Figure 16: Verification of the fast integration method by comparing
$\sigma_{x}$, the root-mean-square change in $x=1/a$, between two integration
methods. ‘Normal’ (black dashed line) means having 3 particles, central star,
companion and comet. ‘Fast’ (red solid line) is the integration method
presented in this article with only 1 particle: a comet. In the production of
this plot, the fast method is faster by about 40 times. This plot can be
compared with isotropic curve Figure 9.3 in Tremaine (2023).

To verify the fast direct integration scheme (Section A) and our
implementation of it in REBOUND, we plot Figure 16. In this Figure, we compare
$\sigma_{x}$, the root-mean-square change in $x=1/a$, between our fast
integration method and the full 3-body simulation. At each $q$, one thousand
comets are initialised isotropically, with an initial semi-major axis at
$a=1~{}000$ AU and eccentricity $e=1-q/a$. The planet is at $5$ AU with mass
$10^{-3}M_{\odot}$. Comets are integrated over one orbital period to calculate
$\sigma_{x}$. We can see that the fast method is able to reproduce well close
encounters between comets and planet. The results in this plot can be compared
with isotropic curve in Figure 9.3 in Tremaine (2023) for further
verification. The fast method is 40 times faster. Finally, note that this fast
integration method can be applied in multiple planet systems in other
applications. We show an additional application of this with Kuiper Belt
objects interactions with the outer Solar System in Appendix C.

##  Appendix C An application of the fast N-body method

The fast REBOUND integration method can be extended to multiple planets. One
major caveat in this case is that we assume that massive objects (planets) do
not interact with each other. In other words, we assume the massive objects to
be “on rails”. This allows us to analytically calculate the position of
bodies, which is required for our fast integration method. That being said,
this method is still a good approximation when the integration timescale is
less than the timescales on which planet-planet interactions begin to become
important (for example, this would be the secular timescale in the Solar
System). It is possible to overcome this limitation by pre-running a
simulation of the massive particles, record their positions, interpolate the
positions at any $t$, and then we can use those interpolated positions for the
fast integration method.

In the example here, we will use the simple version (no position
interpolation). We showcase an integration of 1000 objects with semi-major
axes initially distributed between $[30,50]$ AU and inclinations drawn
randomly from a Rayleigh distribution with $\langle I\rangle=10^{\circ}$. We
include the Sun and the outer Solar System planets. This integration is run
for 100 Jupiter orbits. Figure 17 shows the semi-major axis distribution of
these objects. We show here that the fast method is capable of statistically
reproducing the Kuiper Belt peaks induced by resonance.

Figure 17: The Kuiper Belt semi-major axis distribution reproduced by the fast
integration method (blue histogram) compared with the normal integration
method (orange histogram). The simulation includes effects from the outer
Solar System planets (and the Sun). The simulation is run for 100 Jupiter
orbits.

##  Appendix D Semi-major Axis Rejection Sampling

Figure 18: The efficiency of performing rejection sampling on a broken
powerlaw versus rejection sampling on a uniform distribution. We can see that
the using the broken powerlaw distribution to create powerlaw samples is much
more effective. With the broken powerlaw rejection sampling, the number of
usable samples at any particular $\gamma$ is about half of the total simulated
samples.

The distribution of comet semi-major axis itself follows a density profile
described by a powerlaw relationship $n(a)\propto a^{-\gamma}$. We wish to
study all $\gamma$ between $2\leq\gamma\leq 4$. We could draw samples at fixed
$\gamma$ and run simulation and repeat that over a grid $\gamma$. However,
this increases the computation costs significantly. For example, to make
Figure 9 would require running 20 simulations at different $\gamma$. To
resolve this, we use a rejection sampling scheme.

We initialise comets’ semi-major axes on a broken powerlaw distribution (c.f.
Equation 17):

$\differential N(a)\propto\begin{cases}a^{-\gamma_{\mathrm{max}}}\cdot
a^{2}~{}\differential a,&a_{1}\leq a\leq a_{\mathrm{turnover}}\\\
a^{-\gamma_{\mathrm{min}}}\cdot a^{2}~{}\differential
a,&a_{\mathrm{turnover}}\leq a\leq a_{2}\end{cases}$ (47)

where $\gamma_{\mathrm{min}}=2$ is the minimum $\gamma$ that we can sample and
$\gamma_{\mathrm{max}}=4$ is the maximum. $a_{\mathrm{turnover}}=17~{}000$ AU
is the broken powerlaw turnover point. $a_{\mathrm{turnover}}$ is determined
heuristically to maximise the efficiency of rejection sampling.

We simulate $N_{\mathrm{comets}}^{\mathrm{sim}}$ comets with semi-major axes
drawn from this broken powerlaw distribution. We record their initial and
final positions. Using rejection sampling, we select comets after running the
simulations to construct results for any particular $\gamma$ we are interested
in.

In Figure 18, we show the efficiency of rejection sampling from the broken
powerlaw at various $\gamma$. We also compare that with rejection sampling
from a uniform distribution instead. As shown, the broken powerlaw is much
efficient for rejection sampling into a powerlaw distribution than a uniform
distribution. We find that using the broken powerlaw as proposed gives a 50%
efficiency. That is, to construct results at any particular $\gamma$, the
number of comets representative in that results is
$N_{\mathrm{comets}}^{\mathrm{eff}}\sim
0.5N_{\mathrm{comets}}^{\mathrm{sim}}$. As a consequence, we need to simulate
twice the number of comets we would like to see for any particular result.
However, this is still less than one simulation per $\gamma$.


# Bergman Kernels for a sequence of almost Kähler-Ricci solitons

Wenshuai Jiang  ,  Feng Wang  and  Xiaohua $\text{Zhu}^{*}$ Wenshuai Jiang
School of Mathematical Sciences, Peking University, Beijing, 100871, China
Feng Wang
School of Mathematical Sciences, Peking University, Beijing, 100871, China
Xiaohua Zhu
School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871,
China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we give a lower bound of Bergman kernels for a sequence of
almost Kähler-Einstein Fano manifolds, or more general, a sequence of Fano
manifolds with almost Kähler-Ricci solitons. This generalizes a result by
Donaldson-Sun, Tian for Kähler-Einstein manifolds sequence with positive
scalar curvature. As an application of our result, we prove that the Gromov-
Hausdorff limit of sequence is homomorphic to a log terminal $Q$-Fano variety
which admits a Kähler-Ricci soliton on its smooth part.

###### Key words and phrases:

Kähler-Einstein metrics, almost Kähler-Ricci solitons, Ricci flow,
$\bar{\partial}$-equation.

###### 2000 Mathematics Subject Classification:

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

* Partially supported by the NSFC Grants 11271022 and 11331001

###### Contents

  1. 1 Introduction
  2. 2 Estimates from Kähler Ricci flow
  3. 3 Estimates for holomorphic sections
  4. 4 Construction of locally approximate holomorphic sections
  5. 5 $\bar{\partial}$-equation and construction of holomorphic sections
  6. 6 Proof of Theorem 1.3–I
  7. 7 Proof of Theorem 1.3–II
  8. 8 Proof of Corollary 1.4
  9. 9 Conclusion
  10. 10 Appendix

##  1\. Introduction

Let $(M^{n},g)$ be an $n$-dimensional Fano manifold with its Kähler form
$\omega_{g}$ in $2\pi c_{1}(M)$. Then $g$ induces a Hermitian metric $h$ of
the anti-canonical line bundle $K_{M}^{-1}$ such that ${\rm
Ric}(K_{M}^{-1},h)=\omega_{g}$. Also $h$ induces a Hermitian metric ( for
simplicity, we still use the notation $h$ ) of $l$-multiple line bundle
$K_{M}^{-l}$. As usual, the $L^{2}$-inner product on $H^{0}(M,K_{M}^{-l})$ is
given by

(1.1) $\displaystyle(s_{1},s_{2})=\int_{M}\langle s_{1},s_{2}\rangle_{h}d{\rm
v}_{g},~{}\forall~{}s_{1},s_{2}~{}\in H^{0}(M,K_{M}^{-l}).$

Choosing a unit orthogonal basis $\\{s_{i}\\}$ of $H^{0}(M,K_{M}^{-l})$ with
respect to the inner product $(\cdot,\cdot)$ in (1.1), we define the Bergman
kernel of $(M,K_{M}^{-l},h)$ by

$\displaystyle\rho_{l}(x)=\Sigma_{i}|s_{i}|^{2}_{h}(x).$

Clearly, $\rho_{l}(x)$ is independent of the choice of basis $\\{s_{i}\\}$. In
[22], Tian proposed a conjecture for the existence of uniform lower bound of
$\rho_{l}(x)$:

######  Conjecture 1.1.

Let $\\{(M_{i},g^{i})\\}$ be a sequence of $n$-dimensional Kähler-Einstein
manifolds with constant scalar curvature $n$. Then there exists an integer
number $l_{0}$ such that for any integer $l>0$ there exists a uniform constant
$c_{l}>0$ with property:

(1.2) $\displaystyle\rho_{ll_{0}}(M_{i},g^{i})\geq c_{l}.$

Here $c_{l}$ depends only on $l,n$.

The above conjecture was recently proved by Donaldson-Sun [7] and Tian [24],
independently. 111Phong-Song-Strum extended the result to a sequence of
Kähler-Ricci solitons lately [16]. The main idea in their proofs is to use the
Hörmander $L^{2}$-estimate to construct peak holomorphic sections by solving
$\overline{\partial}$-equation. This idea can go back to Tian’s orginal work
[19] (see also a survey paper by him [22]). In fact he used the idea to prove
the conjecture for the Kähler-Einstein surfaces more than twenty years ago
[20].

The estimate (1.2) is usually called the partial $C^{0}$-estimate. Very
recently, (1.2) was generalized to a sequence of conical Kähler-Einstein
manifolds by Tian [23]. As an application of (1.2) he gives a proof of the
famous Yau-Tian-Donaldson’s conjecture for the existence problem of Kähler-
Einstein metrics with positive scalar curvature. Chen-Donaldson-Sun also gives
a proof of the conjecture independently [4].

######  Theorem 1.2 (Tian, Chen-Donaldson-Sun).

A Fano manifold admits a Kähler-Einstein metric if and only if it is
$K$-stable.

The $K$-stability was first introduced by Tian [21] and it was reformulated by
Donaldson in terms of test-configurations [6].

In this paper, we want to generalize the estimate (1.2) to a sequence of
almost Kähler-Einstein Fano manifolds [26], or more general, a sequence of
almost Kähler-Ricci solitons (see Definition 7.3 in Section 7). Namely, we
prove

######  Theorem 1.3.

Let $\\{(M_{i},g^{i})\\}$ be a sequence of almost Kähler-Einstein Fano
manifolds ( or a sequence of Fano manifolds with almost Kähler-Ricci solitons)
with dimension $n\geq 2$. Then there exists an integer number $l_{0}$ such
that for any integer $l>0$ there exists a uniform constant $c_{l}>0$ with
property:

(1.3) $\displaystyle\rho_{ll_{0}}(M_{i},g^{i})\geq c_{l}.$

Here the constant $c_{l}$ depends only on $l,n$, and some uniform geometric
constants (cf. Section 9).

As in the proof of Theorem 1.2, we need to construct peak holomorphic sections
by solving $\overline{\partial}$-equation to prove Theorem 1.3. Because there
is a lack of local strong convergence of $\\{(M_{i},g^{i})\\}$, we shall
smooth the sequence to approximate the original one by Ricci flow as in [26],
[32]. This approximation will depend on points in the Gromov-Hausdorff limit
space of $\\{(M_{i},g^{i})\\}$, so it depends on the time $t$ in the Ricci
flow. Thus we need to give estimates for the scalar curvatures and Kähler
potentials along the flow for small time $t$ (cf. Section 2, 3, 7). Another
technical part is in the construction of peak holomorphic sections by using
the rescaling method as in [7], [24], which will depend on the choice of
Kähler metrics evolved in the Ricci flow in our case (cf. Section 5, 6, 7).

Together with the main results in [31] and [32], Theorem 1.3 implies

######  Corollary 1.4.

Let $\\{(M_{i},g^{i})\\}$ be a sequence of almost Kähler-Einstein Fano
manifolds ( or a sequence of Fano manifolds with almost Kähler-Ricci solitons)
with dimension $n\geq 2$. Then $\\{(M_{i},g^{i})\\}$ (maybe replaced by a
subsequence of $\\{(M_{i},g^{i})\\}$) converges to a metric space
$(M_{\infty},g_{\infty})$ in Gromov-Hausdorff topology with properties:

i) The codemenison of singularities of $(M_{\infty},g_{\infty})$ is at least
$4$;

ii) $g_{\infty}$ is a Kähler-Einstein metric ( or a Kähler-Ricci soliton) on
the regular part of $M_{\infty}$;

iii) $M_{\infty}$ is homomorphic to a log terminal $Q$-Fano variety.

In case of Kähler-Einstein manifolds with positive scalar curvature, we note
that i) and ii) in Corollary 1.4 follow from the Cheeger-Colding-Tian
compactness theorem [3]. Donaldson-Sun proved the part iii ) except log
terminal property [7] (also see [12]). Since any $Q$-Fano variety, which
admits a Kähler-Einstein metric, is automatically log terminal according to
Proposition 3.8 in [1]222 The result also holds for a $Q$-Fano variety, which
admits a Kähler-Ricci soliton, from the proof of Proposition 3.8., Thus the
part iii) is true. A normal variety $M$ is called $Q$-Fano if the restriction
of $\mathcal{O}_{\mathbb{C}P^{N}}(1)$ is a multiple of $K_{M}^{-1}$ on the
smooth part of $M$. The log terminal means that there exists a resolution
$\pi:~{}\tilde{M}\to M$ such that $K_{\tilde{M}}=\pi^{*}K_{M}+\sum
a_{i}D_{i},$ where $a_{i}>-1$, $\forall~{}i.$

There are important examples of almost Kähler-Einstein metrics and almost
Kähler-Ricci solitons:

1) Tian and B. Wang constructed a family of almost Kähler-Einstein metrics
$g_{t}$ $(t\to 1)$ arising from solutions of certain complex Monge-Ampère
equations on a Fano manifold with the Mabuchi’s $K$-energy bounded below [26].

2) Tian constructed a family of almost Kähler-Einstein metrics $g_{t}$ $(t\to
1)$ modified from conical Kähler-Einstein metrics on a Fano manifold whose
corresponding conical angles go to $2\pi$ [23].

3) F. Wang and Zhu constructed a family of almost Kähler-Ricci solitons
$g_{t}$ $(t\to 1)$ arising from solutions of certain complex Monge-Ampère
equations on a Fano manifold with the modified $K$-energy bounded below [31],
[32].

By Theorem 1.3 and Corollary 1.4, we have

######  Corollary 1.5.

Let $g_{t}$ $(t\to 1)$ be a family of almost Kähler-Einstein metrics ( or
almost Kähler-Ricci solitons ) on a Fano manifold $M$ constructed above 1),
2), 3). Then there exists an integer number $l_{0}$ such that for any integer
$l>0$ there exists a uniform constant $c_{l}>0$ independent of $t$ with
property:

(1.4) $\displaystyle\rho_{ll_{0}}(M,g_{t})\geq c_{l}>0.$

Moreover, there exists a sequence $\\{(M,g_{t})\\}$ which converges to a
metric space $(M_{\infty},g_{\infty})$ in Gromov-Hausdorff topology with
properties i),ii),iii) in Corollary 1.4.

It was proved recently by Li that the lower boundedness of $K$-energy is
equivalent to the $K$-semistablity [L2]. Li’s proof depends on the
construction of test-configurations in Theorem 1.2 by studying conical Kähler-
Einstein metrics. It is reasonable to believe that there is an analogy of Li’s
result to describe the modified $K$-energy in sense of modified
$K$-semistability.

######  Question 1.6.

Is there a direct proof (without using conical Kähler-Einstein metrics as in
the proof of Theorem 1.2) for that the $K$-stability implies the $K$-energy
bounded below?

A same question was proposed by Paul in his recent paper [15]. He proved there
that the $K$-stability is equivalent to the properness of $K$-energy in the
space of Kähler metrics induced by the Bergman Kernels. Thus as pointed by
Tian in [24], [25] (also see [21]), (1.4) will give a new proof of Theorem 1.2
if the answer to Question 1.6 is positive.

Finally, let us to state our organization to the paper. In Section 2, we give
some estimates for scalar curvatures and Kähler potentials along the Ricci
flow, then, in Section 3, we use them to give the $C^{0}$-estimate and the
gradient estimate for holomorphic sections on multiple line bundles of
$K_{M}^{-1}$. Section 4 is devoted to construct almost peak holomorphic
sections by using the trivial bundle on the tangent cone. The peak holomorphic
sections, which depend on time $t$, will be constructed in Section 5. Theorem
1.3 will be proved in Section 6, 7, according to almost Kähler-Einstein
metrics and almost Kähler-Ricci solitons, respectively, while its proof is
completed in Section 9. In Section 8, we prove Corollary 1.4.

Acknowledgements. The authors would like to thank professor G. Tian for his
encourage and many valuable discussions during working on the paper.

