Almost scalar-flat K\"{a}hler metrics on affine algebraic manifolds

TL;DR

This paper constructs complete Kähler metrics with scalar curvature flatness away from a divisor on affine algebraic manifolds, under specific ampleness and ratio conditions of line bundles.

Contribution

It demonstrates the existence of almost scalar-flat Kähler metrics on affine algebraic manifolds using line bundle ampleness and ratio conditions.

Findings

Existence of complete Kähler metrics with flat scalar curvature away from divisors.

Conditions involving very ample line bundles and small ratios ensure metric construction.

Provides new examples of affine algebraic manifolds with special Kähler metrics.

Abstract

Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. In this paper, we show the existence of a complete K\"{a}hler metric on $X \setminus D$ whose scalar curvature is flat away from some divisor if there are positive integers $l(>n),m$ such that the line bundle $K_{X}^{-l} \otimes L_{X}^{m}$ is very ample and the ratio $m/l$ is sufficiently small.

Equations

$$D\in |L_X|.$$

$${\hat{S}}_D:=\frac{{\int }_{D}S({\theta }_D){\theta }_D^{n-1}}{{\int }_D{\theta }_D^{n-1}}=\frac{(n-1)c_1(K_D^{-1})\cup c_1(L_D{)}^{n-2}}{c_1(L_D{)}^{n-1}},$$

$${\hat{S}}_D>0.$$

$${\omega }_0:=\frac{n(n-1)}{{\hat{S}}_D}\sqrt{-1}\partial \overline{\partial }\exp \left(\frac{{\hat{S}}_D}{n(n-1)}t\right)$$

$$\xi :={\sigma }_F\otimes {\sigma }_D^{-m}.$$

$$({\theta }_X+\sqrt{-1}\partial \overline{\partial }\phi {)}^n={\xi }^{-1/l}\wedge {\overline{\xi }}^{-1/l}.$$

$$\left|\frac{{\partial }^2}{\partial z^i\partial {\overline{z}}^j}{\partial }^{\alpha }\phi \right|$$

$$\left|\frac{{\partial }^4}{\partial w_F^2\partial \overline{w_F^2}}\phi \right|$$

$$\left|\frac{{\partial }^4}{\partial w_D^2\partial \overline{w_D^2}}\phi \right|$$

$$\frac{a(n)m}{2l}<\frac{{\hat{S}}_D}{n(n-1)}$$

$$t=\log ||{\sigma }_D|{|}^{-2}$$

$$\Theta (t)=\frac{n(n-1)}{{\hat{S}}_D}\exp \left(\frac{{\hat{S}}_D}{n(n-1)}t\right).$$

$${\omega }_0:=\sqrt{-1}\partial \overline{\partial }\Theta (t)=\frac{n(n-1)}{{\hat{S}}_D}\sqrt{-1}\partial \overline{\partial }\exp \left(\frac{{\hat{S}}_D}{n(n-1)}t\right)$$

$$S({\omega }_0)=O\left(||{\sigma }_D|{|}^{2{\hat{S}}_D/n(n-1)}\right)$$

$$S({\omega }_0)=O\left(||{\sigma }_D|{|}^{2+2{\hat{S}}_D/n(n-1)}\right)$$

$$G_v^{\beta }(b):={\int }_{b_0}^b{\left(\frac{1}{e^{-y}+v}\right)}^{1/\beta }dy$$

$$\sqrt{-1}\partial \overline{\partial }{G}_v^{\beta }(\beta b)$$

$$\frac{e^{-\beta b}}{e^{-\beta b}+v}\sqrt{-1}\partial b\wedge \overline{\partial }b$$

$$S({\gamma }_v^{\beta })=S(\sqrt{-1}\partial \overline{\partial }{G}_v^{\beta }(\beta b))=O((||{\sigma }_F|{|}^{2\beta }+v{)}^{1/\beta })$$

$${\left((\sqrt{-1}\partial \overline{\partial }{G}_v^{\beta }(\beta b)\right)}^n$$

$$\mathrm{Ric}(\sqrt{-1}\partial \overline{\partial }{G}_v^{\beta }(\beta b))$$

$$S(\sqrt{-1}\partial \overline{\partial }{G}_v^{\beta }(\beta b))=O(1).$$

$$\xi:=\sigma_{F}\otimes\sigma_{D}^{-m}.\$$

$$V:={\xi }^{-1/l}\wedge \overline{{\xi }^{-1/l}}.$$

$$V=f{\theta }_X^n$$

$${\int }_{X}V={\int }_{X}f{\theta }_X^n={\int }_X{\theta }_X^n.$$

$$({\theta }_X+\sqrt{-1}\partial \overline{\partial }\phi {)}^n=|s_1{|}^{2k_1}|s_2{|}^{-2k_2}\exp (F){\theta }_X^n$$

$$({\theta }_X+\sqrt{-1}\partial \overline{\partial }\phi {)}^n=f{\theta }_X^n={\xi }^{-1/l}\wedge \overline{{\xi }^{-1/l}}.$$

$${\mathrm{Osc}}_X\phi \le C$$

$$M_{\eta }(t_1,...,t_p)={\int }_{{\mathbb{R}}^p}\max \{t_1+h_1,...,t_p+h_p\}\prod _{1\le j\le p}{\eta }_j^{-1}\rho (h_j/{\eta }_j)d{h}_1...d{h}_p$$

$$\left|\frac{{\partial }^{|\alpha |}{M}_{\eta }}{\partial t^{\alpha }}(t)\right|\le C\min \{{\eta }_j|{\alpha }_j\ne 0\}\prod _{{\alpha }_i\ne 0}{\eta }_i^{-{\alpha }_i}$$

Full text

†† 2010 Mathematics Subject Classification. Primary 53C25; Secondary 32Q15, 53C21. †† Key words and phrases. constant scalar curvature Kähler metrics, complex Monge-Amp$\grave{{\rm e}}$re equations, plurisubharmonic functions, Kähler manifolds.

