H\"older estimates for degenerate complex Monge-Amp\`ere equations

TL;DR

This paper extends H"older continuity results for complex Monge-Amp ext{e}re equations to singular K"ahler varieties, using geometric regularization and partial $C^0$ estimates, with applications to K"ahler-Einstein varieties.

Contribution

It introduces a new approach to establish H"older estimates on singular K"ahler varieties via geometric regularization and partial $C^0$ estimates, advancing the understanding of complex Monge-Amp ext{e}re equations.

Findings

Established uniform H"older continuity for Monge-Amp ext{e}re equations on K"ahler varieties.

Proved local potentials of smoothable K"ahler-Einstein varieties are H"older continuous.

Developed a geometric regularization method based on partial $C^0$ estimates.

Abstract

Uniform $L^\infty$ and H\"older estimates were proved by the Kolodziej for complex Monge-Amp\`ere equations on compact K\"ahler manifolds with $L^p$ volume measure with $p>1$. On the other hand, establishing H\"older estimates on singular K\"ahler varieties has remained open. In this paper, we establish uniform H\"older continuity for a family of complex Monge-Amp\`ere equations on K\"ahler varieties, by developing a geometric regularization based on the partial $C^0$ estimate, i.e., quantitive Kodaira embeddings. As an application, we prove that local potentials of smoothable K\"ahler-Einstein varieties are H\"older continuous.

Equations

$$\omega^{n}\ =\ (\omega_{0}+{{\sqrt{{\scalebox{0.75}[1.0]{$-1$}}}}}\thinspace\partial\bar{\partial}\varphi)^{n}\ =\ e^{F}\omega_{0}^{n},\ \ \ \ \ \omega_{0}+{{\sqrt{{\scalebox{0.75}[1.0]{$-1$}}}}}\thinspace\partial\bar{\partial}\varphi>0,\ \ \ \ \ \sup\varphi=0.$$

$${\int }_X{\omega }_0^n\ge 0.$$

$${\omega }_{\mathrm{FS}}=\frac{1}{k}{T}_k^{\ast }{\Theta }_{\mathrm{FS}}$$

$$\omega_{X}=\omega_{\rm FS}+{{\sqrt{{\scalebox{0.75}[1.0]{$-1$}}}}}\thinspace\partial\bar{\partial}\varphi_{X},~\sup\varphi_{X}=0.$$

$$|{\phi }_X(x)-{\phi }_X(y)|\le C{d}_{\mathrm{FS}}(x,y{)}^{\alpha },$$

$$\pi :\mathcal{X}\subset {\mathbb{P}}^N\to \Delta$$

$$Ric(h_{KE})=\lambda {\omega }_{\mathrm{KE}}$$

$${\omega }_{\mathrm{KE}}=\theta +\partial \bar{\partial }{\phi }_{\mathrm{KE}}.$$

$$|{\phi }_{\mathrm{KE}}(x)-{\phi }_{\mathrm{KE}}(y)|\le C{d}_{{\mathbb{P}}^N}(x,y{)}^{\gamma }$$

$$\Omega =e^f|\eta {|}^{2/m}.$$

$$(\theta_{0}+{{\sqrt{{\scalebox{0.75}[1.0]{$-1$}}}}}\thinspace\partial\bar{\partial}\varphi)^{n}=e^{F}\Omega,~\sup_{X}\varphi=0.$$

$$|\phi (x)-\phi (y)|\le C{d}_{{\mathbb{P}}^N}(x,y{)}^{\gamma },$$

$$Ric(\omega )=-\partial \bar{\partial }\log {\omega }^n=Ric(\Omega )-\partial \bar{\partial }F\ge -C{\theta }_0$$

$$\rho (X,m\omega ,mL,h_L^m)(x):=\sum _{j=0}^N|{\sigma }_j{|}_{h_L^m}^2$$

$$⟨{\sigma }_i,{\sigma }_j{⟩}_{\mathrm{Hilb}({\mathrm{h}}_{\mathrm{L}}^{\mathrm{m}})}:={\int }_X{\sigma }_i{\bar{\sigma }}_j{h}_L^m(m\omega {)}^n={\delta }_{ij}$$

$$A^{-1}\le \rho (X,m{\omega }_X,mL,h_L^m)\le A$$

$${\left({\int }_X|u{|}^{\frac{2n}{n-1}}{\omega }_X^n\right)}^{\frac{n-1}{n}}\le S\left({\int }_X|u{|}^2{\omega }_X+{\int }_X|\nabla u{|}^2{\omega }_X^n\right),\forall u\in W^{1,2}(X)$$

$$U=\sum _{{\alpha }_1,...,{\alpha }_l=0}^{N_k}U({\alpha }_1,...,{\alpha }_l)s_{{\alpha }_1}\cdots s_{{\alpha }_l}$$

$$U({\alpha_{1},...,\alpha_{l}})\in H^{0}(X,(m-kl)L),\$$

$$\Vert U({\alpha }_1,...,{\alpha }_l){\Vert }_{\mathrm{Hilb}({\mathrm{h}}_{\mathrm{L}}^{(\mathrm{m}-\mathrm{lk})})}^2\le \frac{(n+l)!}{n!l!}\frac{(m-kl{)}^n}{m^n}{A}^{2n+2+l}\Vert U{\Vert }_{\mathrm{Hilb}({\mathrm{h}}_{\mathrm{L}}^{\mathrm{m}})}^2$$

$$\underset{m=0}{\overset{\infty }{⨁}}{H}^0(X,mL)$$

$$U\in H^0(X,(n+l+1)G+E+K_X)satisfies{\int }_X\frac{|U{|}^2{e}^{-{\psi }_E}}{|s{|}^{2(n+l+1)}}<\infty$$

$$U_0,...,U_N\in H^0(X,((n+l)G+E+K_X)\mathrm{and}{\int }_X\frac{|U_{\alpha }{|}^2{e}^{-{\psi }_E}}{|s{|}^{2(n+l)}}\le \frac{n+l}{l}{\int }_X\frac{|U{|}^2{e}^{-{\psi }_E}}{|s{|}^{2(n+l+1)}}$$

$$\bf{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}mL}\ =\ {\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(n+l+1)(kL)}\ +\ {\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}((m-(n+l+1)k)L-K_{X})}\ +\ {\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}K_{X}}:={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(n+l+1)G}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}E}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}K_{X}}$$

$${\psi }_E=(m-(n+l+1)k){\phi }_L-\log \frac{{\omega }_X^n}{n!}$$

$${{\sqrt{{\scalebox{0.75}[1.0]{$-1$}}}}}\thinspace\partial\bar{\partial}\psi_{E}\ =\ (m-(n+l+1)k)\omega_{X}\ +\ {\rm Ric}(\omega_{X})\geq\ \ \omega_{X}\ +\ {\rm Ric}(\omega_{X})\ \geq\ \omega_{X}/2$$

$$e^{-\psi }=\frac{e^{-(m-rk)\phi }}{e^{-([n+l+1]-r)k\phi }}\cdot \frac{{\omega }^n}{n!}\mathrm{and}{e}^{-([n+l+1]-r)k\phi }|s{|}^{2([n+l+1]-r)}={\rho }_k^{([n+l+1]-r)}$$

$$U=\sum _{{\alpha }_1=0}^{N_k}U({\alpha }_1;1)s_{{\alpha }_1}\mathrm{and}{\int }_X\frac{|U({\alpha }_1;1){|}^2{e}^{-\psi }}{|s{|}^{2[(n+l+1)-1]}}\le \frac{[n+l+1]-1}{[l+1]-1}{\int }_X\frac{|U{|}^2{e}^{-\psi }}{|s{|}^{2(n+l+1)}}$$

$$U({\alpha }_1;1)=\sum _{\alpha =0}^{N_k}U({\alpha }_1,{\alpha }_2;2)s_{{\alpha }_2}\mathrm{and}{\int }_X\frac{|U({\alpha }_1,{\alpha }_2;2){|}^2{e}^{-\psi }}{|s{|}^{2([n+l+1]-2)}}\le \frac{[n+l+1]-2}{[l+1]-2}{\int }_X\frac{|U(\alpha ;1){|}^2{e}^{-\psi }}{|s{|}^{2([n+l+1]-1)}}$$

$$U(\alpha_{1},...,{\alpha_{r-1}};1)\ =\ \sum_{\alpha_{r}=0}^{N_{k}}U(\alpha_{1},\alpha_{2},...,\alpha_{r};r)s_{\alpha_{r}}\ {\rm and}\ \$$

Full text

1. Introduction

Let $X$ be a compact complex manifold of dimension $n$, and $\omega$ a Kähler metric on $X$. We say that $\omega$ is a Kähler-Einstein metric if it satisfies the equation ${\rm Ric}(\omega)=\lambda\omega$ for some $\lambda\in\{-1,0,1\}$. Uniqueness of such metrics was proved by Calabi [6] in the cases $\lambda=-1,0$ and Bando-Mabuchi [4] in the case $\lambda=1$. Existence was settled much later: The case $\lambda=-1$ was proved by Aubin [1] and Yau [58]. The case $\lambda=0$ was achieved in Yau’s celebrated proof of the Calabi-Yau conjecture [58]. The case $\lambda=1$ was proved in the deep and influential papers of Chen-Donaldson-Sun [7, 8, 9] and Tian [56].

Central to these existence proofs is the establishment of a priori estimates. When $\lambda=0$, a critical first step is the $C^{0}$ a priori estimate, which can be formulated as follows. Consider the complex Monge-Ampère equation

[TABLE]

Here $\omega_{0}$ is a smooth background metric on $X$ and $F$ is a smooth function satisfying $\int_{X}e^{F}\omega_{0}^{n}=\int_{X}\omega_{0}^{n}$. Yau [58] showed that (1.1) admits a smooth solution $\varphi$. The first step in the proof is an a priori estimate which states that $\|\varphi\|_{C^{0}(X)}\leq C(n,\omega_{0},\|F\|_{L^{\infty}(X)})$. A major advance in extending these estimates was achieved by Kołodziej ([28, 29]), who, using techniques of pluripotential theory, was able to establish both $L^{\infty}$ and $C^{\alpha}$ a priori estimates for the solutions under weaker assumptions on $F$. More precisely, for any $p>1$ there exists $\alpha>0$ such that $\|\varphi\|_{C^{\alpha}(X)}\leq C(n,\omega_{0},\|e^{F}\|_{L^{p}(X)})$. The $L^{\infty}$ estimate is extended to the degenerate case for equation (1.1) when $[\omega_{0}]$ is a big and semi-positive class by [16]. Naturally one would also like to extend the Hölder estimate to the degnerate setting.