##  2\. Estimates from Kähler Ricci flow

In this section, we give some necessary estimates for the scalar curvatures
and Kähler potentials along the Kähler-Ricci flow. Let $(M,g)$ be an
$n$-dimensional Fano manifold with its Kähler form $\omega_{g}$ in $2\pi
c_{1}(M)$. Let $g_{t}=g(\cdot,t)$ be a solution of normalized Kähler Ricci
flow,

(2.1) $\displaystyle\left\\{\begin{aligned} &\frac{\partial}{\partial
t}g=-{\rm{Ric}}(g)+g,\\\ &g_{0}=g(\cdot,0)=g.\end{aligned}\right.$

Recall an estimate for Sobolev constants of $g_{t}$ by Zhang [34].

######  Lemma 2.1.

Let $g_{t}$ be the solution of (LABEL:Ricci_flow). Suppose that there exists a
Sobolev constant $C_{s}$ of $g$ such that the following inequality holds,

(2.2)
$\displaystyle(\int_{M}f^{\frac{2n}{n-1}}{\rm{dv}}_{g})^{\frac{n-1}{n}}\leq
C_{s}(\int_{M}f^{2}{\rm{dv}}_{g}+\int_{M}|\nabla
f|^{2}{\rm{dv}}_{g}),~{}\forall~{}f\in C^{1}(M).$

Then there exist two uniform constants $A=A(C_{s},-\inf_{M}R(g),V)$ and
$C_{0}=C_{0}(C_{s},-\inf_{M}R(g),V)$ such that for any $f\in C^{1}(M)$ it
holds

(2.3)
$\displaystyle(\int_{M}f^{\frac{2n}{n-1}}{\rm{dv}}_{g_{t}})^{\frac{n-1}{n}}\leq
A(\int_{M}(|\nabla f|^{2}+(R_{t}+C_{0})f^{2}){\rm{dv}}_{g_{t}},$

where $R_{t}$ are scalar curvatures of $g_{t}$.

By using the Moser iteration method, we have

######  Lemma 2.2.

Let $\Delta=\Delta_{t}$ be Lapalace operators associated to $g_{t}$. Suppose
that $f\geq 0$ satisfies

(2.4) $\displaystyle(\frac{\partial}{\partial t}-\Delta)f\leq
af,~{}\forall~{}t\in(0,1),$

where $a\geq 0$ is a constant. Then for any $t\in(0,1)$, it holds

$\displaystyle\sup_{x\in M}f(x,t)$ (2.5)
$\displaystyle\leq\frac{C}{t^{\frac{n+1}{p}}}(\int_{\frac{t}{2}}^{t}\int_{M}|f(x,\tau)|^{p}\text{{\rm{dv}}}_{g_{\tau}}d\tau)^{\frac{1}{p}},$

where $C=C(a,p,C_{s},-\inf R(g),V)$ , $p\geq 1$ and $C_{s}$ is the Sobolev
constant of $g$ in (2.2).

###### Proof.

Multiplying both sides of (2.4) by $f^{p}$, we have

$\displaystyle\int_{M}f^{p}f^{\prime}_{\tau}\text{{\rm{dv}}}_{g_{\tau}}-\int_{M}f^{p}\Delta
f\text{{\rm{dv}}}_{g_{\tau}}\leq a\int_{M}f^{p+1}.$

Taking integration by parts, we get

$\displaystyle\frac{1}{p+1}\int_{M}(f^{p+1})^{\prime}_{\tau}\text{{\rm{dv}}}_{g_{\tau}}+\frac{4p}{(p+1)^{2}}\int_{M}|\nabla
f^{\frac{p+1}{2}}|^{2}\text{{\rm{dv}}}_{g_{\tau}}$ $\displaystyle\leq
a\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}.$

Since

$\displaystyle\frac{d}{d\tau}\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}=\int_{M}(f^{p+1})^{\prime}_{\tau}\text{{\rm{dv}}}_{g_{\tau}}+\int_{M}f^{p+1}(n-R)\text{{\rm{dv}}}_{g_{\tau}},$

we deduce

$\displaystyle\frac{1}{p+1}\frac{d}{d\tau}\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}+\frac{1}{p+1}\int_{M}Rf^{p+1}\text{{\rm{dv}}}_{g_{\tau}}$
$\displaystyle+\frac{4p}{(p+1)^{2}}\int_{M}|\nabla
f^{\frac{p+1}{2}}|^{2}\leq(a+\frac{n}{p+1})\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}.$

It turns

$\displaystyle\frac{d}{d\tau}\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}+\int_{M}(R+C_{0})f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}+2\int_{M}|\nabla
f^{\frac{p+1}{2}}|^{2}$ (2.6)
$\displaystyle\leq((p+1)a+n+C_{0})\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}.$

For any $0\leq\sigma^{\prime}\leq\sigma\leq 1$, we define

$\displaystyle\psi(\tau)=\begin{cases}0,\tau\leq\sigma^{\prime}t\\\
\frac{\tau-\sigma^{\prime}t}{(\sigma-\sigma^{\prime})t},\sigma^{\prime}t\leq\tau\leq\sigma
t\\\ 1,\sigma t\leq\tau\leq t.\end{cases}$

Then by (2), we have

$\displaystyle\frac{d}{d\tau}(\psi\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}})+\psi\int_{M}[(R+C_{0})f^{p+1}+2|\nabla
f^{\frac{p+1}{2}}|^{2}]\text{{\rm{dv}}}_{g_{\tau}}$
$\displaystyle\leq[\psi((p+1)a+n+C_{0})+\psi^{\prime}]\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}.$

It follows

$\displaystyle\sup_{\sigma t\leq\tau\leq
t}\int_{M}f^{p+1}d\text{{\rm{v}}}_{g_{\tau}}+\int_{\sigma
t}^{t}\int_{M}[(R+C_{0})f^{p+1}+2|\nabla
f^{\frac{p+1}{2}}|^{2}]d\text{{\rm{v}}}_{g_{\tau}}$
$\displaystyle\leq((p+1)a+n+C_{0}+\frac{1}{(\sigma-\sigma^{\prime})t})\int_{\sigma^{\prime}t}^{t}\int_{M}f^{p+1}d\text{{\rm{v}}}_{g_{\tau}}.$

Thus by Lemma 2.1, we get

$\displaystyle\int_{\sigma
t}^{t}\int_{M}f^{(p+1)(1+\frac{1}{n})}\text{{\rm{dv}}}_{g_{\tau}}$
$\displaystyle\leq(\int_{\sigma
t}^{t}\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}})^{\frac{1}{n}}(\int_{M}f^{(p+1)\frac{n}{n-1}})^{\frac{n-1}{n}}$
$\displaystyle\leq[(\sup_{\sigma t\leq\tau\leq
t}\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}}]^{\frac{1}{n}}\int_{\sigma
t}^{t}A\int_{M}[(R+C_{0})f^{p+1}+2|\nabla
f^{\frac{p+1}{2}}|^{2}]\text{{\rm{dv}}}_{g_{\tau}}$ (2.7) $\displaystyle\leq
A((p+1)a+n+C_{0}+\frac{1}{(\sigma-\sigma^{\prime})t})^{\frac{n+1}{n}}(\int_{\sigma^{\prime}t}^{t}\int_{M}f^{p+1}\text{{\rm{dv}}}_{g_{\tau}})^{\frac{n+1}{n}}.$

By choosing
$\sigma^{\prime}=\frac{1}{2}+\frac{1}{4}\sigma_{k},\sigma=\frac{1}{2}+\frac{1}{4}\sigma_{k+1}$,
where $\sigma_{k}=\sum_{l=0}^{k}(\frac{1}{2}))^{l}-1$, and replacing $p$ by
$p_{k+1}=(p_{k}+1)^{\frac{n+1}{n}}-1$ with $p_{0}=p\geq 0$ in (2), then
iterating $k$ we will get the desired estimate (2.2). ∎

By Lemma 2.2, we prove

######  Proposition 2.3.

Let $u=u_{t}$ and $R=R_{t}$ be Ricci potentials and scalar curvatures of
solutions $g_{t}$ in (LABEL:Ricci_flow), respectively. Suppose that $(M,g)$
satisfies

(2.8) $\displaystyle{\rm Ric}(g)\geq-\Lambda^{2}g~{}{\rm and}~{}{\rm
diam}(M,g)\leq D.$

Then there exists a constant $C(n,,\Lambda,D)$ such that

$\displaystyle|\nabla u|^{2}(x,t)$ (2.9)
$\displaystyle\leq\frac{C}{t^{(n+1)(n+\frac{3}{2})}}\int_{\frac{t}{2}}^{t}\int_{M}|R-n|\text{{\rm{dv}}}_{g_{\tau}},~{}~{}\forall~{}0<t\leq
1$

and

$\displaystyle|R-n|(x,t)$ (2.10)
$\displaystyle\leq\frac{C}{t^{(n+1)(n+\frac{3}{2})+n}}\int_{\frac{t}{2}}^{t}\int_{M}|R-n|\text{{\rm{dv}}}_{g_{\tau}},~{}~{}\forall~{}0<t\leq
1.$

###### Proof.

By a direct computation, we have the the following evolution formulas for
$|\nabla u|$ and $R$, respectively,

(2.11) $\displaystyle(\frac{\partial}{\partial t}-\Delta)|\nabla u|^{2}$
$\displaystyle=\Delta|\nabla u|^{2}-|\nabla\nabla
u|^{2}-|\nabla\bar{\nabla}u|^{2}+|\nabla u|^{2}\leq|\nabla u|^{2}$

and

(2.12) $\displaystyle(\frac{\partial}{\partial t}-\Delta)R$
$\displaystyle=\Delta R+R-n+|\text{Ric}(g)-g|^{2}.$

It follows

$\displaystyle(\frac{\partial}{\partial t}-\Delta)(R+n\Lambda+|\nabla u|^{2})$
(2.13) $\displaystyle=R-n-|\nabla\nabla u|^{2}+|\nabla u|^{2}\leq
R+n\Lambda+|\nabla u|^{2}.$

Note that $R(g_{t})+n\Lambda\geq 0$ by the maximum principle. It was proved in
[11] that there exists a uniform constant $C=C(\Lambda,D)$ such that

$\int_{0}^{1}\int_{M}(R+n\Lambda+|\nabla u|^{2})d\text{{\rm{v}}}_{g}dt\leq C.$

Then by Lemma 2.2, we obtain

(2.14) $\displaystyle(R+n\Lambda+|\nabla u|^{2})(x,t)\leq\frac{C}{t^{n+1}}.$

In particular,

(2.15) $\displaystyle|\nabla u|^{2}(x,t)\leq\frac{C}{t^{n+1}},~{}~{}\text{ and
}~{}~{}~{}R\leq\frac{C}{t^{n+1}}.$

Next we estimate the $C^{0}$-norm of $u_{t}$. By Lemma 2.1 we have the Sobolev
inequality,

$\displaystyle(\int_{M}f^{\frac{2n}{n-1}}\text{{\rm{dv}}}_{g_{t}})^{\frac{n-1}{n}}$
$\displaystyle\leq A(\int_{M}(|\nabla
f|^{2}+(R(x,t)+C_{0})f^{2})\text{{\rm{dv}}}_{g_{t}})$ $\displaystyle\leq
A(\int_{M}(|\nabla f|^{2}+\frac{C}{t^{n+1}}f^{2})\text{{\rm{dv}}}_{g_{t}}).$

The inequality implies (cf. [10], [33]),

$\displaystyle{\rm{vol}}(B(x,1))\geq Ct^{n(n+1)},~{}~{}\forall~{}x\in M.$

Since ${\rm{vol}}(M)=V$, it is easy to obtain

$\displaystyle{\rm{diam}}(M,g_{t})\leq\frac{V}{Ct^{n(n+1)}}.$

Thus by (2.15), we get

(2.16)
$\displaystyle\text{osc}_{M}u(x,t)\leq\frac{C}{t^{(n+1)(n+\frac{1}{2})}}.$

By (2.16), we can improve (2.15) to (2.3). In fact, by applying Lemma 2.2 to
(2.11), we have

$\displaystyle|\nabla u|^{2}(x,t)$
$\displaystyle\leq\frac{C}{t^{n+1}}\int_{\frac{t}{2}}^{T}\int_{M}|\nabla
u|^{2}\text{{\rm{dv}}}_{g_{\tau}}d\tau$
$\displaystyle=\frac{C}{t^{n+1}}\int_{\frac{t}{2}}^{t}\int_{M}-u\Delta
u\text{{\rm{dv}}}_{g_{\tau}}d\tau$
$\displaystyle\leq\frac{C}{t^{n+1}}\text{osc}_{(x,\tau)\in
M\times[\frac{t}{2},t]}|u|(x,\tau)\int_{\frac{t}{2}}^{t}\int_{M}|R-n|\text{{\rm{dv}}}_{g_{\tau}}d\tau$
(2.17)
$\displaystyle\leq\frac{C^{\prime}}{t^{(n+1)(n+\frac{3}{2})}}\int_{\frac{t}{2}}^{t}\int_{M}|R-n|\text{{\rm{dv}}}_{g_{\tau}}d\tau,$

where the constant $C^{\prime}$ depends only on $n,~{}\Lambda,~{}D$. This
proves (2.3).

To get (2.3), we use the evolution equation as same as (2),

$\displaystyle(\frac{\partial}{\partial t}-\Delta)(|\nabla u|^{2}+R-n)$
$\displaystyle=R-n-|\nabla\nabla u|^{2}+|\nabla u|^{2}$
$\displaystyle\leq|\nabla u|^{2}+R-n.$

Then applying Lemma 2.2, we see

$\displaystyle(|\nabla u|^{2}+R-n)_{+}$
$\displaystyle\leq\frac{C}{t^{n+1}}\int_{\frac{t}{2}}^{t}\int_{M}||\nabla
u|^{2}+R-n|\text{{\rm{dv}}}_{g_{\tau}}d\tau$
$\displaystyle\leq\frac{C}{t^{(n+1)(n+\frac{3}{2})}}\int_{\frac{t}{2}}^{t}\int_{M}|R-n|\text{{\rm{dv}}}_{g_{\tau}}d\tau.$

Thus by (2.3), it follows

$\displaystyle(R-n)_{+}$ (2.18)
$\displaystyle\leq\frac{C}{t^{(n+1)(n+\frac{3}{2})}}\int_{\frac{t}{2}}^{t}\int_{M}|R-n|d\text{v}_{g_{\tau}}d\tau:=A(t).$

On the other hand, by the evolution equation (2.12) of $R$,

$\displaystyle(\frac{\partial}{\partial
t}-\Delta)R=R-n+|\nabla\bar{\nabla}u|^{2},$

we have

$\displaystyle(\frac{\partial}{\partial t}-\Delta)(A(T)+n-R)\leq A(T)+n-R.$

Hence applying Lemma 2.2 again, we get

$\displaystyle(A(t)+n-R)(x,t)$
$\displaystyle\leq\frac{C^{\prime\prime}}{t^{n+1}}\int_{\frac{t}{2}}^{t}\int_{M}(A(t)+n-R)\text{{\rm{dv}}}_{g_{t}}dt$
$\displaystyle\leq\frac{C^{\prime\prime}}{t^{n+1}}\int_{\frac{t}{2}}^{T}\int_{M}|n-R|\text{{\rm{dv}}}_{g_{\tau}}d\tau+\frac{A(t)VC}{t^{n}}.$

Therefore, inserting (2) into the above estimate, we obtain (2.3). ∎

##  3\. Estimates for holomorphic sections

In this section, we use the estimates in Section 2 to give the
$C^{0}$-estimate and the gradient estimate for holomorphic sections with
respect to $g_{t}$. Let $(M^{n},g)$ be a Fano manifold and $L=K_{M}^{-1}$ its
anti-canonical line bundle with induced Hermitian metric $h$ by $g$. We begin
with the following lemma.

######  Lemma 3.1.

Suppose that the Ricci potential $u$ of $g$ satisfies

(3.1) $\displaystyle\|\nabla u\|_{g}\leq 1.$

Then for $s\in H^{0}(M,L^{l})$ we have

(3.2) $\displaystyle\|s\|_{h}+l^{-\frac{1}{2}}\|\nabla s\|_{h}\leq
C(C_{s},n)l^{\frac{n}{2}}(\int_{M}|s|^{2}{\rm dv}_{g})^{\frac{1}{2}},$

where $C_{s}$ is the Sobolev constant of $(M,g)$.

###### Proof.

Note that

$\displaystyle\Delta|s|_{h}^{2}=|\nabla s|_{h}^{2}-nl|s|_{h}^{2}.$

It follows

(3.3) $\displaystyle-\Delta|s|_{h}^{2}\leq nl|s|_{h}^{2}.$

Thus applying the standard Moser iteration method to (3.3), we get

(3.4) $\displaystyle\|s\|_{h}\leq
C(C_{s},n)l^{\frac{n}{2}}(\int_{M}|s|^{2}{\rm dv}_{g})^{\frac{1}{2}}.$

On the other hand, we have the following Bochner formula,

$\Delta|\nabla s|_{h}^{2}=|\nabla\nabla s|^{2}+|\bar{\nabla}\nabla
s|^{2}-(n+2)l|\nabla s|^{2}+\langle\text{Ric}(\nabla s,.),\nabla s\rangle.$

Then we can also apply the Moser iteration to obtain a $L^{\infty}$-estimate
for $|\nabla s|_{h}^{2}$ as done for $|s|_{h}^{2}$. In fact, it suffices to
deal with the extra integral terms like $\langle\text{Ric}(\nabla s,.),\nabla
s\rangle|\nabla s|^{2p}$. But those terms can be controlled by the integral of
$(|\nabla\nabla s|^{2}+|\bar{\nabla}\nabla s|^{2})|\nabla s|_{h}^{2p}$ by
taking integral by parts with the help of the condition (3.1) (cf. [32],
[27]). As a consequence, we obtain

(3.5) $\displaystyle\|\nabla s\|_{h}\leq
C(C_{s},n)l^{\frac{n}{2}}(\int_{M}|\nabla s|^{2}{\rm
dv}_{g})^{\frac{1}{2}}\leq C(C_{s},n)l^{\frac{n+1}{2}}(\int_{M}|s|^{2}{\rm
dv}_{g})^{\frac{1}{2}}.$

Therefore, combining (3.4) and (3.5), we derive (3.6).