1. 1 Introduction

2. 2 Plurisubharmonic functions with small scalar curvature

3. 2.1 Kähler potential near $D$

4. 2.2 Kähler potential near $F$

5. 2.3 Ricci-flat Kähler metric away from $D\cup F$

6. 2.4 Gluing plurisubharmonic functions

7. 3 Proof of Theorem 1.1

8. 3.1 The $C^{2}$-estimate

9. 3.2 The $C^{2,\epsilon}$-estimate

10. 3.3 The third and the forth order estimates

11. 4 Proof of Theorem 1.2

1

Introduction

Let $(X,L_{X})$ be a polarized manifold of dimension $n$, i.e., $X$ is an $n$-dimensional compact complex manifold and $L_{X}$ is an ample line bundle over $X$. Assume that there is a smooth hypersurface $D\subset X$ with

[TABLE]

Set an ample line bundle $L_{D}:=\mathscr{O}(D)|_{D}=L_{X}|_{D}$ over $D$. Since $L_{X}$ is ample, there exists a Hermitian metric $h_{X}$ on $L_{X}$ which defines a Kähler metric $\theta_{X}$ on $X$, i.e., the curvature form of $h_{X}$ multiplied by $\sqrt{-1}$ is positive definite. Then, the restriction of $h_{X}$ to $L_{D}$ defines also a Kähler metric $\theta_{D}$ on $D$. Let $\hat{S}_{D}$ be the average of the scalar curvature $S(\theta_{D})$ of $\theta_{D}$ defined by

[TABLE]

where $K_{D}^{-1}$ is the anti-canonical line bundle of $D$. Note that $\hat{S}_{D}$ is a topological invariant in the sense that it is representable in terms of Chern classes of the line bundles $K_{D}^{-1}$ and $L_{D}$. In this paper, we treat the following case :

[TABLE]

Let $\sigma_{D}\in H^{0}(X,L_{X})$ be a defining section of $D$ and set $t:=\log||\sigma_{D}||_{h_{X}}^{-2}$. Following [3], we can define a complete Kähler metric $\omega_{0}$ by

[TABLE]

on the noncompact complex manifold $X\setminus D$. This Kähler metric $\omega_{0}$ is of asymptotically conical geometry (see [1]).

In [1], we show that there exists a complete scalar-flat Kähler metric which is of asymptotically conical geometry if the following conditions hold : (1) $n\geq 3$ and there is no nonzero holomorphic vector field on $X$ vanishing on $D$, (2) $\theta_{D}$ is a cscK metric and $0<\hat{S}_{D}<n(n-1)$, (3) the scalar curvature of $\omega_{0}$ is sufficiently small in the weighted Banach space (see Condition 1.2 and Condition 1.3 in [1]). In this paper, we construct a complete Kähler metric on $X\setminus D$ whose scalar curvature can be made small arbitrarily by gluing plurisubharmonic functions.

To show this, we consider a degenerate (meromorphic) complex Monge-Amp$\grave{{\rm e}}$re equation. Take positive integers $l>n$ and $m$ such that the line bundle $K_{X}^{-l}\otimes L_{X}^{m}$ is very ample. Let $F\in|K_{X}^{-l}\otimes L_{X}^{m}|$ be a smooth hypersurface defined by a holomorphic section $\sigma_{F}\in H^{0}(X,K_{X}^{-l}\otimes L_{X}^{m})$ such that the divisor $D+F$ is simple normal crossing. For a defining section $\sigma_{D}\in H^{0}(X,L_{X})$ of $D$, set

[TABLE]

From the result due to Yau [13, Theorem 7], we can solve the following degenerate complex Monge-Amp$\grave{{\rm e}}$re equation:

[TABLE]

Moreover, it follows from a priori estimate due to Kołodziej [11] that the solution $\varphi$ is bounded on $X$. Thus, we can glue plurisubharmonic functions by using the regularized maximum function. To compute the scalar curvature of the glued Kähler metric, we need to study behaviors of higher order derivatives of the solution $\varphi$. So, we give explicit estimates of them near the intersection $D\cap F$ :

Theorem 1.1**.**

Let $(z^{i})_{i=1}^{n}=(z^{1},z^{2},...,z^{n-2},w_{F},w_{D})$ be local holomorphic coordinates such that $\{w_{F}=0\}=F$ and $\{w_{D}=0\}=D$. Then, there exists a positive integer $a(n)$ depending only on the dimension $n$ such that

[TABLE]

as $|w_{F}|,|w_{D}|\to 0$, for any $1\leq i,j\leq n-2$ and multi-index $\alpha=(\alpha_{1},...,\alpha_{n})$ with $0\leq\sum_{i}\alpha_{i}\leq 2$.

By applying Theorem 1.1 and gluing plurisubharmonic functions, we have the following result :

Theorem 1.2**.**

Assume that there exist positive integers $l>n$ and $m$ such that

[TABLE]

and the line bundle $K_{X}^{-l}\otimes L_{X}^{m}$ is very ample. Here, $a(n)$ is the positive integer in Theorem 1.1. Take a smooth hypersuface $F\in|K_{X}^{-l}\otimes L_{X}^{m}|$ such that $D+F$ is simple normal crossing. Then, for any relatively compact domain $Y\Subset X\setminus(D\cup F)$, there exists a complete Kähler metric $\omega_{F}$ on $X\setminus D$ whose scalar curvature $S(\omega_{F})=0$ on $Y$ and is arbitrarily small on the complement of $Y$. In addition, $\omega_{F}=\omega_{0}$ on some neighborhood of $D\setminus(D\cap F)$.

For example, if the anti-canonical line bundle $K_{X}^{-1}$ of the compact complex manifold $X$ is nef (in particular, $X$ is Fano), the assumption (1.2) in Theorem 1.2 holds, i.e., we can always find such integers $l,m$. In this article, we treat the case that $K_{X}^{-1}$ has positivity in the senses of (1.1) and (1.2). From [1], if there exists a complete Kähler metric which is of asymptotically conical geometry and satisfies Condition 1.2 and Condition 1.3, $X\setminus D$ admits a complete scalar-flat Kähler metric. In fact, Theorem 1.2 gives a Kähler metric whose scalar curvature is under control. However, the Kähler metric $\omega_{F}$ in Theorem 1.2 is not of asymptotically conical geometry (near the intersection of $D$ and $F$). This problem will be solved in [2].

This paper is organized as follows. In Section 2, we construct Kähler potentials, i.e., strictly plurisubharmonic functions, whose scalar curvature is under control. In addition, we glue these plurisubharmonic functions by using the regularized maximum function. In Section 3, we prove Theorem 1.1. To show this, we recall the $C^{2,\epsilon}$-estimate of a solution of the degenerate complex Monge-Amp$\grave{{\rm e}}$re equation. In Section 4, we prove Theorem 1.2.

Acknowledgment**.**

The author would like to thank Professor Ryoichi Kobayashi who first brought the problem in this article to his attention, for many helpful comments.

2

Plurisubharmonic functions with small scalar curvature

To prove Theorem 1.2, we prepare Kähler potentials, i.e., strictly plurisubharmonic functions, whose scalar curvature is under control.