Problem 1**.**

Suppose the smooth closed $(1,1)$-form $\omega_{0}$ is nonnegative with

[TABLE]

The equation (1.1) still admits a unique solution $\varphi\in L^{\infty}(X)\cap PSH(X,\omega_{0})$. Is $\varphi$ still Hölder continuous, i.e., $\varphi\in C^{\alpha}(X)$ for some $\alpha>0$?

Equation (1.1) is degenerate when $[\omega]$ is not Kähler. Such degenerate equations are essential to the study of canonical metrics such as Kähler-Einstein metrics on a singular Kähler variety $X$ because the complex Monge-Ampère equations become degenerate after being pulled back on the non-singular model of $X$. In recent years, there has been a growing interest in Kähler-Einstein metrics on singular varieties, motivated in part by their role in the compactification of moduli space and their connection to K-stability (c.f. [54, 15, 7, 8, 9, 56, 52]). The existence theory for Kähler-Einstein currents on singular varieties is fairly complete, However, unlike the case of smooth manifolds, a priori estimates for singular Kähler-Einstein metrics and their behavior near the singular set are largely unknown. The stronger regularity, particularly the Hölder continuity for the Kähler-Einstein potentials with respect to smooth Kähler metrics on $X$, reveals the deep connections between intrinsic (the Kähler-Einstein metrics) and extrinsic (the Bergman metrics via projective embeddings) geometric structures on $X$. For example, the Hölder estimate for complex Monge-Ampère equation reveals the Hölder comparison between the background metric and the Kähler metric induced from the equation [32].

The goal of this paper is to establish various Hölder estimates for complex Monge-Ampère equations on singular Kähler varieties under suitable geometric assumptions. We will first develop a uniform Hölder estimate for polarized manifolds with Ricci curvature lower bound and diameter upper bound. Fix $n,d\geq 1$. We define $\mathcal{F}(n)$ to be the set of all quadruples $(X,\omega_{X},L,h_{L})$ satisfying

1. (1)

$(X,\omega_{X})$ is a compact Kähler manifold of dimension $n$, 2. (2)

${\rm Ric}(\omega_{X})\geq{-1}$, 3. (3)

$L\rightarrow X$ a holomorphic line bundle with metric $h_{L}$ satisfying -{{\sqrt{{\scalebox{0.75}[1.0]{-1}}}}}\thinspace\partial\bar{\partial}\log h_{L}=\omega_{X}.

and ${\mathcal{F}}(n,d)=\{(X,\omega_{X},L,h_{L})\in{\mathcal{F}}(n)\,:\,{\rm diam}(X,\omega_{X})\leq d)$. We also define ${\mathcal{F}}^{0}(n)=\{(X,\omega_{X},L,h_{L})\in{\mathcal{F}}(n)\,:\,{\rm Ric}(\omega_{X})\leq 1\}$ and similarly for ${\mathcal{F}}^{0}(n,d)$

The family $\mathcal{F}(n,D)$ is bounded: According to a theorem of Donaldson-Sun [15] and Liu-Szekelyhidi [36], there exists $k(n,D)\geq 1$ such that the map $T_{k}:X\rightarrow{\mathbb{P}}^{N_{k}}$, given by an orthonormal basis of $H^{0}(X,kL)$, is an embedding. If $\Theta_{\rm FS}$ is the Fubini-Study metric on ${\mathbb{P}}^{N_{k}}$, we define

[TABLE]

and we let $\varphi_{X}$ be defined by the equation

[TABLE]

The following is our first result.

Theorem 1**.**

For any $n$, $D>1$, there exist $k(n,D)>0$, $C(n,D)>0$ and $\alpha(n,D)>0$ such that for any $(X,\omega_{X},L,h_{L})\in\mathcal{F}(n,D)$ and $x,y\in X$,

[TABLE]

where $\varphi_{X}$ is defined in 1.3 and $d_{\rm FS}$ is the extrinsic distance function on $X$ associated to $\omega_{\rm FS}$.

The idea of the proof is the global approximation of $\varphi_{X}$ by Bergman potentials as in [14, 44, 45, 38, 39] combined with the partial $C^{0}$-estimate, i.e., quantitive Kodaira embeddings developed in [54, 15, 36] and a finite generation scheme in [30].

Remark: If $r$ is a positive integer, we have the multiplication map ${\mathcal{F}}(n,d)\rightarrow{\mathcal{F}}(n,rd)$ defined by $r\cdot(X,\omega_{X},L,h_{L})=(X,r\omega_{X},rL,h_{L}^{r})$. Let ${\mathcal{F}}_{r}(n,d)$ be the image of this map. If $r=r(n,d)\geq 1$ we may, in the proof of (1.4), replace ${\mathcal{F}}(n,d)$ by ${\mathcal{F}}_{r}(n,d)$. We will make this type of replacement repeatedly throughout the proof.

We will apply Theorem 1 to Kähler currents on smoothable singular projective varieties. Let us recall some definitions before stating our results.

Definition 1**.**

Let $X$ be an $n$-dimensional projective variety with klt singularities. $X$ is said to be smoothable if there is a flat family

[TABLE]

over the unit disk $\Delta\subseteq{\mathbb{C}}$ such that

1. (1)

$X={\mathcal{X}}_{0}$* is the central fiber and $\pi:{\mathcal{X}}\backslash X_{0}\rightarrow\Delta\backslash\{0\}$ is smooth,* 2. (2)

$\mathcal{X}$* is $\mathbb{Q}$-Gorenstein and $K_{{\mathcal{X}}/D}$ is ${\mathbb{Q}}$-Cartier.*

Definition 2**.**

Let $X$ be a normal projective variety with klt singularities. A closed positive $(1,1)$-current $\omega_{\rm KE}$ with bounded local potentials is said to be a Kähler-Einstein current if it satisfies the following.

1. (1)

$[\omega_{\rm KE}]\in H^{2}(X,\mathbb{Q})$* is an ample $\mathbb{Q}$-Cartier divisor.* 2. (2)

Let $h_{KE}=(\omega_{\rm KE})^{n}$ be the hermitian metric on $-K_{X}$. Then $Ric(h_{KE})=-\partial\bar{\partial}\log h_{KE}$ is a well-defined closed $(1,1)$-current with

[TABLE]

for some $\lambda\in\{-1,0,1\}$.

We say that $(X,\omega_{\rm KE})$ is a projective Kähler-Einstein variety.

For $\lambda=-1,0,1$, $h_{KE}$ corresponds to a negative, flat and positive hermitian metrics. By the smoothing property of the weak Kähler-Ricci flow [50], it is shown in [53], $\omega_{\rm KE}$ is in fact smooth on $X^{\rm reg}$, the regular part of $X$.

Our next result gives the Hölder continuity for Kähler-Einstein potentials on smoothable Kähler-Einstein variety.

Theorem 2**.**

Let $(X,\omega_{\rm KE})$ be a smoothable projective Kähler-Einstein variety as the central fibre the smoothing family $\pi:\mathcal{X}\subset\mathbb{P}^{N}\rightarrow\Delta$. Let $\theta\in[\omega_{\rm KE}]$ be a smooth background metric on $X$ and

[TABLE]

Then there exist $C,\gamma>0$ such that

[TABLE]

where $d_{\mathbb{P}^{N}}$ is the distance between $x$ and $y$ in $\mathbb{P}^{N}$.

Theorem 2 confirms a conjecture in [21] in the case of smoothable Kähler-Einstein varieties. Remark: One can not expect Kähler-Einstein metrics on singular varieties to be more regular than Hölder continuous, as can already be seen in dimension two.

We will now return to Problem 1 and study Hölder continuity for complex Monge-Ampère equation on smoothable projective varieties. Suppose $X$ is an $n$-dimensional smoothable projective variety with klt singularities, where $X=\mathcal{X}_{0}$ in the flat family (1.5). Since $X$ has klt singularities, we can choose an adaptive volume measure $\Omega$ on $X$, i.e., locally there exist $m\in\mathbb{Z}^{+}$, a generator $\eta$ of $mK_{X}$ and a smooth function $f$ such that

[TABLE]

Let $\Theta$ be the Fubini-Study metric on $\mathbb{P}^{N}$ and $\theta_{t}=\Theta|_{\mathcal{X}_{t}}$ for $t\in\Delta$. We can consider the following complex Monge-Ampère equation on $X$

[TABLE]

where $\varphi\in PSH(X,\theta_{0})$ and $\int_{X}e^{F}\Omega=[\theta_{0}]^{n}$. It is proved in [16] that equation (1.7) admits a unique continuous solution if $e^{F}\in L^{p}(X,\Omega)$ for some $p>1$. Naturally, one would hope that $\varphi$ is Hölder continuous with respect to $\theta_{0}$ or $d_{\mathbb{P}^{N}}$ (the distance in $\mathbb{P}^{N}$).

Theorem 3**.**

Let $X$ be an $n$-dimensional smoothable projective variety with klt singularities as the central fibre the smoothing family $\pi:\mathcal{X}\subset\mathbb{P}^{N}\rightarrow\Delta$. Suppose there exists $p>1$ and $A>0$ such that

1. (1)

$e^{F}\in L^{p}(X,\Omega)$, 2. (2)

$-F\in PSH(X,A\theta_{0})$, i.e., A\theta_{0}-{{\sqrt{{\scalebox{0.75}[1.0]{-1}}}}}\thinspace\partial\bar{\partial}F\geq 0.