∎

######  Remark 3.2.

Using the same argument in Lemma 3.1, we can prove: If $(M,g)$ satisfies

${\rm
Ric}(\omega_{g})\geq-\Lambda^{2}\omega_{g}+\sqrt{-1}\partial\bar{\partial}u,$

for some $u$ with $|\nabla u|_{g}\leq A$, then

(3.6) $\displaystyle\|s\|_{h}+l^{-\frac{1}{2}}\|\nabla s\|_{h}\leq
C(C_{s},A,\Lambda)l^{\frac{n}{2}}(\int_{M}|s|^{2}{\rm
dv}_{g})^{\frac{1}{2}},~{}\forall s\in H^{0}(M,L^{l}).$

######  Lemma 3.3.

Let $(M,g)$ be a Fano manifold which satisfies (3.1) as in Lemma 3.1. Let
$\bar{\partial}$-operator be defined for smooth sections on $(M,L^{l})$ (
$l\geq 4n$ ) with the induced metric $h$. Then for any $\sigma\in
C^{\infty}(\Gamma(M,L^{l}))$, there exists a solution $v\in
C^{\infty}(\Gamma(M,L^{l}))$ such that $\bar{\partial}v=\bar{\partial}\sigma$
with property:

(3.7) $\displaystyle\int_{M}|v|^{2}\leq
4l^{-1}\int_{M}|\bar{\partial}\sigma|^{2}.$

###### Proof.

The existence part comes from the Hömander $L^{2}$-theory. We suffice to
verify (3.7), which is equal to prove that the first eigenvalue
$\lambda_{1}(\bar{\partial},L^{l})$ of $\Delta_{\bar{\partial}}$ is greater
than $\frac{l}{4}$, where $\Delta_{\bar{\partial}}$ denotes the Lapalce
operator defined on $L^{2}(T^{*}M\bigotimes L^{l})$.

Note that the following two identities hold for any
$\theta\in\Omega^{0,1}(L^{l})$,

$\Delta_{\bar{\partial}}\theta=\bar{\nabla}^{*}\bar{\nabla}\theta+\text{Ric}(\theta,.)+l\theta$

and

$\Delta_{\bar{\partial}}\theta=\nabla^{*}\nabla\theta-(n-1)l\theta.$

It follows

(3.8)
$\displaystyle\Delta_{\bar{\partial}}\theta=(1-\frac{1}{2n})\bar{\nabla}^{*}\bar{\nabla}+(1-\frac{1}{2n})\text{Ric}(\theta,.)+\frac{1}{2n}\nabla^{*}\nabla\theta+\frac{l}{2}\theta.$

Then with the help of condition (3.1), a direct computation shows

$\displaystyle\int_{M}\langle\Delta_{\bar{\partial}}\theta,\theta\rangle$
$\displaystyle=(1-\frac{1}{2n})\int_{M}|\bar{\nabla}\theta|^{2}+\frac{1}{2n}\int_{M}|\nabla\theta|^{2}+\frac{l}{2}\int_{M}|\theta|^{2}$
$\displaystyle+(1-\frac{1}{2n})\int_{M}(|\theta|^{2}+\langle\nabla\bar{\nabla}u(\theta,.),\theta\rangle)$
$\displaystyle\geq(1-\frac{1}{2n})\int_{M}|\bar{\nabla}\theta|^{2}+\frac{1}{2n}\int_{M}|\nabla\theta|^{2}+\frac{l}{2}\int_{M}|\theta|^{2}$
$\displaystyle+(1-\frac{1}{2n})\int_{M}|\theta|^{2}-(1-\frac{1}{2n})\int_{M}\langle\bar{\nabla}u,(\langle\nabla\theta,\theta\rangle+\langle\theta,\bar{\nabla}\theta\rangle)\rangle$
$\displaystyle\geq(1-\frac{1}{2n})\int_{M}|\bar{\nabla}\theta|^{2}+\frac{1}{2n}\int_{M}|\nabla\theta|^{2}+\frac{l}{2}\int_{M}|\theta|^{2}$
$\displaystyle+(1-\frac{1}{2n})\int_{M}|\theta|^{2}-(1-\frac{1}{2n})\int_{M}[\frac{1}{2n}(|\bar{\nabla}\theta|^{2}+|\nabla\theta|^{2})+n|\theta|^{2}]$
(3.9) $\displaystyle\geq(\frac{l}{2}-n)\int_{M}|\theta|^{2}.$

Now we can choose $l\geq 4n$ to get that
$\lambda_{1}(\bar{\partial},L)\geq\frac{l}{4}$ as required. ∎

######  Remark 3.4.

If the upper bound of $|\nabla u|$ is replaced by a constant $C$, the
coefficient at the last inequality in (3) will be $\frac{l}{2}-nC^{2}$ . Then
by choosing $l\geq 4nC^{2}$, one can also get (3.7). This was proved in [28].

Recall that a sequence of almost Kähler-Einstein Fano manifolds
$(M_{i},J_{i},$ $g^{i})$ satisfy:

$\displaystyle i)~{}{\rm Ric}(g^{i})\geq-\Lambda^{2}g^{i}~{}\text{and}~{}{\rm
diam}(M_{i},g^{i})\leq D;$ $\displaystyle ii)~{}\int_{M_{i}}|{\rm
Ric}(g^{i})-g^{i}|\text{{\rm{dv}}}_{g^{i}}\rightarrow 0;$ (3.10)
$\displaystyle
iii)~{}\int_{0}^{1}\int_{M_{i}}|R(g^{i}_{t})-n|\text{{\rm{dv}}}_{g^{i}_{t}}dt\rightarrow
0,~{}{\rm as}~{}i\to\infty.$

Here $g^{i}$ are normalized so that $\omega_{g^{i}}\in 2\pi c_{1}(M_{i})$ and
$g^{i}_{t}$ are the solutions of (LABEL:Ricci_flow) with the initial metrics
$g^{i}$. We note that $\text{vol}(M_{i},g^{i})=(2\pi)^{n}c_{1}(M_{i})^{n}\geq
V$ for some uniform constant $V$ by the normalization.

Applying Lemma 3.1 and Lemma 3.3 to almost Kähler-Einstein manifolds with the
help of gradient estimate (2.3) in Proposition 2.3, we have the following
proposition.

######  Proposition 3.5.

Let $\\{(M_{i},g^{i})\\}$ be a sequence of almost Kähler Einstein metrics
which satisfy (3). Then for any $t\in(0,1)$ there exists an integer $N=N(t)$
such that for any $i\geq N$ and $l\geq 4n$ it holds,

(3.11) $\displaystyle\|s\|_{h_{t}^{i}}+l^{-\frac{1}{2}}\|\nabla
s\|_{h_{t}^{i}}\leq
Cl^{\frac{n}{2}}(\int_{M}|s|^{2}\text{{\rm{dv}}}_{g_{t}^{i}})^{\frac{1}{2}}$

and

(3.12) $\displaystyle\int_{M_{i}}|v|_{h_{t}^{i}}^{2}\leq
4l^{-1}\int_{M_{i}}|\bar{\partial}\sigma|^{2}.$

Here $s\in H^{0}(M_{i},K_{M_{i}}^{-l})$, the norms of $|\cdot|_{h^{i}_{t}}$
are induced by $g_{t}^{i}$, and $C$ is a uniform constant independent of $t$.

###### Proof.

A well-known result shows that the Sobolev constants $C_{s}$ of $(M_{i},$
$g^{i})$ depend only on the constants $\Lambda,D$ and $V$. Then by (2.3) in
Proposition 2.3, for any $t\in(0,1)$, there exists $N=N(t)$ such that

$\|\nabla u^{i}\|_{h^{i}_{t}}\leq 1,~{}\forall~{}i\geq N,$

where $u^{i}$ are Ricci potentials of $g^{i}_{t}$ . Thus we can apply Lemma
3.1 to get (3.11). Similarly, we can get (3.12) by Lemma 3.3. ∎

##  4\. Construction of locally approximate holomorphic sections

Let $\\{(M_{i},g^{i})\\}$ be a sequence of almost Kähler-Einstein manifolds as
in Section 3 and $(M_{\infty},g_{\infty})$ its Gromov-Hausdorff limit. It was
proved by Tian and B. Wang that the regular part $\mathcal{R}$ of $M_{\infty}$
is an open Kähler manifold and the codimension of singularities of
$M_{\infty}$ is at least 4 [26]. Moreover, according to Proposition 5.1 in
that paper, we have

######  Lemma 4.1.

Let $x\in M_{\infty}$. Then there exist constants $\epsilon=\epsilon(n)$ and
$r_{0}=r_{0}(n,C)$ such that if ${\rm
vol}(B_{x}(r))\geq(1-\epsilon)\omega_{2n}r^{2n}$ for some $r\leq r_{0}$, then
$B_{x}(\frac{r}{2})\subseteq\mathcal{R}$, and

$\displaystyle{\rm Ric}(g_{\infty})=g_{\infty},\,\|\nabla^{l}{\rm
Rm}\|_{C^{0}(B_{x}(\frac{r}{2}))}\leq\frac{C}{r^{l+2}},$

where the constant $C$ depends only on $l$, and the constants $\Lambda$ and
$D$ in (3).

Recall that a tangent cone $C_{x}$ at $x\in M_{\infty}$ is a Gromov-Hausdorff
limit defined by

(4.1)
$\displaystyle(C_{x},g_{x},x)=\lim_{j\rightarrow\infty}(M_{\infty},\frac{g_{\infty}}{r_{j}^{2}},x),$

where $\\{r_{j}\\}$ is some sequence which goes to $0$. Without the loss of
generality, we may assume that $l_{j}=\frac{1}{r_{j}^{2}}$ are integers. Since
$(C_{x},g_{x},x)$ is a metric cone, $g_{x}={\rm hess}\frac{\rho_{x}^{2}}{2}$,
where $\rho_{x}={\rm dist}(x,\cdot)$ is a distance function staring from $x$
in $C_{x}$.

Denote the regular part of $(C_{x},g_{x},x)$ by $\mathcal{CR}$, which consists
of points in $C_{x}$ with flat cones. By Lemma 4.1, we prove

######  Lemma 4.2.

$\mathcal{CR}$ is an open Kähler-Ricci flat manifold. Moreover, for any
compact set $K\subset\mathcal{CR}$, there exist a sequence of
$(K_{j}\subset\mathcal{R},\frac{1}{r_{j}^{2}}g_{\infty})$ which converges to
$K$ in $C^{\infty}$-topology.

###### Proof.

Let $\epsilon$ be a small number chosen as in Lemma 4.1. Then for any
$y\in\mathcal{CR}$, there exists some small $r$ such that
$\hat{B}_{y}(r)\subset C_{x}$ and

$\displaystyle\text{vol
}(\hat{B}_{y}(r))\geq(1-\frac{\epsilon}{2})\omega_{2n}r^{2n}.$

Thus there exists a sequence of $y_{\alpha}\in C_{x}$ such that

$\displaystyle\text{vol
}(B_{y_{\alpha}}(rr_{\alpha}))\geq(1-\epsilon)\omega_{2n}(rr_{\alpha})^{2n},$

where the sequence $\\{r_{\alpha}\\}$ is chosen as in (4.1). By Lemma 4.1, it
follows

$\displaystyle\|{\rm
Rm}(\tilde{g}_{\infty})\|_{C^{l}(\tilde{B}_{y_{\alpha}}(\frac{r}{2}))}\leq\frac{C_{l}}{r^{l+2}},$

where $\tilde{g}_{\infty}=\frac{g_{\infty}}{r_{\alpha}^{2}}$ and
$\tilde{B}_{y_{\alpha}}(\frac{r}{2})\subset M_{\infty}$ is a
$\frac{r}{2}$-geodesic ball with respect to the metric $\tilde{g}_{\infty}$.
Hence, by the Cheeger-Gromov compactness theorem [GW],
$(\tilde{B}_{y_{\alpha}}(\frac{r}{2}),\tilde{g}_{\infty})$ converge to
$(\hat{B}_{y}(\frac{r}{2}),g_{x})$ in $C^{\infty}$-topology. In particular,
$B_{y_{\alpha}}(\frac{r_{\alpha}r}{2}$
$)\subset\mathcal{R}$ and $\hat{B}_{y}(\frac{r}{2})\subset\mathcal{CR}$. This
implies that $\mathcal{CR}$ is an open manifolds. Moreover, $\mathcal{CR}$ is
a Kähler-Ricci flat manifold since each
$(B_{y_{\alpha}}(\frac{r_{\alpha}r}{2}),{g}_{\infty})$ is an open Kähler-
Einstein manifold. If $K$ is a compact set of $\mathcal{CR}$, then by taking
finite small geodesic covering balls, one can find a sequence
$\\{(K_{j}\subset\mathcal{R},\frac{1}{r_{j}^{2}}g_{\infty})\\}$ which
converges to $(K,g_{x})$ in $C^{\infty}$-topology. ∎

Define an open set $V(x;\delta)$ of $\mathcal{CR}$ by

(4.2) $\displaystyle V(x;\delta)=\\{y\in C_{x}|~{}{\rm
dist}(y,S_{x})\geq\delta,d(y,x)\leq\frac{1}{\delta}\\},$

where $S_{x}=C_{x}\setminus\mathcal{CR}$. The following lemma shows that there
exists a “nice” cut-off function on $C_{x}$ which supported on $V(x;\delta)$.

######  Lemma 4.3.

For any $\eta,\delta>0$, there exist some $\delta_{1}<\delta$ and a cut-off
function $\beta$ on $C_{x}$ which supported in $V(x;\delta_{1})$ with
property: $\beta=1,\text{ in }V(x;\delta);$

$\displaystyle\int_{C_{x}}|\nabla\beta|^{2}e^{-\frac{\rho_{x}^{2}}{2}}d{\rm
v}_{g_{x}}\leq\eta.$

Lemma 4.3 is in fact a corollary of following fundamental lemma.

######  Lemma 4.4.

Let $(X^{m},d,\mu)$ be a measured metric space such that

(4.3) $\displaystyle\mu(B_{y}(r)\leq C_{0}r^{m},~{}\forall~{}r\leq 1,~{}y\in
X.$

Let $Z$ be a closed subset of $X$ with $\mathcal{H}^{m-2}(Z)=0$. Suppose that
there exists a nonnegative function $f\leq 1$ on $X$ such that

$\int_{X}fd\mu\leq 1.$

Then for any $x\in X$, $\eta>0$ and $\delta>0$, there exist a positive
$\delta_{1}\leq\delta$ and a cut-off function $\beta\geq 0$, which supported
in $B_{x}(\frac{1}{\delta_{1}})\setminus Z_{\delta_{1}}$ with property:
${\beta}=1,\text{ in }B_{x}(\frac{1}{\delta})\setminus Z_{\delta};$

(4.4) $\displaystyle\int_{X}f|{\rm Lif}(\beta)|^{2}d\mu\leq\eta.$

Here $Z_{\delta_{1}}=\\{x^{\prime}\in X|~{}{\rm
dist}(x^{\prime},Z)\leq\delta_{1}\\}$ and ${\rm
Lip}(\beta)(z)=\sup_{w\rightarrow z}|\frac{f(w)-f(z)}{d(w,z)}|$.

###### Proof.

Let $R\geq\sqrt{\frac{8}{\eta}}+\frac{2}{\delta}$. Since
$\mathcal{H}^{m-2}(Z)=0$, then for any $\kappa>0$, we can take finite geodesic
balls $B_{x_{i}}(r_{i})$ $(r_{i}\leq\delta)$ with $x_{i}\in Z$ to cover
$B_{x}(R)\bigcap Z$ such that

$\displaystyle\Sigma_{i}r_{i}^{m-2}\leq\kappa.$

Let $\zeta:\mathbb{R}\rightarrow\mathbb{R}$ be a cut-off function which
satisfies:

$\displaystyle\zeta(t)=1,\text{ for }t\leq\frac{1}{2};\zeta(t)=0,\text{ for
}t\geq 1;|\zeta^{\prime}(t)|\leq 2.$

Set

$\chi(y)=\min_{i}\\{1-\zeta(\frac{d(y,x_{i})}{r_{i}})\\}$

and

$\beta(y)=\zeta(\frac{\epsilon}{d(y,x)})\zeta(\frac{d(y,x)}{R})\chi(y),$

where $\epsilon\leq\frac{\delta}{2}$. Then it is easy to see that $\beta$ is
supported in $B_{x}(R)\setminus\cup B_{x_{i}}(\frac{r_{i}}{2})$ with
$\beta\equiv 1$ in $B_{x}(\frac{1}{\delta})\setminus Z_{\delta}$. Moreover,

$\displaystyle\int_{X}f|Lif\beta|^{2}d\mu$ $\displaystyle\leq
4C_{0}\Sigma_{i}r_{i}^{-2}r_{i}^{m}+4C_{0}\epsilon^{m-2}+\frac{4}{R^{2}}$
$\displaystyle\leq 4C_{0}\kappa+4C_{0}\epsilon^{2n-2}+\frac{\eta}{2}.$

Thus, if we choose $\epsilon$ and $\kappa$ such that
$4C_{0}\kappa+4C_{0}\epsilon^{2n-2}\leq\frac{\eta}{2}$, then we get (4.4). By
choosing $\delta_{1}\leq\min\\{\frac{\epsilon}{2},\frac{1}{2R}\\}$ such that

$Z_{\delta_{1}}\cap B_{x}(R)\subseteq\cup B_{x_{i}}(\frac{r_{i}}{2}),$

we can also get ${\rm supp}(\beta)\subset B_{x}(\frac{1}{\delta_{1}})\setminus
Z_{\delta_{1}}.$ Hence $\beta$ satisfies all conditions required in the lemma.

