# Complete Kähler manifolds with nonnegative Ricci curvature

Gang Liu  School of Mathematical Sciences, Shanghai Key Laboratory of PMMP,
East China Normal University<EMAIL_ADDRESS>
###### Abstract.

We consider complete Kähler manifolds with nonnegative Ricci curvature. The
main results are: 1\. When the manifold has nonnegative bisectional curvature,
we show that
$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S$
exists. In other words, it depends only on the manifold. This solves a
question of Ni in [47]. Also, we establish estimates among volume growth
ratio, integral of scalar curvature, and the degree of polynomial growth
holomorphic functions. The new point is that the estimates are sharp for _any_
prescribed volume growth rate. As a byproduct, we show that
$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S<\epsilon$
iff the asymptotic volume ratio of the universal cover is almost maximal. 2.
We discover a strong rigidity for complete Ricci flat Kähler metrics. Let
$M^{n}(n\geq 2)$ be a complete Kähler manifold with nonnegative Ricci
curvature and Euclidean volume growth. Assume either the curvature has
quadratic decay, or the Kähler metric is $dd^{c}$-exact with quadratic decay
of scalar curvature. If one tangent cone at infinity is Ricci flat, then $M$
is Ricci flat. In particular, the tangent cone is unique. In other words, _we
can test Ricci flatness of the manifold by checking one single tangent cone_.
This seems unexpected, since apriori, there is no equation on $M$ and the
Bishop-Gromov volume comparison is not sharp on Ricci flat (nonflat)
manifolds. Such result is in sharp contrast to the Riemannian setting: Colding
and Naber [16] showed that tangent cones are quite flexible when $Ric\geq 0$
and $|Rm|r^{2}<C$. This reveals subtle differences between Riemannian case and
Kähler case. The result contains a lot of examples, such as all noncompact
Ricci flat Kähler surfaces of Euclidean volume growth (hyper-Kähler ALE
4-manifolds classified by Kronheimer [32]), higher dimensional examples of
Tian-Yau type [58], as well as an example with irregular cross section [19].
It also covers Ricci flat Kähler metrics of Euclidean volume growth on Stein
manifolds with $b_{2}=0$, such as Ricci flat Kähler metrics on
$\mathbb{C}^{n}$ [33][54][17][12]. Note in this case, the cross section is
singular.

††The author is partially supported by National Key Research and Development
Program of China (No. 2022YFA1005502), NSFC No. 12071140, the New Cornerstone
Science Foundation through the New Cornerstone Investigator Program and the
Xplorer Prize, Science and Technology Commission of Shanghai Municipality (No.
22DZ2229014)

##  1\. Introduction

In Riemannian geometry, it is interesting to study the distribution of scalar
curvature on manifolds with nonnegative Ricci curvature. Conjectures of Yau
[60], Gromov [27], Naber [46] predict bounds of scalar curvature integral. In
this paper, we are interested in the opposite direction. We shall study the
impact of scalar curvature integral on complete Kähler manifolds. It turns out
there are very strong rigidity phenomena, under both $BK\geq 0$ (nonnegative
bisectional curvature) and $Ric\geq 0$.

In [47], Ni proved a remarkable gap theorem on complete Kähler manifolds with
nonnegative bisectional curvature:

######  Theorem 1.

[optimal gap theorem] Let $(M,p)$ be a complete noncompact Kähler manifold
with nonnegative bisectional curvature. If $\lim\limits_{r\to\infty}\inf
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S=0$,
then $M$ is flat. Here $S$ is the scalar curvature,
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int$
is the average.

In the same paper, he asked whether or not
$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S$
exists on complete Kähler manifolds with nonnegative bisectional curvature.
Our first result answers Ni’s question affirmatively:

######  Theorem 2.

Let $(M,p)$ be a complete noncompact Kähler manifold with nonnegative
bisectional curvature. Then
$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S$
exists. In other words, such limit (possibly infinity) depends only on $M$.

Through the works of [48][50][36][38], we know that the following conditions
are equivalent on a complete Kähler manifold with nonnegative bisectional
curvature, if the universal cover does not split:

  * •

$M$ has Euclidean volume growth;

  * •

There exists a nonconstant polynomial growth holomorphic function on $M$;

  * •

$\lim\limits_{r\to\infty}\sup
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S<+\infty$.

However, explicit relations seem ellusive from previous arguments. Our second
result addresses this (it is clear that we just need to handle Euclidean
volume growth case):

######  Theorem 3.

Let $(M^{n},p)$ be a complete noncompact Kähler manifold with nonnegative
bisectional curvature and Euclidean volume growth. Let $d$ be the smallest
degree of polynomial growth holomorphic functions (nonconstant) on $M$. Also
let

$v=\lim\limits_{r\to\infty}\frac{Vol(B(p,r)}{Vol(B_{\mathbb{C}^{n}}(0,r))},$
$\alpha=\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S.$

Then we have the following sharp estimates:

(1)

$4n^{2}(v^{-\frac{1}{n}}-1)\leq\alpha\leq 4n(v^{-1}-1),$

(2)

$1\leq d\leq v^{-\frac{1}{n}},$

(3)

$\lim\limits_{k\to\infty}\frac{dim(\mathcal{O}_{k}(M))}{k^{n}/n!}=v\leq
1=\lim\limits_{k\to\infty}\frac{dim(\mathcal{O}_{k}(\mathbb{C}^{n}))}{k^{n}/n!}.$

When the metric is unitary symmetric [59], the left side of $(1)$ and right
side of $(2)$ become equality. However, the metric is not necessarily unique
in this case. The right side of $(1)$ becomes an equality if and only if $M$
splits off $\mathbb{C}^{n-1}$.

For previous sharp estimates on Kähler manifolds with nonnegative bisectional
curvature, the equality is only valid for Euclidean or splitting case. In our
case, sharp equalities may hold under _any_ asymptotic volume ratio, while the
metric does not split. $(3)$ indicates a connection between (asymptotic)
dimension estimate (compare [48][21]) and Bishop-Gromov volume comparison.
These estimates support the common sense in a quantitative way: the larger the
curvature, the smaller the volume. Another interesting thing is that we obtain
volume ratio lower bound, when the integral of scalar curvature has an upper
bound. This can be regarded as an inverse of volume comparison. Such thing is
not possible if we only assume $Ric\geq 0$. As a byproduct, we obtain a
quantitative version of Ni’s optimal gap theorem:

######  Corollary 1.

Let $(M^{n},p)$ be a complete noncompact Kähler manifold with nonnegative
bisectional curvature. Let $(\tilde{M},\tilde{p})$ be the universal cover of
$M$. Then the smallness of
$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S$
is equivalent to the smallness of
$1-\lim\limits_{r\to\infty}\frac{vol(B(\tilde{p},r))}{\omega_{2n}r^{2n}}$,
where $\omega_{2n}$ is the volume of unit ball in $\mathbb{C}^{n}$.

Now we move to $Ric\geq 0$. A result of Anderson [1] states that if the
asymptotic volume ratio is sufficiently close to that of the Euclidean space,
then Ricci flatness is the same as flatness. We have the following
generalization of Anderson’s result in the Kähler setting:

######  Theorem 4.

There exists $\epsilon=\epsilon(n)>0$ such that the following hold: Assume
$M^{n}$ is a complete noncompact Kähler manifold with nonnegative Ricci
curvature and

$v=\lim\limits_{r\to\infty}\frac{Vol(B(p,r)}{Vol(B_{\mathbb{C}^{n}}(o,r))}\geq
1-\epsilon.$

If

$\lim\limits_{r\to\infty}\inf
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S=0$

Then $M$ is flat.

It is clear that such result cannot be true, if the volume is not close to
Euclidean, e.g., the Eguchi-Hanson metric. Unlike the flat (Euclidean) case,
there is abundance of complete Ricci flat Kähler manifolds, even assuming
Euclidean volume growth and quadratic curvature decay. The reader is referred
to [20] for a description of these manifolds.

######  Definition 1.

Let $(M^{n},\omega)(n\geq 2)$ be a complete noncompact Kähler manifold with
nonnegative Ricci curvature and Euclidean volume growth. We say $M$ satisfies
condition $A$, if $|Rm|\leq\frac{C}{r^{2}}$; $M$ satisfies condition $B$, if
$S\leq\frac{C}{r^{2}}$ and $\omega$ is $dd^{c}$-exact.

Frequently, we shall consider tangent cones of complete manifolds. The complex
structure on a tangent cone is induced from the sequence of rescaled
manifolds. Below, when we talk about convergence, we mean both the metric
(Gromov-Hausdorff distance) and the complex structure in the regular part.

######  Definition 2.

Let $(M^{n},\omega)(n\geq 2)$ be a complete noncompact Kähler manifold with
nonnegative Ricci curvature and Euclidean volume growth. Let $V$ be a tangent
cone of $M$ at infinity. We say $V$ is Ricci flat, if the metric on $V$ is
Ricci flat (hence smooth) away from a real codimension $4$ set, and there
exists a nontrival parallel pluri-canonical form on the metric regular part.

Note if $M$ is a complete Ricci flat Kähler manifold with Euclidean volume
growth, then by Cheeger-Naber [16], any tangent cone is Ricci flat.

The interplay between the space (manifold) and its asymptotic cones (tangent
cone) is fundamental in geometric analysis. On the one hand, uniqueness of
asymptotic cones is expected when there are natural equations (e.g., Einstein
manifolds, minimal surfaces, solitons). On the other hand, one looks for
classification of spaces, according to their asymptotic cones. The following
theorem asserts that when a Kähler manifold has nonnegative Ricci curvature,
one tangent cone at infinity might impose very strong restriction on the
manifold (compare Colding [14]):

######  Theorem 5.

Let $M^{n}$ satisfy either condition $A$ or condition $B$. If one tangent cone
at infinity is Ricci flat, then $M$ is Ricci flat. Hence the tangent cone is
unique.

Such result seems unexpected, since apriori, there is no equation on $M$, and
Bishop-Gromov volume comparison is not sharp on Ricci flat (nonflat)
manifolds. This seems to be the _first_ rigidity result for noncompact Ricci
flat metrics, from metric geometry view point. Note by Theorem $B$ in [28],
such result is not true, if we merely assume the scalar curvature is
nonnegative. The result is in sharp contrast to the Riemannian setting:
according to Theorem $1.1$ in [16] by Colding-Naber, tangent cones are quite
flexible (in Colding-Naber case, one can make $Ric\geq 0$ and $|Rm|r^{2}\leq
C$). This reveals subtle differences between Riemannian case and Kähler case.
Theorem 5 covers all noncompact Ricci flat Kähler surfaces of Euclidean volume
growth (hyper-Kähler ALE 4-manifolds classified by Kronheimer [32]). It also
covers many examples of Tian-Yau type [57][58][5][52][3][18][26][31][13] in
higher dimensions, as well as an example with irregular cross section [19].
Also, it covers complete Ricci flat Kähler metrics with Euclidean volume
growth on $\mathbb{C}^{n}$, e.g., [33][54][17][12]. Note in this case, the
cross section is singular. We also note that given a tangent cone, the Ricci
flat metric on the manifold need not be unique [12].

######  Corollary 2.

Let $(M^{n},\omega)(n\geq 2)$ be a complete noncompact Kähler manifold with
nonnegative Ricci curvature, $S\leq\frac{C}{r^{2}}$ and Euclidean volume
growth. Assume $H^{1}(M,\mathcal{O})=0$ (e.g., $M$ is Stein) and $b_{2}=0$. If
one tangent cone at infinity is Ricci flat, then $M$ is Ricci flat.

