The continuity equation with cusp singularities

TL;DR

This paper investigates the behavior of cusp Kähler-Einstein metrics during the continuity process, focusing on their geometric and algebraic properties near singularities and limits.

Contribution

It introduces a detailed study of cusp singularities in the context of the continuity method for Kähler-Einstein metrics, expanding understanding of their limits and properties.

Findings

Analysis of noncollapsing limits with cusp singularities

Characterization of geometric properties near cusp singularities

Insights into the algebro-geometric structure of the limits

Abstract

In this paper we study a special case of the completion of cusp K\"{a}hler-Einstein metric on the regular part of varieties by taking the continuity method proposed by La Nave and Tian. The differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method with cusp singularities will be investigated.

Equations

$$\mathrm{Ric}(\omega )=-\omega +[D].$$

$$\Phi :\overline{M}\to {\mathbb{CP}}^N$$

$$(1+t)\omega ={\omega }_0-\sqrt{-1}\partial \overline{\partial }\log {\log }^2|s_D{|}_{h_D}^2-t(\mathrm{Ric}(\omega )-[D]),$$

$${\omega }_{cusp}=\frac{\sqrt{-1}d{z}_1\wedge d\overline{z_1}}{|z_1{|}^2{\log }^2|z_1{|}^2}+\sum _{k=2}^{n}d{z}_k\wedge d\overline{z_k}.$$

$$\mathrm{Ric}({\omega }_1)=-{\omega }_1,on\overline{M}\backslash ({\mathcal{S}}_{\overline{M}}\cup D),$$

$${\mathcal{S}}_{\overline{M}}=\bigcap \{E|Eisaneffectivedisiviorsuchthat{K}_{\overline{M}}+D-\rho E>0forsome\rho >0\}.$$

$$B_{+}({\pi }^{\ast }G+F)={\pi }^{-1}(B_{+}(G))\cup Exc(\pi ),$$

$${\mathcal{S}}_{\overline{M}}=B_{+}(K_{\overline{M}}+D)={\Phi }^{-1}(B_{+}(K_{\Phi (\overline{M})}+\Phi (D)))\cup Exc(\Phi )={\Phi }^{-1}(\Phi (\overline{M}{)}_{sing})$$

$$\overline{M}\backslash ({\mathcal{S}}_{\overline{M}}\cup D)={\overline{M}}_{reg}\setminus D,$$

$$\frac{1}{C}{\delta }_{ij}<(g_{i\bar{j}})<C{\delta }_{ij},$$

$$\Big{|}\frac{\partial^{|p|+|q|}}{\partial v^{|p|}\partial\bar{v}^{|q|}}g_{i\bar{j}}\Big{|}<A_{|p|+|q|},\ for\ any\ multiindices\ p\ and\ q\,$$

$$\displaystyle||u||_{k,\lambda}=\sup_{V\in\Gamma}\Big{\{}\sup_{z\in V}\Big{(}\sum_{|p|+|q|\leq k}\Big{|}\frac{\partial^{|p|+|q|}}{\partial v^{|p|}\partial\bar{v}^{|q|}}u(z)\Big{|}\Big{)}$$

$$\displaystyle+\sup_{z,z^{\prime}\in V}\Big{(}\sum_{|p|+|q|=k}|z-z^{\prime}|^{-\lambda}\Big{|}\frac{\partial^{|p|+|q|}}{\partial v^{|p|}\partial\bar{v}^{|q|}}u(z)-\frac{\partial^{|p|+|q|}}{\partial v^{|p|}\partial\bar{v}^{|q|}}u(z^{\prime})\Big{|}\Big{)}\Big{\}}.$$

$$C^{k,\lambda }(M)=\{u\in C^k(M);||u|{|}_{k,\lambda }<\infty \},$$

$$|\tau |=|\tau {|}_h,|\nabla \tau {|}_{h\otimes \omega }=|\nabla \tau |,$$

$$|\nabla \nabla \tau {|}^2=\sum _{i,j}|{\nabla }_i{\nabla }_j\tau {|}^2,|\overline{\nabla }\nabla \tau {|}^2=\sum _{i,j}|{\nabla }_{\bar{i}}{\nabla }_j\tau {|}^2.$$

$${△}_{\omega }|\tau {|}^2=|\nabla \tau {|}^2-|\tau {|}^2\cdot t{r}_{\omega }{\Theta }_h$$

$$\begin{gathered} {△}_{\omega }|\nabla \tau {|}^2=|\nabla \nabla \tau {|}^2+|\overline{\nabla }\nabla \tau {|}^2-{\nabla }_j({\Theta }_h{)}_{i\bar{j}}⟨\tau ,{\nabla }_{\bar{i}}\bar{\tau }⟩-{\nabla }_{\bar{j}}(t{r}_{\omega }{\Theta }_h)⟨{\nabla }_j\tau ,\bar{\tau }⟩ \\ +R_{i\bar{j}}⟨{\nabla }_j\tau ,{\nabla }_{\bar{i}}\bar{\tau }⟩-2({\Theta }_h{)}_{i\bar{j}}⟨{\nabla }_j\tau ,{\nabla }_{\bar{i}}\bar{\tau }⟩-|\nabla \tau {|}^2\cdot t{r}_{\omega }{\Theta }_h \end{gathered}$$

$$\Phi :\overline{M}\to {\mathbb{CP}}^N$$

$$(\widetilde{{\omega }_l}+\sqrt{-1}\partial \overline{\partial }{u}_l{)}^n=e^{\frac{u_l}{l}}\frac{{\Omega }_l}{|s_D{|}_{h_D^l}^2{\log }^2|s_D{|}_{h_D^l}^2},$$

$$(\widetilde{{\omega }_l}+\sqrt{-1}\partial \overline{\partial }{u}_l{)}^n=e^{\frac{u_l}{l}}\frac{{\Omega }_l}{|s_D{|}^2{\log }^2|s_D{|}^2}.$$

$$U=\{v\in C^{k,\lambda }(M)|\frac{1}{C}\widetilde{{\omega }_l}<\widetilde{{\omega }_l}+\sqrt{-1}\partial \overline{\partial }v<C\widetilde{{\omega }_l},forsomepositiveconstantC\},$$

$$(\widetilde{{\omega }_l}+\sqrt{-1}\partial \overline{\partial }{u}_{l,s}{)}^n=e^{\frac{u_{l,s}}{l}}\cdot e^{sF}\cdot {\widetilde{{\omega }_l}}^n,$$

$$S=\{s\in [0,1]|thereisasolution{u}_{l,s}satisfies\Psi (u_{l,s})=e^{sF}\}.$$

$$h\to e^{sF}({△}_{{\omega }_{l,s}}h-\frac{h}{l}),$$

$${△}_{{\omega }_{l,s}}h-\frac{h}{l}=w$$

$${△}_{{\omega }_{l,s}}{h}_i-\frac{h_i}{l}=win{\Omega }_i,$$

$$h_i=0on\partial {\Omega }_i.$$

$$\underset{{\Omega }_i}{\sup }|h_i|\le l\cdot \underset{{\Omega }_i}{\sup }|w|.$$

$$\underset{M}{\sup }|u_{l,s_i}|\le l\cdot s_i\underset{M}{\sup }|F|\le C\underset{M}{\sup }|F|.$$

$$\mathrm{Ric}({\omega }_{l,s_i})=-\frac{1}{l}{\omega }_{l,s_i}+\frac{1}{l}\widetilde{{\omega }_l}-s_i\sqrt{-1}\partial \overline{\partial }F+\mathrm{Ric}(\widetilde{{\omega }_l})\le -\frac{1}{l}{\omega }_{l,s_i}+C\widetilde{{\omega }_l}$$

Full text

1. Introduction

The Yau-Tian-Donaldson conjecture for Fano manifolds has revealed deep connections among complex Monge-Ampère equation, metric geometry and complex algebraic geometry. Some specialists develop many techniques to deal with the celebrated conjecture, see [Ti15] or [CDS1], [CDS2], [CDS3]. These methods also play an important role in studying many other problems. For instance, in [So], J. Song proves that the metric completion of the regular set of Calabi-Yau varieties and canonical models of general type with crepant singularities is a compact length space which homeomorphic to the original variety. In [NT], G. La Nave and G. Tian introduce a new continuity equation to consider the analytic minimal model program. Later in [NTZ], G. La Nave, G. Tian and Z. L. Zhang study the differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity equation. These fundamental results focus on the compact Kähler manifolds. For noncompact case, let us recall some facts. Suppose $\overline{M}$ is a compact complex manifold, $D$ is an effective divisor with only normal crossings and $K_{\overline{M}}+D$ is ample, where $K_{\overline{M}}$ is the canonical line bundle over $\overline{M}$. A well known result achieved by Kobayashi [Ko84] and Tian-Yau [TY87] asserts that there exists a complete negative Kähler-Einstein metric on $\overline{M}\setminus D$. Recently, in [BG], two authors generalize this result. For the convenience, this consequence is stated as following (Theorem C [BG]).

Theorem 1.1**.**

Let $\overline{M}$ be a compact Kähler manifold and $D$ is a simple normal crossings $\mathbb{R}$-divisor on $\overline{M}$ with coefficients in $[-1,+\infty)$ such that $K_{\overline{M}}+D$ is semi-positive and big. Then there exists a unique $\omega$ in $c_{1}(K_{\overline{M}}+D)$ which is smooth on a Zariski open set $U$ of $\overline{M}$ and such that

[TABLE]

More precisely, $U$ can be taken to the $\overline{M}\setminus(D\cup\mathcal{S})$, where $\mathcal{S}$ is the intersetion of all effective $\mathbb{Q}$-divisors $E$ such that $K_{\overline{M}}+D-E$ is ample.

Motivated by [So], a natural problem is to ask what the completion of $(U,\omega)$ is. In this article, a special case is investigated. More precisely, suppose $\overline{M}$ is a projective manifold, $D$ is a smooth hypersurface and $K_{\overline{M}}+D$ is big and semi-ample. According to the Kawamata base point free theorem, there exists an integer $K\in\mathbb{Z}^{+}$ such that an orthonormal basis of $K(K_{\overline{M}}+D)$ gives a holomorphic map

[TABLE]

where $N=\dim H^{0}(\overline{M},K(K_{\overline{M}}+D))-1$. The mainly result in this article is that the completion of $(U,\omega)$ in the sense of Theorem 1 homeomorphic to $\Phi(\overline{M}\setminus D)$. If the divisor $D$ is simple normal crossing with coefficients $1$, then there is a similar result. To dealing with this problem, the continuity method is taken proposed by G. La Nave and G. Tian in [NT].

To begin with, let $\overline{M}$ be a projective manifold with a Kähler metric $\omega_{0}$ and $D$ be a smooth hypersurface in $\overline{M}$ such that $K_{\overline{M}}+D$ is big and semi-ample. $h_{D}$ is denoted by the hermitian metric on $L_{D}$, the associated line bundle of $D$ such that $\omega_{0}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|^{2}_{h_{D}}>0$, where $s_{D}$ is the defining section of $D$. The following $1$-parameter family equations are considered:

[TABLE]

where $[D]$ is the current of integratioin along $D$.