∎

###### Proof of Lemma 4.3.

Applying Lemma 4.4 to $X=C_{x},Z=S_{x},f=e^{-\frac{\rho_{x}^{2}}{2}}$, we get
the lemma. ∎

By Lemma 4.2, we see that for any $\delta>0$ there exists a sequence of
$K_{j}\subset(M_{\infty},r_{j}^{-2}g_{\infty})$ which converge to
$V(x;\delta)$. Let $L_{0}=(C_{x},\mathbb{C})$ be the trivial holomorphic
bundle over $C_{x}$ with a hermitian metric
$h_{0}=e^{-\frac{\rho_{x}^{2}}{2}}$. Then $h_{0}$ induces the Chern connection
$\nabla_{0}$ with its curvature

${\rm Ric}(L_{0},\nabla_{0})=g_{x}.$

In the following we show that a sufficiently large multiple line bundles of
$K^{-1}_{\mathcal{R}}|_{K_{j}}$ will approximate to $L_{0}$ over
$V(x;\delta)$. This is in fact an application of the following fundamental
lemma.

######  Lemma 4.5.

Let $(V,g)$ be a $C^{2}$ open Riemannian manifold and
$U,U^{\prime}\subset\subset V$ are two pre-compact open subsets of $V$ with
$U\subset\subset U^{\prime}$. Then for any positive number $\epsilon$, there
exist a small number $\delta=\delta(U^{\prime},g,\epsilon)$ and a positive
integer $N=N(U,g,\epsilon)$, which depends on the fundamental group of $U$,
the metric $g$ on $U$, and the small $\epsilon$ such that the following is
true: if a hermitian complex line bundle $(L,h)$ over $V$ with associated
connection $\nabla$ satisfies

(4.5) $\displaystyle|{\rm Ric}^{\nabla}|_{g}\leq\delta,~{}{\rm
in}~{}U^{\prime},$

then there exist a positive integer $l\leq N$ and a section $\psi$ of
$L^{\otimes l}$ over $U$ with $|\psi|_{h}\equiv 1$ which satisfies

(4.6) $\displaystyle|D^{\nabla\otimes l}\psi|_{h,g}\leq\epsilon,~{}{\rm
in}~{}U.$

###### Proof.

The proof seems standard. First we show that $(L,U)$ is a flat bundle with
respect to some connection. Let $B_{x_{i}}(r_{i})$ ($r_{i}\leq 1$) be finite
convex geodesic balls in $V$ such that $\bar{U}\subset\cup
B_{x_{i}}(r_{i})\subset U^{\prime}$. Then for $y\in B_{x_{i}}(r_{i})$ there
exists a minimal geodesic curve $\gamma_{y}$ in $B_{x_{i}}(r_{i})$, which
connects $x_{i}$ and $y$. Picking any vector $s_{i}\in L_{x_{i}}$ with
$|s_{i}|=1$ and using the parallel transportation, we define a parallel vector
field by

$\displaystyle e_{i}(y)={\rm Para}_{\gamma_{y}}(s_{i}),~{}\forall~{}y\in
B_{x_{i}}(r_{i}).$

In particular, $De_{i}(x_{i})=0$. Let $T$ be a vector field, which is tangent
to $\gamma_{y}$, and $X$ another vector field with $[T,X]=0$. Then

(4.7) $\displaystyle D_{T}[D_{X}e_{i}]=D_{X}[D_{T}e_{i}]+{\rm
Ric}^{\nabla}(T,X)e_{i}={\rm Ric}^{\nabla}(T,X)e_{i}.$

By the condition (4.5), it follows

(4.8) $\displaystyle|De_{i}|_{h,g}\leq
C({U^{\prime}},g)||\text{Ric}^{\nabla}||_{({U^{\prime}},g)}\leq
C({U^{\prime}},g)\delta,~{}{\rm in}~{}B_{x_{i}}(r_{i}).$

This implies that the transformation function $g_{ij}$ of $L$ is nearly
constant in $B_{x_{i}}\cap B_{x_{j}}$. Since the first Chern class lies in the
secondary integral cohomology group, $L$ is topologically trivial as long as
$\delta$ is small, i.e., $c_{1}(L)=0$. Hence, there exist some complex
functions $f_{i}$ over $B_{x_{i}}(r_{i})$ such that

(4.9) $\displaystyle|Df_{i}|\leq C({U^{\prime}},g)\delta<<1,$

and the transition functions for $\tilde{e_{i}}=f_{i}e_{i}$ are constant. Here
$\|\tilde{e_{i}}\|_{h}=\|e_{i}\|_{h}.$ As a consequence, we can define an
associated connection $\nabla^{\prime}$ on $L$ to $h$ such that

$|\tilde{e_{i}}|_{h}=1~{}{\rm and}~{}D^{\nabla^{\prime}}\tilde{e_{i}}=0.$

In fact, if we set $\nabla^{\prime}=\nabla+\alpha\otimes e_{i}$, then locally,

$D^{\nabla^{\prime}}\tilde{e_{i}}=D^{\nabla}(f_{i}e_{i})+\alpha(\tilde{e_{i}})=f_{i}D^{\nabla}e_{i}+df_{i}\otimes
e_{i}+f_{i}\alpha\otimes e_{i}.$

Thus

$\alpha=-\frac{1}{f_{i}}(df_{i}+\langle
f_{i}D^{\nabla}e_{i},e_{i}\rangle_{h})$

which is uniquely determined by requiring
$D^{\nabla^{\prime}}(\tilde{e_{i}})=0$. Therefore, $(L,\nabla^{\prime})$ is a
flat bundle over $U$ with respect to $\nabla^{\prime}$. Moreover, by (4.8) and
(4.9), we have

(4.10) $\displaystyle\|\nabla-\nabla^{\prime}\|_{(U,g)}=\|\alpha\|_{(U,g)}\leq
C({U^{\prime}},g)||\text{Ric}^{\nabla}||_{({U^{\prime}},g)}\leq
C({U^{\prime}},g)\delta.$

Next we note that the holonomy group of a flat bundle over $U$ is an element
of $Hom(\pi_{1}(U),\mathbb{S}^{1})\cong G\times\mathbb{T}^{k}$ for some finite
group $G$ with order $m_{1}$, where $k$ is the Betti number of $\pi_{1}(U)$.
By the pigeon-hole principle, we see that for any $\gamma$-neighborhood
$W\subseteq\mathbb{T}^{k}$ of the identity there exists a positive integer
$m_{2}=m_{2}(\gamma)$ such that for any element $\rho\in\mathbb{T}^{k}$,
$\rho^{a}\in W$ for some number $a$ $(1\leq a\leq m_{2}).$ As a consequence,
for any element $t\in G\times\mathbb{T}^{k}$, there exists $l~{}(1\leq l\leq
N=m_{1}m_{2})$ such that $t^{l}\in W.$ Hence, there exist $l$ and a smooth
section $\psi$ of $L^{\otimes l}$ over $U$ by perturbing a parallel vector
field in $L^{\otimes l}$ such that

$||\psi|_{h}-1|,\|\psi^{{\nabla^{\prime}}^{\otimes l}}\|_{h,g}\leq
C(U^{\prime},g)\gamma(\delta),{\rm in}~{}U.$

Moreover, By (4.10), we can normalize $\psi$ by $|\psi|_{h}\equiv 1$ so that
$(\ref{psi-nabla})$ is true. The lemma is proved. ∎

######  Proposition 4.6.

Let $x\in M_{\infty}$ and $\delta_{1}>0$. Then for any $\epsilon>0$, there
exist a positive integer $N=N(V(x;\delta_{1}),\epsilon)$ and a large integer
$j_{0}$ such that for $j\geq j_{0}$ there exist $l=l(j)\leq N$, and a sequence
of ${K}_{j}\subseteq M_{\infty}$ and a sequence of pairs of isomorphisms
($\phi_{j},\psi_{j}$) with property:

(4.11)
$\displaystyle\begin{CD}L_{0}@>{{\psi}_{j}}>{}>K_{\mathcal{R}}^{-ll_{j}}|_{{K}_{j}}\\\
@V{}V{}V@V{}V{}V\\\ V(x;\delta_{1})@>{{\phi}_{j}}>{}>{K}_{j},\end{CD}$

which satisfy

$\displaystyle{\phi}_{j}^{*}(ll_{j}g_{\infty})\rightarrow g_{x},~{}{\rm
as}~{}j\to\infty,$

and

$|D\psi_{j}|_{g_{x}}\leq\epsilon,~{}{\rm in}~{}V(x;\delta_{1}).$

###### Proof.

Define an open set $U$ of $\mathcal{CR}$ by

$\displaystyle U=U(x;\epsilon_{1},\epsilon_{2},R)=\\{y\in C_{x}|~{}{\rm
dist}(\bar{y},S_{x})\geq\epsilon_{1},\epsilon_{2}\leq d(y,x)\leq R\\},$

where $\bar{y}$ is the projection to the section $Y$ of $C_{x}=C(Y)$. Then
there exist some $\epsilon_{1},\epsilon_{2}$ and $R$ such that

$V(x;\delta_{1})\subseteq U(x;\epsilon_{1},\epsilon_{2},R).$

Moreover, we can choose a sequence of integers $l_{j}=\frac{1}{r_{j}^{2}}$
such that

$(M_{\infty},l_{j}g_{\infty},x)\rightarrow(C_{x},g_{x},x),~{}{\rm
as}~{}j\to\infty.$

Hence by Lemma 4.2, there exist a sequence of $\tilde{K}_{j}\subseteq
M_{\infty}$ and a sequence of diffeomorphisms $\tilde{\phi}_{j}$ from
$U(x;\epsilon_{1},\frac{\epsilon_{2}}{\sqrt{N}},R)$ to $\tilde{K}_{j}$ such
that $\tilde{\phi}_{j}^{*}(l_{j}g_{\infty})\rightarrow g_{x},$ where
$N=N(U,g_{x},\epsilon)$ is a large integer as determined in Lemma 4.5.

Let $h_{\infty}$ be the induced hermitian metric on $K_{\mathcal{R}}^{-1}$ by
$g_{\infty}$ on the regular part $\mathcal{R}$ of $M_{\infty}$. Let

$(L_{j},h)=\tilde{\phi}_{j}^{*}(K_{\mathcal{R}}^{-l_{j}},h_{\infty}^{\otimes
l_{j}})\otimes(L_{0},h_{0})^{*}$

be product complex line bundle on $U$, where $h$ is an induced hermitian
metric by $h_{\infty}$ and $h_{0}$ with associated connection $\nabla_{j}$ on
$L_{j}$ for each $j$. Clearly,

$\|\text{Ric}^{\nabla_{j}}\|_{(U^{\prime},g_{x})}\leq\delta<<1,$

as long as $j$ is large enough, where $U^{\prime}\subset\subset\mathcal{C}R$
is an open set such that $\bar{U}\subset\subset U^{\prime}$. Applying Lemma
4.5 to $L_{j}$ over $U^{\prime}$, we see that there exist some positive
integer $l=l(j)\leq N$ and a section $\psi^{\prime}$ on $L_{j}^{\otimes l}$
such that

$|D^{\nabla_{j}^{\otimes l}}\psi^{\prime}|_{(U,g_{x})}\leq\epsilon.$

Let $Y_{\epsilon_{1}}=U(x;\epsilon_{1},\epsilon_{2},R)\bigcap Y$ and
$\tilde{\psi}$ an extension section over
$U(x;\epsilon_{1},\frac{\epsilon_{2}}{\sqrt{l}},$
$R)$ of the restriction of $\psi^{\prime}$ on $Y_{\epsilon_{1}}$ by the
parallel transportation along rays from $x$. Clearly,

$\|\tilde{\psi}\|_{\otimes^{l}h}\equiv 1.$

Moreover, by the formula (4.7), it is easy to see

(4.12) $\displaystyle|D^{\nabla_{j}^{\otimes
l}}\tilde{\psi}|_{(U(x;\epsilon_{1},\frac{\epsilon_{2}}{\sqrt{l}},R),g_{x})}\leq\frac{\sqrt{l}}{\epsilon_{2}}(\epsilon+C_{0}R^{2}\delta),$

where the constant $C_{0}$ depends only on $(Y,g_{x})$. Thus we have pairs of
isomorphisms $(\tilde{\phi}_{j},\tilde{\psi}_{j})$ with property:

(4.13)
$\displaystyle\begin{CD}L_{0}^{l}@>{\tilde{\psi}_{j}}>{}>K_{M_{\infty}}^{-l_{j}l}|_{K_{j}}\\\
@V{}V{}V@V{}V{}V\\\
(U(x;\epsilon_{1},\frac{\epsilon_{2}}{\sqrt{l}},R),g_{x})@>{\tilde{\phi}_{j}}>{}>(K_{j},l_{j}g_{\infty}),\end{CD}$

which satisfy

(4.14) $\displaystyle|D\tilde{\psi}_{j}|_{g_{x}}\leq
2\frac{\sqrt{l}}{\epsilon_{2}}\epsilon,$

as long as $j$ is large enough.

Rescaling $U(x;\epsilon_{1},\epsilon_{2},R)$ into
$U(x;\epsilon_{1},\frac{\epsilon_{2}}{\sqrt{l}},R)$ by

$\mu_{l}:~{}y\rightarrow\frac{y}{\sqrt{l}},y\in
U(x;\epsilon_{1},\epsilon_{2},R).$

We have isometrics

$\mu_{l}^{*}L_{0}^{l}\cong L_{0},\mu_{l}^{*}g_{x}=\frac{g_{x}}{l}.$

By (4.13), it follows

(4.15)
$\displaystyle\begin{CD}L_{0}@>{\tilde{\psi}_{j}\circ(\mu_{l}^{*})^{-1}}>{}>K_{M_{\infty}}^{-l_{j}l}|_{K_{j}}\\\
@V{}V{}V@V{}V{}V\\\
(U(x;\epsilon_{1},\epsilon_{2},R),\frac{g_{x}}{l})@>{\tilde{\phi}_{j}\circ\mu_{l}}>{}>(K_{j},l_{j}g_{\infty}).\end{CD}$

Let

${\phi}_{j}=\tilde{\phi}_{j}\circ\mu_{l},~{}{\rm
and}~{}{\psi}_{j}=\tilde{\psi}_{j}\circ(\mu_{l}^{*})^{-1}.$

Note that $V(x;\delta_{1})\subseteq U(x;\epsilon_{1},\epsilon_{2},R).$ Then
$K_{j}={\phi}_{j}(V(x;\delta_{1}))$ is well-defined. Hence, rescaling the
metric $\frac{g_{x}}{l}$ back to $g_{x}$, we get from (4.14),

(4.16) $\displaystyle|D{\psi}_{j}|_{g_{x}}\leq
2\frac{\epsilon}{\epsilon_{2}},~{}{\rm in}~{}V(x;\delta_{1}).$

Replacing $2\frac{\epsilon}{\epsilon_{2}}$ by $\epsilon$, we prove the
proposition. ∎

Proposition 4.6 will be used to construct peak sections of holomorphic line
bundles over a sequence of Kähler manifolds in next section.

##  5\. $\bar{\partial}$-equation and construction of holomorphic sections

In this section, we give a construction of peak holomorphic sections by
solving $\bar{\partial}$-equation on a smoothing sequence of almost Kähler-
Einstein manifolds in [26]. We will use the rescaling method as done for the
Kähler-Einstein manifolds sequence in [7], [24].

######  Proposition 5.1.

Let $\\{(M_{i},g^{i})\\}$ be a sequence of almost Kähler-Einstein Fano
manifolds as in Section 3 and $(M_{\infty},g_{\infty})$ be its Gromov-
Hasusdorff limit. Then for any sequence of $p_{i}\in M_{i}$ which converges to
$x\in M_{\infty}$, there exist two large number $l_{x}$ and $i_{0}$, and a
small time $t_{x}$ such that for any $i\geq i_{0}$ there exists a holomorphic
section $s_{i}\in\Gamma(K_{M_{i}}^{-l_{x}},h^{i}_{t_{x}})$ which satisfies

(5.1) $\displaystyle\int_{M_{i}}|s_{i}|_{h^{i}_{t_{x}}}^{2}{\rm
dv}_{g^{i}_{t_{x}}}\leq 1~{}{\rm
and}~{}|s_{i}|_{h^{i}_{t_{x}}}(p_{i})\geq\frac{1}{8},$

where $g^{i}_{t}$ are solutions of (LABEL:Ricci_flow) with the initial metrics
$g^{i}$ and $h^{i}_{t_{x}}$ are the hermitian metrics of $K_{M_{i}}^{-l_{x}}$
induced by $g^{i}_{t_{x}}$.

