ On the other
hand, since $\overline{\partial}\varphi=0$, we have

$\displaystyle\eta_{\overline{i}\overline{j},\overline{k}}$
$\displaystyle=-4g_{l\overline{i}}\psi^{l}_{\overline{j},\overline{k}}$
$\displaystyle=-4g_{l\overline{i}}\psi^{l}_{\overline{k},\overline{j}}$
$\displaystyle=\eta_{\overline{i}\overline{k},\overline{j}}.$

The lemma is proved. ∎

######  Remark 3.2.

When $(M,\omega,J_{0})$ is a KE manifold, (3.6) in Lemma 3.1 was verified by
Koiso [15]. By using the $h$-harmonic space in (1.3), we can generalize
Koiso’s result for any Fano manifold. The lemma will be used in the
computation of the second variation of $\nu$ below.

###  3.2. The first variation of $\nu(\cdot)$

As in [40], we have the first variation of $\lambda$ at $(M,\omega,J_{0})$ as
a Riemannian manifold,

(3.7) $\displaystyle\delta\lambda=-\frac{1}{2}\int_{M}<{\rm Ric}(g)-g+{\rm
Hess}_{g}(f),\delta g>e^{-f}\omega^{n}.$

Then

(3.8) $\displaystyle\mathcal{N}(g)={\rm Ric}(g)-g+{\rm Hess}_{g}(f)$

can be regarded as the derivative of $\lambda$ which is a map from the space
of $2$-symmetric tensors to itself. But for the restricted entropy
$\nu(\cdot)$, we shall define its derivative from the product space
$\mathcal{U}=\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\times C^{\infty}(M)$ to
itself in the following.

Recall $f_{\psi,\chi}$ be the minimizer of $W$-functional at $g_{\tau,\chi}$
in (3.5). We introduce a map ${\mathbf{R}}:\mathcal{U}\to C^{\infty}(M)$ by

$\displaystyle{\mathbf{R}}(\psi,\chi)$
$\displaystyle=\frac{1}{2}\sqrt{-1}(\bar{\partial}_{J_{\psi}})^{*}_{-f_{\psi,\chi}}\cdot(\partial_{J_{\psi}})^{*}_{-f_{\psi,\chi}}\cdot({\rm
Ric}(\omega_{\tau,\chi})-\omega_{\tau,\chi}+\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}f_{\psi,\chi})e^{-f_{\psi,\chi}-\theta}\frac{\omega_{\tau,\chi}^{n}}{\omega^{n}},$

where $(\overline{\partial}_{J_{\psi}})^{*}_{-f_{\psi,\chi}}$ and
$(\partial_{J_{\psi}})^{*}_{-f_{\psi,\chi}}$ are the dual operators of
$\overline{\partial}_{J_{\psi}}$ and $\partial_{J_{\psi}}$, respectively as
same as $\delta_{h}$ in Section 1 with respect to the following inner product,

(3.9)
$\displaystyle(\chi_{1},\chi_{2})=\int_{M}\chi_{1}\chi_{2}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n},~{}~{}\chi_{1},\chi_{2}\in
L^{2}(M,e^{-f_{\psi,\chi}}).$

Choose an unitary orthogonal basis $\\{e_{1},\ldots,e_{l}\\}$ of
$\mathcal{H}_{\theta}^{0,1}(M,T^{1,0}M)$ with respect to the inner product

$(a,b)_{\theta}=\int_{M}<a,b>_{\omega_{KS}}e^{\theta}\omega_{KS}^{n},~{}\forall~{}a,b\in\mathcal{H}_{\theta}^{0,1}(M,T^{1,0}M).$

We define another map
${\mathbf{Q}}:\mathcal{U}\to\mathcal{H}_{\theta}^{0,1}(M,T^{1,0}M)$ by

${\mathbf{Q}}(\psi,\chi)=\sum_{i=1}^{l}(a_{i}(\psi,\chi)+\sqrt{-1}b_{i}(\psi,\chi))e_{i},$

where

$\displaystyle a_{i}(\psi,\chi)$ $\displaystyle=-\frac{1}{2}\int_{M}({\rm
Ric}(g_{\tau,\chi})-g_{\tau,\chi}+{\rm
Hess}_{g_{\tau,\chi}}(f_{\psi,\chi}),DL_{\psi,\chi}(e_{i},0))e^{-f_{\psi,\chi}}\omega^{n}_{\tau,\chi}$

and

$\displaystyle b_{i}(\psi,\chi)$ $\displaystyle=-\frac{1}{2}\int_{M}({\rm
Ric}(g_{\tau,\chi})-g_{\tau,\chi}+{\rm
Hess}_{g_{\tau,\chi}}(f_{\psi,\chi}),DL_{\psi,\chi}(\sqrt{-1}e_{i},0))e^{-f_{\psi,\chi}}\omega^{n}_{\tau,\chi}.$

######  Definition 3.3.

We call the pair $({\mathbf{Q}},{\mathbf{R}})$ the gradient map $\nabla\nu$ of
$\nu$ on $\mathcal{U}_{\epsilon}$ with the inner product

(3.10)
$\displaystyle(\nabla\nu,(\psi,\chi))_{\theta}=(\mathbf{Q},\psi)_{\theta}+\int_{M}\mathbf{R}\chi
e^{\theta}\omega_{KS}^{n},$

where $(\psi,\chi)\in\mathcal{U}.$

It is easy to verify that $\nabla\nu=({\mathbf{Q}},{\mathbf{R}})$ is real
analytic on $\mathcal{U}_{\epsilon}$. The following lemma gives the first
variation $\nu$ by $\nabla\nu$.

######  Lemma 3.4.

Let $(\psi_{0},\chi_{0})\in\mathcal{U}_{\epsilon}$ and
$(\psi,\chi)\in\mathcal{H}_{\theta}^{0,1}(M,T^{1,0}M)\times C^{\infty}(M)$.
Then

(3.11)
$\displaystyle\frac{d}{ds}|_{s=0}\nu((\psi_{0},\chi_{0})+s(\psi,\chi))=(\nabla\nu(\psi_{0},\chi_{0}),(\psi,\chi))_{\theta}.$

###### Proof.

For convenience, we let $g=L(\psi_{0},\chi_{0})$ and $f=f_{g}$ be the
minimizer of $W$ functional at $g$. By (3.7), we have

$\displaystyle\frac{d\nu((\psi_{0},\chi_{0})+s(\psi,\chi))}{ds}|_{s=0}$
$\displaystyle=-\frac{1}{2}\int_{M}<{\rm Ric}(g)-g+{\rm
Hess}_{g}(f),DL_{(\psi_{0},\chi_{0})}(\psi,\chi)>e^{-f}\omega_{\psi_{0},\chi_{0}}^{n}$
$\displaystyle=-\frac{1}{2}\int_{M}<{\rm Ric}(g)-g+{\rm
Hess}_{g}(f),DL_{(\psi_{0},\chi_{0})}(\psi,0)>e^{-f}\omega_{\psi_{0},\chi_{0}}^{n}$
$\displaystyle-\frac{1}{2}\int_{M}<{\rm Ric}(g)-g+{\rm
Hess}_{g}(f),DL_{(\psi_{0},\chi_{0})}(0,\chi)>e^{-f}\omega_{\psi_{0},\chi_{0}}^{n}.$

It is easy to see that

$\displaystyle I$ $\displaystyle=-\frac{1}{2}\int_{M}<{\rm Ric}(g)-g+{\rm
Hess}_{g}(f),DL_{(\psi_{0},\chi_{0})}(\psi,0)>e^{-f}\omega_{\psi_{0},\chi_{0}}^{n}$
$\displaystyle=(\mathbf{Q}(\psi_{0},\chi_{0}),\psi)_{\theta}$

and

$\displaystyle II$ $\displaystyle=-\frac{1}{2}\int_{M}<{\rm Ric}(g)-g+{\rm
Hess}_{g}(f),DL_{(\psi_{0},\chi_{0})}(0,\chi)>e^{-f}\omega_{\psi_{0},\chi_{0}}^{n}$
$\displaystyle=\frac{d\lambda((\psi_{0},\chi_{0})+s(0,\chi))}{ds}|_{s=0}$
$\displaystyle=-\frac{1}{2}\int_{M}<\sqrt{-1}\partial_{J_{\psi_{0}}}\overline{\partial}_{J_{\psi_{0}}}\chi,{\rm
Ric}(\omega_{\psi_{0},\chi_{0}})-\omega_{\psi_{0},\chi_{0}}+\sqrt{-1}\partial_{J_{\psi_{0}}}\overline{\partial}_{J_{\psi_{0}}}f>e^{-f}\omega_{\psi_{0},\chi_{0}}^{n}$
$\displaystyle=\int_{M}\mathbf{R}(\psi_{0},\chi_{0})\chi
e^{-f}\omega_{\psi_{0},\chi_{0}}^{n}.$

Thus (3.11) is true.

∎

######  Remark 3.5.

From the above, we actually have

$\displaystyle\frac{d}{ds}|_{s=0}\nu((\psi_{0},\chi_{0})+s(\psi,\chi))$
$\displaystyle=((\psi,\chi),\nabla\nu(\psi_{0},\chi_{0}))_{\theta}$ (3.12)
$\displaystyle=-\frac{1}{2}\int_{M}<{\rm Ric}(g)-g+{\rm
Hess}_{g}(f),DL_{(\psi_{0},\chi_{0})}(\psi,\chi)>e^{-f}\omega_{\psi_{0},\chi_{0}}^{n}.$

The $L^{2}$-norm of ${\mathbf{R}}$ can be controlled by
$\nabla\lambda(\cdot)=\mathcal{N}(\cdot)$ as follows.

######  Lemma 3.6.

Let $k>2$ be an integer and $\epsilon$ small enough. Then there exists a
constant $C=C(\epsilon,k)$ such that for any
$(\psi,\chi)\in\mathcal{U}_{\epsilon}$ with
$\|\chi\|_{C^{4,\gamma}}\leq\epsilon$ it holds

(3.13) $\displaystyle\|{\mathbf{R}}(\psi,\chi)\|_{L^{2}}\leq C(\int_{M}|{\rm
Ric}(\omega_{\psi,\chi})-\omega_{\psi,\chi}+\sqrt{-1}\partial\overline{\partial}f_{\psi,\chi}|^{2}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n})^{\frac{k-2}{k-1}}.$

###### Proof.

Let

(3.14) $\displaystyle{\rm
Ric}(\omega_{\psi,\chi})-\omega_{\psi,\chi}+\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}f_{\psi,\chi}=\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}F.$

We extend ${\mathbf{R}}(\psi,\chi)$ to a fourth-order operator on the Hilbert
space $W_{4}^{2}(M,e^{-f_{\psi,\chi}})$ with respect to the inner product
(3.9) by

(3.15) $\displaystyle
S=-(\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}})^{*}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}=-\overline{\partial}_{J_{\psi}}^{*}\cdot\partial_{J_{\psi}}^{*}\cdot\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}.$

We show that $S$ is elliptic and self-adjoint like the Lichnerowicz operator
[4].

Let $\eta,\phi\in C^{\infty}(M)$ and $\Delta\eta=\eta_{i\bar{j}}g^{i\bar{j}}$
associated to the metric $g=L(\psi,\chi)$. Then in local coordinates on
$(M,J_{\varphi})$, we compute

$\displaystyle(S\eta,\phi)$
$\displaystyle=\int_{M}\eta_{i\bar{j}}\overline{\phi_{k\bar{l}}}g^{i\bar{k}}g^{l\bar{j}}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}$
$\displaystyle=-\int_{M}\eta_{i\bar{j}\bar{k}}\overline{\phi_{\bar{l}}}g^{i\bar{k}}g^{l\bar{j}}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}+\int_{M}\eta_{i\bar{j}}\overline{\phi_{\bar{l}}}(f_{\psi,\chi})_{\bar{k}}g^{i\bar{k}}g^{l\bar{j}}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}$
$\displaystyle=-\int_{M}\eta_{i\bar{k}\bar{j}}\overline{\phi_{\bar{l}}}g^{i\bar{k}}g^{l\bar{j}}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}+\int_{M}\eta_{i\bar{j}}\overline{\phi_{\bar{l}}}(f_{\psi,\chi})_{\bar{k}}g^{i\bar{k}}g^{l\bar{j}}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}$
$\displaystyle=-\int_{M}(\Delta\eta)_{\bar{j}}\overline{\phi_{\bar{l}}}g^{l\bar{j}}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}+\int_{M}\eta_{i\bar{j}}\overline{\phi_{\bar{l}}}(f_{\psi,\chi})_{\bar{k}}g^{i\bar{k}}g^{l\bar{j}}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}$
$\displaystyle=\int_{M}[(\Delta^{2}\eta)-(\Delta\eta)_{\bar{j}}(f_{\psi,\chi})_{l}g^{l\bar{j}}]\phi
e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}$
$\displaystyle-\int_{M}[\eta_{i\bar{j}l}(f_{\psi,\chi})_{\bar{k}}g^{i\bar{k}}g^{l\bar{j}}+\eta_{i\bar{j}}(f_{\psi,\chi})_{l\bar{k}}g^{i\bar{k}}g^{l\bar{j}}-\eta_{i\bar{j}}(f_{\psi,\chi})_{\bar{k}}(f_{\psi,\chi})_{l}g^{i\bar{k}}g^{l\bar{j}}]\phi
e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}$
$\displaystyle=\int_{M}[(\Delta^{2}\eta)-(\Delta\eta)_{\bar{j}}(f_{\psi,\chi})_{l}g^{l\bar{j}}]\phi
e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}$
$\displaystyle-\int_{M}[\eta_{l\bar{j}i}(f_{\psi,\chi})_{\bar{k}}g^{i\bar{k}}g^{l\bar{j}}+\eta_{i\bar{j}}(f_{\psi,\chi})_{l\bar{k}}g^{i\bar{k}}g^{l\bar{j}}-\eta_{i\bar{j}}(f_{\psi,\chi})_{\bar{k}}(f_{\psi,\chi})_{l}g^{i\bar{k}}g^{l\bar{j}}]\phi
e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n}.$

It follows that

$\displaystyle(S\eta,\phi)$
$\displaystyle=\int_{M}[\Delta^{2}\eta-(\Delta\eta)_{\bar{j}}(f_{\psi,\chi})_{l}g^{l\bar{j}}-(\Delta\eta)_{l}(f_{\psi,\chi})_{\bar{j}}g^{l\bar{j}}$
$\displaystyle-\eta_{i\bar{j}}(f_{\psi,\chi})_{l\bar{k}}g^{i\bar{k}}g^{l\bar{j}}+\eta_{i\bar{j}}(f_{\psi,\chi})_{l}(f_{\psi,\chi})_{\bar{k}}g^{i\bar{k}}g^{l\bar{j}}]\phi
e^{-f_{\psi,\chi}}\omega_{g}^{n}.$

Thus we derive

$S\eta=\Delta^{2}\eta-(\Delta\eta)_{\bar{j}}(f_{\psi,\chi})_{l}g^{l\bar{j}}-(\Delta\eta)_{l}(f_{\psi,\chi})_{\bar{j}}g^{l\bar{j}}-\eta_{i\bar{j}}(f_{\psi,\chi})_{l\bar{k}}g^{i\bar{k}}g^{l\bar{j}}+\eta_{i\bar{j}}(f_{\psi,\chi})_{l}(f_{\psi,\chi})_{\bar{k}}g^{i\bar{k}}g^{l\bar{j}}.$

Hence, $S$ is an elliptic operator. By the regularity theorem of elliptic
operator, $S$ is a self-adjoint operator on the domain
$D(S)=W^{2}_{4}(M,e^{-f_{\psi,\chi}})$.