We also have results on local tangent cones:

######  Corollary 3.

Let $(M_{i},p_{i})$ be a sequence of complete Kähler manifolds with
$Ric_{M_{i}}\geq 0$, $vol(B(p_{i},1))>v$. Assume there exists $C,v>0$ such
that $|Rm|_{M_{i}}(x)d^{2}(x,p_{i})\leq C$ for all $i$ and $d(x,p_{i})<1$ .
Let $(M_{i},p_{i})$ converge in the pointed Gromov-Hausdorff sense to $(X,o)$.
If one tangent cone at $o\in X$ is Ricci flat, then all tangent cones at $o$
are Ricci flat, and the tangent cone is unique (certainly $X$ need not be
Ricci flat).

######  Corollary 4.

Let $(M_{i},p_{i})$ be a sequence of polarized Kähler manifolds with
$Ric_{M_{i}}\geq-1$, $vol(B(p_{i},1))>v$. Assume there exists $C,v>0$ such
that $S_{M_{i}}(x)d^{2}(x,p_{i})\leq C$ for all $i$ and $d(x,p_{i})<1$ .
Assume $(M_{i},p_{i})$ converges in the pointed Gromov-Hausdorff sense to
$(X,o)$. If one tangent cone at $o\in X$ is Ricci flat, then all tangent cones
at $o$ are Ricci flat, and the tangent cone is unique.

The uniqueness of tangent cones is an important problem in geometric analysis.
A well-known conjecture states that tangent cones are unique for noncollapsed
Einstein manifolds (see [9][15][25] for important progress). However, without
assuming Einstein equation, the uniqueness is not ensured in general
[51][6][16]. In our settings, tangent cones are unique, as long as one tangent
cone is Ricci flat. As we mentioned before, by Colding-Naber’s contruction
[16], such thing cannot be true in the Riemannian setting.

######  Corollary 5.

Let $M^{n}$ satisfy condition $A$. If $\lim\limits_{r\to\infty}\inf
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S=0$,
then $M$ is Ricci flat.

######  Conjecture 1.

Let $M$ be a complete noncompact Kähler manifold with nonnegative Ricci
curvature and Euclidean volume growth. Assume one tangent cone at infinity is
Ricci flat, then $M$ is Ricci flat. Hence, the tangent cone should be unique.
We can also ask about the similar question on local tangent cones.

One can compare Conjecture 1 with a theorem of Colding [14], which says that
if one tangent cone is Euclidean, then the manifold is Euclidean.

######  Conjecture 2.

Let $(M^{n},p)$ be a complete noncompact Kähler manifold with nonnegative
Ricci curvature. If $\lim\limits_{r\to\infty}\inf
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S=0$,
then $M$ is Ricci flat. We can also ask whether or not
$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S$
exists (possibly infinity).

######  Corollary 6.

Let $M^{n}$ satisfy condition $A$. If the scalar curvature has faster than
quadratic decay, then $M$ is Ricci flat.

This can be regarded as a gap theorem for Ricci flat metrics. In fact, one
only needs to assume that scalar curvature has faster than quadratic decay
along a sequence of annuli with fixed ratio.

######  Corollary 7.

Let $M^{n}$ satisfy condition $A$. If $\int_{B(p,r)}S^{n}=o(\log r)$, then $M$
is Ricci flat.

Both the power and $o(\log r)$ are sharp.

######  Corollary 8.

Let $M^{n}$ satisfy condition $A$. If $\int_{B(p,r)}|Rm|^{n}=o(\log r)$, then
$M$ is Ricci flat.

In this case, the tangent cone is an orbifold with flat metric. By [53], the
metric converges in polynomial rate to the tangent cone. Thus, curvature
decays at a polynomial rate greater than $2$. Thus $\int_{M}|Rm|^{n}<+\infty$.
So Corollary 8 can also be regarded as a gap result. These manifolds have been
studied in [4][56][31][30].

Now we describe the strategies in the proof. For $BK\geq 0$, recall in [38],
we used Gromov compactness to derive the existence of polynomial growth
holomorphic functions. However, the method seems too soft to achieve sharp
estimates. The new point is to use Cheeger-Colding-Tian [14][6][8], [42][43]
that the tangent cone is biholomorphic to $\mathbb{C}^{n}$, as well as the
three circles lemma [34]. The key thing is to express the volume ratio and
scalar curvature integral in terms of degrees of polynomial growth holomorphic
functions. Thus the degrees are deeper underlying invariants of the manifold.
In the course of the proof, we have to overcome the difficulty on non-
smoothness of the metric on tangent cones. When these connections are built,
the proof for $BK\geq 0$ follows by arithmetic-geometric inequality. For
$Ric\geq 0$, a difficulty is that usual monotonicity formulae (e.g., Bishop-
Gromov volume comparison) are not sharp for Ricci flat (nonflat) metrics. We
use a new monotonicity, $\int Ric^{n}$. The important thing is that such
quantity is rescale invariant. Also, it behaves well under Gromov-Hausdorff
convergence (under certain assumptions) and provides connections between
different tangent cones. With such aid, we are able to show that if one
tangent cone is Ricci flat, then all tangent cones are Ricci flat. Then
holomorphic spectra for Ricci flat tangent cones are algebraic numbers
[44][25], hence rigid. We apply algebraic compactification in [25][35].
Eventually, we just need to test the integral of Ricci form (use scale
invariance of Ric) along an algebraic curve. If $M$ is not Ricci flat,
integration by parts gives a contradiction.

Notation: Unless otherwise stated, $\Psi(x_{1},..,x_{n})$ means quantities
converging to zero, when $x_{1},..,x_{n}$ are converging to zero. We also let
$dd^{c}=\sqrt{-1}\partial\overline{\partial}$.

Acknowledgments

The author thanks Professors A. Naber, L. Ni, G. Tian for interests and
helpful discussions. He also thanks Professors J. Lott and J.-P Wang for
answering several questions.

##  2\. The case of nonnegative bisectional curvature

Proof of Theorem 2:

Note $n=1$ case follows from the usual Cohn-Vossen inequality [23]. Below we
let $n\geq 2$. We claim that if the limit is independent of $r$, then it is
independent of the base point. Assume
$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p_{1},r)}S=a$
(possibly infinity). For any $p_{2}\in M$, let $b=d(p_{1},p_{2})$. Then

$\int_{B(p_{1},r-b)}S\leq\int_{B(p_{2},r)}S\leq\int_{B(p_{1},r+b)}S.$

Also by Bishop-Gromov volume comparsion,

$1<\frac{vol(B(p_{1},r+b))}{vol(B(p_{1},r-b))}\leq(\frac{r+b}{r-b})^{2n}.$

Putting these two inequalities together, we obtain that

$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p_{2},r)}S=a.$

Below we fix the base point $p$. There are two cases:

  * •

$\lim\limits_{r\to\infty}\sup
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S=\infty$;

  * •

$\lim\limits_{r\to\infty}\sup
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S<\infty$.

In the first case, we can apply [47, Theorem $4.1$] (take $\rho=Ric$) to
conclude that

$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S=\infty.$

Thus we are left with the second case. Let us first assume that the universal
cover does not split off a flat $\mathbb{C}$ factor. It is known from [36,
Theorem $2$] that $M$ has Euclidean volume growth.

Consider a sequence $r_{i}\to\infty$, assume
$(M_{i},p_{i},d_{i})=(M,p,\frac{d}{r_{i}})$ converges in the pointed-Gromov-
Hausdorff sense to a tangent cone $(V,o)$. As is known by [42] [43,
Proposition $6.1$], $V$ is smooth in the complex analytic sense and $V$ is
biholomorphic to $\mathbb{C}^{n}$. Moreover, there exists a holomorphic
coordinate $(z_{1},..,z_{n})$ on $V$ and a reeb vector field given by
$J(r\frac{\partial}{\partial r})=\sum
Re(\sqrt{-1}d_{j}z_{j}\frac{\partial}{\partial z_{j}})$, where $d_{j}\geq 1$
corresponds to weighted degree of the homogeneous function $z_{j}$.

Under the pointed Gromov-Hausdorff convergence $(M_{i},p_{i})\to(V,o)$, we can
find holomorphic functions on $M_{i}$ converging uniformly on each compact set
to $z_{j}$ on $V$. Such functions serve as local holomorphic coordinates
around $p_{i}$. For simplicity, we still call these functions
$z_{1},..,z_{n}$, defined on $M_{i}$. In other words, over $M_{i}$ and $V$, we
have the same holomorphic coordinate (different metrics). Let the rescaled
Kähler form on $M_{i}$ be $\omega_{i}$. As $Ric\geq 0$, $\log|dz_{1}\wedge
dz_{2}\wedge\cdot\cdot\cdot\wedge dz_{n}|_{\omega_{i}}^{2}$ is psh. Hence by
passing to a subsequence, we may assume it converges to a psh function on $V$
(we use the same holomorphic chart). Let us call it $\log|dz|^{2}$.

######  Claim 1.

Under the homothetic vector field $r\frac{\partial}{\partial r}$,
$|dz|^{2}=e^{\log|dz|^{2}}$ is homogeneous of degree $\sum d_{j}-n$.

###### Proof.

Let $\varphi_{i}=\log|dz_{1}\wedge\cdot\cdot\wedge
dz_{n}|_{\omega_{i}}^{2},\varphi=\log|dz|^{2}$. Then $\varphi_{i}\to\varphi$
in $L_{loc}^{1}$ sense. Pick an arbitrary bounded domain $\Omega\subset V$.
Under the same holomorphic coordinate $(z_{1},..,z_{n})$, we take
$\Omega_{i}=\Omega$, where $\Omega_{i}\subset M_{i}$. Let $dz\wedge
d\overline{z}$ be the Euclidean measure on $\mathbb{C}^{n}$.

######  Lemma 1.

$vol(\Omega)=\int_{\Omega}e^{-\varphi}dz\wedge d\overline{z}$.

###### Proof.

As $\varphi_{i}\to\varphi$ in $L_{loc}^{1}$, $e^{-\varphi_{i}}\to
e^{-\varphi}$ almost everywhere. Note

$vol(\Omega_{i})=\int_{\Omega_{i}}e^{-\varphi_{i}}dz\wedge d\overline{z}.$

By Cheeger-Colding [7],

$vol(\Omega_{i})\to vol(\Omega).$

Then

$\int_{\Omega}e^{-\varphi}dz\wedge d\overline{z}\leq Vol(\Omega).$

Assume strict inequality holds. This occurs only when

$\lim\limits_{i\to\infty}\inf\int_{\Omega_{i}\cap\\{\varphi_{i}\leq-C\\}}e^{-\varphi_{i}}dz\wedge
d\overline{z}\geq\epsilon_{0},$

where $\epsilon_{0}>0$ is fixed and $C$ is arbitrarily large. Then we can find
a sequence $C_{i}\to\infty$ slowly such that

$E_{i}=\Omega_{i}\cap\\{\varphi_{i}\leq-C_{i}\\}$

has measure

$m(E_{i})=vol(E_{i})\geq\epsilon_{0},$

for all large $i$. By passing to a subsequence, we may assume $E_{i}$
converges to $E\subset\Omega$ in the weak sense. Then $m(E)\geq\epsilon_{0}$.
Note regular points on $\Omega$ has full Hausdorff measure [7]. Then there
exists a regular point $x\in\Omega$ and a sequence $c_{k}\to 0$ such that

$\frac{m(E\cap B(x,c_{k}))}{m(B(x,c_{k}))}\geq 1/2.$

Recall by [40, Theorem $1.5$], around $x$, we have a holomorphic chart
$(w^{k}_{1},..,w^{k}_{n})$ on $B(x,c_{k})$ which is a
$\Psi(\frac{1}{k})c_{k}$-Gromov-Hausdorff approximation to its image in
$\mathbb{C}^{n}$. We can pull back the holomorphic coordinate to $M_{i}$, say
$x_{i}\to x$, $w^{ki}_{s}\to w^{k}_{s}$ (fix $k$). Then

(2.1) $\frac{m(E_{i}\cap B(x_{i},c_{k}))}{m(B(x_{i},c_{k}))}\geq 1/3.$

By Cheeger-Colding [6], we have

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x_{i},c_{k})}|\langle
dw^{ki}_{s},d\overline{w}^{ki}_{t}\rangle-\delta_{st}|^{2}<\Psi(\frac{1}{k}).$

Let

$E^{\prime}_{i,k}=\\{y\in B(x_{i},c_{k})||dw^{ki}_{1}\wedge
dw^{ki}_{2}\wedge\cdot\cdot\cdot\wedge dw^{ki}_{n}(y)|>0.5\\}.$

So if $k$ is large enough (fixed), then for all large $i$,

(2.2) $m(E^{\prime}_{i,k})>0.9m(x_{i},c_{k}).$

Note the holomorphic chart $(w^{ki}_{1},...,w^{ki}_{n})$ has fixed size. Hence
by Cauchy estimate, on $E^{\prime}_{i,k}$,

$|dz_{1}\wedge dz_{2}\wedge\cdot\cdot\cdot\wedge dz_{n}|>\frac{1}{C(k)}.$

If $i$ is large enough, there is a contradiction between (2.1) and (2.2).

