Smooth approximation of twisted Kähler-Einstein metrics

Lize Jin; Feng Wang

doi:10.1515/ans-2022-0032

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Smooth approximation of twisted Kähler-Einstein metrics

Lize Jin and Feng Wang

Published/Copyright: January 18, 2023

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From the journal Advanced Nonlinear Studies Volume 23 Issue 1

Abstract

In this article, we prove the existence of smooth approximations of twisted Kähler-Einstein metrics using the variational method.

Keywords: twisted Kahler-Einstein metrics; complex Monge-Ampère equation

MSC 2010: 32Q20; 32Q26; 32W20

1 Introduction

Let ( M , ω 0 ) be a compact Kähler manifold and T be a closed positive current. Assume that c 1 ( M , T ) ≔ 2 π c 1 ( M ) − [ T ] is a positive class and ω ∈ c 1 ( M , T ) . We say that ω is a twisted Kähler-Einstein metric if

Ric ω = ω + T

holds as currents. Twisted Kähler-Einstein metric can be considered as a generalization of Kähler-Einstein metric. The twisted term can be a current in general. If the current is the Dirac measure along a smooth divisor, the metric is the conic Kähler-Einstein metric. The existence of twisted Kähler-Einstein metric is proved in [3,8,16]. The metric ω is obtained using the variational method, so there is little information of the metric geometry of ω . As a first step, we want to study the smooth approximation of metric ω as shown in [13,14].

We always assume that T is a closed positive current with klt singularities. By choosing a smooth (1,1)-form θ in the same cohomological class of T , we obtain

(1) T = θ + − 1 ∂ ∂ ¯ ψ ,

where ψ is a quasi-psh function such that e − ψ ∈ L p ( M , ω 0 ) for some p > 1 . Then the following holds.

Theorem 1.1

Let ω 0 be a smooth Kähler metric and ω = ω 0 + − 1 ∂ ∂ ¯ φ be a twisted Kähler-Einstein metric such that φ is bounded. If T is smooth on an open set U, then φ is smooth on U. Moreover, if T has analytic singularity and Aut 0 ( X , T ) = 0 , there exists a sequence of smooth metric ω i with Ricci curvature bounded from below such that ω i converges to ω smoothly outside the singularity of T .

The smoothness of ω on the regular part of T is proved in Proposition 2.1. This result is essentially proved in [11] (see also Appendix B in [1]). The existence of smooth approximation is proved in Proposition 3.1 using the perturbation method in [14].

2 Regularity of twisted Kähler-Einstein metric

In this section, we prove the smoothness of φ in the region where T is smooth.

Proposition 2.1

Let ( M , ω 0 ) be a compact Kähler manifold and T be a closed positive current, c 1 ( M , T ) = [ ω 0 ] . Assume that there exists a twisted Kähler-Einstein metric ω φ = ω 0 + − 1 ∂ ∂ ¯ φ with bounded potential. If for the neighborhood U of x ∈ M , T ∣ U is smooth, then ω φ is smooth on U .

Since ω φ is a twisted Kähler-Einstein metric, it satisfies

(2) Ric ( ω φ ) = ω φ + T .

For c 1 ( M , T ) = [ ω 0 ] , there is a smooth function h such that

ω 0 = Ric ( ω 0 ) + − 1 ∂ ∂ ¯ h − θ .

So we obtain

Ric ( ω φ ) = Ric ( ω 0 ) + − 1 ∂ ∂ ¯ ( h + φ + ψ ) ,

which is equivalent to

(3) ( ω 0 + − 1 ∂ ∂ ¯ φ ) n = e − h − φ − ψ ω 0 n

by adding a constant to φ . We only need to prove that φ is smooth in the region where ψ is smooth. First, we give the C 0 -estimate. Since e − ψ ∈ L p and φ is bounded, we obtain f = e − h − ψ − φ ∈ L p , so C 0 -estimate is obtained by Corollary 6.9 in [9].

Next we show the C 2 -estimate. By Theorem 9.1 in [6], we know the following:

Theorem 2.2

Let ϕ be a quasi-psh function on compact Kähler manifold ( M , ω 0 ) such that for a smooth (1,1) form θ

− 1 ∂ ∂ ¯ ϕ ≥ θ .