Since $S$ is non-negative, by the spectral theorem we see that for integer
$k>0$, $x\in D(S)$ it holds

(3.16)
$\displaystyle(S^{k}x,x)=\int_{0}^{\infty}\lambda^{k}dE_{x,x}(\lambda),$

where $E_{x,x}(\lambda)$ is the spectral decomposition defined by $S$ [31]. By
the Hölder inequality and (3.16) it follows that

(3.17)
$\displaystyle(S^{2}x,x)\leq(Sx,x)^{\frac{k-2}{k-1}}(S^{k}x,x)^{\frac{1}{k-1}}.$

Thus by (3.14), we get

$\displaystyle\|{\mathbf{R}}(\psi,\chi)\|_{L^{2}}$ $\displaystyle\leq
C((\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}})^{*}\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}F,(\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}})^{*}\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}F)$
$\displaystyle=C(SF,SF)$ $\displaystyle=C(S^{2}F,F)$ $\displaystyle\leq
C(SF,F)^{\frac{k-2}{k-1}}(S^{k}F,F)^{\frac{1}{k-1}}$ (3.18) $\displaystyle\leq
C^{\prime}(\int_{M}|{\rm
Ric}(\omega_{\psi,\chi})-\omega_{\psi,\chi}+\sqrt{-1}\partial\overline{\partial}f_{\psi,\chi}|^{2}e^{-f_{\psi,\chi}}\omega_{\psi,\chi}^{n})^{\frac{k-2}{k-1}}(S^{k}F,F)^{\frac{1}{k-2}}.$

Note that

$SF=(\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}})^{*}({\rm
Ric}(\omega_{\psi,\chi})-\omega_{\psi,\chi}+\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}f_{\psi,\chi}).$

Then

$\displaystyle(S^{k}F,F)$ $\displaystyle=(S^{k-2}SF,SF)$
$\displaystyle=(S^{k-2}(\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\varphi}})^{*}({\rm
Ric}(\omega_{\psi,\chi})-\omega_{\psi,\chi}+\sqrt{-1}\partial_{J_{\psi}}\overline{\partial}_{J_{\psi}}f_{\psi,\chi}).$

Hence

(3.19) $\displaystyle(S^{k}F,F)^{\frac{1}{k-2}}<C(\epsilon)\to 0,~{}{\rm
as}~{}\epsilon\to 0.$

Therefore, we get (3.13) immediately by (3.2). ∎

###  3.3. The second variation of $\nu(\cdot)$

We will calculate the second valuation of $\nu$ at
$(0,0)\in\mathcal{U}_{\epsilon}$. Let $(g,f)$ be a pair of $W$-functional in
(3.3) while $f$ is a smooth solution of (3.5). Denote $\eta=\delta g,k=\delta
f$ to be the variations of $g$ and $f$ respectively. Then it has been shown in
[5, 40] that

(3.20) $\displaystyle\delta({\rm Ric}(g)-g+{\rm
Hess}(f))=-\Delta_{-f}(\eta)-{\rm Rm}(\eta,\cdot)-{\rm div}^{*}_{-f}{\rm
div}_{-f}(\eta)-\frac{1}{2}{\rm Hess}(v_{\eta}),$

where the function $v_{\eta}={\rm tr}\eta-2k$. Moreover, by differentiating
(3.5), we get (cf. [17]),

(3.21) $\displaystyle\Delta_{-f}(v_{\eta})+\frac{v_{\eta}}{2}=-\frac{1}{2}{\rm
div}_{-f}{\rm div}_{-f}(\eta).$

For convenience, we write the right term in (3.20) as

$N(\eta)=-\Delta_{-f}(\eta)-{\rm Rm}(\eta,\cdot)-{\rm div}^{*}_{-f}{\rm
div}_{-f}(\eta)-\frac{1}{2}{\rm Hess}(v_{\eta}).$

In case that $-f=\theta$, namely $g$ is a KR soliton, if

(3.22) $\displaystyle{\rm div}_{\theta}\eta=0,$

then

$\Delta_{\theta}(v_{\eta})+\frac{v_{\eta}}{2}=0.$

Since the first non-zero eigenvalue of $-\Delta_{\theta}$ is not less than one
[40], we get

$v_{\eta}=0.$

Thus

(3.23) $\displaystyle N(\eta)=-\Delta_{\theta}(\eta)-{\rm Rm}(\eta,\cdot),$

which is an anti-hermitian symmetric 2-tensor.

Now we restrict the Kähler metric $g=g_{\tau,\chi}$ in
$\mathcal{U}_{\epsilon}$. Let

$(\psi,\chi),(\psi^{\prime},\chi^{\prime})\in\mathcal{U}=\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\times
C^{\infty}(M).$

Then by (3.5), we have

$\displaystyle\frac{d}{ds}|_{s=0}(\nabla\nu(s(\psi,\chi)),(\psi^{\prime},\chi^{\prime}))_{\theta}$
$\displaystyle=-\frac{1}{2}\int_{M}<\frac{d}{ds}_{s=0}({\rm
Ric}(g_{s\psi,s\chi})+{\rm
Hess}(f_{s\psi,s\chi})-g_{s\psi,s\chi}),DL_{(0,0)}(\psi^{\prime},\chi^{\prime})>e^{\theta}\omega_{KS}^{n}$
(3.24)
$\displaystyle=-\frac{1}{2}\int_{M}<N(DL_{(0,0)}(\psi,\chi),DL_{(0,0)}(\psi^{\prime},\chi^{\prime})>e^{\theta}\omega_{KS}^{n}.$

######  Lemma 3.7.

$\int_{M}<N(DL_{(0,0)}(\psi,0),DL_{(0,0)}(0,\chi^{\prime})>e^{\theta}\omega_{KS}^{n}=0.$

###### Proof.

We note that $DL_{(0,0)}(\psi,0)$ is anti-hermitian symmetric and (3.22) holds
by Lemma 3.1. Thus $N(DL_{(0,0)}(\psi,0))$ is an anti-hermitian symmetric
2-tensor by (3.23). On the other hand, $DL_{(0,0)}(0,\chi^{\prime})$ is
hermitian symmetric. Hence the lemma is true. ∎

Analogous to the $Q$-map in Section 3.1, we introduce a map

$H_{1}:\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\mapsto\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$

by

$H_{1}(\psi)=\sum_{i=1}^{l}(c_{i}(\psi)+\sqrt{-1}d_{i}(\psi))e_{i},$

where

$c_{i}(\psi)=-\frac{1}{2}\int_{M}<N(DL_{(0,0)}(\psi,0),DL_{(0,0)}(\delta_{i},0)>e^{\theta}\omega_{KS}^{n},i=1,2,\ldots,l,$

and

$d_{i}(\psi)=-\frac{1}{2}\int_{M}<N(DL_{(0,0)}(\psi,0),DL_{(0,0)}(\sqrt{-1}\delta_{i},0)>e^{\theta}\omega_{KS}^{n},i=1,2,\ldots,l.$

Recall a fourth order non-positively elliptic operator $H_{2}$ on
$C^{\infty}(M)$ introduced in [40] by

$H_{2}=[P_{0}^{-1}(\overline{L}^{\prime}_{1}L^{\prime}_{1})(\overline{L}_{1}L_{1})],$

where

$P_{0}\chi=2\triangle_{\theta}\chi+\chi-(X+\overline{X})(\chi),$
$L_{1}\chi=\triangle_{\theta}\chi+\psi-X(\chi)$

and

$L_{1}^{\prime}\chi=\triangle_{\theta}\chi-X(\chi).$

######  Definition 3.8.

Define a map $H_{\psi,\chi}:\mathcal{U}\to\mathcal{U}$ by

$H_{\psi,\chi}=D_{\psi,\chi}\nabla\nu$

and write $H=H_{0,0}$ for convenience. We call $H$ the second variation
operator of $\nu(\cdot)$ at $\omega_{KS}$.

By the above definition and (3.3), we see that for any
$(\psi,\chi)\in\mathcal{U}$, it holds

$\displaystyle\delta^{2}\nu_{(0,0)}((\psi,\chi),(\psi,\chi))$
$\displaystyle=\int_{M}<H(\psi,\chi),(\psi,\chi)>e^{\theta}\omega_{KS}^{n}$
(3.25) $\displaystyle=(H(\psi,\chi),(\psi,\chi))_{\theta}.$

Moreover, by Lemma 3.7 we get

######  Proposition 3.9.

$H=H_{1}\oplus H_{2}$, namely, $H(\psi,\chi)=H_{1}(\psi)+H_{2}(\chi).$

By a result in [43] (also see [18]), we have the following explicit formula
for $H_{1}$.

######  Lemma 3.10.

(3.26)
$\displaystyle(H_{1}(\psi),\psi)_{\theta}=\int_{M}|\psi|^{2}\widetilde{\theta}e^{\theta}\omega_{KS}^{n},~{}\forall~{}\psi\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M),$

where $\widetilde{\theta}$ differs from $\theta$ with a constant such that

(3.27) $\displaystyle\int_{M}\widetilde{\theta}e^{\theta}\omega_{KS}^{n}=0.$

###### Proof.

By Lemma 3.1, $\eta=DL_{(0,0)}(\psi,0)$ satisfies that

$\eta_{i\overline{j}}=0,{\rm div}_{\theta}\eta=0~{}{\rm
and}~{}\eta_{\bar{i}\bar{j},\bar{k}}=\eta_{\bar{i}\bar{k},\bar{j}}.$

For such a variation of Kähler metrics, $\delta^{2}\lambda$ has been computed
by the following formula [43, 18],

$\displaystyle\delta^{2}\nu((\psi,0),(\psi,0))=\delta^{2}\lambda(\eta,\eta)=\frac{1}{2}\int_{M}|\eta|^{2}(\nu(\omega_{KS})-2m+\theta)e^{\theta}\omega_{KS}^{n}.$

Notice that $|\eta|^{2}=2|\psi|^{2}$ and

$\int_{M}(\nu(\omega_{KS})-2m+\theta)e^{\theta}\omega_{KS}^{n}=0.$

This implies that (3.26) holds. ∎

##  4\. A new version of $\delta^{2}\nu$

In this section, we give a new version of $H_{1}$ and then describe the
geometry of the kernel space $Z={\rm ker}(H_{1})$. As an application, we are
able to prove the local maximality of $\lambda(\cdot)$ on the space of Kähler
metrics associated to the complex structures determined by $Z$.

###  4.1.  Kernel space $Z$ of $H_{1}$

We begin with the following technical lemma.

######  Lemma 4.1.

Suppose that $\varphi,\psi\in A^{0,1}(M,T^{1,0}M)$ satisfy

$\left\\{\begin{array}[]{ll}\overline{\partial}\varphi=\overline{\partial}\psi=0\\\
\varphi\lrcorner\omega=\psi\lrcorner\omega=0.\end{array}\right.$

Let $v$ be a $(1,0)$-VF. Then

$v<\varphi,\psi>-<[v,\varphi],\psi>={\rm
div}_{\theta}(v\lrcorner\overline{\psi}\lrcorner\varphi)-(v\lrcorner\overline{\varphi})\lrcorner{\rm
div}_{\theta}\varphi.$

###### Proof.

Let $v=v^{i}\frac{\partial}{\partial
z^{i}},\varphi=\varphi^{l}_{\overline{k}}d\overline{z}^{k}\otimes\frac{\partial}{\partial
z^{l}},\psi=\psi^{s}_{\overline{t}}d\overline{z}^{t}\otimes\frac{\partial}{\partial
z^{s}}$. Then

$\displaystyle[v,\varphi]$
$\displaystyle=(v^{i}\frac{\partial\varphi_{\overline{k}}^{l}}{\partial
z^{i}}-\varphi_{\overline{k}}^{i}\frac{\partial v^{l}}{\partial
z^{i}})d\overline{z}^{k}\otimes\frac{\partial}{\partial z^{l}}$
$\displaystyle=(v^{i}\varphi_{\overline{k},i}^{l}-\varphi_{\overline{k}}^{i}v^{l}_{,i})d\overline{z}^{k}\otimes\frac{\partial}{\partial
z^{l}}$

and

$<[v,\varphi],\psi>=(v^{i}\varphi_{\overline{k},i}^{l}-\varphi_{\overline{k}}^{i}v^{l}_{,i})\overline{\psi^{s}_{\overline{t}}}g^{t\overline{k}}g_{l\overline{s}}.$

Similarly,

$<\varphi,\psi>=\varphi^{l}_{\overline{k}}\overline{\psi^{s}_{\overline{t}}}g^{t\overline{k}}g_{l\overline{s}}$

and

$v<\varphi,\psi>=v^{i}\varphi^{l}_{\overline{k},i}\psi^{s}_{\overline{t}}g^{t\overline{k}}g_{l\overline{s}}+v^{i}\varphi^{l}_{\overline{k}}\overline{\psi^{s}_{\overline{t}\overline{i}}}g^{t\overline{k}}g_{l\overline{s}}.$

On the other hand, by the condition,

$\varphi\lrcorner\omega=\psi\lrcorner\omega=0,$

we have

$\varphi^{l}_{\overline{k}}g^{t\overline{k}}=\varphi^{t}_{\overline{k}}g^{l\overline{k}},\psi^{s}_{\overline{t}}g_{s\overline{l}}=\psi^{s}_{\overline{l}}g_{s\overline{t}},$

and by the condition,

$\overline{\partial}\psi=0,$

we have

$\psi^{s}_{\overline{t},\overline{i}}=\psi^{s}_{\overline{i},\overline{t}}.$

Thus

$\displaystyle v<\varphi,\psi>-<[v,\varphi],\psi>$
$\displaystyle=v^{i}\varphi^{l}_{\overline{k}}\overline{\psi^{s}_{\overline{t},\overline{i}}}g^{t\overline{k}}g_{l\overline{s}}+\varphi_{\overline{k}}^{i}v^{l}_{,i}\overline{\psi^{s}_{\overline{t}}}g^{t\overline{k}}g_{l\overline{s}}$
$\displaystyle=v^{i}\varphi^{l}_{\overline{k}}\overline{\psi^{s}_{\overline{i},\overline{t}}}g^{t\overline{k}}g_{l\overline{s}}+\varphi_{\overline{k}}^{i}v^{l}_{,i}\overline{\psi^{s}_{\overline{t}}}g^{t\overline{k}}g_{l\overline{s}}$
$\displaystyle=v^{l}\varphi^{i}_{\overline{k}}\overline{\psi^{s}_{\overline{l},\overline{t}}}g^{t\overline{k}}g_{i\overline{s}}+\varphi_{\overline{k}}^{i}v^{l}_{,i}\overline{\psi^{s}_{\overline{l}}}g^{t\overline{k}}g_{t\overline{s}}$
$\displaystyle=v^{l}\varphi^{t}_{\overline{k}}\overline{\psi^{s}_{\overline{l},\overline{t}}}g^{i\overline{k}}g_{i\overline{s}}+\varphi_{\overline{k}}^{i}v^{l}_{,i}\overline{\psi^{s}_{\overline{l}}}g^{t\overline{k}}g_{t\overline{s}}$
$\displaystyle=v^{l}\varphi^{t}_{\overline{k}}\overline{\psi^{k}_{\overline{l},\overline{t}}}+\varphi_{\overline{k}}^{i}v^{l}_{,i}\overline{\psi^{k}_{\overline{l}}}$
$\displaystyle=v^{l}\varphi^{i}_{\overline{k}}\overline{\psi^{k}_{\overline{l},\overline{i}}}+\varphi_{\overline{k}}^{i}v^{l}_{,i}\overline{\psi^{k}_{\overline{l}}}$
$\displaystyle=(v^{l}\varphi^{i}_{\overline{k}}\overline{\psi^{k}_{\overline{l}}})_{i}-v^{l}\varphi^{i}_{\overline{k},i}\overline{\psi^{k}_{\overline{l}}}$
$\displaystyle=(v^{l}\varphi^{i}_{\overline{k}}\overline{\psi^{k}_{\overline{l}}})_{i}+(v^{l}\varphi^{i}_{\overline{k}}\overline{\psi^{k}_{\overline{l}}})\theta_{i}-v^{l}(\varphi^{i}_{\overline{k},i}+\varphi^{i}_{\overline{k}}\theta_{i})\overline{\psi^{k}_{\overline{l}}}$
$\displaystyle={\rm
div}_{\theta}(v\lrcorner\overline{\psi}\lrcorner\varphi)-(v\lrcorner\overline{\psi})\lrcorner{\rm
div}_{\theta}\varphi.$

∎

If $v\in\eta_{r}(M,J_{0})$ is a HVF on $(M,J_{0},\omega_{KS})$, there is a
potential function $P$ of $v$ such that

(4.1) $\displaystyle
v=\nabla_{\overline{j}}Pg^{i\overline{j}}\frac{\partial}{\partial z^{i}}={\rm
grad}^{1,0}P.$

We may normalize $P$ by the following equation (cf. [35]),

$g^{i\overline{j}}\nabla_{\overline{j}}\nabla_{i}P+g^{i\overline{j}}\nabla_{\overline{j}}P\nabla_{i}\theta=-P.$

This means that

(4.2) $\displaystyle{\rm div}_{\theta}(v)=v^{i}_{,i}+v(\theta)=-P.$

######  Lemma 4.2.

Let $v={\rm grad}^{1,0}P$ as in (4.1) and
$\varphi,\psi\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$. Then

(4.3)
$\displaystyle\int_{M}<[v,\varphi],\psi>e^{\theta}\omega_{KS}^{n}=\int_{M}P<\varphi,\psi>e^{\theta}\omega_{KS}^{n}.$

###### Proof.

By Corollary 1.3, we see that

$\left\\{\begin{array}[]{ll}\overline{\partial}\varphi=\overline{\partial}\psi=0\\\
\varphi\lrcorner\omega_{KS}=\psi\lrcorner\omega_{KS}=0.\end{array}\right.$

Note that we also have

${\rm div}_{\theta}\varphi=0.$

Thus by Lemma 4.1 and Stoke’s formula, we get

$\displaystyle\int_{M}<[v,\varphi],\psi>e^{\theta}\omega_{KS}^{n}$
$\displaystyle=\int_{M}v<\varphi,\psi>e^{\theta}\omega_{KS}^{n}.$

Hence, by (4.2), we prove

$\displaystyle\int_{M}<[v,\varphi],\psi>e^{\theta}\omega_{KS}^{n}$
$\displaystyle=-\int_{M}{\rm
div}_{\theta}(v)<\varphi,\psi>e^{\theta}\omega_{KS}^{n}$
$\displaystyle=\int_{M}P<\varphi,\psi>e^{\theta}\omega_{KS}^{n}.$

∎

######  Remark 4.3.

A similar relation (4.3) is obtained in [6] when $\omega_{KS}$ is a KE metric.