Then there exists a unique Hölder continuous (with respect to $d_{\mathbb{P}^{N}}$) solution $\varphi$ to the equation (1.7). More precisely, there exist $C>0$ and $\gamma>0$ such that for any $x,y\in X$,

[TABLE]

where $C$ and $\gamma$ depend on $X$, $\theta_{0}$, $\Omega$, $p$, $A$ and $||e^{F}||_{L^{p}(X,\Omega)}$.

Let $\omega=\theta_{0}+\partial\bar{\partial}\varphi$ for the solution $\varphi$ to equation (1.7). In addition to the algebraic smoothing condition for the underlying variety $X$, Theorem 3 imposes a positivity assumption for the Monge-Ampère measure $\omega^{n}$, i.e.,

[TABLE]

for some $C>0$ since $Ric(\Omega)=-\partial\bar{\partial}\log\Omega$ is bounded below by some negative propotion of $\theta_{0}$. The positivity assumption (1.9) is equivalent to a uniform lower bound for the Ricci curvature of $\omega$.

Corollary 1**.**

Let $X$ be an $n$-dimensional smoothable projective variety with klt singularities with klt singularities as the central fibre the smoothing family $\pi:\mathcal{X}\subset\mathbb{P}^{N}\rightarrow\Delta$. Suppose $F\in C^{\infty}(X)$. Then there exists a unique Hölder continuous (with respect to $d_{\mathbb{P}^{N}}$) solution $\varphi$ to the equation (1.7).

There are various extensions of Theorem 3 that will be investigated in forthcoming works. For example, the smoothable assumption in Theorem 3 can be removed in complex dimension $3$ as well as in high dimensions for normal varieties that admit special resolution of singularities.

2. Partial $C^{0}$ estimate and effective finite generation

We shall make use of the partial $C^{0}$ estimate of Zhang [59], which is extends earlier work of [55, 15, 36].

We will generally use $s$ to denote a section of $L$ and $\sigma$ to denote of section of $mL$ for $m>1$. After replacing ${\mathcal{F}}(n)$ by ${\mathcal{F}}_{r}(n)$ for an appropriate $r\geq 1$, we may assume $L\rightarrow X$ is very ample for all $(X,\omega_{X},L,h_{L})\in{\mathcal{F}}(n)$. Let $(X,\omega_{X},L,h_{L})\in\mathcal{F}(n)$ and define, for each $m\geq 1$ the Bergman kernel

[TABLE]

where $\{\sigma_{0},,,,.\sigma_{N_{m}}\}$ is an orthonormal basis of $H^{0}(X,mL)$, i.e.

[TABLE]

Theorem 4**.**

([59]).* Given $n\geq 1$ and $S>0$ there exist $r=r(n,S)\in\mathbb{N}$ and $A(n,S)>1$ with the following property. Let $(X,\omega_{X},L,h_{L})\in\mathcal{F}_{r}(n)$ and assume $C(\omega_{X})\leq S$ where $C(\omega_{X})$ is the Sobolev constant of $\omega_{X}$. Then for any $m\in\mathbb{N}$ we have*

[TABLE]

The condition $C(\omega_{X})\leq S$ means

[TABLE]

Corollary 2**.**

Let $n,d>0$. Then there exists $A=A(n,d)>1$ and $r=r(n,d)>1$ satisfying (2.1) for all $(X,\omega_{X},L,h_{L})\in{\mathcal{F}}_{r}(n,d)$.

This follows from Croke’s theorem [11] which guarantees that the Sobolev constants of $m\omega_{X}$ are uniformly bounded for $(X,\omega_{X},L,h_{L})\in{\mathcal{F}}(n,d)$ and $m\geq 1$.

Next we recall a theorem of Li [30].

Theorem 5**.**

[30]. Let $k\geq 1,l\geq 1$ and $m\geq(n+2+l)k$. Let $(X,\omega_{X},L,h_{L})\in\mathcal{F}(n,d)$, and $s_{0}\cdots s_{N_{k}}$ an orthonormal basis of $H^{0}(X,kL)$ with respect to ${\rm Hilb(h_{L}^{k})}$. In particular, $N_{k}+1=\dim H^{0}(X,kL)$. Then for $U\in H^{0}(X,mL)$, we can write

[TABLE]

where

[TABLE]

[TABLE]

Corollary 3**.**

Let $p>0$ be a multiple of $(n+2)$ and let $(X,L)\in{\mathcal{F}}_{p}(n)$. Then

[TABLE]

is generated, as a ${\mathbb{C}}$-algebra, by $H^{0}(X,L)$

Remarks. Li’s normalization of Hilb is a bit different, so in his formulation, the factor ${(m-kl)^{n}\over m^{n}}$ is missing. Also, in his proof of Theorem 5, he assumes ${\rm Ric}(\omega)>0$ and proves a slightly stronger estimate. The proof in the case ${\rm Ric}(\omega)\geq-\omega/2$ requires very minor modifications. For the sake of completeness, we provide it below. First we recall Siu’s global version of the Skoda division theorem.

Theorem 6**.**

([40])* Let $X$ be a compact projective manifold of dimension $n$, let $G\rightarrow X$ and $E\rightarrow X$ be holomorphic line bundles and $e^{-\psi_{E}}$ a hermitian metric on $E$ with $\partial\bar{\partial}\psi_{E}>0$. Let $l\geq 1$ and $s_{0},...,s_{N}\in H^{0}(X,G)$. Let $|s|^{2}=\sum_{\alpha=0}^{N}|s_{\alpha}|^{2}$. If*

[TABLE]

then $U=\sum_{\alpha=0}^{N}U_{\alpha}s_{\alpha}$ with

[TABLE]

Here we define $|s|^{2(n+l+1)}:=\left(\sum_{\alpha=0}^{N}|s_{\alpha}|^{2}\right)^{(n+l+1)}$.

Proof of Theorem 2. Let $(X,\omega_{X},L,h_{L})\in\mathcal{F}(n,d)$ and write (locally) $h_{L}=e^{-\varphi_{L}}$ and \omega_{X}={{\sqrt{{\scalebox{0.75}[1.0]{-1}}}}}\thinspace\partial\bar{\partial}\varphi_{L}. Then we can write

[TABLE]

[TABLE]

Then

[TABLE]

since $(m-(n+l+1)k)\geq 1$ by assumption. Note that if $0\leq r\leq l$ we have

[TABLE]

where $\rho_{k}=\rho(X,k\omega_{X},h_{L}^{k},kL)$. Let $U\in H^{0}(X,mL)$ and let $s_{0},...,s_{N_{k}}$ be an orthonormal basis of $kL$. Then Theorem 6 immplies that

[TABLE]

where $U(\alpha_{1};1)\in H^{0}(X,(m-k)L)$ for $0\leq\alpha_{1}\leq N_{k}$. Applying Theorem 6 again:

[TABLE]

with $U(\alpha_{1},\alpha_{2};2)\in H^{0}(X,(m-2k)L)$. Continuing in this fashion, we obtain

[TABLE]

[TABLE]

Applying this step for $r=1,2,...,l$ and using (2.3) we obtain

[TABLE]

Finally, estimate $\rho_{k}^{n+1}$ on the left by $A^{n+1}$ and $\rho_{k}^{n+l+1}$ on the right by $A^{-(n+l+1)}$ ∎Proof of Corollary 3. If $(X,L^{\prime})\in{\mathcal{F}}_{p}(n,d)$ there exists $(X,L)\in{\mathcal{F}}(n,d)$ such that $L^{\prime}~=~pL$. Then our goal becomes the following: Let $r\geq 1$ and $U\in H^{0}(X,(r+1)L^{\prime})=H^{0}((r+1)pL)$. We must show that $U$ is a linear combination of terms of the form $U_{0}U_{1}\cdots U_{r}$ where $U_{j}\in H^{0}(X,L^{\prime})=H^{0}(X,pL)$.

Choose $k$ such that $p=(n+2)k$ and let $l=(n+2)r$. Then $m:=(r+1)p=(n+2+l)k$. In particular, $U\in H^{0}(X,mL)$ and $m-lk=p$.

Now by Theorem 5 we can write

[TABLE]

Since $l=(n+2)r$ we can write $s_{\alpha_{1}}\cdots s_{\alpha_{l}}\ =\ U_{1}\cdots U_{r}$ with $U_{j}\in H^{0}(X,pL)$. ∎

3. Weak effective finite generation

In Theorem 5 the coefficients $U({\alpha_{1},...,\alpha_{l}})$ in (2.2) are sections and the bounds they satisfy are uniform and effective. We need another version of this result in which we allow the coefficients to be real numbers, and with bounds that are uniform but not effective.

Proposition 1**.**

Let $n,d,m\geq 1$. Then there exist $\lambda_{m}>0$ and $r=r(n,d)$ with the following properties. Let $(X,\omega_{X},L,h_{L})\in\mathcal{F}_{r}(n,d)$, and $s_{0}\cdots s_{N}$ an orthonormal basis of $H^{0}(X,L)$ with respect to ${\rm Hilb}(h_{L})$. Let $U\in H^{0}(X,mL)$. Then we can write

[TABLE]

where

[TABLE]

[TABLE]

The proof will be via contradiction and makes use some results of [15] which we now recall.

