Gromov–Hausdorff convergence of constant scalar curvature Kähler metrics near the canonical class

Frederick Tsz-Ho Fong and Junbang Liu Address: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Email address: frederick.fong@ust.hk Email address: junbangliu@ust.hk

Abstract.

Let $X$ be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class $[{\omega}]$ , Jian–Shi–Song [12] proved that there exists a unique cscK metric in the class $2\pi c_{1}(K_{X})+\varepsilon[{\omega}]$ when $\varepsilon\in(0,\varepsilon_{0}]$ , based on the properness criterion of Chen–Cheng [2]. We prove that the cscK metrics in $2\pi c_{1}(K_{X})+\varepsilon[{\omega}]$ converge, as $\varepsilon\downarrow 0$ , in the Gromov–Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of $X$ as conjectured by Jian–Shi–Song in [12]. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee–Tosatti–Zhang [14] on the Gromov-Hausdorff convergence of the normalized Kähler–Ricci flow on $X$ .

1. Introduction

Let $X$ be a connected compact Kähler manifold of complex dimension $n$ with semi-ample canonical bundle $K_{X}$ . Fix a Kähler class $[{\omega}]$ . Jian–Shi–Song [12, Theorem 1.1] proved that $-2\pi c_{1}(X)+\varepsilon[{\omega}]$ contains a unique cscK metric for all sufficiently small $\varepsilon>0$ based on the properness criterion of Chen–Cheng [2]. They conjectured that these metrics converge to the canonical metric on the pluricanonical base, both locally away from singular fibers and globally in the Gromov–Hausdorff topology [12, Conjecture 1.1].

There are two questions in the degeneration: local convergence of the metrics on the regular part of the fibration, and identification of the global metric limit. In the big and nef case, W. Liu [15, Theorem 1.3] established smooth convergence outside an effective divisor to the singular negative Kähler–Einstein metric. For an arbitrary semi-ample canonical bundle, Guo–Jian–Shi–Song [9, Theorems 1.1 and 2.1] obtained uniform bounds for potentials, normalized volume densities and diameters, together with lower bounds for normalized volumes of metric balls. These imply Gromov–Hausdorff precompactness. The goal is to identify every limit with the canonical metric completion.

The corresponding degeneration has also been studied through the Kähler–Ricci flow. Tosatti–Weinkove–Yang [23] established local convergence to the canonical metric and convergence of the rescaled fiber metrics. Related geometric estimates and metric convergence results were obtained in [7, 16, 21]. Lee–Tosatti–Zhang [14, Theorem 1.2] proved the global Gromov–Hausdorff convergence of the normalized Kähler–Ricci flow.

We now describe the setting. Since $K_{X}$ is semi-ample, $\ell K_{X}$ is base-point free for some $\ell\geq 1$ , and the linear system $|\ell K_{X}|$ for $\ell$ sufficiently divisible gives a fiber space (the Iitaka fibration of $X$ ): $f:X\to Y\subset\mathbb{C}P^{N}$ with $\ell K_{X}=f^{*}(\mathcal{O}_{\mathbb{C}P^{N}}(1)|_{Y})$ , onto a normal projective variety $Y$ . Put $m=\dim_{\mathbb{C}}Y=\kappa(X)$ , which is the Kodaira dimension. The generic fibers of $f$ are Calabi–Yau $d$ -folds with $d=n-m$ . For $m>0$ , let

Y^{\circ}=Y_{\rm reg}\setminus\{\text{discriminant locus of }f\},\qquad X^{\circ}=f^{-1}(Y^{\circ}).

Then $f:X^{\circ}\to Y^{\circ}$ is a proper holomorphic submersion with connected $d$ -dimensional Calabi–Yau fibers. The twisted Kähler-Einstein metric $\omega_{\rm can}$ on $Y^{\circ}$ satisfies

\Ric(\omega_{\rm can})=-\omega_{\rm can}+\omega_{\rm WP},

where $\omega_{\rm WP}$ is the Weil–Petersson form of the smooth Calabi–Yau fibration. Let $d_{\rm can}$ be the distance induced by $\omega_{\rm can}$ on $Y^{\circ}$ and $\nu_{\rm can}$ be the normalized volume form for the canonical metric. Its metric completion $(\overline{Y^{\circ}},d_{\rm can},\nu_{\rm can})$ is a compact $RCD(-1,2m)$ space and homeomorphic to $Y$ , as shown by Székelyhidi [22]. The main result in this paper is

Theorem 1.1.

Let $X$ be a connected compact Kähler manifold with semi-ample canonical bundle, and let $[{\omega}]$ be any Kähler class. For all sufficiently small $\varepsilon>0$ , let $\omega_{\varepsilon}\in 2\pi c_{1}(K_{X})+\varepsilon[{\omega}]$ be the unique cscK metric. If $m=0$ , the metrics converge to a point in the Gromov-Hausdorff topology. If $m>0$ , then:

(1)

For any compact $K\Subset Y^{\circ}$ ,

\|\omega_{\varepsilon}-(f^{*}\omega_{\rm can}+\varepsilon\omega_{\rm SRF})\|_{f^{*}\omega_{\rm can}+\varepsilon{\omega}}\to 0\quad\text{uniformly on }f^{-1}(K).

(1.1)

(2)

Let $d_{\omega_{\varepsilon}}$ be the distance function induced by the metric $\omega_{\varepsilon}$ on $X$ ,

$(X,d_{\omega_{\varepsilon}})\longrightarrow(\overline{Y^{\circ}},d_{\rm can})\qquad\text{in the Gromov--Hausdorff topology}.$ (1.2)

The meaning of $\omega_{\rm SRF}$ will be explained in the next section. For $m=0$ , choose $\chi=0$ . The cscK equations give

\underline{R}_{\varepsilon}=0,\qquad\Delta_{\omega_{\varepsilon}}F_{\varepsilon}=0,\qquad\Ric(\omega_{\varepsilon})=0.

If $\omega_{\rm RF}\in[{\omega}]$ is the Ricci-flat metric, uniqueness gives

\omega_{\varepsilon}=\varepsilon\omega_{\rm RF},\qquad\operatorname{diam}(X,\omega_{\varepsilon})=\sqrt{\varepsilon}\operatorname{diam}(X,\omega_{\rm RF})\to 0.

Hence, we assume $m>0$ in the rest of the paper.

Main ideas of the proof. The starting point is Guo–Jian–Shi–Song’s uniform $L^{\infty}$ estimates for the potential and the volume density. They established uniform noncollapsing estimates and a uniform Sobolev inequality for this family of cscK metrics. These estimates enable us to establish a $C^{2}$ -estimate in the base direction and a gradient estimate for the logarithm of the volume density:

\tr_{\omega_{\varepsilon}}\chi\leq C,\qquad|\nabla F_{\varepsilon}|_{\omega_{\varepsilon}}\leq C.

The method is inspired by the groundbreaking work of Chen–Cheng [2], but adapted to the fibration structure of the canonical model here. The next step is to show that the subsequential weak limit of the volume density $F_{\varepsilon}$ is exactly the pullback of the canonical one. The strategy for this step is to use the minimizing property of the Mabuchi energy for cscK metrics and a variational characterization of the Monge-Ampère measures. Next, after identifying the weak limit, the uniform Sobolev inequality enables us to improve the weak convergence to $L^{\infty}$ -convergence away from the singular fibers. This is done by local Moser iteration, which is available thanks to the uniform Sobolev inequality obtained by Guo–Phong–Song–Sturm [10].

After identifying the limit of the normalized volume form, one can show an integral estimate for the difference of $\omega_{\varepsilon}$ and the metric in the same class from the normalized Kähler–Ricci flow. With local Moser iteration, one can improve the previous bound $\tr_{\omega_{\varepsilon}}\chi\leq C$ to the sharp Schwarz-type estimate $\omega_{\varepsilon}\geq(1+o(1))f^{*}\omega_{\rm can}$ .

With the sharp Schwarz-type estimate, we are able to show that the rescaled fiber potential of $\omega_{\varepsilon}$ converges uniformly to the potential of $\omega_{\rm SRF}$ . Then a blow-up argument, together with the crucial $W^{2,p}$ -estimate of Cheng–Xu [3], and the local estimates for cscK metrics of Chen–Cheng [2] proves the metric convergence away from singular fibers.

After obtaining the metric convergence away from singular fibers, the full Gromov-Hausdorff convergence can be reduced to a one-sided estimate of the distance functions (by the work of Lee–Tosatti–Zhang [14]):

\sup_{x,x^{\prime}\in X}\big(d_{\rm can}(f(x),f(x^{\prime}))-d_{\omega_{\varepsilon}}(x,x^{\prime})\big)_{+}\longrightarrow 0.

In the setting of the Kähler–Ricci flow, an important ingredient in Lee–Tosatti–Zhang’s proof is Perelman’s monotonicity of the reduced volume. Here we develop another method based on purely analytic propositions about Lipschitz functions. We show that the above estimate on the distance functions can be reduced to the following Lipschitz lift proposition: for any $v\in\Lip_{d_{\rm can}}(Y)$ with $\Lip(v)\leq 1$ , and $\delta>0$ , there exists $v_{\varepsilon}\in C^{\infty}(X)$ such that $\limsup\|v_{\varepsilon}-f^{*}v\|_{L^{\infty}}\leq\delta$ and $\Lip_{d_{\omega_{\varepsilon}}}(v_{\varepsilon})\leq 1+\delta.$ The RCD structure of $Y$ and volume noncollapsing play an important role here. Finally, we show that this method can also be applied to the normalized Kähler–Ricci flow to give an independent proof of the theorem of Lee–Tosatti–Zhang.

Acknowledgements. The second author is sincerely grateful to Prof. Xiuxiong Chen for his continued encouragement and inspiring discussions on cscK metrics. He would also like to thank Hanbing Fang for helpful discussions on the convergence problem for the family of cscK metrics. This research is partially supported by the Hong Kong General Research Fund #16305625 of the Hong Kong Research Grants Council.

Declaration on the use of AI. The key $C^{2}$ -estimate in the base direction and gradient bound for the logarithm of the volume density are due to the authors. We then decomposed the proof into several steps and lemmas and asked ChatGPT 6 to produce counterexamples or proofs in special cases. The feedback inspired ideas that helped us adjust our approach. We arrived at the appropriate lemmas after several rounds of intensive discussion. The entire proof was written by the authors.

2. Basic facts

We give a preliminary discussion on the family of cscK metrics, known results for uniform estimates along this family, and known results for the normalized Kähler–Ricci flow.

We use the conventions

dd^{c}=\sqrt{-1}\partial\bar{\partial},\qquad\Delta_{\omega}u=\tr_{\omega}dd^{c}u,\qquad|\nabla u|_{\omega}=|\partial u|_{\omega}=\frac{1}{\sqrt{2}}|\nabla u|_{g_{\omega}},

where $g_{\omega}$ is the Riemannian metric corresponding to the Kähler form $\omega.$ The distance and Lipschitz constants are defined using the Riemannian metric $g_{\omega}$ , and hence $\Lip_{d_{\omega}}(u)=\sqrt{2}\||\nabla u|_{\omega}\|_{L^{\infty}}$ .

Choose a sufficiently large $\ell$ such that $\ell K_{X}=f^{*}\mathcal{O}_{Y}(1)$ and the pluricanonical map has connected fibers. Set

\chi_{Y}=\ell^{-1}\omega_{FS}|_{Y},\qquad\chi=f^{*}\chi_{Y},\qquad\chi_{\varepsilon}=\chi+\varepsilon{\omega}.

Here ${\omega}$ is a fixed smooth Kähler form in the Kähler class $[{\omega}]$ . Let $m$ be the Kodaira dimension, which is equal to the dimension of $Y$ here. Let $d=n-m$ , $V_{\varepsilon}=\int_{X}\chi_{\varepsilon}^{n},d\mu_{\varepsilon}=V_{\varepsilon}^{-1}\omega_{\varepsilon}^{n},$ where $\omega_{\varepsilon}$ is the unique cscK metric in $[\chi_{\varepsilon}]$ . We fix a smooth positive volume form $\Omega$ on $X$ with $\int_{X}\Omega=1$ , and $\Ric(\Omega)=-\chi$ . Let $D\subsetneq Y$ be the singular locus together with the critical values of $f$ . Then

Y^{\circ}=Y\setminus D,\qquad X^{\circ}=f^{-1}(Y^{\circ}),\qquad f:X^{\circ}\longrightarrow Y^{\circ}\quad\text{is a proper submersion with connected fibers}.

For $m>0$ , Song–Tian’s canonical metric [17, 18] $\omega_{\rm can}=\chi_{Y}+dd^{c}\varphi_{\rm can}$ is the unique solution of the twisted Kähler-Einstein equation

\Ric(\omega_{\rm can})=-\omega_{\rm can}+\omega_{WP},

where $\omega_{WP}\geq 0$ is the Weil–Petersson form of the Calabi–Yau fibration $X^{\circ}\to Y^{\circ}$ . The potential $\varphi_{\rm can}$ satisfies $\varphi_{\rm can}\in\PSH(Y,\chi_{Y})\cap L^{\infty}(Y)\cap C^{\infty}(Y^{\circ})$ , and $\omega_{\rm can}$ extends as a positive current on $Y$ .

Set

{\omega_{y}}={\omega}|_{X_{y}},\quad V_{F}=\int_{X_{y}}{\omega_{y}}^{d},\quad V_{Y}=\int_{Y}\chi_{Y}^{m}.

When $d=0$ , $V_{F}:=1$ . For $d>0$ , on $X^{\circ}$ , we have the semi-Ricci-flat form

\omega_{\rm SRF}={\omega}+dd^{c}\rho,\qquad\int_{X_{y}}\rho{\,\omega_{y}^{d}}=0,\qquad\omega_{{\rm SRF},y}:=\omega_{\rm SRF}|_{X_{y}}>0,\qquad\Ric(\omega_{{\rm SRF},y})=0.

The form $\omega_{\rm SRF}$ is smooth but need not be positive on $X^{\circ}$ . For $d=0$ , set $\rho=0$ and $\omega_{\rm SRF}={\omega}$ . We have

e^{f^{*}\varphi_{\rm can}}\Omega=\frac{(f^{*}\omega_{\rm can})^{m}\wedge\omega_{\rm SRF}^{d}}{V_{F}V_{Y}}.

(2.1)

and

(\chi_{Y}+dd^{c}\varphi_{\rm can})^{m}=V_{Y}e^{\varphi_{\rm can}}f_{*}\Omega

(2.2)

Let $d_{\rm can}$ be the distance on $Y^{\circ}$ induced by the canonical metric $\omega_{\rm can}$ . The canonical measure $\nu_{\rm can}$ is defined as $\nu_{\rm can}:=\frac{\omega_{\rm can}^{m}}{V_{Y}}=e^{\varphi_{\rm can}}f_{*}\Omega.$

Recall first the following theorem of Székelyhidi on the RCD structure of the canonical model.

Theorem 2.1 (Székelyhidi [21, Theorem 15]).

For the semi-ample canonical fibration with $m>0$ , the completion $\overline{Y^{\circ}}=\overline{(Y^{\circ},d_{\rm can})}$ satisfies:

(1)

$\overline{Y^{\circ}}$ is compact and canonically homeomorphic to $Y$ .
(2)

$(Y,d_{\rm can},\nu_{\rm can})$ is a noncollapsed $\operatorname{RCD}(-1,2m)$ space.
(3)

$\dim_{\mathcal{H}}(Y\setminus Y^{\circ})\leq 2m-2$ .

Now we recall the equations of cscK metrics. The cscK equation can be expressed as

\begin{cases}\omega_{\varepsilon}^{n}=V_{\varepsilon}e^{F_{\varepsilon}}\Omega\\ \Delta_{\omega_{\varepsilon}}F_{\varepsilon}=-\underline{R}_{\varepsilon}-\tr_{\omega_{\varepsilon}}\chi\end{cases}

We also set $\omega_{\varepsilon}=\chi_{\varepsilon}+dd^{c}\varphi_{\varepsilon}$ with $\sup_{X}\varphi_{\varepsilon}=0$ . The average scalar curvature can be computed as

\underline{R}_{\varepsilon}=-n\frac{A\cdot(A+\varepsilon B)^{n-1}}{(A+\varepsilon B)^{n}},\quad\text{ where }A=2\pi c_{1}(K_{X}),B=[{\omega}].