Combining Lemma 4.2 with Lemma 3.10, we have the following new version of the
operator $H_{1}$.

######  Theorem 4.4.

Let $\xi={\rm Im}(X)$, where $X$ is a soliton HVF of $(M,J_{0},\omega_{KS}).$
Then

(4.4) $\displaystyle
H_{1}(\psi)=2\sqrt{-1}\mathcal{L}_{\xi}\psi,\forall~{}\psi\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M).$

###### Proof.

Note that

$X={\rm grad}^{1,0}\widetilde{\theta}$

and

$g^{i\overline{j}}\nabla_{\overline{j}}\nabla_{i}\widetilde{\theta}+g^{i\overline{j}}\nabla_{\overline{j}}\widetilde{\theta}\nabla_{i}\widetilde{\theta}=-\widetilde{\theta}.$

Then by Lemma 4.2, for any
$\psi^{\prime}\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$, we have

$\int_{M}<[X,\psi],\psi^{\prime}>e^{\theta}\omega_{KS}^{n}=\int_{M}\widetilde{\theta}<\psi,\psi^{\prime}>e^{\theta}\omega_{KS}^{n}.$

It follows that

(4.5)
$\displaystyle\int_{M}<\mathcal{L}_{X}\psi,\psi^{\prime}>e^{\theta}\omega^{n}=\int_{M}\widetilde{\theta}<\psi,\psi^{\prime}>e^{\theta}\omega_{KS}^{n}.$

Let $(z^{1},\ldots,z^{n})$ be holomorphic local coordinates on $(M,J_{0})$.
Write

$\psi=\psi^{i}_{\overline{k}}d\overline{z}^{k}\otimes\frac{\partial}{\partial
z^{i}},~{}X=X^{i}\frac{\partial}{\partial z^{i}}.$

Then

$\mathcal{L}_{\overline{X}}\psi=(\frac{\partial\overline{X^{k}}}{\partial\overline{z}^{j}}\psi^{i}_{\overline{k}}+\overline{X}(\psi^{i}_{\overline{j}}))d\overline{z}^{j}\otimes\frac{\partial}{\partial
z^{i}}.$

Since

$\overline{\partial}\psi=0,$
$\frac{\partial\psi^{i}_{\overline{j}}}{\partial\overline{z}^{k}}=\frac{\partial\psi^{i}_{\overline{k}}}{\partial\overline{z}^{j}}.$

Thus

$\displaystyle\overline{\partial}(\overline{X}\lrcorner\psi)$
$\displaystyle=\frac{\partial\overline{X^{k}}\psi^{i}_{\overline{k}}}{\partial\overline{z}^{j}}d\overline{z}^{j}\otimes\frac{\partial}{\partial
z^{i}}$
$\displaystyle=(\frac{\partial\overline{X^{k}}}{\partial\overline{z}^{j}}\psi^{i}_{\overline{k}}+\overline{X^{k}}\frac{\partial\varphi^{i}_{\overline{k}}}{\partial\overline{z}^{j}})d\overline{z}^{j}\otimes\frac{\partial}{\partial
z^{i}}$
$\displaystyle=(\frac{\partial\overline{X^{k}}}{\partial\overline{z}^{j}}\psi^{i}_{\overline{k}}+\overline{X^{k}}\frac{\partial\psi^{i}_{\overline{j}}}{\partial\overline{z}^{k}})d\overline{z}^{j}\otimes\frac{\partial}{\partial
z^{i}}$
$\displaystyle=(\frac{\partial\overline{X^{k}}}{\partial\overline{z}^{j}}\psi^{i}_{\overline{k}}+\overline{X}(\psi^{i}_{\overline{j}}))d\overline{z}^{j}\otimes\frac{\partial}{\partial
z^{i}}$ (4.6) $\displaystyle=\mathcal{L}_{\overline{X}}\psi.$

Hence, by the fact that

$\delta_{\theta}\psi=0,$

we derive

(4.7)
$\displaystyle\int_{M}<\mathcal{L}_{\overline{X}}\psi,\psi^{\prime}>e^{\theta}\omega_{KS}^{n}=\int_{M}<\bar{X}\lrcorner\psi,\delta_{\theta}\psi^{\prime}>e^{\theta}\omega^{n}_{KS}=0.$

By (4.5) and (4.7), we have

$\int_{M}<\mathcal{L}_{\xi}\psi,\psi^{\prime}>e^{\theta}\omega_{KS}^{n}=-\frac{\sqrt{-1}}{2}\int_{M}\widetilde{\theta}<\psi,\psi^{\prime}>e^{\theta}\omega_{KS}^{n}.$

By Lemma 3.10, it follows that

(4.8)
$\displaystyle\int_{M}<\mathcal{L}_{\xi}\psi,\psi^{\prime}>e^{\theta}\omega_{KS}^{n}=-\frac{\sqrt{-1}}{2}\int_{M}<H_{1}(\psi),\psi^{\prime}>e^{\theta}\omega_{KS}^{n}.$

On the other hand, for any $\sigma\in K_{X}$, where $K_{X}$ is the compact
1-ps of holomorphic transformations generated by $\xi$, $\sigma$ preserves the
Kähler form $\omega_{KS}$ and the function $\theta$. Then $\sigma$ maps
$\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ to itself. As a consequence,
$\mathcal{L}_{\xi}$ maps $\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ to itself.
Thus we prove (4.4) by (4.8) since $\psi^{\prime}$ is an arbitrary element in
$\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$.

∎

By Theorem 4.4, we have

(4.9) $\displaystyle{\rm
Ker}(H_{1})=Z=\\{\psi\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)|~{}\mathcal{L}_{\xi}\psi=0\\}.$

Moreover, we have the following character for the kernel $Z$.

######  Corollary 4.5.

$\psi\in Z$ if only if
$\xi_{J_{\Phi(\tau\psi)}}=J_{\Phi(\tau\psi)}\xi+\sqrt{-1}\xi$ is a HVF on
$(M,J_{\Phi(\tau\psi)})$ for some small $\tau$.

###### Proof.

Recall that $K_{X}=e^{s\xi}$ is the 1-ps of holomorphic transformations on
$(M,J_{0})$ generated by $\xi$. Then $\xi$ is holomorphic on
$(M,J_{\Phi(\tau\psi)})$ if and only if

$\displaystyle(e^{s\xi})^{*}J_{\Phi(\tau\psi)}=J_{\Phi(\tau\psi)},~{}\forall
s\in\mathbb{R},$

which is equivalent to

$\displaystyle J_{(e^{s\xi})^{*}\Phi(\tau\psi)}=J_{\Phi(\tau\psi)},~{}\forall
s\in\mathbb{R}.$

The latter is also equivalent to

(4.10) $\displaystyle(e^{s\xi})^{*}\Phi(\tau\psi)=\Phi(\tau\psi),~{}\forall
s\in\mathbb{R}.$

Note that $e^{s\xi}$ preserves the Kähler form $\omega_{KS}$ and $\theta$.
Then

$(e^{s\xi})^{*}\psi\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M).$

Moreover, for any $(1,1)$-form $\varphi_{k}$ $(k\geq 2)$ defined as in (2.9)
for $\tau\psi$, we have

$\delta_{\theta}((e^{s\xi})^{*}\varphi_{k})=0,$

Thus by the uniqueness of Kuranishi’s solutions we get

$\displaystyle(e^{s\xi})^{*}\Phi(\tau\psi)=\Phi(\tau(e^{s\xi})^{*}\psi).$

Since the Kuranishi map is injective, by (4.10), we derive

(4.11) $\displaystyle(e^{s\xi})^{*}\psi=\psi~{}\forall s\in\mathbb{R}.$

It follows that

$\mathcal{L}_{\xi}\psi=0.$

By Theorem 4.4, we prove that $\psi\in Z$ from (4.9). The inverse is also
true. In fact, if $\psi\in Z$, then (4.10) holds for any small $\tau$. Hence,
$\psi\in Z$ if only if $e^{s\xi}$ is a family of holomorphisms on
$(M,J_{\Phi(\tau\psi)})$. The corollary is proved.

∎

######  Remark 4.6.

It has been shown [40] that the kernel of $H_{2}$ is finitely dimensional,
which is isomorphic to the linear space generalized by the real and imaginary
parts on HVFs of $(M,J_{0})$. Thus by Proposition 3.9,

${\rm Ker}(H)={\rm Ker}(H_{1})\oplus{\rm Ker}(H_{2})$

is also finitely dimensional.

###  4.2. Index of $H_{1}$

Since $H_{2}$ is always non-positively elliptic operator, the index (the
number of positive eigenvalues) of $H$ depends only on the Lie operator
$\mathcal{L}_{\xi}$ on $\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ by Theorem
4.4. In the following, we will show that it just depends on the cohomology
group $H^{1}(M,J_{0},\Theta)$.

Suppose that $e_{1},\ldots,e_{l}$ are the eigenvectors of $\mathcal{L}_{\xi}$.
Namely there exists $\lambda_{i},i=1,2,\ldots,l$ such that

$\mathcal{L}_{\xi}e_{i}=\lambda_{i}e_{i}.$

By (4.4) we have

$H_{1}e_{i}=2\sqrt{-1}\lambda_{i}e_{i}.$

Thus $2\sqrt{-1}\lambda_{i}$ is real. Denote
$\lambda_{i}=-\sqrt{-1}\delta_{i},\delta_{i}\in\mathbb{R}$ and let

$\theta_{i}=e_{i},~{}\theta_{l+i}=\sqrt{-1}e_{i},~{}i=1,2,\ldots,l.$

It follows that

$H_{1}\theta_{i}=2\delta_{i}\theta_{i},~{}H_{1}\theta_{l+i}=2\delta_{i}\theta_{l+i}.$

Hence, we need to show that the number $\delta_{i}$ is independent of choice
of representation of $[\theta_{i}]$.

The following lemma can be found in the book of Kodaira [14].

######  Lemma 4.7.

If $\varphi\in A^{0,p}(M,T^{1,0}(M)),\psi\in A^{0,q}(M,T^{1,0}(M))$, we have

(4.12)
$\displaystyle\overline{\partial}[\varphi,\psi]=[\overline{\partial}\varphi,\psi]+(-1)^{p}[\varphi,\overline{\partial}\psi].$

In particular, if $Y$ is a HVF, and $\varphi\in A^{0,p}(M,T^{1,0}(M))$ then

(4.13)
$\displaystyle\overline{\partial}\mathcal{L}_{Y}(\varphi)=\mathcal{L}_{Y}\overline{\partial}(\varphi).$

######  Lemma 4.8.

Let $\varphi\in A^{0,1}(M,T^{1,0}(M))$. Then the following is true:

$\displaystyle\overline{\partial}\mathcal{L}_{\xi}\varphi=0,~{}{\rm
if}~{}\overline{\partial}\varphi=0;$ (4.14)
$\displaystyle\mathcal{L}_{\xi}\varphi=\frac{\overline{\partial}(\mathcal{L}_{X}Y-\overline{X}\lrcorner\varphi)}{2\sqrt{-1}},~{}{\rm
if}~{}\varphi=\overline{\partial}Y,Y\in T^{1,0}M.$

###### Proof.

By (4.1), we have

(4.15)
$\displaystyle\bar{\partial}\mathcal{L}_{\bar{X}}\varphi=0,~{}\forall~{}\varphi\in
H^{1}(M,J_{0},\Theta).$

By (4.13), we also have

(4.16) $\displaystyle\bar{\partial}\mathcal{L}_{X}\varphi=0.$

Thus,

$\overline{\partial}\mathcal{L}_{\xi}\varphi=0.$

By (4.13) we have

$\mathcal{L}_{X}\bar{\partial}Y=\bar{\partial}\mathcal{L}_{X}Y.$

Together with (4.1) we see that

$\displaystyle\mathcal{L}_{\xi}\bar{\partial}Y$
$\displaystyle=\frac{\mathcal{L}_{X}\bar{\partial}Y-\mathcal{L}_{\bar{X}}\bar{\partial}Y}{2\sqrt{-1}}$
$\displaystyle=\frac{\bar{\partial}\mathcal{L}_{X}Y-\bar{\partial}(\bar{X}\lrcorner\bar{\partial}Y)}{2\sqrt{-1}}.$

Hence (4.8) is true. ∎

By (4.8), we see that for any $\varphi=\overline{\partial}Y,Y\in T^{1,0}M$, it
holds

$[\mathcal{L}_{\xi}\varphi]=0.$

Thus

(4.17) $\displaystyle\mathcal{L}_{\xi}[\varphi]=[\mathcal{L}_{\xi}\varphi].$

This means that the eigenvalue $\theta_{i}$ depends only on the operator
$\mathcal{L}_{\xi}$ on $H^{1}(M,J_{0},\Theta)$. Hence, we prove

######  Proposition 4.9.

The index of $H$ depends only on the operator $\mathcal{L}_{\xi}$ on
$H^{1}(M,J_{0},\Theta)$.

###  4.3. Maximality of $\lambda(\cdot)$ associated to $Z$

First we recall a formula computed for the $W$-functional on a Fano manifold
$(M,J)$ in [39]. Let $Y$ be any HVF with ${\rm Im}(Y)$ generating a compact
1-ps $K_{Y}$ of holomorphic transformations on $(M,J)$. Then for any
$K_{Y}$-invariant Kähler form $\omega_{g}\in 2\pi c_{1}(M,J)$, the potential
$\theta_{Y}(\omega_{g})$ of $Y$ in (3.1) associated to $\omega_{g}$ is real.
Define an invariant for $Y$ by

$N_{Y}(c_{1}(M,J))=\int_{M}\theta_{Y}(\omega_{g})e^{\theta_{Y}(\omega_{g})}\omega_{g}^{n},$

which is independent of choice of $K_{Y}$-invariant $\omega_{g}\in 2\pi
c_{1}(M,J)$ [39]. Moreover, for any $K_{Y}$-invariant $\omega_{g}\in 2\pi
c_{1}(M,J)$, we have the following formula,

(4.18) $\displaystyle W(g,-\theta_{Y}(\omega_{g}))=(2\pi)^{-n}(nV-
N_{Y}(c_{1}(M,J))-F_{Y}^{J}(Y)),$

where $F_{\cdot}^{J}(\cdot)$ is the modified Futaki-invariant on $(M,J)$
introduced in [36]. $N_{Y}(c_{1}(M,J))+F_{Y}^{J}(Y)$ is also called the
$H(Y)$-invariant on $\eta_{r}(M,J)$ [39].