3.1. Gromov-Hausdorff Limits of Kähler Manifolds

We state here some results from [15] that will be needed in the proof of Proposition 1. There exists $r=r(n,d)$ with the following properties. Let $(X_{i},\omega_{i},L_{i},h_{i})\in{\mathcal{F}}_{r}^{0}(n,d)$. After passing to a subsequence, the sequence converges, in the Gromov-Hausdorff sense, to a limit $(X_{\infty},\omega_{\infty},L_{\infty},h_{\infty})$. Here $X_{\infty}$ is a metric space whose regular part $X_{\infty}^{\rm reg}\subseteq X_{\infty}$ is an open dense complex manifold of dimension $n$, $\omega_{\infty}$ is a Kähler metric on $X_{\infty}^{\rm reg}$ and $L_{\infty}$ is a holomorphic line bundle on $X^{\rm reg}$ with metric $h_{\infty}$ and curvature $\omega_{\infty}$. The meaning of the convergence $(X_{i},\omega_{i},L_{i},h_{i})\rightarrow(X_{\infty},\omega_{\infty},L_{\infty},h_{\infty})$ is as follows:

There are open sets $U_{i}\subseteq U_{i+1}\subseteq\cdots\subseteq X^{\rm reg}_{\infty}$, exhausting $X^{\rm reg}$ with relatively compact containments, open sets $V_{i}\subseteq X_{i}$, and diffeomorphisms $\Gamma_{i}:U_{i}\rightarrow V_{i}$ which are approximate Kähler isometries, in the sense that $\omega_{i}\rightarrow\omega_{\infty}$ or, more precisely, $\|\Gamma_{i}^{*}\omega_{i}\rightarrow\omega_{\infty}\|_{C^{\infty}(\Omega)}\rightarrow 0$ for any relatively compact open set $\Omega\subseteq U_{i}$. Similarly $J_{i}\rightarrow J_{\infty}$ and $g_{i}\rightarrow g_{\infty}$. We will usually omit $\Gamma_{i}$ in what follows to lighten the notation and similarly we will identify $U_{i}$ with $V_{i}$. Furthermore, $\Omega\subseteq U_{i}$ will always denote a relatively compact open set. We also have $L_{i}\rightarrow L_{\infty}$. This means there is a locally finite open cover $X^{\rm reg}_{\infty}=\cup_{\alpha}A_{\alpha}$ and holomorphic sections $s^{\infty}_{\alpha}\in H^{0}(A_{\alpha},L_{\infty})$ and $s^{i}_{\alpha}\in H^{0}((A_{\alpha}\cap U_{i}),L_{i})$ such that the sequence of smooth cocycles $[(s^{i}_{\alpha})(s^{i}_{\beta})^{-1}]\circ\Gamma_{i}$ converges uniformly to the holomorphic cocycle $[(s^{\infty}_{\alpha})(s^{\infty}_{\beta})^{-1}]$ on $A_{\alpha}\cap A_{\beta}\cap\Omega$. Similarly, if $\sigma_{\infty}\in H^{0}(X_{\infty},mL_{\infty})$ and $\sigma_{i}\in H^{0}(X_{i},mL_{i})$ we say $\sigma_{i}\rightarrow\sigma_{\infty}$ if $\sigma_{i}/(s^{i}_{\alpha})^{m}\rightarrow\sigma_{\infty}/(s^{\infty}_{\alpha})^{m}$ uniformly on $A_{\alpha}\cap\Omega$. If $\sigma_{i}\rightarrow\sigma_{\infty}\in H^{0}(X_{\infty},mL_{\infty})$ and $\tau_{i}\rightarrow\tau_{\infty}\in H^{0}(X_{\infty},lL_{\infty})$ one easily sees

[TABLE]

There is an isometry $Q_{i}^{m}:H^{0}(X_{\infty},mL_{\infty})\rightarrow H^{0}(X_{i},mL_{i})$ such that $Q_{i}^{m}(s)\rightarrow s$ as $i\rightarrow\infty$. In particular, if $\{s_{0}^{\infty},...,s_{N}^{\infty}\}$ is an orthonormal basis of $H^{0}(X_{\infty},L_{\infty})$, then $\{s_{0}^{i},...,s_{N}^{i}\}$ is an orthonormal basis of $H^{0}(X_{i},L_{i})$, where $s_{\alpha}^{i}:=Q_{i}(s_{\alpha}^{\infty})$.

For $1\leq i\leq\infty$ we let $T_{i}:X_{i}\hookrightarrow{\mathbb{P}}^{N}$ be the embedding of $H^{0}(X_{i},L_{i})$ with respect to a orthonormal basis (so $T_{i}$ is only defined up to the action of $U(N+1)$). Then there is a compact metric space $(X_{\infty},d_{\infty})$ such that, after passing to a subsequence if necessary, $(X_{i},\omega_{i})\rightarrow(X_{\infty},d_{\infty})$ in the Gromov-Hausdorff topology. In addition, $T_{i}(X_{i}):=W_{i}$ is a normal sub-variety of ${\mathbb{P}}^{N}$ (smooth if $i<\infty$), such that $W_{i}\rightarrow W_{\infty}$ algebraically, i.e. the corresponding point converge in the Hilbert scheme. Moreover, we have $T_{i}(x_{i})\rightarrow T_{\infty}(x_{\infty})$ in ${\mathbb{P}}^{N}$ if $x_{i}\rightarrow x_{\infty}$. The map $T_{\infty}:X_{\infty}\rightarrow W_{\infty}$ is a homeomorphism - in fact it is a biholomorphic map with respect to the natural structure of complex variety on $X_{\infty}$.

We summarize this discussion with the following diagram:

[TABLE]

Here the vertical arrows represent convergence in the metric (Cheeger-Colding) sense and the the algebraic (Hilbert scheme) sense respectively. The horizontal arrows isomorphisms: $T_{i}$ is an algebraic isomorphism, and $T_{\infty}$ is a holomorphic isomorphism. For $1\leq i\leq\infty$, the maps $W_{i}\hookrightarrow{\mathbb{P}}^{N}$ are inclusions.

3.2. Proof of Proposition 1

We will give the argument for ${\mathcal{F}}_{r}^{0}(n,d)$. We don’t need to use the fact that the singular sets of $X_{\infty}$ and $W_{\infty}$ match, so proof of the proposition for ${\mathcal{F}}_{r}(n,d)$ is the same, using instead the results of [36]. Assume not. Let $\{s_{0}^{\infty},...,s_{N}^{\infty}\}$ be an orthonormal basis for $H^{0}(X_{\infty},L_{\infty})$ and $s_{p}^{i}=Q_{i}(s_{p}^{\infty})$. Then there exists $m\geq 1$ and $(X_{j},\omega_{j},L_{j},h_{j})\in\mathcal{F}(n,d)$ and $U_{j}\in H^{0}(X,mL)$ with $\|U_{j}\|_{\rm Hilb(h^{m})}=1$ satisfying the following. If

[TABLE]

for some $U_{j}({\alpha_{1},...,\alpha_{m}})\in{\mathbb{R}}$, then $\max_{\alpha}\{|U_{j}(\alpha)|\}\geq j$.

After passing to a subsequence, the theorem of [15] says that $X_{i}\rightarrow X_{\infty}$ in the Gromov-Hausdorff topology, and if for $1\leq i\leq\infty$, $T_{i}:X_{i}\hookrightarrow{\mathbb{P}}^{N}$ is an embedding by the orthonormal basis $\{s_{0}^{\infty},...,s_{N}^{i}\}$ of $H^{0}(X_{i},L_{i})$, and $W_{i}\rightarrow W_{\infty}$ in the algebraic sense.

After passing to a further subsequence, there exists $U_{\infty}\in H^{0}(X,mL)$ such that $U_{j}\rightarrow U_{\infty}$. This holds since after scaling and a unitary transformation we may assume $U_{j}=\sigma_{0}^{j}$ is the first element in an orthonormal basis for $H^{0}(X_{j},mL_{j})$.

Now choose a subset $A\subseteq\{(\alpha_{0},...,\alpha_{m})\,:\,1\leq\alpha_{j}\leq N\}$ with $N_{m}+1$ elements such that $\{s_{\alpha}^{\infty}\,:\,\alpha\in A\}$ is a basis of $H^{0}(X_{\infty},mL_{\infty})$. This is possible since $\oplus_{m=0}^{\infty}H^{0}(X_{\infty},mL)$ is generated by $H^{0}(X_{\infty},L)$. Write $U_{\infty}$ as a linear combination of basis elements:

[TABLE]

with $U^{\alpha}_{\infty}\in{\mathbb{R}}$. Then

[TABLE]

Note that ${\rm det}\langle s_{\alpha}^{\infty},s_{\beta}^{\infty}\rangle_{{\rm Hilb}(h_{\infty})}\ \not=0$ since $\{s_{\alpha}^{\infty}:\alpha\in A\}$ is a basis. Since $s_{\alpha}^{j}\rightarrow s_{\alpha}^{\infty}$ by virtue of (3.3), we see that ${\rm det}\langle s_{\alpha}^{j},s_{\beta}^{j}\rangle_{{\rm Hilb}(h_{j})}\not=0$ for $j$ sufficiently large. Moreover, $U^{\infty}$ is defined so that $U^{j}\rightarrow U^{\infty}$. Cramer’s rule now implies that $U_{j}^{\alpha}\rightarrow U_{\infty}^{\alpha}$. But this contradicts our assumption, namely $\max_{\alpha}\{|U_{j}^{\alpha}|\}\geq j$. ∎

4. Proof of Theorem 1

After replacing ${\mathcal{F}}(n,d)$ by ${\mathcal{F}}_{r}(n,d)$ for some fixed highly divisible positive integer $r$, we may assume $m_{0}(n,d)=1$ (defined in section 1) and $r(n,S)=1$ (defined in Theorem 4). Also, without loss of generality we may assume $K=1$ so that all elements in ${\mathcal{F}}(n,d)$ have the same Hilbert polynomial $p(x)$. In other words, if $(X,\omega_{X},L,h_{L})\in{\mathcal{F}}(n,d)$ we may assume

[TABLE]

We let $N_{m}+1=p(m)$ and $N=N_{1}$. If $(X,\omega_{X},L,h_{L})\in{\mathcal{F}}(n,d)$ then $T_{X}:X\hookrightarrow{\mathbb{P}}^{N}$ is an embedding, where

[TABLE]

Here $\{s_{0,X},s_{1,X},...,s_{N,X}\}$ is an orthonormal basis of $H^{0}(X,L)$ with respect to ${\rm Hilb}_{h_{L}}$. In this way we may view $X$ as a submanifold of ${\mathbb{P}}^{N}$, for all $(X,\omega_{X},L,h_{L})\in{\mathcal{F}}(n,d)$ so \theta_{X}=\Theta_{\rm FS}\big{|}_{X} where $\Theta_{\rm FS}$ is the Fubini-Study metric on ${\mathbb{P}}^{N}$ and L=O_{{\mathbb{P}}^{N}}(1)\big{|}_{X} and h_{X}=H_{\rm FS}\big{|}_{L}. With these identifications we have

[TABLE]

as well as

[TABLE]

Note that if $\{E_{0},...,E_{N}\}$ is the basis of $H^{0}({\mathbb{P}}^{N},O(1))$, given by $E_{j}(x_{0},...,x_{N})=x_{j}$, then we may write

[TABLE]

In other words, $E_{j}$ is a section of $H^{0}({\mathbb{P}}^{N},O(1))$, independent of $X$, and is simultaneously an extension of $s_{j,X}$ from $H^{0}(X,L)$ to $H^{0}({\mathbb{P}}^{N},O(1))$ for all $(X,L)\in{\mathcal{F}}(n,d)$. Now we fix $(X,\omega_{X},L,h_{L})\in{\mathcal{F}}(n,d)$. For $m\geq 1$ we define the Bergman potential

[TABLE]

where $\{\sigma_{0},...,\sigma_{N_{m}}\}$ is an orthonormal basis of $H^{0}(X,mL)$ and $h_{L}=h_{X}e^{-\varphi_{X}}$. Note that (4.2) implies $\varphi_{1}=0$. Corollary 2 implies

[TABLE]

In particular, if $(X,L)\in{\mathcal{F}}(n,d)$ then

[TABLE]

We will also make use of the following. If $\sigma\in H^{0}(X,mL)$ then

[TABLE]

since

[TABLE]

and $-mA\leq m\varphi_{X}\leq 0$.