Recall that $\omega$ is sald to have cusp singularities along $D$ if, whenever $D$ is locally given by $(z_{1}=0)$, $\omega$ is quasi-isometric to the cusp metric

[TABLE]

Since $\omega_{0}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|^{2}_{h_{D}}$ is a Kähler metric on $\overline{M}\setminus D$ with cusp singularities, the equation (1.2) essentially state the variation of cusp Kähler metric along $t$. Therefore, the equation (1.2) is called the cusp continuity equation.

Theorem 1.3**.**

The cusp continuity equation (1.2) is solvable for all $t\in[0,+\infty)$.

$\omega_{t}$ is denoted by the solution of (1.2), then we have the following convergence result.

Theorem 1.4**.**

$\omega_{t}$* converge to a unique weakly Kähler metric $\omega_{1}$ such that $\omega_{1}$ is smooth on $\overline{M}\setminus(D\cup\mathcal{S})$ and satisfies*

[TABLE]

where

[TABLE]

If $G$ is a big divisor, we denoted $B_{+}(G)$ by the intersection of all effective $\mathbb{Q}$-divisors $E$ such that $G-E$ is ample. Then $\mathcal{S}_{\overline{M}}$ appeared in Theorem 1.4 is $B_{+}(K_{\overline{M}}+D)$. Observing that $\Phi:(\overline{M},D)\rightarrow(\Phi(\overline{M}),\Phi(D))$ can be viewed as a resolution of $(\Phi(\overline{M}),\Phi(D))$ and $K_{\overline{M}}+D=\Phi^{*}(K_{\Phi(\overline{M})}+\Phi(D))$. According to Theorem 0-3-12 [Kw], $K_{\Phi(\overline{M})}+\Phi(D)$ is ample. Let $\Phi(\overline{M})_{reg}$ be the regular part of $\Phi(\overline{M})$ and $\Phi(\overline{M})_{sing}$ be the singular part of $\Phi(\overline{M})$. The following Proposition illustrate the connection between $B_{+}(K_{\overline{M}}+D)$ and $\Phi(\overline{M})_{sing}$, due to Proposition 2.3 [Bo].

Proposition 1.5**.**

Let $\pi:X\rightarrow Y$ be a birational morphism between normal projective varieties. For any big $\mathbb{R}$-divisor $G$ on $Y$ and any effective $\pi$-exceptional divisor $\mathbb{R}$-divisor $F$ on $X$, then we have

[TABLE]

where $Exc(\pi)\subset X$ is the set of points $x\in X$ such that $\pi$ is not biregular.

From Theorem 1.4 and Proposition 1.5, the following Corollary is derived immediately.

Corollary 1.6**.**

[TABLE]

and

[TABLE]

where $\overline{M}_{reg}$ represents $\Phi^{-1}(\Phi(\overline{M})_{reg})$. Furthermore the metric $\omega_{1}$ is smooth on $\overline{M}_{reg}\setminus D$.

Remark 1.7**.**

If the codimension of $\Phi(D)$ is not $1$, then $D$ is an exceptional divisor of the resolution $\Phi$. Thus, $\overline{M}\backslash(\mathcal{S}_{\overline{M}}\cup D)=\overline{M}_{reg}$.

$M$ is denoted by $\overline{M}\setminus D$. The next result states that the limit space $(M,\omega_{t})$ converge to in the Gromov-Hausdorff topology has more regular properties, such as metric structure, algebraic structure.

Theorem 1.8**.**

The following results are hold.

1. (1)

$(M,\omega_{t},x)$* converges in the Gromov-Hausdorff topology to a length space $(M_{1},d_{1},x_{1})$ which is the metric completion $(\overline{M}_{reg}\setminus D,\omega_{1})$.* 2. (2)

$M_{1}=\mathcal{R}\cup\mathcal{S}$* and $\mathcal{R}=\overline{M}_{reg}\setminus D$, where $\mathcal{R}$ is the regular part and $\mathcal{S}$ is the singular part.* 3. (3)

$\mathcal{R}$* is geodesically convex and $\mathcal{S}$ is closed set which has codimension $\geq 2$.* 4. (4)

$M_{1}$* homeomorphic to a normal quasi-subvariety $\Phi(\overline{M}\setminus D)$.*

**Acknowledgement:**The author would like to thank Professor Gang Tian for his constant help, support and encouragement.

2. Preliminaries

In this section, we list some fundamental definitions and results which will be used in the later.

Definition 2.1**.**

Let $V$ be an open set in $\mathbb{C}^{n}$. A holomorphic map from $V$ into a complex manifold $M$ of complex dimension $n$ is called a quasi-coordinate map if it is of maximal rank everywhere in $V$. This open set $V$ is called a local quasi-coordinate of $M$.

Definition 2.2**.**

Let $M$ be a complete Kähler manifold and $\omega$ is the Kähler form. $(M,\omega)$ is called bounded geometry if there is a quasi-coordinates $\Gamma=\{(V;v^{1},\cdot\cdot\cdot,v^{n})\}$ which satisfies the following three conditions:

1. (1)

$M$* is covered by the image of $(V;v^{1},\cdot\cdot\cdot,v^{n})$.* 2. (2)

The complement of some open neighborhood of $D$ is covered by a finite of $(V;v^{1},\cdot\cdot\cdot,v^{n})$ which are local coordinates in the usual sense. 3. (3)

There exist positive constants $C$ and $A_{k}$ $(k=0,1,2,\cdot\cdot\cdot)$ independent of $\Gamma$ such that at each $(V;v^{1},\cdot\cdot\cdot,v^{n})$, the inequalities

[TABLE]

[TABLE]

hold, where $g_{i\bar{j}}$ denote the component of $\omega$ with respect to $V$.

Now we define the Hölder space of $C^{k,\lambda}$-functions on a complete Kähler manifold $(M,\omega)$ which cover by the image of quasi-coordinates. For a nonnegative integer $k$ , $\lambda\in(0,1)$ and $u\in C^{k}(M)$, we define

[TABLE]

The function space $C^{k,\lambda}(M)$ is, by definition,

[TABLE]

which is a Banach space with respect to the norm $||\cdot||_{k,\lambda}$.

Next we state the generalized maximum principle, due to Yau (Proposition 1.6 [CY]).

Theorem 2.3**.**

Suppose $(M,\omega)$ is a complete Kähler manifold with bounded geometry and $f$ is a function on $M$ which is bounded from above. Then there exists a sequence ${x_{i}}$ in $M$ such that $\lim_{i\rightarrow\infty}f(x_{i})=\sup f$, $\lim_{i\rightarrow\infty}|\nabla f(x_{i})|=0$ and $\overline{\lim}_{i\rightarrow\infty}{\rm Hess}(f)(x_{i})\leq 0$, where the Hessian is taken with respect to $\omega$.

Now we introduce the Bochner formula on a general line bundle. Let $(M,\omega)$ be a Kähler manifold of dimension $n$ and $(L,h)$ be a Hermitian Line bundle over $M$. Let $\Theta_{h}$ be the Chern curvature form of $h$. Let $\nabla$ and $\overline{\nabla}$ denote the $(1,0)$ and $(0,1)$ part of a connection respectively. The connection appeared in this paper is usually known as the Chern connection or Levi-Civita connection.

For a holomorphic section $\tau\in H^{0}(M,L)$ we write for simplicity

[TABLE]

and

[TABLE]

By direct computation we have

Lemma 2.4**.**

(Bochner formulas). For any $\tau\in H^{0}(M,L)$ one has

[TABLE]

and

[TABLE]

where $R_{i\bar{j}}$ is the Ricci curvature of $\omega$, $\langle,\rangle$ is the inner product defined by $h$.

3. Existence and uniqueness of cusp continuity equation

This section is devoted to prove the Theorem 1.3. When $t=0$ the equation (1.2) has a solution $\omega(0)=\omega_{0}-\sqrt{-1}\partial\bar{\partial}\log\log^{2}|s_{D}|^{2}_{h_{D}}$. For a fixed $t\neq 0$, we reduce (1.2) to a scalar equation. First, the background metric will be constructed. Since $K_{\overline{M}}+D$ is semi-ample, the Kawamata base point free Theorem claims that there exists an integer $K_{0}$ such that $K_{0}(K_{\overline{M}}+D)$ has no base point. Then a basis of $H^{0}(\overline{M},K_{0}(K_{\overline{M}}+D))$ gives a holomorphic map

[TABLE]

where $N=\dim H^{0}(\overline{M},K_{0}(K_{\overline{M}}+D))-1$. $\omega_{FS}$ is denoted by the Fubini-Study metric on $\mathbb{CP}^{N}$. Set $\eta_{1}=\frac{1}{K_{0}}\Phi^{*}(\omega_{FS})$. Since $\eta_{1}\in c_{1}(K_{\overline{M}}+D)$, there exist a smooth volume form $\Omega$ on $\overline{M}$ and hermitian metric $h_{D}^{t}$ on $L_{D}$ such that $\eta_{1}=-{\rm Ric}(\Omega)+\Theta_{h_{D}^{t}}$ and $\frac{1}{1+t^{\prime}}\omega_{0}+\frac{t^{\prime}}{1+t^{\prime}}\eta_{1}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|_{h_{D}^{t}}>0$ for $t^{\prime}\in[0,t]$, where $\Theta_{h_{D}^{t}}$ is the curvature form of $L_{D}$ with the metric $h_{D}^{t}$. Set $l=\frac{t}{1+t}$, then $\widetilde{\omega_{l}}:=(1-l)\omega_{0}+l\eta_{1}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|_{h_{D}^{t}}$ is chosen as the background meric. Therefore the equation (1.2) is reduced to the following scalar equation

[TABLE]

where $|s_{D}|^{2}_{h_{D}^{l}}$ is denoted by $|s_{D}|^{2}_{h_{D}^{t}}$ and \Omega_{l}=\Omega\Big{(}\frac{\log^{2}|s_{D}|^{2}_{h_{D}}}{\log^{2}|s_{D}|^{2}_{h_{D}^{l}}}\Big{)}^{\frac{1-l}{l}} is a smooth volume form on $\overline{M}$. For the convenience, we simplified the notation of the above equation as following

[TABLE]

To get a complete metric, we define an open subset $U$ in $C^{k,\lambda}(M)$ by

[TABLE]

where $M=\overline{M}\setminus D$. If $u_{l}$ belongs to $U$ and satisfies (3.1), then $\widetilde{\omega_{l}}+\sqrt{-1}\partial\overline{\partial}u_{l}$ is a complete Kähler metric.