Then there exists a decreasing sequence ϕ ε ∈ C ∞ ( M ) having the following properties:

There exists a constant C such that
− 1 ∂ ∂ ¯ ϕ ε ≥ θ − C ω 0 .
lim ε → 0 ϕ ε ( x ) = ϕ ( x ) for all x ∈ M .

So we have the decreasing sequences of smooth quasi-psh functions { φ ˜ ε } , { ψ ε } converging to φ , ψ , respectively. Since φ is continuous, { φ ˜ ε } converge to φ in C 0 -topology. And since

∣ e − ψ − e − ψ ε ∣ ≤ e − ψ , e − ψ ∈ L p ,

e − ψ ε converges to e − ψ in L p norm by dominated convergence theorem. By the result of Yau [15], the equation

( ω 0 + − 1 ∂ ∂ ¯ φ ) n = e − h − φ ˜ ε − ψ ε ω 0 n

has smooth solution φ ε .

Proposition 2.3

Assume φ ε satisfies

(4) ( ω 0 + − 1 ∂ ∂ ¯ φ ε ) n = e − h − φ ˜ ε − ψ ε ω 0 n ,

then Δ φ ε = O ( e − ψ ε ) .

Proof

Write ( Δ , tr ) and ( Δ ω ε , tr ω ε ) as the Laplace operator and trace with respect to ω 0 , ω ε , and

ω ε = ω 0 + − 1 ∂ ∂ ¯ φ ε .

We only need to prove

tr ( ω ε ) ≤ A e − ψ ε .

Recall the Laplace inequality for the second-order estimate in [12].

Lemma 2.4

If τ and τ ′ are two Kähler forms on a complex manifold, then there exists a constant B > 0 only depending on a lower bound for the holomorphic bisectional curvature of τ such that

Δ τ ′ log tr τ ( τ ′ ) ≥ − tr τ Ric ( τ ′ ) tr τ ( τ ′ ) − B tr τ ′ ( τ ) .

It follows that

Δ ω ε log tr ( ω ε ) ≥ − tr Ric ( ω ε ) tr ( ω ε ) − B tr ω ε ( ω 0 ) .

On the other hand, by applying − 1 ∂ ∂ ¯ log to (4), we obtain

− Ric ( ω ε ) = − Ric ( ω 0 ) − − 1 ∂ ∂ ¯ ( h + φ ˜ ε + ψ ε ) ≥ − A ω 0 − − 1 ∂ ∂ ¯ ( φ ˜ ε + ψ ε ) ,

then

(5) Δ ω ε log tr ( ω ε ) ≥ − A n + Δ ( φ ˜ ε + ψ ε ) tr ( ω ε ) − B tr ω ε ( ω 0 ) .

Since ψ ε , φ ˜ ε are quasi-psh functions, we have

(6) 0 ≤ A ω 0 + − 1 ∂ ∂ ¯ ( ψ ε + φ ˜ ε ) ≤ tr ω ε ( A ω 0 + − 1 ∂ ∂ ¯ ( ψ ε + φ ˜ ε ) ) ω ε ⇒ A n + Δ ( ψ ε + φ ˜ ε ) ≤ ( A tr ω ε ( ω 0 ) + Δ ω ε ( ψ ε + φ ˜ ε ) ) tr ω ε ⇒ Δ ω ε ( ψ ε + φ ˜ ε ) ≥ A n + Δ ( ψ ε + φ ˜ ε ) tr ω ε − A tr ω ε ( ω 0 ) .

Actually, constants A for two inequalities can be chosen as the same. Combining (5) and (6), we obtain

(7) Δ ω ε ( log tr ( ω ε ) + ψ ε + φ ˜ ε ) ≥ − A tr ω ε ( ω 0 ) .

We have ω ε = ω 0 + − 1 ∂ ∂ ¯ φ ε , hence,

n = tr ω ε ( ω 0 ) − Δ ω ε φ ε .

We deduce from (7) that

(8) Δ ω ε ( log tr ( ω ε ) + ψ ε + φ ˜ ε − A 1 φ ε ) ≥ tr ω ε ( ω 0 ) − A 2 .

on M , with constants A 1 and A 2 . Set

H = log tr ( ω ε ) + ψ ε + φ ˜ ε − A 1 φ ε .

Since ω ε is smooth on X , H achieves its maximum at some x 0 belongs to smooth part, and (8) yields

tr ω ε ( ω 0 ) ( x 0 ) ≤ A 2 .