Fix $x\in X$ and assume that $\sigma_{i}(x)=0$ if $i>0$ so $\rho_{h_{L}^{m}}(x)=|\sigma_{0}|^{2}_{h_{L}^{m}}$ (this can be achieved by a unitary transformation). Let $\nabla$ be the Chern connection on $L$ for $h_{X}$. The induced connection on $mL$ for $h_{X}^{m}$ will be denoted by $\nabla$ as well. Now we compute

[TABLE]

[TABLE]

where have made use of $\rho_{{}_{h_{L}^{m}}}(x)=|\sigma_{0}|^{2}_{h_{L}^{m}}(x)$ in the last identity. Here we have defined $|\nabla\sigma_{0}|^{2}_{\theta_{X},h_{L}^{m}}:=\theta_{X}^{p\bar{q}}(\nabla_{p}\sigma_{0})(\nabla_{\bar{q}}\bar{\sigma}_{0})h_{L}^{m}$. Now our goal is to find a good estimate for $|\nabla\sigma_{0}|^{2}_{\theta_{X},h_{L}^{m}}$.

Lemma 1**.**

Let $m\geq 1$. There exists $\mu_{m}(n,d)>0$ with the following property. For every $(X,\omega_{X},L,h_{L})\in\mathcal{F}(n,d)$ and for every $\sigma\in H^{0}(X,mL)$ we have

[TABLE]

Proof. The first estimate follows from the partial $C^{0}$ estimate. As for the second, we must show that

[TABLE]

It suffices to show that

[TABLE]

making use of (4.5) and $h_{L}^{m}=h_{X}^{m}e^{-m\varphi_{X}}\leq e^{mA}h_{X}$.

We first prove (4.9) in the case where $m=1$. Let $(X,\omega_{X},L,h_{L})\in\mathcal{F}(n,d)$ and let $s=\sigma\in H^{0}(X,L)$. After scaling, we may assume $\|s\|_{\rm Hilb(h_{L}))}=1$. After transformation by a matrix in $U(N+1)$, we may assume that $s$ is the first element in the orthonormal basis that defines $T:X\rightarrow{\mathbb{P}}^{N}$. In other words, $s$ is the restriction of $E_{0}$, the first of the $N+1$ coordinate functions on ${\mathbb{C}}^{N+1}$. Thus we can write (4.3) as

[TABLE]

Let $\Theta_{\rm FS}$ denote the Fubini-Study metric on ${\mathbb{P}}^{N}$ and $H_{\rm FS}$ the Fubini-Study metric on $O(1)$. Then $s,\theta_{X}$ and $h_{X}$ are the restrictions of $E_{0},\Theta_{\rm FS}$ and $H_{\rm FS}$ from $H^{0}({\mathbb{P}}^{N},O(1)),{\mathbb{P}}^{N}$ and $O(1)$ to $H^{0}(X,L),X$ and $L$. Let $\tilde{\nabla}$ be the connection on $O(1)$ for $H_{\rm FS}$. Then, for $x\in X$ and $V\in T_{x}X\subseteq T_{x}{\mathbb{P}}^{N}$, we have

[TABLE]

Thus if $V_{1},..,V_{n}$ is an orthonormal frame for $T_{x}X$ and $V_{1},..,V_{n},V_{n+1},...,V_{N}$ and orthonormal frame for $T_{x}{\mathbb{P}}^{N}$, we have

[TABLE]

So we see

[TABLE]

since $|\tilde{\nabla}E_{0}|^{2}_{\Theta_{\rm FS},H_{\rm FS}}$ is a continuous function on the compact set ${\mathbb{P}}^{N}$ and is thus bounded by a constant $\mu_{1}=\mu_{1}(N)$. The proves (4.9) in the case $m=1$. Now assume $m>1$ and let $\sigma\in H^{0}(X,mL)$. Proposition (1) implies

[TABLE]

where

[TABLE]

[TABLE]

Thus

[TABLE]

[TABLE]

where $\mu_{1}$ is defined in (4.11).

Lemma 2**.**

There exists $B>0$ with the following property. For all $(X,\omega,L,h_{L})\in\mathcal{F}(n,d)$ and for all $\sigma\in H^{0}(X,mL)$ we have

[TABLE]

Proof. We must show

[TABLE]

Proposition 1 gives us a the following weaker statement. For each $m\geq 1$ there exists $\mu_{m}>0$ such that

[TABLE]

i.e., $|\nabla\sigma|^{2}_{\theta_{X},h_{L}^{m}}\ \leq\ \mu_{m}\|\sigma\|_{\rm Hilb(h_{L}^{m})}$ (this is weaker since we must further show $\mu_{m}=B^{m}$). In order to prove (4.15) we apply Theorem 5 with

[TABLE]

After scaling, we may assume $\|\sigma\|_{\rm Hilb(h_{L}^{m})}=1$ so, we obtain

[TABLE]

[TABLE]

[TABLE]

where $c_{1}(n,d)={(n+2)^{n}}A^{n}$. Now

[TABLE]

Noting that $U(\alpha_{1},...,\alpha_{l})\in H^{0}((n+2)L)$ and $|s_{\alpha_{j}}|^{2}\leq A$ and applying (4.16) we have

[TABLE]

[TABLE]

Where $B=N_{1}\mu_{n+1}c_{1}(n,d)A^{3}\mu_{1}$. Here we use the inequality $A^{m}lA^{l}\leq A^{m}A^{m}A^{m}$. This proves Lemma 2.

Lemma 3**.**

Let $n,d\geq 1$. Then there exists $C=C(n,d)>0$ and $\kappa=\kappa(n,d)>0$ with the following properties. Let $r\geq 2$ and $m=(n+2)^{r}$. Let $U\in H^{0}(X,mL)$. Then

[TABLE]

Proof. Let

[TABLE]

Thus $m=(n+2)k+kl=(n+2+l)k\geq(n+2+l)k$ so we may apply Theorem 5. Fix $x\in X$ and choose the orthonormal basis $\{s_{0},...,s_{N_{k}}\}$ of $H^{0}(X,kL)$ such that $s_{0}(x)\not=0$, $s_{1},...,s_{n}$ vanishing at $x$ to first order and the remaining sections vanishing to at least second order. Then Theorem 5 implies

[TABLE]

where

[TABLE]

Applying $\nabla$ to (4.17) and evaluating at $x$, we get

[TABLE]

where $l_{p}^{*}=l$ if $p=0$ and $l_{p}^{*}=1$ otherwise. Thus

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

We claim that for all $r\geq 1$ and all $u\in H^{0}(X,(n+1)^{r}L)$ that

[TABLE]

We proceed by induction on $r$ and $t$. We know (4.19) is true for $t=1$ by the previous estimate. Now assume it holds for $t$ and try to show it holds for $t+1$. Recall that $u_{p}\in H^{0}((n+2)^{r-1}L)$ and $s_{\alpha_{\alpha_{j}}}\in H^{0}((n+2)^{r-2}L)$. Thus if we apply (4.19) to $u_{p}$ and $s_{\alpha_{j}}$ in (4.18) we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Letting $t=r-2$ we get

[TABLE]

Since $m=(n+2)^{r}$ we see $r={\log m\over\log(n+2)}$ so

[TABLE]

where $P(n,d)=D^{1\over\log(n+2)}$ and $\kappa=\log P$. Applying this to (4.6) we obtain

[TABLE]

Now we can complete the proof of Theorem 1. Let $x,y\in X$. Then

[TABLE]

[TABLE]

where $C_{0}={2\log A}$. Choosing $r\geq 2$ such that

[TABLE]

and setting $\alpha={1\over\kappa}$ we see that

[TABLE]

[TABLE]

[TABLE]

where $C=C_{0}+C_{1}$ and

[TABLE]

5. Proof of Theorem 2

In order to deduce Theorem 2 from Theorem 1, we shall express $(X_{\infty},\omega_{\infty}):=(X,\omega_{\rm KE})$ as the limit of smooth Kähler manifolds $(X_{j},\omega_{j})$, apply (1.4) to each $X_{j}$, and take the limit as $j\to\infty$ in order to obtain (1.6). More precisely, if $x_{\infty},y_{\infty}\in X$, we write $x_{\infty}=\lim_{j\to\infty}x_{j}$ and $y_{\infty}=\lim_{j\to\infty}y_{j}$. If \omega_{\rm FS}+{{\sqrt{{\scalebox{0.75}[1.0]{-1}}}}}\thinspace\partial\bar{\partial}\varphi_{j} is the Kähler metric on $X_{j}$, and $d_{X_{j}}$ the intrinsic Fubini-Study metric, then (1.4) yields

[TABLE]

If $j\rightarrow\infty$, then, by the smoothability assumption, the left side of (5.1) converges to $|\varphi_{X}(x)-\varphi_{X}(y)|$, which is the left side of (1.6). What remains is to show that

[TABLE]

for some $C,\mu$, independent of $x,y$, and this is the content of Proposition 2 below. Before stating Proposition 2 we need first to recall some basic defnitions.