Put $a_{j}=\int_{X}\chi^{j}\wedge{\omega}^{n-j}$ . Then

(1)

$a_{m}>0$ , $\chi^{m+1}=0$ , and $V_{\varepsilon}=\sum_{j=0}^{m}\binom{n}{j}a_{j}\varepsilon^{n-j}.$

(2)

If $m>0$ , then

	$\displaystyle V_{\varepsilon}$	$\displaystyle=\binom{n}{m}a_{m}\varepsilon^{d}+\binom{n}{m-1}a_{m-1}\varepsilon^{d+1}+O(\varepsilon^{d+2}),$
	$\displaystyle\underline{R}_{\varepsilon}$	$\displaystyle=-m+\frac{ma_{m-1}}{(d+1)a_{m}}\varepsilon+O(\varepsilon^{2}).$

(3)

If $m=0$ , then $V_{\varepsilon}=a_{0}\varepsilon^{n}$ and $\underline{R}_{\varepsilon}=0$ .

We also recall the uniform estimates of Guo–Jian–Shi–Song.

Theorem 2.2 (Guo–Jian–Shi–Song [9, Theorems 1.1 and 2.1]).

There are constants $C,\varepsilon_{0},a_{\rm vol}>0$ such that for $0<\varepsilon<\varepsilon_{0}$ :

(1)

$\|\varphi_{\varepsilon}\|_{\infty}+\|F_{\varepsilon}\|_{\infty}\leq C$ , and hence $C^{-1}\Omega\leq{d\mu_{\varepsilon}}\leq C\Omega$ .
(2)

$\diam(X,\omega_{\varepsilon})\leq C$ .
(3)

$\mu_{\varepsilon}(B_{\omega_{\varepsilon}}(x,r))\geq C^{-1}r^{a_{\rm vol}}$ for $x\in X$ , $0<r<1$ . Consequently the family is GH precompact.

To prove higher-order estimates, we also need the uniform Sobolev inequality obtained by Guo–Phong–Song–Sturm.

Theorem 2.3 (Guo–Phong–Song–Sturm [10, Theorem 2.1]).

Let ${(X,\omega)}$ be a fixed compact Kähler manifold. For a Kähler form ${\eta}$ , set

\color[rgb]{0,0,0}V_{\eta}=\int_{X}\eta^{n},\qquad d\mu_{\eta}=\frac{\eta^{n}}{V_{\eta}},\qquad I_{\eta}=[\eta]\cdot[\omega]^{n-1},\qquad V_{\omega}=\int_{X}\omega^{n}.

Fix ${p>n}$ and constants ${K,c>0}$ . Suppose that

\color[rgb]{0,0,0}\int_{X}\left|\log\frac{\eta^{n}/V_{\eta}}{\omega^{n}/V_{\omega}}\right|^{p}d\mu_{\eta}\leq K,\qquad\frac{\eta^{n}/V_{\eta}}{\omega^{n}/V_{\omega}}\geq c.

Then there exist constants ${q=q(n,p)>1}$ and ${C_{S}>0}$ , depending only on the fixed background, ${n,p,K,c}$ , and ${q}$ , such that, for every ${u\in W^{1,2}(X,\eta)}$ ,

\color[rgb]{0,0,0}\left(\int_{X}|u-\bar{u}_{\eta}|^{2q}d\mu_{\eta}\right)^{1/q}\leq C_{S}I_{\eta}\int_{X}|\nabla u|_{\eta}^{2}d\mu_{\eta},\qquad\bar{u}_{\eta}=\int_{X}u\,d\mu_{\eta}.

This is the positive-constant lower-density special case of the cited theorem. For ${\eta=\omega_{\varepsilon}}$ , Theorem 2.2 and the smooth positivity of ${\Omega}$ give both hypotheses uniformly, while ${I_{\omega_{\varepsilon}}}$ is uniformly bounded. Thus the Sobolev constant for the cscK family is uniform.

We will also need the convergence results of normalized Kähler–Ricci flow,

\begin{cases}\partial_{t}{\tilde{\omega}}(t)=-\Ric({\tilde{\omega}}(t))-{\tilde{\omega}}(t),\\ {\tilde{\omega}}(0)={\omega}.\end{cases}

It can be written as a scalar equation:

\begin{cases}\partial_{t}{\tilde{\varphi}}=\log\frac{e^{dt}(e^{-t}{\omega}+(1-e^{-t})\chi+dd^{c}{\tilde{\varphi}})^{n}}{\binom{n}{d}V_{F}V_{Y}\Omega}-{\tilde{\varphi}},\\ {\tilde{\varphi}}(0)=0.\end{cases}

We recall the following facts for the above normalized Kähler–Ricci flow, see [23] and the references therein.

Theorem 2.4.

(1)

$|{\tilde{\varphi}}(t)|+|\partial_{t}{\tilde{\varphi}}(t)|\leq C$ and $C^{-1}e^{-dt}{\omega}^{n}\leq({\tilde{\omega}}(t))^{n}\leq Ce^{-dt}{\omega}^{n}$ . See Song–Tian [19, Lemma 2.1 and Proposition 2.1]; the volume bounds follow from the scalar flow equation and these potential bounds.
(2)

${\tilde{\varphi}}(t)\to f^{*}\varphi_{\rm can}$ and $\partial_{t}{\tilde{\varphi}}\to 0$ uniformly on every compact set of $X^{\circ},$ and in $L^{p}(X,\Omega)$ for any $1\leq p<\infty$ . For convergence of the potential, see Song–Tian [18, Proposition 5.4]; for convergence of the time derivative on the regular locus, see Tosatti–Weinkove–Yang [23, Lemma 3.1(ii), (iv)]. The finite- $L^{p}$ convergence follows from part (1) and dominated convergence.
(3)

$\|{\tilde{\omega}}(t)-(e^{-t}\omega_{\rm SRF}+(1-e^{-t})f^{*}\omega_{\rm can})\|_{{\tilde{\omega}}(t)}\to 0$ on every compact subset of $X^{\circ}$ . See Tosatti–Weinkove–Yang [23, Section 3.3, proof of Theorem 1.2(i), immediately after equation (3.22)], where this relative-norm convergence is established using Lemma 2.6.

We set

t(\varepsilon)=\log\frac{1+\varepsilon}{\varepsilon},\qquad{\tilde{\omega}}_{\varepsilon}=(1+\varepsilon){\tilde{\omega}}(t(\varepsilon)),\qquad\psi_{\varepsilon}=(1+\varepsilon){\tilde{\varphi}}(t(\varepsilon)).

This time reparametrization and rescaling adjust the Kähler–Ricci flow metric to lie in the same Kähler class as the cscK metric $\omega_{\varepsilon}$ . Indeed, ${\tilde{\omega}}_{\varepsilon}=\chi+\varepsilon{\omega}+dd^{c}\psi_{\varepsilon}\in 2\pi c_{1}(K_{X})+\varepsilon[{\omega}]$ . The above convergence and estimates give

\psi_{\varepsilon}\to f^{*}\varphi_{\rm can}\text{ in }L^{p}(X,\Omega),\qquad\|\psi_{\varepsilon}\|_{L^{\infty}}\leq C,

and

\frac{{\tilde{\omega}}_{\varepsilon}^{n}}{V_{\varepsilon}}=\frac{\binom{n}{d}V_{F}V_{Y}(1+\varepsilon)^{m}\varepsilon^{d}}{V_{\varepsilon}}e^{({\tilde{\varphi}}+{\dot{\tilde{\varphi}}})(t(\varepsilon))}\Omega\longrightarrow e^{f^{*}\varphi_{\rm can}}\Omega\quad\text{in total variation}.

3. Uniform second-order estimate in base direction for the cscK metrics

In this section, we show that the second-order estimate in the base direction for the path of cscK metrics. This is analogous to the parabolic Schwarz lemma of Song–Tian [17], whose proof uses the parabolic maximum principle. The proof is inspired by the work of Chen-Cheng [2], and adapted to our fibration structure over the canonical model here.

Fix $\varepsilon$ , set

\varphi=\varphi_{\varepsilon},\quad\omega_{\varphi}=\omega_{\varepsilon},\quad F=F_{\varepsilon},\quad\underline{R}=\underline{R}_{\varepsilon},\quad\Delta_{\varphi}=\Delta_{\omega_{\varphi}},\quad d\mu=\frac{\omega_{\varphi}^{n}}{\int_{X}\omega_{\varphi}^{n}}.

We write $\tr_{\varphi}\chi=\tr_{\omega_{\varphi}}\chi$ and use $\nabla_{\varphi}$ for covariant differentiation with respect to $g_{\varphi}$ . In this section we prove the $C^{2}$ -estimate in the base direction in the following order:

\|\tr_{\varphi}\chi\|_{L^{p}(\mu)}\leq C_{p}\quad(p<\infty)\ \Longrightarrow\ \|\nabla_{\varphi}F\|_{\infty}\leq C\ \Longrightarrow\ \|\tr_{\varphi}\chi\|_{\infty}\leq C.

At a prescribed point, choose holomorphic normal coordinates for $g_{\varphi}$ and diagonalize $\chi$ :

g_{\varphi,i\bar{j}}=\delta_{ij},\qquad\partial_{k}g_{\varphi,i\bar{j}}=0,\qquad\chi_{i\bar{j}}=\lambda_{i}\delta_{ij},\qquad\lambda_{i}\geq 0.

(3.1)

The following lemma provides the first step above: uniform bounds for the trace of the pullback metric in every finite ${L^{p}}$ space. We formulate it for a holomorphic map into a fixed Kähler target. In our application, we take ${Z=\mathbb{C}P^{N}}$ , equipped with ${\omega_{Z}=\ell^{-1}\omega_{FS}}$ , and ${\Phi=\iota\circ f}$ , where ${\iota:Y\hookrightarrow\mathbb{C}P^{N}}$ is the inclusion. Thus ${\Phi}$ is the pluricanonical map supplied by the semi-ampleness of $K_{X}$ , and its pullback metric is precisely $\chi$ .

Lemma 3.1.

Let $\Phi:X\to Z$ be a holomorphic map from a compact complex manifold $X$ to a fixed compact Kähler manifold $(Z,\omega_{Z})$ , and put $\chi=\Phi^{*}\omega_{Z}$ . Suppose that $\omega_{\varphi}$ , $F$ , and a constant $\underline{R}$ satisfy

\Ric(\omega_{\varphi})=-\chi-dd^{c}F,\qquad\Delta_{\varphi}F=-\underline{R}-\tr_{\varphi}\chi.

(3.2)

If $|\underline{R}|+\operatorname{osc}_{X}F\leq C_{0}$ , then, for every finite $p\geq 1$ ,

\int_{X}(\tr_{\varphi}\chi)^{p}\,d\mu\leq C_{p}.

(3.3)

The constant $C_{p}$ depends only on $p,C_{0}$ and an upper bound for the holomorphic bisectional curvature of $\omega_{Z}$ .

Proof.

Integration of (3.2) gives $-\underline{R}=\int_{X}\tr_{\varphi}\chi\,d\mu\geq 0$ . Choose source coordinates satisfying (3.1) at $x_{0}$ , and normal target coordinates at $\Phi(x_{0})$ , and write $\omega_{Z}=\sqrt{-1}h_{\alpha\bar{\beta}}dw^{\alpha}\wedge d\bar{w}^{\beta}$ . Then

\chi_{i\bar{j}}=\sum_{\alpha,\beta}h_{\alpha\bar{\beta}}(\Phi)\Phi_{i}^{\alpha}\overline{\Phi_{j}^{\beta}},\qquad\tr_{\varphi}\chi=\sum_{i,\alpha}|\Phi_{i}^{\alpha}|^{2}=\sum_{i}\lambda_{i}.

With $R_{i\bar{j}k\bar{l}}=-\partial_{k}\partial_{\bar{l}}g_{\varphi,i\bar{j}}$ at the normal point, differentiation gives

	$\displaystyle(\tr_{\varphi}\chi)_{k}$	$\displaystyle=\sum_{i,\alpha}\Phi_{ik}^{\alpha}\overline{\Phi_{i}^{\alpha}},$		(3.4)
	$\displaystyle\Delta_{\varphi}\tr_{\varphi}\chi$	$\displaystyle=\sum_{i,k,\alpha}\|\Phi_{ik}^{\alpha}\|^{2}+\sum_{i}\lambda_{i}\Ric(\omega_{\varphi})_{i\bar{i}}-\sum_{i,k,\alpha,\beta,\gamma,\delta}R^{\omega_{Z}}_{\alpha\bar{\beta}\gamma\bar{\delta}}\Phi_{i}^{\alpha}\overline{\Phi_{i}^{\beta}}\Phi_{k}^{\gamma}\overline{\Phi_{k}^{\delta}}.$		(3.5)

By Cauchy–Schwarz inequality,

|\nabla_{\varphi}\tr_{\varphi}\chi|_{\varphi}^{2}\leq(\tr_{\varphi}\chi)\sum_{i,k,\alpha}|\Phi_{ik}^{\alpha}|^{2}.

By (3.2),

\sum_{i}\lambda_{i}\Ric(\omega_{\varphi})_{i\bar{i}}=-\sum_{i}\lambda_{i}^{2}-\sum_{i}\lambda_{i}F_{i\bar{i}}.

Choose $K\geq 1$ so that the bisectional curvature of $\omega_{Z}$ is at most $K-1$ . Raising both indices of $\chi$ with $g_{\varphi}$ , we obtain

\Delta_{\varphi}\tr_{\varphi}\chi\geq\sum_{i,k,\alpha}|\Phi_{ik}^{\alpha}|^{2}-K(\tr_{\varphi}\chi)^{2}-\sum_{i,j}\chi^{i\bar{j}}F_{i\bar{j}}.

(3.6)

Here $\chi^{i\bar{j}}=g_{\varphi}^{i\bar{b}}\chi_{a\bar{b}}g_{\varphi}^{a\bar{j}}$ , so $\chi^{i\bar{j}}=\lambda_{i}\delta_{ij}$ at the normal point.

For $0<\delta\leq 1$ , put $v=\tr_{\varphi}\chi+\delta$ . Since $v>0$ ,

\Delta_{\varphi}v\geq\frac{|\nabla_{\varphi}v|_{\varphi}^{2}}{v}-Kv^{2}-\sum_{i,j}\chi^{i\bar{j}}F_{i\bar{j}},\qquad\Delta_{\varphi}F=-\underline{R}+\delta-v.

(3.7)

The closedness of $\chi$ gives, at the normal point,

\sum_{i}\nabla_{\varphi,i}\chi^{i\bar{j}}=\sum_{i}\nabla_{\varphi,i}\chi_{j\bar{i}}=\sum_{i}\nabla_{\varphi,j}\chi_{i\bar{i}}=v_{j}.