###### Proof.

As in Section 4, we let

$\displaystyle(C_{x},\omega_{x},x)=\lim_{j\rightarrow\infty}(M_{\infty},\frac{g_{\infty}}{r_{j}^{2}},x).$

Choose a $\delta$ so that $\delta\leq(2\pi)^{-\frac{n}{2}}\frac{C_{1}}{64}$,
where $C_{1}$ is a constant chosen as in (3.11). We consider the
$\overline{\partial}$-equation for sections on the trivial line bundle
$L_{0}=(V(x;\delta),\mathbb{C})$,

$\displaystyle\bar{\partial}\sigma=f,~{}\forall~{}f\in\Gamma^{\infty}((TV^{*})^{(0,1)}\otimes
L_{0}).$

Then the standard $C^{0}$-estimate for the elliptic equation shows

(5.2) $\displaystyle|\sigma|_{C^{0}(V(x;2\delta))}\leq
C_{2}(|f|_{C^{0}(V(x;\delta))}+\delta^{-n}[\int_{V(x;\delta)}|\sigma|^{2}d{\rm
v}g_{x}]^{\frac{1}{2}}),$

where the constant $C_{2}$ depends on the metric $g_{x}$.

Let $0<\eta\leq\frac{\delta^{2n}}{1000C_{2}^{2}}$ and $\beta$ a cut-off
function supported in $V(x;\delta_{1})$ constructed in Lemma 4.3. Let $K_{j}$
be the sequence of open sets in $M_{\infty}$ which converge to
$V(x;\delta_{1})$ and $\psi_{j}$ be the sequence of isomorphisms from $L_{0}$
to $K_{\mathcal{R}}^{-ll_{j}}|_{{K}_{j}}$ constructed in Proposition 4.6,
where $l=l(l_{j})\leq N=N(V(x;\delta_{1}),\epsilon)$ and
$l_{j}=\frac{1}{r_{j}^{2}}$. Set $\tau_{j}=\psi_{j}(\beta e)$, where $e$ is a
unit basis of $L_{0}$. Then $\\{\tau_{j}\\}$ is a sequence of smooth sections
of $K_{\mathcal{R}}^{-l_{j}l}$ supported in $\psi_{j}(V(x;\delta_{1}))$.
Moreover, $\tau_{j}$ satisfies the following property as long as $j$ is large
enough:

$\displaystyle i)~{}\|\tau_{j}\|^{2}_{C^{0}(\phi_{i}(V(x;\delta)\bigcap
B_{x}(3\delta)))}\geq\frac{3}{4}e^{-3\delta^{2}}\geq\frac{1}{2};$
$\displaystyle ii)~{}\int_{M_{\infty}}|\tau_{j}|^{2}{\rm
dv}_{g_{\infty}}\leq\frac{3}{2}\frac{r_{j}^{2n}}{l^{n}}(2\pi)^{n};$
$\displaystyle
iii)~{}\bar{\partial}_{J\infty}\tau_{j}\leq\frac{\eta}{8},~{}{\rm
in}~{}V(x;\delta);$ (5.3) $\displaystyle
iv)~{}\int_{M_{\infty}}|\bar{\partial}_{J_{\infty}}\tau_{j}|^{2}{\rm
dv}_{g_{\infty}}\leq\frac{3}{2}r_{j}^{2n-2}\frac{\eta}{l^{n-1}}.$

On the other hand, from the proof of Lemma 4.2, we see that there exists
$t_{0}$, which depends on $V(x;\delta_{1})$ such that for any sufficiently
large $j$ it holds

${\rm vol}(B_{y}(\sqrt{t_{0}}\frac{r_{j}}{\sqrt{l}})\geq(1-\epsilon){\rm
vol}(B_{0}(\sqrt{t_{0}}\frac{r_{j}}{\sqrt{l}}),~{}\forall~{}y\in K_{j},$

where $\epsilon$ is a small constant chosen as in Lemma 4.1. Then by the
pseudo-locality theorem in [26], there exist a $t_{0}^{\prime}\leq t_{0}$, and
a sequence of sets $B_{i}\subseteq M_{i}$ and a sequence of diffeomorphisms
$\varphi_{i}:{K}_{j}\rightarrow B_{i}$ such that

$\displaystyle\varphi_{i}^{*}g^{i}(t_{0}^{\prime}\frac{r_{j}^{2}}{l})\rightarrow
g_{\infty},$ $\displaystyle\varphi_{i}^{*}J_{i}\rightarrow J_{\infty},$
$\displaystyle\varphi_{i}^{*}K_{M_{i}}^{-1}\rightarrow K_{\mathcal{CR}}^{-1},$

in $C^{\infty}$-topology, where $g^{i}(t)=g^{i}_{t}$. Thus, if we let
$v_{i}=(\varphi_{i})_{*}\tau_{j_{0}}\in\Gamma(M_{i},$ $\newline
K_{M_{i}}^{-ll_{j_{0}}})$ for some large integer
$l_{j_{0}}=\frac{1}{r_{j_{0}}^{2}}$ and $l=l(l_{j_{0}})\leq N$, then there
exists a large integer $i_{0}$ such that for any $i\geq i_{0}$ it holds:

$\displaystyle i^{\prime})~{}|v_{i}|_{h^{i}_{t_{x}}}\geq\frac{3}{8},~{}{\rm
in}~{}(\varphi_{i}\circ\psi_{j_{0}})(V(x;2\delta)\bigcap B_{x}(3\delta)));$
$\displaystyle ii^{\prime})\int_{M_{i}}|v_{i}|_{h^{i}_{t_{x}}}^{2}{\rm
dv}_{g_{t_{x}}^{i}}\leq\frac{5}{2}r_{j_{0}}^{2n-2}\frac{\eta}{l^{n-1}};$
$\displaystyle
iii^{\prime})~{}|\bar{\partial}_{J_{i}}v_{i}|_{h^{i}_{t_{x}}}\leq\frac{1}{4}\eta,~{}{\rm
in}~{}(\varphi_{i}\circ\psi_{j_{0}})(V(x;\delta));$ (5.4) $\displaystyle
iv^{\prime})~{}\int_{M_{i}}|\bar{\partial}_{J_{i}}v_{i}|^{2}_{h^{i}_{t_{x}}}{\rm
dv}_{g_{t_{x}}^{i}}\leq\frac{5}{4}r_{j_{0}}^{2n-2}\frac{\eta}{l^{n-1}}.$

Here $t_{x}=t_{0}^{\prime}r_{j_{0}}^{2}/l$ and $h^{i_{j}}_{t_{x}}$ are
hermitian metrics of $K_{M_{i}}^{-ll_{j_{0}}}$ induced by $g^{i}_{t_{x}}$.

By solving $\bar{\partial}$-equations for $K_{M_{i}}^{-ll_{j_{0}}}$-valued
(0,1)-form $\sigma_{i}$,

$\displaystyle\bar{\partial}\sigma_{i}=\bar{\partial}v_{i},~{}{\rm
in}~{}M_{i},$

we get the $L^{2}$-estimates from (3.7) and $iv^{\prime})$ in (5),

(5.5)
$\displaystyle\|\sigma_{i}\|^{2}_{L^{2}(M_{i},g^{i}_{t_{x}})}\leq\frac{4}{ll_{j_{0}}}\int_{M_{i}}|\bar{\partial}_{J_{i}}v_{i}|^{2}{\rm
dv}_{g_{t_{x}}^{i}}\leq\frac{5\eta}{l^{n}l_{j_{0}}^{n}}.$

Hence, by (5.2) and $iii^{\prime})$ in (5), we derive

$\displaystyle|\sigma_{i}|_{h^{i}_{t_{x}}}(q)$ $\displaystyle\leq
2C_{2}(\sup_{(\varphi_{i}\circ\psi_{j_{0}})(V(x;\delta))}|\bar{\partial}v_{i}|_{h^{i}_{t_{x}}}$
$\displaystyle+\delta^{-n}[(ll_{j_{0}})^{n}\int_{(\varphi_{i}\circ\psi_{j_{0}})(V(x;\delta))}|\sigma_{i}|^{2}_{h^{i}_{t_{x}}}{\rm
dv}_{g^{i}_{t_{x}}}]^{\frac{1}{2}})$ $\displaystyle\leq
2C_{2}(\frac{1}{4}\eta+\delta^{-n}[(ll_{j_{0}})^{n}\int_{M_{i}}|\sigma_{i}|^{2}{\rm
dv}_{g^{i}_{t_{x}}}]^{\frac{1}{2}})$ $\displaystyle\leq
2C_{2}(\frac{1}{4}\eta+\delta^{-n}[(ll_{j_{0}})^{n}\frac{5\eta}{l^{n}l_{j_{0}}}r_{j_{0}}^{2n-2}]^{\frac{1}{2}})$
(5.6) $\displaystyle\leq
5C_{2}(\frac{1}{4}\eta+\delta^{-n}\sqrt{\eta})\leq\frac{1}{8},~{}\forall~{}q\in(\varphi_{i}\circ\psi_{j_{0}})(V(x;2\delta)).$

Let $s_{i}=v_{i}-\sigma_{i}$. Then $s_{i}$ is a holomorphic section of
$K_{M_{i}}^{-ll_{j_{0}}}$. By $i^{\prime})$ in (5) and (5), we have

$|s_{i}|_{h^{i}_{t_{x}}}(q_{1})\geq\frac{3}{8}-\frac{1}{8}=\frac{1}{4},~{}\forall~{}q_{1}\in(\varphi_{i}\circ\psi_{j_{0}})(V(x;2\delta)\bigcap
B_{x}(3\delta)).$

Moreover, by $ii^{\prime})$ in (5), it is easy to see that

$\displaystyle\int_{M_{i}}|s_{i}|_{h_{t_{x}}^{i}}^{2}{\rm dv}_{g^{i}_{t_{x}}}$
$\displaystyle\leq 2(\int_{M_{i}}|v_{i}|_{h_{t_{x}}^{i}}^{2}{\rm
dv}_{g^{i}_{t_{x}}}+\int_{M_{i}}|\sigma_{i}|_{h_{t_{x}}^{i}}^{2}{\rm
dv}_{g^{i}_{t_{x}}})$ (5.7) $\displaystyle\leq
4(2\pi)^{n}\frac{r_{j_{0}}^{2n}}{l^{n}}.$

Thus by the estimate (3.11), we get

$\displaystyle\|\nabla
s_{i}\|_{h^{i}_{t_{x}}}\leq\sqrt{4(2\pi)^{n}}C_{1}\sqrt{l}r_{j_{0}}^{-1}.$

Since $d(p_{i},q_{1})\leq 4\frac{r_{j_{0}}}{\sqrt{l}}\delta$, we deduce

$\displaystyle|s_{i}(p_{i})|_{h^{i}_{t_{x}}}\geq|s_{i}(q_{1})|-4\frac{r_{j_{0}}}{\sqrt{l}}\delta\|\nabla
s_{i}\|_{h^{i}_{t_{x}}}$
$\displaystyle\geq|s_{i}|_{h^{i}_{t_{x}}}(q_{1})-8\sqrt{(2\pi)^{n}}C_{1}\delta\geq\frac{1}{8}.$

This proves the theorem while $l_{x}$ is chosen by $ll_{j_{0}}$.

∎

##  6\. Proof of Theorem 1.3–I

In this section, we use the estimate in Section 5 to give a lower bound of
$\rho_{ll_{0}}(x)$ for a sequence of almost Kähler-Einstein manifolds.

######  Theorem 6.1.

Let $(M_{i},g^{i})$ be a sequence of almost Kähler-Einstein manifolds as in
Section 3 and $(M_{\infty},g_{\infty})$ be its Gromov-Hasusdorff limit. Then
there exists an integer $l_{0}>0$, which depends only on
$(M_{\infty},g_{\infty})$ such that for any integer $l>0$ there exists a
uniform constant $c_{l}>0$ with property:

(6.1) $\displaystyle\rho_{ll_{0}}(M_{i},g^{i})\geq c_{l}.$

The proof of Theorem 6.1 depends on the following lemma.

######  Lemma 6.2.

Let $(M,g)$ be a Fano manifold with $\omega_{g}\in 2\pi c_{1}(M)$ which
satisfies

(6.2) $\displaystyle{\rm Ric}(g)\geq-\Lambda^{2}g~{}{\rm and}~{}{\rm
diam}(M,g)\leq D.$

Let $g_{t}$ be a solution of (LABEL:Ricci_flow) with the initial metric $g$.
Then there exists a small $t_{0}=t_{0}(l,\Lambda,D)$ such that the following
is true: if $s\in\Gamma(M,K_{M}^{-l})$ is a holomorphic section with
$\int_{M}|s|^{2}_{h_{t}}{\rm dv}_{g_{t}}=1$ for some $t\leq t_{0}$ which
satisfies

$|s|^{2}_{h_{t}}(p)\geq c>0,$

then

(6.3) $\displaystyle|s|^{2}_{h}(p)\geq c^{\prime}>0~{}{\rm
and}~{}\int_{M}|s|^{2}_{h}{\rm dv}_{g}\leq c^{\prime\prime}.$

Here ${h_{t}}$ and $h$ are hermitian metrics of $K_{M_{i}}^{-l}$ induced by
$g_{t}$ and $g$, respectively, and $c^{\prime},c^{\prime\prime}>0$ are uniform
constants depending only on $c,l,\Lambda$ and $D$.

###### Proof.

Let $\omega_{g_{t}}=\omega_{g}+\sqrt{-1}\partial\bar{\partial}\phi$. Namely,
$\phi$ are potentials of $g_{t}$. Then $\phi=\phi(x,t)$ satisfies

(6.4) $\displaystyle\frac{\partial}{\partial
t}\phi=\log\frac{(\omega_{g}+\sqrt{-1}\partial\bar{\partial}\phi)^{n}}{\omega_{g}^{n}}+\phi-
f_{g},$

where $f_{g}$ is the Ricci potential of $g$ normalized by

$\int_{M}f_{g}{\rm dv}_{g}^{n}=0.$

Since

$\Delta f_{g}=\rm{R}(g)-n\geq-(n-1)\Lambda^{2}-n,$

by using the Green formula, we see

$f_{g}(x)\leq-\int_{M}G(x,\cdot)\Delta f_{g}\leq C(\Lambda,D).$

Thus applying the maximum principle to (6.4), it follows

$\phi\geq-C(\Lambda,D).$

On the other hand, integrating both sides of (6.4), we have

$\displaystyle\frac{d}{dt}\int_{M}\phi{\rm dv}_{g}$
$\displaystyle=\int_{M}\log\frac{(\omega_{g}+\sqrt{-1}\partial\bar{\partial}\phi)^{n}}{\omega_{g}^{n}}{\rm
dv}_{g}+\int_{M}\phi{\rm dv}_{g}-\int_{M}f_{g}{\rm dv}_{g}$
$\displaystyle\leq\int_{M}\phi{\rm dv}_{g}+C,$

It follows

$\int_{M}\phi{\rm dv}_{g}\leq Ce^{t}\leq eC.$

Hence by using the Green formula to $\phi$, we can also get

$\phi\leq C^{\prime}(\lambda,D).$

As a consequence, we derive

(6.5) $\displaystyle
e^{-C^{\prime}l}|\cdot|_{h}\leq|\cdot|_{h_{t}}=e^{-l\phi}|\cdot|_{h}\leq
e^{Cl}|\cdot|_{h}.$

Therefore to prove Proposition 6.2, we suffice to prove

######  Claim 6.3.

Let $s\in\Gamma(M,K_{M}^{-l})$ be a holomorphic section. Suppose that

$\displaystyle\int_{M}|s|^{2}_{h}{\rm{dv}}_{g}=1.$

Then

(6.6) $\displaystyle\int_{M}|s|^{2}_{h_{t}}{\rm{dv}}_{g_{t}}\geq
c(l,\Lambda,D)>0.$

Since

$\displaystyle\frac{\partial}{\partial
t}\frac{(\omega_{g}+\sqrt{-1}\partial\bar{\partial}\phi)^{n}}{\omega_{g}^{n}}$
$\displaystyle=\Delta^{\prime}\frac{\partial\phi}{\partial t}$
$\displaystyle=-R(g_{t})+n\leq\lambda=\lambda(\Lambda),$ (6.7)
$\displaystyle\text{vol }_{g_{t}}(\Omega)\leq e^{\lambda t}\text{vol
}_{g}(\Omega),~{}\forall~{}\Omega\subset M.$

It follows

$\displaystyle\text{vol }_{g_{t}}(\Omega)$ $\displaystyle=V-\text{vol
}_{g_{t}}(M\setminus\Omega)\geq V-e^{\lambda t}\text{vol
}_{g}(M\setminus\Omega)$ (6.8) $\displaystyle\geq\text{vol
}_{g}(\Omega)-2V\lambda t.$

By the estimate (3.4), we see

$|s(x)|_{h}^{2}\leq H=H(\Lambda,D).$

Then

$\displaystyle\int_{0}^{H}{\rm vol}_{g}\\{x\in M|~{}|s(x)|_{h}^{2}\geq
s\\}ds=\int_{M}|s|^{2}_{h}{\rm dv}_{g}.$

Hence, by using (6.5), and (6.7) and (6), we get

$\displaystyle\int_{M}|s|^{2}_{h_{t}}{\rm dv}_{g_{t}}\geq$
$\displaystyle\int_{0}^{H}{\rm vol}_{g_{t}}\\{x\in M|~{}|s(x)|_{h_{t}}^{2}\geq
s\\}ds$ $\displaystyle\geq\int_{0}^{H}{\rm vol}_{g_{t}}\\{x\in
M|~{}|s(x)|_{h}^{2}\geq e^{C^{\prime}l}s\\}ds$ $\displaystyle\geq
e^{-C^{\prime}l}\int_{0}^{e^{C^{\prime}l}H}[{\rm vol}_{g}\\{x\in
M|~{}|s(x)|_{h}^{2}\geq s\\}-2V\lambda t]ds$ $\displaystyle\geq
e^{-C^{\prime}l}(1-2\lambda VHe^{C^{\prime}l}t).$

Therefore, by choosing $t_{0}\leq(4\lambda VHe^{C^{\prime}l})^{-1}$, we derive
(6.6). The claim is proved.