If $\gamma:[0,1]\rightarrow{\mathbb{P}}^{N}$ is piecewise $C^{1}$ curve, we define

[TABLE]

Here $\gamma^{\prime}(t)\in T_{\gamma(t)}({\mathbb{P}}^{N})$ and $|\gamma^{\prime}(t)|_{\rm FS}$ its length with respect to the Fubini-Study metric.

Let $X\subseteq{\mathbb{P}}^{N}$ be a normal variety and let $x,y\in X$. We define $d_{\rm FS}(x,y)$ the extrinsic Fubini-Study distance, and $d_{X}(x,y)$, the intrinsic Fubini-Study distance, as follows.

[TABLE]

[TABLE]

Here we require all curves $\gamma$ to be $C^{1}$.

Next we recall the definition of algebraic convergence. Let $X_{1},X_{2},...\subseteq{\mathbb{P}}^{N}$ be a sequence of smooth projective manifolds with the same Hilbert polynomial $P\in{\mathbb{Q}}[x]$. This means that for each $j$, we have $\dim(H^{0}(X_{j},O(k))=P(k):=N_{k}+1$ for $k\geq k_{0}$. One can show that $k_{0}$ is independent of $j$. By compactness of the Hilbert scheme, after passing to a subsequence we have algebraic convergence $X_{j}\rightarrow X_{\infty}$. Here $X_{\infty}\subseteq{\mathbb{P}}^{N_{k}}$ is a projective variety (possibly singular) with Hilbert polynomial P. To say that the convergence is algebraic is equivalent to saying that the corresponding sequence of points in the Hilbert scheme converge. More concretely, for each $1\leq j\leq\infty$, there exists $m,d>0$, and $f_{j}^{(1)},f_{j}^{(2)},...f_{j}^{(m)}\in{\mathbb{C}}[T_{0},...,T_{N}]$, homogeneous of degree $d$, such that $X_{j}=\{x\in{\mathbb{P}}^{N}\,:\,f_{j}^{(k)}(x)=0,\ 1\leq k\leq m\}$ and satisfying the following property. For each $k$, the coefficients of $f_{j}^{(k)}$ converge to those of $f_{\infty}^{(k)}$. This implies that for every $\varepsilon>0$ there exists $j_{0}>0$ such that for $j\geq j_{0}$, the variety $X_{j}$ lies in an $\varepsilon$-neighborhood of $X_{\infty}$. Thus, for every $x_{\infty}\in X_{\infty}$ there exist $x_{j}\in X_{j}$ such that $x_{j}\rightarrow x$ in the topology of ${\mathbb{P}}^{N}$.

Proposition 2**.**

Fix $k\geq k_{0}$ and let $X_{j},X_{\infty}\subseteq{\mathbb{P}}^{N_{k}}$, be as above. Assume $X_{j}\rightarrow X_{\infty}$ algebraically and that $X_{\infty}$ is a normal variety. Let $x_{j},y_{j}\in X_{j}$ and assume $x_{j}\rightarrow x_{\infty}\in X_{\infty}$ and $y_{j}\rightarrow y_{\infty}\in X_{\infty}$. Then there are $C,\mu>0$ such that

[TABLE]

The method of proof is as follows. First we use a theorem of Łojasiewicz to show there exists $C,\mu>0$ with the following property. For any $x_{\infty},y_{\infty}\in X_{\infty}$, and for every $\varepsilon>0$, there is a piecewise analytic arc $\gamma_{\infty}:[0,1]\rightarrow X_{\infty}$ joining $x_{\infty}$ to $y_{\infty}$, such that $\gamma_{\infty}(t)\in X^{\rm reg}$ for all $t\in[0,1]$ and $\ell_{X_{\infty}}(\gamma_{\infty})\leq Cd_{\rm FS}(x_{\infty},y_{\infty})^{\mu}_{\rm FS}+\varepsilon$. Then we use [15] to construct curves $\gamma_{j}:[0,1]\rightarrow X_{j}$ joining $x_{j}$ to $y_{j}$ such that $\ell_{X_{j}}(\gamma_{j})\leq\ell_{X_{\infty}}(\gamma_{\infty})+\varepsilon$ for $j$ sufficiently large. Thus

[TABLE]

Here, by abuse of notation, we write $\mu$ in place of $\mu/\alpha$.

Definition 3**.**

Let $M$ be a real analytic manifold and $E\subseteq M$ a subset. We say that $E$ is semianalytic if for every $p\in E$ there is an open set $p\in V\subseteq M$ such that

[TABLE]

where $g_{ij},f_{i}$ are real analytic functions on $V$. A function $f:M\rightarrow{\mathbb{R}}$ is semianalytic if its graph in $M\times{\mathbb{R}}$ is semianalytic.

Proof of Proposition 2. Since $X:=X_{\infty}$ a connected analytic set (and hence a connected semianalytic set), we can apply Łojasiewicz’s theorem [37] (see also Theorem 6.10 of [4]), and conclude that there exist $C,\mu>0$ with the following property. For every $x,y\in X$, there is a semianalytic arc $\eta:[0,1]\rightarrow X$ joining $x$ to $y$, such that $\ell_{X}(\eta)\leq Cd_{\rm FS}(x,y)^{\mu}$.

Let $\pi:\tilde{X}\rightarrow X$ be a resolution of singularities, and fix an embedding of $\tilde{X}$ into projective space. Let $\tilde{\omega}_{\rm FS}$ be the restriction of the Fubini-Study metric to $\tilde{X}$.

Lemma 4**.**

There exist $\tilde{x},\tilde{y},\in\tilde{X}$ and a piecewise analytic curve $\tilde{\eta}$ joining $\tilde{x}$ to $\tilde{y}$ with the following properties.

1. (1)

$\tilde{\eta}$* has finite length $L$.* 2. (2)

$\pi(\tilde{x})=x$* and $\pi(\tilde{y})=y$.* 3. (3)

$\ell_{X}(\pi\circ\tilde{\eta})\ \leq\ \ell_{X}(\eta)\ \leq\ Cd_{\rm FS}(x,y)^{\mu}$**

Proof of Lemma 4. Since $[0,1]$ is one dimensional, Theorem 1 of section 23 in [37] implies that $C\subseteq X$, the image of $\eta$, is a semianalytic set.

Let $\tilde{Z}=\pi^{-1}(C)\subseteq\tilde{X}$. Then $\tilde{Z}\subseteq\tilde{X}$ is a semianalytic set since it is the inverse image of a semianalytic set. Since $X$ is normal and $C$ is connected, we may apply Zariski’s connectedness theorem to conclude $\tilde{Z}$ is connected. Fix $\tilde{x}^{*}\in\pi^{-1}(x)$ and $\tilde{y}^{*}\in\pi^{-1}(y)$. Since $\tilde{Z}$ is a connected semianalytic set, we my apply Łojasiewicz’s theorem again to obtain a semianalytic arc $\tilde{\eta}^{*}:[0,1]\rightarrow\tilde{Z}$ of finite length joining $\tilde{x}^{*}$ to $\tilde{y}^{*}$. Let $\tilde{x}=\tilde{\eta}^{*}(t_{1})$ where $t_{1}=\sup\{t\in[0,1]\,:\,\tilde{\eta}^{*}(t)\in\pi^{-1}(x)\}$. Similarly, let $\tilde{y}=\tilde{\eta}^{*}(t_{2})$ where $t_{2}=\inf\{t\in[0,1]\,:\,\tilde{\eta}^{*}(t)\in\pi^{-1}(y)\}$. Case 1. The graph $\tilde{\Gamma}^{*}=\{(t,\tilde{\xi})\in[0,1]\times\tilde{X}\,:\,\tilde{\xi}=\tilde{\eta}^{*}(t)\}$ is contained in an analytic curve $\tilde{K}\subseteq{\mathbb{R}}\times\tilde{X}$ which is smooth at all points for which $0<t<1$.

Let $a_{0}=0$ and $b_{0}=\sup\{b\in[0,1]\,:\,\tilde{\eta}^{*}:[a_{0},b]\rightarrow\tilde{X}$ is injective $\}$. Let $a_{1}=\sup\{a\in[b_{0},1]\,:\,\tilde{\eta}^{*}(a)=\tilde{\eta}^{*}(b_{0})\}$. Let $b_{1}=\sup\{b\in[a_{1},1]\,:\,\tilde{\eta}^{*}:[a_{1},b]\rightarrow\tilde{X}$ is injective $\}$. This process must terminate after a finite number of steps since $\tilde{\Gamma}^{*}$ is contained in an analytic curve. If $m$ is the number of steps we obtain a sequence

[TABLE]

Now we define $\tilde{\eta}$ as follows. If $t\in[a_{j},b_{j}]$ then $\tilde{\eta}(t)=\tilde{\eta}^{*}(t)$. Since all fibers of $\pi$ are connected, we may define $\tilde{\eta}:[b_{j},a_{j+1}]\rightarrow\ \pi^{-1}(\pi(b_{j}))$ to be any curve of finite length connecting $\tilde{\eta}^{*}(b_{j})$ to $\tilde{\eta}^{*}(a_{j+1})$. Then $\ell_{X}(\pi\circ\tilde{\eta})\ \leq\ \ell_{X}(\eta)$.