Now we take the continuity method to solve the equation (3.1). Consider the following equations

[TABLE]

where $F=\frac{\Omega_{l}}{\widetilde{\omega_{l}}^{n}|s_{D}|^{2}\log^{2}|s_{D}|^{2}}$. We consider

We consider the $C^{0}$ map $\Psi:C^{k,\lambda}(M)\rightarrow C^{k-2,\lambda}(M)$ defined by $\Psi(v)=e^{-\frac{u_{l}}{l}}\cdot\frac{(\widetilde{\omega_{l}}+\sqrt{-1}\partial\overline{\partial}v)^{n}}{\widetilde{\omega_{l}}^{n}}$. Define

[TABLE]

Obviously, $0\in S$. To prove $1\in S$, it is sufficient to show that $S$ is open and closed. The inverse mapping theorem implies the openness. The Fréchet derivative $\Psi^{\prime}(u_{l,s}):C^{k,\lambda}(M)\rightarrow C^{k-2,\lambda}(M)$ at $u_{l,s}\in U$ is given by

[TABLE]

where $\omega_{l,s}=\widetilde{\omega_{l}}+\sqrt{-1}\partial\overline{\partial}u_{l,s}$. Due to Kobayashi [Ko84], $F\in C^{k-2,\lambda}(M)$. Therefore, we have to show that, for any $w\in C^{k-2,\lambda}(M)$,

[TABLE]

can be solved for $h\in C^{k,\lambda}(M)$ and that $|h|_{C^{k,\lambda}}\leq C|w|_{C^{k-2,\lambda}}$ for some constant $C$ independent of $w$.

We first to show that there is at most one function $h$ in $C^{k,\lambda}(M)$ solving the equation (3.3). It suffices to verify that $\triangle_{\omega_{l,s}}h-\frac{h}{l}=0$ and $h\in C^{k,\lambda}(M)$ imply $h\equiv 0$. Note that $\omega_{l,s}$ is complete Kähler metric with bounded geometry due to Lemma 2 [Ko84] and Proposition 1.4 [CY]. For such a metric we can use the generalized maximum principle. Suppose $h\in C^{k,\lambda}(M)$, $h$ is in particular bounded. The generalized maximum principle implies that there exists a sequence of points $\{x_{i}\}$ in $M$ such that $\lim_{i\rightarrow\infty}h(x_{i})=\sup h$ and $\overline{\lim}_{i\rightarrow\infty}\triangle_{\omega_{l,s}}h(x_{i})\leq 0$. We immediately see that $\sup h\leq 0$ according to the equation $\triangle_{\omega_{l,s}}h-\frac{h}{l}=0$. Similarly, $\inf h\geq 0$ and $h\equiv 0$.

Now we prove the existence of $h$. Let $\{\Omega_{i}\}$ be an exhaustion of $M$ by compact subdomains. Suppose $w\in C^{k-2,\lambda}(M)$ and let $h_{i}$ be the unique solution to

[TABLE]

[TABLE]

The maximum principle applied to $\Omega_{i}$ shows that

[TABLE]

Interior Schauder estimates shows that a sequence of $h_{i}$ converge to some $h\in C^{k,\lambda}(M)$ which solves the equation (3.3) and that the estimate $|h|_{C^{k,\lambda}}\leq C|w|_{C^{k-2,\lambda}}$.

Next, it remains to show that $S$ is closed. Assume that $\{s_{i}\}\subset E$ is a sequence with $\lim_{i\rightarrow\infty}s_{i}=\bar{s}$ and $u_{l,s_{i}}$ is the solution of (3.2) with $s=s_{i}$. We want to prove $\bar{s}\in E$. It amounts to getting a prior $C^{k,\lambda}(M)$-estimate for each $u_{l,s_{i}}$. By applying the generalized maximum principle to (3.2), we have

[TABLE]

So we have the $C^{0}$-estimate due to Lemma 1 [Ko84]. For the $C^{2}$-estimate, since $(M,\widetilde{\omega_{l}})$ is a bounded geometry, by the standard calculation we have

[TABLE]

and

[TABLE]

Then

[TABLE]

where $-a$ is the lower bound of holomorphic bisectional curvature of metric $\widetilde{\omega_{l}}$. Note that

[TABLE]

Let $H=\log tr_{\widetilde{\omega_{l}}}\omega_{l,s_{i}}-(a+1)u_{l,s_{i}}$, then

[TABLE]

By the generalized maximum principle, there exists a sequence $\{x_{i}\}$ such that $\lim_{i\rightarrow\infty}H(x_{i})=\sup H$ and $\overline{\lim}_{i\rightarrow\infty}\sqrt{-1}\partial\overline{\partial}H(x_{i})\leq 0$. So we have a subsequence also denoted by $\{x_{i}\}$ such that

[TABLE]

Nota that $(tr_{\widetilde{\omega_{l}}}\omega_{l,s_{i}})^{\frac{1}{n-1}}\leq C^{\prime}tr_{\omega_{l,s_{i}}}\widetilde{\omega_{l}}$. Then we get

[TABLE]

If

[TABLE]

then we see

[TABLE]

Otherwise

[TABLE]

then by (3.4) we have

[TABLE]

Therefore, $H\leq C$. This implies $tr_{\widetilde{\omega_{l}}}\omega_{l,s_{i}}\leq C$. Furthermore by a standard inequality, we get $C^{-1}\widetilde{\omega_{l}}\leq\omega_{l,s_{i}}\leq C\widetilde{\omega_{l}}$.

For the $3$-order estimate, let $T=|\nabla_{\widetilde{g}_{l}}\partial\overline{\partial}u_{l,s_{i}}|^{2}_{g_{l,s_{i}}}$, where $\widetilde{g}_{l}$ and $g_{l,s_{i}}$ represent Riemannian metrics associated with Kähler forms $\widetilde{\omega}_{l}$ and $\omega_{l,s_{i}}$. By a standard computations (c.f. Proposition 4.3 [CY]), we have

[TABLE]

By the Laplace estimate of $u_{l,s_{i}}$ and generalized maximum principle, we get $T\leq C$. Thus, by taking a subsequence if necessary, $u_{l,s_{i}}$ $C^{2,\lambda}$-converge to a solution with $s=\bar{s}$. This implies $S$ is closed.

Next we prove the uniqueness of equation (3.1). Suppose that $u_{l,1}$ and $u_{l,2}$ are solutions to (3.1). Set $\omega_{2}=\widetilde{\omega_{l}}+\sqrt{-1}\partial\overline{\partial}u_{l,2}$, then we have

[TABLE]

Since $(M,\omega_{2})$ is a complete Kähler manifold with bounded geometry (c.f. Proposition 1.4 [CY]), applying the generalized maximum principle, there exists a sequence $\{x_{i}\}$ such that $\lim_{i\rightarrow\infty}(u_{l,1}-u_{l,2})(x_{i})=\sup_{M}(u_{l,1}-u_{l,2})$ and $\overline{\lim}_{i\rightarrow\infty}{\rm Hess}(u_{l,1}-u_{l,2})(x_{i})\leq 0$. Furthermore, we obtain $u_{l,1}\leq u_{l,2}$. By the same argument, we have $u_{l,1}\geq u_{l,2}$. Therefore, the equation (3.1) has only one solution. Finally, the cusp continuity equation is solvable for all $t\in[0,\infty)$ i.e., $l\in[0,1)$.

4. Convergence of cusp continuity equation

In this section we investigate the regular properties of limit metric.

Lemma 4.1**.**

Let $F$ be a divisor on $\overline{M}$. If $F$ is nef and big, then there is an effective divisor $E=\sum_{i}a_{i}E_{i}$ such that $F-\epsilon E>0$ for all sufficiently small $\epsilon>0$.

By the assumption that $K_{\overline{M}}+D$ is big and semi-ample, there exists an effective divisor $E=\sum_{i}a_{i}E_{i}$ such that $K_{\overline{M}}+D-\epsilon E>0$ for all sufficiently small $\epsilon>0$ according to Lemma 4.1. Thus we choose a volume form $\Omega$, a hermitian metric $h^{\prime}_{D}$ on $L_{D}$ and hermitian metrics $h_{E_{i}}$ such that

[TABLE]

where $\Theta^{\prime}_{D}$ and $\Theta_{E_{i}}$ represent curvature forms of line bundles $L_{D}$ and $L_{E_{i}}$ associated with metrics $h^{\prime}_{D}$ and $h_{E_{i}}$ respectively. $s_{D}$ and $s_{E_{i}}$ are denoted by the defining sections of $L_{D}$ and $L_{E_{i}}$. For simplicity, we write $\log|s_{E}|^{2}=\sum_{i}a_{i}\log|s_{E_{i}}|^{2}$. By taking appropriate $\Omega$, $h^{\prime}_{D}$ and $h_{E_{i}}$, we can assume that

[TABLE]

and

[TABLE]

for $l\in[\frac{1}{2},1]$. Let $\widetilde{\omega_{l}}:=(1-l)\omega_{0}+l(-{\rm Ric}(\Omega)+\Theta_{D})-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|^{2}_{h_{D}}$ (may not be a metric), where the hermitian metric $h_{D}$ is defined as $\omega(0)=\omega_{0}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|^{2}_{h_{D}}>0$ and $\Theta_{D}$ is the curvature form of $h_{D}$. Then the equation (1.2) is written as

[TABLE]

This equation is also equivalent to

[TABLE]

where $w_{l}=u_{l}-l\epsilon\log|s_{E}|^{2}+l\log\frac{|s_{D}|^{2}_{h^{\prime}_{D}}}{|s_{D}|^{2}_{h_{D}}}+\log\frac{\log^{2}|s_{D}|^{2}_{h^{\prime}_{D}}}{\log^{2}|s_{D}|^{2}_{h_{D}}}$ and \Omega_{l}=\Omega\Big{(}\frac{\log^{2}|s_{D}|^{2}_{h_{D}}}{\log^{2}|s_{D}|^{2}_{h^{\prime}_{D}}}\Big{)}^{\frac{1-l}{l}}.

Lemma 4.4**.**

There exists a constant $C$ independent of $l$ such that $-C\leq w_{l}\leq C-l\epsilon\log|s_{E}|^{2}$.

Proof.

For the lower bound, we note that $\widetilde{\omega_{l,E}}=(1-l)\omega_{0}+l(-{\rm Ric}(\Omega)+\Theta^{\prime}_{D}-\sum_{i}\epsilon a_{i}\Theta_{E_{i}})-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|^{2}_{h^{\prime}_{D}}$ is a complete Kähler metric with bounded geometry on $M$. Applying the generalized maximum principle to (4.3), we get $w_{l}\geq-C-l\epsilon\log|s_{E}|^{2}\geq-C$.

For the upper bound, we differentiate $l$ at both side of equation (4.2), then

[TABLE]

where $\omega_{l}=\widetilde{\omega_{l}}+\sqrt{-1}\partial\overline{\partial}u_{l}$.

By the simple calculation, we get

[TABLE]

According to the generalized maximum principle, $(\frac{u_{l}}{l}-n\log l)$ decrease when $l$ tends to $1$. Therefore, there exists a constant $C$ such that $u_{l}\leq C$. By the definition of $w_{l}$, we see $w_{l}\leq C-l\epsilon\log|s_{E}|^{2}$. ∎

Lemma 4.5**.**

There exist two constants $C$ and $a$ independent of $l$ such that $C^{-1}|s_{E}|^{2l\epsilon(a+1)}\widetilde{\omega_{l,E}}\leq\omega_{l}:=\widetilde{\omega_{l}}+\sqrt{-1}\partial\overline{\partial}u_{l}\leq C|s_{E}|^{-2l\epsilon(a+1)(n-1)}\widetilde{\omega_{l,E}}$.