(3.8)

Fix $p\geq 1$ , and choose $b\in\mathbb{R}$ later. Testing (3.7) with $e^{bF}v^{p-1}$ ,

	$\displaystyle\int_{X}e^{bF}v^{p-1}\Delta_{\varphi}v\,d\mu$	$\displaystyle=-\int_{X}e^{bF}v^{p}\sum_{i}\left[(p-1)\left\|\frac{v_{i}}{v}\right\|^{2}+b\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)\right]d\mu,$
	$\displaystyle\int_{X}e^{bF}v^{p-2}\|\nabla_{\varphi}v\|_{\varphi}^{2}\,d\mu$	$\displaystyle=\int_{X}e^{bF}v^{p}\sum_{i}\left\|\frac{v_{i}}{v}\right\|^{2}d\mu,$
	$\displaystyle-\int_{X}e^{bF}v^{p-1}\chi^{i\bar{j}}F_{i\bar{j}}\,d\mu$	$\displaystyle=\operatorname{Re}\int_{X}\nabla_{\varphi,i}(e^{bF}v^{p-1}\chi^{i\bar{j}})F_{\bar{j}}\,d\mu.$

Repeated indices in the last line are summed. Expanding its derivative and using (3.8) gives

		$\displaystyle\operatorname{Re}\sum_{i,j}\nabla_{\varphi,i}(e^{bF}v^{p-1}\chi^{i\bar{j}})F_{\bar{j}}$
		$\displaystyle=e^{bF}v^{p}\sum_{i}\left[b\frac{\lambda_{i}}{v}\|F_{i}\|^{2}+(p-1)\frac{\lambda_{i}}{v}\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)+\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)\right].$

Consequently,

\begin{split}&\int_{X}e^{bF}v^{p}\sum_{i}\left[p\left|\frac{v_{i}}{v}\right|^{2}+\left((p-1)\frac{\lambda_{i}}{v}+b+1\right)\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)+b\frac{\lambda_{i}}{v}|F_{i}|^{2}\right]d\mu\\ &\hskip 128.0374pt\leq K\int_{X}e^{bF}v^{p+1}\,d\mu.\end{split}

(3.9)

Testing the equation $\Delta_{\varphi}F=-\underline{R}+\delta-v$ with $e^{bF}v^{p}$ similarly gives

\begin{split}&\int_{X}e^{bF}v^{p}\sum_{i}\left[p\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)+b|F_{i}|^{2}\right]d\mu\\ &\qquad=\int_{X}e^{bF}v^{p+1}\,d\mu-(-\underline{R}+\delta)\int_{X}e^{bF}v^{p}\,d\mu.\end{split}

(3.10)

Subtract $\Lambda$ times (3.10) from (3.9):

\begin{split}\int_{X}e^{bF}v^{p}\mathcal{Q}\,d\mu+(\Lambda-K)\int_{X}e^{bF}v^{p+1}\,d\mu\leq\Lambda(-\underline{R}+\delta)\int_{X}e^{bF}v^{p}\,d\mu,\end{split}

(3.11)

where

\begin{split}\mathcal{Q}=\sum_{i}\bigg[&p\left|\frac{v_{i}}{v}\right|^{2}+\left((p-1)\frac{\lambda_{i}}{v}+b+1-\Lambda p\right)\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)\\ &+b\left(\frac{\lambda_{i}}{v}-\Lambda\right)|F_{i}|^{2}\bigg].\end{split}

We seek $\Lambda>K$ and $\mathcal{Q}\geq 0$ . At the endpoint $\lambda_{i}/v=1$ , requiring the summand to be $p|v_{i}/v-\alpha F_{i}|^{2}$ gives

\begin{cases}p+b-\Lambda p=-2p\alpha,\\ b(1-\Lambda)=p\alpha^{2}.\end{cases}\quad\Longrightarrow\quad(\Lambda-1-\alpha)^{2}=0,\qquad\Lambda=\alpha+1,\quad b=-p\alpha.

For these choices, set $u=e^{-\alpha F}v$ . Then

(\log u)_{i}=\frac{v_{i}}{v}-\alpha F_{i},\qquad 0\leq\frac{\lambda_{i}}{v}\leq 1,

and the coefficients of the remaining terms satisfy

	$\displaystyle(p-1)\frac{\lambda_{i}}{v}+b+1-\Lambda p$	$\displaystyle=-2p\alpha-(p-1)\left(1-\frac{\lambda_{i}}{v}\right),$
	$\displaystyle b\left(\frac{\lambda_{i}}{v}-\Lambda\right)$	$\displaystyle=p\alpha^{2}+p\alpha\left(1-\frac{\lambda_{i}}{v}\right).$

Substitution, followed by $v_{i}/v=(\log u)_{i}+\alpha F_{i}$ , yields

	$\displaystyle\mathcal{Q}$	$\displaystyle=\sum_{i}p\left\|\frac{v_{i}}{v}-\alpha F_{i}\right\|^{2}+\sum_{i}\left(1-\frac{\lambda_{i}}{v}\right)\left[-(p-1)\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)+p\alpha\|F_{i}\|^{2}\right]$
		$\displaystyle=\sum_{i}p\|(\log u)_{i}\|^{2}+\sum_{i}\left(1-\frac{\lambda_{i}}{v}\right)\left[-(p-1)\operatorname{Re}((\log u)_{i}F_{\bar{i}})+\alpha\|F_{i}\|^{2}\right].$

For each $i$ , the last bracket equals

\alpha\left|F_{i}-\frac{p-1}{2\alpha}(\log u)_{i}\right|^{2}-\frac{(p-1)^{2}}{4\alpha}|(\log u)_{i}|^{2}.

Hence one gets the complete square identity

\begin{split}\mathcal{Q}={}&\sum_{i}\alpha\left(1-\frac{\lambda_{i}}{v}\right)\left|F_{i}-\frac{p-1}{2\alpha}(\log u)_{i}\right|^{2}\\ &+\sum_{i}\left[p-\frac{(p-1)^{2}}{4\alpha}\left(1-\frac{\lambda_{i}}{v}\right)\right]|(\log u)_{i}|^{2}.\end{split}

(3.12)

Choose

\alpha=K+p,\qquad\Lambda-K=p+1,\qquad p-\frac{(p-1)^{2}}{4\alpha}\left(1-\frac{\lambda_{i}}{v}\right)\geq\frac{p}{2}.

Thus (3.11) gives

\begin{split}&\frac{p}{2}\int_{X}e^{-p\alpha F}v^{p}|\nabla_{\varphi}\log u|_{\varphi}^{2}\,d\mu+(p+1)\int_{X}e^{-p\alpha F}v^{p+1}\,d\mu\\ &\qquad\leq(\alpha+1)(-\underline{R}+\delta)\int_{X}e^{-p\alpha F}v^{p}\,d\mu.\end{split}

(3.13)

Hölder’s inequality gives

\int_{X}e^{-p\alpha F}v^{p}\,d\mu\leq\left(\int_{X}e^{-p\alpha F}v^{p+1}\,d\mu\right)^{p/(p+1)}\left(\int_{X}e^{-p\alpha F}\,d\mu\right)^{1/(p+1)}.

Dropping the first nonnegative term in (3.13) yields

\int_{X}e^{-p\alpha F}v^{p+1}\,d\mu\leq\left(\frac{(\alpha+1)(-\underline{R}+\delta)}{p+1}\right)^{p+1}\int_{X}e^{-p\alpha F}\,d\mu.

Since $\mu(X)=1$ , multiplying by $e^{p\alpha\sup F}$ gives

\int_{X}(\tr_{\varphi}\chi+\delta)^{p+1}\,d\mu\leq\left(\frac{(\alpha+1)(-\underline{R}+\delta)}{p+1}\right)^{p+1}e^{p\alpha\operatorname{osc}F},\qquad\alpha=K+p.

(3.14)

Let $\delta\downarrow 0$ . This proves (3.3) for $p\geq 2$ ; for $1\leq p<2$ , use $\|\tr_{\varphi}\chi\|_{p}\leq\|\tr_{\varphi}\chi\|_{2}$ . ∎

For the canonical family, take $Z=\mathbb{\mathbb{C}}P^{N}$ and the scaled Fubini–Study metric whose pullback is $\chi$ . Theorem 2.2 give $|\underline{R}_{\varepsilon}|+\operatorname{osc}F_{\varepsilon}\leq C$ . The target is smooth even when $Y$ is singular, so (3.3) holds uniformly for every fixed finite $p$ .

Now we prove the gradient bound of $F$ .

Lemma 3.2.

Under the same assumptions in Lemma 3.1,

\sup_{X}|\nabla_{\varphi}F|_{\varphi}^{2}\leq C^{\prime}.

(3.15)

Proof.

Put $w=|\nabla_{\varphi}F|_{\varphi}^{2}$ . The Bochner formula gives

\Delta_{\varphi}w=\sum_{i,j}(|F_{ij}|^{2}+|F_{i\bar{j}}|^{2})-2\operatorname{Re}\sum_{i}(\tr_{\varphi}\chi)_{i}F_{\bar{i}}-\sum_{i}\lambda_{i}|F_{i}|^{2}-\sum_{i,j}F_{i\bar{j}}F_{j}F_{\bar{i}}.

(3.16)

Direct computation gives

w_{i}=\sum_{j}(F_{ji}F_{\bar{j}}+F_{j}F_{i\bar{j}}),

e^{-bF}\Delta_{\varphi}e^{bF}=b(-\underline{R}-\tr_{\varphi}\chi)+b^{2}w,

2e^{-bF}\operatorname{Re}\sum_{i}(e^{bF})_{i}w_{\bar{i}}=2b\operatorname{Re}\sum_{i,j}F_{ij}F_{\bar{i}}F_{\bar{j}}+2b\sum_{i,j}F_{i\bar{j}}F_{j}F_{\bar{i}},

\sum_{i,j}|F_{ij}|^{2}+2b\operatorname{Re}\sum_{i,j}F_{ij}F_{\bar{i}}F_{\bar{j}}+b^{2}w^{2}=\sum_{i,j}|F_{ij}+bF_{i}F_{j}|^{2}.

Thus

\begin{split}e^{-bF}\Delta_{\varphi}(e^{bF}w)=&\sum_{i,j}|F_{ij}+bF_{i}F_{j}|^{2}+\sum_{i,j}|F_{,i\bar{j}}|^{2}+(2b-1)\sum_{i,j}F_{i\bar{j}}F_{j}F_{\bar{i}}-2\operatorname{Re}\sum_{i}(\tr_{\varphi}\chi)_{i}F_{\bar{i}}\\ &-\sum_{i}\lambda_{i}|F_{i}|^{2}+b(-\underline{R}-\tr_{\varphi}\chi)w.\end{split}

The mixed Hessian has no sign; its coefficient vanishes for $b=\frac{1}{2}$ . This is [2, equations (4.3)–(4.4)], with reference Ricci form $-\chi$ . Set

u=1+e^{F/2}|\nabla_{\varphi}F|_{\varphi}^{2}.

Then

\begin{split}\Delta_{\varphi}u=&e^{F/2}\sum_{i,j}\left(|F_{ij}+\tfrac{1}{2}F_{i}F_{j}|^{2}+|F_{i\bar{j}}|^{2}\right)-2e^{F/2}\operatorname{Re}\sum_{i}(\tr_{\varphi}\chi)_{i}F_{\bar{i}}-e^{F/2}\sum_{i}\lambda_{i}|F_{i}|^{2}\\ &+\tfrac{1}{2}(-\underline{R}-\tr_{\varphi}\chi)(u-1).\end{split}

(3.17)

Since $0\leq\lambda_{i}\leq\tr_{\varphi}\chi$ , dropping the Hessian squares gives

\Delta_{\varphi}u\geq-2e^{F/2}\operatorname{Re}\langle\nabla_{\varphi}\tr_{\varphi}\chi,\nabla_{\varphi}F\rangle_{\varphi}-\left(\tfrac{1}{2}\underline{R}+\tfrac{3}{2}\tr_{\varphi}\chi\right)(u-1).

(3.18)

For $p\geq 2$ , testing (3.18) with $u^{p-1}$ gives

\begin{split}&(p-1)\int_{X}u^{p-2}|\nabla_{\varphi}u|_{\varphi}^{2}\,d\mu\\ &\quad\leq 2\int_{X}e^{F/2}u^{p-1}\operatorname{Re}\langle\nabla_{\varphi}\tr_{\varphi}\chi,\nabla_{\varphi}F\rangle_{\varphi}\,d\mu\\ &\qquad+\int_{X}\left(\tfrac{1}{2}\underline{R}+\tfrac{3}{2}\tr_{\varphi}\chi\right)(u-1)u^{p-1}\,d\mu.\end{split}

(3.19)

For the first integral on the right,

\begin{split}&2\int_{X}e^{F/2}u^{p-1}\operatorname{Re}\langle\nabla_{\varphi}\tr_{\varphi}\chi,\nabla_{\varphi}F\rangle_{\varphi}\,d\mu\\ &\quad=-2\int_{X}(\tr_{\varphi}\chi)e^{F/2}u^{p-1}(-\underline{R}-\tr_{\varphi}\chi)\,d\mu\\ &\qquad-\int_{X}(\tr_{\varphi}\chi)(u-1)u^{p-1}\,d\mu\\ &\qquad-2(p-1)\int_{X}(\tr_{\varphi}\chi)e^{F/2}u^{p-2}\operatorname{Re}\langle\nabla_{\varphi}u,\nabla_{\varphi}F\rangle_{\varphi}\,d\mu.\end{split}

(3.20)

Because $e^{F/2}|\nabla_{\varphi}F|_{\varphi}=e^{F/4}\sqrt{u-1}\leq C\sqrt{u}$ , Young’s inequality gives

\begin{split}&2(p-1)\left|\int_{X}(\tr_{\varphi}\chi)e^{F/2}u^{p-2}\operatorname{Re}\langle\nabla_{\varphi}u,\nabla_{\varphi}F\rangle_{\varphi}\,d\mu\right|\\ &\quad\leq\frac{p-1}{2}\int_{X}u^{p-2}|\nabla_{\varphi}u|_{\varphi}^{2}\,d\mu+C(p-1)\int_{X}(\tr_{\varphi}\chi)^{2}u^{p-1}\,d\mu.\end{split}

The terms without $\nabla_{\varphi}u$ combine to

\begin{split}&2\underline{R}\int_{X}(\tr_{\varphi}\chi)e^{F/2}u^{p-1}\,d\mu+2\int_{X}(\tr_{\varphi}\chi)^{2}e^{F/2}u^{p-1}\,d\mu\\ &\quad+\frac{1}{2}\int_{X}(\tr_{\varphi}\chi)(u-1)u^{p-1}\,d\mu+\frac{\underline{R}}{2}\int_{X}(u-1)u^{p-1}\,d\mu\\ &\qquad\leq C\int_{X}\bigl(1+(\tr_{\varphi}\chi)^{2}\bigr)u^{p}\,d\mu,\end{split}

using $u\geq 1$ . Absorbing the gradient term and using $|\nabla_{\varphi}u^{p/2}|_{\varphi}^{2}=\frac{p^{2}}{4}u^{p-2}|\nabla_{\varphi}u|_{\varphi}^{2}$ yields

\int_{X}|\nabla_{\varphi}u^{p/2}|_{\varphi}^{2}\,d\mu\leq Cp^{2}\int_{X}\bigl(1+(\tr_{\varphi}\chi)^{2}\bigr)u^{p}\,d\mu.

(3.21)

Let $\theta=q/[r(q-1)]\in(0,1)$ , where $q>1$ . Hölder and interpolation inequality give

\begin{split}\int_{X}\bigl(1+(\tr_{\varphi}\chi)^{2}\bigr)u^{p}\,d\mu&\leq C\|u^{p/2}\|_{L^{2r/(r-1)}}^{2}\leq C\|u^{p/2}\|_{L^{2q}}^{2\theta}\|u^{p/2}\|_{L^{2}}^{2(1-\theta)}\\ &\leq\tau\|u^{p/2}\|_{L^{2q}}^{2}+C\tau^{-\theta/(1-\theta)}\|u^{p/2}\|_{L^{2}}^{2},\qquad 0<\tau\leq 1.\end{split}

(3.22)

Apply the uniform Sobolev inequality 2.3 to $u^{p/2}$ . Choosing $\tau=(2Cp^{2})^{-1}$ in (3.22) and absorbing half the left side gives

\|u\|_{L^{pq}}\leq\bigl(Cp^{2/(1-\theta)}\bigr)^{1/p}\|u\|_{L^{p}},\qquad p\geq 2.

(3.23)

For $p_{j}=2q^{j}$ , both $\sum_{j}p_{j}^{-1}$ and $\sum_{j}p_{j}^{-1}\log p_{j}$ converge. Iteration and interpolation yield

\|u\|_{L^{\infty}}\leq C\|u\|_{L^{2}}\leq C\|u\|_{L^{\infty}}^{1/2}\|u\|_{L^{1}}^{1/2},\qquad\|u\|_{L^{\infty}}\leq C^{2}\|u\|_{L^{1}}.

Finally, by (3.2),

\int_{X}wd\mu=-\int_{X}F(-\underline{R}-\tr_{\varphi}\chi)d\mu\leq 2|\underline{R}|\|F\|_{L^{\infty}},\qquad\|u\|_{L^{1}}\leq 1+e^{\|F\|_{L^{\infty}}/2}\int_{X}wd\mu\leq C.

This proves (3.15). ∎

Now we can prove the desired $C^{2}$ -estimates in the base direction.

Lemma 3.3.

Under the assumption of Lemma 3.1,

\tr_{\varphi}\chi\leq C^{\prime},\qquad\omega_{\varphi}\geq(C^{\prime})^{-1}\chi.

Proof.

Set $v=1+\tr_{\varphi}\chi$ . Taking $\delta=1$ , $b=0$ in (3.9) gives, for $p\geq 2$ ,

\int_{X}v^{p}\sum_{i}\left[p\left|\frac{v_{i}}{v}\right|^{2}+\left((p-1)\frac{\lambda_{i}}{v}+1\right)\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)\right]d\mu\leq K\int_{X}v^{p+1}\,d\mu.