Case 2. The graph $\tilde{\Gamma}^{*}$ is contained in an analytic curve $\tilde{K}\subseteq{\mathbb{R}}\times\tilde{X}$. Let $\Sigma\subseteq\tilde{K}$ be the singular set and $0\leq t_{0}<t_{1}<\cdots<t_{r}\leq 1$ the projection of $\Sigma$ to $[0,1]$. We let $x_{j}=\tilde{\eta}^{*}(t_{j-1})$ and $y_{j}=\tilde{\eta}^{*}(t_{j})$. Then we can apply Case 1 to obtain continuous curves $\tilde{\eta}_{j}:[t_{j-1},t_{j}]\rightarrow M$ which are analytic on $(t_{j-1},t_{j})$ such that $\sum_{j}\ell_{X}(\pi\circ\tilde{\eta}_{j})\ \leq\ \ell_{X}(\eta)$. To define $\tilde{\eta}$ we join $\tilde{\eta}_{j}(t_{j})$ to $\tilde{\eta}_{j+1}(t_{j})$ by a piecewise analytic curve that lies in their common fiber so that

[TABLE]

In general, $\tilde{\eta}^{*}:[0,1]\rightarrow\tilde{X}$ is semianalytic. Since the graph $\tilde{\Gamma}^{*}$ of $\tilde{\eta}^{*}$ is a one dimensional semianalytic set, we may apply the stratification theorem of [37] (see also Theorem 2.13 of [4]) to conclude that $\Gamma$ is contained in an analytic curve $\tilde{K}\subseteq{\mathbb{R}}\times\tilde{X}$. Thus we are reduced to Case 2, and this completes the proof of the lemma. ∎

Continuation of the proof of Proposition 2. Write $I=\cup_{j=1}^{A}I_{j}$ where $I_{j}=[t_{j-1},t_{j}]$ is such that $\tilde{\eta}$ is continuous on $[t_{j-1},t_{j}]$ and analytic on $(t_{j-1},t_{j})$, and $\tilde{\eta}([t_{j-1},t_{j}])\subseteq V_{j}$. Here $V_{j}$ is a coordinate neighborhood with coordinate function $\varphi_{j}:V_{j}\rightarrow U_{j}\subseteq{\mathbb{C}}^{n}$ and $U_{j}$ is chosen so that the exceptional divisor intersected with $V_{j}$ is contained in the set $\varphi_{j}^{-1}(\{z_{1}z_{2}\cdots z_{n}=0\})$.

Let $L=\ell_{\tilde{X}}(\tilde{\eta})$ and $g_{j}$ the euclidean metric on $U_{j}$. Choose $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}<{\varepsilon\over A}$ and $\delta>0$ such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for all $x\in U_{j}$ and for all $\rho\in{\mathbb{C}}^{n}$ with $|\rho|<\delta$.

Define $\tilde{\gamma}$ on the interval $[t_{j-1}+\delta,t_{j}-\delta]$ by the formula

[TABLE]

and define it on $[t_{j}-\delta,t_{j}+\delta]$ so that it avoids the coordinate hyperplanes and has length at most ${\varepsilon_{2}\over A}$. Here $\rho\in{\mathbb{C}}^{n}$ is chosen so that $|\rho|<\delta$ and $\tilde{\gamma}_{j}$ does not intersect the coordinate axes.

Define $\gamma:=\,\pi\circ\tilde{\gamma}$. Then

[TABLE]

[TABLE]

where $\gamma^{j}$ and $\eta^{j}$ are the restrictions to $I_{j}$. If we consider the restriction to $[t_{j-1}+\delta,t_{j}-\delta]$, each term on the right is bounded by

[TABLE]

The length over $[t_{j}-d,t_{j}+\delta]$ is bounded by

[TABLE]

so the sum from $j=1$ to $j=A$ is bounded by $\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}<\varepsilon$.

Proof of Proposition 2. Let $x_{\infty},y_{\infty}\in X$ and choose $x_{j},y_{j}\in X_{j}$ such that $x_{j}\rightarrow x_{\infty}$ and $y_{j}\rightarrow y_{\infty}$. Choose $\gamma_{\infty}:[0,1]\rightarrow X_{\infty}^{\rm reg}$ whose length is at most $Cd_{\rm FS}(x_{\infty},y_{\infty})^{\mu}$, and let $Z_{\infty}\subseteq X_{\infty}$ be its image. We want to show that there exist curves $\gamma_{j}:[0,1]\rightarrow X_{j}$ joining $x_{j}$ to $y_{j}$ such that $\ell_{X_{j}}(\gamma_{j})\leq\ Cd_{\rm FS}(x_{j},y_{j})^{\mu}+2\varepsilon$ for $j$ sufficiently large. Let $V_{\infty}\subseteq X_{\infty}^{\rm reg}$ be an small open set and let $\Omega\subseteq{\mathbb{P}}^{N}$ be an open set such that $\Omega\cap X_{\infty}=V_{\infty}$. There exist homogeneous polynomials $f^{(\infty)}_{1},...,f^{(\infty)}_{N-n}$ such that, for $z_{\infty}\in\Omega$ we have $z_{\infty}\in V_{\infty}$ if and only if $f^{(\infty)}_{m}(z_{\infty})=0$ for $1\leq m\leq N-n$. The implicit function theorem implies that after possibly shrinking $V_{\infty}$, reordering the variables and working in affine coordinates so that $V_{\infty}\subseteq{\mathbb{C}}^{N}$, there is an open $U\subseteq{\mathbb{C}}^{n}$ and a biholomorphic map $\psi^{\infty}:U\rightarrow V_{\infty}\subseteq{\mathbb{C}}^{N}$ of the form $(z_{1},...,z_{n})\mapsto(z_{1},...,z_{n},\psi^{\infty}_{n+1}(z),...,\psi^{\infty}_{N}(z))$. Here $\psi^{\infty}_{m}:{\mathbb{C}}^{n}\rightarrow{\mathbb{C}}$ is a holomorphic function and $\varphi^{\infty}:V_{\infty}\rightarrow U$ is given by the projection map. In other words, $V_{\infty}$ is a graph over the domain $U$. Similarly,  we construct $\psi^{j}:U\rightarrow V_{j}\subseteq X_{j}$ where $V_{j}\rightarrow V_{\infty}$. If we choose $f_{m}^{j}$ for $1\leq m\leq N-n$ in such a way that $f_{m}^{j}\rightarrow f_{m}^{(\infty)}$, the proof of the implicit function theorem implies $\psi_{\alpha}^{(j)}\rightarrow\psi_{\alpha}^{\infty}$ on compact sets. In particular, $[{\psi^{j}}]^{*}\omega_{\rm FS}\rightarrow[\psi^{\infty}]^{*}\omega_{\rm FS}$ smoothly. If $V_{\infty}$ covers $\gamma_{\infty}([0,1])$ we define $\gamma_{j}=\psi^{j}\circ\varphi^{\infty}\circ\gamma_{\infty}$, and we obtain $\gamma_{j}$ joining $x_{j}$ to $y_{j}$ and satisfying $d_{\rm FS}(x_{j},y_{j})\leq\ell(\gamma_{j})\rightarrow\ell(\gamma_{\infty})\leq d_{\rm FS}(x_{\infty},y_{\infty})+\varepsilon$. If $V_{\infty}$ does not cover $\gamma_{\infty}([0,1])$, then we choose a finite cover and construct $\gamma_{j}$ using a partition of unity. Hence

[TABLE]

Proof of Theorem 2. Let $(X,\omega_{\rm KE})$ be a Kähler-Einstein variety and $(X_{i},\omega_{i})$ smooth Kähler manifolds such that $(X_{i},\omega_{i})\xrightarrow{\text{GH}}(X,\omega_{\rm KE})$. We write \omega_{i}=\omega_{\rm FS}+{{\sqrt{{\scalebox{0.75}[1.0]{-1}}}}}\thinspace\partial\bar{\partial}\varphi_{i} and \omega_{\rm KE}=\omega_{\rm FS}+{{\sqrt{{\scalebox{0.75}[1.0]{-1}}}}}\thinspace\partial\bar{\partial}\varphi_{\rm KE} with $\sup_{X_{i}}\varphi_{i}=\sup_{X}\varphi=0$. Theorem 1 implies that the $\varphi_{i}$ are uniformly Hölder, so Proposition 2 implies that, after passing to a subsequence, they converge to $\varphi_{\infty}$ satsifying

[TABLE]

for all $x,y\in X$. On the other hand, We know from [15] that $\omega_{i}\rightarrow\omega_{\rm KE}$ smoothly on $X^{\rm reg}$. This implies $\varphi_{\infty}=\varphi_{\rm KE}$ and completes the proof of Theorem 2.

6. Proof of Theorem 3

Suppose an $n$-dimensional smoothable projective variety $X$ with klt singularities in the central fibre $X=\mathcal{X}_{0}$ in the flat family

[TABLE]

As in Theorem 3, we assume that $K_{\mathcal{X}}$ is $\mathbb{Q}$-Gorenstein and $K_{\mathcal{X}/\Delta}$ is $\mathbb{Q}$-Cartier. By inversion of adjunction, $\mathcal{X}$ is locally klt and we choose an adaptive volume measure $\Omega_{\mathcal{X}}$ on $\mathcal{X}$ after possibly shrinking $\Delta$. Let $\Omega_{t}$ be the relative volume measure defined by

[TABLE]

for $t\in\Delta$. We can view $\Omega_{t}$ as an adaptive volume measure on $\mathcal{X}_{t}$ for each $t\in\Delta$ and let

[TABLE]

Let $\theta_{t}$ be the restriction of the Fubini-Study metric $\Theta$ on $\mathbb{P}^{N}$ restricted to $\mathcal{X}_{t}$.

We consider the following complex Monge-Ampère equation on $X$

[TABLE]

where $F$ satisfies

[TABLE]

for some $A>0$. Equation (6.4) admits a unique bounded solution. The goal of this section is to prove that the solution $\varphi$ to equation (6.4) is Hölder continuous with respect to $d_{\mathbb{P}^{N}}$. This will be achieved by regularizing the $\varphi$ via the smoothing $\pi:\mathcal{X}\rightarrow\Delta$ of $X$.

We begin our proof with the following lemma due to [19, 20] (c.f. Lemma 3.5 [20]).

Lemma 5**.**

There exists $p>1$ and $C>0$ such that for all $t\in\Delta$,

[TABLE]

Since $\Omega_{\mathcal{X}}$ is a smooth adaptive volume measure on $\mathcal{X}$, there exists $B>0$ such that

[TABLE]

on $\mathcal{X}$ and for all $t\in\Delta$,

[TABLE]

on $\mathcal{X}_{t}$.

The following extension result for $F$ follows directly from the global extension theorem of [10] for quasi-plurisubharmonic functions on a normal projective variety.