Proof.

Since ${\rm Ric}(\omega_{l})\geq-\frac{1}{l}\omega_{l}$, by Yau’s Schwarz Lemma [Y], we have

[TABLE]

where $a$ is a positive upper bound of the holomorphic bisectional curvature of $\widetilde{\omega_{l,E}}$ for $l\in[0,1]$. Put $H=\log tr_{\omega_{l}}\widetilde{\omega_{l,E}}-(a+1)w_{l}$, then we get

[TABLE]

By the generalized maximum principle, there exists a sequence $\{x_{i}\}$ such that $\lim_{i\rightarrow\infty}H(x_{i})=\sup_{M}H$ and $\overline{\lim}_{i\rightarrow\infty}\triangle_{\omega_{l}}H(x_{i})\leq 0$. Thus by the Lemma 4.4 we have $H\leq C$. This implies

[TABLE]

Note that

[TABLE]

Hence this Lemma is proved. ∎

According to Lemma 4.5, we know that for any compact subset $K\subset\overline{M}\setminus(D\cup{\rm Supp}E)$, there exists a constant $C_{K}>0$ independent of $l$ such that $C^{-1}_{K}\omega_{0}\leq\omega_{l}\leq C_{K}\omega_{0}$, i.e., $|\triangle_{\omega_{0}}u_{l}|\leq C_{K}$. By Theorem 17.14 in [GT], we have $|u_{l}|_{C^{2,\lambda}}\leq C^{\prime}_{K}$ on $K\times[\frac{1}{2},1]$. Furthermore, by the standard bootstrapping argument, for any $m>0$, $|u_{l}|_{C^{m,\lambda}}\leq C_{K,m}$ on $K\times[\frac{1}{2},1]$. By the standard diagonal argument and passing to a subsequence $\{l_{i}\}$ such that $u_{l_{i}}$ $C^{\infty}$-converge to a function on each compact $K$ when $l_{i}$ tends to $1$. The monotonicity of $(\frac{u_{l}}{l}-n\log l)$ implies that $u_{l}$ $C^{\infty}$-converge to a function on each compact $K$ when $l$ tends to $1$. Therefore, the Theorem 1.4 is proved.

5. Algebraic structure of the limit space

5.1. Gromov-Hausdorff convergence: global convergence

In this subsection we consider a family of manifolds $(M,\omega_{l})$ on which the lower bound of Ricci curvature can be controlled, i.e. $Ric(\omega_{l})\geq-\frac{1}{l}\omega_{l}$ for $l\in[\frac{1}{2},1)$. By Gromov precompactness theorem [Cheeger], passing to a subsequence $l_{i}\rightarrow 1$ and fix $x_{0}\in\overline{M}_{reg}\setminus D$, we may assume that

[TABLE]

The limit $(M_{1},d_{1})$ is a complete length metric space. It has a regular/singular decomposition $M_{1}=\mathcal{R}\cup\mathcal{S}$, a point $x\in\mathcal{R}$ iff the tangent cone at $x$ is the Euclidean space $\mathbb{R}^{2n}$. The following lemma is the same as Lemma 3.3 in [NTZ].

Lemma 5.1**.**

There is a sufficiently small constant $\delta>0$ such that for any $l\in[\frac{1}{2},1)$, if a metric ball $B_{\omega_{l}}(x,r)$ satisfies

[TABLE]

where ${\rm Vol}(B^{0}_{r})$ is the volume of a metric ball of radius $r$ in $2n$-Euclidean space, then

[TABLE]

Lemma 5.2**.**

The regular set $\mathcal{R}$ is open in the limit space $(M_{T},d_{T},x_{T})$.

Proof.

If $x\in\mathcal{R}$, then by Colding’s volume convergence theorem [Co] , there exists $r=r(x)>0$ such that $\mathcal{H}^{2n}(B_{d_{1}}(x,r))\geq(1-\frac{\delta}{2}){\rm Vol}(B^{0}_{r})$, where $\mathcal{H}^{2n}$ denotes the Hausdorff measure. Suppose $x_{i}\in M$ satisfying $x_{i}\xrightarrow{d_{GH}}x$, then by the volume convergence theorem again, ${\rm Vol}_{\omega_{l_{i}}}(B_{\omega_{l_{i}}}(x_{i},r))\geq(1-\delta){\rm Vol}(B^{0}_{r})$ for sufficiently large $i$. According to Lemma 5.1 and Anderson’s harmonic radius estimate [An], there is a constant $\delta^{\prime}=\delta^{\prime}(\alpha)$ for any $0<\alpha<1$ such that the $C^{1,\alpha}$ harmonic radius at $x_{i}$ is bigger than $\delta^{\prime}\delta r$. Passing to the limit, it gives a harmonic coordinate on $B_{d_{1}}(x,\delta^{\prime}\delta r)$. This implies $B_{d_{1}}(x,\delta^{\prime}\delta r)\subset\mathcal{R}$. So $\mathcal{R}$ is open with a $C^{1,\alpha}$ Kähler metric $\overline{\omega_{1}}$; moreover $\omega_{l_{i}}$ converges to $\overline{\omega_{1}}$ in $C^{1,\alpha}$ topology on $\mathcal{R}$. ∎

Since $\mathcal{R}$ is dense in $M_{1}$, so we have the following Lemma.

Lemma 5.3**.**

$(M_{1},d_{1})=\overline{(\mathcal{R},\overline{\omega_{1}}})$, the metric completion of $(\mathcal{R},\overline{\omega_{1}})$.

Lemma 5.4**.**

$\mathcal{R}$* is geodesically convex in $M_{1}$ in the sense that any minimal geodesic with endpoints in $\mathcal{R}$ lies in $\mathcal{R}$.*

Proof.

It is simply a consequence of Colding-Naber’s Hölder continuity of tangent cones along a geodesic in $M_{1}$ [CN]. Actually, if $x,y\in\mathcal{R}$, then for any minimal geodesic connecting $x$ and $y$, a neighborhood of endpoints lies in $\mathcal{R}$, so the geodesic will never touch the singular set. ∎

Let $\overline{D}$ be any divisor in $\overline{M}$ such that $D\cup\mathcal{S}_{\overline{M}}\subset\overline{D}$. Define the Gromov-Hausdorff limit of $\overline{D}$

[TABLE]

Proposition 5.5**.**

$(M_{1},d_{1})$* is isometric to $\overline{(\overline{M}\backslash\overline{D},\omega_{1})}$, where $\omega_{1}$ is defined as Theorem 1.4.*

Proof.

First, we prove the following Claim.

Claim 5.6*.*

$\overline{D_{1}}\backslash\mathcal{S}$ is a subvariety of dimension $(n-1)$ if it is not empty.

Proof.

Let $x\in\overline{D_{1}}\backslash\mathcal{S}$ and $x_{i}\in\overline{D}$ such that $x_{i}\xrightarrow{d_{GH}}x$. By the $C^{1,\alpha}$ convergence of $\omega_{l_{i}}$ around $x$, there are $C,r>0$ independent of $i$ and a sequence of harmonic coordinates in $B_{\omega_{l_{i}}}(x_{i},r)$ such that $C^{-1}\omega_{E}\leq\omega_{l_{i}}\leq C\omega_{E}$ where $\omega_{E}$ is the Euclidean metric in the coordinates. Furthermore, according to Lemma 3.11 [TZ2], any $x_{i}\in M$ converging to $x$ has a holomorphic coordinate $(z_{i}^{1},z_{i}^{2},\cdot\cdot\cdot,z_{i}^{n})$ on $B_{\omega_{l_{i}}}(x_{i},r)$ such that $C^{-1}\omega_{E}(\frac{\partial}{\partial z_{i}^{k}},\frac{\partial}{\partial\bar{z}_{i}^{l}})\leq\omega_{l_{i}}(\frac{\partial}{\partial z_{i}^{k}},\frac{\partial}{\partial\bar{z}_{i}^{l}})\leq C\omega_{E}(\frac{\partial}{\partial z_{i}^{k}},\frac{\partial}{\partial\bar{z}_{i}^{l}})$. Since the total volume of $\overline{D}$ is uniformly bounded for any $\omega_{l_{i}}$, the local analytic $\overline{D}\cap B_{\omega_{l_{i}}}(x_{i},r)$ have a uniform bound of degree and so converge to an analytic set $\overline{D_{1}}\cap B_{d_{1}}(x,r)$. ∎

From the above Claim we know that $dim_{\mathbb{R}}(\overline{D_{1}})=dim_{\mathbb{R}}(\mathcal{S}\cup(\overline{D_{1}}\backslash\mathcal{S}))\leq 2n-2$. By the argument of [RZ], $(M_{1}\backslash\overline{D_{1}},\overline{\omega_{1}})$ homeomorphic and locally isometric to $(\overline{M}\backslash\overline{D},\omega_{1})$. Since $M_{1}$ is a length space and $dim_{\mathbb{R}}(\overline{D_{1}})\leq 2n-2$, $(M_{1}\backslash\overline{D_{1}},\overline{\omega_{1}})$ isometric to $(\overline{M}\backslash\overline{D},\omega_{1})$. Furthermore, we have

[TABLE]

∎

A direct corollary is

Corollary 5.7**.**

$(M,\omega_{l},x_{0})$* converges globally to $(M_{1},d_{1},x_{1})$ in the Gromov-Hausdorff topology as $l\rightarrow l$.*

Corollary 5.8**.**

Let $M_{reg}$=$\overline{M}_{reg}\setminus D$, then $\omega_{1}$ is smooth on $M_{reg}$. $(M_{1},d_{1})$ is isometric to $\overline{(M_{reg},\omega_{1})}$.

Proof.

Note that $M_{reg}\backslash(\overline{M}\backslash\overline{D})=M_{reg}\cap\overline{D}$ has real codimension larger than $2$ in $(M_{reg},\omega_{1})$. So $\overline{M}\backslash\overline{D}$ is dense in $M_{reg}$. We conclude

[TABLE]

∎

Proposition 5.9**.**

$M_{reg}=\mathcal{R}$, the regular set of $M_{1}$.

Proof.