Since $0\leq\lambda_{i}/v\leq 1$ ,

\left|\sum_{i}\left((p-1)\frac{\lambda_{i}}{v}+1\right)\operatorname{Re}\left(\frac{v_{i}}{v}F_{\bar{i}}\right)\right|\leq p\frac{|\nabla_{\varphi}v|_{\varphi}}{v}|\nabla_{\varphi}F|_{\varphi}\leq\frac{p}{2}\frac{|\nabla_{\varphi}v|_{\varphi}^{2}}{v^{2}}+Cp.

Absorption gives

\frac{p}{2}\int_{X}v^{p-2}|\nabla_{\varphi}v|_{\varphi}^{2}\,d\mu\leq K\int_{X}v^{p+1}\,d\mu+Cp\int_{X}v^{p}\,d\mu,

and hence

\int_{X}|\nabla_{\varphi}v^{p/2}|_{\varphi}^{2}\,d\mu\leq Cp^{2}\int_{X}(1+\tr_{\varphi}\chi)v^{p}\,d\mu.

(3.24)

For fixed $r>q/(q-1)$ , the coefficient $1+\tr_{\varphi}\chi$ has a uniform $L^{r}$ bound. The interpolation and iteration in (3.22)–(3.23) therefore give

\|1+\tr_{\varphi}\chi\|_{L^{\infty}}\leq C\|1+\tr_{\varphi}\chi\|_{L^{2}}\leq C^{\prime}.

Finally, $\lambda_{i}\leq\sum_{j}\lambda_{j}=\tr_{\varphi}\chi\leq C^{\prime}$ , so $\chi\leq C^{\prime}\omega_{\varphi}$ . ∎

4. Uniform convergence of the normalized volume density

In this section, we first show that one can always take limit(up to subsequence) of $\varphi_{\varepsilon}$ and $F_{\varepsilon}$ in $L^{p}$ for any finite $p$ . Then we analyze the convergence of the Mabuchi energy along the cscK path. Using the minimizing property of the cscK metrics to identify the limit of $F_{\varepsilon}$ as the canonical volume density $f^{*}\varphi_{\rm can}$ . Then based on the uniform Sobolev inequality of Guo–Phong–Song–Sturm [10], we apply the local Moser iteration to improve the $L^{p}$ convergence of $F_{\varepsilon}$ to $L^{\infty}$ convergence away from the singular fibers.

With the estimates in previous section, we observe that one can take some weak limit up to subsequence.

Lemma 4.1.

The functions $F_{\varepsilon}$ , $\varphi_{\varepsilon}$ , have uniformly bounded normalized Dirichlet energy:

\int_{X}|\nabla F_{\varepsilon}|_{\omega_{\varepsilon}}^{2}\,d\mu_{\varepsilon}+\int_{X}|\nabla\varphi_{\varepsilon}|_{\omega_{\varepsilon}}^{2}\,d\mu_{\varepsilon}\leq C.

Proof.

$\chi\geq 0$ gives

-\underline{R}_{\varepsilon}=\int_{X}\operatorname{tr}_{\omega_{\varepsilon}}\chi\,d\mu_{\varepsilon}\geq 0.

Note that $\underline{R}_{\varepsilon}\to-m$ . Integration by parts yields

\int_{X}|\nabla F_{\varepsilon}|_{\omega_{\varepsilon}}^{2}\,d\mu_{\varepsilon}=-\int_{X}F_{\varepsilon}(-\underline{R}_{\varepsilon}-\operatorname{tr}_{\omega_{\varepsilon}}\chi)\,d\mu_{\varepsilon}\leq 2|\underline{R}_{\varepsilon}|\|F_{\varepsilon}\|_{\infty}.

Likewise,

\Delta_{\omega_{\varepsilon}}\varphi_{\varepsilon}=n-\operatorname{tr}_{\omega_{\varepsilon}}\chi_{\varepsilon},\qquad\int_{X}\operatorname{tr}_{\omega_{\varepsilon}}\chi_{\varepsilon}\,d\mu_{\varepsilon}=n,

where the last identity follows from $[\chi_{\varepsilon}]=[\omega_{\varepsilon}]$ . Both $\chi_{\varepsilon}$ and $\omega_{\varepsilon}$ are positive. Thus

\int_{X}|\nabla\varphi_{\varepsilon}|_{\omega_{\varepsilon}}^{2}\,d\mu_{\varepsilon}\leq 2n\|\varphi_{\varepsilon}\|_{\infty}.

∎

Lemma 4.2.

Suppose

\|u_{\varepsilon}\|_{L^{\infty}}\leq C,\qquad\int_{X}|\nabla u_{\varepsilon}|_{\omega_{\varepsilon}}^{2}\,d\mu_{\varepsilon}\leq C.

Then:

(1)

$\|u_{\varepsilon}\|_{W^{1,1}(X,{\omega})}\leq C$ . Hence every sequence has a subsequence converging in every finite $L^{p}(X,\Omega)$ .

(2)

For any compact subset $K\Subset Y^{\circ}$ , put

\bar{u}_{\varepsilon}(y)=V_{F}^{-1}\int_{X_{y}}u_{\varepsilon}{\omega_{y}}^{d}.

Then

	$\displaystyle\int_{f^{-1}(K)}\|\nabla_{V}u_{\varepsilon}\|_{{\omega_{y}}}\Omega$	$\displaystyle\leq C_{K}\sqrt{\varepsilon},$
	$\displaystyle\int_{f^{-1}(K)}\|u_{\varepsilon}-f^{*}\bar{u}_{\varepsilon}\|\Omega$	$\displaystyle\leq C_{K}\sqrt{\varepsilon}.$

Here $\nabla_{V}u$ is the covariant derivative in the fiber direction. Hence every limit is constant on almost every smooth connected fiber. For $d=0$ , the fibers on $Y^{\circ}$ are points and this last assertion is automatic.

Proof.

1. By Cauchy–Schwarz inequality

|\nabla u_{\varepsilon}|_{{\omega}}^{2}\leq|\nabla u_{\varepsilon}|_{\omega_{\varepsilon}}^{2}\tr_{{\omega}}\omega_{\varepsilon}.

Since $\|F_{\varepsilon}\|_{\infty}\leq C$ , $\mu_{\varepsilon},\Omega,{\omega}^{n}$ are all equivalent,

	$\displaystyle\int_{X}\tr_{{\omega}}\omega_{\varepsilon}\,\Omega$	$\displaystyle\leq Cn\int_{X}\omega_{\varepsilon}\wedge{\omega}^{n-1}=Cn(2\pi c_{1}(K_{X})+\varepsilon[{\omega}])\cdot[{\omega}]^{n-1}\leq C,$
	$\displaystyle\int_{X}\|\nabla u_{\varepsilon}\|_{{\omega}}\Omega$	$\displaystyle\leq C\left(\int_{X}\|\nabla u_{\varepsilon}\|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}\right)^{1/2}\left(\int_{X}\tr_{{\omega}}\omega_{\varepsilon}\,\Omega\right)^{1/2}\leq C.$

Rellich compactness gives $h_{\varepsilon_{j}}\to h$ in $L^{1}$ ; then

\|h_{\varepsilon_{j}}-h\|_{L^{p}}^{p}\leq(2C)^{p-1}\|h_{\varepsilon_{j}}-h\|_{L^{1}}\longrightarrow 0.

2. Take $K\Subset K^{\prime}\Subset Y^{\circ}$ . For $d>0$ ,

[\omega_{\varepsilon}|_{X_{y}}]=\varepsilon[{\omega_{y}}],\qquad\int_{X_{y}}\tr_{{\omega_{y}}}(\omega_{\varepsilon}|_{X_{y}}){\omega_{y}}^{d}=d\varepsilon V_{F}.

On $f^{-1}(K^{\prime})$ , $\Omega,{\omega}^{d}\wedge f^{*}\chi_{Y}^{m}$ are equivalent; hence fiber integration yields

\int_{f^{-1}(K)}\tr_{{\omega_{y}}}(\omega_{\varepsilon}|_{X_{y}})\Omega\leq C_{K}\varepsilon.

The dual norm decreases on restriction, so

	$\displaystyle\|\nabla_{V}u_{\varepsilon}\|_{{\omega_{y}}}^{2}$	$\displaystyle\leq\|\nabla h_{\varepsilon}\|_{\omega_{\varepsilon}}^{2}\tr_{{\omega_{y}}}(\omega_{\varepsilon}\|_{X_{y}}),$
	$\displaystyle\int_{f^{-1}(K)}\|\nabla_{V}u_{\varepsilon}\|_{{\omega_{y}}}\Omega$	$\displaystyle\leq C_{K}\left(\int_{X}\|\nabla u_{\varepsilon}\|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}\right)^{1/2}\varepsilon^{1/2}\leq C_{K}\sqrt{\varepsilon}.$

The smooth compact family of connected fibers has a uniform $L^{1}$ Poincaré inequality:

\int_{X_{y}}|u_{\varepsilon}-\bar{u}_{\varepsilon}(y)|{\omega_{y}}^{d}\leq C_{K}\int_{X_{y}}|\nabla_{V}u_{\varepsilon}|_{{\omega_{y}}}{\omega_{y}}^{d}.

Integrating proves the second estimate. Passing to distributions, $\nabla_{V}u=0$ ; connectedness gives fiberwise constancy. ∎

Applying the energy and compactness estimates to the two functions in the cscK system gives the following subsequential limits.

Proposition 4.3.

Every sequence $\varepsilon_{j}\downarrow 0$ has a subsequence, still denoted $\varepsilon_{j}$ , with the following properties.

(1)

F_{\varepsilon_{j}}\longrightarrow F_{\infty},\qquad\varphi_{\varepsilon_{j}}\longrightarrow\varphi_{\infty}\quad\text{ in every }L^{p}(X,\Omega)\quad(p<\infty).

On $X^{\circ}$ , the two limits are almost everywhere pullbacks of bounded measurable functions on $Y^{\circ}$ .

(2)

\frac{\omega_{\varepsilon_{j}}^{n}}{V_{\varepsilon_{j}}}\longrightarrow e^{F_{\infty}}\Omega\quad\text{in total variation},\qquad\int_{X}F_{\varepsilon_{j}}e^{F_{\varepsilon_{j}}}\Omega\longrightarrow\int_{X}F_{\infty}e^{F_{\infty}}\Omega.

(3)

The weak current limit $\chi+dd^{c}\varphi_{\infty}$ is positive and has bounded local potentials.

Proof.

Apply Lemmas 4.1 and 4.2 and a common subsequence. Exponentiation and the function $x\mapsto xe^{x}$ are Lipschitz on the common bounded range. Hence their $L^{1}$ convergence follows from that of $F_{\varepsilon_{j}}$ , proving total variation and entropy convergence. Positivity follows by passing to the current limit in $\chi+\varepsilon_{j}{\omega}+dd^{c}\varphi_{\varepsilon_{j}}\geq 0$ . The uniform potential bound passes to an almost everywhere representative of the limit, whose quasi-plurisubharmonic regularization has the corresponding bounded local potentials. ∎

We recall some facts about bounded psh functions on singular space.

(1)

For a bounded $\chi$ -psh function $u$ , one has

$u|_{X_{y}}\in\PSH(X_{y})\Longrightarrow u|_{X_{y}}\equiv h(y),\qquad h\in\PSH(Y^{\circ},\chi_{Y})\cap L^{\infty}.$

By [5, Chatper I, Theorem 5.24], $h$ can be extended to a bounded psh function $Y_{\rm reg}$ . Thus $h$ is a weak psh function on a normal space $Y$ . By [8, Theorem 1.2], the extension $g(y)=\limsup_{Y^{\circ}\ni y^{\prime}\to y}h(y^{\prime})$ is psh on $Y$ . The extension does not increase the $L^{\infty}$ -norm of the psh function.
(2)

Let $h\in\PSH(Y,\chi_{Y})\cap L^{\infty}$ . There exists a sequence of smooth functions $h_{j}\in\PSH(\mathbb{C}P^{N},\ell^{-1}\omega_{FS})$ which decrease pointwisely on $\mathbb{C}P^{N}$ so that $\lim h_{j}=h$ on $Y$ . See [4, Corollary C].

(3)

Let $\pi:\widetilde{Y}\to Y$ be a resolution which is obtained by composition of blowups with smooth centers. Then the pullback identidies the bounded psh spaces:

\PSH(Y,\chi_{Y})\cap L^{\infty}\longleftrightarrow\PSH(\widetilde{Y},\pi^{*}\chi_{Y})\cap L^{\infty}.

See [6]. Moreover, by monotone convergence theorem of Bedford-Taylor, for bounded $u_{j}\downarrow u\in\PSH(Y,\chi_{Y})\cap L^{\infty}$ ,

\MA_{Y}(u_{j})\rightharpoonup\MA_{Y}(u),\quad E_{Y}(u_{j})\downarrow E_{Y}(u),

where

\MA_{Y}(u)=V_{Y}^{-1}(\chi_{Y}+dd^{c}u)^{m},\qquad E_{Y}(u)=\frac{1}{(m+1)V_{Y}}\sum_{j=0}^{m}\int_{Y}u(\chi_{Y}+dd^{c}u)^{j}\chi_{Y}^{m-j}.

The convergence also holds for mixed products. And with this identification of psh functions, we can regard all functionals on $Y$ as those on its resolutions.

(4)

For $\varphi\in\PSH(\widetilde{Y},\pi^{*}\chi_{Y})\cap L^{\infty}$ , $g\in C^{0}(\widetilde{Y})$ , and the envelope $P_{\pi^{*}\chi_{Y}}(h):=\sup\{u\in\PSH:u\leq h\}$ ,

\left.\frac{d}{dt}\right|_{t=0}E_{\pi^{*}\chi_{Y}}(P_{\pi^{*}\chi_{Y}}(\varphi+tg))=\int_{\widetilde{Y}}g\MA_{\pi^{*}\chi_{Y}}(\varphi).

This is proved in [1, Lemma 4.2].

We want to use the variational method to show that $F_{\varepsilon}\to f^{*}\varphi_{\rm can}.$ Let $\nu=e^{F}f_{*}\Omega$ , where $F$ is a bounded function. If $u$ is a bounded psh function that maximizes

E_{Y}(\varphi)-\int_{Y}\varphi d\nu,

then $MA_{Y}(u)=\nu$ . To see this, lift $\nu$ to $\widetilde{Y}$ with zero measure on the exceptional set. Set $\tilde{u}=\pi^{*}u\in\PSH(\widetilde{Y},\pi^{*}\chi_{Y})\cap L^{\infty}$ , and $w_{s}=P_{\pi^{*}\chi_{Y}}(\tilde{u}+sg)$ for a bounded continuous $g$ . Hence $w_{s}\in\PSH(\widetilde{Y},\pi^{*}\chi_{Y})\cap L^{\infty}$ . Since $\tilde{u}$ maximizes the functional, we get

E_{\pi^{*}\chi_{Y}}(w_{s})-\int_{\widetilde{Y}}\tilde{u}d\nu-s\int_{\widetilde{Y}}gd\nu\leq E_{\chi_{Y}}(u)-\int_{Y}ud\nu.

Apply the variational formula, we get $\MA_{Y}(u)=\nu$ .

In the following, we suppose a subsequence $\varphi_{\varepsilon}\to f^{*}u$ , and $F_{\varepsilon}\to f^{*}F$ in $L^{p}(X,\Omega)$ , and set $\nu=e^{F}f_{*}\Omega$ ,

E_{\varepsilon}(\varphi)=\frac{1}{(n+1)V_{\varepsilon}}\sum_{j=0}^{n}\int_{X}\varphi(\chi_{\varepsilon}+dd^{c}\varphi)^{j}\chi_{\varepsilon}^{n-j}.

Lemma 4.4.

(1)

$E_{\varepsilon}(\varphi_{\varepsilon})\to E_{Y}(u)$ .
(2)

$u$ maximizes the functional $E_{Y}(\varphi)-\int_{Y}\varphi d\nu$ .
(3)

$\MA_{Y}(u)=\nu$ .

Proof.