2.1

Kähler potential near $D$

In this subsection, we consider a Kähler potential near $D$ and study the scalar curvature of it. Recall that

[TABLE]

and $\theta_{X}=\sqrt{-1}\partial\overline{\partial}t=\sqrt{-1}\partial\overline{\partial}\log||\sigma_{D}||^{-2}$ on $X\setminus D$. Set

[TABLE]

Following [3], we can define a complete Kähler metric by

[TABLE]

on $X\setminus D$. Following [1], recall the asymptotic behavior of the scalar curvature of $\omega_{0}$.

Lemma 2.1**.**

The scalar curvature $S(\omega_{0})$ can be estimated as follows :

[TABLE]

as $\sigma_{D}\to 0$.

Remark 2.2**.**

Moreover, from Theorem 1.1 in [1], if $\theta_{D}$ is cscK, we have the following strong result :

[TABLE]

as $\sigma_{D}\to 0$.

2.2

Kähler potential near $F$

In this subsection, we construct a Kähler metric on $X$ whose scalar curvature is small near the smooth hypersurface $F\in|K_{X}^{-l}\otimes L_{X}^{m}|$. Here, $l,m$ are positive integers such that the line bundle $K_{X}^{-l}\otimes L_{X}^{m}$ is very ample. For a fixed Hermitian metric on $K_{X}^{-l}\otimes L_{X}^{m}$, set $b:=\log||\sigma_{F}||^{-2}$. Since the holomorphic line bundle $K_{X}^{-l}\otimes L_{X}^{m}$ is very ample, we may assume that $\sqrt{-1}\partial\overline{\partial}b$ is a Kähler metric on $X$. For parameters $v>0$ and $\beta\in\mathbb{Z}_{>0}$, define a function by

[TABLE]

for some fixed $b_{0}\in\mathbb{R}$. Note that $G_{v}^{\beta}(b)$ is defined smoothly outside $F$ and $\lim_{b\to\infty}G_{v}^{\beta}(b)=+\infty$ for any $v>0$.

Lemma 2.3**.**

For $\mathbb{Z}\ni\beta\geq 1$, $\gamma_{v}^{\beta}:=\sqrt{-1}\partial\overline{\partial}G_{v}^{\beta}(\beta b)$ defines a Kähler metric on $X$.

Proof..

In fact,

[TABLE]

Note that the last term

[TABLE]

is defined smoothly on $X$ from the assumption that $\mathbb{Z}\ni\beta\geq 1$. Since $\sqrt{-1}\partial\overline{\partial}b$ is a Kähler metric on $X$, we finish the proof. ∎

Next, the scalar curvature of $\gamma_{v}^{\beta}$ is given by

Lemma 2.4**.**

For $\beta\geq 3$, we obtain

[TABLE]

as $||\sigma_{F}||\to 0$.

Proof..

This lemma follows from the similar way in the computation of the scalar curvature of $\omega_{0}$. In fact, since

[TABLE]

we have

[TABLE]

Note that second and last terms above are zero on $F$. Thus, when we consider the scalar curvature $S(\gamma_{v}^{\beta})$, it is enough to see the term $1/(e^{-\beta b}+v)^{1/\beta}\sqrt{-1}\partial\overline{\partial}b$ and the Ricci form ${\rm Ric}(\sqrt{-1}\partial\overline{\partial}b)$. Therefore the desired result is obtained. ∎

Remark 2.5**.**

If the value of the function $e^{-\beta b}=||\sigma_{F}||^{2\beta}$ is compatible with $v$, i.e., $||\sigma_{F}||^{2\beta}\approx v$, we have the following estimate of $S(\sqrt{-1}\partial\overline{\partial}G_{v}^{\beta}(\beta b))$ :

[TABLE]

However, we will consider the case that $||\sigma_{F}||^{2\beta}\approx v^{k}$ for sufficiently large $k\in\mathbb{N}$ which will be specified later. Namely, it suffices to consider a sufficiently small neighborhood of $F$ defined by the inequality $||\sigma_{F}||^{2\beta}\leq v^{k}$ and Lemma 2.4 holds on this region.

2.3

Ricci-flat Kähler metric away from $D\cup F$

In this subsection, we study an incomplete Ricci-flat Kähler metric away from the support of the divisor $D+F$. Recall the setting in Theorem 1.2. Let $l>n$ and $m$ be positive integers such that there exists a holomorphic section $\sigma_{F}\in H^{0}(K_{X}^{-l}\otimes L_{X}^{m})$ which defines a smooth hypersurface $F\subset X$, i.e., $(\sigma_{F})_{0}=F$. It follows from the hypothesis of the average value $\hat{S}_{D}$ of the scalar curvature that divisors $D$ and $F$ intersect to each other. Set

[TABLE]

Note that $\xi$ is a meromorphic section of $K_{X}^{-l}$. Then, define a singular and degenerate volume form $V$ by

[TABLE]

From the construction above, $V$ has finite volume on $X$ and its curvature form, i.e., the Ricci form, is zero on the complement of $D\cup F$. For the Kähler metric $\theta_{X}$ on $X$, write

[TABLE]

for some non-negative function $f$ on $X$ with the normalized condition

[TABLE]

We know that $f$ is smooth away from $D\cup F$. From the result due to Yau [13, Theorem 7], recall the solvability of a meromorphic complex Monge-Amp$\grave{{\rm e}}$re equation :

Theorem 2.6**.**

Let $L_{1}$ and $L_{2}$ be holomorphic line bundles over a compact Kähler manifold $(X,\theta_{X})$. Let $s_{1}$ and $s_{2}$ be nonzero holomorphic sections of $L_{1}$ and $L_{2}$, respectively. Let $F$ be a smooth function on $X$ such that $\int_{X}|s_{1}|^{2k_{1}}|s_{2}|^{-2k_{2}}\exp(F)\theta_{X}^{n}={\rm Vol}(X)$, where $k_{1}\geq 0$ and $k_{2}\geq 0$. Suppose that $\int_{X}|s_{2}|^{-2nk_{2}}<\infty$ for $n=\dim X$. Then, we can solve the following equation

[TABLE]

so that $\varphi$ is smooth outside divisors of $s_{1}$ and $s_{2}$ with $\sup_{X}\varphi<+\infty$.