∎

Let $\alpha>0$ be arbitrary. Let $\Omega^{\alpha}$ be the homothetic action of
$\Omega$ (induced by $r\frac{\partial}{\partial r}$) by multiplying $\alpha$.
Let $F_{\alpha}$ be such homothetic action. Then by Lemma 1,

$vol(\Omega^{\alpha})=\int_{\Omega^{\alpha}}e^{-\varphi}dz\wedge
d\overline{z}=\alpha^{n}\int_{\Omega}e^{-\varphi}dz\wedge d\overline{z}.$

Therefore, by pulling back coordinate via the biholomorphism $F_{\alpha}$, we
obtain that

$\int_{\Omega}e^{-\varphi}dz\wedge d\overline{z}=\alpha^{\sum
d_{j}-n}\int_{\Omega}e^{-F_{\alpha}^{*}(\varphi)}dz\wedge d\overline{z}.$

That is to say,

$\int_{\Omega}(e^{-\varphi}-\alpha^{\sum
d_{j}-n}e^{-F_{\alpha}^{*}(\varphi)})dz\wedge d\overline{z}=0.$

Since $\alpha$ and $\Omega$ are arbitrary, we concludes the proof of Claim 1.
∎

######  Proposition 1.

$\psi=\log r^{2}$ is psh on $V$.

###### Proof.

By [25] and $(2.4)$ in [39], we can consider approximation of $r^{2}$ by
$\rho_{i}$ on $M_{i}$, where $dd^{c}\rho_{i}=\omega_{i}$ on $B(p_{i},10)$. By
Bochner formula,

$\Delta(|\nabla\rho_{i}|^{2}-4\rho_{i})\geq 0.$

Let $\hat{u}=\max(|\nabla\rho_{i}|^{2}-4\rho_{i},0)$. Then $\hat{u}$ is
subharmonic. Furthermore, by Cheeer-Colding [6],

$\int_{B(p_{i},9)}\hat{u}^{2}\leq\int_{B(p_{i},9)}(|\nabla\rho_{i}|^{2}-4\rho_{i})^{2}<\Psi(\frac{1}{i}).$

Then by mean value inequality, on $B(p_{i},5)$,

$|\nabla\rho_{i}|\leq 2r+\Psi(\frac{1}{i}).$

Thus

$dd^{c}\log\rho_{i}=\frac{\rho_{i}dd^{c}\rho_{i}-d\rho_{i}\wedge
d^{c}\rho_{i}}{\rho_{i}^{2}}\geq-\Psi(\frac{1}{i})\omega_{i}.$

The proposition follows by letting $i\to\infty$.

∎

Let $u$ be a smooth function on $V\backslash\\{o\\}$ which defines a smooth
Kähler cone metric $\omega_{0}$ ($\omega_{0}=dd^{c}u$) with reeb vector
$J(r\frac{\partial}{\partial r})=\sum
Re(\sqrt{-1}d_{j}z_{j}\frac{\partial}{\partial z_{j}})$. We also require that
$u$ is invariant under reeb. Such $u$ exists by [44]. We would like to
approximate the limit metric on $V$ by a family of smooth Kähler cone metrics
in the potential sense, without changing reeb. The argument is similar to [29,
Lemma $3.1$]. For reader’s convenience, we provide the details. Below
$\tau(k)$ represents a positive sequence converging to zero. The value might
change from line to line. Now consider an annulus in $V$ which is a complex
manifold, say $A=B(o,C)\backslash B(o,1/C)$, where $C$ is large so that $A$
contains the level set $\\{u=1\\}$. By [2] and Proposition 1, we can
approximate $\psi$ by $\psi_{k}$ on $A$, where $\psi_{k}$ is almost psh in the
sense that $dd^{c}\psi_{k}\geq-\tau(k)\omega_{0}$. Now by averaging torus
integration (use invariant measure) generated by the reeb, we find that

(2.3) $dd^{c}\psi_{k}\geq-\tau(k)\omega_{0}$

still holds. Let ′ be the derivative along $r\frac{\partial}{\partial r}$.

######  Claim 2.

$(\psi^{\prime}_{k})^{\prime}\geq-\tau(k)$.

###### Proof.

This follows by considering $dd^{c}\psi_{k}$ along the holomorphic direction
generated by reeb, using that $\psi_{k}$ is reeb invariant,
$\frac{\partial}{\partial logr}=r\frac{\partial}{\partial r}$ and
$\log|z|+i\theta$ as a local chart on the holomorphic curve. ∎

Recall $\psi_{k}\to\log r^{2}$ pointwise. By Claim 2 and an ode argument, we
find that

(2.4) $|\psi^{\prime}_{k}-2|<\tau(k).$

Define

$H_{k}=\psi_{k}-\log u.$

Then $H_{k}^{\prime}$ is almost 0. Let us define a function $h_{k}$ which is
constant along $r\frac{\partial}{\partial r}$ and the value equals $H_{k}$ on
the level set $\Sigma$ where $u=1$. Given an arbitrary $p\in\Sigma$, we can
find a local holomorphic chart $(w_{1},w_{2},...,w_{n})$ such that
$\frac{\partial}{\partial w_{1}}=r\frac{\partial}{\partial
r}-\sqrt{-1}Jr\frac{\partial}{\partial r}$. Hence $h_{k}$ is constant on each
$w_{1}$-slice. We may adjust the coordinate (without changing
$\frac{\partial}{\partial w_{1}}$) such that $\frac{\partial}{\partial w_{1}}$
is orthogonal to $\frac{\partial}{\partial w_{i}}(i\geq 2)$ at $p$, with
repect to the Kähler cone metric $\omega_{0}$. Let $e\in T_{p}^{1,0}V$,
$e=a+b$ with $|e|_{\omega_{0}}=1$, where $a\in$ span
$\\{\frac{\partial}{\partial w_{1}}\\}$, $b\in$ span
$\\{\frac{\partial}{\partial w_{2}},...,\frac{\partial}{\partial w_{n}}\\}$.
As $h_{k}$ is constant along $\frac{\partial}{\partial
w_{1}},\frac{\partial}{\partial\overline{w_{1}}}$,

$(h_{k})_{e\overline{e}}=(h_{k})_{b\overline{b}}=(h_{k})_{b\overline{b}}|_{\Sigma}.$

By using (2.3), (2.4) and $(\log u)^{\prime}=2$, we find

$\displaystyle(h_{k})_{b\overline{b}}|_{\Sigma}$
$\displaystyle=(\psi_{k})_{b\overline{b}}|_{\Sigma}-(\log
u)_{b\overline{b}}|_{\Sigma}$
$\displaystyle\geq(\psi_{k})_{b\overline{b}}-(\log u)_{b\overline{b}}-\tau(k)$
$\displaystyle\geq-(\log u)_{b\overline{b}}-\tau(k)$ $\displaystyle=-(\log
u)_{e\overline{e}}-\tau(k).$

In other words, on $\Sigma$, we have

$dd^{c}(h_{k}+\log u)\geq-\tau(k)\omega_{0}.$

Thus, on the local leaf of reeb foliation, one can find constants $s_{k}\to 1$
such that

$d_{B}d_{B}^{c}(s_{k}h_{k})+\omega^{T}>0,$

where $d_{B}$ is the exterior differential on local leaf space, $\omega^{T}$
is the metric on local leaf space induced by $\omega_{0}$. So
$u_{k}=\exp(s_{k}h_{k}+\log u)$ defines a smooth Kähler cone metric with the
same reeb as $\omega_{0}$. Also the potential converges to $r^{2}$.

Define

$\lambda=\sum d_{j}-n.$

Set $1>a>0$ as a constant. Let $\chi_{k}$ be a smooth nonnnegative cut-off
function on $\mathbb{R}^{+}$ such that $\chi_{k}=1$ on $[a^{2},1]$ and the
supported in $(a^{2}-1/k,1+1/k)$. Let $S_{i}$ be the scalar curvature on
$M_{i}$, $\omega=dd^{c}r^{2}$ on $(V,o)$. By Chern-Levine-Nirenberg [24, pp.
146],

$\displaystyle
r_{i}^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r_{i})}S$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p_{i},1)}S_{i}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p_{i},1)}dd^{c}\log|dz_{1}\wedge
dz_{2}\wedge\cdot\cdot\cdot\wedge dz_{n}|^{2}\wedge\omega_{i}^{n-1}$
$\displaystyle\to\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}dd^{c}\log|dz|^{2}\wedge\omega^{n-1}.$

Let $\tilde{\omega}_{k}$ be a sequence of smooth Kähler cone metrics over
$(V,o)$ (same reeb) so that the potential $\rho_{k}$ is converging to $r^{2}$.
Let us evaluate the integral over annulus
$A(o,1,a)=B(o,1)\backslash\overline{B(o,a)}$.