Fix a smooth $\varphi\in\PSH(Y,\chi_{Y})$ , one has

E_{\varepsilon}(f^{*}\varphi)=\frac{1}{V_{\varepsilon}}\int_{X}\int_{0}^{1}f^{*}\varphi(\chi+\varepsilon{\omega}+tdd^{c}f^{*}\varphi)^{n}dt

Since powers of a pull-back form above degree $m$ vanish, uniformly for $t\in[0,1]$ ,

		$\displaystyle\int_{X}f^{}\varphi\,\bigl(\chi+tdd^{c}f^{}\varphi+\varepsilon{\omega}\bigr)^{n}$
		$\displaystyle=\sum_{k=d}^{n}\binom{n}{k}\varepsilon^{k}\int_{X}f^{*}\bigl(\varphi(\chi_{Y}+tdd^{c}\varphi)^{n-k}\bigr)\wedge{\omega}^{k}$
		$\displaystyle=\binom{n}{d}\varepsilon^{d}V_{F}\int_{Y}\varphi(\chi_{Y}+tdd^{c}\varphi)^{m}+O_{\varphi}(\varepsilon^{d+1}),$
		$\displaystyle V_{\varepsilon}=\binom{n}{d}\varepsilon^{d}V_{F}V_{Y}+O(\varepsilon^{d+1}).$

Here we used that $f_{*}({\omega}^{d})=V_{F}$ , so division cancels the factor $\binom{n}{d}\varepsilon^{d}V_{F}$ :

	$\displaystyle E_{\varepsilon}(f^{*}\varphi)$	$\displaystyle=\frac{1}{V_{Y}}\int_{0}^{1}\int_{Y}\varphi\bigl((1-t)\chi_{Y}+t\chi_{\varphi}\bigr)^{m}\,dt+O_{\varphi}(\varepsilon)$
		$\displaystyle=\frac{1}{(m+1)V_{Y}}\sum_{j=0}^{m}\int_{Y}\varphi\chi_{\varphi}^{j}\wedge\chi_{Y}^{m-j}+O_{\varphi}(\varepsilon)=E_{Y}(\varphi)+O_{\varphi}(\varepsilon).$

A similar argument also shows that

\frac{(\chi+\varepsilon{\omega}+dd^{c}f^{*}\varphi)^{n}}{V_{\varepsilon}}\to\frac{{\omega}^{d}\wedge f^{*}(\chi_{Y}+dd^{c}\varphi)^{m}}{V_{F}V_{Y}},

and integration along fiber gives

f_{*}\left(\frac{{\omega}^{d}\wedge f^{*}(\chi_{Y}+dd^{c}\varphi)^{m}}{V_{F}V_{Y}}\right)=V_{Y}^{-1}(\chi_{Y}+dd^{c}\varphi)^{m}.

Note that the energy $E_{\varepsilon}$ is concave, so

		$\displaystyle E_{\varepsilon}(\varphi_{\varepsilon})\leq E_{\varepsilon}(f^{}\varphi)+\int_{X}(\varphi_{\varepsilon}-f^{}\varphi)\frac{(\chi+\varepsilon{\omega}+dd^{c}f^{*}\varphi)^{n}}{V_{\varepsilon}}$
		$\displaystyle E_{\varepsilon}(\varphi_{\varepsilon})\geq E_{\varepsilon}(f^{}\varphi)+\int_{X}(\varphi_{\varepsilon}-f^{}\varphi)d\mu_{\varepsilon}.$

Taking $\varepsilon\to 0$ , we have

		$\displaystyle\limsup_{\varepsilon\to 0}E_{\varepsilon}(\varphi_{\varepsilon})\leq E_{Y}(\varphi)+\int_{Y}(u-\varphi)\MA_{Y}(\varphi),$
		$\displaystyle\liminf_{\varepsilon\to 0}E_{\varepsilon}(\varphi_{\varepsilon})\geq E_{Y}(\varphi)+\int_{Y}(u-\varphi)d\nu.$

Here, we used

	$\displaystyle\left\|\int_{X}(\varphi_{\varepsilon}-f^{*}u)\,d\mu_{\varepsilon}\right\|$	$\displaystyle\leq C\\|\varphi_{\varepsilon}-f^{*}u\\|_{L^{1}(\Omega)}\longrightarrow 0,$
	$\displaystyle\left\|\int_{X}f^{}(u-\varphi)\bigl(e^{F_{\varepsilon}}-e^{f^{}F}\bigr)\Omega\right\|$	$\displaystyle\leq\\|u-\varphi\\|_{\infty}\\|e^{F_{\varepsilon}}-e^{f^{*}F}\\|_{L^{1}(\Omega)}\longrightarrow 0,$

Since $\varphi_{\varepsilon}$ and $F_{\varepsilon}$ are uniformly bounded.

Now we choose a sequence of bounded smooth $\varphi_{j}\in\PSH(\mathbb{CP}^{N},\ell^{-1}\omega_{FS})$ that $\varphi_{j}\downarrow u$ . First let $\varepsilon\to 0$ with $j$ fixed. Since $\varphi_{j}-u\geq 0$ , we have

	$\displaystyle E_{Y}(\varphi_{j})+\int_{Y}(u-\varphi_{j})\,d\nu$	$\displaystyle\leq\liminf_{\varepsilon\to 0}E_{\varepsilon}(\varphi_{\varepsilon})$
		$\displaystyle\leq\limsup_{\varepsilon\to 0}E_{\varepsilon}(\varphi_{\varepsilon})$
		$\displaystyle\leq E_{Y}(\varphi_{j})+\int_{Y}(u-\varphi_{j})\MA_{Y}(\varphi_{j})\leq E_{Y}(\varphi_{j}).$

Let $j\to\infty$ ,

E_{Y}(\varphi_{j})\downarrow E_{Y}(u),\qquad\int_{Y}(u-\varphi_{j})d\nu\to 0.

Hence $E_{\varepsilon}(\varphi_{\varepsilon})\to E_{Y}(u)$ . Moreover, we have

E_{Y}(u)\geq E_{Y}(\varphi)+\int_{Y}(u-\varphi)d\nu

for any smooth $\varphi\in\PSH(Y,\chi_{Y})$ . Apply the decreasing approximation again, it follows that the above inequality holds for any bounded $\varphi\in\PSH(Y,\chi_{Y})$ . The equation follows from the variational formula discussed before.

∎

Lemma 4.5.

(1)

On $Y^{\circ}$ ,

$f_{*}\left(\frac{\omega_{\varepsilon}^{n-1}}{V_{\varepsilon}}\right)\rightharpoonup\frac{m}{nV_{Y}}(\chi_{Y}+dd^{c}u)^{m-1}.$ (4.1)

(2)

On $Y$ ,

f_{*}\left(\frac{\chi\wedge\omega_{\varepsilon}^{n-1}}{V_{\varepsilon}}\right)\rightharpoonup\frac{m}{nV_{Y}}\chi_{Y}\wedge(\chi_{Y}+dd^{c}u)^{m-1}.

(4.2)

Proof.

If $n=1$ , then $m=1$ , $d=0$ , and $f$ is an isomorphism of smooth curves. Both conclusions follow from $V_{\varepsilon}\to V_{Y}$ . For the rest of the proof assume $n\geq 2$ . Fix smooth $\varphi$ with $\chi_{Y}+dd^{c}\varphi\geq\delta\chi_{Y}$ , put $\widehat{\omega}=\chi+\varepsilon{\omega}+dd^{c}f^{*}\varphi$ , $h=\varphi_{\varepsilon}-f^{*}\varphi$ , and

I_{\varepsilon}=V_{\varepsilon}^{-1}\int_{X}h(\widehat{\omega}^{n}-\omega_{\varepsilon}^{n})=V_{\varepsilon}^{-1}\sum_{k=0}^{n-1}\int_{X}i\partial h\wedge\bar{\partial}h\wedge\omega_{\varepsilon}^{k}\wedge\widehat{\omega}^{n-1-k}.

We claim that

I_{\varepsilon}\to I_{Y}(u,\varphi):=\int_{Y}(u-\varphi)(\operatorname{MA}_{Y}(\varphi)-\operatorname{MA}_{Y}(u)).

(4.3)

Recall the binomial expansion in Lemma 4.4:

\frac{\widehat{\omega}^{n}}{V_{\varepsilon}}=\frac{{\omega}^{d}\wedge f^{*}(\chi_{Y}+dd^{c}\varphi)^{m}}{V_{F}V_{Y}}+O_{\varphi}(\varepsilon)\Omega\leq C_{\varphi}\Omega.

Write $h=(\varphi_{\varepsilon}-f^{*}u)+(f^{*}u-\varphi)$ . Then

		$\displaystyle\left\|\frac{1}{V_{\varepsilon}}\int_{X}h\,\widehat{\omega}^{n}-\int_{Y}(u-\varphi)\MA_{Y}(\varphi)\right\|$
		$\displaystyle\leq\left\|\frac{1}{V_{\varepsilon}}\int_{X}(\varphi_{\varepsilon}-f^{}u)\widehat{\omega}^{n}\right\|+\left\|\int_{X}f^{}(u-\varphi)\left(\frac{\widehat{\omega}^{n}}{V_{\varepsilon}}-\frac{{\omega}^{d}\wedge f^{*}\chi_{\varphi}^{m}}{V_{F}V_{Y}}\right)\right\|$
		$\displaystyle\leq C_{\varphi}\\|\varphi_{\varepsilon}-f^{*}u\\|_{L^{1}(\Omega)}+C_{\varphi}\varepsilon\\|u-\varphi\\|_{\infty}\longrightarrow 0.$

For the second integral, Lemma 4.4 has already proved $\MA_{Y}(u)=e^{F}f_{*}\Omega$ . Thus

		$\displaystyle\left\|\frac{1}{V_{\varepsilon}}\int_{X}h\,\omega_{\varepsilon}^{n}-\int_{Y}(u-\varphi)\MA_{Y}(u)\right\|$
		$\displaystyle\leq\left\|\int_{X}(\varphi_{\varepsilon}-f^{}u)e^{F_{\varepsilon}}\Omega\right\|+\left\|\int_{X}f^{}(u-\varphi)(e^{F_{\varepsilon}}-e^{f^{*}F})\Omega\right\|$
		$\displaystyle\leq C\\|\varphi_{\varepsilon}-f^{}u\\|_{L^{1}(\Omega)}+\\|u-\varphi\\|_{L^{\infty}}\\|e^{F_{\varepsilon}}-e^{f^{}F}\\|_{L^{1}(\Omega)}\longrightarrow 0.$

The last limit uses $e^{F_{\varepsilon}}\leq C$ and $\|e^{F_{\varepsilon}}-e^{f^{*}F}\|_{L^{1}}\leq C\|F_{\varepsilon}-f^{*}F\|_{L^{1}}\to 0$ . Subtracting these two limits proves the claim.

Next, for smooth compactly supported $(1,1)$ test form $\Xi$ on $Y^{\circ}$ , pair it with $f_{*}(V_{\varepsilon}^{-1}(\widehat{\omega}^{n-1}-\omega_{\varepsilon}^{n-1}))$ . The pairing is a sum of the following terms

V_{\varepsilon}^{-1}\int_{X}i\partial h\wedge\bar{\partial}f^{*}\Xi\wedge T_{k},\qquad T_{k}=\omega_{\varepsilon}^{k}\wedge\widehat{\omega}^{n-2-k}.

All covectors of $\bar{\partial}f^{*}\Xi$ are horizontal with bounded coefficients in $\chi$ . Cauchy–Schwarz inequality gives

\left|V_{\varepsilon}^{-1}\int i\partial h\wedge\bar{\partial}f^{*}\Xi\wedge T_{k}\right|\leq C_{\Xi}\left(V_{\varepsilon}^{-1}\int i\partial h\wedge\bar{\partial}h\wedge\chi\wedge T_{k}\right)^{1/2}\left(V_{\varepsilon}^{-1}\int\chi^{2}\wedge T_{k}\right)^{1/2}.

(4.4)

Since $\widehat{\omega}\geq\delta\chi$ , the first squared factor is at most $\delta^{-1}I_{\varepsilon}$ . The other squared factor is

	$\displaystyle V_{\varepsilon}^{-1}\int_{X}\chi^{2}\wedge T_{k}$	$\displaystyle=\frac{[\chi]^{2}[\chi_{\varepsilon}]^{n-2}}{V_{\varepsilon}}$
		$\displaystyle=\frac{\binom{n-2}{d}\varepsilon^{d}V_{F}V_{Y}+O(\varepsilon^{d+1})}{\binom{n}{d}\varepsilon^{d}V_{F}V_{Y}+O(\varepsilon^{d+1})}\longrightarrow\frac{m(m-1)}{n(n-1)}\qquad(m\geq 2).$

If $m=1$ , this factor is identically zero because $\chi^{2}=0$ . Therefore

\left|\left\langle\Xi,f_{*}\frac{\omega_{\varepsilon}^{n-1}-\widehat{\omega}_{\varepsilon,\varphi}^{n-1}}{V_{\varepsilon}}\right\rangle\right|\leq C_{\Xi}\delta^{-1/2}\sqrt{I_{\varepsilon}}.

(4.5)

For $0<\delta<1$ , put $v_{\delta}=(1-\delta)u$ , so that $\chi_{v_{\delta}}=(1-\delta)\chi_{u}+\delta\chi_{Y}\geq\delta\chi_{Y}$ . For bounded psh functions, we have

\begin{split}I_{Y}(u,v_{\delta})&=\frac{\delta^{2}}{V_{Y}}\sum_{k=0}^{m-1}\int_{Y}i\partial u\wedge\bar{\partial}u\wedge\chi_{u}^{k}\wedge\chi_{v_{\delta}}^{m-1-k}\\ &\leq 2m\|u\|_{L^{\infty}}\delta^{2}.\end{split}

(4.6)

Indeed, for $T=\chi_{u}^{k}\wedge\chi_{v_{\delta}}^{m-1-k}$ , bounded Bedford–Taylor integration by parts on the resolution yields

	$\displaystyle 0\leq\int_{Y}i\partial u\wedge\bar{\partial}u\wedge T$	$\displaystyle=-\int_{Y}u\,dd^{c}u\wedge T=\int_{Y}u(\chi_{Y}-\chi_{u})\wedge T$
		$\displaystyle\leq\\|u\\|_{\infty}\int_{Y}(\chi_{Y}+\chi_{u})\wedge T=2\\|u\\|_{L^{\infty}}V_{Y}.$

This proves the uniform bound in (4.6).

Choose smooth $u_{j}\downarrow u$ , and set $\varphi_{\delta,j}=(1-\delta)u_{j}$ . The three limits are taken in the order $\varepsilon\to 0$ , then $j\to\infty$ , then $\delta\to 0$ . For fixed $\delta,j$ , direct computation gives

f_{*}\left(\frac{(\chi+\varepsilon{\omega}+dd^{c}f^{*}\varphi_{j,\delta})^{n-1}}{V_{\varepsilon}}\right)\longrightarrow\frac{\binom{n-1}{d}V_{F}}{\binom{n}{d}V_{F}V_{Y}}(\chi_{Y}+dd^{c}\varphi_{\delta,j})^{m-1}=\frac{m}{nV_{Y}}(\chi_{Y}+dd^{c}\varphi_{\delta,j})^{m-1}.

Hence (4.3) and (4.5) imply

\limsup_{\varepsilon\to 0}\left|\left\langle\Xi,\,f_{*}\frac{\omega_{\varepsilon}^{n-1}}{V_{\varepsilon}}-\frac{m}{nV_{Y}}\chi_{\varphi_{\delta,j}}^{m-1}\right\rangle\right|\leq C_{\Xi}\delta^{-1/2}\sqrt{I_{Y}(u,\varphi_{\delta,j})}.

For fixed $\delta$ , monotone convergence of the Bedford-Taylor products implies

I_{Y}(u,\varphi_{\delta,j})\longrightarrow I_{Y}(u,v_{\delta}),\qquad(\chi_{Y}+dd^{c}{\varphi_{\delta,j}})^{m-1}\rightharpoonup(\chi_{Y}+dd^{c}{v_{\delta}})^{m-1}.

Letting $j\to\infty$ in the preceding estimate and using (4.6), we obtain

\limsup_{\varepsilon\to 0}\left|\left\langle\Xi,\,f_{*}\frac{\omega_{\varepsilon}^{n-1}}{V_{\varepsilon}}-\frac{m}{nV_{Y}}(\chi_{Y}+dd^{c}{v_{\delta}})^{m-1}\right\rangle\right|\leq C_{\Xi}\sqrt{\delta}.

Finally,

(\chi_{Y}+dd^{c}{v_{\delta}})^{m-1}=\sum_{k=0}^{m-1}\binom{m-1}{k}(1-\delta)^{k}\delta^{m-1-k}(\chi_{Y}+dd^{c}u)^{k}\wedge\chi_{Y}^{m-1-k}\rightharpoonup(\chi_{Y}+dd^{c}u)^{m-1}.