By Corollary 4.5, we know that
$X=\xi_{J_{\psi_{\tau}}}=J_{\psi_{\tau}}\xi+\sqrt{-1}\xi$ is a HVF on
$(M,J_{\psi_{\tau}})$ for any $\tau<<1$, where $\psi_{\tau}=\tau\psi\in
B(\epsilon)\cap Z$ and $J_{\psi_{\tau}}$ is associated to
$\varphi_{\tau}=\Phi(\psi_{\tau})$. Thus by (4.18), for any
$\omega_{g}=\omega_{g_{\tau,\chi}}$ with $\xi(\chi)=0$ in (3.2), it holds

(4.19) $\displaystyle W(g,-\theta_{X}(\omega_{g}))=(2\pi)^{-n}(nV-
N_{X}(c_{1}(M,J_{\psi_{\tau}}))-F_{X}^{J_{\psi_{\tau}}}(X)).$

Notice that $\theta=\theta_{X}(\omega_{KS})$ is independent of $\psi_{\tau}$.
Thus

$\displaystyle N_{X}(c_{1}(M,J_{\psi_{\tau}}))$
$\displaystyle=\int_{M}\theta_{X}(\omega_{KS})e^{\theta_{X}(\omega_{KS})}\omega_{KS}^{n}$
$\displaystyle=\int_{M}\theta e^{\theta}\omega_{KS}^{n}$ (4.20)
$\displaystyle=N_{X}(c_{1}(M,J_{0}))$

is independent of $\psi_{\tau}$.

Next we show that $F_{X}^{J_{\psi_{\tau}}}(X)$ is also independent of
$\psi_{\tau}$. In fact, we prove

######  Lemma 4.10.

(4.21) $\displaystyle
F_{X}^{J_{\psi_{\tau}}}(X)=F_{X}^{J_{0}}(X)=0,~{}\forall~{}\psi_{\tau}\in
B(\epsilon)\cap Z.$

###### Proof.

We use an argument in [19] to prove the lemma. Let
$\mathcal{J}_{K_{X}}(M,\omega_{KS})$ be a set of almost $K_{X}$-invariant
complex structures which are compatible with $\omega_{KS}$. Then for any
$J\in\mathcal{J}_{K_{X}}(M,\omega_{KS})$ it induces a Hermitian metric
$g_{J}$. As in [19], we define a modified Hermitian scalar curvature function
on $\mathcal{J}_{K_{X}}(M,\omega_{KS})$ by

$s_{\xi}(J)=s(J)-n+2\Box_{J}{\theta}-X({\theta})-{\theta},~{}\forall~{}J\in\mathcal{J}_{K_{X}}(M,\omega_{KS}),$

where $s(J)$ is the Hermitian scalar curvature of $g_{J}$ (cf. [12]) and
$\Box_{J}$ is the Lapalace operator induced by the Chern connection associated
to $g_{J}$. Now we consider a family of
$J_{\psi_{\tau}}\in\mathcal{J}_{K_{X}}(M,\omega_{KS})$. By [19, Proposition
3.1], we have the formula,

(4.22)
$\displaystyle\frac{d}{ds}\int_{M}2s_{\xi}(J_{\psi_{\tau}}){\theta}e^{{\theta}}\omega_{KS}^{n}=\Omega_{\xi}(\mathcal{L}_{\xi}J_{\psi_{\tau}},\dot{J}_{\psi_{\tau}}).$

where $\Omega_{\xi}$ is a non-degenerate 2-form on $\mathcal{J}_{T}(M,\omega)$
(cf. [12, 19]).

By Corollary 4.5, we have

$\mathcal{L}_{\xi}J_{\psi_{\tau}}=0$

and so we get

$\Omega_{\xi}(\mathcal{L}_{\xi}J_{\psi_{\tau}},\dot{J}_{\psi_{\tau}})=0.$

Thus by (4.22), the quantity

$\int_{M}2s_{\xi}(J_{\psi_{\tau}}){\theta}e^{{\theta}}\omega_{KS}^{n}$

is independent of $\tau$. As a consequence, for the normalized
$\widetilde{\theta}$ as in (3.27),

$\int_{M}2s_{\xi}(J_{\psi_{\tau}}){\tilde{\theta}}e^{{\tilde{\theta}}}\omega_{KS}^{n}$

is also independent of $\tau$. On the other hand, it is know in [36] that

$\int_{M}s_{\xi}(J_{\psi_{\tau}})\widetilde{\theta}e^{\widetilde{\theta}}\omega_{KS}^{n}=-F_{X}^{J_{\psi_{\tau}}}(X).$

Note that

$F_{X}^{J_{0}}(X)=0$

since $(\omega_{KS},X)$ is the KR soliton on $(M,J_{0})$. Hence,

$\int_{M}s_{\xi}(J_{\psi_{\tau}})\widetilde{\theta}e^{\widetilde{\theta}}\omega_{KS}^{n}=0,~{}\forall\tau<<1.$

As a consequence, we get

$F_{X}^{J_{\psi_{\tau}}}(X)=-\int_{M}s_{\xi}(J_{\psi_{\tau}})\widetilde{\theta}e^{\widetilde{\theta}}\omega_{KS}^{n}=0.$

∎

######  Proposition 4.11.

Let $\psi_{\tau}\in B(\epsilon)\cap Z$. Then for any $\chi\in C^{\infty}(M)$
with $\xi(\chi)=0$ it holds

$\lambda(g_{\tau,\chi})\leq\lambda(g_{KS}).$

###### Proof.

By (4.18) and (4.21), we have

$\displaystyle\lambda(g_{\tau,\chi})$ $\displaystyle\leq
W(g_{\tau,\chi},-\theta(\omega_{g_{\tau,\chi}}))$
$\displaystyle=(2\pi)^{-n}(nV-N_{X}(c_{1}(M,J_{\varphi_{\tau}})).$

Thus by (4.3), we get

$\displaystyle\lambda(g_{\tau,\chi})$ $\displaystyle\leq(2\pi)^{-n}(nV-
N_{X}(c_{1}(M,J_{0})))$ $\displaystyle=\lambda(g_{KS}).$

∎

Proposition 4.11 can be also proved by using the equivalent formula for the
modified Futaki-invariant in [47] as follows.

###### Another proof of Proposition 4.11.

According to the above proof of Proposition 4.11, we need to show that (4.21)
holds for $J_{\psi_{\tau}}$. In fact, as in the proof of [21, Lemma 2.1], by
the partial $C^{0}$-estimate, there is a large integer $L_{0}$ such that for
any integer $k$ the family of Fano manifolds $(M,g_{\tau,0})$
$(\tau\leq\epsilon$) can be embed into an ambient projection space
$\mathbb{C}P^{N_{k}}$ by normal orthogonal bases of
$H^{0}(M,kK_{M}^{-L_{0}},g_{\tau,0})$. Then by introducing two equivariant
Riemann-Roch formulas of $S_{1}$ and $S_{2}$ with $G=(S^{1})^{2}$-action by

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

where $X^{k}=(X_{\alpha}^{k})$ is the induced HVF as an element of Lie algebra
$sl(N_{k}+1,\mathbb{C})$, $F_{X}^{J_{\psi_{\tau}}}(X)$ is the leading term
$F_{0}$ (which is a multiple of $F_{1}$ for a smooth Fano variety) in the
following expansion of

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

Here

${\rm trace}(e^{\frac{sX^{k}+tv^{k}}{k}})=\int_{M}{\rm ch}^{G}(kL){\rm
Td}^{G}(M),$

and ${\rm ch}^{G}(kL)$ is the $G$-equivalent Chern character of multiple line
bundle $kL=kK_{M}^{-L_{0}}$ and ${\rm Td}^{G}(M)$ is the $G$-equivalent Chern
character of $M$ [1]. Since $K_{M_{\tau}}^{-1}$ is the restriction of
$\frac{1}{kL_{0}}\mathcal{O}(-1)$, where
$M_{\tau}=(M,J_{\tau})\subset\mathbb{C}P^{N_{k}}$, $S_{1}$ (or $S_{2}$ ) as a
restriction of derivatives of ${\rm trace}(e^{\frac{sX^{k}+tv^{k}}{k}})$ on
$kL$ is same for any $\tau$ as long as $X$ can be lifted as a HVF on
$M_{\tau}$. Thus we get

$F_{X}^{J_{\psi_{\tau}}}(X)=F_{X}^{J_{0}}(X)=0.$

∎

##  5\. Lojasiewicz inequality on the space of Kähler metrics

In this section we prove an inequality of Lojasiewicz type for the functional
$\nu(\cdot)$ on $\mathcal{U}_{\epsilon}$. We note that $\nu(\cdot)$ can be
defined on the $C^{k+4,\gamma}$ continuous space,

$W^{k+4,\gamma}(M)=\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\times
C^{k+4,\gamma}(M),$

where $k\geq 0$ is any integer. Clearly, there is a natural
$C^{k+4,\gamma}$-norm on $W^{k+4,\gamma}(M)$ by

$\|(\psi,\chi)\|_{C^{k+4,\gamma}}=\|\psi\|_{\theta}+\|\chi\|_{C^{k+4,\gamma}}.$

Let

$W_{\epsilon}^{k+4,\gamma}(M)=\\{(\psi,\chi)\in W^{k+4,\gamma}(M)|~{}\psi\in
B(\epsilon)\\}.$

Set an $\epsilon$-neighborhood of $(0,0)$ in $W_{\epsilon}^{k+4,\gamma}(M)$ by

$\mathcal{V}_{\epsilon}=\\{(\psi,\chi)\in
W_{\epsilon}^{k+4,\gamma}(M)|~{}\psi\in
B(\epsilon),\|\chi\|_{C^{k+4,\gamma}}<\epsilon\\}.$

We prove

######  Proposition 5.1.

There are $\epsilon>0$ and $\alpha\in[\frac{1}{2},1)$ such that for any
$a=(\varphi,\chi)\in\mathcal{V}_{\epsilon}$ it holds

(5.1) $\displaystyle\|\nabla\nu(a)\|_{L^{2}}\geq
c_{0}|\nu(a)-\nu(0)|^{\alpha},$

where $\|\cdot\|_{L^{2}}$ is taken for $\nabla\nu=(Q,R)\in W^{k,\gamma}(M)$
and $c_{0}>0$ is some small constant.

###### Proof.

We will follow the argument in [32]. By Remark 4.6, $W_{0}={\rm ker}(H)$ is
finitely dimensional. Then both of $W_{0}={\rm ker}(H)$ and

$W^{\perp}=\\{a\in W^{k+4,\gamma}(M)|~{}(a,b)_{\theta}=0,~{}\forall b\in
W_{0}\\}$

are closed sets of $W^{k+4,\gamma}(M)$. Since $H_{2}$ is elliptic, there is a
constant $C>0$ such that

(5.2) $\displaystyle\parallel H(\eta)\parallel_{L^{2}}\geq
C\parallel\eta\parallel_{L^{2}_{4}},~{}\forall~{}\eta\in W^{\perp}.$

Consider the project map,

$\Phi={\rm pr}_{W^{\perp}}\nabla\nu:\mathcal{U_{\epsilon}}\to W^{\perp}.$

Then $\Phi$ is analysis and it satisfies that

$\displaystyle\Phi(0,0)=0,D_{(0,0)}\Phi={\rm pr}_{W^{\perp}}\circ H.$

Thus

$D_{(0,0)}\Phi(\cdot):0\oplus W^{\perp}\to 0\oplus W^{\perp}$

is an isomorphic. By the implicity function theorem, there is a neighborhood
$U$ of $0$ in $W_{0}$ and a map $G(x):U\to W^{\perp}$ such that

$\nabla\nu(x+G(x))\in W_{0},~{}\forall~{}x\in U.$

Define a functional on $U$ by

$F(x)=\nu(x+G(x)).$

Then for any $x\in U$, $z\in W_{0}$, we have

$\frac{dG(x+tz)}{dt}|_{t=0}\in W^{\perp}.$

It follows that

$\displaystyle\frac{dF(x+tz+G(x+tz))}{dt}|_{t=0}$
$\displaystyle=(z+\frac{dG(x+tz)}{dt}|_{t=0},\nabla\nu(x+G(x)))_{\theta}$
$\displaystyle=(z,\nabla\nu(x+G(x)))_{\theta}$

Thus

$\nabla F(x)=\nabla\nu(x+G(x))\in W_{0},~{}\forall x\in U.$

Since $F(x)$ is analytic, by the classic Lojasiewicz inequality on $W_{0}$
(cf. [8, 32]), there is an $\alpha\in[\frac{1}{2},1)$ such that

$\displaystyle|\nabla\nu(x+G(x))|_{L^{2}}\geq c_{1}|\nabla\nu(x+G(x))|$ (5.3)
$\displaystyle\geq c_{2}|\nu(x+G(x))-\nu(0)|^{\alpha},~{}\forall~{}|x|<<1.$

By Definition 3.8, we have

$\|H_{(\psi,\chi)}\|_{C^{k,\gamma}}\leq
C,~{}\forall~{}(\psi,\chi)\in\mathcal{V}_{\epsilon}.$

Moreover,

(5.4) $\displaystyle\|H-H_{(\psi,\chi)}\|_{C^{k,\gamma}}<<1,~{}{\rm
as~{}long~{}as}~{}\|(\psi,\chi)\|_{C^{k+4,\gamma}}<<1.$

For any $a=(\psi,\chi)\in\mathcal{V}_{\epsilon}$, we write $a=x+G(x)+y$ for
some $y\in W^{\perp}$. Thus there exists a small $\epsilon$ such that for any
$a=(\psi,\chi)\in\mathcal{V}_{\epsilon}$ it holds

$\displaystyle\nabla\nu(a)$ $\displaystyle=\nabla\nu(x+G(x)+y)$
$\displaystyle=\nabla\nu(x+G(x))+\int_{0}^{1}\delta_{y}\nabla\nu(x+G(x)+sy)ds$
$\displaystyle=\nabla\nu(x+G(x))+\delta_{y}\nabla\nu(x+G(x))+\int_{0}^{1}[\delta_{y}\nabla\nu(x+G(x)+sy)-\delta_{y}\nabla\nu(x+G(x))]ds$
$\displaystyle=\nabla
F(x)+\delta_{y}\nabla\nu(0)+\int_{0}^{1}[\delta_{y}\nabla\nu(x+G(x)+sy)-\delta_{y}\nabla\nu(x+G(x))]ds+o(\parallel
y\parallel_{W^{2}_{4}})$ (5.5) $\displaystyle=\nabla F(x)+H(y)+o(\parallel
y\parallel_{W^{2}_{4}}).$

Since $\nabla F(x)$ is perpendicular to $H(y)$ with respect to
$(\cdot,\cdot)_{\theta}$, and $H$ is nondegenerate on $W^{\perp}$ by (5.2), we
get

(5.6) $\displaystyle\|\nabla\nu(a)\|^{2}_{L^{2}}\geq\|\nabla
F(x)\|^{2}_{L^{2}}+c_{3}\|y\|^{2}_{W^{2}_{4}},$

where $c_{3}>0$ is some small constant.