$\displaystyle\frac{1}{vol(B_{V}(o,1))}\int_{A(o,1,a)}dd^{c}\log|dz|^{2}\wedge\omega^{n-1}$
$\displaystyle=\lim\limits_{k\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}\chi_{k}(\rho_{k})dd^{c}\log|dz|^{2}\wedge\tilde{\omega}_{k}^{n-1}$
$\displaystyle=\lim\limits_{k\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}\log|dz|^{2}dd^{c}\chi_{k}(\rho_{k})\wedge\tilde{\omega}_{k}^{n-1}$
$\displaystyle=\lim\limits_{k\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}(\log|dz|^{2}-\lambda\log\rho_{k})dd^{c}\chi_{k}(\rho_{k})\wedge\tilde{\omega}_{k}^{n-1}$
$\displaystyle+\lim\limits_{k\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}\lambda\log\rho_{k}dd^{c}\chi_{k}(\rho_{k})\wedge\tilde{\omega}_{k}^{n-1}.$

As $\tilde{\omega}_{k}$ is smooth, by direct computation, we find

$dd^{c}\chi_{k}(\rho_{k})\wedge\tilde{\omega}_{k}^{n-1}=F_{k}(\rho_{k})\tilde{\omega}_{k}^{n}=G_{k}(\rho_{k})ds_{k}dr_{k},$

where in the last step, we used polar coordinate. Then

$\displaystyle\lim\limits_{k\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}(\log|dz|^{2}-\lambda\log\rho_{k})dd^{c}\chi_{k}(\rho_{k})\wedge\tilde{\omega}_{k}^{n-1}$
$\displaystyle=\lim\limits_{k\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}(\log|dz|^{2}-\lambda\log\rho_{k})G_{k}(\rho_{k})dr_{k}ds_{k}$
$\displaystyle=0,$

where we used that $\log|dz|^{2}-\lambda\log\rho_{i}$ is constant along
$r\frac{\partial}{\partial r}$ (ensured by Claim 1) and the Fubini theorem. As
$\tilde{\omega}_{k}$ is a smooth Kähler cone metric, by direct computation, we
find that

$\lim\limits_{k\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}\lambda\log\rho_{k}dd^{c}\chi_{k}(\rho_{k})\wedge\tilde{\omega}_{k}^{n-1}=\lambda
C(n,a),$

where $C(n,a)$ is an explicit number. Then

$\displaystyle\int_{B(o,1)}dd^{c}\log|dz|^{2}\wedge\omega^{n-1}$
$\displaystyle=\int_{A(o,1,a)}dd^{c}\log|dz|^{2}\wedge\omega^{n-1}+\int_{B(o,a)}dd^{c}\log|dz|^{2}\wedge\omega^{n-1}$
$\displaystyle=C(n,a)\lambda
vol(B(o,1))+\int_{B(o,a)}dd^{c}\log|dz|^{2}\wedge\omega^{n-1}.$

By the assumption

$\lim\limits_{r\to\infty}\sup
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S<\infty,$

we have

$\int_{B(o,a)}dd^{c}\log|dz|^{2}\wedge\omega^{n-1}\leq Ca^{2n-2}.$

As $n\geq 2$, by letting $a\to 0$, we find that

(2.5)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(o,1)}dd^{c}\log|dz|^{2}\wedge\omega^{n-1}=C(n)\lambda=C(n)(\sum
d_{j}-n),$

where $C(n)$ is a positive constant depending only on $n$.

######  Claim 3.

$d_{1},..,d_{n}$ (counting multiplicity) are independent of tangent cones. In
other words, they depend only on $M$.

###### Proof.

Assume $a_{1}<a_{2}<\cdot\cdot\cdot<a_{k}$, where $a_{j}$ are the distinct
values of $d_{1},d_{2},...,d_{n}$. Let $b_{1},...,b_{k}$ be the multiplicities
of $a_{1},...,a_{k}$. By [37, Proposition $4.1$], for all $d>0$,

$dim(\mathcal{O}_{d}(M))=dim(\mathcal{O}_{d}(V)).$

That is, as a function of $d$, $dim(\mathcal{O}_{d}(V))$ is independent of
$V$. Recall that $V$ is isomorphic to $\mathbb{C}^{n}$. We claim that
$a_{j},b_{j}$ are determined by $dim(\mathcal{O}_{d}(V))$. Observe that

$a_{1}=\min\\{d|dim(\mathcal{O}_{d}(V))>1\\},b_{1}=dim(\mathcal{O}_{a_{1}}(V))-1.$

Let $\\{1,h^{1}_{1},..,h^{1}_{b_{1}}\\}$ be a basis for
$\mathcal{O}_{a_{1}}(V)$. Let

$\mathcal{O}^{1}_{d}(V)=\mathbb{C}[h^{1}_{1},...,h^{1}_{b_{1}}]\cap\mathcal{O}_{d}(M).$

Then $dim(\mathcal{O}^{1}_{d}(V))$ is the number of solutions to
$\sum\limits_{t=1}^{b_{1}}a_{1}x_{1t}\leq d$, where $x_{1t}$ are nonnegative
integers. Hence $a_{1},b_{1},dim(\mathcal{O}^{1}_{d}(V))$ are independent of
$V$. Inductively, for $j\geq 2$, assume
$a_{j-1},b_{j-1},dim(\mathcal{O}^{j-1}_{d}(V))$ are independent of $V$. Then
we have

$a_{j}=\min\\{d|dim(\mathcal{O}_{d}(V))>dim(\mathcal{O}^{j-1}_{d}(V))\\},b_{j}=dim(\mathcal{O}_{a_{j}}(V))-dim(\mathcal{O}^{j-1}_{a_{j}}(V)).$

Let $\\{1,h^{j}_{1},..,h^{j}_{s_{j}}\\}$ be a basis for
$\mathcal{O}_{a_{j}}(V)$. Define

$\mathcal{O}^{j}_{d}(V)=\mathbb{C}[h^{j}_{1},...,h^{j}_{s_{j}}]\cap\mathcal{O}_{d}(M).$

Then $dim(\mathcal{O}^{j}_{d}(V))$ is the number of solutions to
$\sum\limits_{s=1}^{j}a_{s}(\sum\limits_{t=1}^{b_{s}}x_{st})\leq d$, where
$x_{st}$ are nonnegative integers. This depends only on
$a_{1},b_{1},...,a_{j},b_{j},d$. Thus,
$a_{j},b_{j},dim(\mathcal{O}^{j}_{d}(V))$ are independent of $V$. This
completes the proof.

∎

Putting (2.5) and Claim 3 together, we obtain the proof of Theorem 2, if the
universal cover of $M$ does not split. If the universal cover of $M$ splits,
by $\lim\limits_{r\to\infty}\sup
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S<\infty$
and [50, Theorem $1.2$], we can find a psh function $u$ of logarithmic growth
such that $dd^{c}u=Ric$. We can lift $u$ to the universal cover $\tilde{M}$,
thus

$dd^{c}u=Ric$

on $\tilde{M}.$ Assume $\tilde{M}$ is biholomorphic and isometric to
$M_{1}\times M_{2}\times\cdot\cdot\cdot\times M_{k}\times\mathbb{C}^{l}$,
where each $M_{j}$ is nonflat and non-splitting. Pick $q_{j}\in M_{j}$. Then
$v_{j}=u|_{(q_{1},q_{2},...,q_{j-1},M_{j},q_{j+1},...,q_{k})}$ satisfies
$dd^{c}v_{j}=Ric(M_{j})$. As $v_{j}$ is of logarithmic growth, by [36],
$M_{j}$ is of Euclidean volume growth. Let $N=M_{1}\times
M_{2}\times\cdot\cdot\cdot\times M_{k}$.

######  Claim 4.

Let $G$ be the holomorphic isometry group of $N$. Then $G$ is compact.
Moreover, there exists a common fixed point of $G$.

###### Proof.

Pick $q^{\prime}\in N$. If $G$ is not compact, there exists $g_{i}\in G$ such
that $r_{i}=d(g_{i}(q^{\prime}),q^{\prime})\to\infty$. By passing to a
subsequence, $(N_{i},q^{\prime}_{i})=(N,q^{\prime},\frac{r}{r_{i}})$ converges
in the pointed Gromov-Hausdorff sense to a tangent cone. By Cheeger-Colding-
Tian [8], such tangent cone splits off $\mathbb{C}$. By three circles lemma
[34], we find a linear growth holomorphic function on $N$. Then by [49,
Theorem $0.3$], $N$ itself splits isometrically with a $\mathbb{C}$ factor.
This contradicts the assumption that $N$ has no flat factor. Let
$a_{1}<a_{2}<\cdot\cdot\cdot<a_{k}$, $b_{1},...,b_{k}$ be defined as in Claim
3 for $N$. According to the bottom of page $55$ in [37], we can find
polynomial growth holomorphic functions $f_{ij}$ ($1\leq i\leq k,1\leq j\leq
b_{i}$) so that $f_{ij}$ has degree $a_{i}$ (i.e.,
$a_{i}=\inf\\{d|f_{ij}\in\mathcal{O}_{d}(N)\\}$) and that $\\{f_{ij}\\}$
serves as a biholomorphism from $N$ to $\mathbb{C}^{m}$ ($m=dim(N))$. For any
$g\in G$, as $g$ preserves the degree of polynomial growth holomorphic
functions, for $1\leq s\leq b_{1}$,

$g(f_{1s})=\sum\limits_{1\leq t\leq b_{1}}a_{st}f_{1t}+C_{s}.$

We may regard $x_{s}=f_{1s}$ as variables. Set
$X=(x_{1},...,x_{b_{1}})^{t}\in\mathbb{C}^{b_{1}}$. Then there exists a matrix
$A_{g}$ and a vector $Y_{g}$ such that $gX=A_{g}X+Y_{g}$. In other words, $G$
acts as affine transformations on $\mathbb{C}^{b_{1}}$. As $G$ is compact, $G$
admits a Haar measure $dg$ with total volume $1$. set $X_{0}=\int_{G}g(X)dg$.
Then $g(X_{0})=X_{0}$ for any $g\in G$. By shifting the coordinates of
$\mathbb{C}^{b_{1}}$ (we shift $f_{1s}$ accordingly), we may assume $X_{0}=0$.
Therefore, the common zeros of $f_{1s}$, say $N_{1}$, is invariant under $G$
action. Recall $f_{1s}$ forms a part of global coordinate on $N$ isomorphic to
$\mathbb{C}^{m}$, then $N_{1}$ is identified with $\mathbb{C}^{m_{1}}$, where
$m_{1}=m-b_{1}$. Then we can consider the action of $G$ on polynomial growth
holomorphic functions with degree $a_{2}$, restricted to $N_{1}$. Note
$f_{1s}$ all vanish on $N_{1}$. As before, we obtain affine action $G$ acting
on $\mathbb{C}^{b_{2}}$. Then we can find a fixed point of $G$ on
$\mathbb{C}^{b_{2}}$, which defines $N_{2}\subset N_{1}$. By repetition, we
obtain a fixed point of $G$ on $N$.

∎

Let $a\in N$ be a fixed point of $G$ ensured by Claim 4. Let $q=(a,0)\in
N\times\mathbb{C}^{l}$. Let $G^{\prime}=\pi_{1}(M)$. Then $G^{\prime}$ acts as
deck transformations on $N\times\mathbb{C}^{l}$. Define

$\Sigma=(a,\mathbb{C}^{l})\subset N\times\mathbb{C}^{l}.$

By projection, $G^{\prime}$ also acts on $N$ holomorphically and
isometrically. As $a$ is a fixed point, $G^{\prime}(\Sigma)=\Sigma$. Hence,
the action of $G^{\prime}$ on $\Sigma$ is free and discrete. Then
$M^{\prime}=\Sigma/G^{\prime}$ is a flat manifold. Let

$L=\\{x\in\Sigma|d(x,q)<d(x,g(q)),\forall g\in G^{\prime},g\neq Id\\}$

be the fundamental domain of $G^{\prime}$ on $\Sigma$. Let $q_{0}\in M$ be the
image of $q$ under the covering map $N\times\mathbb{C}^{l}\to M$. For large
$r$, consider $B(q_{0},r)\subset M$. Up to a zero measure set, $B(q_{0},r)$ is
identified with

$\Omega_{r}=\\{(y,x)\in B(q,r)\subset
N\times\mathbb{C}^{l}|d((y,x),q)<d((y,x),g(q)),\forall g\in G^{\prime},g\neq
Id\\}.$

Because $g(a)=a$ for all $g$,

$\Omega_{r}=\\{(y,x)|x\in L,d(y,a)<\sqrt{r^{2}-|x|^{2}}\\},$

where $|x|$ is the Euclidean distance to $0$ in $\mathbb{C}^{l}$. Let

$L_{r}=L\cap\\{x||x|<r\\}.$

When $r$ is fixed, for each $x\in L$, let

$\Sigma_{x}=\Omega_{r}\cap(N\times\\{x\\}).$

Then $\Sigma_{x}$ can be identified with $B(a,\sqrt{r^{2}-|x|^{2}})\subset N$.
For any $\epsilon>0$,