Then, we can solve the following complex Monge-Amp$\grave{{\rm e}}$re equation

[TABLE]

with $\varphi\in C^{\infty}(X\setminus D\cup F)$. Thus, we obtain a Ricci-flat Kähler metric $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$ on the complement of $D\cup F$. For this solution $\varphi$, we obtain the following a priori estimate due to Kołodziej [11] (see also [9]):

Theorem 2.7**.**

If $f$ is in $L^{p}(\theta_{X}^{n})$ for some $p>1$, we have

[TABLE]

for some $C>0$ depending only on $\theta_{X}$ and $||f||_{L^{p}}$.

2.4

Gluing plurisubharmonic functions

In this subsection, following [6, Chapter I], we consider gluing Kähler potentials, i.e., plurisubharmonic functions, obtained in previous subsections. Let $\rho\in C^{\infty}(\mathbb{R},\mathbb{R})$ be a nonnegative function with support in $[-1,1]$ such that $\int_{\mathbb{R}}\rho(h)dh=1$ and $\int_{\mathbb{R}}h\rho(h)dh=0$.

Lemma 2.8** (the regularized maximum).**

For arbitrary $\eta=(\eta_{1},...,\eta_{p})\in(0,+\infty)^{p}$, the function

[TABLE]

called the regularized maximum possesses the following properties $:$

a)

$M_{\eta}(t_{1},...,t_{p})$* is non decreasing in all variables, smooth and convex on $\mathbb{R}^{p}\hskip 2.0pt;$*

b)

$\max\{t_{1},...,t_{p}\}\leq M_{\eta}(t_{1},...,t_{p})\leq\max\{t_{1}+\eta_{1},...,t_{p}+\eta_{p}\}\hskip 2.0pt;$**

c)

$M_{\eta}(t_{1},...,t_{p})=M_{(\eta_{1},...,\hat{\eta_{j}},...,\eta_{p})}(t_{1},...,\hat{t_{j}},...,t_{p})$* if $t_{j}+\eta_{j}\leq\max_{k\neq j}\{t_{k}-\eta_{k}\}\hskip 2.0pt;$*

d)

$M_{\eta}(t_{1}+a,...,t_{p}+a)=M_{\eta}(t_{1},...,t_{p})+a\hskip 2.0pt;$**

e)

if $u_{1},...,u_{p}$ are plurisubharmonic and satisfy $H(u_{j})_{z}(\xi)\geq\gamma_{z}(\xi)$ where $z\mapsto\gamma_{z}$ is a continuous hermitian form on $TM$, then $u=M_{\eta}(u_{1},...,u_{p})$ is a plurisubharmonic and satisfies $Hu_{z}(\xi)\geq\gamma_{z}(\xi)$.

Remark 2.9**.**

Lemma 2.8 is a key in the proof of Richberg theorem (see [6, p.43]). In our case, we have already prepared three plurisubharmonic functions and must compute the Ricci form of the glued Kähler metric later. Therefore, we need the explicit formula of the glued function.

In addition, we obtain

Lemma 2.10**.**

There exists a constant $C>0$ such that

[TABLE]

for any multi index $\alpha=(\alpha_{i})_{i}$ with $1\leq|\alpha|\leq 4$.

Recall that the Kähler potential of $\omega_{0}$ is given by

[TABLE]

For $\kappa\in(0,1)$, set

[TABLE]

This constant $\kappa$ will be specified later. For this Kähler potential, we have

Lemma 2.11**.**

For the complete Kähler metric $\sqrt{-1}\partial\overline{\partial}\tilde{G}_{v}^{\beta}(b)$ on $X\setminus D$, we have

[TABLE]

Proof..

First, we study the behavior of the scalar curvature near $D$. Since

[TABLE]

is a smooth volume form on $X$, the Ricci form of $\sqrt{-1}\partial\overline{\partial}(\tilde{G}_{v}^{\beta}(b))$ given by

[TABLE]

is defined smoothly on X. Recall that

[TABLE]

As $\omega_{0}$ is of asymptotically conical geometry, we have the desired result near $D$. Similarly, the volume form

[TABLE]

is smooth near $F\setminus(D\cap F)$. Then, the following identity

[TABLE]

implies the desired result near $F$. ∎

In summary, we have prepared the three strictly plurisubharmonic functions $\Theta(t)=(n(n-1)/\hat{S}_{D})\exp((\hat{S}_{D}/n(n-1))t),\tilde{G}_{v}^{\beta}(b)=G_{v}^{\beta}(\beta b)+\kappa\Theta(t),t+\varphi=\log||\sigma_{D}||^{-2}+\varphi$ whose scalar curvature is under control. From Lemma 2.8, we immediately have

Proposition 2.12**.**

For parameters $c,v,\eta$ and $\kappa\in(0,1)$, a function defined by

[TABLE]

is a strictly plurisubharmonic function on $X\setminus(D\cup F)$. Here, the functions above are defined in (2.1), (2.2), (2.3), (2.4) and (2.5).

Remark 2.13**.**

From a priori estimate due to Kołodziej [11], the solution $\varphi$ is bounded on $X$. Thus, by taking $c>0$ sufficiently large, $\varphi$ can be ignored when we consider the value of $M_{c,v,\eta}$.

By taking a sufficiently large $c>0$, we have

[TABLE]

Set

[TABLE]

The reason why we consider the second Kähler potential which contains the term $\kappa\Theta(t)$ is that we want to make $\omega_{c,v,\eta}$ complete on $X\setminus D$. The function $M_{c,v,\eta}$ is defined on $X\setminus(D\cup F)$. On the other hand, Lemma 2.3 implies that $\omega_{c,v,\eta}$ is defined on $X\setminus D$ since the Kähler metric $\gamma_{v}^{\beta}$ is a smooth Kähler metric on $X$. From (2.12), we know that the scalar curvature of $\omega_{c,v,\eta}$ is small on three regions above (in particular, away from $D$ and $F$, $S(\omega_{c,v,\eta})=0$ since $t+\varphi+c$ is a Kähler potential whose Ricci form is zero).

The explicit formula of $\omega_{c,v,\eta}$ is written as

[TABLE]

Thus, when we compute the scalar curvature of $\omega_{c,v,\eta}$, higher order derivatives of $\varphi$ arise in the components of the Ricci tensor of $\omega_{c,v,\eta}$. So, we must study the behavior of higher order derivatives of $\varphi$ near $D\cup F$.