Let $\delta\to 0$ to obtain (4.1). These formulas also hold when $m=1$ .

Proof of (2). The bounds for $F_{\varepsilon}$ and $\tr_{\omega_{\varepsilon}}\chi$ imply

0\leq\frac{\chi\wedge\omega_{\varepsilon}^{n-1}}{V_{\varepsilon}}=\frac{1}{n}(\tr_{\omega_{\varepsilon}}\chi)e^{F_{\varepsilon}}\Omega\leq C\Omega.

Thus, for every bounded nonnegative measurable $b$ on $Y$ ,

\displaystyle 0\leq\frac{1}{V_{\varepsilon}}\int_{X}f^{*}b\,\chi\wedge\omega_{\varepsilon}^{n-1}

\displaystyle\leq C\int_{X}f^{*}b\,\Omega=C\int_{Y}b\,f_{*}\Omega.

Equivalently, as positive measures on $Y$ ,

0\leq f_{*}\!\left(\frac{\chi\wedge\omega_{\varepsilon}^{n-1}}{V_{\varepsilon}}\right)\leq Cf_{*}\Omega.

For $b\in C^{\infty}_{c}(Y^{\circ})$ , apply (i) with the test form $\Xi=b\chi_{Y}$ . Since $\chi=f^{*}\chi_{Y}$ , this gives

	$\displaystyle\frac{1}{V_{\varepsilon}}\int_{X}f^{*}b\,\chi\wedge\omega_{\varepsilon}^{n-1}$	$\displaystyle=\left\langle b\chi_{Y},\,f_{*}\frac{\omega_{\varepsilon}^{n-1}}{V_{\varepsilon}}\right\rangle$
		$\displaystyle\longrightarrow\frac{m}{nV_{Y}}\int_{Y^{\circ}}b\,\chi_{Y}\wedge(\chi_{Y}+dd^{c}u)^{m-1}.$

For nonnegative $b$ , the preceding bound passes to this limit. Consequently,

0\leq\frac{m}{nV_{Y}}\chi_{Y}\wedge(\chi_{Y}+dd^{c}u)^{m-1}\leq Cf_{*}\Omega\quad\text{on }Y^{\circ}.

Recall $D=Y\setminus Y^{\circ}$ . Its inverse image is proper analytic, and bounded Bedford–Taylor products give no mass to proper analytic sets on a resolution. Therefore

(f_{*}\Omega)(D)=\int_{f^{-1}(D)}\Omega=0,\qquad\int_{D}\chi_{Y}\wedge(\chi_{Y}+dd^{c}u)^{m-1}=0.

The same measure inequality thus holds on all of $Y$ .

It remains to allow test functions without compact support in $Y^{\circ}$ . Fix $b\in L^{1}(Y,f_{*}\Omega)$ . The restriction of $f_{*}\Omega$ to $Y^{\circ}$ is a smooth positive volume form; it has finite total mass and gives no mass to $D$ . Smooth approximation and exhaustion of $Y^{\circ}$ therefore give

b_{j}\in C^{\infty}_{c}(Y^{\circ}),\qquad\|b_{j}-b\|_{L^{1}(f_{*}\Omega)}\longrightarrow 0.

The two measure bounds imply

		$\displaystyle\left\|\frac{1}{V_{\varepsilon}}\int_{X}f^{}(b-b_{j})\,\chi\wedge\omega_{\varepsilon}^{n-1}\right\|\leq C\\|b-b_{j}\\|_{L^{1}(f_{}\Omega)},$
		$\displaystyle\left\|\frac{m}{nV_{Y}}\int_{Y}(b-b_{j})\,\chi_{Y}\wedge(\chi_{Y}+dd^{c}u)^{m-1}\right\|\leq C\\|b-b_{j}\\|_{L^{1}(f_{*}\Omega)}.$

First let $\varepsilon\to 0$ with $j$ fixed, using the convergence already proved for $b_{j}$ . The triangle inequality gives

		$\displaystyle\limsup_{\varepsilon\to 0}\left\|\frac{1}{V_{\varepsilon}}\int_{X}f^{*}b\,\chi\wedge\omega_{\varepsilon}^{n-1}-\frac{m}{nV_{Y}}\int_{Y}b\,\chi_{Y}\wedge(\chi_{Y}+dd^{c}u)^{m-1}\right\|$
		$\displaystyle\leq 2C\\|b-b_{j}\\|_{L^{1}(f_{*}\Omega)}\xrightarrow{j\to\infty}0.$

This proves (4.2).

∎

A similar concavity argument as in Lemma 4.4, together with the above convergence of mixed products, shows convergence of the $J$ -functional as follows. The $J$ -functional is defined as

	$\displaystyle J_{\chi,\varepsilon}(\varphi)$	$\displaystyle:=\frac{1}{nV_{\varepsilon}}\sum_{j=0}^{n-1}\int_{X}\varphi\chi(\chi_{\varepsilon}+dd^{c}\varphi)^{j}\chi_{\varepsilon}^{n-1-j},$
	$\displaystyle J_{\chi_{Y},Y}(\varphi)$	$\displaystyle:=\frac{1}{mV_{Y}}\sum_{j=0}^{m-1}\int_{Y}\varphi\chi_{Y}(\chi_{Y}+dd^{c}\varphi)^{j}\chi_{Y}^{m-1-j}.$

Lemma 4.6.

$nJ_{\chi,\varepsilon}(\varphi_{\varepsilon})\to mJ_{\chi_{Y},Y}(u)$ .

Proof.

Direct computation shows

\frac{d}{dt}|_{t=0}J_{\chi,\varepsilon}(\varphi+th)=V_{\varepsilon}^{-1}\int_{X}h\chi\wedge(\chi+\varepsilon{\omega}+dd^{c}\varphi)^{n-1},

and

\frac{d}{dt}|_{t=0}J_{\chi_{Y},Y}(\varphi+th)=V_{Y}\int_{Y}h\chi_{Y}\wedge(\chi_{Y}+dd^{c}\varphi)^{m-1}.

Moreover

\frac{d^{2}}{dt^{2}}|_{t=0}J_{\chi,\varepsilon}\leq 0,\qquad\frac{d^{2}}{dt^{2}}|_{t=0}J_{\chi_{Y},Y}\leq 0.

Then similar expansion in the proof of Lemma 4.4 gives, for fixed smooth psh function $\varphi\in\PSH(Y,\chi_{Y})$ ,

J_{\chi,\varepsilon}(f^{*}\varphi)\to\frac{m}{n}J_{\chi_{Y},Y}(\varphi).

Note that $|F_{\varepsilon}|,\tr_{\omega_{\varepsilon}}\chi$ are uniformly bounded,

\left|\frac{1}{V_{\varepsilon}}\int_{X}(\varphi_{\varepsilon}-f^{*}u)\chi\wedge\omega_{\varepsilon}^{n-1}\right|\leq C\|\varphi_{\varepsilon}-f^{*}u\|_{L^{2}(\Omega)}\|\tr_{\omega_{\varepsilon}}\chi\|_{L^{2}(\mu_{\varepsilon})}\to 0.

By Lemma 4.5,

\frac{1}{V_{\varepsilon}}\int_{X}f^{*}(u-\varphi)\chi\wedge\omega_{\varepsilon}^{n-1}\to\frac{m}{nV_{Y}}\int_{Y}(u-\varphi)\chi_{Y}\wedge(\chi_{Y}+dd^{c}u)^{m-1}.

Hence, a similar concavity argument as in Lemma 4.4 gives,

\limsup_{\varepsilon\to 0}nJ_{\chi,\varepsilon}(\varphi_{\varepsilon})\leq mJ_{\chi_{Y},Y}(\varphi)+\frac{m}{V_{Y}}\int_{Y}(u-\varphi)\chi_{Y}\wedge(\chi_{Y}+dd^{c}\varphi)^{m-1},

\liminf_{\varepsilon\to 0}nJ_{\chi,\varepsilon}(\varphi_{\varepsilon})\geq mJ_{\chi_{Y},Y}(\varphi)+\frac{m}{V_{Y}}\int_{Y}(u-\varphi)\chi_{Y}\wedge(\chi_{Y}+dd^{c}u)^{m-1}.

We choose $\varphi_{j}\downarrow u$ , with $\varphi_{j}\in\PSH(\mathbb{C}P^{N},\ell^{-1}\omega_{FS})$ being smooth. Monotone convergence of the mixed energies gives the desired convergence. ∎

Now, we use the Mabuchi energy to characterize the weak limit of the volume density.

Lemma 4.7.

Any subsequential limit $f^{*}F$ satisfies $F=\varphi_{\rm can}$ $f_{*}\Omega$ -almost everywhere. Thus $F_{\varepsilon}\to f^{*}\varphi_{\rm can}$ in $L^{p}(X,\Omega)$ for any finite $p.$

Proof.

For a smooth $\chi_{\varepsilon}$ -psh function $\varphi$ , the normalized Mabuchi energy is defined as

\mathcal{M}_{\varepsilon}(\varphi):=\frac{1}{V_{\varepsilon}}\int_{X}\log{\frac{(\chi_{\varepsilon}+dd^{c}\varphi)^{n}}{V_{\varepsilon}\Omega}}(\chi_{\varepsilon}+dd^{c}\varphi)^{n}+nJ_{\chi,\varepsilon}(\varphi)+\underline{R}_{\varepsilon}E_{\varepsilon}(\varphi)

(4.7)

The Mabuchi energy is convex along geodesics, and cscK metrics minimize it. Now we apply this to two family of metrics:

(1)

For ${\tilde{\omega}}_{\varepsilon}$ , the potential $\psi_{\varepsilon}\to f^{*}\varphi_{\rm can}$ , and $\nu_{\rm can}=e^{f^{*}\varphi_{\rm can}}f_{*}\Omega$ , and $\tr_{{\tilde{\omega}}_{\varepsilon}}\chi\leq C$ .
(2)

For $\omega_{\varepsilon}$ , by passing to subsequence, the potential $\varphi_{\varepsilon}\to f^{*}u$ , and $\nu_{u}=e^{F}f_{*}\Omega=\MA_{Y}(u)$ , and $\tr_{\omega_{\varepsilon}}\chi\leq C$ .

On $Y$ , one has

mJ_{\chi_{Y},Y}(u)=(m+1)E_{Y}(u)-\int_{Y}u\MA_{Y}(u).

Since $\underline{R}_{\varepsilon}\to-m$ and the bounded $\log$ -density converges in $L^{1}$ , we get

\mathcal{M}_{\varepsilon}(\varphi_{\varepsilon})\to\int_{Y}Fe^{f^{*}F}f_{*}\Omega+mJ_{\chi_{Y},Y}(u)-mE_{Y}(u)=\int_{Y}Fe^{f^{*}F}f_{*}\Omega+E_{Y}(u)-\int_{Y}u\MA_{Y}(u).

The same argument in Lemma 4.4,4.5,4.6 also works for ${\tilde{\omega}}_{\varepsilon}$ , and we get

\mathcal{M}_{\varepsilon}(\psi_{\varepsilon})\to\int_{Y}\varphi_{\rm can}e^{\varphi_{\rm can}}f_{*}\Omega+E_{Y}(\varphi_{\rm can})-\int_{Y}\varphi_{\rm can}\MA_{Y}(\varphi_{\rm can}).

Since $\omega_{\varepsilon}$ is the unique cscK metric, we have $\mathcal{M}_{\varepsilon}(\varphi_{\varepsilon})\leq\mathcal{M}_{\varepsilon}(\psi_{\varepsilon})$ .

Note that from concavity, we also have

E_{Y}(\varphi_{\rm can})\leq E_{Y}(u)+\int_{Y}(\varphi_{\rm can}-u)\MA_{Y}(u).

Therefore

	$\displaystyle 0\geq$	$\displaystyle\int_{Y}(Fe^{f^{}F}-\varphi_{\rm can}e^{\varphi_{\rm can}})f_{}\Omega+E_{Y}(u)-E_{Y}(\varphi_{\rm can})-\int_{Y}u\MA_{Y}(u)+\int_{Y}\varphi_{\rm can}\MA_{Y}(\varphi_{\rm can})$
	$\displaystyle=$	$\displaystyle\int_{X}(F-\varphi_{\rm can})e^{F}f_{*}\Omega+E_{Y}(u)-E_{Y}(\varphi_{\rm can})-\int_{Y}(u-\varphi_{\rm can})\MA_{Y}(u)$
	$\displaystyle\geq$	$\displaystyle\int_{Y}(F-\varphi_{\rm can})e^{F}f_{*}\Omega.$

Here we used that $\MA_{Y}(u)=e^{F}f_{*}\Omega$ by Lemma 4.4, and the equation of the canonical metric on the base $\MA_{Y}(\varphi_{\rm can})=e^{\varphi_{\rm can}}f_{*}\Omega$ .

Now the difference of the entropy term

\int_{Y}(F-\varphi_{\rm can})e^{F}f_{*}\Omega=\int_{Y}(F-\varphi_{\rm can})e^{F-\varphi_{\rm can}}\MA_{Y}(\varphi_{\rm can})=\int_{Y}(s\log s-s+1)\MA_{Y}(\varphi_{\rm can})\geq 0,

where we let $s=e^{F-\varphi_{\rm can}}$ and used the facts that $s\log s-s+1\geq 0$ and $=$ attains only at $s=1$ . Therefore, every inequality is equality. In particular, $s=1$ almost everywhere, i.e.

F=\varphi_{\rm can}.

∎

Next, we show that the base direction of $\omega_{\varepsilon}$ converges to $f^{*}\omega_{\rm can}$ in the following weak $L^{p}$ -sense. We started with a general results concerning entropy convergence in the same cohomology class.

Lemma 4.8.

Let $\omega,\widehat{\omega}$ be cohomologous Kähler forms on a compact complex $n$ -fold. Set $V=\int_{X}\omega^{n}$ , $\mu=\omega^{n}/V$ , and $\nu=\widehat{\omega}^{n}/V$ .

(1)

\int_{X}|\widehat{\omega}-\omega|_{\omega}\,d\mu\leq 2\sqrt{n\Ent_{\nu}(\mu)},\qquad\text{ where }\Ent_{\nu}(\mu)=\int_{X}\log\frac{d\mu}{d\nu}d\mu.

(2)

Suppose $\mu_{j}=\rho_{j}\Omega$ , $\nu_{j}=\widehat{\rho}_{j}\Omega$ , where $C^{-1}\leq\rho_{j},\widehat{\rho}_{j}\leq C$ and $\|\rho_{j}-\widehat{\rho}_{j}\|_{L^{1}(\Omega)}\to 0$ . Then

\int_{X}|\widehat{\omega}_{j}-\omega_{j}|_{\omega_{j}}\,d\mu_{j}\to 0,\qquad\int_{X}|\omega_{j}-\widehat{\omega}_{j}|_{\widehat{\omega}_{j}}\,d\nu_{j}\to 0.

For every $\eta>0$ , outside a set of $\Omega$ -measure tending to zero,

(1-\eta)\omega_{j}\leq\widehat{\omega}_{j}\leq(1+\eta)\omega_{j}.

Proof.

Note the following elementary inequality

(\sqrt{t}-1)^{2}\leq t-1-\log t,\qquad(\sqrt{t}+1)^{2}\leq 2(t+1),\qquad|t-1|=|\sqrt{t}-1|(\sqrt{t}+1).

It follows that

|\widehat{\omega}-\omega|_{\omega}^{2}\leq 2(\tr_{\omega}\widehat{\omega}+n)\left(\tr_{\omega}\widehat{\omega}-n-\log\frac{\widehat{\omega}^{n}}{\omega^{n}}\right).

The first factor is bounded due to the cohomology class

\int_{X}\tr_{\omega}\widehat{\omega}d\mu=\frac{n}{V}\int_{X}\widehat{\omega}\wedge\omega^{n-1}=n,

and

\int_{X}\left(\tr_{\omega}\widehat{\omega}-n-\log\frac{\widehat{\omega}^{n}}{\omega^{n}}\right)d\mu=\Ent_{\nu}(\mu).

By Cauchy–Schwarz inequality, one gets (i).