(2.6) $\displaystyle\frac{r^{2}}{vol(B(q_{0},r))}\int_{B(q_{0},r)}S$
$\displaystyle=\frac{r^{2}}{vol(B(q_{0},r))}\int_{L_{r}}\int_{\Sigma_{x}}Sd\Sigma_{x}dx$
$\displaystyle=\frac{r^{2}}{vol(B(q_{0},r))}\int_{L_{r}}(\int_{B(a,\sqrt{r^{2}-|x|^{2}})}S(y)dy)dx$
$\displaystyle=\frac{r^{2}}{vol(B(q_{0},r))}\int_{L_{r}\backslash
L_{(1-\epsilon)r}}(\int_{B(a,\sqrt{r^{2}-|x|^{2}})}S(y)dy)dx$
$\displaystyle+\frac{r^{2}}{vol(B(q_{0},r))}\int_{L_{(1-\epsilon)r}}(\int_{B(a,\sqrt{r^{2}-|x|^{2}})}S(y)dy)dx.$

Since $L$ is convex,

$\frac{vol(L_{(1-\epsilon)r})}{vol(L_{r})}\geq(1-\epsilon)^{l}.$

Then from the existence of
$A=\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(a,r)}S$,
we find

$\frac{r^{2}}{vol(B(q_{0},r))}\int_{L_{r}\backslash
L_{(1-\epsilon)r}}(\int_{B(a,\sqrt{r^{2}-|x|^{2}})}S(y)dy)dx\leq\frac{r^{2}\int_{B(a,r)}S(y)dy}{vol(B(q_{0},r))}\int_{L_{r}\backslash
L_{(1-\epsilon)r}}dx<\Psi(\epsilon).$

In order to estimate

$\frac{r^{2}}{vol(B(q_{0},r))}\int_{L_{(1-\epsilon)r}}(\int_{B(a,\sqrt{r^{2}-|x|^{2}})}S(y)dy)dx,$

we can replace the integrand $\int_{B(a,\sqrt{r^{2}-|x|^{2}})}S(y)dy$ by
$A(r^{2}-|x|^{2})^{-1}vol(B(a,\sqrt{r^{2}-|x|^{2}}))$, up to
$\Psi(\frac{1}{r\epsilon})$ error. This is approximately
$vA(r^{2}-|x|^{2})^{m-1}$, where $v$ is the asymptotic volume ratio for $N$.
Therefore, in order to show that the limit of (2.6) exists, it suffices to
show that for $t<1$,

$\lim\limits_{r\to\infty}\frac{vol(L_{tr})}{vol(L_{r})}$

exists. Recall $M^{\prime}=\Sigma/G^{\prime}$ is a flat manifold. $L_{r}$ is
just $B_{M^{\prime}}(0,r)$, up to zero measure set. Then we are left to show
that
$\lim\limits_{r\to\infty}\frac{vol(B_{M^{\prime}}(0,tr))}{vol(B_{M^{\prime}}(0,r))})$
exists. By a theorem of Cheeger-Gromoll [10], such $M^{\prime}$ has a soul
$K$, which is a compact flat manifold of fundamental group $G^{\prime}$. The
normal bundle to the soul is a flat orthogonal vector bundle over $K$, which
is given by a representation $G^{\prime}\rightarrow O(s)$. The normal
exponential map is an isometric diffeomorphism. In particular, up to a
codimension $1$ set, $M^{\prime}$ is isometric to $K\times\mathbb{R}^{s}$.
Therefore,

$\lim\limits_{r\to\infty}\frac{vol(B_{M^{\prime}}(0,tr))}{vol(B_{M^{\prime}}(0,r))}=t^{s}.$

Therefore, we can find $C(m,s)$ depending only on $m$ and $s$ so that

$\lim\limits_{r\to\infty}\frac{r^{2}}{vol(B(q_{0},r))}\int_{B(q_{0},r)}S=C(m,s)A.$

By considering a special case, e.g., $M=N^{m}\times\mathbb{C}^{s}$, we find

$C(m,s)=\frac{2(m+s)-1}{2m-1}.$

This completes the proof of Theorem 2.

Through the proof, we also obtained the following:

######  Corollary 9.

Let $(M^{n},p)$ be a complete noncompact Kähler manifold with nonnegative
bisectional curvature. If
$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S$
is finite, then $\pi_{1}(M)$ is isomorphic to $\pi_{1}(M^{\prime})$, where
$M^{\prime}$ is a complete flat Kähler manifold of dimension at most $n$.

Proof of Theorem 3:

Let $d_{1},d_{2},...,d_{n}$ be the degrees appeared in the proof of Theorem 2.
Consider a special Kähler manifold $M^{\prime}$ given by $S_{1}\times
S_{2}\times\cdot\cdot\cdot\times S_{n}$, where each $S_{j}$ is a complete
surface with nonnegative curvature and quadratic volume growth, isomorphic to
$\mathbb{C}$. We also require that the degree of $z$ on $S_{j}$ is $d_{j}$.
The tangent cone of $M^{\prime}$ and the tangent cone of $M$ have the same
reeb vector field, up to an isomorphism. The volume of unit ball on tangent
cone of $M^{\prime}$ is computed as $\frac{\omega_{2n}}{d_{1}\cdot\cdot\cdot
d_{n}}$. Such metric is singular. However, as we see in the proof of Theorem
2, it can be approximated by smooth Kähler cone metrics with the same reeb
vector field. By Chern-Levine-Nirenberg and the convergence of Kähler
potentials, we find that the volume of unit ball of the smooth cone metric
converges to $\frac{\omega_{2n}}{d_{1}\cdot\cdot\cdot d_{n}}$. By [44, Section
$3.1$], such volume depends only on the reeb vector field. So volume of unit
ball for smooth Kähler cone metric is $\frac{\omega_{2n}}{d_{1}\cdot\cdot\cdot
d_{n}}$. We also approximate a tangent cone of $M$ by smooth Kähler cone
metrics. Then the volume of unit ball in tangent cone of $M$ is also
$\frac{\omega_{2n}}{d_{1}\cdot\cdot\cdot d_{n}}$.

Inequalities $(1)$, $(2)$ follow by the fact that $d_{i}\geq 1$ and the
arithmetic-geometric inequality. Equality $(3)$ follows from straightforward
dimension counting, using the fact that it corresponds to the number of
solutions to $\sum d_{j}a_{j}\leq d$, where $a_{j}\geq 0$ are nonnegative
integers. The rigidity part also follows. In one extremal case,
$d_{1}=d_{2}=\cdot\cdot\cdot=d_{n-1}=1$. By [49, Theorem $0.3$], we find that
$M$ splits off $\mathbb{C}^{n-1}$. In another case,
$d_{1}=d_{2}=\cdot\cdot\cdot=d_{n}$. This is the case when $M$ is unitary
symmetric [59]. However, the metric is not necessarily unique: if the
bisectional curvature is strictly positive, then we can do local perturbation,
which does not affect the holomorphic spectrum. This completes the proof of
Theorem 3.

Proof of Corollary 1:

By the last part in the proof of Theorem 2, we find that the universal cover
$\tilde{M}$ has Euclidean volume growth. Assume

$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S<\delta<<1,$

Then

$\lim\limits_{r\to\infty}r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(\tilde{p},r)}S<C(n)\delta<<1.$

Then by Theorem 3,

$\lim\limits_{r\to\infty}\frac{vol(B(\tilde{p},r))}{\omega_{2n}r^{2n}}\geq
1-\Psi(\delta).$

The opposite direction holds similarly. This completes the proof of Corollary
1.

Proof of Theorem 4: Recall by [40, Proposition $2.7$], the manifold is
biholomorphic to $\mathbb{C}^{n}$. By adapting the argument in [43,
Proposition $6.1$] and [40, Theorem $1.5$], we can show that the tangent cone
is also biholomorphic to $\mathbb{C}^{n}$. We can still apply the argument as
in the proof of Theorem 2. The difference is that, we do not have three
circles lemma. So apriori, $d_{1},...,d_{n}$ could vary for different tangent
cones. However, this does not affect the argument for a single tangent cone.
If $\lim\limits_{r\to\infty}\inf
r^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r)}S=0$,
then for some tangent cone, the coordinate functions are of linear growth and
the volume of unit ball is the same as the Euclidean case. Thus by Colding’s
volume continuity [14], $M$ is flat. This completes the proof of Theorem 4.

##  3\. The case of nonnegative Ricci curvature

Proof of Theorem 5:

Let us first assume $M$ satisfies condition $B$.

######  Lemma 2.

Let $M^{n}$ satisfy condition $B$. Then there exists $C_{1}>0$ such that for
all $1\leq k\leq n$, for all $r>0$,

$r^{2k-2n}\int_{B(p,r)}Ric^{k}\wedge\omega^{n-k}<C_{1}.$

Furthermore, if we consider a tangent cone $(V,o)$ and a sequence of rescaled
manifolds $(M_{i},p_{i})=(M,p,\frac{r}{r_{i}})\to(V,o)$ in the pointed Gromov-
Hausdorff sense, then for $k<n$,

$\lim\limits_{i\to\infty}r_{i}^{2k-2n}\int_{B(p,r_{i})}Ric^{k}\wedge\omega^{n-k}=\int_{B_{V}(o,1)}Ric_{V}^{k}\wedge\omega_{V}^{n-k},$

where $Ric_{V}$ is understood in the current sense.

###### Proof.

Assume by contradiction that there exists a sequence $r_{i}\to\infty$ such
that

(3.1) $r_{i}^{2k-2n}\int_{B(p,r_{i})}Ric^{k}\wedge\omega^{n-k}\to\infty.$

Set $(M_{i},p_{i})=(M,p,\frac{r}{r_{i}})$. By passing to a subsequence, we may
assume $(M_{i},p_{i})\to(V,o)$. According to [25] and page $1161$ of [55],
there exists $u_{i}$ on $B(p_{i},10)$ such that

(3.2)
$dd^{c}u_{i}=\omega_{i},|u_{i}-d^{2}(p_{i},\cdot)|<\Psi(\frac{1}{i}),|u_{i}|+|\nabla
u_{i}|\leq C.$

We can find holomorphic functions $f_{1},...,f_{n}$ on $V$ so that
$df_{1}\wedge\cdot\cdot\cdot\wedge df_{n}$ is not identically zero. Note
$df_{1}\wedge\cdot\cdot\cdot\wedge df_{n}$ could have divisor. By solving
$\overline{\partial}$ equation, $f_{j}$ can be lifted from tangent cone to
rescaled manifolds $B(p_{i},10)$, say $f_{ij}\to f_{j}$. Let

$s_{i}=df_{i1}\wedge\cdot\cdot\cdot\wedge df_{in}.$

Let $\chi$ be a nonnegative smooth function on $\mathbb{R}^{+}$ so that
$\chi=1$ on $[0,1]$, with compact support in $[0,1.5)$. By Poincaré-Lelong, we
have

$\displaystyle 2\pi\int\chi(u_{i})Ric_{i}^{k}\wedge\omega_{i}^{n-k}$
$\displaystyle\leq\int\chi(u_{i})dd^{c}\log|s_{i}|^{2}\wedge
Ric_{i}^{k-1}\wedge\omega_{i}^{n-k}$
$\displaystyle=\int\log|s_{i}|^{2}dd^{c}\chi(u_{i})\wedge
Ric_{i}^{k-1}\wedge\omega_{i}^{n-k}$ $\displaystyle\leq
C\int_{B(p_{i},2)\backslash B(p_{i},1)}|\log|s_{i}|^{2}|\omega_{i}^{n},$

using $Ric\leq C\omega$ on $B(p_{i},2)\backslash B(p_{i},1)$. Then it suffices
to find uniform bound of $\int\log|s_{i}|^{2}\omega_{i}^{n}$ on an annulus.
Absolute value is not needed, as $\log|s_{i}|^{2}$ is uniformly bounded from
above. Without loss of generality, we may assume $\log|s_{i}|^{2}\leq 0$.
Since there may be singularity on cross section of $V$, we consider
projection. According to the local parametrization theorem in [24] (Theorem
$4.19$ on page $95$), for any point $q\in\partial B(o,1)$, we can locally
properly project $B(q,\delta)$ to a domain in $\mathbb{C}^{n}$. Say the
projection is given by $P$. Then under the Gromov-Hausdorff convergence, we
can also project $B(q_{i},\delta)$, where $q_{i}\to q$, say the projection is
$P_{i}$ and $P_{i}\to P$. Recall $\omega_{i}=dd^{c}u_{i}$. Assume
$P_{i}(q_{i})=0,P(q)=0$ and there exists $\delta_{1}>0$ with
$P_{i}^{-1}(B(0,\delta_{1}))\subset\subset B(q_{i},0.5\delta)$. For any $y\in
B(0,\delta_{1})$, define