On the other hand, a trivial inequality shows that

tr τ ( τ ′ ) ≤ τ ′ τ n tr τ ′ ( τ ) n − 1

for any two Kähler forms τ , τ ′ . Hence,

log tr ( ω ε ) ≤ log ( e − h − ψ ε − φ ˜ ε ) + ( n − 1 ) log tr ω ε ( ω 0 ) ≤ A 3 + A 4 ( log tr ω ε ( ω 0 ) ) − ( ψ ε + φ ˜ ε ) ,

then

H ≤ sup M H = H ( x 0 ) ≤ A 3 + A 4 ( log tr ω ε ( ω 0 ) ) − A 1 φ ε ≤ A 0

on M , which means that

log tr ( ω ε ) + ψ ε + φ ˜ ε − A 1 φ ε ≤ A 0 .

For φ is bounded and φ ˜ ε converges in C 0 -topology, we infer

tr ( ω ε ) ≤ A e − ψ ε . □

Since we have e − h − ψ ε − φ ˜ ε → e − h − ψ − φ in L p , it follows that φ ε converges as ε → 0 to the solution φ of

( ω 0 + − 1 ∂ ∂ ¯ φ ) n = λ e − h − ψ − φ ω 0 n .

So we know that φ satisfies as well ∣ φ ∣ C 1 , 1 ≤ A e − ψ . Thus, for any neighborhood U with ψ is smooth, we have

∣ φ ∣ C 1 , 1 ≤ C .

By the Evans-Krylov theory, there is some α ∈ ( 0 , 1 ) such that

∣ φ ∣ C 2 , α ≤ C ′ .

By applying ∂ l to equation (3), we obtain

a i j ¯ ∂ i j ¯ ( ∂ l φ ) + ( ∂ l φ ) = f ,

where f = − ∂ l ( h + ψ ) is smooth on U . Through Schauder interior estimate and bootstrap argument, we obtain the regularity of φ on U . Proposition 2.1 is proved.

3 Approximate metrics with uniform Ricci lower bound

In this section, we prove the second part of Theorem 1.1 when ψ has analytic singularity, i.e., ψ is equal to u + ∑ i = 1 m ∣ f i ∣ 2 locally, where u is a smooth function and f i ( 1 ≤ i ≤ m ) are some analytic functions. It is easy to see that ( e ψ + δ ) − 1 is a smooth function for any real number δ > 0 or positive smooth function δ . So we can perturb equation (3) by

(9) ( ω 0 + − 1 ∂ ∂ ¯ φ δ ) n = λ e − h − φ δ ( e ψ + δ e − K φ δ ) − 1 ω 0 n .

We will use the variational method to solve (9) as shown in [14].

Proposition 3.1

Assume Aut 0 ( M , T ) = 1 , and θ + K ω 0 ≥ 0 . Then there are constants a , b , δ 0 > 0 depending on ( M , ω 0 , ψ ) , such that for δ < δ 0 (9) has a smooth solution ω δ with some λ ∈ [ a , b ] , which converges to ω φ for δ approaching 0 outside the singularity of ψ . Moreover, the Ricci curvature of ω δ is greater than 1 − K uniformly.

As shown in [4], define

PSH full ( M , ω 0 ) = { φ ∈ PSH ( M , ω 0 ) ∣ lim j → ∞ ∫ φ ≤ − j ( ω 0 + − 1 ∂ ∂ ¯ max { φ , − j } ) n = 0 } ,

and the Monge-Ampère energy on PSH full ( M , ω 0 ) :

E ( φ ) = 1 ( n + 1 ) V ∑ i = 0 n ∫ M φ ω 0 i ∧ ω φ n − i .

Set

ℰ 1 ( M , ω 0 ) = { φ ∈ PSH full ( M , ω 0 ) ∣ E ( φ ) > − ∞ }

and

ℰ C 1 ( M , ω 0 ) = { φ ∈ ℰ 1 ( M , ω 0 ) , sup M φ ≤ C and E ( φ ) ≥ − C } ,

which is weakly compact for each C > 0 .

Then, we define

Q = { φ ∈ ℰ 1 ( M , ω 0 ) ∣ ∫ M h δ ( e − φ ) ω 0 n = ∫ M h δ ( 1 ) ω 0 n } ,

where

h δ ( x ) = ∫ 0 x e − h ( e ψ + δ t K ) − 1 d t .