∎

###### Proof of the Theorem 6.1.

By Proposition 5.1, we see that for any $x\in M_{\infty}$ and a sequence
$\\{p_{i}\subset M_{i}\\}$ which converges to $x$, there exist two large
number $l_{x}$ and $i_{0}$, a small time $t_{x}$ such that there exists a
holomorphic section $s_{i}\in\Gamma(K_{M_{i}}^{-l_{x}},h^{i}_{t_{x}})$ for any
$i\geq i_{0}$ with $\int_{M_{i}}|s_{i}|_{h^{i}_{t_{x}}}^{2}\rm{dv}_{g^{i}}\leq
1$ which satisfies

$\displaystyle|s_{i}|_{h_{t_{x}}^{i}}(p_{i})\geq\frac{1}{8},$

where $h^{i}_{t_{x}}$ is the hermitian metric of $K_{M_{i}}^{-l_{x}}$ induced
by $g^{i}_{t_{x}}$. By Lemma 6.2, it follows that there exists a constant
$c(l_{x},\Lambda,D)$ and a holomorphic section
$\hat{s}_{i}\in\Gamma(K_{M_{i}}^{-l_{x}},h_{i})$ for any $i\geq i_{0}$ with
$\int_{M_{i}}|\hat{s}_{i}|_{h_{i}}^{2}\rm{dv}_{g^{i}}=1$ which satisfies

$\displaystyle|\hat{s}_{i}|_{h_{i}}(p_{i})\geq c_{x}=c(l_{x},\Lambda,D),$

where $h_{i}$ is the hermitian metric of $K_{M_{i}}^{-l_{x}}$ induced by
$g^{i}$.

Let $C=C(C_{S},n)$ be the constant as in (3.6), which depending only on
$\Lambda$ and $D$. For each $x$, we choose
$r_{x}=\frac{c_{x}}{2}l_{x}^{-\frac{n+1}{2}}C.$ Then by the estimate (3.6), we
get

$\displaystyle|\hat{s}_{i}|_{h_{i}}(q)\geq\frac{c_{x}}{2},~{}\forall~{}q\in
B_{p^{i}}(r_{x}).$

Take $N$ balls $B_{x_{\alpha}}(\frac{r_{x_{\alpha}}}{2})$ to cover
$M_{\infty}$. Then it is easy to see that there exists $i_{1}\geq i_{0}$ such
that $\cup_{\alpha}B_{p_{\alpha}^{i}}(r_{x_{\alpha}})=M_{i}$ for any $i\geq
i_{1}$, where $\\{p_{\alpha}^{i}\\}$ is a set of $N$ points in $M_{i}$. This
shows that for any $q\in M_{i}$ $(i\geq i_{1})$ there exist a ball
$B_{p_{\alpha}^{i}}(r_{x_{\alpha}})$ and a holomorphic section
$s^{i}_{\alpha}\in\Gamma(K_{M_{i}}^{-l_{x_{\alpha}}},h_{i})$ such that $q\in
B_{p_{\alpha}}(r_{x_{\alpha}}^{i})$, and
$\int_{M_{i}}|s^{i}_{\alpha}|_{h_{i}}^{2}\rm{dv}_{g^{i}}=1$ and

(6.9) $\displaystyle|s_{\alpha}^{i}|_{h_{i}}(q)\geq
c=\min_{\alpha}\\{c_{x_{\alpha}}\\}>0.$

Set $l_{0}=\prod_{\alpha}l_{x_{\alpha}}$. Then by using a standard method (cf.
[7], [23]), for any $q\in M_{i}$ $(i\geq i_{1})$, one can construct another
holomorphic section $s\in\Gamma(K_{M_{i}}^{-l_{0}},h_{i})$ based on
holomorphic sections $s_{\alpha}^{i}$ such that
$\int_{M^{i}}|s|_{h_{i}}^{2}\rm{dv}_{g^{i}}=1$ and

$\displaystyle|s|_{h_{i}}(q)\geq c^{\prime}>0,$

where $c^{\prime}=c^{\prime}(l_{0},c)$. This proves the theorem for $l=1$. One
can also prove the theorem for general multiple $l\geq 1$ as above. ∎

##  7\. Proof of Theorem 1.3–II

In this section, we prove Theorem 1.3 in case of almost Kähler-Ricci solitons.
We assume that a Fano manifold $(M,g)$ admits a non-trivial holomorphic vector
field $X$, where $X$ lies in an reductive Lie subalgebra $\eta_{r}$ of space
of holomorphic vector fields, and $g$ is $K_{X}$-invariant with $\omega_{g}\in
2\pi c_{1}(M)$ [29]. We also suppose that $g$ satisfies the following
geometric conditions:

$\displaystyle i)~{}{\rm Ric}(g)+L_{X}g\geq-\Lambda^{2}g,~{}|X|_{g}\leq
A~{}\text{and}~{}\text{diam }(M,g)\leq D;$ (7.1) $\displaystyle
ii)~{}R(g)\geq-C_{0}.$

In particular, under the condition i), $g$ has a uniform $L^{2}$-Sobolev
constant $C_{s}=C_{s}(\Lambda,A,D)$ (cf. [32]). We note that the volume of
$(M,g)$ is uniformly bounded below by the normalized condition $\omega_{g}\in
2\pi c_{1}(M)$ and it is uniformly bounded above by the volume comparison
theorem [30].

Now we consider the following modified Kähler-Ricci flow with the above
initial Kähler metric $g$,

(7.2) $\displaystyle\left\\{\begin{aligned} &\frac{\partial}{\partial
t}g=-{\rm{Ric}}(g)+g+L_{X}g,\\\ &g_{0}=g(\cdot,0)=g.\end{aligned}\right.$

Clearly, solutions $g_{t}$ $(t\in(0,\infty))$ of (LABEL:modified-Ricci-flow)
are all $K_{X}$-invariant.

Since the Sobolev constant $g$ is uniformly bounded below, by Zhang’s result
[34], we have an analogy to Lemma 2.1 as follows.

######  Lemma 7.1.

All solutions $g_{t}$ of (LABEL:modified-Ricci-flow) have Sobolev constants
$C_{s}=C_{s}(\Lambda,A,$
$D)$ uniformly bounded below. Namely, the following inequalities hold,

$\displaystyle(\int_{M}f^{\frac{2n}{n-1}}d{\rm{v}}_{g_{t}})^{\frac{n-1}{n}}\leq
C_{s}(\int_{M}f^{2}(R+\hat{C}_{0})d{\rm{v}}_{g_{t}}+\int_{M}|\nabla
f|^{2}d{\rm{v}}_{g_{t}}),$

where $f\in C^{1}(M)$ and $\hat{C}_{0}$ is a uniform constant depending only
on the lower bound $C_{0}$ of scalar curvature $R$ of $g$.

######  Lemma 7.2.

Let $\Delta=\Delta_{t}$ be the Lapalace operator associated to $g_{t}$.
Suppose that $f\geq 0$ satisfies

(7.3) $\displaystyle(\frac{\partial}{\partial t}-(\Delta+X))f\leq af,$

where $a$ is a constant. Then for any $t\in(0,1)$, we have

$\displaystyle\sup_{x\in M}f(x,t)$ (7.4)
$\displaystyle\leq\frac{C_{1}(\Lambda,A,D,C)}{t^{\frac{n+1}{p}}}(\int_{\frac{t}{2}}^{t}\int_{M}|f(x,\tau)|^{p}d\text{{\rm{v}}}_{g_{\tau}}d\tau)^{\frac{1}{p}}.$

###### Proof.

As in the proof of Lemma 2.2 , multiplying both sides of (7.3) by $f^{p}$, we
have

$\displaystyle\int_{M}f^{p}f^{\prime}_{\tau}{\rm
dv}_{g_{\tau}}+p\int_{M}|\partial f|^{2}f^{p-1}{\rm
dv}_{g_{\tau}}-\int_{M}\langle\partial\theta,\partial f\rangle f^{p}{\rm
dv}_{g_{\tau}}$ $\displaystyle\leq a\int_{M}f^{p+1}{\rm dv}_{g_{\tau}}.$

On the other hand, by (LABEL:modified-Ricci-flow), it is easy to see

$\displaystyle\int_{M}f^{p}f^{\prime}_{\tau}{\rm
dv}_{g_{\tau}}=\frac{1}{p+1}\frac{d}{d\tau}(\int_{M}f^{p+1}{\rm
dv}_{g_{\tau}})+\frac{1}{p+1}\int_{M}(R-n-\Delta\theta)f^{p+1}d\text{v}_{g_{\tau}}.$

Thus we get

$\displaystyle\frac{1}{p+1}\frac{d}{d\tau}(\int_{M}f^{p+1}{\rm
dv}_{g_{\tau}})+\frac{1}{p+1}\int_{M}(R-n)f^{p+1}d\text{v}_{g_{\tau}}+p\int_{M}|\partial
f|^{2}f^{p-1}{\rm dv}_{g_{\tau}}$ $\displaystyle\leq a\int_{M}f^{p}{\rm
dv}_{g_{\tau}}.$

It follows

$\displaystyle\frac{d}{d\tau}\int_{M}f^{p+1}{\rm
dv}_{g_{\tau}}+\int_{M}(R+\hat{C}_{0})f^{p+1}{\rm
dv}_{g_{\tau}}+2\int_{M}|\nabla f^{\frac{p+1}{2}}|^{2}$ (7.5)
$\displaystyle\leq((p+1)a+n+C_{0})\int_{M}f^{p+1}{\rm dv}_{g_{\tau}}.$

Note that (7) is similar to (2). Therefore, we can follow the argument in the
proof of Lemma 2.2 to obtain (7.2).

∎

Recall that according to [32] a sequence of weak almost Kähler-Ricci solitons
$(M_{i},J_{i},g^{i},X_{i})$ $(i\to\infty)$ satisfy the condition i) in (7) and

(7.6) $\displaystyle iii)~{}\int_{M_{i}}|{\rm
Ric}(g^{i})-g^{i}-L_{X_{i}}g^{i}|{\rm dv}_{g^{i}}^{n}\rightarrow
0,~{}\text{as}~{}i\to\infty.$

As in [32], we shall further assume that the solutions $g^{i}_{t}$ of (7) with
the initial metrics $g^{i}$ satisfy

$\displaystyle iii)~{}|X^{i}|_{g^{i}_{t}}\leq\frac{B}{\sqrt{t}};$ (7.7)
$\displaystyle
vi)~{}\int_{0}^{1}dt\int_{M_{i}}|R(g^{i}_{t})-\Delta\theta_{g^{i}_{t}}-n|{\rm
dv}_{g^{i}_{t}}^{n}\rightarrow 0,~{}\text{as}~{}i\to\infty,$

where $B$ is a uniform constant. It was proved that under the conditions $i)$
of (7), and (7.6) and (7) there exists a subsequence of
$\\{(M_{i},J_{i},g^{i},X_{i})\\}$ which converges to a Kähler-Ricci soliton
away from singularities of Gromov-Hausdorff limit with codimension 4.

######  Definition 7.3.

$\\{(M_{i},J_{i},g^{i},X_{i})\\}$ are called a sequence of almost Kähler-Ricci
solitons if (7), (7.6) and (7) are satisfied.

######  Lemma 7.4.

Let $\\{(M_{i},J_{i},g^{i},X_{i})\\}$ be a sequence of almost Kähler-Ricci
solitons. Then there exists a uniform constant $C=C(\Lambda,D,B,C_{0})$ such
that for any $t\in(0,1)$ there exists $N=N(t)$ such that for any $i\geq N$ it
holds

$|\nabla h^{i}_{t}|\leq C~{}\text{and}~{}|R^{i}_{t}|\leq C.$

###### Proof.

By

$\displaystyle(\frac{\partial}{\partial t}-(\Delta+X))|\nabla(h-\theta)|^{2}$
$\displaystyle=-|\nabla\bar{\nabla}(h-\theta)|^{2}-|\nabla\nabla(h-\theta)|^{2}+|\nabla(h-\theta)|^{2}$
(7.8) $\displaystyle\leq|\nabla(h-\theta)|^{2},$

we apply Lemma 7.2 to get

$\displaystyle|\nabla(h-\theta)|^{2}$
$\displaystyle\leq\frac{C}{t^{n+1}}\int_{\frac{t}{2}}^{t}\int_{M}|\nabla(h-\theta)|^{2}{\rm
dv}_{g_{\tau}}d\tau$
$\displaystyle=\frac{C}{t^{n+1}}\int_{\frac{t}{2}}^{t}\int_{M}(\theta-h)\Delta(h-\theta){\rm
dv}_{g_{\tau}}d\tau$
$\displaystyle\leq\frac{C}{t^{n+1}}\int_{\frac{t}{2}}^{t}\int_{M}osc_{M}(h-\theta)|R-n-\Delta\theta|{\rm
dv}_{g_{\tau}}d\tau.$

By (2.16), it follows

(7.9)
$\displaystyle|\nabla(h-\theta)|^{2}\leq\frac{C}{t^{(n+1)(n+\frac{3}{2})}}\int_{\frac{t}{2}}^{t}\int_{M}|R-n-\Delta\theta|{\rm
dv}_{g_{\tau}}d\tau.$

On the other hand, by the evolution equation of $(\Delta+X)(h-\theta)$ [5],

$\displaystyle(\frac{\partial}{\partial t}-(\Delta+X))[(\Delta+X)(h-\theta)]$
$\displaystyle=(\Delta+X)(h-\theta)+|\nabla\bar{\nabla}(h-\theta)|^{2},$

we have

$\displaystyle(\frac{\partial}{\partial
t}-(\Delta+X))[(\Delta+X)(h-\theta)+|\nabla(h-\theta)|^{2}]$
$\displaystyle\leq(\Delta+X)(h-\theta)+|\nabla(h-\theta)|^{2}.$

Then applying Lemma 7.2, we get

$\displaystyle(\Delta+X)(h-\theta)+|\nabla(h-\theta)|^{2}$ (7.10)
$\displaystyle\leq\frac{C}{t^{n+1}}\int_{\frac{t}{2}}^{t}\int_{M}|(\Delta+X)(h-\theta)+|\nabla(h-\theta)|^{2}|{\rm
dv}_{g_{\tau}}d\tau.$

Note that by iii) in (7) we have

$\displaystyle\int_{\frac{t}{2}}^{t}\int_{M}|X(h-\theta)|{\rm
dv}_{g_{\tau}}d\tau$ $\displaystyle\leq B{\rm
vol}(M)[\int_{\frac{t}{2}}^{t}\int_{M}|\nabla(h-\theta)|^{2}{\rm
dv}_{g_{\tau}}d\tau]^{\frac{1}{2}}.$

It follows from (2.16),

$\displaystyle\int_{\frac{t}{2}}^{t}\int_{M}|(\Delta+X)(h-\theta)+|\nabla(h-\theta)|^{2}|{\rm
dv}_{g_{\tau}}$
$\displaystyle\leq\int_{\frac{t}{2}}^{t}\int_{M}|R-n-\Delta\theta|{\rm
dv}_{g_{\tau}}$ $\displaystyle+CB{\rm
vol}(M)\frac{1}{t^{\frac{1}{2}(n+1)(n+\frac{1}{2})}}[\int_{\frac{t}{2}}^{t}\int_{M}|R-n-\Delta\theta|{\rm
dv}_{g_{\tau}}]^{\frac{1}{2}}$
$\displaystyle+C\frac{1}{t^{(n+1)(n+\frac{1}{2})}}\int_{\frac{t}{2}}^{t}\int_{M}|R-n-\Delta\theta|{\rm
dv}_{g_{\tau}}.$

Thus inserting the above inequality into (7), we derive

$\displaystyle(\Delta+X)(h-\theta)+|\nabla(h-\theta)|^{2}$
$\displaystyle\leq\frac{C}{t^{(n+1)(n+\frac{3}{2})}}(\int_{\frac{t}{2}}^{t}\int_{M}|R-n-\Delta\theta|{\rm
dv}_{g_{\tau}}d\tau$ (7.11)
$\displaystyle+[\int_{\frac{t}{2}}^{t}\int_{M}|R-n-\Delta\theta|{\rm
dv}_{g_{\tau}}d\tau]^{\frac{1}{2}}).$

Combining (7.9) and (7), we see that for any $t\in(0,1)$ there exists $N=N(t)$
such that

(7.12) $\displaystyle|\frac{1}{\sqrt{t}}\nabla(h-\theta)|\leq
1~{}\text{and}~{}R-n-\Delta\theta\leq 1,~{}\forall~{}i\geq N(t).$

It follows

$\displaystyle\Delta\theta=-|\nabla\theta|^{2}-X(h-\theta)-\theta\leq C.$

As a consequence, we get $R\leq C$, and so $|R|\leq C$.