$h_{i}(y)=\sum\limits_{x\in P_{i}^{-1}(y)}\log|s_{i}(x)|^{2}\leq
0,v_{i}(y)=\sum\limits_{x\in P_{i}^{-1}(y)}u_{i}(x).$

Then $h_{i},v_{i}$ are psh and $v_{i}$ is uniformly bounded. We may assume
$P_{i}(B(q_{i},\delta_{2}))\subset B(0,0.5\delta_{1})$. Then

$\int_{B(q_{i},\delta_{2})}-\log|s_{i}|^{2}\omega_{i}^{n}\leq-\int_{B(0,0.5\delta_{1})}h_{i}(dd^{c}v_{i})^{n}\leq
C.$

In the last step, we used Proposition $3.11$ on page $158$ of [24] and that
$h_{i}$ has a convergent subsequence as psh function, hence the $L^{1}$ norm
(w.r.t Euclidean volume) is uniformly bounded. Then by a standard covering
argument, we find a contradiction to (3.1).

Now we come to the convergence of integral for $k<n$. As we proved that

$\int_{B(p,r)}Ric^{k}\wedge\omega^{n-k}\leq Cr^{2n-2k},$

it sufficies to prove the convergence on an annulus (the integral on the small
ball is controlled by $Cr^{2n-2k}$). Away from singular set, as we have
$C^{1,\alpha}$ convergence [1], the potentials of $Ric_{i}$ and $\omega_{i}$
converge. Then the proof follows from Chern-Levine-Nirenberg. Near the
singular set, as $Ric\leq C\omega$, then the integral is bounded by a constant
times the volume of singular set. So the contribution is negligible. This
completes the proof of Lemma 2. ∎

Consider one tangent cone $(W,o^{\prime})$ which is Ricci flat. Let us
consider a sequence $r_{i}\to\infty$, such that
$(M_{i},p_{i})=(M,p,\frac{r}{r_{i}})$ converges in the pointed-Gromov-
Hausdorff sense to $(W,o^{\prime})$. Let $s$ be the parallel pluri-
anticanonical form on $W$ (dual of the pluri-canonical form), say $s\in
H^{0}(W_{reg},K^{-m})$. Let $\delta_{0}>0$ be a small number. Let
$\Sigma^{\prime}$ be the metric singular set of $W$. We can lift
$\Sigma^{\prime}$ to $B(p_{i},9)$ by Gromov-Hausdorff approximation. Say the
corresponding set is $\Sigma_{i}^{\prime}$. By pulling $s$ back to
$B(p_{i},9)$, we obtain smooth sections $s^{\prime\prime}_{i}\to s$ uniformly
on each compact set of $B(p_{i},9)\backslash\Sigma_{i}^{\prime}$. Set
$S_{i}=s_{i}^{\prime\prime}\eta_{i1}\eta_{i2}$, where $\eta_{i1}$ vanishes on
$B(p_{i},2\delta_{0})$ and equals to $1$ outside $B(p_{i},3\delta_{0})$,
$\eta_{i2}$ vanishes on $B(\Sigma_{i}^{\prime},\delta_{0})$, and equals to $1$
outside $B(\Sigma_{i}^{\prime},2\delta_{0})$. Let $u_{i}$ be defined as in
(3.2). Let $\Omega_{i}$ be the connected component of $u_{i}^{-1}([0,5))$
containing $B(p_{i},1)$. Then

$B(p_{i},3)\supseteq\Omega_{i}\supseteq B(p_{i},2).$

Since $n\geq 2$ and $\Sigma^{\prime}$ has real codimension at least $4$ by
Cheeger-Naber [16], we obtain $S_{i}$ satisfies

$\int_{\Omega_{i}}|\overline{\partial}S_{i}|^{2}e^{-2u_{i}}\leq\Psi(\delta_{0},\frac{1}{i}).$

Note here we crucially used that $s$ is parallel, hence the norm is constant.
Let $(g^{\prime})^{i}_{m}$ be the metric on $K_{M_{i}}^{-m}$ induced by the
Kähler metric. As $Ric\geq 0$, $(g^{\prime})^{i}_{m}$ has nonnegative
curvature. Let the new metric be $g^{i}_{m}=e^{-u_{i}}(g^{\prime})^{i}_{m}$.
We solve $\overline{\partial}\hat{s}_{i}=\overline{\partial}S_{i}$, by using
$(g^{\prime})^{i}_{m}$. Hörmander $L^{2}$ estimate of $\overline{\partial}$
gives that

$\int_{\Omega_{i}}|\hat{s}_{i}|^{2}e^{-2u_{i}}\leq
10\int_{\Omega_{i}}|\overline{\partial}S_{i}|^{2}e^{-2u_{i}}\leq\Psi(\delta_{0},\frac{1}{i}).$

Let $s_{i}=S_{i}-\hat{s}_{i}$. Then $s_{i}\in H^{0}(\Omega_{i},K^{-m})$ and

(3.3) $\int_{\Omega_{i}}|s_{i}|^{2}<C.$

Recall the scalar curvature has quadratic decay. Hence, on
$B(p_{i},2)\backslash B(p_{i},1)$, we have:

$\frac{1}{2\pi}dd^{c}\log|s_{i}|^{2}\geq-mRic_{i}-\omega_{i}\geq-C\omega_{i}.$

Therefore

$\Delta\log(e^{Cu_{i}}|s_{i}|^{2})\geq 0.$

This implies that

$\Delta(e^{Cu_{i}}|s_{i}|)\geq 0).$

Then (3.3) and mean inequality imply the pointwise upper bound for $|s_{i}|$
on $B(p_{i},2)\backslash B(p_{i},1)$. Let $\delta_{0}\to 0$ slowly as
$i\to\infty$, we obtain that $s_{i}$ converges uniformly on each compact set
of the regular part to $s$. We claim $s_{i}$ has no zero point on
$B(p_{i},2)\backslash B(p_{i},1)$, for large $i$. Assume not. It is clear the
zero set is a divisor. One can apply the Poincaré-Lelong equation and
projection as in Lemma 2, to conclude that the projection of the divisor to
$\mathbb{C}^{n}$ has uniform volume (w.r.t Euclidean measure) upper bound.
Hence, there exists a subsequence so that the divisor converges to a divisor
(as current). However, the projection of the singular set has codimension $4$,
this means there exists a point on the limit divisor away from the metric
singular set. Hence for large $i$, there exists a point on divisor and
uniformly away from singular set. This contradicts that $s_{i}\to s$ on
regular part. Once such section has no zero point, we can find the dual of
$s_{i}$ in pluri-canonical bundle, say $\mu_{i}$, which is holomorphic. Then
by multiplying the metric by $e^{Cu_{i}}$, we may assume $\log|\mu_{i}|^{2}$
is psh. We claim that $\log|\mu_{i}|^{2}$ is uniformly bounded from above.
Assume not, there exists a sequence of points $B(p_{i},2)\backslash
B(p_{i},1)\ni q_{i}\to q$ so that $\log|\mu_{i}(q_{i})|^{2}\to\infty$. Again
we can apply local parametrization theorem to project via $P_{i}$ to a ball on
$\mathbb{C}^{n}$, and define $\nu_{i}(y)=\sum\limits_{x\in
P_{i}^{-1}(y)}\log|\mu_{i}(x)|^{2}$. Since $|s_{i}|^{2}$ has uniform upper
bound, $\log|\mu_{i}|^{2}$ has uniform lower bound. Hence
$\nu_{i}(P_{i}(q_{i}))\to\infty$. Note the metric singular set
$\Sigma^{\prime}$ is a proper analytic set. Now pick a generic complex line
$l$ through $P(q)$ ($P$ is the limit of $P_{i}$), then locally $l$ intersects
$P(\Sigma^{\prime})$ at only finitely many points. In particular, a small
(fixed) circle on $l$ around $P(q)$ has no intersection with
$P(\Sigma^{\prime})$. Now we can shift the complex line $l$ slightly so that
it passes through $P_{i}(q_{i})$. By maximum principle for psh functions, the
supremum of $\nu_{i}$ on that circle is at least $\nu_{i}(P_{i}(q_{i}))$. But
that small circle has fixed distance to $P(\Sigma^{\prime})$. This is a
contradiction. Thus there exists a constant $C$ (independent of $i$) such that
on $B(p_{i},2)\backslash B(p_{i},1)$,

$|\Delta\log|s_{i}|^{2}|\leq C,|\log|s_{i}|^{2}|<C.$

We claim that $|\nabla\log|s_{i}|^{2}|$ is uniformly bounded. Let $G_{R}(x,y)$
be Green’s function with boundary value $0$ on a geodesic ball in
$B(p_{i},2)\backslash B(p_{i},1)$. Then

$\log|s_{i}(x)|^{2}+\int G_{R}(x,y)\Delta\log|s_{i}(y)|^{2}dy$

is harmonic and bounded in the interior. Hence the gradient is bounded. We can
take gradient for each term. By the estimate for $|\nabla G_{R}(x,y)|$, we
find that $|\nabla\log|s_{i}(x)|^{2}|$ is uniformly bounded. Arzela-Ascoli
theorem implies $|s_{i}|$ on $B(p_{i},2)\backslash B(p_{i},1)$ converges
uniformly to constant $s$ on $B(o^{\prime},2)\backslash B(o^{\prime},1)$.
Since $\omega$ is $dd^{c}$-exact, $M$ has no compact subvariety of positive
dimension. Hence, $|s_{i}|^{2}$ is nonvanishing anywhere. Then

(3.4) $Ric_{M_{i}}=dd^{c}\log|s_{i}|^{-\frac{2}{m}}.$

Certainly such argument works, if $B(p_{i},10)$ is Gromov-Hausdorff close to
$B(o,10)$. Therefore, if a tangent cone $V$ sufficiently close to $W$, we can
consider a sequence $r^{\prime}_{i}\to\infty$ so that
$(M^{\prime}_{i},p^{\prime}_{i})=(M,\frac{r}{r^{\prime}_{i}},p)$ converges in
the pointed-Gromov-Hausdorff sense to $(V,o)$. We may assume that
$r^{\prime}_{i}\geq 10r_{i}$ for all $i$. Let
$\Omega^{\prime}_{i},u^{\prime}_{i}$ be defined similarly as
$\Omega_{i},u_{i}$. Then we find $s^{\prime}_{i}$ converging to $s_{V}\in
H^{0}(V_{reg},K^{-m})$. Also $s_{V}$ is nowhere vanishing. As
$r^{\prime}_{i}\geq 10r_{i}$, by naturally identifying $M^{\prime}_{i}$ and
$M_{i}$ with $M$, we find $\overline{\Omega_{i}}\subset B(p,r^{\prime}_{i})$.
Then we have