3

Proof of Theorem 1.1

In this section, we prove Theorem 1.1. Firstly, we use the $C^{2}$-estimate due to P$\check{{\rm a}}$un [12] (see also [7], [9, p.366, Theorem 14.3]) for the solution $\phi$ of the complex Monge-Amp$\grave{{\rm e}}$re equation (2.4) in the previous section to obtain the estimate of the ellipticity. i.e., the maximal ratio of the maximal eigenvalue to the minimal eigenvalue, of the Kähler metric $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$. Secondly, we study how the $C^{2,\epsilon}$-estimate of $\varphi$ depends on the ellipticity of $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$ on a fixed relatively compact domain in $X\setminus(D\cup F)$. Finally, we estimate the higher order derivatives of $\varphi$ by using the Schauder estimate.

3.1 The $C^{2}$-estimate

To study the behavior of the higher order derivatives of $\varphi$, the elliptic operator defined by the Kähler metric $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$ plays an important role. To obtain the ellipticity of $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$, we use the $C^{2}$-estimate due to P$\check{{\rm a}}$un [12] (see also [7], [9, p.366, Theorem 14.3]).

Theorem 3.1**.**

Let $dV$ be a smooth volume form. Assume that $\varphi\in{\rm PSH}(X,\theta_{X})$ satisfies

[TABLE]

with $\displaystyle\int_{X}\varphi\theta_{X}^{n}=0$. Here, $\psi_{+},\psi_{-}$ are quasi-plurisubharmonic functions on $X$. Assume that we are given $C>0$ and $p>1$ such that

(i)

$\sqrt{-1}\partial\overline{\partial}\psi_{+}\geq-C\theta_{X}$* and $\sup_{X}\psi_{+}\leq C$.*

(ii)

$\sqrt{-1}\partial\overline{\partial}\psi_{-}\geq-C\theta_{X}$* and $||e^{-\psi_{-}}||_{L^{p}}\leq C$.*

Then there exists $A>0$ depending only on $\theta_{X}$, $p$ and $C$ such that

[TABLE]

Set $\psi_{+}:=\log||\sigma_{D}||^{2m/l}$ and $\psi_{-}:=\log||\sigma_{F}||^{2/l}$. Then, Theorem 3.1 implies the following inequality

[TABLE]

Recall that the singular and degenerate volume form

[TABLE]

vanishes along $D$ with order $2m/l$ and has a pole along $F$ of order $2/l$. So, we obtain the behavior of the product of the eigenvalues of the Kähler metric $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$. From (3.1) and (3.2), we can estimate the eigenvalues of $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$. Namely, the maximal eigenvalue $\Lambda$ and the minimal eigenvalue $\lambda$ of the Kähler metric $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$ are estimated as follows :

[TABLE]

In the next subsection, to consider the third and the forth order derivatives, we recall the $C^{2,\epsilon}$-estimate of $\varphi$.

3.2

The $C^{2,\epsilon}$-estimate

This subsection follows from [9, Chapter 14]. In this subsection, we study the relation between the ellipticity of $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$ and the $C^{2,\epsilon}$-estimate of $\varphi$. This subsection is the core of the proof of Theorem 1.1 because the estimate of the higher order derivatives of the solution $\varphi$ are obtained by the $C^{2,\epsilon}$-estimate and the Schauder estimate.

Let $\mathcal{H}$ be the set of all $n\times n$ Hermitian matrices and set

[TABLE]

In addition, for $0<\lambda<\Lambda<\infty$, let $S(\lambda,\Lambda)$ be the subset of $\mathcal{H}_{+}$ whose eigenvalues lie in the interval $[\lambda,\Lambda]$. First, recall the following result from linear algebra (see [8, p.454, Lemma 17.13], [9, p.372, Lemma 14.10]):

Lemma 3.2**.**

One can find unit vectors $\zeta_{1},...,\zeta_{N}\in\mathbb{C}^{n}$ and $0<\lambda_{*}<\Lambda_{*}<\infty$, depending only on $n,\lambda$ and $\Lambda$, such that every $A\in S(\lambda,\Lambda)$ can be written as

[TABLE]

where $\beta_{k}\in[\lambda_{*},\Lambda_{*}]$. The vectors $\zeta_{1},...,\zeta_{N}\in\mathbb{C}^{n}$ can be chosen so that they contain a given orthonormal basis of $\mathbb{C}^{n}$.

Remark 3.3**.**

In the proof of Lemma 3.2, they use the following covering

[TABLE]

of the compact subset $S(\lambda/2,\Lambda)$ (see [8, p.454, Lemma 17.13], [9, p.372, Lemma 14.10]). Here, $\zeta_{1},...,\zeta_{n^{2}}\in\mathbb{C}^{n}$ are unit vectors such that the matrices $\zeta_{k}\otimes\overline{\zeta}_{k}$ span $\mathcal{H}$ over $\mathbb{R}$. Thus, it follows from the form of the covering $U(\zeta_{1},...,\zeta_{n^{2}})$ that the number $N$ in Lemma 3.2 is depending only on the dimension $n$. In particular, $N$ is independent of the ellipticity of $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$.

Take local holomoriphic coordinates $(z^{i})_{i=1}^{n}=(z^{1},z^{2},...,z^{n-2},w_{F},w_{D})$ such that $\{w_{F}=0\}=F$ and $\{w_{D}=0\}=D$. On this coordinate chart, we can write $t=a+\log|w_{D}|^{-2}$ for some smooth plurisubharmonic function $a$. Since $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi=\sqrt{-1}\partial\overline{\partial}(a+\varphi)$ on this coordinate chart, it is enough to consider the following complex Monge-Amp$\grave{{\rm e}}$re equation

[TABLE]

on an open subset $\Omega\Subset\mathbb{C}^{n}\setminus(D\cup F)$ by setting

[TABLE]

It follows from our construction that we may assume that the function $f$ is a form of

[TABLE]

Fix an unit vector $\zeta\in\mathbb{C}^{n}$. Differentiating the following equation :

[TABLE]

we have

[TABLE]

Here we use the standard Einstein convention and the notation $(u^{i,\overline{j}})=((u_{i,\overline{j}})^{t})^{-1}$. Set

[TABLE]

Then, for any $i$, we have

[TABLE]

Thus, we obtain

[TABLE]

Note that $u_{\zeta,\overline{\zeta}}$ is a subsolution of the equation $Lv=0$, where $Lv:=\sum_{i,j}(a^{i,\overline{j}}v_{i})_{\overline{j}}$. The assumption of $u$ and the later lemma ensure that the operator $L$ is uniformly elliptic (in the real sense). Then, we have the following estimate (see [8, Theorem 8.18]).

Lemma 3.4**.**

The weak Harnack inequality

[TABLE]

holds. Here, $B_{4r}:=B(z_{0},4r)\subset\Omega$ with $d(z_{0},\partial\Omega)>4r$. Moreover, in our case, we have the following estimate of the constant $C_{H}$ in Harnack inequality :

[TABLE]

Proof..