By (7.12), we have

$\Delta\theta\geq R-n-1\geq-C.$

Thus

(7.13) $\displaystyle|\nabla\theta|^{2}=-X(h-\theta)-\theta-\Delta\theta\leq
C.$

Again by (7.12), we prove that $|\nabla h|\leq C.$

∎

By Lemma 7.1 and the scalar curvature estimate in Lemma 7.4, we see that for
any $t\in(0,1)$ there exists an integer $N=N(t)$ such that the Sobolev
constant $C_{s}$ of $g_{t}^{i}$ is uniformly bounded for any $i\geq N$. Then
by the gradient estimate of Kähler potentials in Lemma 7.4, we can follow the
arguments in Lemma 3.1 and Lemma 3.3 (also see Remark 3.2 and Remark 3.4) to
get an analogy of Proposition 3.5.

######  Proposition 7.5.

Let $(M_{i},g^{i})$ be a sequence of Fano manifolds with almost Kähler-Ricci
solitons which satisfy (7), (7.6) and (7). Then for any $t\in(0,1)$ there
exist integers $N=N(t)$ such that for any $i\geq N$ and $l\geq l_{0}$ it
holds,

(7.14) $\displaystyle\|s\|_{h_{t}^{i}}+l^{-\frac{1}{2}}\|\nabla
s\|_{h^{i}_{t}}\leq Cl^{\frac{n}{2}}(\int_{M_{i}}|s|^{2}{\rm
dv}_{g^{i}_{t}})^{\frac{1}{2}}$

and

(7.15) $\displaystyle\int_{M^{i}}|v|_{h_{t}^{i}}^{2}\leq
4l^{-1}\int_{M_{i}}|\bar{\partial}\sigma|_{h_{t}^{i}}^{2}.$

Here $s\in H^{0}(M_{i},K_{M_{i}}^{-l})$, the norms of $|\cdot|_{h^{i}_{t}}$
are induced by $g_{t}^{i}$, and the integer $l_{0}$ and the uniform constant
$C$ are independent of $t$.

By Proposition 7.5, we can follow the arguments in Proposition 5.1 and Theorem
6.1 to prove

######  Theorem 7.6.

Let $(M_{i},g^{i})$ be a sequence of Fano manifolds with almost Kähler-Ricci
solitons and $(M_{\infty},g_{\infty})$ be their Gromov-Hasusdorff limit. Then
there exists an integer $l_{0}>0$ which depending only on
$(M_{\infty},g_{\infty})$ such that for any integer $l>0$ there exists a
uniform constant $c_{l}>0$ with property:

(7.16) $\displaystyle\rho_{ll_{0}}(M_{i},g^{i})\geq c_{l}.$

###### Proof.

We give a sketch of proof of Theorem 7.6.

Step 1. By the rescaling method as in proof of Proposition 5.1 with the helps
of Proposition 7.5 and the pseudo-locallity theorem in [32], we have an
analogy of Proposition 5.1: For any sequence of $p_{i}\in M_{i}$ which
converge to $x\in M_{\infty}$, there exist two large number $l_{x}$ and
$i_{0}$, and a small time $t_{x}$ such that for any $i\geq i_{0}$ there exists
a holomorphic section $s_{i}\in\Gamma(K_{M_{i}}^{-l_{x}},h_{t_{x}}^{i})$ which
satisfies

(7.17) $\displaystyle\int_{M_{i}}|s_{i}|_{h^{i}_{t_{x}}}^{2}{\rm
dv}_{g^{i}_{t_{x}}}\leq 1~{}{\rm
and}~{}|s_{i}|_{h^{i}_{t_{x}}}(p_{i})\geq\frac{1}{8},$

where $g^{i}_{t}$ is a solution of (LABEL:modified-Ricci-flow) with the
initial metric $g^{i}$ and $h^{i}_{t_{x}}$ is the hermitian metric of
$K_{M_{i}}^{-l_{x}}$ induced by $g^{i}_{t_{x}}$.

Step 2. We can compare the $C^{0}$-norm of holomorphic sections with respect
to the varying metrics $g_{t}$ evolved in the flow (LABEL:modified-Ricci-
flow). In fact, we have

######  Lemma 7.7.

Let $(M,g)$ be a Fano manifold with $\omega_{g}\in 2\pi c_{1}(M)$ which
satisfies (7), and $g_{t}$ a solution of (LABEL:modified-Ricci-flow) with the
initial metric $g$. Then there exists a small $t_{0}=t_{0}(l,\Lambda,D)$ such
that the following is true: if $s\in\Gamma(M,K_{M}^{-l})$ is a holomorphic
section with

(7.18) $\displaystyle\int_{M}|s|^{2}_{h_{t}}{\rm{dv}}_{g_{t}}=1$

for some $t\leq t_{0}$ which satisfies

(7.19) $\displaystyle|s|_{h_{t}}(p)\geq c>0,$

then there is a holomorphic section $s^{\prime}$ of $K_{M}^{-l}$ which
satisfies

$\displaystyle|s^{\prime}|_{h}(p)\geq c^{\prime}>0~{}{\rm
and}~{}\int_{M}|s^{\prime}|^{2}_{h}{\rm dv}_{g}\leq c^{\prime\prime},$

where ${h_{t}}$ and $h$ are the hermitian metrics of $K_{M}^{-l}$ induced by
$g_{t}$ and $g$, respectively, and the constants $c^{\prime}$ and
$c^{\prime\prime}$ depend only on $c,l,\Lambda$, $A$, $C_{0}$ and $D$.

###### Proof of Lemma 7.7.

Let $\Phi_{t}$ be a one-parameter subgroup generated by $-X$. Then
$\Phi_{t}^{*}g_{t}$ is a solution of (LABEL:Ricci_flow). It is clear that
(7.18) also holds for $\Phi_{t}^{*}s,\Phi_{t}^{*}g_{t},\Phi_{t}^{*}h_{t}$ and
the condition (7.19) is equivalent to
$|\Phi_{t}^{*}s|_{\Phi_{t}^{*}h_{t}}(\Phi_{-t}(p))\geq c.$ Since the Green
functions associated to the metric $g$ is bounded below under the condition
$i)$ of (7) (cf. [14], [5]), we can follow the argument in Lemma 6.2 for the
metrics $\Phi_{t}^{*}g_{t}$ to obtain

$\displaystyle|\Phi_{t}^{*}s|_{h}(\Phi_{-t}(p))\geq\tilde{c}~{}{\rm
and}~{}\int_{M}|\Phi_{t}^{*}s|^{2}_{h}{\rm dv}_{g}\leq c^{\prime\prime},$

where the constant $\tilde{c}$ depends only on $c,l,\Lambda$, $A$ and $D$. Let
$s^{\prime}=\Phi_{t}^{*}s$. Then by the gradient estimate of $|\nabla
s^{\prime}|\leq C(l,\Lambda,D,C_{0},A)$, we have

$\displaystyle|s^{\prime}|_{h}(p)\geq|s^{\prime}|_{h}(\Phi_{-t}(p))-C(\Lambda,D,C_{0},A)At\geq
c^{\prime}.$

This proves Lemma 7.7. ∎

Step 3. By using the covering argument as in Theorem 6.1 together with the
results in Step 1 and Step 2, we can finish the proof of Theorem 7.6.

∎

##  8\. Proof of Corollary 1.4

In this section, for simplicity, we just give a proof of Corollary 1.4 in case
of almost Kähler-Einstein manifolds with dimension $n\geq 2$. We assume that a
sequence of almost Kähler-Einstein manifolds $(M_{i},g^{i})$ with a limit
$(M_{\infty},g_{\infty})$ in Goromov-Hausdorff topology satisfies the partial
$C^{0}$-estimate,

(8.1) $\displaystyle\rho_{l}(M_{i},g^{i})\geq c_{l}>0,$

for some integer $l$. Then, as an application of (8.1), we have

(8.2) $\displaystyle H^{0}(M_{i},K_{M_{i}}^{-m})\subseteq
H^{0}(M_{i},K_{M_{i}}^{-(m-l)})\otimes H^{0}(M,K_{M_{i}}^{-l}),$

where $m\geq l(n+2+[\Lambda^{2}])$ is any integer and the constant
$-\Lambda^{2}$ is a uniform lower bound of Ricci curvature of $(M_{i},g^{i})$
(cf. Proposition 7, [12])333There is a generalization of (8.2) under the
Bakry-Eméry Ricci curvature condition in Appendix.

We need a strong version of (8.1) as follows.

######  Lemma 8.1.

For two different points $x,y\in M_{\infty}$, there exist
$\ell=\ell(n,\Lambda,D,$
$x,y)$, which is a multiple of $l$, and two sections $s_{x},s_{y}\in
H^{0}(M_{i},K_{M_{i}}^{-\ell})$ such that

(8.3) $\displaystyle|s_{x}(p_{i})|_{h_{i}}=|s_{y}(q_{i})|_{h_{i}}=1~{}{\rm
and}~{}s_{x}(q_{i})=s_{y}(p_{i})=0,$

where $p_{i}\rightarrow x,q_{i}\rightarrow y$.

###### Proof.

As in the proof of Proposition 5.1, we can choose two compact sets
$V(x;\delta_{1}^{x}),V(y;\delta_{1}^{y})$ in $C_{x}$ and $C_{y}$,
respectively, such that $\phi_{i}\circ\psi_{j}(V(x;\delta_{1}^{x}))$ and
$\phi_{i}\circ\psi_{j}(V(y;\delta_{1}^{y}))$ are disjoint as long as $j$ and
$i$ are large enough. Let
$v_{i}^{x},\sigma_{i}^{x},s^{i}_{x}\in\Gamma(M_{i},K_{M_{i}}^{-l_{x}})$ and
$v_{i}^{y},\sigma_{i}^{y},s^{i}_{y}\in\Gamma(M_{i},K_{M_{i}}^{-l_{y}})$ be
sections associated $x$ and $y$, respectively. We may assume that
$l_{x}=l_{y}=\ell$ for a multiple of $l$. Moreover, by the $C^{0}$-estimate of
$\sigma_{i}^{x}$ in $V(x;\delta^{x})$ in (5), we see that $|s^{i}_{x}(q_{i})|$
is small. Similarly, $|s^{i}_{y}(p_{i})|$ is also small. Now we define
holomorphic sections

(8.4)
$\displaystyle\tilde{s}^{i}_{x}=s^{i}_{x}-\frac{s^{i}_{x}(q_{i})}{s^{i}_{y}(q_{i})}s^{i}_{y}~{}{\rm
and}~{}\tilde{s}^{i}_{y}=s^{i}_{y}-\frac{s^{i}_{y}(p_{i})}{s^{i}_{x}(p_{i})}s^{i}_{x}.$

Clearly, $\tilde{s}_{x}(q_{i})=\tilde{s}_{y}(p_{i})=0.$ Then
$s_{x}=\frac{\tilde{s}^{i}_{x}}{|\tilde{s}^{i}_{x}(p_{i})|_{h_{i}}}$ and
$s_{y}=\frac{\tilde{s}^{i}_{y}}{|\tilde{s}^{i}_{y}(q_{i})|_{h_{i}}}$ will
satisfy (8.3).

∎

By Lemma 8.1, we prove

######  Proposition 8.2.

Let $\\{(M_{i},g^{i})\\}$ be a sequence of Fano manifolds with Ricci bounded
from below and diameter bounded from above, and $(M_{\infty},g_{\infty})$ its
limit in Goromov-Hausdorff topology. Suppose that (8.1) and (8.3) in Lemma 8.1
hold. Then $M_{\infty}$ is homeomorphic to an algebraic variety.

###### Proof.

By (8.1), for any $k$, we can define holomorphisms

$\displaystyle T_{kl,i}:M_{i}\rightarrow\mathbb{C}P^{N},$

where $N+1={\rm dim}H^{0}(M_{i},K_{M_{i}}^{-kl})$ is constant if $i$ is large
enough. Since $T_{kl,i}$ is uniformly Lipschitz by (3.6), we get a limit map

$\displaystyle T_{kl,\infty}:M_{\infty}\rightarrow\mathbb{C}P^{N}.$

On the other hand, the images $W^{kl}_{i}$ of $T_{kl,i}$ have a chow limit
$W^{kl}$, which coincides with the image of the map $T_{kl,\infty}$. Thus
$T_{kl,\infty}$ maps $M_{\infty}$ onto $W^{kl}=T_{kl,\infty}(M_{\infty})$. We
claim that $T_{(n+2+[\Lambda^{2}])l,\infty}$ is injective, so the proposition
is proved.

By Lemma 8.3, for any $x,y\in M_{\infty}$, there are $p_{i}\rightarrow x$ and
$q_{i}\rightarrow y$, and $s_{x},s_{y}\in
H^{0}(M_{\infty},K_{M_{i}}^{-k_{1}l})$ for some $k_{1}$ such that

(8.5) $\displaystyle|s_{x}|_{h_{i}}(p_{i})=|s_{y}|_{h_{i}}(q_{i})|=1~{}{\rm
and}~{}s_{x}(q_{i})=s_{y}(p_{i})=0.$

This means $T_{k_{1}l,\infty}(x)\neq T_{k_{1}l,\infty}(y)$. We further show
that

(8.6) $\displaystyle T_{(n+2+[\Lambda^{2}])l,\infty}(x)\neq
T_{(n+2+[\Lambda^{2}])l,\infty}(y).$

In fact, if (8.6) is not true, it is easy to see
$T_{il,\infty}(x)=T_{il,\infty}(y)$ for any $i\leq n+2+[\Lambda^{2}]$. Then by
(8.2), it follows

$T_{kl,\infty}(x)=T_{kl,\infty}(y),~{}\forall~{}k,$

which is contradict to (8.5). Thus (8.6) is true. Hence
$T_{(n+2+[\Lambda^{2}])l,\infty}$ must be injective.

∎

###### Proof of Corollary 1.4.

By the Gromov compactness theorem, there exists a subsequence
$\\{(M_{i_{k}},g^{i_{k}})\\}$ of $\\{(M_{i},g^{i})\\}$, which converges to
$(M_{\infty},g_{\infty})$. Then i) and ii) in Corollary 1.4 follow from a
generalized Cheeger-Colding-Tian compactness theorem for a sequence of almost
Kähler-Einstein manifolds [26] ( or a sequence of Fano manifolds with almost
Kähler-Ricci solitons [32]). Thus we suffice to prove the part iii). By
Proposition 8.2, we know that $M_{\infty}$ is homomorphic to an algebraic
variety $W^{k_{0}l}$, where $k_{0}=n+2+[\Lambda^{2}]$. We further show that
$W^{k_{0}l}$ is a log terminal $Q$-Fano variety.

Let $H^{0}(M_{\infty},K_{M_{\infty}}^{-k_{0}l})$ be a space of bounded
holomorphic sections of $K_{\mathcal{R}}^{-k_{0}l}$ with respect to the
induced metric $g_{\infty}$. Then for any compact set
$K\subseteq\mathcal{R}\subseteq M_{\infty}$, we know that there are $t_{K}>0$
and $K_{i}\subseteq M_{i}$ such that $(K_{i},g_{i}(t_{K}))$ converge to
$(K,g_{\infty})$ smoothly. Thus by the argument in Proposition 5.1 and Lemma
6.2, we can identify $H^{0}(M_{\infty},K_{M_{\infty}}^{-k_{0}l})$ with the
limit of $H^{0}(M_{i},K_{M_{i}}^{-k_{0}l})$. But, from the proof in
Proposition 8.2, the latter is the same as $H^{0}(W^{k_{0}l},$
$\mathcal{O}_{\mathbb{C}P^{N}}(1))$. This implies that $M_{\infty}$ is
homeomorphic to the normalization of $W^{k_{0}l}$ since the codimension of
singularities of $W^{k_{0}l}$ is at least $2$ (cf. [23] and [7]). Hence
$W^{kl_{0}}$ is normal. By [1], it remains to prove that $W^{k_{0}l}$ is a
$Q$-Fano variety.

Let $\mathcal{S}={\rm Sing}(M_{\infty})$,
$\mathcal{\hat{S}}=T_{k_{0}l,\infty}(\mathcal{S})$, and let
$W_{s}\subset\mathcal{\hat{S}}$ be the singular set of $W^{k_{0}l}$. Then both
$W_{s}$ and $\mathcal{\hat{S}}$ lie in a subvariety of $W^{k_{0}l}$ with
codimension at least 2. Thus we suffice to prove that
$W_{s}=\mathcal{\hat{S}}$ since
$(W^{k_{0}l},\mathcal{O}_{\mathbb{C}P^{N}}(1))=$ $\newline
K^{-k_{0}l}_{W^{k_{0}l}\setminus\mathcal{\hat{S}}}$. In the following, we give
a proof for the general limit Kähler-Ricci soliton $(M_{\infty},g_{\infty})$
in Section 7 by using PDE method as in [7]. Namely, $g_{\infty}$ satisfies an
equation,

(8.7) $\displaystyle{\rm
Ric}(g_{\infty})-g_{\infty}-L_{X_{\infty}}g_{\infty}=0,~{}{\rm
in}~{}M_{\infty}\setminus\mathcal{S},$

where $X_{\infty}$ is the limit holomorphic vector field of $(M_{i},X_{i})$ on
$M_{\infty}\setminus\mathcal{S}$ [32].