Since $M_{reg}$ has smooth structure in $M_{1}$, we have $M_{reg}\subset\mathcal{R}$. Next we show the converse. We argue by contradiction. Suppose $p\in\mathcal{R}\setminus M_{reg}$, then there exists a family of points $p_{l}\in M_{sing}$ such that $p_{l}\xrightarrow{d_{GH}}p$, where $M_{sing}=(\Phi^{-1}(\Phi(\overline{M})_{sing}))\backslash D$. By $C^{1,\alpha}$ convergence on $\mathcal{R}$, there exist $C,r>0$ independent of $l$ and a sequence of harmonic coordinates on $B_{\omega_{l}}(p_{l},r)$ such that $C^{-1}\omega_{E}\leq\omega_{l}\leq C\omega_{E}$ where $\omega_{E}$ is the Euclidean metric in this coordinate. Furthermore, the sequence of harmonic coordinate can be perturbed to a holomorphic coordinate on $B_{\omega_{l}}(p_{l},r)$ [TZ2]. Denote $m=dim_{\mathbb{C}}(M_{sing})$. Then

[TABLE]

which has a uniform lower bound. However, this contradicts with the degeneration of the limit metric $\eta_{1}$ along $M_{sing}$:

[TABLE]

which tends to [math] as $l\rightarrow 1$, where the last equality bases on a Lemma ([Ko84] P410). So we have $M_{reg}\supset\mathcal{R}$. ∎

5.2. $L^{\infty}$ estimate and gradient estimate to holomorphic sections

In this subsection we obtain the $L^{\infty}$ estimate and gradient estimate to holomorphic section $s\in H^{0}(\mathcal{R},k(K_{\overline{M}}+D))$. $h=\omega_{1}^{-nk}$ is chosen as the Hermitian metric of line bundle $k(K_{\overline{M}}+D)$, where $k\in\mathbb{Z}$. The curvature form $\Theta_{h}$ of Hermitian metric $h=\omega_{1}^{-nk}$ is $k\omega_{1}$. By Lemma 2.4, we have the following formulas.

Lemma 5.10**.**

For $s\in H^{0}(\mathcal{R},k(K_{\overline{M}}+D))$, there exists a constant $C$ such that

[TABLE]

and

[TABLE]

Proof.

Since on $\mathcal{R}$, ${\rm Ric}(\omega_{1})=-\omega_{1}$. So these formulas are directly derived from Lemma 2.4. ∎

In order to applying Moser iteration, the Sobolev inequality on $\mathcal{R}$ is needed. The following two Lemmas are due to Song (Lemma 3.7 and 4.6 [So]).

Lemma 5.11**.**

There is a family of cut-off functions $\rho_{\epsilon}\in C_{0}^{\infty}(\mathcal{R})$ with $0<\rho_{\epsilon}<1$ such that $\rho^{-1}_{\epsilon}(1)$ forms an exhaustion of $\mathcal{R}$ and

[TABLE]

Lemma 5.12**.**

Fix any $0<r<R$, the Sobolev constant on $B_{\omega_{l}}(x,r)$ is uniformly bounded by a constant $C_{S}$ depending on upper bound of $R$, $R^{-1}$ and $(R-r)^{-1}$. More precisely, for any $l\in[\frac{1}{2},1)$ and $f\in C_{0}^{1}(B_{\omega_{l}}(x,r))$,

[TABLE]

Fix $0<r<R$ such that $B_{\omega_{1}}(x,r)\subset B_{\omega_{1}}(x,2r)\subset B_{\omega_{1}}(x,R)$.

Lemma 5.13**.**

If $f\in C_{0}^{1}(B_{\omega_{1}}(x,r)\cap\mathcal{R})$, then there exists a constant $C$ depending on $R$, $R^{-1}$ and $(R-r)^{-1}$ such that

[TABLE]

Proof.

Let $f_{\epsilon}=\rho_{\epsilon}f$, where $\rho_{\epsilon}$ is constructed as Lemma 5.11 and $\Omega_{\epsilon}={\rm Supp}f_{\epsilon}$. Then $\omega_{l}$ uniformly converge to $\omega_{1}$ on $\Omega_{\epsilon}$ as $l$ tends to $1$ for a fixed $\epsilon$. Therefore $\Omega_{\epsilon}\subset B_{\omega_{l}}(x,r)$ for $l$ sufficiently close to $1$. By Lemma 5.12, we have

[TABLE]

Let $l\rightarrow 1$, the above inequality gives

[TABLE]

Note that by letting $\epsilon\rightarrow 0$, we get

[TABLE]

and

[TABLE]

By some calculations we have

[TABLE]

which tends to [math]. So this Lemma is proved. ∎

Lemma 5.14**.**

There exists a constant $C$ independent of $k$ such that if $s\in H^{0}(\mathcal{R},k(K_{\overline{M}}+D))$, then

[TABLE]

and

[TABLE]

Proof.

Let $\vartheta\in C_{0}^{\infty}(B_{\omega_{1}}(x,\frac{15}{8}r)\cap\mathcal{R})$ be any cut-off function such that $0\leq\vartheta\leq 1$, $|\nabla\vartheta|\leq 10r^{-2}$ and $\vartheta=1$ on $B_{\omega_{1}}(x,\frac{7}{4}r)\cap\mathcal{R}$, then by Bochner formula we have

[TABLE]

Note that

[TABLE]

Therefore,

[TABLE]

For the second inequality, also by the Bochner formula

[TABLE]

Note that

[TABLE]

and

[TABLE]

Summing up these estimates we have

[TABLE]

Applying the first inequality we obtain the second estimate. ∎

Proposition 5.15**.**

There exists a constant $C(R,r)$ independent of $k$ such that if $s\in H^{0}(\mathcal{R},k(K_{\overline{M}}+D))$, then

[TABLE]

and

[TABLE]

Proof.

Choose a cut-off function $\vartheta\in C_{0}^{\infty}(B_{\omega_{1}}(x,2r)\cap\mathcal{R})$. Then for any $p\geq\frac{n}{n-1}$, by Lemma 5.10, we have

[TABLE]

By Cauchy-Schwarz inequality,

[TABLE]

Therefore

[TABLE]

By Lemma 5.13

[TABLE]

Put $p_{j}=\nu^{j+1}$ for $j\geq 0$, where $\nu=\frac{n}{n-1}$. Define a family of radius inductively by $r_{0}=\frac{3}{2}r$ and $r_{j}=r_{j-1}-2^{-j-1}r$. $B_{j}$ is denoted by $B_{\omega_{1}}(x,r_{j})\cap\mathcal{R}$. We choose a family of cut-off functions $\vartheta_{j}\in C^{\infty}_{0}(B_{j})$ such that

[TABLE]

Thus (5.16) gives, by setting $\vartheta=\vartheta_{j}$

[TABLE]

By the iteration argument, we see

[TABLE]

According to Lemma 5.14, we get the first estimate.

Next we prove the second inequality. Let $\vartheta$ and $p$ as above. By Lemma 5.10, we have

[TABLE]

The term $\int_{\mathcal{R}}k\nabla_{j}(\omega_{1})_{i\bar{j}}\langle s,\nabla_{\bar{i}}\bar{s}\rangle\cdot\vartheta^{2}|\nabla s|^{2(p-1)}\omega_{1}^{n}$ can be estimate by integration by parts as follows

[TABLE]

Note that

[TABLE]

Summing up these estimates we conclude

[TABLE]

Applying the Lemma 5.13

[TABLE]

Put $p_{j}=\nu^{j+1}$ for $j\geq 0$, where $\nu=\frac{n}{n-1}$. Define a family of radius inductively by $r_{0}=\frac{3}{2}r$ and $r_{j}=r_{j-1}-2^{-j-1}r$. $B_{j}$ is denoted by $B_{\omega_{1}}(x,r_{j})\cap\mathcal{R}$. We choose a family of cut-off functions $\vartheta_{j}\in C^{\infty}_{0}(B_{j})$ such that

[TABLE]

By setting $\vartheta=\vartheta_{j}$, the above inequality gives

[TABLE]

Case 1: If \Big{(}\int_{B_{j}}|\nabla s|^{2p_{j}}\omega_{1}^{n}\Big{)}^{\frac{1}{p_{j}}}\geq k\Big{(}\int_{B_{j}}|s|^{2p_{j}}\omega_{1}^{n}\Big{)}^{\frac{1}{p_{j}}} for all $j\geq 0$. Then

[TABLE]

Then (5.17) gives

[TABLE]

By iteration argument we get

[TABLE]

By Lemma 5.13 and a cut-off argument, we have

[TABLE]

According to Lemma 5.14 we get

[TABLE]

Case2: There exists $j_{0}$ such that \Big{(}\int_{B_{j}}|\nabla s|^{2p_{j}}\omega_{1}^{n}\Big{)}^{\frac{1}{p_{j}}}\geq k\Big{(}\int_{B_{j}}|s|^{2p_{j}}\omega_{1}^{n}\Big{)}^{\frac{1}{p_{j}}} for all $j>j_{0}$, but

[TABLE]

Then

[TABLE]

By the iteration argument and (5.17), we have

[TABLE]

The supermum of $|\nabla s|$ follows from

[TABLE]

Case 3: If \Big{(}\int_{B_{j}}|\nabla s|^{2p_{j}}\omega_{1}^{n}\Big{)}^{\frac{1}{p_{j}}}\leq k\Big{(}\int_{B_{j}}|s|^{2p_{j}}\omega_{1}^{n}\Big{)}^{\frac{1}{p_{j}}} for infinite $i$, then

[TABLE]

∎

5.3. $L^{2}$ estimate

In order to construct global holomorphic section on line bundle $k(K_{\overline{M}}+D)$, we need the following version of $L^{2}$-estimate due to Demailly (Theorem 5.1 [De]).

Theorem 5.18**.**

Let $(M,\omega)$ be a $n$-dimensional complete Kähler manifold and $L$ be a holomorphic line bundle over $M$ equipped with a smooth hermitian metric such that $\Theta_{h}\geq\delta\omega$. Then for every $L$-value $(n,1)$ form $\tau$ satisfying

[TABLE]

there exists a $L$-valued $(n,0)$ form $u$ such that $\bar{\partial}u=\tau$ and

[TABLE]

For the singular hermitian metric $h$ on $L$, by the approximation argument, we have

Corollary 5.19**.**

Let $(M,\omega)$ be a $n$-dimensional complete Kähler manifold and $L$ be a holomorphic line bundle over $M$ equipped with a singular hermitian metric such that $\Theta_{h}\geq\delta\omega$ in the current sense. Then for every $L$-value $(n,1)$ form $\tau$ satisfying

[TABLE]

there exists a $L$-valued $(n,0)$ form $u$ such that $\bar{\partial}u=\tau$ and

[TABLE]

Proposition 5.20**.**

$(\mathcal{R}=M_{reg},k\omega_{1})$* is a Kähler manifold (not complete). $k(K_{\overline{M}}+D)$ is a holomorphic line bundle over $\mathcal{R}$. Choosing a hermitian metric $h=\omega_{1}^{-nk}$, then the curvature form $\Theta_{h}=k\omega_{1}$. For any smooth $k(K_{\overline{M}}+D)$-valued $(0,1)$ form $\tau$ satisfying*

[TABLE]

there exists a $k(K_{\overline{M}}+D)$-valued section $\varsigma$ such that $\bar{\partial}\varsigma=\tau$ and

[TABLE]

Proof.