It suffices to show the estimate of the constant $C_{H}$. In our case, we will only consider the behavior of $\varphi$ in the neighborhood of $D\cup F$ and the $C^{2}$-estimate of $\varphi$ implies that

[TABLE]

as $||\sigma_{F}||\to 0$ and $||\sigma_{D}||\to 0$. Thus, the weak Harnack inequality implies that the lemma follows. ∎

Remark 3.5**.**

From the proof of [8, Theorem 8.18], we know that the optimal Harnack constant $C_{H}$ is estimated by

[TABLE]

where $C_{n}$ depends only on $n$.

Set $U:=(u_{i,\overline{j}})$. For $x,y\in B_{4r}$, we obtain

[TABLE]

In particular, $a^{i,\overline{j}}(y)u_{i,\overline{j}}(y)=nf(y)$. Since $\det(f(y)^{1/n}U(y)^{-1})=1$, we have

[TABLE]

Here, we have used the following lemma (see [9, Lemma 5.8]) :

Lemma 3.6**.**

For any $A\in\mathcal{H}_{+}$, we have

[TABLE]

Therefore, for any $x,y\in B_{4r}$ and $\epsilon\in(0,1)$, we have

[TABLE]

where

[TABLE]

and \mbox{ H\ddot{{\rm o}}l}_{\epsilon,\Omega} denotes an $\epsilon$-Hölder constant. In this case, the following estimates

[TABLE]

implies that we have

[TABLE]

Remark 3.7**.**

In [9, p.375], they used the Lipscitz constant of $f$. But in our case, it is enough to use the Hölder constant of $f$ for sufficiently small $\epsilon$.

Set $\lambda,\Lambda>0$ so that the eigenvalues of $(a^{i,\overline{j}}(y))$ lie in the interval $[\lambda,\Lambda]$. Then, Lemma 3.2 implies that we can find unit vectors $\zeta_{1},...,\zeta_{N}\in\mathbb{C}^{n}$ such that for any $x,y\in\Omega$,

[TABLE]

where $\beta_{k}(y)\in[\lambda_{*},\Lambda_{*}]$ and $\lambda_{*},\Lambda_{*}>0$.

Thus, we have

[TABLE]

Set

[TABLE]

and

[TABLE]

To establish the Hölder condition

[TABLE]

for some $0<\tilde{\epsilon}<1$, we need the following lemma from [8, p.201, Lemma 8.23] :

Lemma 3.8**.**

Let $\eta$ and $\sigma$ be non-decreasing functions defined on the interval $(0,R_{0}]$ such that there exist $\tau,\alpha\in(0,1)$ satisfying

[TABLE]

for all $r\in(0,R_{0}]$. Then, for any $\mu\in(0,1)$, we have

[TABLE]

So, it suffices to show that

[TABLE]

where $\delta,\epsilon\in(0,1)$ and $r_{0}>0$.

For fixed $k$, Harnack inequality implies that

[TABLE]

For $x\in B_{4r}$ and $y\in B_{r}$, we have

[TABLE]

Thus, for all $y\in B_{r}$, we have

[TABLE]

Therefore,

[TABLE]

Using Harnack inequality again, we have

[TABLE]

Summing over $k$, we have

[TABLE]

Thus, we obtain

[TABLE]

Since we can take arbitrary $\lambda^{*}N<\lambda$ and $\Lambda^{*}>\Lambda$, we may assume that $\lambda^{*}N=\lambda$ and $\Lambda^{*}=\Lambda$. Thus, we have

Lemma 3.9**.**

By taking $\epsilon\leq 2/l$, there exists $0<\tilde{\epsilon}<\epsilon$ with

[TABLE]

Proof..

In order to show this lemma, we apply Lemma 3.8 to the inequality (3.7). Set

[TABLE]

where this is the coefficient of $\eta(4r)$ in (3.7). Then, we have the following estimates :

[TABLE]

Here, we have used the fact that the number $N$ depends only on the dimension $n$ (Remark 3.3). Define a non-decreasing function $\sigma$ by

[TABLE]

Here, this is the second term in the right hand side of the inequality (3.7). Recall the estimate (3.6)

[TABLE]

and Lemma 3.4. The assumption that $\epsilon\leq 2/l$ implies that we have the following

[TABLE]

Lemma 3.8 implies that we have

[TABLE]

for any $\mu\in(0,1)$. Take $\mu\in(0,1)$ so that

[TABLE]

Thus, we have

[TABLE]

Set $\tilde{\epsilon}:=\epsilon\mu<\epsilon$. From the interior Hölder estimate for solutions of Poisson’s equation [8, Theorem 4.6], we finish the proof. ∎

Recall the relation (3.3) between $u$ and $\varphi$. Lemma 3.4 implies

Proposition 3.10**.**

For the domain $\Omega\Subset X\setminus(D\cup F)$, we have

[TABLE]

as $\sigma_{D},\sigma_{F}\to 0$.

3.3

The third and the forth order estimates

In this subsection, we prove Theorem 1.1. This subsection also follows from [9, Chapter 14]. To consider higher order estimates, we recall the Schauder estimate with respect to the elliptic linear operator defined by the Kähler metric $\theta_{X}+\sqrt{-1}\partial\overline{\partial}\varphi$. The complex Monge-Amp$\grave{{\rm e}}$re operator

[TABLE]

is elliptic if the $2n\times 2n$ real symmetric matrix $A:=(\partial F/\partial u_{p,q})$ is positive (we denote here by $u_{p,q}$ the element of the real Hessian $D^{2}u$). The matrix $A$ is determined by

[TABLE]

From [4] (see also [9, Exercise 14.8]), we have

Lemma 3.11**.**

One has

[TABLE]

where $\lambda_{{\rm min}}(\partial F/\partial u_{p,q})$ and $\lambda_{{\rm max}}(\partial F/\partial u_{p,q})$ denote minimal and maximal eigenvalue of the matrix $(\partial F/\partial u_{p,q}))_{p.q}$ respectively.

Then, we can estimate the ellipticity in the real sense. We apply the standard elliptic theory to the equation

[TABLE]

For a fixed unit vector $\zeta$ and small $h>0$, we consider

[TABLE]

and

[TABLE]

Thus, we have

[TABLE]

From the definition of $a_{h}^{p,q}$, we obtain

[TABLE]

for sufficiently small $h>0$.