$\frac{1}{2\pi}dd^{c}\log|s^{\prime}_{i}|^{-\frac{2}{m}}=Ric_{M^{\prime}_{i}}=Ric_{M}.$

Therefore, on $B(p,r_{i})$,

(3.5) $dd^{c}\log|s^{\prime}_{i}|^{2}=dd^{c}\log|s_{i}|^{2}.$

Then by Lemma 2 (take $k=n$), we find that

(3.6)
$\int_{B(p^{\prime}_{i},2)}(dd^{c}\log|s^{\prime}_{i}|^{-\frac{2}{m}})^{n}\to\int_{M}Ric^{n},\int_{B(p_{i},2)}(dd^{c}\log|s_{i}|^{-\frac{2}{m}})^{n}\to\int_{M}Ric^{n}.$

Let $\chi_{i}=h(u^{\prime}_{i})$, where $h$ is a smooth function on $[0,2)$,
$h\equiv 1$ on $[0,1]$, and $h$ has compact support. Then

(3.7) $\int\chi_{i}Ric_{i}^{n}=\int_{B(p^{\prime}_{i},2)\backslash
B(p^{\prime}_{i},1)}\log|s^{\prime}_{i}|^{-\frac{2}{m}}dd^{c}\chi_{i}\wedge(dd^{c}\log|s^{\prime}_{i}|^{-\frac{2}{m}})^{n-1}.$

Note on $B(p^{\prime}_{i},2)\backslash B(p^{\prime}_{i},1)$,
$|dd^{c}\chi_{i}|\leq C\omega^{\prime}_{i}$, $Ric_{M^{\prime}_{i}}\leq
C\omega^{\prime}_{i}$, also $|\log|s_{i}|^{2}|$ is uniformly bounded. Assume
$\chi_{i}\to\chi$. Recall $s^{\prime}_{i}\to s_{V}\in H^{0}(V_{reg},K^{-m})$.
We assert that

(3.8)
$\displaystyle\lim\limits_{i\to\infty}\int\log|s^{\prime}_{i}|^{-\frac{2}{m}}dd^{c}\chi_{i}\wedge(dd^{c}\log|s^{\prime}_{i}|^{-\frac{2}{m}})^{n-1}$
$\displaystyle=\int\log|s_{V}|^{-\frac{2}{m}}dd^{c}\chi\wedge(dd^{c}\log|s_{V}|^{-\frac{2}{m}})^{n-1}$
$\displaystyle=\int_{V}\chi(dd^{c}\log|s_{V}|^{-\frac{2}{m}})^{n}.$

Certainly the potential convergence holds on each compact set of regular
parts, hence the integral converges by Chern-Levine-Nirenberg. In a small
neighborhood of singular part, note the integrand is bounded by
$C\omega_{M^{\prime}_{i}}^{n}$, hence the contribution is negligible. This
proves (3.8). We can also apply (3.8) to the sequence $s_{i}\to s$. Putting
(3.6), (3.7) and (3.8) together, we find

$\int_{V}\chi(dd^{c}\log|s_{V}|^{-\frac{2}{m}})^{n}=0.$

Let $\Omega\in H^{0}(V_{reg},K^{m})$ be the dual of $s_{V}$, which is nowhere
vanishing. Then $\log|\Omega|^{2}$ is psh and

(3.9) $\int_{B(o,2)}\chi(dd^{c}\log|\Omega|^{2})^{n}=0.$

As in [44] and Lemma $6.1$ in [22], by multiplying by a nowhere vanishing
holomorphic function, we may assume $\Omega$ is homogeneous with respect to
$r\frac{\partial}{\partial r}$. We remark that $\log|\Omega|$ can not be
strictly decreasing along $r\frac{\partial}{\partial r}$. Otherwise, for some
$c>0$, we may assume $\log|\Omega|+c\log r$ is homogeneous and the norm is
constant along $r\frac{\partial}{\partial r}$. Such function is psh, and by
maximum principle, it is constant. Hence, the psh function
$\log|\Omega|=c_{1}-c\log r$. This is not possible when $n\geq 2$.

Let $u=\log|\Omega|^{\frac{2}{m}}$, $v=c\log r^{2}$, where $c$ is chosen so
that they have the same degree along $r\frac{\partial}{\partial r}$. Let
$A=B(o,2)\backslash B(o,0.1)$. Then on $A_{reg}$,

(3.10) $0\leq dd^{c}u\leq C\omega,0\leq dd^{c}v\leq C\omega,|u|+|v|\leq
C,|du|+|dv|\leq C.$

######  Lemma 3.

(3.11)
$\int_{A_{reg}}udd^{c}\chi\wedge(dd^{c}u)^{n-1}=\int_{A_{reg}}vdd^{c}\chi\wedge(dd^{c}v)^{n-1}.$

###### Proof.

Let $\eta$ (depending on $\epsilon$) be a smooth cut-off function which is
supported on $A\backslash B_{\epsilon}(\Sigma)$, where $\Sigma$ is the metric
singular set, $0\leq\eta\leq 1$. Also assume $\eta\equiv 1$ on $A\backslash
B_{2\epsilon}(\Sigma)$, $|d\eta|\leq C/\epsilon$. According to (3.10),

$\lim\limits_{\epsilon\to 0}\int_{A}\eta
udd^{c}\chi\wedge(dd^{c}u)^{n-1}=\int_{A_{reg}}udd^{c}\chi\wedge(dd^{c}u)^{n-1}.$

We have

(3.12) $\displaystyle\int_{A}\eta
udd^{c}\chi\wedge(dd^{c}u)^{n-1}-\int_{A}\eta
vdd^{c}\chi\wedge(dd^{c}u)^{n-1}$ $\displaystyle=\int_{A}\eta d(v-u)\wedge
d^{c}\chi\wedge(dd^{c}u)^{n-1}+\int_{A}(v-u)d\eta\wedge
d^{c}\chi\wedge(dd^{c}u)^{n-1}.$

For any $q\in A_{reg}$, choose a holomorphic chart containing $q$ so that
$\frac{\partial}{\partial z_{1}}=r\frac{\partial}{\partial
r}-\sqrt{-1}Jr\frac{\partial}{\partial r}$. Now in such coordinate, if we set
$x_{1}=Rez_{1}$, then

$u=kx_{1}+f_{1}(z_{2},..,z_{n}),v=kx_{1}+f_{2}(z_{2},..,z_{n}),v-u=g(z_{2},..,z_{n}).$

Now assume in addition that $u,v$ are smooth. Under the local coordinate
above, for all $s$, we have $u_{1\overline{s}}=v_{s\overline{1}}=0$, also
$(v-u)_{1}=(v-u)_{\overline{1}}=0$, hence

$d(v-u)\wedge d^{c}\chi\wedge(dd^{c}u)^{k}\wedge(dd^{c}v)^{n-k-1}\equiv 0.$

The nonsmoothness of $u,v$ can be handled by equivariant local smooth
approximation on the chart above. Hence the first term in the second line of
(3.12) vanishes identically. The second term approaches zero as $\epsilon\to
0$, since we have (3.10) and that the singular set has real codimension $4$.
We claim that

$\int_{A_{reg}}vdd^{c}\chi\wedge(dd^{c}u)^{k}\wedge(dd^{c}v)^{n-1-k}=\int_{A_{reg}}udd^{c}\chi\wedge(dd^{c}u)^{k-1}\wedge(dd^{c}v)^{n-k}.$

This directly follow from integration by parts, if $A$ is smooth. In the
general case,

$\displaystyle\int_{A_{reg}}vdd^{c}\chi\wedge(dd^{c}u)^{k}\wedge(dd^{c}v)^{n-1-k}$
$\displaystyle=\lim\limits_{\epsilon\to 0}\int_{A_{reg}}\eta
vdd^{c}\chi\wedge(dd^{c}u)^{k}\wedge(dd^{c}v)^{n-1-k}$
$\displaystyle=-\lim\limits_{\epsilon\to 0}\int_{A_{reg}}d(\eta v)\wedge
dd^{c}\chi\wedge d^{c}u\wedge(dd^{c}u)^{k-1}\wedge(dd^{c}v)^{n-1-k}$
$\displaystyle=-\lim\limits_{\epsilon\to 0}\int_{A_{reg}}(\eta
dv+vd\eta)\wedge dd^{c}\chi\wedge
d^{c}u\wedge(dd^{c}u)^{k-1}\wedge(dd^{c}v)^{n-1-k}$
$\displaystyle=-\lim\limits_{\epsilon\to 0}\int_{A_{reg}}\eta dv\wedge
dd^{c}\chi\wedge d^{c}u\wedge(dd^{c}u)^{k-1}\wedge(dd^{c}v)^{n-1-k}$
$\displaystyle=-\lim\limits_{\epsilon\to 0}\int_{A_{reg}}ud^{c}(\eta dv)\wedge
dd^{c}\chi\wedge(dd^{c}u)^{k-1}\wedge(dd^{c}v)^{n-1-k}$
$\displaystyle=\int_{A_{reg}}udd^{c}\chi\wedge(dd^{c}u)^{k-1}\wedge(dd^{c}v)^{n-k}.$

Therefore, we have

$\displaystyle\int_{A_{reg}}udd^{c}\chi\wedge(dd^{c}u)^{n-1}$
$\displaystyle=\int_{A_{reg}}vdd^{c}\chi\wedge(dd^{c}u)^{n-1}$
$\displaystyle=\int_{A_{reg}}udd^{c}\chi\wedge(dd^{c}u)^{n-2}\wedge dd^{c}v$
$\displaystyle=\int_{A_{reg}}vdd^{c}\chi\wedge(dd^{c}u)^{n-2}\wedge dd^{c}v$
$\displaystyle=\cdot\cdot\cdot=\int_{A_{reg}}vdd^{c}\chi\wedge(dd^{c}v)^{n-1}.$

∎

######  Lemma 4.

$\log|\Omega|$ is constant along $r\frac{\partial}{\partial r}$.

###### Proof.

Assume $\log|\Omega|$ is not constant along $r\frac{\partial}{\partial r}$.
Choose $c>0$ such that $\log|\Omega|^{2}-c\log r^{2}$ is constant along
$r\frac{\partial}{\partial r}$. By Lemma 3 and integration by parts, we find

$\int_{B(o,1)}(dd^{c}\log|\Omega|^{2})^{n}=c^{n}\int_{B(o,1)}(dd^{c}\log
r^{2})^{n}.$

According to formula $5.5$ (page 158) of [24],

$\int_{B(o,1)}(dd^{c}\log
r^{2})^{n}=\int_{B(o,1)}(dd^{c}r^{2})^{n}=vol(B(o,1))>0.$

Therefore,

$\int_{B(o,1)}(dd^{c}\log|\Omega|^{2})^{n}=c^{n}vol(B(o,1))>0.$

This contradicts (3.9).

∎

As $\log|\Omega|$ is psh, by maximum principle, $|\Omega|$ is constant. This
means $V$ is Ricci flat in the current sense. By the regularity theory of
complex Monge-Ampère equations, we see that $V$ is Ricci flat in the
differential geometric sense, and the metric is $C^{\infty}$ on regular part.
Also $\Omega$ is a parallel pluri-canonical section: by parallel transport, we
obtain a pluri-canonical form $\gamma$ locally, then $\frac{\gamma}{\Omega}$
is holomorphic with constant norm, hence $\frac{\gamma}{\Omega}$ is constant.
This proves the openness of Ricci flat tangent cones with a parallel pluri-
canonical section. Closedness follows from convergence theory by Anderson [1]
and Cheeger-Naber [16]. By connectness of tangent cones, all tangent cones are
Ricci flat.