By Lemma 6.4 of [2], we obtain

Lemma 3.2

The map

ℰ 1 ( M , ω 0 ) → L 1 ( M , ω 0 ) : φ → e − φ

is continuous. Thus, Q is a closed subset of ℰ 1 ( M , ω 0 ) .

We have the following two functionals on ℋ :

J ( φ ) = 1 V ∫ M φ ω 0 n − E ( φ ) ,

F δ ( φ ) = − E ( φ ) − log ∫ M h δ ( e − φ ) ω 0 n .

It is easy to see that

F δ ( φ ) = − E ( φ ) + F δ ( 0 ) , F δ ( 0 ) = − log ∫ M h δ ( 1 ) ω 0 n .

For δ < 1 , F δ ( 0 ) is uniformly bounded by a constant depending on ( M , ω 0 , ψ , h ) .

Lemma 3.3

J ( φ ) is lower semi-continuous on Q .

Proof

Actually, by Proposition 2.10 in [4], we know that J ( φ ) is lsc on ℰ 1 ( M , ω 0 ) . Since ℋ is closed subset of ℰ 1 ( M , ω 0 ) , the lemma is proved.□

Now we prove the proposition. Since Aut 0 ( M , T ) = 1 , by Theorem 2.18 in [3], we know that Ding functional

F 0 ( φ ) = − E ( φ ) − log ∫ M e − h − φ − ψ ω 0 n

is coercive, i.e., there are some positive constants A and B , such that

F 0 ( φ ) ≥ A J ( φ ) − B .

Clearly, F δ ≥ F 0 , so F δ is also coercive. Choose a minimizing sequence { φ j } of F δ satisfying:

lim j → ∞ F δ ( φ j ) = inf φ ∈ Q F δ ( φ ) .

For j large sufficiently, we have

(10) J ( φ j ) ≤ 1 A ( F δ ( φ j ) + B ) ≤ 1 A ( F δ ( 0 ) + B ) + 1 .

Hence,

(11) 1 V ∫ M φ j ω 0 n ≤ ∣ J ( φ j ) ∣ + ∣ F δ ( φ j ) ∣ + ∣ F δ ( 0 ) ∣ ≤ C ( A , B , F δ ( 0 ) ) .

So we obtain

(12) ∣ sup ( φ j ) ∣ ≤ C ( A , B , F δ ( 0 ) ) .

From (10) and (12), we know that φ j lies in a weakly compact subset ℰ C 1 ( M , ω 0 ) of ℰ 1 ( M , ω 0 ) . Hence, by taking a subsequence of { φ j } , we can assume that φ j converge to a limit φ δ in ℰ 1 ( M , ω 0 ) . From Lemma 3.3, we know that the functional − E ( ϕ ) is lower semi-continuous. Thus, F δ is lower semi-continuous. It follows that φ δ is a minimizer of F δ . As the proof of Theorem 4.1 in [2], we can show that φ δ is a solution of (9) for some λ .

Then, we give the estimate of λ . By (11), we know that

∫ M ∣ φ j ∣ ω 0 n ≤ C ( A , B , F δ ( 0 ) , V ) .

Hence,

∣ { e − φ j ≥ C 1 } ∣ = ∣ { φ j ≤ − ln C 1 } ∣ ≤ ∫ M ∣ φ j ∣ ω 0 n ln C 1 ≤ C ( A , B , F δ ( 0 ) , V ) ln C 1 .

So we can choose C 1 > 0 , such that

∣ { e − φ j ≥ C 1 } ∣ ≤ V 4 .

And we also can choose ε > 0 , such that

∣ { e ψ ≤ ε } ∣ ≤ V 4 .

Set

N = { e − φ j ≤ C 1 } ∩ { e ψ ≥ ε } ,

then

∣ N ∣ ≥ V 2 .

On N , there is a δ 0 ( M , ω 0 , ψ ) such that for any δ ≤ δ 0 , we have

1 ≤ e − φ j ≤ C 1

and

( e ψ + δ e − K φ j ) − 1 ≥ 1 2 e − ψ .

So we obtain

∫ N e − h − φ δ ( e ψ + δ e − K φ δ ) − 1 ω 0 n ≥ C 2 ( M , ω 0 , ψ , h ) .

Combining with perturbed equation, we obtain

λ ≤ V C 2 ( M , ω 0 , ψ , h ) .