On contrary, we suppose that $W_{s}\neq\mathcal{\hat{S}}$. Then there exists
some $x\in\mathcal{S}$ such that $p=T_{k_{0}l,\infty}(x)\in
W^{k_{0}l}\setminus W_{s}$, a smooth point in $W^{k_{0}l}$. Thus there exists
a small ball $B$ around $p$ in $W^{k_{0}l}$ with the standard holomorphic
coordinates such that the induced Kähler form
$\omega_{0}=\frac{1}{k_{0}l}\omega_{g_{FS}}$ by the Fubini-Study metric
$g_{KS}$ of the projective space is smooth on $B$. We may assume that
$\omega_{0}=\sqrt{-1}\partial\bar{\partial}v$ for some Kähler potential $v$ on
$B$.

Let $\rho_{\infty}$ be the limit of $\rho_{k_{0}l}(M_{i},g^{i})$ (perhaps
replaced by a subsequence of $\rho_{k_{0}l}(M_{i},g^{i})$) on
$(M_{\infty}\setminus\mathcal{S},g_{\infty})$. Then $\rho_{\infty}$ and
$|\nabla\rho_{\infty}|_{g_{\infty}}$ are both uniformly bounded since
$\rho_{k_{0}l}(M_{i},g^{i})$ and $|\nabla\rho_{k_{0}l}(M_{i},g^{i})|_{g^{i}}$
are all uniformly bounded by (3.6). Clearly, $\rho_{\infty}$ satisfies

$\omega_{g_{\infty}}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\rho_{\infty},~{}{\rm
in}~{}W^{k_{0}l}\setminus\mathcal{\hat{S}}.$

Let $u=v+\rho_{\infty}$. Then by (8.7), we see that $u$ satisfies

$\displaystyle\sqrt{-1}\partial\bar{\partial}(\log{\rm
det}(u_{i\bar{j}})+X_{\infty}(u)+u)=0,~{}{\rm
in}~{}B\setminus\mathcal{\hat{S}}.$

It follows

(8.8) $\displaystyle\log{\rm det}(u_{i\bar{j}})+X_{\infty}(u)+u=const.,~{}{\rm
in}~{}B\setminus\mathcal{\hat{S}}.$

We claim that there exists a uniform $C$ such that

(8.9) $\displaystyle C^{-1}\delta_{i\overline{j}}\leq u_{i\overline{j}}\leq
C\delta_{i\overline{j}},~{}{\rm in}B\setminus\mathcal{\hat{S}}.$

Since the basis in $H^{0}(M_{\infty},K_{M_{\infty}}^{-k_{0}l})$, which gives
the embedding $T_{k_{0}l,\infty}$, is uniformly $C^{1}$-bounded, we have

$\omega_{0}\leq C\omega_{g_{\infty}},~{}{\rm in}~{}M_{\infty}.$

On the other hand, by (7.13),

$|X_{\infty}(\rho_{\infty})|\leq|X_{\infty}|_{g_{\infty}}|\nabla\rho_{\infty}|_{g_{\infty}}\leq
C,~{}{\rm in}~{}M_{\infty}.$

Then $X_{\infty}(u)$ is uniformly bounded. Thus by (8.8), we see that
$\log{\rm det}(u_{i\bar{j}})$ is uniformly positive and bounded. This implies
(8.9).

By the above claim, we can apply the following lemma to show that $u$ is a
smooth function in a small neighborhood of $p$. But this is impossible by
$x\in\mathcal{S}$. Hence $W^{k_{0}l}$ must be a $Q$-Fano variety.

∎

######  Lemma 8.3.

Let $u$ be a smooth solution of (8.8) in $B\setminus\mathcal{\hat{S}}$, where
$B$ is a ball in the euclidean space in $\mathbb{C}^{n}$ and
$\mathcal{\hat{S}}$ is a closed subset in $\mathbb{C}^{n}$ with real dimension
less than $2n-1$. Suppose that $u$ satisfies (8.9). Then $u$ can be extended
to a smooth function on $\frac{1}{4}B$.

###### Proof.

By the Schaulder estimate for the equation (8.8), we suffices to get a
$C^{2,\alpha}$-regularity of $u$ in $\frac{1}{4}B$. We first do the
$C^{1,1}$-estimate.

For any $0<\epsilon<\frac{1}{8}$ and any unit vector $v$, we let the
difference quotient

$w=w_{\epsilon}=\frac{u(x+\epsilon v)+u(x-\epsilon v)-2u(x)}{\epsilon^{2}}.$

Then by the convexity of $\log{\rm det}$, we get from (8.8),

(8.10) $\displaystyle u^{i\bar{j}}w_{i\bar{j}}\geq e^{g}\frac{g(x+\epsilon
v)+g(x-\epsilon v)-2g(x)}{\epsilon^{2}},$

where $g=-u-X_{\infty}(u)$. Denote $(a_{\alpha\beta})$ to be the $2n\times 2n$
matrix of Riemannian metric of $g_{\infty}$ and $(a^{\alpha\beta})={\rm
det}(a_{\delta\gamma})(a_{\alpha\beta})^{-1}.$ It is clear that (8.10) is
equivalent to

(8.11) $\displaystyle(a_{\alpha\beta}w_{\beta})_{\alpha}\geq
l(x)+\frac{h(x+\epsilon v)-h(x)}{\epsilon},~{}{\rm
in}~{}\frac{3}{4}B\setminus\mathcal{\hat{S}},$

where $l=f(x+\epsilon v)\frac{e^{g}(x+\epsilon
v)-e^{g}(x)}{\epsilon},h=e^{g}f$ and $f=\frac{g(x)-g(x-\epsilon
v)}{\epsilon}$. Note that $X_{\infty}$ can be extended to a holomorphic vector
field on $B$. Then by (8.9), $w$ can be regarded as a weak sub-solution in
(8.11) in whole $\frac{3}{4}B$. Thus by the $L^{\infty}$-estimate arising from
the Moser iteration, we have,

(8.12) $\displaystyle\sup_{\frac{1}{2}B}(w_{\epsilon})\leq
C(|w_{\epsilon}|_{L^{p}(\frac{3}{4}B)}+|l|_{L^{\frac{q}{2}}(B)}+|h|_{L^{q}(B)}),$

where $C$ depends only on $(a_{\alpha\beta})$, $p\geq 1$ and $q>2n$. In fact,
by Theorem 8.17 in [8], the estimate (8.12) holds for sub-solution $w$ as
follows,

$(a_{\alpha\beta}w_{\beta})_{\alpha}\geq l+<v,Dh>.$

But Theorem 8.17 is also true when the term $<v,Dh>$ is replaced by the
difference quotient $\frac{h(x+\epsilon v)-h(x)}{\epsilon}$.

Since $g$ is uniformly Lipschitz in $B\setminus\mathcal{\hat{S}}$, $l,h$ are
$L^{\infty}$-functions in $B$. On the other hand, by (8.9), $u\in
W^{2,p}(\frac{3}{4}B)$ for any $p\geq 1$, and so
$|w_{\epsilon}|_{L^{p}(\frac{3}{4}B)}$ is uniformly bounded. Thus the (8.12)
implies that $w_{\epsilon}$ is uniformly bounded above. As a consequence,
$C^{1,1}$-derivative $u_{vv}$ is uniformly bounded above. By (8.9), we can
also get a uniform lower bound of $u_{vv}$. Hence $C^{1,1}$-norm of $u$ is
uniformly bounded in $\frac{1}{2}B$.

Next to get $C^{2,\alpha}$-estimate of $u$ in (8.8), we can apply Evans-Krylov
theorem, Theorem 17.14 in [8] to $C^{1,1}$-solution of (8.8) in $\frac{1}{2}B$
directly. This is because (8.8) is strictly elliptic in $B$ and
$-u-X_{\infty}(u)$ is Lipschitz. Thus the lemma is proved.

∎

##  9\. Conclusion

In the proofs of Theorem 6.1 and Theorem 7.6, the constants $c_{l}$ in the
estimates (6.1) and (7.6) may depend on the limit $(M_{\infty},g_{\infty})$.
In this section, we show that $c_{l}$ just depends on $n,l_{0}$ and $l$, and
the geometric uniform constants $\Lambda$ and $D$ in $i)$ of (3), or the
constants $\Lambda,D,C_{0}$ and $B$ in (7) and $iii)$ of (7). Thus we complete
the proof of Theorem 1.3. For simplicity, we just consider the case of almost
Kähler-Einstein Fano manifolds below.

Set a class of Fano manifolds by

$\mathcal{K}_{\Lambda,D}=\\{(M^{n},g)|~{}\omega_{g}\in 2\pi c_{1}(M),{\rm
Ric}(g)\geq-(n-1)\Lambda^{2},{\rm diam}(M,g)\leq D\\}.$

It is known that $\mathcal{K}_{\Lambda,D}$ is precompact in Gromov-Hausdorff
topology. Moreover, by Cheeger-Colding theory in [2], any Gromov-Hausdorff
limit $M_{\infty}$ in $\mathcal{K}_{\Lambda,D}$ contains singularities with
codimension at least 2 and each tangent cone at $x\in M_{\infty}$ is a metric
cone $C_{x}$, which also contains singularities with codimension at least 2.

Let $\mathcal{K}_{\Lambda,D}^{0}$ be a subset of $\mathcal{K}_{\Lambda,D}$
such that ${\mathcal{H}}^{2n-2}({\rm Sing}(C_{x}))=0$ for any $x\in
M_{\infty}$, where $M_{\infty}$ is any Gromov-Hausdorff limit in
$\mathcal{K}_{\Lambda,D}^{0}$. Then according to the proofs in Proposition 5.1
and Theorem 6.1, we have

######  Proposition 9.1.

Let $(M,g)\in\mathcal{K}_{\Lambda,D}^{0}$ and $g_{t}$ a solution of
(LABEL:Ricci_flow) with the initial metric $g$. Then there exist a small
number $\delta=\delta(\Lambda,D,n)$ and a large integer
$l_{0}=l_{0}(n,\Lambda,D)$ such that the following is true: if $g$ satisfies

(9.1) $\displaystyle\int_{0}^{1}\int_{M}|R-n|{\rm dv}_{g_{t}}dt\leq\delta,$

then for any integer $l$ there exists a uniform constant
$c=c(n,l,\Lambda,D)>0$ such that

(9.2) $\displaystyle\rho_{ll_{0}}(M,g)\geq c.$

###### Proof.

By Theorem 6.1, we see that for any $Y\in\bar{\mathcal{K}}_{\Lambda,D}^{0}$,
there exist a small number $\delta_{Y}>0$, a large integer $l_{Y}$ and a
uniform constant $c_{Y}>0$ such that if $M\in\mathcal{K}_{\Lambda,D}$
satisfies

$\displaystyle{\rm
d}_{GH}((M,g),(Y,g_{Y}))\leq\delta_{Y},\int_{0}^{1}\int_{M}|R-n|{\rm
dv}_{g_{t}}dt\leq\delta_{Y},$

then

$\rho_{l_{Y}}(M,g)\geq c_{Y}.$

Since $\bar{\mathcal{K}}_{\Lambda,D}$ is compact, we can cover it by finite
balls $B_{Y_{i}}(\delta_{Y_{i}})(1\leq i\leq N)$ in Gromov-Hausdroff topology.
Putting $l_{0}=\Pi l_{Y_{i}},\delta=\min\\{\delta_{Y_{i}}\\}$ and
$c=\min\\{c_{Y_{i}}\\}$. Then we get (9.2) for $l=1$, if $(M,g)$ satisfies
(9.1). (9.2) is also true for general $l$ as in the proof of Theorem 6.1. ∎

(1.3) in Theorem 1.3 follows from (9.2).

##  10\. Appendix

In this appendix, we use the following Siu’s lemma to generalize the finite
generation formula (8.2) under the Bakry-Eméry Ricci curvature condition [17].

######  Lemma 10.1.

Let $(M^{n},g)$ be a compact complex manifold, $G$ a holomorphic line bundle,
$E$ a holomorphic line bundle with a hermitian metric $e^{-\psi}$ whose Ricci
curvature is positive. Let $\\{s_{i}\\}_{1\leq i\leq p}$ be a basis of
$H^{0}(M,G)$ and $|s|^{2}=\Sigma_{i=1}^{p}|s_{i}|^{2}$. Then for any $f\in
H^{0}(M,(n+k+1)G+E+K_{M})$ which satisfies

$\int_{M}\frac{|f|^{2}e^{-\psi}}{|s|^{2(n+k+1)}}d{\rm v}_{g}<+\infty,$

there are some $h_{i}\in H^{0}(M,(n+k)G+E+K_{M})$ ($k\geq 1$) such that
$f=\Sigma_{i=1}^{p}h_{j}\otimes s_{j}$ and each $h_{i}$ satisfies

$\int_{M}\frac{|h_{j}|^{2}e^{-\psi}}{|s|^{2(n+k)}}d{\rm
v}_{g}\leq\frac{n+k}{k}\int_{M}\frac{|f|^{2}e^{-\psi}}{|s|^{2(n+k+1)}}d{\rm
v}_{g}.$

######  Proposition 10.2.

Let $(M,g)$ be a Kähler manifold with

${\rm Ric}(g)+{\rm Hess}\,u\geq-Cg,$

where $X=\nabla u$ is a holomorphic vector field and $|u|\leq A$. Assume that

(10.1) $\displaystyle c^{\prime}\geq\rho_{l}(M,g)\geq c>0$

for some $l\in\mathbb{N}$. Then for any $s\in H^{0}(M,K_{M}^{-m})$ with
$m\geq(n+2)l+C+1$, there are $u_{i}\in H^{0}(M,K_{M}^{-(m-l)})$ such that
$s=\Sigma_{i=0}^{N}u_{i}\otimes s_{i}$, where $\\{s_{i}\\}$ is an orthonormal
basis of $H^{0}(M,K_{M}^{-l})$. Moreover, each $u_{i}$ satisfies

(10.2) $\displaystyle\int_{M}|u_{i}|^{2}_{h^{\otimes m-l}}{\rm
dv}_{g}\leq(n+1)e^{2A}(\frac{c^{\prime}}{c})^{\frac{m}{l}}\int_{M}|s|^{2}_{h^{\otimes
m}}{\rm dv}_{g}.$

###### Proof.

Putting $L=K_{M}^{-1}$ and $m-C-1=(n+k+1)l+r$ $(0\leq r<l)$, we decompose $mL$
as

$mL=(n+k+1)(lL)+((m-(n+k+1)l)L-K_{M})+K_{M}.$

Let $h$ and $\omega_{g}^{n}$ be two hermitian metrics on $L$ such that

${\rm Ric}(L,h)=g,\,{\rm Ric}(L,\omega_{g}^{n})={\rm Ric}(g).$

Denote the line bundle $(m-(n+k+1)l)L-K_{M}$ by $E$. Then $h_{1}=h^{\otimes
m-(n+k+1)l}\otimes e^{-u}\otimes\omega_{g}^{n}$ is a hermitian metric on $E$.
It is easy to see

${\rm Ric}(E,h_{1})=(m-(n+k+1)l)\omega_{g}+{\rm
Ric}(g)+\sqrt{-1}\partial\bar{\partial}u\geq\omega_{g}.$

Now applying the above lemma to $G=lL$, $s_{i}$, $E$ and $f=s$, we see that
there are $u_{i}\in H^{0}(M,(n+k)G+E+K_{M})$ such that

$\int_{M}\frac{|u_{i}|^{2}_{h^{\otimes(n+k)l}\otimes
h_{1}}}{(\Sigma_{i=0}^{N}|s_{i}|^{2}_{h^{\otimes l}})^{n+k}}{\rm dv}_{g}\\\
\leq\frac{n+k}{k}\int_{M}\frac{|s|^{2}_{h^{\otimes(n+k+1)l}\otimes
h_{1}}}{(\Sigma_{i=0}^{N}|s_{i}|^{2}_{h^{\otimes l}})^{n+k+1}}{\rm dv}_{g}.$

The above is equivalent to

$\displaystyle\int_{M}\frac{|u_{i}|^{2}_{h^{\otimes
m-l}}}{(\Sigma_{i=0}^{N}|s_{i}|^{2}_{h^{\otimes l}})^{n+k}}e^{-u}{\rm
dv}_{g}\leq\frac{n+k}{k}\int_{M}\frac{|s|^{2}_{h^{\otimes
m}}}{(\Sigma_{i=0}^{N}|s_{i}|_{h^{\otimes l}}^{2})^{n+k+1}}e^{-u}{\rm
dv}_{g}.$

By (10.1), it follows

$\displaystyle\frac{1}{e^{2A}c^{\prime n+k}}\int_{M}|u_{i}|^{2}_{h^{\otimes
m-l}}{\rm dv}_{g}\leq\frac{n+k}{kc^{n+k+1}}\int_{M}|s|^{2}_{h^{\otimes m}}{\rm
dv}_{g},$

which implies (10.2) immediately. ∎

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