Since $K_{\overline{M}}+D$ is big and semi-ample over $\overline{M}$, by Lemma 4.1, there exists an effective divisor $E$ on $\overline{M}$ such that $K_{\overline{M}}+D-\epsilon E$ is ample for all sufficiently small $\epsilon>0$. Let $s_{E}$ be the defining section of $E$ and $h_{E}$ be a smooth hermitian metric satisfying $\eta_{1}-\epsilon\Theta_{E}>0$, where $\eta_{1}$ is constructed as section 3 and $\Theta_{E}$ is the curvature form. We consider the following Monge-Ampère equation

[TABLE]

Fixed a small $\alpha>0$, this equation is rewritten as

[TABLE]

By the same argument of subsection 5.5, we know that $\omega_{1,\epsilon}=\eta_{1}-\epsilon\Theta_{E}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|^{2}+\sqrt{-1}\partial\overline{\partial}u_{1,\epsilon}$ $C_{loc}^{\infty}(M_{reg})$-converge to $\omega_{1}$ as $\epsilon$ tends to [math]. Now we define a family of hermitian metric

[TABLE]

By a direct calculation, ${\rm Ric}(h_{\epsilon})\geq k\omega_{1,\epsilon}$ in the current sense. $\tau$ has compact support and

[TABLE]

So by the above corollary, there exists $\varsigma_{\epsilon}$ on $M$ such that

[TABLE]

for each $\epsilon$. This also implies

[TABLE]

Hence we can take a subsequence of $\varsigma_{\epsilon}$ converging weakly in $L^{2}(M,(k\omega_{1})^{n})$ to $\varsigma$ and

[TABLE]

on $M$. The proof is complete after pushing $\varsigma$ to $M_{reg}$. ∎

5.4. local separation of points

Recall that $\Phi:\overline{M}\rightarrow\Phi(\overline{M})$ is defined as in section 3. Naturally, $\Phi$ induce a map $\Phi:\mathcal{R}\rightarrow\Phi(\overline{M}\backslash D)$. If $s\in H^{0}(\mathcal{R},k(K_{\overline{M}}+D))$, then by Proposition 5.15, we know that $s$ is local bounded and local Lipschitz. So $s$ can be continuous extended to the limit space $M_{1}$. Furthermore, the map $\Phi_{1}:(\mathcal{R},\omega_{1})\rightarrow(\Phi(\overline{M}\backslash D),\omega_{FS})$ defined by $\Phi$ can be continuously extend to $\Phi_{1}:(M_{1},d_{1})\rightarrow(\Phi(\overline{M}\backslash D),\omega_{FS})$. This subsection is devoted to demonstrate that this map is injective. First we recall some notations and results which originate from [DS].

Definition 5.21**.**

We consider the following data $(p_{*},O,U,J,g,L,h,A)$ satisfying

1. (1)

$(p_{*},O,U,J,g)$* is an open Kähler manifold with a complex structure $J$, a Riemannian metric $g$ and a base point $p_{*}\in O\subset\subset U$ for an open set $O$.* 2. (2)

$L\rightarrow U$* is a hermitian line bundle equipped with a hermitian metric $h$ and $A$ is the connection induced by the hermitian metric $h$ on $L$, with its curvature $\Theta(A)=\omega$ which is a Kähler form of $g$.*

The data $(p_{*},O,U,J,g,L,h,A)$ is said to satisfy the $H$-condition if there exist a constant $C$ and a compactly supported smooth section $\sigma:U\rightarrow L$ such that

1. (1)

$H_{1}:||\sigma||_{L^{2}(U)}\leq(2\pi)^{\frac{n}{2}}$, 2. (2)

$H_{2}:|\sigma(p_{*})|>\frac{3}{4}$, 3. (3)

$H_{3}$: for any smooth section $\tau$ of $L$ over $O$, we have

[TABLE] 4. (4)

H_{4}:||\bar{\partial}\sigma||_{L^{2}(U)}<\min\big{(}\frac{1}{8\sqrt{2}C},10^{-20}\big{)}, 5. (5)

$H_{5}:||\bar{\partial}\sigma||_{L^{2n+1}(O)}<\frac{1}{8C}$.

Fix any point $p$ in $M_{1}$, $(M_{1},p,kd_{1})$ converges in pointed Gromov-Hausdorff topology to a tangent cone $C(Y)$ over the cross section when $k\rightarrow\infty$. We still use $p$ for the vertex of $C(Y)$. Let $Y_{reg}$ and $Y_{sing}$ be the regular part and singular part of $Y$ respectively. By [Cheeger], $Y_{sing}$ has Hausdorff dimension equal or less than $2n-2$. $C(Y_{reg})\backslash\{p\}$ has a natural complex structure induced from the Gromov-Hausdorff limit and the cone metric $g_{C}$ on $C(Y)$ is given by

[TABLE]

where $r$ is the distance function for any point $z\in C(Y)\backslash p$. According to Proposition 5.5, the singular set $\mathcal{S}$ of $M_{1}$ must be a locally analytic set by taking the limit of a divisor on $\overline{M}$. So we also have the following cut-off function on $Y$.

Proposition 5.22**.**

For any $\epsilon>0$, there exists a cut-off function $\gamma$ on $Y$ such that

1. (1)

$\gamma\in C^{\infty}(Y_{reg})$* and $0\leq\gamma\leq 1$,* 2. (2)

$\gamma$* is supported in the $\epsilon$ neighborhood of $Y_{sing}$,* 3. (3)

$\gamma=1$* on a neighborhood of $Y_{sing}$,* 4. (4)

$||\nabla\gamma||_{L^{2}(Y,g_{C})}<\epsilon$.

We consider the trivial line bundle $L_{C}$ on $C(Y)$ equipped the hermitian metric $h_{C}=e^{-|z|^{2}}$, where $|z|^{2}=r^{2}$. Then the curvature coincides with $\omega_{C}$. $A_{C}$ is denoted by the connection of $L_{C}$ with metric $h_{C}$.

Lemma 5.23**.**

Let $p_{*}\in C(Y_{reg})\backslash\{p\}$ such that $\frac{3}{4}<e^{-|p_{*}|^{2}}$. Then there exists $U\subset\subset C(Y_{reg})\backslash\{p\}$ and an open neighborhood $O\subset\subset U$ of $p_{*}$ such that $(p_{*},O,U,J_{C},g_{C},L_{C},h_{C},A_{C})$ satisfies the $H$-condition.

From the construction in [DS], $U$ is a product in $C(Y_{reg})\backslash\{p\}$ i.e., there exists $U_{Y}\subset Y_{reg}$ such that $U=\{z=(y,r)\in C(Y)|y\in U_{Y},r\in(r_{U},R_{U})\}$. For $m\in\mathbb{Z}^{+}$ defined as [DS] (P79), we define

[TABLE]

For any integer $t$ and $1\leq t\leq m$, $\mu_{t}:U\rightarrow U(m)$ is defined by $\mu_{t}(z)=t^{-\frac{1}{2}}z$. The following proposition is due to [DS].

Proposition 5.24**.**

Suppose $(p_{*},O,U(m),J_{C},g_{C},L_{C},h_{C},A_{C})$ constructed as in Lemma 5.23 satisfies the $H$-condition. If $(p_{*},O,U(m),J,g,L,h,A)$ satisfies $H$-condition and there exists a small constant $\epsilon>0$ such that

[TABLE]

Then we can find some $1\leq t\leq m$ such that $(p_{*},O,U,\mu^{*}_{t}J,\mu^{*}_{t}(tg),\mu^{*}_{t}(L^{t}),\mu^{*}_{t}(h^{t}),\mu^{*}_{t}(A^{t}))$ satisfies $H$-condition.

Fix any point $p$, we can assume that $(M_{1},k_{v}^{\frac{1}{2}}d_{1},p)$ converge to a tangent cone $C(Y_{p})$ for some sequence $k_{v}$ in pointed Gromov-Hausdorff topology. For an open set $U\subset\subset C(Y_{reg})\backslash\{p\}$, there is an embedding $\chi_{k_{v}}:U\rightarrow\mathcal{R}=M_{reg}$. Note that $d_{1}|_{\mathcal{R}}=\omega_{1}$. The following Lemma follows from the convergence of $(M_{1},k_{v}^{\frac{1}{2}}d_{1},p)$.

Lemma 5.25**.**

There exist $v$ and $\epsilon>0$ such that we can find an embedding $\chi_{k_{v}}$ which satisfies

1. (1)

$\frac{1}{2}|z|\leq k_{v}^{\frac{1}{2}}d_{1}(p,\chi_{k_{v}}(z))\leq 2|z|$, 2. (2)

$||\chi_{k_{v}}^{*}(k_{v}\omega_{1})-\omega_{C}||_{C^{0}(U)}+||\chi_{k_{v}}^{*}(J_{\mathcal{R}})-J_{C}||_{C^{0}(U)}<\epsilon$.

Proposition 5.26**.**

For any two distinct point $p$ and $q$ in $M_{1}$, we have

[TABLE]

Proof.

Step 1: For any two distinct points $p$ and $q$, there exist $r$ and $R$ such that $p,q\in B_{d_{1}}(x_{1},r)\subset B_{d_{1}}(x_{1},2r)\subset B_{d_{1}}(x_{1},R)$. Suppose $C(Y_{p})$ and $C(Y_{q})$ are two tangent cones of $p$ and $q$ after rescaling $(M_{1},d_{1})$ at $p$ by $k_{v_{p}}\rightarrow\infty$ and at $q$ by $k_{v_{q}}\rightarrow\infty$. Then according to Lemma 5.23, we can construct two collection of data $(p_{*},O_{p},U_{p}(m_{p}),J_{p},g_{p},L_{p},h_{p},A_{p})$ and $(q_{*},O_{q},U_{q}(m_{q}),J_{q},g_{q},L_{q},h_{q},A_{q})$ which satisfy the $H$-condition, where $U_{p}(m_{p})\subset C(Y_{p})$ and $U_{q}(m_{q})\subset C(Y_{q})$. In addition, we can always assume that

1. (1)

the constant $C$ appeared in the $H$-condition for $U_{p}(m_{p})$ and $U_{q}(m_{q})$ are the same, 2. (2)

$k_{v_{p}}=k_{v_{q}}=k_{v_{p,q}}$, 3. (3)

$r_{p_{*}}:=d_{C(Y_{p})}(p,p_{*})$ and $r_{q_{*}}:=d_{C(Y_{q})}(q,q_{*})$ are small enough which definite below.

Step 2: From Lemma 5.25, there exist $k_{v_{p,q}}$ such that $\chi_{p,k_{v_{p,q}}}:U_{p}(m_{p})\rightarrow\mathcal{R}$ and $\chi_{q,k_{v_{p,q}}}:U_{q}(m_{q})\rightarrow\mathcal{R}$ satisfy the following:

1. (1)

$\frac{1}{2}|z|\leq k_{v_{p,q}}^{\frac{1}{2}}d_{1}(p,\chi_{p}(z))\leq 2|z|$, 2. (2)

$\frac{1}{2}|z|\leq k_{v_{p,q}}^{\frac{1}{2}}d_{1}(q,\chi_{q}(z))\leq 2|z|$, 3. (3)

$\chi_{p}(U_{p}(m_{p}))\cap\chi_{q}(U_{q}(m_{q}))=\varnothing$, 4. (4)

$||\chi_{p}^{*}(k_{v_{p,q}}\omega_{1})-\omega_{p}||_{C^{0}(U_{p}(m_{p}))}+||\chi_{p}^{*}(J_{\mathcal{R}})-J_{p}||_{C^{0}(U_{p}(m_{p}))}<\epsilon$, 5. (5)

$||\chi_{q}^{*}(k_{v_{p,q}}\omega_{1})-\omega_{q}||_{C^{0}(U_{q}(m_{q}))}+||\chi_{q}^{*}(J_{\mathcal{R}})-J_{q}||_{C^{0}(U_{q}(m_{q}))}<\epsilon$.

where for the convenience, $\chi_{p,k_{v_{p,q}}}$ and $\chi_{q,k_{v_{p,q}}}$ are denoted by $\chi_{p}$ and $\chi_{q}$ respectively.