Similarly, by (5.4), we have

$\displaystyle\nu(a)$ $\displaystyle=\nu(x+G(x)+y)$
$\displaystyle=\nu(x+G(x))+\int_{0}^{1}<\nabla\nu(x+G(x)+sy),y>ds$
$\displaystyle=\nu(x+G(x))+\frac{1}{2}<\delta_{y}\nabla\nu(x+G(x)+y),y>$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}s<\delta_{y}\nabla\nu(x+G(x)+sty)-\delta_{y}\nabla\nu(x+G(x)+y),y>dtds$
$\displaystyle=F(x)+\frac{1}{2}<H(y),y>$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}s<\delta_{y}\nabla\nu(x+G(x)+sty)-H(y),y>dtds+o(\parallel
y\parallel_{W^{2}_{4}}^{2})$ (5.7)
$\displaystyle=F(x)+\frac{1}{2}<H(y),y>+o(\parallel
y\parallel_{W^{2}_{4}}^{2}).$

Notice that

$\displaystyle|<H(y),y>|$ $\displaystyle\leq\parallel
y\parallel_{L^{2}}|\parallel H(y)\parallel_{L^{2}}$ $\displaystyle\leq
C_{1}\parallel y\parallel_{L^{2}}\parallel y\parallel_{C^{k+4,\gamma}}$
$\displaystyle\leq C_{2}\parallel y\parallel_{L^{2}}^{2}$ $\displaystyle\leq
C\parallel y\parallel_{W^{2}_{4}}^{2}.$

It follows that

(5.8) $\displaystyle|\nu(a)-\nu(0)|\leq|F(x)-F(0)|+C\parallel
y\parallel_{W^{2}_{4}}^{2}.$

Hence combining (5.6) and (5.8) together with (5), we obtain

$\displaystyle\parallel\nabla\nu(a)\parallel_{L^{2}}^{2}$ $\displaystyle\geq
c_{2}|F(x)-F(0)|^{2\alpha}+c_{3}\parallel y\parallel_{W^{2}_{4}}^{2}$
$\displaystyle\geq c_{4}(|F(x)-F(0)|+\parallel
y\parallel_{W^{2}_{4}}^{2})^{2\alpha}$ $\displaystyle\geq
c_{0}|\nu(a)-\nu(0)|^{2\alpha}.$

This proves (5.1).

∎

Recall the operator $\mathcal{N}(\cdot)$ in (3.8) from ${\rm Sym}^{2}(T^{*}M)$
to itself. Then we can rewrite (3.5) as

$(\nabla\nu(a_{0}),(\psi,\chi))_{\theta}=-\frac{1}{2}(\mathcal{N}(L(a_{0})),DL_{a_{0}}((\psi,\chi)))_{L^{2}(L(a_{0}))},~{}\forall~{}a_{0}\in\mathcal{U}_{\epsilon}.$

Thus there is a dual operator $(DL_{a_{0}})^{*}$ of $DL_{a_{0}}$ with respect
to the inner product $(\cdot,\cdot)_{\theta}$ such that

$((DL_{a_{0}})^{*}\mathcal{N}(L(a_{0})),(\psi,\chi))_{\theta}=(\mathcal{N}(L(a_{0}),DL_{a_{0}}((\psi,\chi)))_{L^{2}(L(a_{0}))}.$

As a consequence, we have

$\nabla\nu(a_{0})=-\frac{1}{2}(DL_{a_{0}})^{*}\mathcal{N}(L(a_{0})).$

Hence,

(5.9)
$\displaystyle(\nabla\nu(a_{0}),\nabla\nu(a_{0}))_{\theta}=\frac{1}{4}(\mathcal{N}(L(a_{0}),DL_{a_{0}}(DL_{a_{0}})^{*}\mathcal{N}(L(a_{0})))_{L^{2}(L(a_{0}))}.$

The following is a generalization of Lemma 3.6.

######  Lemma 5.2.

Let $k>2$ be an integer, and $\epsilon$ a small constant. Then there exists a
constant $C=C(\epsilon)>0$ such that

(5.10)
$\displaystyle((DL_{a})^{*}\mathcal{N}(L(a)),(DL_{a})^{*}\mathcal{N}(L(a)))_{\theta}\leq
C(\mathcal{N}(L(a),\mathcal{N}(L(a))))^{\gamma}_{L^{2}(L(a))},~{}\forall
a\in\mathcal{V}_{\epsilon}.$

Here $\gamma=\frac{k-2}{k-1}<1$.

###### Proof.

Let $S=(DL_{a})^{*}(DL_{a})$ be an operator on the Hilbert space

$\mathcal{H}=\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\times L^{2}(M),$

whose domain $D(S)$ contains $\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\times
C^{\infty}(M)$. As same as the operator in (3.15), $S$ is a self-adjoint
fourth-order non-negative elliptic operator. Thus by the spectral theorem, for
integer $k>0$, $x\in D(S)$, it holds

(5.11)
$\displaystyle(S^{k}x,x)_{\theta}=\int_{0}^{\infty}\lambda^{k}dE_{x,x}(\lambda).$

By the Hölder inequality and (5.11) we get

(5.12)
$\displaystyle(S^{2}x,x)_{\theta}\leq(Sx,x)_{\theta}^{\frac{k-2}{k-1}}(S^{k}x,x)_{\theta}^{\frac{1}{k-1}}.$

For any $a\in\mathcal{V}_{\epsilon}$, we decompose $\mathcal{N}(L(a))$ into

(5.13) $\displaystyle\mathcal{N}(L(a))=DL_{a}x+y,$

where $x\in\mathcal{H}^{0,1}(M,T^{1,0}M)\times C^{\infty}(M)$ and $y\in{\rm
Im}(DL_{a})^{\perp}$. Since

$(z,(DL_{a})^{*}(y))_{\theta}=(DL_{a}(z),y)_{L^{2}(L(a))}=0,~{}\forall~{}z\in\mathcal{H}^{0,1}(M,T^{1,0}M)\times
C^{\infty}(M),$ $(DL_{a})^{*}(y)=0.$

Thus by (5.12) and the fact that $S$ is self-adjoint, we have

$\displaystyle((DL_{a})^{*}\mathcal{N}(L(a)),(DL_{a})^{*}\mathcal{N}(L(a)))_{\theta}$
$\displaystyle=((DL_{a})^{*}DL_{a}x,(DL_{a})^{*}DL_{a}x)_{\theta}$
$\displaystyle=(Sx,Sx)_{\theta}$ $\displaystyle=(S^{2}x,x)_{\theta}$
$\displaystyle\leq(Sx,x)_{\theta}^{\gamma}(S^{k}x,x)_{\theta}^{1-\gamma}$
$\displaystyle=(DL_{a}x,DL_{a}x)_{L^{2}(L(a))}^{\gamma}(S^{k}x,x)_{\theta}^{1-\gamma}$
(5.14)
$\displaystyle\leq\|\mathcal{N}(L(a)))\|_{L^{2}(L(a))}(S^{k}x,x)_{\theta}^{1-\gamma}.$

On the other hand,

$\displaystyle(S^{k}x,x)_{\theta}$ $\displaystyle=(S^{k-1}x,Sx)_{\theta}$
$\displaystyle=(S^{k-2}(DL_{a})^{*}DL_{a}x,(DL_{a})^{*}DL_{a}x)_{\theta}$
$\displaystyle=(S^{k-2}(DL_{a})^{*}\mathcal{N}(L(a)),(DL_{a})^{*}\mathcal{N}(L(a)))_{\theta}.$

Then

(5.15) $\displaystyle(S^{k}x,x)_{\theta}^{1-\gamma}\leq C,$

as long as $a\in\mathcal{V}_{\epsilon}$. Hence, combining (5) and (5.15), we
derive (5.10). ∎

######  Remark 5.3.

In order to apply the spectral theorem, the fourth-order operator $S$ should
be self-adjoint as in (3.15). In the other words, we shall define the domain
$D(S)$ of S. By decomposing $S$ into

$S(\varphi,\chi)=(S_{11}\varphi+S_{12}\chi,S_{21}\varphi+S_{22}\chi),$

it suffices to consider the domains $D(S_{12})$ and $D(S_{22})$. On the other
hand, by (3.15),

$S_{22}=(\sqrt{-1}\partial_{J_{\varphi}}\bar{\partial}_{J_{\varphi}})^{*}(\sqrt{-1}\partial_{J_{\varphi}}\bar{\partial}_{J_{\varphi}})$

is a fourth-order elliptic self-adjoint operator. Thus
$D(S_{22})=W^{2}_{4}(M)$. Note that $S_{12}$ is a second order operator.
Hence,

$D(S)=\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\times W^{2}_{4}(M)$

and so $S$ is self-adjoint on $D(S)$.

######  Corollary 5.4.

There are constant $c_{0}>0$ and number $\alpha^{\prime}\in(\frac{1}{2},1)$
such that

(5.16) $\displaystyle\|\mathcal{N}(L(a))\|_{L^{2}(L(a))}\geq
c_{0}\|\lambda(L(a))-\lambda(L(0))\|^{\alpha^{\prime}},\forall
a\in\mathcal{V}_{\varepsilon}.$

###### Proof.

By (5.9), Lemma 5.2 and Proposition 5.1, we have

$\displaystyle\|\mathcal{N}(L(a))\|_{L^{2}(L(a))}$ $\displaystyle\geq
C\|\nabla\nu(a)\|_{L^{2}}^{\frac{1}{\gamma}}$ $\displaystyle\geq
C\|\nu(a)-\nu(0)\|^{\frac{\alpha}{\gamma}}$
$\displaystyle=C\|\lambda(L(a))-\lambda(L(0))\|^{\frac{\alpha}{\gamma}}.$

Set $\alpha^{\prime}=\frac{\alpha}{\gamma}$. Then
$\alpha^{\prime}\in(\frac{1}{2},1)$ when $k>>1$. Hence, (5.16) is true. ∎

##  6\.  Proof of Theorem 0.1

In this section, we prove Theorem 0.1. Let $\widetilde{f}(t)$ be the minimizer
of $W$-functional as in (3.5) for the solution $\widetilde{g}(t)$ of KR flow
(0.1). Let $X(t)$ be a family of gradient vector fields defined by

$X(t)=\frac{1}{2}{\rm grad}_{\widetilde{g}(t)}\widetilde{f}(t),$

which generates a family of differential transformations $F(t)\subseteq{\rm
Diff}(M)$. Then $g(t)=F(t)^{*}\widetilde{g}(t)$ is a solution of the following
modified Ricci flow,

(6.1) $\displaystyle\frac{\partial g(t)}{\partial t}=-{\rm
Ric}(g(t))+g(t)+{\rm Hess}_{t}(f(t)),$

where $f(t)=F(t)^{*}\widetilde{f}(t)$ is the minimizer of $W$-functional for
${g}(t)$. It follows that

(6.2) $\displaystyle\|\dot{g}(t)\|_{L^{2}(t)}=\|\mathcal{N}(t)\|_{L^{2}(t)}.$

Here $\mathcal{N}(t)={\rm Ric}(g(t))-g(t)+{\rm Hess}_{t}(f(t))$ just as one in
(3.8).

Let $\omega(t)=F(t)^{*}\widetilde{\omega}(t)$. Then $\omega(t)$ is just the
Kähler form of $g(t)$. Thus by (6.1), we get

$\displaystyle\frac{d\omega(t)}{dt}=-{\rm
Ric}(\omega(t))+\omega(t)+\sqrt{-1}\partial_{J(t)}\bar{\partial}_{J(t)}(f(t))$

and

(6.3) $\displaystyle\omega(t)(\cdot,\frac{dJ(t)}{dt})=D_{t}f(t),$

where $D_{t}f$ denotes the anti-Hermitian part of ${\rm Hess}(f)$. Hence,

$\displaystyle\|\dot{J}(t)\|_{L^{2}(t)}$
$\displaystyle=\|D_{t}f(t)\|_{L^{2}(t)}$
$\displaystyle\leq\|\mathcal{N}(t)\|_{L^{2}(t)}$ (6.4)
$\displaystyle=\|\dot{g}(t)\|_{L^{2}(t)}.$

We first prove the following convergence theorem.

######  Theorem 6.1.

There exists a small $\epsilon$ such that for any
$(\tau,\chi)\in\mathcal{V}_{\epsilon}$ with $\psi_{\tau}=\sum\tau_{i}e_{i}\in
Z$ and $\xi(\chi)=0$ the flow (6.1) with the initial metric $g=L(\tau,\chi)$
converges smoothly to a KR soliton $(M,J_{\infty},g_{\infty})$. Moreover, the
convergence is fast in the polynomial rate.

###### Proof.

We need to prove that the normalized flow $g(t)$ of (6.1) is uniformly bounded
in $C^{k+2,\gamma}$-norm. Fix a small number $\delta_{0}$ we consider

(6.5) $\displaystyle
T=T_{\tau,\chi}=\sup\\{t|~{}\|g(t)-g_{KS}\|_{C^{k+2,\gamma}}<\delta_{0},~{}\|J(t)-J_{0}\|_{C^{k+2,\gamma}}<\delta_{0}\\}.$

Then it suffices to show that $T=\infty$. By the stability of Ricci flow for
the short time, $T_{\tau,\chi}\geq T_{0}>0.$ In fact, for the KR flow (6.1),
$T_{0}$ can be made any large as long as $\tau$ and $\chi$ are small enough.

By the Kuranishi theorem for the completeness of deformation space of complex
structures [25], for any $t<T$, there is a $K(t)\in{\rm Diff(M)}$ closed to
the identity such that

(6.6) $\displaystyle
K(t)^{*}J(t)=J_{\varphi_{t}},~{}\varphi_{t}=\Phi(t)=~{}\Phi(\psi_{t})$

for some $\psi_{t}\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ with
$\|\psi_{t}\|<\epsilon_{0}$, where $\epsilon_{0}=\epsilon$ as chosen in (2.6)
and (5.16) in Corollary 5.4. In fact, by the construction of $K(t)$ in [25,
Theorem 3.1], we have the estimate

$\displaystyle\|\psi_{t}\|_{L^{2}(M)}=O(\|\varphi_{t}\|_{L^{2}(M)})\leq\epsilon_{0},$
(6.7) $\displaystyle
K(t)^{*}\omega(t)=(1+O(\|\varphi_{t}\|_{L^{2}(M)}))\omega(t).$

Thus there is a smooth function $\chi_{t}$ such that

(6.8)
$\displaystyle\omega_{K(t)^{*}g(t)}=K(t)^{*}\omega(t)=\omega_{KS}+\sqrt{-1}\partial_{J_{\varphi_{t}}}\bar{\partial}_{J_{\varphi_{t}}}\chi_{t}.$

Here $\chi_{t}$ can be normalized by

$\int_{M}\chi_{t}\omega_{KS}^{n}=0.$

It follows that

$\displaystyle\|\Delta_{g_{KS}}\chi_{t}\|_{C^{k+2,\gamma}}$ $\displaystyle\leq
2\|\Delta_{g_{\psi_{t},0}}\chi_{t}\|_{C^{k+2,\gamma}}$ $\displaystyle=2\|{\rm
trace}_{\omega_{KS}}(K(t)^{*}\omega(t))-n\|_{C^{k+2,\gamma}}$ (6.9)
$\displaystyle\leq 2\|(1+\delta_{1}){\rm
trace}_{g_{KS}}(g(t))-n\|_{C^{k+2,\gamma}}\leq\delta_{2}(\delta_{0})<<1.$

Hence, we get

(6.10) $\displaystyle\|\chi_{t}\|_{C^{k+4,\gamma}(M)}\leq\epsilon_{0}$

as long as $\delta_{2}$ is chosen small enough. Note that
$\|\mathcal{N}(g)\|_{L^{2}(g)}$ is invariant under the action of ${\rm
Diff}(M)$. Therefore, by Corollary 5.4, we obtain the following Lojasiewicz
inequality,

(6.11)
$\displaystyle\|\mathcal{N}(g(t))\|_{L^{2}(g(t))}=\|\mathcal{N}(L(\psi_{t},\chi_{t}))\|_{L^{2}(L(\psi_{t},\chi_{t}))}\geq
c_{0}\|\lambda(g(t))-\lambda(g_{KS})\|^{\alpha},$

where $\alpha\in(\frac{1}{2},1)$.