For (2),

0\leq\Ent_{\nu_{j}}(\mu_{j})\leq C^{2}\|\rho_{j}-\widehat{\rho}_{j}\|_{L^{1}(\Omega)}\to 0,

and likewise with the measures interchanged. Finally,

|\widehat{\omega}_{j}-\omega_{j}|_{\omega_{j}}\leq\eta\quad\Longrightarrow\quad-\eta\omega_{j}\leq\widehat{\omega}_{j}-\omega_{j}\leq\eta\omega_{j};

Markov’s inequality gives the last assertion. ∎

Corollary 4.9.

For every smooth $(1,0)$ -form $\alpha$ on $Y^{\circ}$ , and $K\Subset Y^{\circ}$

|f^{*}\alpha|_{\omega_{\varepsilon}}^{2}\longrightarrow f^{*}|\alpha|_{\omega_{\rm can}}^{2}\quad\text{in }L^{p}(f^{-1}(K),\Omega),\quad p<\infty.

Proof.

Apply Lemma 4.8 to $\omega_{\varepsilon}$ and ${\tilde{\omega}}_{\varepsilon}$ . Their normalized densities are uniformly bounded above and below and have the same total-variation limit. The uniform convergence of the metric in regular part in Theorem 2.4 gives

(1-o_{K}(1))(f^{*}\omega_{\rm can}+\varepsilon\omega_{\rm SRF})\leq{\tilde{\omega}}_{\varepsilon}\leq(1+o_{K}(1))(f^{*}\omega_{\rm can}+\varepsilon\omega_{\rm SRF}).

Use the $\omega_{\rm SRF}$ -orthogonal complement of the vertical tangent bundle. The reference form has zero mixed term; its horizontal restriction is $f^{*}\omega_{\rm can}+O_{K}(\varepsilon)$ and its vertical restriction is $\varepsilon\omega_{{\rm SRF},y}$ . Since $f^{*}\alpha$ vanishes on vertical vectors, duality gives

|f^{*}\alpha|_{{\tilde{\omega}}_{\varepsilon}}^{2}\longrightarrow f^{*}|\alpha|_{\omega_{\rm can}}^{2}\quad\hbox{uniformly on }f^{-1}(K).

When $d=0$ , this follows directly from convergence of positive metrics on $f^{-1}(K)$ . For $0<\eta<1$ , outside a set of measure tending to zero,

(1-\eta)\omega_{\varepsilon}\leq{\tilde{\omega}}_{\varepsilon}\leq(1+\eta)\omega_{\varepsilon},

so duality gives

(1-\eta)|f^{*}\alpha|_{{\tilde{\omega}}_{\varepsilon}}^{2}\leq|f^{*}\alpha|_{\omega_{\varepsilon}}^{2}\leq(1+\eta)|f^{*}\alpha|_{{\tilde{\omega}}_{\varepsilon}}^{2}.

The comparison-family limit gives convergence in measure as $\eta\downarrow 0$ . Since $|f^{*}\alpha|_{\omega_{\varepsilon}}^{2}\leq C_{\alpha}\tr_{\omega_{\varepsilon}}\chi\leq C_{\alpha}$ , the convergence holds in every finite $L^{p}$ . ∎

Next we use the Moser iteration to improve the $L^{p}$ convergence of $F_{\varepsilon}$ to $L^{\infty}$ -convergence away from singular fibers. We summarize it as a lemme here.

Fix a holormorhic coordinate $z$ on a neighborhood of $B_{2r}\Subset Y^{\circ}$ , where $B_{2r}=\{|z|\leq 2r\}$ . Set $\omega_{E}=dd^{c}|z|^{2}$ and assume $\omega_{E}\leq C\chi_{Y}$ . Fix a smooth function $0\leq\eta\leq 1$ on $\mathbb{R}$ , with $\eta=1$ on $(-\infty,0]$ and $\eta=0$ on $[\frac{1}{2},\infty)$ . For $r\leq s<t\leq 2r$ , set $\eta_{s,t}(z)=\eta\left(\frac{|z|^{2}-s^{2}}{t^{2}-s^{2}}\right)$ , and $\xi_{s,t}=f^{*}\eta_{s,t}.$ Then

\eta_{s,t}\in C^{\infty}_{c}(B_{t}),\quad\eta_{s,t}=1\text{ on }B_{s},\quad|\nabla\eta|^{2}_{\omega_{E}}\leq\frac{C}{(t-s)^{2}},

and on the pullback,

|\nabla\xi_{s,t}|_{\omega_{\varepsilon}}^{2}\leq\frac{C}{(t-s)^{2}}\tr_{\omega_{\varepsilon}}f^{*}\omega_{E}\leq\frac{C}{(t-s)^{2}}.

The following estimates may depend on the fixed coordinate chart but uniform on $\varepsilon$ . Recall that we use $q$ to denote the exponent of uniform Sobolev inequality obtained by Guo–Phong–Song–Strum [10].

Lemma 4.10.

Fix $q/(q-1)<p_{*}<\infty$ . Every smooth real function $U$ on $f^{-1}(B_{2r})$ satisfies

\|U\|_{L^{\infty}(f^{-1}(B_{r}))}\leq C\left(\|U\|_{L^{2}(f^{-1}(B_{2r}),\mu_{\varepsilon})}+\|\Delta_{\omega_{\varepsilon}}U\|_{L^{p_{*}}(f^{-1}(B_{2r}),\mu_{\varepsilon})}\right).

(4.8)

The constant depends only on $p_{*}$ and the fixed chart.

Proof.

Suppress $\varepsilon$ and use $\mu_{\varepsilon}$ for all integrals and norms. We first estimate $U_{+}=\max\{U,0\}$ . For $\delta>0$ , set

v=U_{+}+\|\Delta_{\omega}U\|_{L^{p_{*}}(f^{-1}(B_{2r}))}+\delta,\qquad\left\|\frac{|\Delta_{\omega}U|}{v}\right\|_{L^{p_{*}}(f^{-1}(B_{2r}))}\leq 1.

$-\Delta_{\omega}U_{+}\leq|\Delta_{\omega}U|$ in the weak sense, hence $-\Delta_{\omega}v\leq|\Delta_{\omega}U|$ . For $p\geq 2$ and $\xi=\xi_{s,t}$ , use the nonnegative test function $\xi^{2}v^{p-1}$ . All integrals in the next display are over $f^{-1}(B_{t})$ , and all differential norms use $\omega_{\varepsilon}$ . Integration by parts and Young’s inequality give

	$\displaystyle(p-1)\int\xi^{2}v^{p-2}\|\nabla v\|^{2}$	$\displaystyle\leq 2\int\xi v^{p-1}\|\nabla\xi\|\,\|\nabla v\|+\int\frac{\|\Delta_{\omega}U\|}{v}\,\xi^{2}v^{p}$
		$\displaystyle\leq\frac{p-1}{2}\int\xi^{2}v^{p-2}\|\nabla v\|^{2}+\frac{2}{p-1}\int v^{p}\|\nabla\xi\|^{2}+\int\frac{\|\Delta_{\omega}U\|}{v}\,\xi^{2}v^{p}.$

The cutoff bound and the uniform Sobolev inequality imply

\|\xi v^{p/2}\|_{L^{2q}}^{2}\leq Cp^{2}\left[\bigl(1+(t-s)^{-2}\bigr)\int_{f^{-1}(B_{t})}v^{p}+\int_{f^{-1}(B_{t})}\frac{|\Delta_{\omega}U|}{v}(\xi v^{p/2})^{2}\right].

Put $p_{*}^{\prime}=p_{*}/(p_{*}-1)<q$ and $\theta=q/[p_{*}(q-1)]\in(0,1)$ . For $h=\xi v^{p/2}$ , Hölder’s inequality and interpolation give

	$\displaystyle\int\frac{\|\Delta_{\omega}U\|}{v}h^{2}$	$\displaystyle\leq\\|h\\|_{L^{2p_{*}^{\prime}}}^{2}\leq\\|h\\|_{L^{2q}}^{2\theta}\\|h\\|_{L^{2}}^{2(1-\theta)}$
		$\displaystyle\leq\tau\\|h\\|_{L^{2q}}^{2}+C_{\theta}\tau^{-\theta/(1-\theta)}\\|h\\|_{L^{2}}^{2}\qquad(\tau>0).$

Choose $\tau=(2Cp^{2})^{-1}$ and absorb the first term. Since $\|h\|_{L^{2}}^{2}\leq\int_{f^{-1}(B_{t})}v^{p}$ and $\xi=1$ on $f^{-1}(B_{s})$ ,

\|v\|_{L^{pq}(f^{-1}(B_{s}))}\leq\left[Cp^{2/(1-\theta)}\bigl(1+(t-s)^{-2}\bigr)\right]^{1/p}\|v\|_{L^{p}(f^{-1}(B_{t}))}.

(4.9)

Set $r_{l}=r(1+2^{-l})$ and $p_{l}=2q^{l}$ . Taking $(s,t,p)=(r_{l+1},r_{l},p_{l})$ in (4.9) yields

\|v\|_{L^{p_{l+1}}(f^{-1}(B_{r_{l+1}}))}\leq\left(C_{r}4^{l}p_{l}^{2/(1-\theta)}\right)^{1/p_{l}}\|v\|_{L^{p_{l}}(f^{-1}(B_{r_{l}}))}.

(4.10)

Consequently,

\prod_{l=0}^{\infty}\left(C_{r}4^{l}p_{l}^{2/(1-\theta)}\right)^{1/p_{l}}<\infty,\qquad\|v\|_{L^{\infty}(f^{-1}(B_{r}))}\leq C\|v\|_{L^{2}(f^{-1}(B_{2r}))}.

Since $U_{+}\leq v$ and $\mu_{\varepsilon}(X)=1$ ,

\|U_{+}\|_{L^{\infty}(f^{-1}(B_{r}))}\leq C\left(\|U_{+}\|_{L^{2}(f^{-1}(B_{2r}))}+\|\Delta_{\omega}U\|_{L^{p_{*}}(f^{-1}(B_{2r}))}+\delta\right).

Let $\delta\downarrow 0$ . The same argument for $-U$ estimates $U_{-}=(-U)_{+}$ and proves the desired estimate. ∎

Lemma 4.11.

For any compact $K\Subset X^{\circ}$ , $\|F_{\varepsilon}-f^{*}\varphi_{\rm can}\|_{L^{\infty}(K)}\to 0$ .

Proof.

Fix $q/(q-1)<p_{*}<\infty$ . On any coordinate ball $B_{2r}\Subset Y^{\circ}$ , set $U=F_{\varepsilon}-f^{*}\varphi$ . Then one has

\Delta_{\omega_{\varepsilon}}U=-\underline{R}_{\varepsilon}-\tr_{\omega_{\varepsilon}}\chi-\tr_{\omega_{\varepsilon}}(f^{*}(\omega_{\rm can}-\chi_{Y}))=-\underline{R}_{\varepsilon}-\tr_{\omega_{\varepsilon}}f^{*}\omega_{\rm can}.

Choose a smooth $\omega_{\rm can}$ -unitary $(1,0)$ -froms $\alpha_{1},...,\alpha_{m}$ on a neighborhood of $B_{2r}$ . Then by Lemma 4.9, and $|F_{\varepsilon}|\leq C$ ,

\omega_{\rm can}=\sqrt{-1}\sum_{i}\alpha_{j}\wedge\bar{\alpha}_{j},\quad\tr_{\omega_{\varepsilon}}f^{*}\omega_{\rm can}\to m\text{ in }L^{p^{*}}(f^{-1}(B_{2r}),\mu_{\varepsilon})

Combined with $\underline{R}_{\varepsilon}\to-m$ , it follows that

\|\Delta_{\omega_{\varepsilon}}U\|_{L^{p^{*}}(f^{-1}(B_{2r}),\mu_{\varepsilon})}\leq|\underline{R}_{\varepsilon}+m|+\|\tr_{\omega_{\varepsilon}}f^{*}\omega_{\rm can}-m\|_{L^{p^{*}}(f^{-1}(B_{2r}),\mu_{\varepsilon})}\to 0.

Moreover, we showed in Lemma 4.7 that $F_{\varepsilon}\to f^{*}\varphi\ _{\rm can}$ in $L^{p}$ for any finite $p$ . Hence $\|U\|_{L^{p^{*}}(f^{-1}(B_{2r}),\mu_{\varepsilon})}\to 0$ . For any given $K\Subset X^{\circ}$ , choose finitely many coordinate balls such that

f(K)\subset\cup_{j=1}^{M}B_{r_{j}},\quad B_{2r_{j}}\Subset Y^{\circ}.

Applying Lemma 4.10, we get the desired $L^{\infty}$ convergence. ∎

5. Sharp Schwarz-type estimate

Lemma 5.1.

For every $K\Subset X^{\circ}$ , $\omega_{\varepsilon}\geq(1-o(1))f^{*}\omega_{\rm can}$

Proof.

Work over $B_{2r}\Subset Y^{\circ}$ , with base coordinates $y^{1},\ldots,y^{m}$ . For $a=\sum_{\alpha=1}^{m}a_{\alpha}\,dy^{\alpha}$ with constant coefficients and $|a|_{\omega_{\rm can}}\leq 1$ on $B_{2r}$ , set

\omega_{a}=\sqrt{-1}a\wedge\bar{a},\qquad u_{\varepsilon}=\tr_{\omega_{\varepsilon}}f^{*}\omega_{a},\qquad u_{0}=\tr_{\omega_{\rm can}}\omega_{a}.

We have

0\leq u_{\varepsilon}\leq C,\qquad\|u_{\varepsilon}-f^{*}u_{0}\|_{L^{2}(f^{-1}(B_{2r}),\mu_{\varepsilon})}\longrightarrow 0.

Suppress $\varepsilon$ and put

h=\sum_{\alpha=1}^{m}a_{\alpha}(y^{\alpha}\circ f),\qquad f^{*}a=\partial h,\qquad f^{*}\omega_{a}=\sqrt{-1}\,\partial h\wedge\bar{\partial}\bar{h}.

Fix $x\in f^{-1}(B_{2r})$ and choose holomorphic coordinates $z^{1},\ldots,z^{n}$ normal for $\omega$ at $x$ :

\omega=\sqrt{-1}g_{i\bar{j}}\,dz^{i}\wedge d\bar{z}^{j},\qquad g_{i\bar{j}}(x)=\delta_{ij},\qquad\partial_{k}g_{i\bar{j}}(x)=0.

Write $h_{i}=\partial_{i}h$ , $h_{ik}=\partial_{k}\partial_{i}h$ , with $\sum_{j}g^{i\bar{j}}g_{k\bar{j}}=\delta^{i}_{k}$ . Since $h$ is holomorphic,

u=\sum_{i,j}g^{i\bar{j}}h_{i}\overline{h_{j}},\qquad\partial_{\bar{l}}h_{i}=0,\qquad\partial_{k}\overline{h_{j}}=0.

Straightforward computation gives

\partial_{k}u=\sum_{i,j}(\partial_{k}g^{i\bar{j}})h_{i}\overline{h_{j}}+\sum_{i,j}g^{i\bar{j}}h_{ik}\overline{h_{j}}.

u_{k\bar{l}}=\sum_{i,j}(\partial_{k}\partial_{\bar{l}}g^{i\bar{j}})h_{i}\overline{h_{j}}+\sum_{i}h_{ik}\overline{h_{il}}.

At $x$ , we have

\displaystyle\partial_{k}\partial_{\bar{l}}g^{i\bar{j}}

\displaystyle=-\partial_{k}\partial_{\bar{l}}g_{j\bar{i}}=R_{j\bar{i}k\bar{l}}.

Thus, using $\Ric(\omega)=-\chi-dd^{c}F$ ,

	$\displaystyle\Delta_{\omega}u$	$\displaystyle=\sum_{i,k}\|h_{ik}\|^{2}+\sum_{i,j}\Ric(\omega)_{j\bar{i}}h_{i}\overline{h_{j}}$
		$\displaystyle=\sum_{i,k}\|h_{ik}\|^{2}-\sum_{i,j}\chi_{j\bar{i}}h_{i}\overline{h_{j}}-\sum_{i,j}F_{j\bar{i}}h_{i}\overline{h_{j}}.$

Thus

\begin{split}\Delta_{\omega}u&=|\nabla^{1,0}_{\omega}(f^{*}a)|_{\omega}^{2}-\langle f^{*}\omega_{a},\chi\rangle_{\omega}\\ &\qquad-\langle f^{*}\omega_{a},dd^{c}F\rangle_{\omega}.\end{split}

(5.1)

It is also (3.5) for the holomorphic map $h$ to flat $\mathbb{C}$ . A unitary change of the normal coordinates diagonalizes $\chi$ , with eigenvalues $\lambda_{i}\geq 0$ , so

0\leq\langle f^{*}\omega_{a},\chi\rangle_{\omega}=\sum_{i}\lambda_{i}|h_{i}|^{2}\leq\left(\sum_{i}\lambda_{i}\right)\left(\sum_{i}|h_{i}|^{2}\right)=(\tr_{\omega}\chi)u\leq Cu.