Step 3: By the Proposition 5.24 and sufficiently small $\epsilon$ in Step 2, there exists $1\leq t_{p}\leq m_{p}$ such that $(p_{*},O_{p},U_{p},\mu_{t_{p}}^{*}\chi_{p}^{*}(J_{\mathcal{R}}),\mu_{t_{p}}^{*}\chi_{p}^{*}(k_{v_{p,q}}\omega_{1}),\mu_{t_{p}}^{*}\chi_{p}^{*}(L_{p}^{t_{p}}),\mu_{t_{p}}^{*}\chi_{p}^{*}(h_{p}^{t_{p}}),\mu_{t_{p}}^{*}\chi_{p}^{*}(A_{p}^{t_{p}}))$ satisfies the $H$-condition. Thus there is a compactly smooth section $\sigma_{p}$ such that $\sigma_{p}$ has properties $H_{1}$, $H_{2}$, $H_{4}$ and $H_{5}$. By the same argument, there exists $1\leq t_{q}\leq m_{q}$ such that $(q_{*},O_{q},U_{q},\mu_{t_{q}}^{*}\chi_{q}^{*}(J_{\mathcal{R}}),\mu_{t_{q}}^{*}\chi_{q}^{*}(k_{v_{p,q}}\omega_{1}),\mu_{t_{q}}^{*}\chi_{q}^{*}(L_{q}^{t_{q}}),$ $\mu_{t_{q}}^{*}\chi_{q}^{*}(h_{q}^{t_{q}}),\mu_{t_{q}}^{*}\chi_{q}^{*}(A_{q}^{t_{q}}))$ satisfies the $H$-condition. Thus there is a compactly smooth section $\sigma_{q}$ such that $\sigma_{q}$ has properties $H_{1}$, $H_{2}$, $H_{4}$ and $H_{5}$.

Step 4: There is an embedding from $(\mu_{t_{p}}^{*}\chi_{p}^{*}(L_{p}^{t_{p}}),U_{p})$ to $(k_{p}(K_{\overline{M}}+D),\mathcal{R})$, where $k_{p}=t_{p}k_{v_{p,q}}$. So $\sigma_{p}$ can be viewed as a compactly smooth section of $k_{p}(K_{\overline{M}}+D)$. We now apply Proposition 5.20 to $\tau_{p}=\bar{\partial}\sigma_{p}$. Then there exists a $(k_{p}(K_{\overline{M}}+D))$ valued section $\varsigma_{p}$ solving the $\bar{\partial}$ equation $\bar{\partial}\varsigma_{p}=\tau_{p}$ with

[TABLE]

Let $z_{p_{*}}=\chi_{p}(p_{*})$, then from $H_{3}$ and $H_{5}$,

[TABLE]

Set $\sigma^{\prime}_{p}=\sigma_{p}-\varsigma_{p}$. Then $\sigma^{\prime}_{p}$ is a holomorphic section of $k_{p}(K_{\overline{M}}+D)$ over $\mathcal{R}$ and from Proposition 5.15, $\sigma^{\prime}_{p}$ can be continuously extended to $M_{1}$. By the $H$-condition, we have the following relations:

1. (1)

$|\sigma^{\prime}_{p}(z_{p_{*}})|>\frac{1}{2}$, 2. (2)

$||\sigma^{\prime}_{p}||_{L^{2}(\mathcal{R},k_{p}\omega_{1},h_{p}^{k_{p}})}\leq 2(2\pi)^{\frac{n}{2}}$, 3. (3)

||\sigma^{\prime}_{p}||_{L^{2}(\mathcal{R}\backslash U_{p},k_{p}\omega_{1},h_{p}^{k_{p}})}=||\varsigma_{p}||_{L^{2}(\mathcal{R}\backslash U_{p},k_{p}\omega_{1},h_{p}^{k_{p}})}\leq\min\Big{(}\frac{1}{8\sqrt{2}C},10^{-20}\Big{)}.

Then by Proposition 5.15,

[TABLE]

when $r_{p_{*}}$ is sufficiently small.

Now we restrict $\sigma^{\prime}_{p}$ on $U_{q}$. By $H_{3}$,

[TABLE]

Similarly,

[TABLE]

when $r_{p_{*}}$ is sufficiently small.

Step 5: By the same argument of Step 4, let $k_{q}=t_{q}k_{v_{p,q}}$, we construct a holomorphic section $\sigma^{\prime}_{q}$ such that

[TABLE]

Step 6: Set $K=t_{q}k_{p}=t_{p}k_{q}$. Then $(\sigma^{\prime}_{p})^{t_{q}}$ and $(\sigma^{\prime}_{q})^{t_{p}}$ are holomorphic section of $K(K_{\overline{M}}+D)$ which can be continuously extended to $M_{1}$. Modifying the constant $10^{-20}$ as small as enough, we have $|(\sigma^{\prime}_{p})^{t_{q}}(p)|>>|(\sigma^{\prime}_{q})^{t_{p}}(p)|$ and $|(\sigma^{\prime}_{q})^{t_{p}}(q)|>>|(\sigma^{\prime}_{p})^{t_{q}}(q)|$. Therefore, we conclude that $\Phi_{1}$ is injective. ∎

5.5. Surjectivity of $\Phi_{1}$

In this subsection we will complete the proof of Theorem 1.8. Let $u_{1}$ be the solution to the following equation in the current sense

[TABLE]

Since $K_{\overline{M}}+D$ is big and semi-ample, there exists an effective divisor $E=\sum_{i}a_{i}E_{i}$ such that $K_{\overline{M}}+D-\epsilon E>0$ for all sufficiently small $\epsilon>0$.

Let $p\in{\rm Supp}E\backslash D$ and $\pi:\widetilde{\overline{M}}\rightarrow\overline{M}$ be the blow up at $p$ with exceptional divisor $\pi^{-1}(p)=F$. Set $\widetilde{D}=\pi^{-1}(D)$ and $\widetilde{E}=\sum_{i}a_{i}\widetilde{E_{i}}$, where $\widetilde{E_{i}}=\overline{\pi^{-1}(E_{i})-F}$. $s_{\widetilde{E_{i}}}$, $s_{F}$ and $s_{\widetilde{D}}$ are denoted by the defining sections of line bundles $L_{\widetilde{E_{i}}}$, $L_{F}$ and $L_{\widetilde{D}}$ respectively. Let $\chi$ be fixed Kähler metric on $\widetilde{\overline{M}}$. We choose appropriate hermitian metrics $h_{\widetilde{E_{i}}}$ and $h_{F}$ such that

[TABLE]

for some small constants $\delta$ and $\mu$. Note that $\widetilde{\Omega}=\frac{\pi^{*}\Omega}{|s_{F}|^{2(n-1)}}$ defines a smooth volume form on $\widetilde{\overline{M}}$. We consider the following Monge-Ampère equation on $\widetilde{\overline{M}}$

[TABLE]

where $\widetilde{\eta_{1}}=\pi^{*}\eta_{1}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{\widetilde{D}}|^{2}_{h_{\widetilde{D}}}$ and $h_{\widetilde{D}}=\pi^{*}h_{D}$. By Theorem 1 of [Ko84], the equation has a unique smooth solution $\varphi_{\epsilon}$ for each $\epsilon$; moreover

[TABLE]

is a smooth complete Kähler metric on $\widetilde{\overline{M}}\backslash\widetilde{D}$.

Lemma 5.28**.**

For any $\delta$ and $\epsilon$, there exist two constants $C(\delta)$ and $C$ independent of $\epsilon$ such that

[TABLE]

Proof.

For the upper bound, let

[TABLE]

so we have $V_{1}\geq V_{\epsilon}\geq V_{0}$. Hence $V_{\epsilon}$ is uniformly bounded. Denote $(\epsilon^{2}+|s_{F}|^{2})^{n-1}\frac{\widetilde{\Omega}}{|s_{\widetilde{D}}|^{2}_{h_{\widetilde{D}}}\log^{2}|s_{\widetilde{D}}|^{2}_{h_{\widetilde{D}}}}$ by $\widetilde{\Omega_{\epsilon}}$, then we have the following calculation

[TABLE]

where the third equality bases on a Lemma ([Ko84] P410). Since $\varphi_{\epsilon}-\log\log^{2}|s_{\widetilde{D}}|^{2}_{h_{\widetilde{D}}}\in PSH(\widetilde{\overline{M}},\pi^{*}\eta+\epsilon\chi)$, the mean inequality implies

[TABLE]

For the lower bound, we set $\varphi_{\epsilon,\delta}=\varphi_{\epsilon}-\delta\log|s_{F}|^{2}-\delta\sum_{i}a_{i}\log|s_{\widetilde{E_{i}}}|^{2}$ and denote $|s_{\widetilde{D}}|^{2}_{h_{\widetilde{D}}^{\delta}}=|s_{\widetilde{D}}|^{2}_{\delta}$, then the equation (5.27) is equivalent to

[TABLE]

where $\eta^{\delta}_{1}=\pi^{*}\eta_{1}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{\widetilde{D}}|^{2}_{\delta}$ satisfying $\eta^{\delta}_{1}+\delta\sqrt{-1}\partial\overline{\partial}\log|s_{F}|^{2}+\delta\sum_{i}a_{i}\sqrt{-1}\partial\overline{\partial}\log|s_{\widetilde{E_{i}}}|^{2}>0$ and $\widetilde{\Omega}^{\prime}=\frac{|s_{\widetilde{D}}|^{2}_{\delta}}{|s_{\widetilde{D}}|^{2}}\widetilde{\Omega}$. We introduce the following equation

[TABLE]

By the generalized maximum principle, there exists a sequence $\{x_{i}\}$ such that $\lim_{i\rightarrow\infty}\psi_{\epsilon,\delta}(x_{i})=\inf\psi_{\epsilon,\delta}$ and $\overline{\lim}_{i\rightarrow\infty}\sqrt{-1}\partial\overline{\partial}\psi_{\epsilon,\delta}(x_{i})\geq 0$. Then we have

[TABLE]

Set $H_{\epsilon,\delta}=\varphi_{\epsilon,\delta}-\psi_{\epsilon,\delta}$ and $v_{\epsilon}^{\delta}=\eta^{\delta}_{1}-\delta\sum_{i}\Theta_{\widetilde{E_{i}}}-\delta\Theta_{F}+\epsilon\chi$, then

[TABLE]

By the generalized maximum principle again

[TABLE]

Note that $\log\frac{\log^{2}|s_{\widetilde{D}}|^{2}_{\delta}}{\log^{2}|s_{\widetilde{D}}|^{2}}$ is a smooth function on $\widetilde{\overline{M}}$, so it can be bounded by $C(\delta)$. Moreover we get the lower bound of $\varphi_{\epsilon}$. ∎

Lemma 5.29**.**

There exists a constant $C$ independent of $\epsilon$ such that on $\widetilde{\overline{M}}\backslash\widetilde{D}$, we have

[TABLE]

Proof.