Lemma 6**.**

There exists $-\mathcal{F}\in PSH(\mathbb{P}^{N},A\Theta)$ such that

[TABLE]

We can immediately approximate $\mathcal{F}$ by the standard regularization for quasi-plurisubharmonic functions.

Lemma 7**.**

There exist a sequence of $-\mathcal{F}_{j}\in C^{\infty}(\mathbb{P}^{N})\cap PSH(\mathbb{P}^{N},A\Theta)$ such that $\mathcal{F}_{j}$ decreasingly converge to $\mathcal{F}$.

We let

[TABLE]

for $t\in\Delta$ and consider the following complex Monge-Ampère equation on $\mathcal{X}_{t}$

[TABLE]

where $c_{t,j}$ is the normalizing constant satisfying $\int_{X}e^{F_{t,j}+c_{t,j}}\Omega_{t}=[\theta_{t}]^{n}$. Obviously, for each fixed $j$, $c_{t,j}$ is uniformly bounded for all $t\in\Delta$. Furthermore, for fixed $j>0$, $\limsup_{j\rightarrow\infty}\lim_{t\rightarrow 0}c_{t,j}=0$ since $\int_{\mathcal{X}_{0}}e^{F}\Omega_{0}=[\theta_{0}]^{n}$.

Obviously there exists a unique bounded solution $\varphi_{t,j}$ to equation (6.5) for all $t\in\Delta$. In particular, $\varphi_{t,j}$ is smooth if $t\neq 0$. We will obtain some uniform estimates for $\varphi_{t,j}$.

Lemma 8**.**

There exists $C>0$ such that for any $j>0$, there exists $\delta_{j}>0$ such that for all $t\in\Delta$ with $|t|<\delta_{j}$, we have

[TABLE]

Proof.

We first notice that $F_{0,j}\leq F$ on $X$ and so $\|F_{0,j}\|_{L^{p}(X,\Omega_{0})}$ is uniformly bounded. For each fixed $j$, $F_{t,j}$ is uniformly bounded above for all $t\in\Delta$ and it converges smoothly to $F_{0,j}$ on $X^{\rm reg}$. Since the singular set of $X$ has measure [math] with respect to $\Omega_{0}$, we obtain uniform $L^{p}$ bound for $F_{t,j}$ for all $t\in\Delta$ sufficiently close to $t=0$. The estimate follows as for each fixed $j$, $|c_{t,j}|$ is uniformly bounded for sufficiently small $|t|$. ∎

Lemma 9**.**

There exist $q>1$ and $C>0$ such that for any $j>0$, there exists $\delta_{j}>0$ such that for all $t\in\Delta$ with $|t|<\delta_{j}$, we have

[TABLE]

Proof.

The lemma immediately follows from Lemma 5 and Lemma 8 combined with Hölder inequality. ∎

Lemma 10**.**

There exists $C>0$ such that for $j>0$, there exists $\delta_{j}>0$ such that for all $t\in\Delta$ with $|t|<\delta_{j}$ we have

[TABLE]

Proof.

The lemma is proved in [19]. The key ingredient of the proof is to obtain the uniform bound on Tian’s $\alpha$-invariant on $(\mathcal{X}_{t},\theta_{t})$ established in [19, 33] (c.f Theorem A [19]). Then one can proceed with the same argument in [28, 16] to achieve the $L^{\infty}$ estimate due to the uniform $L^{p}$ bound on the volume measure from Lemma 9. ∎

Let $\omega_{t,j}=\theta_{t}+\partial\bar{\partial}\varphi_{t,j}$. Then the Ricci curvature of $\omega_{t,j}$ is given by

[TABLE]

The following Schwarz lemma is achieved by the same technique as in [48].

Lemma 11**.**

There exists $\delta>0$ such that for all $j>0$, there exists $\delta_{j}>0$ such that all $t\in\Delta\setminus\{0\}$ with $|t|<\delta_{j}$, we have

[TABLE]

on $\mathcal{X}_{t}$,

Proof.

We consider the quantity

[TABLE]

for a fixed sufficiently large $K>0$. By the Ricci curvature estimate (6.6) for $\omega_{t,j}$ and the uniform $L^{\infty}$-estimate for $\varphi_{t,j}$ in Lemma 10, we can immediately apply the Chern-Lu type estimate by the same calculation in [48, 49] to achieve (6.7). ∎

We immediately obtain a uniform lower Ricci bound for $\omega_{t,j}$.

Corollary 4**.**

There exist $\Lambda>0$ and $D$ such that for all $j>0$, there exists $\delta_{j}>0$ such that all $t\in\Delta\setminus\{0\}$ with $|t|<\delta_{j}$, we have

[TABLE]

on $\mathcal{X}_{t}$ and

[TABLE]

Proof.

Direct calculations show that

[TABLE]

by (6.7). The Ricci lower bound is then proved by letting $\Lambda=\delta^{-1}(A+B)$. The diameter bound immediately follows from the general geometric estimates established in [22] (c.f. [22]) due to the uniform $L^{p}$ bound of Lemma 9 for the volume measure $(\omega_{t,j})^{n}$. ∎

The above Schwarz lemma gives local 2nd order estimates for $\varphi_{t,j}$ away from the singular set of $X$. In particular, for fixed $j>0$, one can obtain the local $C^{\infty}$ convergence of $\varphi_{t,j}$ to $\varphi_{0,j}$ on $X^{\rm reg}$. Hence $\varphi_{0,j}$ is smooth on $C^{\infty}(X^{\rm reg})$.

Let $L_{t}$ be the ample line bundle with $c_{1}(L_{t})=[\theta_{t}]$. We can choose the hermtian metric $h_{t,j}$ on $L_{t}$ with

[TABLE]

We can further obtain the following global Gromov-Hausdorff convergence.

Lemma 12**.**

For each $j>0$, $(\mathcal{X}_{t},\omega_{t,j})$ converge in Gromov-Hausdorff topology as $t\rightarrow 0$ to a compact metric space $(Y_{j},d_{j})$ satisfying

1. (1)

$(Y_{j},d_{j})$* coincides with the metric completion of $(X^{\rm reg},\omega_{0,j})$,* 2. (2)

$(Y_{j},d_{j})$* is homeomorphic to projective variety $X$ itself.* 3. (3)

$\omega_{t,j}$* converges smoothly to $\omega_{0,j}$ on $X^{\rm reg}$.*

Proof.

By the uniform Ricci lower bound and diameter bounds for $(\mathcal{X}_{t,j},\omega_{t,j})$ for $|t|<\delta_{j}$, one can immediately obtain uniform partial $C^{0}$-estimate by [15, 36] for $(\mathcal{X}_{t,j},\omega_{t,j}),kL_{t},h^{k}_{t,j})$ for sufficiently large $k\in\mathbb{Z}^{+}$. The lemma follows from the fact that $\varphi_{t,j}$ converges smoothly to $\varphi_{0,j}$. ∎

Lemma 13**.**

There exist $\gamma>0$ and $C>0$ such that for any $j>0$, there exists $\delta_{j}>0$ such that for any $t\in\Delta\setminus\{0\}$ with $|t|<\delta_{j}$, we have

[TABLE]

for all $x,y\in\mathcal{X}_{t}$.

Proof.

By Lemma 12, the partial $C^{0}$ estimate holds for all $(\mathcal{X}_{t},\omega_{t,j})$ with $t\in\Delta\setminus\{0\}$ and $|t|<\delta_{j}$. Then there exist $k>0$ such that the orthonormal basis of $H^{0}(\mathcal{X}_{t},kL_{t})$ with respect to $h_{t,j}^{k}$ and $k\omega_{t,j}$ induces a projective embedding

[TABLE]

for a fixed projective ambient space $\mathbb{P}^{M}$. We can assume $\mathcal{X}$ is also embedded in $\mathbb{P}^{M}$ after embedding $\mathbb{P}^{N}$ into $\mathbb{P}^{M}$ by choosing a sufficiently large $k$. Let $\Theta_{\mathbb{P}^{M}}$ be the Fubini-Study metric on $\mathbb{P}^{M}$ and

[TABLE]

One can also rewrite

[TABLE]

We immediately apply Theorem 1 to obtain the Hölder estimate for $\varphi_{t,j}$, i.e., there exist $\gamma>0$ and $C>0$ such that for any $j>0$, there exists $\delta_{j}>0$ such that for any $t\in\Delta\setminus\{0\}$ with $|t|<\delta_{j}$, we have

[TABLE]

for all $x,y\in\mathcal{X}_{t}$.

On the other hand, we can extend

[TABLE]

to

[TABLE]

Since $\varphi_{t,j}$ and $\psi_{t,j}$ are both uniformly bounded in $L^{\infty}(\mathcal{X}_{t})$ for all $j$ and $|t|<\delta_{j}$, $\tilde{T}_{t,j}$ must lie in a bounded subset set of $SL(M+1,\mathbb{C})$ for all $j>0$ and $|t|<\delta_{j}$. In particular, this implies that $\theta_{t}$ and $\omega_{FS,t,j}$ are uniformly equivalent for all $j>0$ and $t\in\Delta\setminus\{0\}$ with $|t|<\delta_{j}$. This completes the proof of the lemma. ∎

Finally we are able to prove Theorem 3.

Proposition 3**.**

There exists $\gamma>0$ and $C>0$ such that for all $x,y\in X$, we have

[TABLE]

Proof.

By the uniform Hölder estimate in Lemma 13, for each $j>0$, we can pass the Hölder estimate to the limiting space $X$. More precisely, there exists $\gamma>0$ and $C>0$ independent of $j>0$, such that for all $x,y\in X$ we have

[TABLE]

Since \lim_{j\rightarrow\infty}\left\|e^{F_{0,j}+c_{0,j}}-e^{F}\right\|_{L^{1}(X,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Omega^{t}})}=0, the stability theorem for degenerate complex Monge-Ampère equation of [12] implies that

[TABLE]

The proposition then immediately follows from (6.12) and (6.13). ∎

Acknowledgements: This project was initiated during the AIM workshop on ‘PDE methods in complex geometry’ in August, 2024. The authors would like to thank the generous support and the hospitality of American Institute of Mathematics. This work is supported in part by the National Science Foundation under grants DMS-2203607 and DMS-2303508.