Note that $\widetilde{g}(t)$ are all $K_{X}$-invariant for any $t\geq 0$. Then
by Proposition 4.11, we have

$\displaystyle\lambda(\widetilde{g}(t))\leq\lambda(g_{KS}),~{}t>0.$

and so,

(6.12) $\displaystyle\lambda(g(t))\leq\lambda(g_{KS}),~{}t>0.$

Thus by (6.1), for any $\beta>2-\frac{1}{\alpha}$, we get

$\displaystyle\frac{d}{dt}[\lambda(g_{KS})-\lambda(g(t))]^{1-(2-\beta)\alpha}$
$\displaystyle=-(1-(2-\beta)\alpha)[\lambda(g_{KS})-\lambda(g(t))]^{-(2-\beta)\alpha}\frac{d}{dt}\lambda(g(t))$
(6.13)
$\displaystyle=-\frac{(1-(2-\beta)\alpha)}{2}[\lambda(g_{KS})-\lambda(g(t))]^{-(2-\beta)\alpha}\int_{M}<\mathcal{N}(t),\mathcal{N}(t)>e^{-f_{t}}\omega_{t}^{n}.$

By (6.11), it follows that

$\displaystyle\frac{d}{dt}[\lambda(g_{KS})-\lambda(g(t))]^{1-(2-\beta)\alpha}$
$\displaystyle\leq-C(\int_{M}<\mathcal{N}(t),\mathcal{N}(t)>e^{-f_{t}}\omega_{t}^{n})^{\frac{\beta}{2}}$
$\displaystyle=-C\parallel\dot{g}(t)\parallel^{\beta}_{L^{2}(t)}.$

As a consequence,

(6.14)
$\displaystyle\int_{t_{1}}^{t_{2}}\parallel\dot{g}(t)\parallel^{\beta}_{L^{2}(t)}dt$
$\displaystyle\leq
C(\beta)[\lambda(g_{KS})-\lambda(g(t_{1}))]^{1-(2-\beta)\alpha}$

Hence, for any $(\tau,\chi)\in\mathcal{V}_{\epsilon}$, we prove

$\int_{1}^{T}\|\dot{g}(t)\|_{L^{2}(t)}^{\beta}dt\leq
C(\beta)[\lambda(g_{KS})-\lambda(g(0))]^{1-(2-\beta)\alpha}\leq\delta_{3}(\epsilon).$

Similarly, it also holds

$\int_{1}^{T}\|\dot{J}(t)\|_{L^{2}(t)}^{\beta}dt<\delta_{4}(\epsilon).$

By (6) and (6.11), we see that

$\frac{d}{dt}[\lambda(g_{KS})-\lambda(g(t))]^{1-(2-\beta)\alpha}\leq-C^{\prime}(\lambda(g_{KS})-\lambda(g(t)))^{\alpha\beta}.$

Then

$\frac{d}{dt}[\lambda(g_{KS})-\lambda(g(t))]^{1-2\alpha}\geq C>0.$

It follows that

(6.15) $\displaystyle\lambda(g_{KS})-\lambda(g(t))\leq
C^{\prime\prime}(t+1)^{-\frac{1}{2\alpha-1}}.$

On the other hand, by the interpolation inequalities for tensors, for
$\beta\in(2-\frac{1}{\alpha},1)$ and any integer $p\geq 1$, there exists
$N(p)$ which is independent of $t$, such that

$\displaystyle\|\dot{g}(t)\|_{L^{2}_{p}(t)}$ $\displaystyle\leq
C(p)\|\dot{g}(t)\|^{\beta}_{L^{2}(t)}\||\mathcal{N}(t)|^{1-\beta}_{L^{2}_{N(p)}(t)}$
$\displaystyle\leq
C(p)\parallel\dot{g}(t)\parallel^{\beta}_{L^{2}(t)},~{}\forall~{}t<T.$

Similarly, we have

(6.16) $\displaystyle\|\dot{J}(t)\|_{L^{2}_{p}(t)}\leq
C(p)\parallel\dot{J}(t)\parallel^{\beta}_{L^{2}(t)},~{}\forall~{}t<T.$

Hence, by (6.14) and (6.15), we derive

$\displaystyle\parallel g(t_{1})-g(t_{2})\parallel_{C^{k+2,\gamma}}$
$\displaystyle\leq\int_{t_{1}}^{t_{2}}\parallel\dot{g}(t)\parallel_{C^{k+2,\gamma}}$
$\displaystyle\leq
C\int_{t_{1}}^{t_{2}}\parallel\dot{g}(t)\parallel_{L^{2}_{p}(t)}$
$\displaystyle\leq
C\int_{t_{1}}^{t_{2}}\parallel\dot{g}(t)\parallel^{\beta}_{L^{2}(t)}$
$\displaystyle\leq
C(\beta)[\lambda(g_{KS})-\lambda(g(t_{1}))]^{1-(2-\beta)\alpha}$ (6.17)
$\displaystyle\leq
C(\beta)(t_{1}+1)^{-\frac{1-(2-\beta)\alpha}{2\alpha-1}},~{}\forall~{}t_{2}>t_{1}.$

Therefore, we can choose a large $T_{0}$ such that

$\|g(t)\|_{C^{k+2,\gamma}}<\delta_{0},~{}\forall~{}t\geq T_{0}.$

Similarly we can use (6) and (6.16) to prove that

(6.18) $\displaystyle\|J(t)\|_{C^{k+2,\gamma}}<\delta_{0}.~{}\forall~{}t\geq
T_{0}.$

As a consequence, we prove that $T=\infty$. Furthermore, we can show that the
limit of $g(t)$ is a KR soliton by (6.2) (cf. [37, 38]). The convergence speed
comes from (6).

∎

######  Remark 6.2.

By (6.15) and a result of Dervan-Székelyhidi [11] (also see [45]), we know
that

(6.19) $\displaystyle\sup_{\omega_{g}\in 2\pi
c_{1}(M,J_{\psi_{\tau}})}\lambda(g)=\lim_{t}\lambda(g(t))=\lambda(g_{KS}).$

In particular, (6.12) holds for KR flow (0.1) with any initial metric in $2\pi
c_{1}(M,J_{\psi_{\tau}})$. Thus the condition $\xi(\chi)=0$ can be removed in
Theorem 6.1 according to the above proof 333The convergence part also comes
from the Hamilton-Tian conjecture and the uniqueness result in [16, 45]. .

We also remark that the limit complex structure $J_{\infty}$ in Theorem 6.1
may be different with the original one $J_{\psi_{\tau}}$. But the soliton VF
of $(M_{\infty},J_{\infty})$ must be conjugate to $X$. In fact, we have the
following analogy of [46, Proposition 5.10] for the uniqueness of soliton VFs
.

######  Lemma 6.3.

There is a small $\delta_{0}$ such that for any KR soliton
$(M,J^{\prime},g_{KS}^{\prime})$ with

(6.20) $\displaystyle{\rm
dist}_{CG,C^{3}}((M^{\prime},g_{KS}^{\prime}),(M,g_{KS}))\leq\delta_{0},$

$X$ can be lifted to a HVF on $(M,J^{\prime})$ so that it is a soliton VF of
$(M,J^{\prime},g_{KS}^{\prime})$. Moreover,

(6.21) $\displaystyle\lambda(g_{KS}^{\prime})=\lambda(g_{KS}),$

###### Proof.

As in the second proof of Proposition 4.11, by the partial $C^{0}$-estimate,
one can embed any KR soliton $(M,J^{\prime};\omega_{KS}^{\prime},X^{\prime})$
with satisfying (6.20) into an ambient projection space $\mathbb{C}P^{N}$.
Then the lemma turns to prove that there is some $\sigma\in{\rm
U}(N+1,\mathbb{C})$ such that

$X^{\prime}=\sigma\cdot X\cdot\sigma^{-1}.$

Thus one can use the argument in [46, Proposition 5.10] to get the compactness
of soliton VFs in ${\rm sl}(N+1,\mathbb{C})$ and prove the lemma by the
uniqueness result of soliton VFs [46, Proposition 2.2].

On the other hand, by (6.20), there is a $F\in{\rm Diff(M)}$ such that

$\|F^{*}J^{\prime}-J_{0}\|_{C^{3}}\leq\delta_{0}^{\prime}.$

Then by the Kuranishi’s theorem, there are $K\in{\rm Diff(M)}$ and
$\tau^{\prime}\in B(\epsilon)$ such that

$K^{*}(F^{*}J^{\prime})=J_{\psi_{\tau^{\prime}}}.$

Thus, we may assume that $(M,\omega_{KS}^{\prime})$ is a KR soliton with
respect to $J^{\prime}=J_{\psi_{\tau^{\prime}}}$. Since both of
$\theta_{X}(\omega_{KS}^{\prime})$ and $\theta_{X}(\omega_{KS})$ are
minimizers of the $W$-functional respect to $\omega_{KS}^{\prime}$ and
$\omega_{KS}$, by (4.19)-(4.21) in Section 4, we obtain

$\lambda(g_{KS}^{\prime})=W(g_{KS}^{\prime},-\theta_{X}(\omega_{KS}^{\prime}))=W(g_{KS},-\theta_{X}(\omega_{KS}))=\lambda(g_{KS}).$

∎

By Lemma 6.3, we can finish the proof of Theorem 0.1.

###### Proof of Theorem 0.1.

By the uniqueness of limits of KR flow [16, 45], we may assume that
$\widetilde{\omega}_{0}=g=L(\tau,\chi)$ with
$(\tau,\chi)\in\mathcal{U}_{\epsilon}$ and $\xi(\chi)=0$. Thus the convergence
part of theorem comes from Theorem 6.1. Moreover, by Lemma 6.3, $X$ can be
lifted to a HVF on $(M,J_{\infty})$ so that it is a soliton VF of
$(M,J_{\infty},g_{\infty})$ and (0.2) is satisfied. (0.2) also comes from
(6.15) as well as the convergence speed of $g(t)$ with the polynomial rate
comes from (6).

∎

##  7\. Applications of Theorem 0.1

In this section, we first prove Theorem 0.2. Recall that a special
degeneration  on a Fano manifold $M$ is a normal variety $\mathcal{M}$ with a
$\mathbb{C}^{*}$-action which consists of three ingredients [34]:

  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 $\mathcal{L}$ on $\mathcal{M}$ such that
$\mathcal{L}|_{\pi^{-1}(t)}$ is isomorphic to $K_{M}^{-r}$ for some integer
$r>0$ and any $t\neq 0$;

  3. (3)

a center $M_{0}=\pi^{-1}(t)$ which is a $Q$-Fano variety.

The following definition can be found in [47] (also see [42, 3, 10], etc.).

######  Definition 7.1.

A Fano manifold $M$ is modified $K$-semistable with respect to a HVF $X$ in
$\eta(M)$ if the modified Futaki-invariant $F(\cdot)\geq 0$ for any special
degeneration $\mathcal{M}$ associated to a 1-ps $\sigma_{t}$ induced by a
$\mathbb{C}^{*}$\- action, which communicates with the one-parameter subgroup
$\sigma_{t}^{X}$ associated to the lifting of $X$ on $\mathcal{M}$. In
addition that $F(\cdot)=0$ if only if $\mathcal{M}\cong M\times\mathbb{C}$,
$M$ is called modified $K$-polystable with respect to $X$.

For a Fano manifold $(M,J_{\psi_{\tau}})$ in the deformation space in Theorem
0.1, we introduce

######  Definition 7.2.

A Fano manifold $(M,J_{\psi_{\tau}})$ is called modified $K$-semistable
(modified $K$-polystable) in the deformation space of complex structures on
$(M,J_{0})$ which admits a KR soliton $(g_{KS},X)$ if $X$ can be lifted to a
HVF on $(M,J_{\psi_{\tau}})$ and $(M,J_{\psi_{\tau}})$ is modified
$K$-semistable (modified $K$-polystable) with respect to $X$.

###### Proof of Theorem 0.2.

By Lemma 6.3, the soliton VF on $(M,J_{\psi_{\tau}})$ is conjugate to $X$.
Thus we need to prove the sufficient part since the modified $K$-polystability
is a necessary condition for the existence of KR solitons (cf. [3]).

By Theorem 0.1 and Lemma 6.3, the KR flow $g(t)$ converges smoothly to a KR
soliton $(M,J_{\infty},g_{\infty})$ with respect to $X$ for the initial Kähler
metric $g_{\tau,0}$ on $(M,J_{\psi_{\tau}})$. Then as in the proof of [21,
Lemma 2.1], by the partial $C^{0}$-estimate, there is a large integer $L_{0}$
such that the family of Kähler manifolds $(M,g(t))$ and the limit manifold
$(M,J_{\infty},g_{\infty})$ can be embed into an ambient projection space
$\mathbb{C}P^{N}$ by normal orthogonal bases of $H^{0}(M,K_{M}^{-L_{0}},g(t))$
and the images $\tilde{M}_{t}$ of $(M,g(t))$ converges smoothly to the image
$\tilde{M}_{\infty}$ of $(M,J_{\infty},g_{\infty})$ in $\mathbb{C}P^{N}$. Thus
there is a family of group $\sigma_{t}\in{\rm SL}(N+1,\mathbb{C})$ such that

$\sigma_{t}(\tilde{M}_{t_{0}})=\tilde{M}_{t},~{}\forall t\geq t_{0}.$

Moreover, we may assume that

$\sigma_{t}\cdot\sigma_{s}^{X}=\sigma_{s}^{X}\cdot\sigma_{t},$

by modifying the base of $H^{0}(M,K_{M}^{-L_{0}},g(t))$ after a transformation
in ${\rm SL}(N+1,\mathbb{C})$ (cf. [10]). Here $X$ can be lifted to a HVF in
$\mathbb{C}P^{N}$ such that it is tangent to each $\tilde{M}_{t}$.

Set a reductive subgroup of ${\rm SL}(N+1,\mathbb{C})$ by

$G=\\{\sigma\in{\rm
SL}(N+1,\mathbb{C})|~{}\sigma\cdot\sigma_{s}^{X}=\sigma_{s}^{X}\cdot\sigma\\}.$

Then

$G_{c}=\\{\sigma\in
G|~{}\sigma(\tilde{M}_{\infty})=\tilde{M}_{\infty}\\}\subset{\rm
Aut}_{r}(\tilde{M}_{\infty}),$

where ${\rm Aut}_{r}(\tilde{M}_{\infty})$ is a reductive subgroup of ${\rm
Aut}(\tilde{M}_{\infty})$ which contains $\sigma_{s}^{X}$. Note that
$M_{\infty}$ admits a KR soliton and $X$ is a center of
$\eta_{r}(\tilde{M}_{\infty})$ by [36]. Thus $G_{c}={\rm
Aut}_{r}(\tilde{M}_{\infty})$ is a reductive subgroup of $G$. By GIT, there is
a $\mathbb{C}^{*}$-action in $G$ which induces a smooth degeneration
$\mathcal{M}$ on the Fano manifold $\tilde{M}_{t_{0}}$ with the center
$\tilde{M}_{\infty}$. Clearly, this $\mathbb{C}^{*}$-action communicates with
$\sigma_{s}^{X}$. It follows that the corresponding modified Futaki-invariant
$F(\cdot)=0$ by the fact $\tilde{M}_{\infty}$ admitting a KR soliton. Hence
$\mathcal{M}$ must be a trivial degeneration and we get $M\cong M_{\infty}$.
This proves that $M$ admits a KR soliton.

∎

Let $T$ be a torus subgroup of ${\rm Aut}_{r}(M,J_{0})$ which contains the
soliton VF $X$. Then by Proposition 4.9 and Corollary 4.5, we see that any
$T$-equivalent subspace $H_{X}^{1}(M,J_{0},\Theta)\subset
H^{1}(M,J_{0},\Theta)$ is included in $Z$, which is invariant under ${\rm
Aut}_{r}(M,J_{0})$. Thus by Theorem 0.2 we actually prove the following
existence result for KR solitons in the $T$-equivalent deformation space.