On the other hand, we have

e − h − φ δ ( e ψ + δ e − K φ δ ) − 1 ≤ h δ ( e − φ δ ) .

Hence,

∫ M e − h − φ δ ( e ψ + δ e − K φ δ ) − 1 ω 0 n ≤ ∫ M h δ ( e − φ δ ) ω 0 n = ∫ M h δ ( 1 ) ω 0 n .

So we obtain

λ ≥ V ∫ M h δ ( 1 ) ω 0 n .

Next, we establish the regularity of φ δ .

Lemma 3.4

For some α ∈ ( 0 , 1 ) , ∣ φ δ ∣ C α ( M , ω 0 ) ≤ C , where C depends on ( M , ω 0 , ψ ) .

Proof

From above, we know that

φ δ ∈ ℰ C 1 ( M , ω 0 ) ⊂ PSH full ,

where ℰ C 1 ( M , ω 0 ) is a weak compact subset. By Proposition 1.4 of [1], there is q > 1 and ∣ e − φ δ ∣ L q is uniformly bounded by constant C ( q ) . Indeed, the map

ℰ 1 → L q ( M , ω 0 ) : φ δ → e − φ δ

is continuous. Since e − ψ ∈ L p , so

∣ ( e ψ + δ e − K φ δ ) − 1 ∣ L p ≤ ∣ e − ψ ∣ L p ≤ C ( M , ω 0 , ψ , p ) .

Then for any p 0 ∈ ( 1 , p ) and some constant independent of δ satisfies

∣ e − φ δ ⋅ e − h ⋅ ( e ψ + δ e − K φ δ ) − 1 ∣ L p 0 ≤ C .

By Theorem 2.1 of [10], we have ∣ φ δ ∣ C α ( M , ω 0 ) ≤ C .□

Proposition 3.5

There exists δ l → 0 such that φ δ l converges to φ + c in the C 0 -topology for some constant c.

Proof

By Lemma 3.4, we can choose a subsequence φ δ l , which converges to a continuous function φ 0 . Moreover, some λ for φ 0 satisfy

( ω 0 + − 1 ∂ ∂ ¯ φ 0 ) n = λ e − h − ψ − φ 0 ω 0 n .

Then, through the unique result of Proposition 8.2 of [5], we know that φ 0 = φ + c .□

Now we show φ δ is a smooth function. We need a special case of Proposition 2.1 in [7]:

Proposition 3.6

Let φ be a solution of

ω φ n = e ψ + − ψ − ω 0 n ,

where ω φ = ω 0 + − 1 ∂ ∂ ¯ φ and ψ ± are smooth functions.

Further, we assume that there exists C > 0 such that:

∣ φ ∣ ≤ C ;
∣ ψ ± ∣ ≤ C and − 1 ∂ ∂ ¯ ψ ± ≥ − C ω 0 ;
Ric ( ω 0 ) is bounded from below by − C .

Then there exists a constant A > 0 depending only on C, such that

1 A ω 0 ≤ ω φ ≤ A ω 0 .

Choose a sequence of smooth ω 0 − p s h functions φ ˜ j , which converges to φ δ in C 0 norm.

Lemma 3.7

If φ j is any solution of

(13) ( ω 0 + − 1 ∂ ∂ ¯ φ j ) n = λ e ( K − 1 ) φ ˜ j − h ( e ψ + K φ ˜ j + δ ) − 1 ω 0 n ,

then for some C = C ( M , δ , ∣ φ δ ∣ C 0 ) ,

∣ Δ φ j ∣ ≤ C .

Proof

First, we observe that for any smooth f > 0 ,

− 1 ∂ ∂ ¯ log ( f + δ ) ≥ f ( f + δ ) − 1 ∂ ∂ ¯ log f .

Let

u j = log ( e ψ + K φ ˜ j + δ ) .

Then,

(14) − 1 ∂ ∂ ¯ u j ≥ e ψ + K φ ˜ j e ψ + K φ ˜ j + δ − 1 ∂ ∂ ¯ ( ψ + K φ ˜ j ) ≥ − e ψ + K φ ˜ j e ψ + K φ ˜ j + δ ( θ + K ω 0 ) ≥ − ( θ + K ω 0 ) .

Since θ is smooth, then

− 1 ∂ ∂ ¯ u j ≥ − C ω 0 .