The Schauder estimate implies

Proposition 3.12**.**

There exists $C_{S}>0$ such that

[TABLE]

for any $h>0$.

Therefore, we can obtain the estimate of derivatives of the solution $\varphi$ in the desired direction by taking a suitable vector $\zeta$ and $h\to 0$. The constant $C_{S}$ in Proposition 3.12 also depends on the maximal ratio of the eigenvalues $\Lambda/\lambda$ and the dimension $n$. By examining the proof of [8, Lemma 6.1 and Theorem 6.2], there is a positive constant $s(n)$ depending only on the dimension $n$ such that

[TABLE]

As $h\to 0$, we have the following third order estimates of $\varphi$ :

Proposition 3.13**.**

For any multi-index $\alpha=(\alpha_{1},...,\alpha_{n})$ satisfying $\sum_{i}\alpha_{i}=2$, we have

[TABLE]

[TABLE]

[TABLE]

as $|w_{D}|,|w_{F}|\to 0$.

From the discussion above, we can prove Theorem 1.1.

Proof of Theorem 1.1 Let $\dot{a}_{h}^{p,q}$ be a differential of $a_{h}^{p,q}$ in some direction. From the definition of $a_{h}^{p,q}$, we know that

[TABLE]

Thus, by differentiating the equation $a_{h}^{p,q}(x)u^{h}_{p,q}(x)=f^{h}(x)$, Schauder estimate implies again the following inequality:

[TABLE]

Thus, we finish the proof pf Theorem 1.1 by taking a suitable vector $\zeta$ and $h\to 0$.  $\square$

Remark 3.14**.**

By examining the proof of [8, Lemma 6.1 and Theorem 6.2] and the discussion above, we can find that

[TABLE]

4

Proof of Theorem 1.2

In this section, we prove Theorem 1.2. To compute the scalar curvature of the Kähler metric $\omega_{c,v,\eta}$, we have to consider the inverse matrix (see Lemma 3.4 in [1]). Since we assume that the divisor $D+F$ is simple normal crossing, we can choose block matrices in suitable directions in local holomorphic coordinates defining hypersurfaces $D$ and $F$. To prove Theorem 1.2, we consider the case that the parameter $\eta=(\eta_{1},\eta_{2},\eta_{3})$ depends on $c>0$. More precisely, we set $\eta_{i}:=a_{i}c$ for $i=1,2$ for $a_{i}\in(0,1)$ and $\eta_{3}$ a fixed positive real number. We use many parameters, i.e., $c,v,\beta,\kappa,\eta,a_{i}$. When we want to make the scalar curvature $S(\omega_{c,v,\eta})$ small, we take sufficiently large $c$ and sufficiently small $v$. On the other hand, we don’t make other parameters $\beta,\kappa,a_{i}$ close to $\infty$, 0 or 1. Namely, the parameters $\beta,\kappa,a_{i}$ are bounded in this sense. Settings of these bounded parameters will be given later.

Proof of Theorem 1.2. Take a relatively compact domain $Y\Subset X\setminus(D\cup F)$. Recall that the function $G_{v}^{\beta}(\beta b)$ is defined by

[TABLE]

Immediately, we have $G_{v}^{\beta}(\beta b)<\beta e^{b}$ and $G_{v}^{\beta}(\beta b)\to\beta e^{b}$ as $v\to 0$. So, we can find a sufficiently large number $c_{0}=c_{0}(Y)>0$ so that

[TABLE]

for any $v>0$. Here, $\tilde{G}_{v}^{\beta}(b)=G_{v}^{\beta}(\beta b)+\kappa\Theta(t)$. For simplicity, we write $\varphi+c_{0}$ by the same symbol $\varphi$.

Recall that the property d) of the regularized maximum in Lemma 2.8. If the following inequality

[TABLE]

holds for some $k$, we have $M_{\eta}(t)=t_{k}$. For instance, in our case, if we consider the region defined by the following inequality

[TABLE]

we have $M_{c,v,\eta}=\Theta(t)$. Note that this region is contained in a sufficiently small neighborhood of $D$. In this case, we don’t have to estimate the scalar curvature $S(\omega_{c,v,\eta})$ since $S(\omega_{c,v,\eta})=S(\omega_{0})$ on this region and the estimate of $S(\omega_{0})$ have been obtained in Lemma 2.1 before. Similarly, if the value of $M_{c,v,\eta}$ corresponds to one of the other variables $\tilde{G}_{v}^{\beta}(b),t+\varphi+c$, Lemma 2.11 and the Ricci-flatness of the Kähler metric $\sqrt{-1}\partial\overline{\partial}(t+\varphi)$ implies that $S(\omega_{c,v,\eta})$ is under control on such regions. Thus, it suffices for us to study the $S(\omega_{c,v,\eta})$ on the other regions defined by the inequalities

[TABLE]

for $i,j\neq k$ and

[TABLE]

So we have to study $S(\omega_{c,v,\eta})$ on four regions defined by the inequalities above.

Directly, we have

[TABLE]

It follows from the convexity of $M_{\eta}$ that the last term is semi-positive. When we compute the scalar curvature of $\omega_{c,v,\eta}$, the difficulty comes from terms $\partial\Theta(t)\wedge\overline{\partial}\Theta(t)$ and $\partial\tilde{G}_{v}^{\beta}(b)\wedge\overline{\partial}\tilde{G}_{v}^{\beta}(b)$. For these terms, since functions $t$ and $b$ are defined by Hermitian norms of holomorphic sections, it suffices to focus on derivatives in normal directions of smooth hypersurfaces $D$ and $F$ by taking suitable local trivializations of line bundles $L_{X}$ and $K_{X}^{-l}\otimes L_{X}^{m}$ respectively. The reason why scalar curvatures of two Kähler metrics $\omega_{0},\gamma_{v}^{\beta}$ are under control near these hypersurfaces $D,F$ is that Ricci curvatures are bounded and Kähler metrics grow asymptotically near these hypersurfaces. Thus, it suffices for us to focus on derivatives of $\varphi$ and $M_{\eta}$ arising in Ricci tensors. The higher order derivatives of $\varphi$ are estimated in the previous section (Theorem 1.1). In addition, the definition of a parameter $\eta=(\eta_{i})=(a_{1}c,a_{2}c,\eta_{3})$ and Lemma 2.10 imply that the higher order derivatives in the first or the second variable of $M_{\eta}$ are estimated by some negative power of $c>0$. To estimate $S(\omega_{c,v,\eta})$ on each region, we divide the proof of Theorem 1.2 into the following four claims.