Ricci flatness of all tangent cones imply that the holomorphic spectrum is
rigid [44][25]. Then by [25], $M$ is identified with an affine algebraic
variety $X$ in $\mathbb{C}^{N}$ (we only need that the holomorphic spectrums
are the same). Let $(z_{1},...,z_{N})$ be the holomorphic coordinate on
$\mathbb{C}^{N}$.

Let $\hat{X}$ be the compactification of
$X\subset\mathbb{C}^{N}\subset\mathbb{P}^{N}$. For $N$ large, we can find a
sequence of automorphism $\tau_{i}$ of $\mathbb{P}^{N}$ and a sequence
$r_{i}\to\infty$ such that if we define $(M_{i},p_{i})=(M,p,\frac{r}{r_{i}})$,
then $\tau_{i}((M_{i},p_{i}))$ converges in both intrinsic (the pointed-
Gromov-Hausdorff sense) and extrinsic (current) sense to the tangent cone
$(V,o)$. Also, there exists $c>0$ such that

$B_{V}(o,c)\subset B_{\mathbb{C}^{N}}(0,1)\cap V\subset
B_{\mathbb{C}^{N}}(0,2)\cap V\subset B_{V}(0,\frac{1}{c}).$

Therefore, for large $i$,

$B_{M_{i}}(p_{i},0.5c)\subset
B_{\mathbb{C}^{N}}(0,1)\cap\tau_{i}(M_{i})\subset
B_{\mathbb{C}^{N}}(0,2)\cap\tau_{i}(M_{i})\subset
B_{M_{i}}(p_{i},\frac{2}{c}).$

Let us fix such $c$. Let the Fubini-Study metric on $\mathbb{P}^{N}$ be given
by

$\omega_{FS}=dd^{c}\log(1+|z_{1}|^{2}+\cdot\cdot\cdot+|z_{N}|^{2}).$

Assume $M$ is not Ricci flat. Then there exists $q\in M$ and $e\in T_{q}M$
such that $Ric(e,e)>0$. Consider an algebraic curve $L\subset X$, through $q$
with tangent proportional to $e$. Let $\hat{L}$ be the compactified curve.
Then it has finite degree. Under the sequence of automorphism of
$\mathbb{P}^{N}$, the degree is fixed. Hence, the volume w.r.t $\omega_{FS}$
is uniformly bounded.

Since we have a nowhere zero section of $K_{V}^{m}$ with constant norm $1$, we
can lift such the pluri-anticanonical dual section by $L^{2}$ method. Say the
section is $s_{i}$. Then $s_{i}$ is nowhere vanishing. Also, on the annulus
$B(p_{i},\frac{2}{c})\backslash B(p_{i},0.5c)$, $|s_{i}|^{2}$ is converging
uniformly to $1$. Thus by scale invariance of $Ric$, on $L_{i}=\tau_{i}(L)$,

$\frac{1}{m}dd^{c}(-\log|s_{i}|^{2})=Ric_{i}=Ric.$

Let $h$ be a smooth function on $\mathbb{R}^{+}$ satisfying $h=1$ on $[0,1]$,
$h\geq 0$, $h$ has compact support on $[0,1.5)$. Let
$\chi=h(\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+\cdot\cdot\cdot+|z_{N}|^{2}})$. Now

$\int_{L_{i}}Ric\chi=\int_{L_{i}}\frac{1}{m}dd^{c}(-\log|s_{i}|^{2})\chi=\int_{L_{i}\cap(B(p_{i},\frac{2}{c})\backslash
B(p_{i},0.5c))}\frac{1}{m}(-\log|s_{i}|^{2})dd^{c}\chi.$

Recall that $L_{i}$ has uniform area bound with respect to $\omega_{FS}$. As
$\chi$ is supported on $B_{\mathbb{C}^{N}}(0,1.5)$,

$|dd^{c}\chi|_{\omega_{FS}}<C.$

Also $\log|s_{i}|^{2}$ converges uniformly to $0$ on
$B(p_{i},\frac{2}{c})\backslash B(p_{i},0.5c)$. Putting these together, we
find

(3.13) $\lim\limits_{i\to\infty}\int_{L_{i}}Ric\chi=0.$

On the other hand,

(3.14) $\int_{L_{i}}Ric\chi\geq\int_{L\cap B(q,1)}Ric>\epsilon_{0}>0$

for some fixed $\epsilon_{0}$, since $Ric$ is positive at the tangent of $L$
at $q$. Then (3.14) contradicts (3.13). The uniqueness of tangent cone follows
from [25]. This completes the proof of Theorem 5 when $M$ satisfies condition
$B$.

If $M$ satisfies condition $A$, the argument is pretty much the same. Let us
point out the minor differences. Note for the Ricci flat tangent cone, the
cross section is smooth with $Ric=2n-2$. Then by Bonnet-Meyers theorem, the
fundamental group is finite. Hence the parallel pluri-canonical section exists
automatically. Let $r_{i}$ be a sequence such that $r_{i}\to\infty$. Set
$(M_{i},p_{i})=(M,p,\frac{r}{r_{i}})$. By passing to a subsequence, we may
assume $M_{i}\to V$. By [35], $B(p_{i},1)$ is $1$-convex and
$F_{i}=(f_{i1},..,f_{iN})$ defines a holomorphic map from
$B(p_{i},1)\to\mathbb{C}^{N}$. $F_{i}$ is an isomorphism except finitely many
compact subvarieties (let us call it $E$). Moreover, $f_{ij}=0$ at $p_{i}$,
$f_{ij}\to f_{j}$ on $V$ locally uniformly. $(f_{1},...,f_{N})$ is a
holomorphic embedding from $B_{V}(o,1)\to\mathbb{C}^{N}$. As $\omega$ need not
be $dd^{c}$-exact, one should replace (3.2) by a psh function $u_{i}$ on
$B(p_{i},10)$ such that on $B(p_{i},10)\backslash B(p_{i},\delta_{0})$,

$dd^{c}u_{i}\geq 0.5\omega_{i}.$

Also on $B(p_{i},10)$,

$|u_{i}-d^{2}(p_{i},\cdot)|<\Psi(\frac{1}{i}),|u_{i}|+|\nabla u_{i}|\leq C.$

These can be ensured by [35, Proposition $2.5$]. Note that (3.4) holds on
$B(p_{i},2)\backslash E$. However, (3.5) still holds. The reason is that
$\frac{s^{\prime}_{i}}{s_{i}}$ is holomorphic away from $E$, after contraction
of $E$, $\frac{s^{\prime}_{i}}{s}$ can be extended as a holomorphic function
which is nowhere vanishing (use $n\geq 2$). Correspondingly, (3.6) should be
replaced by

(3.15)
$\displaystyle\int_{B(p^{\prime}_{i},2)}(dd^{c}\log|s^{\prime}_{i}|^{-\frac{2}{m}})^{n}$
$\displaystyle=\int_{B(p,2r^{\prime}_{i})}(dd^{c}\log|s^{\prime}_{i}|^{-\frac{2}{m}})^{n}$
$\displaystyle=\int_{B(p,2r_{i})}(dd^{c}\log|s_{i}|^{-\frac{2}{m}})^{n}+\int_{B(p,2r^{\prime}_{i})\backslash
B(p,2r_{i})}Ric^{n}$
$\displaystyle=\int_{B(p_{i},2)}(dd^{c}\log|s_{i}|^{-\frac{2}{m}})^{n}+\int_{B(p,2r^{\prime}_{i})\backslash
B(p,2r_{i})}Ric^{n}.$

Also, under condition $A$, Lemma 2 can be proved similarly. For reader’s
convenience, let us state it as

######  Lemma 5.

Let $M^{n}$ satisfy condition $A$. Then there exists $C_{1}>0$ such that for
all $1\leq k\leq n$, for all $r>0$,

$r^{2k-2n}\int_{B(p,r)}Ric^{k}\wedge\omega^{n-k}<C_{1}.$

Furthermore, if we consider a tangent cone $(V,o)$ and a sequence of rescaled
manifolds $(M_{i},p_{i})=(M,p,\frac{r}{r_{i}})\to(V,o)$ in the pointed Gromov-
Hausdorff sense, then for $k<n$,

$\lim\limits_{i\to\infty}r_{i}^{2k-2n}\int_{B(p,r_{i})}Ric^{k}\wedge\omega^{n-k}=\int_{B_{V}(o,1)}Ric_{V}^{k}\wedge\omega_{V}^{n-k},$

where $Ric_{V}$ is understood in the current sense.

Then we can show that

$\int_{V}\chi(dd^{c}\log|s_{V}|^{-\frac{2}{m}})^{n}=0.$

Then the argument proceeds as before. We can show that all tangent cones are
Ricci flat. Let $X$ be the blow down contraction of exceptional locus for $M$.
Then one can apply [35] to prove that $X$ is isomorphic to an affine algebraic
variety in $\mathbb{C}^{N}$. The rest of the argument is the same as before.
Uniqueness of tangent cones follows from [15][25][35]. The proof of Theorem 5
is complete. As a side remark, Lemma 5 implies that the condition
$\int_{M}Ric^{n}<+\infty$ in [45] holds automatically (take $k=n$).

Proof of Corollary 3:

We claim that the limit space $(X,o)$ is homeomorphic to a normal complex
space. Note $(X,o)$ is complex analytically smooth away from $o$, as
$|Rm|\leq\frac{C}{r^{2}}$. Let $r_{0}$ be small so that for all $r<r_{0}$,
$B(o,r)$ is Gromov-Hausdorff close to a metric cone, after rescaling the
metric so that the radius is $1$. Then we can show $1$-convexity as before.
Then, for some $c>0$, we can find a holomorphic embedding from
$B(o,cr_{0})\to\mathbb{C}^{N}$, up to contraction of finitely many compact
subvarities to points. Note inside $B(o,cr_{0})$, there is no compact variety
of positive dimension. Otherwise, one can rescale the metric to the scale of
the compact variety, then we find a contradiction by $1$-convexity. This shows
$(X,o)$ is homeomorphic to a normal complex space. The rest of the argument is
similar to the noncompact case. The uniqueness of tangent cone follows from
the argument in [25].

Proof of Corollary 4: The argument is almost identical to noncompact case. One
difference is this: when solving $\overline{\partial}$-equation, we have to
multiply the metric on line bundle by $e^{-Cu}$, where $dd^{c}u=\omega$. This
kills the negative parts of Ric.

Proof of Corollary 5:

Assume $r_{i}\to\infty$ such that
$\lim\limits_{i\to\infty}r_{i}^{2}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p,r_{i})}S=0$.
Define $(M_{i},p_{i})=(M,p,\frac{r}{r_{i}})$. We assume $(M_{i},p_{i})$
converges in the pointed Gromov-Hausdorff sense to $(V,o)$. Then by applying
the convergence result in Lemma 5, we find $V$ is Ricci flat on the regular
part in the current sense. By the regularity theory of complex Monge-Ampère
equations, we see that $V$ is Ricci flat in the differential geometric sense.
Then apply Theorem 5.

Proof of Corollary 6

This is straightforward, since the tangent cone is Ricci flat.

Proof of Corollary 7

There exists a sequence $r_{i}\to\infty$, such that
$\int_{B(p,2r_{i})\backslash B(p,r_{i})}S^{n}\to 0$. Define
$(M_{i},p_{i})=(M,p,\frac{r}{r_{i}})$. We assume $(M_{i},p_{i})$ converges in
the pointed Gromov-Hausdorff sense to $(V,o)$. Then Hölder inequality implies

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p_{i},2)\backslash
B(p_{i},1)}S_{i}\leq(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(p_{i},2)\backslash
B(p_{i},1)}S^{n}_{i})^{\frac{1}{n}}\to 0.$

This means $Ric_{i}$ converges to zero in the current sense on $V$. Then apply
Theorem 5.

Proof of Corollary 8

This follows from Corollary 7.

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