######  Corollary 7.3.

Let $(M,J_{0},\omega)$ be a Fano manifold which admits a KR soliton
$(\omega_{KS},X)$. Then there exists a small $\epsilon$-ball
$B_{\epsilon}(0)\subset H_{T}^{1}(M,J_{0},\Theta)$ such that for any
$\psi_{\tau}\in B_{\epsilon}(0)$ the Fano manifold $(M,J_{\psi_{\tau}})$
admits a KR soliton if and only if $(M,J_{\psi_{\tau}})$ is modified
$K$-polystable.

######  Remark 7.4.

By using the deformation theory, Inoue proved the sufficient part of Corollary
7.3 in sense of the GIT-polystability via the group ${\rm Aut}(M,J_{0})$ [19,
Proposition 3.8]. But it is still unknown whether the necessary part is true
in sense of the GIT-polystability [19, Postscript Remark 1].

Next we prove the following uniqueness result for KR-solitons in the closure
of the orbit by diffeomorphisms.

######  Theorem 7.5.

Let $\\{\omega_{i}^{1}\\}$ and $\\{\omega_{i}^{2}\\}$ be two sequences of
Kähler metrics in $2\pi c_{1}(M,J)$ which converge to KR-solitons
$(M_{\infty}^{1},\omega_{KS}^{1})$ and $(M_{\infty}^{2},\omega_{KS}^{2})$ in
sense of Cheeger-Gromov, respectively. Suppose that

(7.1)
$\displaystyle\lambda(\omega_{KS}^{1})=\lambda(\omega_{KS}^{2})=\sup\\{\lambda(\omega_{g^{\prime}})|~{}\omega_{g^{\prime}}\in
2\pi c_{1}(M,J)\\}.$

Then $M_{\infty}^{1}$ is biholomorphic to $M_{\infty}^{2}$ and
$\omega_{KS}^{1}$ is isometric to $\omega_{KS}^{2}$.

Theorem 7.5 generalizes the uniqueness result of Tian-Zhu for KR-solitons [35]
as well as a recent result of Wang-Zhu [45, Theorem 0.4] in sense of
diffeomorphisms orbit where both of ${\rm Aut}_{0}(M_{\infty}^{1})$ and ${\rm
Aut}_{0}(M_{\infty}^{2})$ are assumed to be reductive444In fact, we expect to
generalize Theorem 7.5 for the singular KR-solitons of
$(M_{\infty}^{1},\omega_{KR}^{1})$ and $(M_{\infty}^{2},\omega_{KR}^{2})$ as
in [45, Theorem 6.7], and it is also a generalization of uniqueness result of
Chen-Sun for KE-metrics orbit [8] (also see [23]). We also note that the
assumption (7.1) is necessary according to Pasquier’s counter-example of
horospherical variety [45, Remark 6.5].

Recall [46]

######  Definition 7.6.

Let $(M,J)$ be a Fano manifold. A complex manifold $(M^{\prime},J^{\prime})$
is called a canonical smooth deformation of $(M,J)$ if there are a sequence of
Kähler metrics $\omega_{i}$ in $2\pi c_{1}(M,J)$ and diffeomorphisms
$\Psi_{i}:M^{\prime}\to M$ such that

(7.2) $\displaystyle\Psi_{i}^{*}\omega_{i}\stackrel{{\scriptstyle
C^{\infty}}}{{\longrightarrow}}\omega^{\prime},~{}\Psi_{i}^{*}J\stackrel{{\scriptstyle
C^{\infty}}}{{\longrightarrow}}J^{\prime},~{}{\rm on}~{}M^{\prime}.$

In addition that $J^{\prime}$ is not conjugate to $J$, $J^{\prime}$ is called
a jump of $J$.

Theorem 7.5 is a direct corollary of following convergence result of KR flow
together with the uniqueness result of Han-Li [16] (cf. [45]).

######  Proposition 7.7.

Let $(M^{\prime},J^{\prime})$ be a canonical smooth jump of a Fano manifold
$(M,J)$. Suppose that $(M^{\prime},J^{\prime})$ admits a KR soliton
$\omega_{KS}$ such that

(7.3) $\displaystyle\lambda(\omega_{KS})=\sup\\{\lambda(g)|~{}\omega_{g}\in
2\pi c_{1}(M,J)\\}.$

Then for any initial metric $\widetilde{\omega}_{0}\in 2\pi c_{1}(M,J)$ the
flow $(M,J,\widetilde{\omega}(t))$ of (0.1) is uniformly
$C^{\infty}$-convergent to $(M^{\prime},J^{\prime},\omega_{KS})$.

###### Proof.

By the assumption, there is a sequence of Kähler metrics $\omega_{i}$ in $2\pi
c_{1}(M,J)$ such that

$\lim_{i}{\rm dist}_{CG}((M,\omega_{i}),(M^{\prime},\omega^{\prime}))=0,$

where $\omega^{\prime}\in 2\pi c_{1}(M^{\prime},J^{\prime})$. Note that the KR
flow of (0.1) with the initial $\omega^{\prime}$ converges to the KR soliton
$\omega_{KS}$ [39, 11]. Then by the stability of (0.1) for the finite time,
there is a sequence of $\tilde{\omega}_{i}^{t_{i}}$ such that

$\tilde{\omega}_{i}^{t_{i}}=\omega_{KS}+\sqrt{-1}\partial_{J}\bar{\partial}_{J}\chi_{i}$

and

$\lim_{i}{\rm
dist}_{CG,C^{3}}((M,\tilde{\omega}_{i}^{t_{i}}),(M^{\prime},\omega_{KS}))=0,$

where $\tilde{\omega}_{i}^{t}$ is the solution of flow (0.1) with the initial
metric $\omega_{i}$. It follows that there are diffeomorphisms $\Phi_{i}$ such
that

$\lim_{i}\|\Phi_{i}^{*}J-J^{\prime}\|_{C^{4,\gamma}}=0.$

Thus there are diffeomorphisms $K_{i}$ (cf. (6.6)) such that

$J_{\varphi_{i}}=K_{i}^{*}(\Phi_{i}^{*}J)~{}{\rm
and}~{}\|\varphi_{i}\|_{C^{k+4,\gamma}(M)}<<1,~{}{\rm as}~{}i>>1.$

Hence, as in the proof of Theorem 6.1 555(6.12) always holds by (7.3)., we
conclude that the sequence $\\{g_{i}(t)\\}$ of modified flows of (6.1)
uniformly converges to the soliton flow generated by $\omega_{KS}$. Since the
limit of $g_{i}(t)$ is independent of the initial metrics
$\tilde{\omega}_{i}^{t_{i}}$ [16, 45], each $g_{i}(t)$ must converge to
$\omega_{KS}$. Again by the uniqueness result in [16, 45], we prove the
proposition.

∎

By Proposition 7.7, we in particular prove the following uniqueness result for
the centers of smooth degenerations with admitting KR solitons.

######  Corollary 7.8.

Let $(M,J)$ be a Fano manifold. Suppose there is a smooth degeneration on
$(M,J)$ with its center $(M^{\prime},J^{\prime})$ admitting a KR soliton
$g_{KS}$ which satisfies

(7.4) $\displaystyle\lambda(g_{KS})=\sup\\{\lambda(g)|~{}\omega_{g}\in 2\pi
c_{1}(M,J)\\}.$

Then $(M^{\prime},J^{\prime})$ is unique.

###### Proof.

By Corollary 2.2 and Kuranishi’s completeness theorem [25], there is a smooth
path $\tau(s)\in B(\epsilon)$ $(s<<1)$ such that $g_{0}=g_{KS}$. Then

$g_{\tau(s)}={\omega}(\cdot,J_{\tau(s)}\cdot)$

induces a smooth jump of $(M,J)$ to $(M^{\prime},J^{\prime})$. Here $\omega$
is the Kähler form of $g_{KS}$. Thus the corollary comes from Proposition 7.7
immediately.

∎

##  8\. Generalization of Theorem 0.1

In this section, we generalize the argument in the proof of Theorem 0.1 to
prove Theorem 0.3. We will use a different way to obtain the entropy estimate
(6.12) in the proof of Theorem 6.1. We first consider the special case of
$H_{1}<0$, i.e., the stable case, and prove the jump of KR flow (0.1) to the
KR soliton $(M,J_{0},\omega_{KS})$.

###  8.1. Linearly stable case

######  Theorem 8.1.

Let $(M,J_{0})$ be a Fano manifold which admits a KR soliton $\omega_{KS}\in
2\pi c_{1}(M,J_{0})$. Suppose that for any $\psi\neq 0~{}{\rm
in}~{}\mathcal{H}^{0,1}_{h}(M,T^{1,0}M)$ it holds

(8.1) $\displaystyle H_{1}(\psi)<0.$

Then there exists a small $\epsilon$ such that for any $\tau\in B(\epsilon)$
the flow (6.1) with an initial metric $g=L(\tau,\chi)$ converges smoothly to
the KR soliton $(M,J_{0},\omega_{KS})$. Furthermore, the convergence is fast
in the polynomial rate.

###### Proof.

By the Hamilton-Tian conjecture and the uniqueness result in [16, 45], we need
to prove the convergence of flow $g(t)$ of (6.1) with the initial metric
$g_{\tau}=L(\tau,0)$ for any $\tau\in B(\epsilon)$. Moreover by (8.1), we may
assume that there is $\psi\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ such that

(8.2)
$\displaystyle\|\psi\|=1,\psi_{\tau}=\sum\tau_{i}e_{i}=\tau\psi=\tau(\psi_{1}+\psi^{\prime})$

with the property:

$\displaystyle\|\psi_{1}\|\geq\frac{1}{l}>0,H_{1}(\psi_{1})=-\delta_{1}\psi_{1};$
(8.3) $\displaystyle<H_{1}(\psi^{\prime}),\psi^{\prime}>\leq
0,~{}\|\psi^{\prime}\|\leq 1.$

Here $\delta_{1}\geq 0$. Without loss of generality, $-\delta_{1}$ may be
regarded as the largest negative eigenvalue of $H_{1}$. Thus for any $\chi$
with $\|\chi\|_{C^{4,\gamma}}\leq\tau N_{0}$, where $N_{0}$ is fixed, the
following is true:

$\displaystyle(\psi_{\tau},\chi)=\tau(\psi,\chi^{\prime}),~{}\|\chi^{\prime}\|_{C^{4,\gamma}}\leq
N_{0},$ (8.4)
$\displaystyle\lambda(g_{\tau,\chi})\leq\lambda(g_{KS})-\tau^{2}(\frac{\delta_{1}}{l^{2}}+o(1))<\lambda(g_{KS}).$

Let $T=T_{\tau,0}$ be the set defined as in (6.5) with $\delta_{0}=O(\tau)$.
Let $\tilde{\psi}=\psi_{t}\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ with
$\|\tilde{\psi}\|<\epsilon_{0}=O(\tau)$ defined as in (6.6) such that

$K(t)^{*}J(t)=J_{\tilde{\varphi}},~{}\tilde{\varphi}=\Phi(\tilde{\psi}).$

Then as in (8.2) and (8.1), there is
$\tilde{\psi}_{1}\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ with
$\|\tilde{\psi}_{1}\|\geq\frac{1}{l}$ such that

$\displaystyle\|\tilde{\psi}\|=\tau^{\prime}<\epsilon_{0};$
$\displaystyle\tilde{\psi}=\tau^{\prime}(\tilde{\psi}_{1}+\tilde{\psi}^{\prime}),\|\tilde{\psi}_{1}+\tilde{\psi}^{\prime}\|=1;$
(8.5) $\displaystyle
H_{1}(\tilde{\psi}_{1})=-\delta_{1}^{\prime}\tilde{\psi}_{1}~{}{\rm
and}~{}<H_{1}(\tilde{\psi}^{\prime}),\tilde{\psi}^{\prime}>\leq 0,$

where $\delta_{1}^{\prime}\geq\delta_{1}.$

On the other hand, we may assume that $\omega_{g(t)}$ is same as
$\omega_{KS}$. Otherwise, by Moser’s theorem, there is a diffeomorphism
$\hat{K}(t)\in{\rm Diff(M)}$ such that $\omega_{\hat{g}(t)}=\omega_{KS},$
where $\hat{g}(t)=\hat{K}(t)^{*}g(t)$. Then we can replace $g(t)$ by
$\hat{g}(t)$. Thus, there are $K(t)\in{\rm Diff(M)}$ and $\tilde{\chi}\in
C^{\infty}(M)$ as in (6.8) such that

(8.6)
$\displaystyle\omega_{K(t)^{*}g(t)}==K(t)^{*}\omega_{KS}=\omega_{KS}+\sqrt{-1}\partial_{J_{\tilde{\varphi}}}\bar{\partial}_{J_{\tilde{\varphi}}}\tilde{\chi},~{}\int_{M}\tilde{\chi}\omega_{KS}^{n}=0.$

Moreover, as in (6), we have

$K(t)^{*}\omega_{KS}=(1+O(\|\tilde{\varphi}_{t}\|_{L^{2}(M)}))\omega_{KS}.$

It follows that (cf. (6))

$\|\Delta\tilde{\chi}\|_{C^{4,\gamma}}\leq\tau^{\prime}N_{0}.$

Consequently, by (8.1), we derive

(8.7)
$\displaystyle\lambda(g(t))=\lambda(L{\tilde{\psi},\tilde{\chi}})\leq\lambda(g_{KS})-(\tau^{\prime})^{2}(\frac{\delta_{1}}{l^{2}}+o(1))\leq\lambda(g_{KS}).$

Hence, we can continue the argument in the proof of Theorem 6.1 to show that
the flow (6.1) converges smoothly to a KR soliton
$(M,J_{\infty},\omega_{\infty})$ such that

$\lambda(g_{\infty})=\lambda(g_{KS}).$

Moreover, by (8.7), $\tau^{\prime}$ must go to zero. Therefore, $\tilde{\psi}$
goes to zero and $g_{\infty}$ is same as $g_{KS}$ by the uniqueness of KR
solitons [35].

For a general initial metric in $2\pi c_{1}(M,J_{\tau})$, we only remark how
to get the decay estimate. In fact, we can choose a large $T_{0}$ such that

${\rm dist}_{CG,C^{k}}((M,g(T_{0})),(M,g_{\infty}))\leq\epsilon$

and there is an auto-diffeomorphism $\sigma$ such that

$\|\sigma^{*}J({T_{0}})-J_{0}\|_{C^{k}}\leq\epsilon^{\prime}.$

Then as in (8.6), there is an auto-diffeomorphism $K(T_{0})$ and
$\psi\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ with
$\varphi=\Phi^{\prime}(\psi)$ for the metric $\sigma^{*}g(T_{0})$ such that

$\|\psi\|=\tau<<1~{}{\rm and}\|\Delta\chi\|_{C^{4,\gamma}}\leq\tau N_{0}.$

Hence, by the above argument, the KR flow with the initial metric $g(T_{0})$
converges to $g_{\infty}$ fast in the polynomial rate. The proof is complete.

∎

Theorem 8.1 means that any KR flow with an initial metric in $2\pi
c_{1}(M,J_{\tau})$ with $\tau<<1$ under the condition (7.6) will jump upto the
original KR soliton $(M,J_{0},\omega_{KS})$ and also show that the KR soliton
$(M,J_{0},\omega_{KS})$ is isolated in the deformation space of complex
structures in this case. But upto now, we do not know whether there is such an
example of Fano manifold with admitting an isolated KR soliton around her
deformation space of complex structures.