Claim 1**.**

On the region defined by

[TABLE]

we can make the scalar curvature $S(\omega_{c,v})$ small arbitrarily by taking a sufficiently large $c$.

Proof..

On this region, we can write as

[TABLE]

To prove this claim, we need the following lemma.

Lemma 4.1**.**

Take a point $p\in D\cap F$ and local holomorphic coordinates $(z^{1},...,z^{n-2},w_{F},w_{D})$ centered at $p$ satisfying $D=\{w_{D}=0\}$ and $F=\{w_{F}=0\}$. By taking suitable local trivializations of $L_{X}$ and $K_{X}^{-l}\otimes L_{X}^{m}$, we may assume that if $(z^{1},...,z^{n-2},w_{F},w_{D})=(0,...,0,w_{F},w_{D})$, we have

[TABLE]

From the definition of this region, we obtain

[TABLE]

as $w_{D},w_{F}\to 0$.

In particular, coefficients $g_{i,\overline{j}}$ for $1\leq i,j\leq n-2$ come from Kähler metrics $\omega_{0}$ and $\gamma_{v}^{\beta}$. Thus,

[TABLE]

For other blocks, we similarly have

[TABLE]

From Lemma 3.4 in [1], we have

[TABLE]

as $w_{D},w_{F}\to 0$. Since metric tensors $g^{i,\overline{j}}$ with $i,j\neq n-1,n$ come from Kähler metrics $\omega_{0}$ and $\gamma_{v}^{\beta}$ whose scalar curvature have been already known. Thus, it is enough to study the case that $i=n-1,n$ and $j=n-1,n$. Recall that the components of the Ricci tensor are defined by $R_{i,\overline{j}}:=-g^{p,\overline{q}}\partial^{2}g_{p,\overline{q}}/\partial z^{i}\partial\overline{z}^{j}+g^{k,\overline{q}}g^{p,\overline{l}}(\partial g_{k,\overline{l}}/\partial z^{i})(\partial g_{p,\overline{q}}/\partial\overline{z}^{j})$. So, the Ricci form ${\rm Ric}(\omega_{c,v,\eta})$ is written as

[TABLE]

as $w_{D},w_{F}\to 0$ and the other components of the Ricci tensor $R_{i,\overline{j}}$ for $1\leq i\leq n-2$ are under control.

By taking the trace, we obtain the following:

[TABLE]

∎

Remark 4.2**.**

On the region in the previous claim, there are the terms $\partial\Theta(t)\wedge\overline{\partial}\Theta(t)$ and $\partial G_{v}^{\beta}(\beta b)\wedge\overline{\partial}G_{v}^{\beta}(\beta b)$ in the complete Kähler metric $\omega_{c,v,\eta}$. Thus, $(X\setminus D,\omega_{c,v,\eta})$ is not of asymptotically conical geometry and we can’t use the analysis in Section 5 of [1] with respect to this Kähler metric $\omega_{c,v,\eta}$. This problem will be solved in [2].

We proceed to the estimate of $S(\omega_{c,v,\eta})$ on another region.

Claim 2**.**

Consider the region defined by

[TABLE]

Take parameters $\eta,\kappa$ so that

[TABLE]

for any $c>0$. Then, we can make the scalar curvature $S(\omega_{c,v})$ small arbitrarily by taking a sufficiently large $c$.

Proof..

On this region, since

[TABLE]

from Lemma 2.8, we have

[TABLE]

From the hypothesis of this claim, we have

[TABLE]

By taking a small $v>0$ and a suitable $b_{0}$ in the definition of the function $G_{v}^{\beta}(\beta b)$, we may assume that

[TABLE]

From a priori estimate due to Kołodziej [11] again, $\varphi$ is bounded on $X$. So, on this region, we have the following inequality:

[TABLE]

for some constant $C>0$ depending only on the $C^{0}$-norm of $\varphi$. By taking $\kappa$ close to $1$ which depends on $m,l$ and $a=a(n)$ in Theorem 1.1, we may assume that

[TABLE]

Thus, on this region, the growth of derivatives of $\varphi$ can be controlled by the Kähler metric $\omega_{0}$. Take a point in $D\setminus(D\cap F)$ and local holomorphic coordinates $(z^{i})_{i=1}^{n}=(z^{1},...,z^{n-1},w_{D})$ satisfying $D=\{w_{D}=0\}$. Then, we have

[TABLE]

if $1\leq i,j\leq n-1$ and

[TABLE]

Similarly, we have

Lemma 4.3**.**

By taking a suitable local holomorphic trivialization of $L_{X}$, we may assume that if $(z^{1},...,z^{n-1},w_{D})=(0,...,0,w_{D})$, we have

[TABLE]

Recall the hypothesis

[TABLE]

So, Theorem 1.1 implies that the growth of the Kähler metric $\omega_{c,v,\eta}$ is greater then the growth of the higher order derivatives of $\varphi$. Thus, Lemma 3.4 in [1] shows that higher order derivatives including $\partial^{4}\varphi/\partial w^{2}\partial\overline{w}^{2}$ are controlled by taking the trace with respect to $\omega_{c,v,\eta}$. Therefore, we can ignore derivatives of $\varphi$ arising in the components of the Ricci tensor and we have

[TABLE]

∎

We proceed to the estimate of $S(\omega_{c,v,\eta})$ the following region.

Claim 3**.**

Consider the region defined by

[TABLE]

By choosing sufficiently small number $v>0$ so that

[TABLE]

holds on this region, we can make the scalar curvature $S(\omega_{c,v})$ small arbitrarily by taking a sufficiently large $c$.

Proof..

The reason why we can find a sufficiently small number $v>0$ satisfying the statement in this claim is that $\min\{||\sigma_{F}||\}$ on this region increase as $v\to 0$ and $4am/l<4$. In order to prove this Claim, we need the following lemma.

Lemma 4.4**.**

By taking a suitable local trivialization of $K_{X}^{-l}\otimes L_{X}^{m}$, we may assume that if $(z^{1},...,z^{n-2},w_{F},z_{n})=(0,...,0,w_{F},0)$, we have

[TABLE]

Thus, we can prove this claim by using the same way in the previous claim. ∎

The remained case is the following claim.

Claim 4**.**

On the region defined by

[TABLE]

we can make the scalar curvature $S(\omega_{c,v,\eta})$ small arbitrarily by taking a sufficiently large $c$.

Proof..

On this region, we can show that $S(\omega_{c,v,\eta})=O(c^{-2})$ similarly. Thus, we have finished proving Theorem 1.2. ∎