We observe some following consequences:

1. (1)

$\pi^{*}\eta_{1}\leq C\chi$, 2. (2)

Since $\widetilde{\Omega}$ is a smooth volume form, ${\rm Ric}(\widetilde{\Omega})\leq C\chi$, 3. (3)

$\Theta_{h_{\widetilde{D}}}\geq-C\chi$, 4. (4)

$\sqrt{-1}\partial\overline{\partial}\log(\epsilon^{2}+|s_{F}|^{2})\geq-C\chi$.

Thus by a simple calculation we get the Lemma. ∎

Set $\chi^{\prime}=\chi-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{\widetilde{D}}|^{2}_{{}^{\prime}}$, then by a calculation we have

[TABLE]

Take an appropriate hermitian metric $|\cdot|_{{}^{\prime}}$, we can assume that

[TABLE]

So, by Lemma 5.29, we have

[TABLE]

On the other hand, we can choose a sufficiently large $A^{\prime}$, a sufficiently small $\alpha$ and a hermitian metric $|\cdot|$ such that

[TABLE]

From now on we always assume that the hermitian metric $|\cdot|$ on $L_{\widetilde{D}}$ satisfy $A^{\prime}\pi^{*}\eta_{1}-\Theta_{F}-\sum_{i}a_{i}\Theta_{\widetilde{E_{i}}}-A^{\prime}\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{\widetilde{D}}|^{2}\geq\alpha\chi^{\prime}$.

Lemma 5.30**.**

There exist $C$ and $\lambda$ independent of $\epsilon$ such that

[TABLE]

Proof.

By Yau’s Schwarz Lemma [Y] and Lemma 5.29, we have

[TABLE]

There is a fact that is

[TABLE]

Let $H=\log(|s_{\widetilde{D}}|^{2A}\cdot|s_{F}|^{2A^{2}}\cdot\prod_{i}|s_{\widetilde{E_{i}}}|^{2A^{2}}\cdot tr_{\chi^{\prime}}\widetilde{\omega_{\epsilon}})-A^{2}A^{\prime}\varphi_{\epsilon}$, where $A^{\prime}$ is chosen as above and $A$ is defined below. Then on $\widetilde{\overline{M}}\backslash(\widetilde{D}\cup F\cup{\rm Supp}E)$, we have

[TABLE]

When $A$ is sufficiently large we observed that

[TABLE]

Therefore

[TABLE]

By the generalized maximum principle, there exists a sequence $\{x_{i}\}$ such that $\lim_{i\rightarrow\infty}H(x_{i})=\sup H$ and $\overline{\lim}_{i\rightarrow\infty}\sqrt{-1}\partial\overline{\partial}H(x_{i})\leq 0$. Thus,

[TABLE]

Since

[TABLE]

then we have

[TABLE]

Furthermore,

[TABLE]

If

[TABLE]

then

[TABLE]

Otherwise

[TABLE]

From (5.31) we know

[TABLE]

In general we have

[TABLE]

By the definition of $H$ and Lemma 5.28 we have

[TABLE]

when choosing $A>>A^{\prime}$ and sufficiently small $\delta$. So we get this Lemma from the upper bound of $\varphi_{\epsilon}$. ∎

Let $B$ be a disk in $\overline{M}\backslash D$ centered at $p$. Denote $f_{1},f_{2},\cdot\cdot\cdot,f_{N}$ by the defining functions of divisors $\widetilde{E_{1}},\widetilde{E_{2}},\cdot\cdot\cdot,\widetilde{E_{N}}$ on $\widetilde{B}=\pi^{-1}(B)$. From Lemma 5.30, we obtain the following corollary.

Corollary 5.32**.**

There exist $C$ and $\lambda$ independent of $\epsilon$ such that

[TABLE]

Let $\hat{\chi}$ be the pull back of the Euclidean metric $\sqrt{-1}\sum_{j}dz_{j}\wedge d\bar{z_{j}}$ on $B$. Then $\hat{\chi}$ is a smooth closed nonnegative $(1,1)$ form and is a Kähler metric on $\widetilde{B}\backslash F$. The following Lemma is due to Song [So].

Lemma 5.33**.**

There exist a constant $C>0$, a sufficiently small $\epsilon_{0}>0$ and a smooth hermitian metric $h^{\prime}_{F}$ on $L_{F}$ such that on $\widetilde{B}$

[TABLE]

and

[TABLE]

Lemma 5.34**.**

There exist $0<\beta<1$, $C>0$ and $\Lambda>0$ independent of $\epsilon$ such that

[TABLE]

Moreover, we have

[TABLE]

where $\omega_{1}=\eta_{1}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{D}|^{2}+\sqrt{-1}\partial\overline{\partial}u_{1}$.

Proof.

Let H=\log\Big{(}|s_{F}|^{2(1+r)}_{h^{\prime}_{F}}\cdot\prod_{i}|f_{i}|^{2\lambda^{2}}\cdot tr_{\hat{\chi}}\widetilde{\omega_{\epsilon}}\Big{)}-A\varphi_{\epsilon} for some sufficiently large $A$ and sufficiently small $r$. There are some facts on $\widetilde{B}\backslash(F\cup{\rm Supp}E)$:

1. (1)

$\triangle_{\widetilde{\omega_{\epsilon}}}\log|s_{F}|^{2}_{h^{\prime}_{F}}=-tr_{\widetilde{\omega_{\epsilon}}}\Theta_{h^{\prime}_{F}}$, 2. (2)

$\triangle_{\widetilde{\omega_{\epsilon}}}\log\prod_{i}|f_{i}|^{2\lambda^{2}}=0$, 3. (3)

$\triangle_{\widetilde{\omega_{\epsilon}}}\varphi_{\epsilon}=n-tr_{\widetilde{\omega_{\epsilon}}}(\pi^{*}\eta_{1}-\sqrt{-1}\partial\overline{\partial}\log\log^{2}|s_{\widetilde{D}}|^{2})-\epsilon tr_{\widetilde{\omega_{\epsilon}}}\chi$, 4. (4)

$\triangle_{\widetilde{\omega_{\epsilon}}}\log tr_{\hat{\chi}}\widetilde{\omega_{\epsilon}}\geq-\frac{tr_{\hat{\chi}}({\rm Ric}(\widetilde{\omega_{\epsilon}}))}{tr_{\hat{\chi}}\widetilde{\omega_{\epsilon}}}\geq 1-\frac{C}{|s_{F}|^{2}_{h^{\prime}_{F}}tr_{\hat{\chi}}\widetilde{\omega_{\epsilon}}}$.

Thus, on $\widetilde{B}\backslash(F\cup{\rm Supp}E)$, we have

[TABLE]

where the last inequality bases on Lemma 5.33 and choosing sufficiently large $A$ and small $r$. By Yau’s Schwarz Lemma [Y],

[TABLE]

Let G=H+\frac{c}{2C_{1}}\log\big{(}\prod_{i}|f_{i}|^{2\lambda^{2}+2}\cdot tr_{\chi^{\prime}}\widetilde{\omega_{\epsilon}}\big{)}. Then

[TABLE]

For a fixed sufficiently large $\lambda>0$, there exists a constant $C>0$ such that

[TABLE]

from the estimate in Corollary 5.32 and Lemma 5.28.

So we can assume that

[TABLE]

for some $p_{max}\in\widetilde{B}\backslash(F\cup{\rm Supp}E)$. Then at $p_{max}$, we have

[TABLE]

Note that

[TABLE]

Then according to the boundedness of $\frac{1}{\log^{2}|s_{\widetilde{D}}|^{2}}$ in $\widetilde{B}$, we get

[TABLE]

If $(tr_{\chi^{\prime}}\widetilde{\omega_{\epsilon}})^{\frac{1}{n-1}}(p_{max})\leq\frac{3\bar{C}}{C^{\prime}}$, then $G$ is bounded from above by a uniform constant.

Otherwise $(tr_{\chi^{\prime}}\widetilde{\omega_{\epsilon}})^{\frac{1}{n-1}}(p_{max})\geq\frac{3\bar{C}}{C^{\prime}}$, i.e., $\bar{C}\leq\frac{C^{\prime}}{3}(tr_{\chi^{\prime}}\widetilde{\omega_{\epsilon}})^{\frac{1}{n-1}}(p_{max})$. Then by equation (5.35) we get

[TABLE]

i.e.

[TABLE]

According to the definition of $G$, Lemma 5.28 and Lemma 5.30 we have $G\leq C$ when we choose large $C_{1}$.

In sum, in all cases, we have $G\leq C$. Then

[TABLE]

Note that $tr_{\hat{\chi}}\widetilde{\omega_{\epsilon}}\geq C^{-1}tr_{\chi^{\prime}}\widetilde{\omega_{\epsilon}}$, then we observe

[TABLE]

If we choose $r=\frac{c}{10C_{1}}$, then $1-\beta=\frac{1+r}{1+\frac{c}{2c_{1}}}$ for some $\beta\in(0,1)$. Furthermore, there exists a constant $\Lambda>0$ such that

[TABLE]

∎

From now on we turn to the Gromov-Hausdorff convergence. Recall

[TABLE]

where $\overline{D}$ is a divisor such that $D\cup\mathcal{S}_{\overline{M}}\subset\overline{D}$. By the Proposition 5.5, $(M_{1}\backslash\overline{D_{1}},\overline{\omega_{1}})$ isometric to $(\overline{M}\backslash\overline{D},\omega_{1})$.

Lemma 5.36**.**

$\Phi_{1}:M_{1}\setminus\overline{D_{1}}\rightarrow\Phi(\overline{M}\setminus\overline{D})$* is bijective.*

Proof.

Note that $(M_{1}\setminus\overline{D_{1}})\subset\mathcal{R}=M_{reg}$ and $\Phi|_{M_{reg}}$ is biholomorphic, so $\Phi_{1}$ is bijective. ∎

Lemma 5.37**.**

$\Phi_{1}:\overline{D_{1}}\rightarrow\Phi(\overline{D}\backslash D)$* is surjective.*

Proof.

For any $x^{\prime}\in\overline{D}\backslash D$, there exists a curve $\gamma:[0,1]\rightarrow\overline{M}\backslash D$ with $\gamma(0)=x^{\prime}$ and $\gamma((0,1])\subset\overline{M}\backslash\overline{D}$ such that $\int_{0}^{1}|\dot{\gamma}|_{\omega_{1}}dt<\infty$ by Lemma 5.34. The curve $\gamma(t)$ gives a curve $\bar{\gamma}(t)$ for $0<t\leq 1$ through an isometry from $(\overline{M}\backslash\overline{D},\omega_{1})$ to $(M_{1}\backslash\overline{D_{1}},\overline{\omega_{1}})$. Hence there is a limit $x^{\prime\prime}=\lim_{t\rightarrow 0}\bar{\gamma}(t)$ in $M_{1}$. Then

[TABLE]

Therefore, $\Phi_{1}$ is surjective. ∎