Moreover we know ω 0 − p s h function φ ˜ j satisfies

− 1 ∂ ∂ ¯ ( K φ ˜ j ) ≥ − K ω 0 .

The right-hand side of (13) can be written as e ψ j + − ψ j − , where

ψ j + = K φ ˜ j ; ψ j − = u j + φ ˜ j + h .

As mentioned earlier, for some constant C > 0 , we have

− 1 ∂ ∂ ¯ ψ j ± ≥ − C ω 0 .

Hence, by Proposition 3.6, we have ∣ Δ φ j ∣ ≤ C δ . □

It follows from the uniqueness theorem for complex Monge-Ampère equations that φ j converges to φ δ + c for some constant c , so we have

∣ φ δ ∣ C 1 , 1 ( M , ω 0 ) ≤ C δ .

By Evans-Krylov theory, we know that for some α ∈ ( 0 , 1 ) ,

∣ φ δ ∣ C 2 , α ( M , ω 0 ) ≤ C δ ′ ,

where C δ ′ depends on δ . And higher order estimates are obtained by bootstrap. So φ δ is a smooth function.

Now we can calculate the Ricci curvature.

Proposition 3.8

Assume ω δ is smooth metric that satisfies (9), then

Ric ( ω δ ) ≥ ( 1 − K ) ω δ .

Proof

Write (9) as follows:

( ω 0 + − 1 ∂ ∂ ¯ φ δ ) n = λ e ( K − 1 ) φ δ − h ( e ψ + K φ δ + δ ) − 1 ω 0 n .

Then the Ric ( ω δ ) is equal to

− 1 ( ( 1 − K ) ∂ ∂ ¯ φ δ + ∂ ∂ ¯ h + ∂ ∂ ¯ log ( e ψ + K φ δ + δ ) ) + Ric ( ω 0 ) ≥ ( 1 − K ) − 1 ∂ ∂ ¯ φ δ + e ψ + K φ δ e ψ + K φ δ + δ − 1 ∂ ∂ ¯ ( ψ + K φ δ ) + ω 0 + θ ≥ ω δ − δ K e ψ + K φ δ + δ − 1 ∂ ∂ ¯ φ δ + δ e ψ + K φ δ + δ θ = ω δ − δ K e ψ + K φ δ + δ ω δ + δ ( K ω 0 + θ ) e ψ + K φ δ + δ ≥ ( 1 − K ) ω δ . □

Lemma 3.9

There exists C = C ( M , ω 0 , ∣ φ δ ∣ C 0 , ∣ h ∣ C 0 ) such that

1 C ω 0 ≤ ω δ ≤ C ⋅ e − ψ ω 0 .

Proof

Since the Ricci curvature of ω δ is bounded below by ( 1 − K ) ω δ , by the Chern-Lu inequality, we have

Δ ω δ log tr ω δ ω 0 ≥ ( K − 1 ) − B tr ω δ ω 0 ,

where B is the upper bounded of the bisectional curvature of ω 0 . Then we have

Δ ω δ ( log tr ω δ ω 0 − ( B + 1 ) φ δ ) ≥ tr ω δ ω 0 − n ( B − 1 ) + ( K − 1 ) .

So by the maximum principle, we obtain

tr ω δ ω 0 ≤ n ( B − 1 ) − ( K − 1 ) ≤ C .

Moreover, combined with (9)

tr ω 0 ω δ ≤ tr ω δ ω 0 ⋅ ω δ n ω 0 n ≤ C ⋅ e − ψ .

Then, we obtain both the upper and lower bound of ω δ .□

Now ω δ is a sequence of smooth metrics such that Ric ( ω δ ) ≥ ( 1 − K ) ω δ and the potential φ δ converges to φ in C 0 norm. By Lemma 3.9, we have a uniform C 2 estimate of φ δ outside the singularity of ψ . Together with the Evans-Krylov theory, we know that φ δ converges to φ smoothly in the regular part. The proof of Proposition 3.1 is complete.

Funding information: This study was partially supported by NSFC Grants 11971423 and 12031017.
Conflict of interest: Authors state no conflict of interest.

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Received: 2022-06-17

Revised: 2022-08-24

Accepted: 2022-09-20

Published Online: 2023-01-18

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https://doi.org/10.1515/ans-2022-0032

Keywords for this article

twisted Kahler-Einstein metrics; complex Monge-Ampère equation

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