###  8.2. Linearly semistable case

In this subsection, we generalize Theorem 8.1 with the existence of non-
trivial kernel $Z$ of operator $H_{1}$ to prove Theorem 0.3. Let $K$ be a
maximal compact subgroup of ${\rm Aut}_{r}(M,J_{0})$ which contains $K_{X}$.
Then $(\omega_{KS},J_{0})$ is $K$-invariant [35], and so $\sigma$ maps
$\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ to itself for any $\sigma\in K$. The
latter means that $\sigma(\psi)$ is still $\theta$-harmonic whenever $\psi$ is
$\theta$-harmonic. Thus we can extend the action ${\rm Aut}_{r}(M,J_{0})$ on
$\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ so that
$\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$ is an ${\rm
Aut}_{r}(M,J_{0})$-invariant space. Since

$K_{X}\cdot\sigma=\sigma\cdot K_{X},~{}\forall\sigma\in{\rm
Aut}_{r}(M,J_{0}),$

we have

$\mathcal{L}_{\xi}\sigma^{*}\psi=0,~{}\forall~{}\psi\in Z.$

Hence, $Z$ is also an ${\rm Aut}_{r}(M,J_{0})$-invariant subspace of
$\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$. Furthermore, we prove

######  Lemma 8.2.

Suppose that $(M,J_{\psi_{\tau}})$ admits a KR soliton for any $\psi_{\tau}\in
B(\epsilon)\cap Z$. Then

(8.8) $\displaystyle\sigma^{*}{\psi}=\psi_{,}~{}\forall~{}\sigma\in{\rm
Aut}_{r}(M,J_{0}),\psi\in Z.$

###### Proof.

Since the union of maximal tori is dense in a reductive complex group, we need
to prove (8.8) for any torus 1-ps $\sigma_{t}$ and $\psi\in Z$. On the
contrary, there are $\sigma\in{\rm Aut}_{r}(M,J_{0})$ and $\psi\in Z$ such
that $\sigma_{t}=\sigma^{-1}\cdot\rho_{t}\cdot\sigma$ with some nontrivial
weights for $\psi$, where

$\rho_{t}:\mathbb{C}^{*}\mapsto{\rm Aut}_{r}(M,J_{0})$

is a diagonal torus action on $Z$. Since $Z$ is ${\rm
Aut}_{r}(M,J_{0})$-invariant, there is a basis of $Z$,
$\\{e_{1},\ldots,e_{s}\\}$ $(s={\rm dim}Z)$ such that
$\sigma(\psi)=\sum_{i\leq s}e_{i}$ and

$\sigma_{t}(\psi)=\sum_{i}t^{a_{i}}\sigma^{-1}(e_{i}),t\in\mathbb{C}^{*},$

where $a_{i}\in\mathbb{Z}$ are weights of $\sigma_{t}$ with some $a_{i}\neq
0$. Without loss of generality, we may assume $a_{1}>0$.

Now we consider a degeneration of $\rho_{t}(e_{1})$
$(|t|<\epsilon,|e_{1}|<<1)$ in $Z$. Note that

$J_{\rho_{t}^{*}\Phi(e_{1})}=\rho_{t}^{*}J_{\Phi(e_{1})}.$

Then by the Kuranishi theorem, there is a $K_{t}\in{\rm Diff(M)}$ such that

$J_{\Phi(\rho_{t}(e_{1}))}=K_{t}^{*}(\rho_{t}^{*}J_{\Phi(e_{1})}).$

Thus, for any small $t_{1}$ and $t_{2}$ it holds

$(M,J_{\Phi(t_{1}e_{1})})\cong(M,J_{\Phi(t_{2}e_{1})}).$

Hence, the family of $(M,J_{\Phi(te_{1})})$ $(|t|<\epsilon)$ is generated by a
$C^{*}$-action on $(M,J_{\Phi(e_{1})})$, which degenerates to $(M,J_{0})$.
Moreover, it communicates with 1-ps $\sigma_{s}^{X}$ and its modified Futaki-
invariant is zero. Therefore, we get a contradiction with the modified
K-polystability of $(M,J_{\Phi(e_{1})})$ which admits a KR soliton. The proof
is finished.

∎

######  Remark 8.3.

In case of $\omega_{KS}=\omega_{KE}$, $Z=\mathcal{H}^{0,1}(M,T^{1,0}M)$. Then
the property (8.8) turns to hold for any $\sigma\in{\rm
Aut}(M,J_{0}),\psi\in\mathcal{H}^{0,1}(M,T^{1,0}M)$. Thus Lemma 8.2 is
actually a generalization of [6, Theorem 3.1] in the case of KE metrics.

By Lemma 8.2, we prove

######  Lemma 8.4.

Under the assumption in Lemma 8.2, there exists $\delta>0$ such that for any
$(\tau,\chi)\in\mathcal{V}_{\delta}$ it holds

$\lambda(g_{\tau,\chi})\leq\lambda(g_{KS}),$

if $H_{1}(\cdot)\leq 0$ on $\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M).$

###### Proof.

Let $\mathfrak{k}$ be the Lie algebra of $K$, and $\eta_{u}$ the nilpotent
part of $\eta(M,J_{0};\mathbb{R})$. Then

$\eta(M,J_{0};\mathbb{R})=\eta_{r}\oplus\eta_{u}=\mathfrak{k}\oplus
J_{0}\mathfrak{k}\oplus\eta_{u}.$

Fix a small ball $U(\epsilon_{0})\subseteq\mathfrak{k}\oplus\eta_{u}$ centered
at the original. By the Kuranishi theorem there exists $\tau^{\prime}\in
B(\epsilon)$ and $\gamma\in{\rm Diff}(M)$ for any $\tau\in
B(\epsilon),(v_{1},v_{2})\in U(\epsilon_{0})$ such that

(8.9)
$\displaystyle(e^{J_{\tau}v_{1}+v_{2}})^{*}J_{\tau}=\gamma^{*}J_{\tau^{\prime}},$

where $J_{\tau}$ and $J_{\tau^{\prime}}$ are complex structures associated to
$\psi_{\tau}$ and $\psi_{\tau^{\prime}}$, respectively. We define a smooth map
$\Phi:U(\epsilon)\times\mathcal{V}_{\epsilon}\mapsto\mathcal{V}_{\epsilon}$ by

$\Phi((v_{1},v_{2}),(\tau,\chi))=(\tau^{\prime},\chi^{\prime}),$

where the Kähler potential $\chi^{\prime}$ is determined by

(8.10)
$\displaystyle(\gamma^{-1})^{*}(e^{J_{\tau}v_{1}+v_{2}})^{*}(\omega+\sqrt{-1}\partial_{J_{\tau}}\bar{\partial}_{J_{\tau}}\chi)=\omega+\sqrt{-1}\partial_{J_{\tau^{\prime}}}\bar{\partial}_{J_{\tau^{\prime}}}\chi^{\prime}~{}{\rm
and}~{}\int_{M}\chi^{\prime}\omega_{KS}=0.$

Let $\Psi:U(\epsilon)\times\mathcal{V}_{\epsilon}\mapsto{\rm ker}(H_{2})$ be a
projection of $\Phi$ such that

$\Psi={\rm Pr}_{{\rm ker}(H_{2})}\circ{\rm Pr}_{2}\circ\Phi.$

Namely,

$\Psi(((v_{1},v_{2})(\tau,\chi)))={\rm Pr}_{{\rm ker}(H_{2})}(\chi^{\prime}).$

A direct calculation shows that

$D_{(0,0)}\Psi((v_{1},v_{2}),0)=\theta_{Y}+\overline{\theta_{Y}},$

where $\theta_{Y}$ is a potential of HVF,

$Y=\frac{1}{2}\sqrt{-1}(v_{1}-\sqrt{-1}J_{0}v_{1})+\frac{1}{2}(v_{2}-\sqrt{-1}J_{0}v_{2}).$

Thus by [40], we see that

$D_{(0,0)}\Psi(\cdot,0):\mathfrak{k}\oplus\eta_{u}\mapsto{\rm ker}(H_{2})$

is an isomorphism. By the implicit function theorem, there are
$\delta<\epsilon$ and map $G:\mathcal{V}_{\delta}\to U(\varepsilon_{0})$ such
that

$\displaystyle{\rm Pr}_{2}\circ\Phi(G(\tau,\chi),(\tau,\chi))\in{\rm
ker}(H_{2})^{\perp}.$

It follows that

(8.11)
$\displaystyle(\tau^{\prime},\chi^{\prime})=\Phi(G(\tau,\chi),(\tau,\chi))\in\mathcal{V}_{\delta^{\prime}},$

where $\chi^{\prime}\in{\rm ker}(H_{2})^{\perp}$ and
$\delta^{\prime}(\delta)<<1.$

Let

$\tilde{\mathcal{V}}_{\epsilon}=\\{(\tau_{0},\widetilde{\chi})\in\mathcal{V}_{\epsilon}|~{}\tau_{0}\in
Z,~{}\xi(\widetilde{\chi})=0\\}.$

Then the restricted map $\widetilde{\Psi}$ of $\Psi$ on
$\mathfrak{k}\times\tilde{\mathcal{V}}_{\epsilon}$ is also smooth and

$D_{(0,0)}\widetilde{\Psi}(\cdot,0):\mathfrak{k}\mapsto\\{\theta_{Y}+\overline{\theta_{Y}}\\}$

is an isomorphic map. Here
$Y=\frac{1}{2}\sqrt{-1}(v-\sqrt{-1}J_{0}v),~{}v\in\mathfrak{k}.$ We claim:

(8.12) $\displaystyle{\rm
ker}(H_{2})\cap\\{\chi|~{}\xi(\chi)=0\\}=\\{\theta_{Y}+\overline{\theta_{Y}}\\}.$

On contrary, there is a nontrivial $v^{\prime}~{}\in\eta_{u}$ such that

$\xi(\theta_{Y^{\prime}}+\overline{\theta_{Y^{\prime}}})=0,$

where $Y^{\prime}=v^{\prime}-\sqrt{-1}J_{0}v^{\prime}$. Then for any
$\sigma\in K_{X}$,

$\sigma^{*}(\theta_{Y^{\prime}}+\overline{\theta_{Y^{\prime}}})=\theta_{Y^{\prime}}+\overline{\theta_{Y^{\prime}}}.$

It follows that

$\sigma^{*}Y^{\prime}=Y^{\prime},~{}\forall~{}\sigma\in K_{X}.$

This implies $Y^{\prime}\in\eta_{r}(M,J_{0})$, which is a contradiction!

By (8.12),

$D_{(0,0)}\widetilde{\Psi}(\cdot,0):\mathfrak{k}\mapsto{\rm
ker}(H_{2})\cap\\{\chi|~{}\xi(\chi)=0\\}$

is an isomorphism. On the other hand, by Lemma 8.4,

$(e^{J_{\tau_{0}}v})^{*}J_{\tau_{0}}=J_{\tau_{0}},~{}\forall~{}\psi_{\tau_{0}}\in
Z,v\in\mathfrak{k}.$

Hence, applying the implicit function theorem to $\widetilde{\Psi}$ at
$(\tau_{0},\widetilde{\chi})\in\tilde{\mathcal{V}}_{\epsilon}$, there are
$v\in\mathfrak{k}$ and Kähler potential $\chi_{0}\in{\rm ker}(H_{2})^{\perp}$
such that $(\tau_{0},\chi_{0})\in\tilde{\mathcal{V}}_{\delta^{\prime}}$ and

$(e^{J_{\tau}v})^{*}(\omega+\sqrt{-1}\partial_{J_{\tau_{0}}}\bar{\partial}_{J_{\tau_{0}}}\widetilde{\chi})=\omega+\sqrt{-1}\partial_{J_{\tau_{0}}}\bar{\partial}_{J{\tau_{0}}}\chi_{0}.$

Note that a Kähler potential $\widetilde{\chi}$ of KR soliton with respect to
$X$ on $(M,J_{\tau_{0}})$ satisfies $\xi(\widetilde{\chi})=0$ [35]. Therefore,
there is a KR soliton
$\omega+\sqrt{-1}\partial_{J_{\tau_{0}}}\bar{\partial}_{J{\tau_{0}}}\chi_{0}$
on $(M,J_{\tau_{0}})$ such that

(8.13) $\displaystyle\chi_{0}\in{\rm ker}(H_{2})^{\perp}~{}{\rm
and}~{}(\tau_{0},\chi_{0})\in\tilde{\mathcal{V}}_{\delta^{\prime}}.$

Decompose $\psi_{\tau^{\prime}}$ in (8.11) as

(8.14)
$\displaystyle\psi_{\tau^{\prime}}=\psi_{\tau_{0}}+\psi_{\tau_{1}},~{}\psi_{\tau_{0}}\in
Z,~{}\psi_{\tau_{1}}\in Z^{\perp}.$

Then by (8.13), there are Kähler potentials $\chi_{0}$ and $\chi_{1}\in{\rm
ker}(H_{2})^{\perp}$ such that

(8.15) $\displaystyle\chi^{\prime}=\chi_{0}+\chi_{1},$

where
$\omega+\sqrt{-1}\partial_{J_{\tau_{0}}}\bar{\partial}_{J{\tau_{0}}}\chi_{0}$
is a KR soliton. Thus, we get the following expansion for the reduced entropy
$\nu$,

(8.16)
$\displaystyle\nu(\tau^{\prime},\chi^{\prime})=\nu(\tau_{0},\chi_{0})+\frac{1}{2}(D^{2}_{(\tau_{0},\chi_{0})+\zeta(\tau_{1},\chi_{1})}\nu)((\tau_{1},\chi_{1}),(\tau_{1},\chi_{1})).$

Here $\zeta\in(0,1)$. Since $D^{2}_{(0,0)}\nu(0,\cdot)$ is strictly negative
on ${\rm ker}(H_{2})^{\perp}$ [40], by the assumption in Lemma 8.4, it is
strictly negative on ${\rm ker}(H_{1})^{\perp}\times{\rm ker}(H_{2})^{\perp}$.
It follows that $D^{2}_{(\tau_{0},\chi_{0})+\zeta(\tau_{1},\chi_{1})}\nu$ is
strictly negative on ${\rm ker}(H_{1})^{\perp}\times{\rm ker}(H_{2})^{\perp}$.
Hence, we derive

(8.17)
$\displaystyle\nu(\tau^{\prime},\chi^{\prime})\leq\nu(\tau_{0},\chi_{0}).$

As a consequence, we prove

$\lambda(g_{\tau,\chi})\leq\lambda(g_{\tau_{0},\chi_{0}})=\lambda(g_{KS}).$

∎

By Lemma 8.4, we can use the argument in the proof of Theorem 6.1 (or Theorem
8.1) to finish the proof of Theorem 0.3. We leave it to the reader.

We end this subsection by the following two remarks to Theorem 0.3.

######  Remark 8.5.

In Theorem 0.3, we can actually prove that the convergence is of polynomial
rate of any order. This is because we can improve the Lojasiewicz inequality
of (5.1) for $\nu(\cdot)$ in Proposition 5.1 as follows: For any
$a=(\varphi,\chi)\in\mathcal{V}_{\epsilon}$ it holds

(8.18) $\displaystyle\|\nabla\nu(a^{\prime})\|_{L^{2}}\geq
c_{0}|\nu(a^{\prime})-\nu(0)|^{\frac{1}{2}},$

where
$a^{\prime}=(\tau^{\prime},\chi^{\prime})\in\mathcal{V}_{\delta^{\prime}}$ is
determined in (8.11). It follows that the Lojasiewicz inequality of (6.11) for
$\lambda(\cdot)$ along the flow holds for any $\alpha^{\prime}>\frac{1}{2}.$
Hence, by (6), the convergence is of polynomial rate of any order. (8.18) can
be proved together with (5) and (5) since the functional $F$ there can vanish
by the decompositions (8.14) and (8.15).

######  Remark 8.6.

We call the KR flow $\widetilde{\omega}(t)$ of (0.1) is stable on
$(M,J_{\psi_{\tau}})$ if it converges smoothly to a KR soliton
$(M_{\infty},J_{\infty},\omega_{\infty})$ with
$\lambda(\omega_{\infty})=\lambda(\omega_{KS})$. In Theorem 0.3, we show that
the flow (0.1) is always stable for any complex structure in the deformation
space under the conditions of (1) and (2). By the above argument (also see
Remark 6.2), it is easy to see that the flow $\widetilde{\omega}(t)$ is stable
on $(M,J_{\psi_{\tau}})$ if and only if the energy level
$L([M,J_{\psi_{\tau}}])$ of flow is equal to $\lambda(\omega_{KS}).$ Here

(8.19) $\displaystyle
L([M,J_{\psi_{\tau}}])=\lim_{t}\lambda(\widetilde{\omega}(t)),$

which is independent of flow $\widetilde{\omega}(t)$ with the initial metric
$\widetilde{\omega}_{0}\in 2\pi c_{1}(M,J_{\psi_{\tau}})$ [38, 11, 45].

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