In particular,

\Delta_{\omega}u\geq-Cu-\langle f^{*}\omega_{a},dd^{c}F\rangle_{\omega}.

We retain the square in (5.1) below. The smoothness of $u_{0}$ on the fixed chart gives

|\nabla f^{*}u_{0}|_{\omega}^{2}\leq C\tr_{\omega}\chi\leq C,\qquad|\Delta_{\omega}f^{*}u_{0}|=|\tr_{\omega}f^{*}(dd^{c}u_{0})|\leq C\tr_{\omega}\chi\leq C.

Consequently,

\Delta_{\omega}(u-f^{*}u_{0})\geq|\nabla^{1,0}_{\omega}(f^{*}a)|_{\omega}^{2}-C-\langle f^{*}\omega_{a},dd^{c}F\rangle_{\omega}.

(5.2)

Set $v=(u-f^{*}u_{0})_{+}$ , $\xi=\xi_{s,t}$ and $\psi=\xi^{2}v^{p-1}$ , where $p\geq 3$ . All following integrals use $d\mu_{\varepsilon}$ on $f^{-1}(B_{t})$ , and all differential norms use $\omega$ . Testing (5.2) with $\psi$ ,

		$\displaystyle(p-1)\int\xi^{2}v^{p-2}\|\nabla v\|^{2}+\int\psi\|\nabla^{1,0}_{\omega}(f^{*}a)\|^{2}$
		$\displaystyle\leq 2\int\xi v^{p-1}\|\nabla\xi\|\,\|\nabla v\|+C\int\psi+\int\psi\langle f^{*}\omega_{a},dd^{c}F\rangle_{\omega}.$

To estimate the last term, note that at the normal point

\nabla_{k}(f^{*}\omega_{a})_{i\bar{j}}=h_{ik}\overline{h_{j}},\qquad\nabla_{\bar{k}}(f^{*}\omega_{a})_{i\bar{j}}=h_{i}\overline{h_{jk}}.

Hence

|\nabla(f^{*}\omega_{a})|_{\omega}\leq Cu^{1/2}|\nabla^{1,0}_{\omega}(f^{*}a)|_{\omega}.

Integration by parts, and the bounds $u+|\nabla F|_{\omega}\leq C$ from Lemma 3.2 and the trace estimate in base give

	$\displaystyle\left\|\int\psi\langle f^{*}\omega_{a},dd^{c}F\rangle_{\omega}\right\|$	$\displaystyle\leq C\int\bigl(u\|\nabla\psi\|+\psi\|\nabla(f^{*}\omega_{a})\|\bigr)\|\nabla F\|$
		$\displaystyle\leq\frac{1}{2}\int\psi\|\nabla^{1,0}_{\omega}(f^{*}a)\|^{2}+C\int(\|\nabla\psi\|+\psi).$

Here the square is absorbed using

C\psi|\nabla^{1,0}_{\omega}(f^{*}a)|\leq\frac{1}{2}\psi|\nabla^{1,0}_{\omega}(f^{*}a)|^{2}+C\psi.

Since

|\nabla\psi|\leq 2\xi v^{p-1}|\nabla\xi|+(p-1)\xi^{2}v^{p-2}|\nabla v|,

the remaining terms satisfy

	$\displaystyle 2\xi v^{p-1}\|\nabla\xi\|\,\|\nabla v\|$	$\displaystyle\leq\frac{p-1}{4}\xi^{2}v^{p-2}\|\nabla v\|^{2}+\frac{C}{p-1}v^{p}\|\nabla\xi\|^{2},$
	$\displaystyle C\xi v^{p-1}\|\nabla\xi\|$	$\displaystyle\leq Cv^{p}\|\nabla\xi\|^{2}+C\xi^{2}v^{p-2},$
	$\displaystyle C(p-1)\xi^{2}v^{p-2}\|\nabla v\|$	$\displaystyle\leq\frac{p-1}{4}\xi^{2}v^{p-2}\|\nabla v\|^{2}+C(p-1)\xi^{2}v^{p-2}.$

After absorption, the cutoff bound and $0\leq v\leq C$ imply

\int|\nabla(\xi v^{p/2})|^{2}+\int\xi^{2}v^{p}\leq Cp^{2}\bigl(1+(t-s)^{-2}\bigr)\int_{f^{-1}(B_{t})}v^{p-2}.

(5.3)

The uniform Sobolev inequality therefore yields

\|v\|_{L^{pq}(f^{-1}(B_{s}))}^{p}\leq Cp^{2}\bigl(1+(t-s)^{-2}\bigr)\|v\|_{L^{p-2}(f^{-1}(B_{t}))}^{p-2}.

Then Moser iteration implies

\|v\|_{L^{\infty}(f^{-1}(B_{r}))}\leq C\|v\|_{L^{2}(f^{-1}(B_{2r}),\mu_{\varepsilon})}^{(q-1)/(2q-1)}\longrightarrow 0.

Write $u_{\varepsilon}(a),u_{0}(a)$ to denote the covector dependence. Since $\tr_{\omega_{\varepsilon}}\chi\leq C$ , for $|a|,|b|\leq 1$ ,

|u_{\varepsilon}(a)-u_{\varepsilon}(b)|+f^{*}|u_{0}(a)-u_{0}(b)|\leq C|a-b|.

For a finite $\delta$ -net of the Euclidean unit covectors, the convergence of $v_{+}\to 0$ is uniform. Hence

\limsup_{\varepsilon\to 0}\sup_{\begin{subarray}{c}x\in f^{-1}(B_{r})\\ |a|=1\end{subarray}}\bigl(u_{\varepsilon}(a)-f^{*}u_{0}(a)\bigr)\leq C\delta.

Let $\delta\downarrow 0$ . Since $u_{0}(a)\geq c_{r}|a|^{2}$ on $\overline{B_{r}}$ , homogeneity gives

|f^{*}\alpha|_{\omega_{\varepsilon}}^{2}\leq(1+o_{r}(1))f^{*}|\alpha|_{\omega_{\rm can}}^{2}\quad\text{on }f^{-1}(B_{r}),\qquad\alpha\in T^{*1,0}Y^{\circ}.

For $Z\in T^{1,0}X$ , duality then gives

|df(Z)|_{\omega_{\rm can}}=\sup_{|\alpha|_{\omega_{\rm can}}=1}|f^{*}\alpha(Z)|\leq(1+o_{r}(1))^{1/2}|Z|_{\omega_{\varepsilon}}.

Finally choose $K\subset\bigcup_{j=1}^{N}B_{r_{j}}$ with $\overline{B_{2r_{j}}}\Subset Y^{\circ}$ . The maximum of the finitely many errors $o_{r_{j}}(1)$ tends to zero. The preceding inequality is therefore equivalent to the desired sharp Schwarz-type estimate on $f^{-1}(K)$ . ∎

6. Fiber collapsing and convergence away from singular fibers

The goal of this section is to prove the metric convergence on the regular part of $X$ . In the first four lemmas assume $d>0$ . All estimates are over a fixed $K\Subset Y^{\circ}$ , enlarged when necessary to $K^{\prime}\Subset Y^{\circ}$ with $K\Subset\operatorname{int}K^{\prime}$ . The smooth proper family $f^{-1}(K^{\prime})\to K^{\prime}$ has uniform Sobolev, Poincaré, and Green-function bounds for ${\omega_{y}}={\omega}|_{X_{y}}$ .

Lemma 6.1.

For any $K\Subset Y^{\circ}$ , $\operatorname{osc}_{X_{y}}\varphi_{\varepsilon}\leq C_{K}\varepsilon$ , for all $y\in K$ .

Proof.

Since $\chi|_{X_{y}}=0$ , one has

\varepsilon^{-1}\omega_{\varepsilon}|_{X_{y}}={\omega_{y}}+dd^{c}_{V}v_{\varepsilon,y},\quad h_{\varepsilon,y}:=\frac{(\varepsilon^{-1}\omega_{\varepsilon}|_{X_{y}})^{d}}{{\omega_{y}}^{d}},

where $v_{\varepsilon,y}:=\varepsilon^{-1}\left(\varphi_{\varepsilon}|_{X_{y}}-V_{F}^{-1}\int_{X_{y}}\varphi_{\varepsilon}{\omega_{y}}^{d}\right),$ and $dd^{c}_{V}$ means the differential on the fiber(vertical) direction. Moreover,

\chi^{m}\wedge\omega_{\varepsilon}^{d}=\varepsilon^{d}h_{\varepsilon,y}\chi^{m}\wedge{\omega}^{d}.

$0\leq\chi\leq\tr_{\omega_{\varepsilon}}\chi\omega_{\varepsilon}\leq C\omega_{\varepsilon}$ , $\chi^{m}\wedge\omega_{\varepsilon}^{d}\leq C^{m}\omega_{\varepsilon}^{n}$ . Note that $V_{\varepsilon}\leq C\varepsilon^{d}$ and $F_{\varepsilon}\leq C$ . Over a compact $f^{-1}(K)\Subset X^{\circ}$ ,

\omega_{\varepsilon}^{n}\leq C\varepsilon^{d}\Omega\leq C_{K}\varepsilon^{d}\chi^{m}\wedge{\omega}^{d}.

Hence

0<h_{\varepsilon,y}=\frac{\chi^{m}\wedge\omega_{\varepsilon}^{d}}{\varepsilon^{d}\chi^{m}\wedge{\omega}^{d}}\leq\frac{C^{m}\omega_{\varepsilon}^{n}}{\varepsilon^{d}\chi^{m}\wedge{\omega}^{d}}\leq C_{K}.

Standard $L^{\infty}$ -estimate for complex Monge-Ampère equation gives the desired estimate for $v_{\varepsilon,y}$ . ∎

Next we show that the normalized fiber potential $v_{\varepsilon,y}:=\varepsilon^{-1}\left(\varphi_{\varepsilon}|_{X_{y}}-V_{F}^{-1}\int_{X_{y}}\varphi_{\varepsilon}{\omega_{y}}^{d}\right)$ indeed converges the the potential of $\omega_{\rm SRF}.$

Lemma 6.2.

$\sup_{y\in K}\|v_{\varepsilon,y}-\rho_{y}\|_{L^{\infty}}\to 0$ , where ${\omega}+dd^{c}\rho_{y}$ is the unique Ricci flat metric in class $[{\omega}|_{y}]$ with $\int_{X_{y}}\rho_{y}{\omega_{y}}^{d}=0$ .

Proof.

By Lemma 5.1, we can choose $0<\delta_{\varepsilon}\to 0$ such that $\omega_{\varepsilon}-(1-\delta_{\varepsilon})f^{*}\omega_{\rm can}\geq 0$ . Thus

	$\displaystyle\omega_{\varepsilon}^{n}\geq$	$\displaystyle\binom{n}{m}(1-\delta_{\varepsilon})^{m}(f^{}\omega_{\rm can})^{m}\wedge(\omega_{\varepsilon}-(1-\delta_{\varepsilon})f^{}\omega_{\rm can})^{d}$
	$\displaystyle=$	$\displaystyle\binom{n}{m}(1-\delta_{\varepsilon})^{m}(f^{*}\omega_{\rm can})^{m}\omega_{\varepsilon}^{d}.$

Let $h_{{\rm RF},y}=\omega_{{\rm SRF},y}^{d}/{\omega_{y}}^{d}$ . Then

\frac{h_{\varepsilon,y}}{h_{{\rm RF},y}}=\frac{(f^{*}\omega_{\rm can})^{m}\wedge\omega_{\varepsilon}^{d}}{\varepsilon^{d}(f^{*}\omega_{\rm can})^{m}\wedge\omega_{\rm SRF}^{d}}\leq\frac{\omega_{\varepsilon}^{n}}{(1-\delta_{\varepsilon})^{m}\binom{n}{m}\varepsilon^{d}(f^{*}\omega_{\rm can})^{m}\wedge\omega_{\rm SRF}^{d}}=1+o(1).

Here we used that $F_{\varepsilon}\to f^{*}\varphi_{\rm can}$ in $L^{\infty}$ away from singlar fiber(Lemma 4.11). Consequently,

0<h_{\varepsilon,y}\leq(1+\eta_{\varepsilon})h_{{\rm RF},y},\quad\eta_{\varepsilon}\to 0.

From stability estimate for complex Monge-Ampère equation, we get the desired $L^{\infty}$ -closeness of the potentials. To see the convergence is uniform on $K$ , we sketch a proof here. Set $a=v_{\varepsilon,y}-\rho_{y}$ , and $\omega_{\varepsilon,y}^{\prime}=\omega_{{\rm SRF},y}+dd^{c}a$ . Both forms are positive, $\|a\|_{\infty}\leq C_{K}$ , and $(\omega_{\varepsilon,y}^{\prime})^{d}/\omega_{{\rm SRF},y}^{d}\leq C_{K}$ . The smooth reference metrics $\omega_{{\rm SRF},y}$ over $K^{\prime}$ have common Sobolev and Poincaré constants and are uniformly equivalent to ${\omega_{y}}$ . Since $\int a\,{\omega_{y}}^{d}=0$ , integration by parts gives

\begin{split}\int_{X_{y}}\sqrt{-1}\partial a\wedge\bar{\partial}a\wedge\omega_{{\rm SRF},y}^{d-1}&\leq\int_{X_{y}}\sqrt{-1}\partial a\wedge\bar{\partial}a\wedge\sum_{j=0}^{d-1}(\omega_{\varepsilon,y}^{\prime})^{j}\wedge\omega_{{\rm SRF},y}^{d-1-j}\\ &=-\int_{X_{y}}a((\omega_{\varepsilon,y}^{\prime})^{d}-\omega_{{\rm SRF},y}^{d})\leq C_{K}\eta_{\varepsilon},\\ \|a\|_{L^{2}(\omega_{{\rm SRF},y}^{d})}&\leq C_{K}\eta_{\varepsilon}^{1/2}.\end{split}

Let $u=a_{+}$ , from difference of two equations we get

\begin{split}(p-1)\int u^{p-2}\sqrt{-1}\partial u\wedge\bar{\partial}u\wedge\omega_{{\rm SRF},y}^{d-1}&\leq C_{K}\int u^{p-1}\omega_{{\rm SRF},y}^{d},\\ \|u\|_{L^{pq}(\omega_{{\rm SRF},y}^{d})}^{p}&\leq C_{K}p^{2}\int u^{p-1}\omega_{{\rm SRF},y}^{d}.\end{split}

Here $q>1$ is a common fiber Sobolev exponent, and the zero-order Sobolev term is absorbed using $0\leq u\leq C_{K}$ . Moser iteration gives

\|u\|_{L^{\infty}}\leq C_{K}\|u\|_{L^{2}}^{\beta},\qquad\beta=\lim_{k\to\infty}\frac{2q^{k}}{p_{k}}=\frac{2}{2+q/(q-1)}>0.

Applying this to both signs proves $\|a\|_{L^{\infty}}\leq C_{K}\eta_{\varepsilon}^{\beta/2}\to 0$ .

∎

Lemma 6.3.

The form

T_{\varepsilon}=\frac{f_{*}(\omega_{\varepsilon}^{d+1}-\varepsilon^{d+1}\omega_{\rm SRF}^{d+1})}{(d+1)\varepsilon^{d}V_{F}}

is smooth and closed on $Y^{\circ}$ . For any compact $K\Subset Y^{\circ}$ , $T_{\varepsilon}\to\omega_{\rm can}$ uniformly on $K$ .

Proof.

As before, choose $0<\delta_{\varepsilon}\to 0$ such that $\gamma_{\varepsilon}:=\omega_{\varepsilon}-(1-\delta_{\varepsilon})f^{*}\omega_{\rm can}\geq 0$ . Then $R_{\varepsilon}:=f_{*}(\gamma_{\varepsilon}^{d+1})/((d+1)\varepsilon^{d}V_{F})\geq 0$ .

\omega_{\varepsilon}^{d+1}-\gamma_{\varepsilon}^{d+1}=(1-\delta_{\varepsilon})f^{*}\omega_{\rm can}\sum_{i=0}^{d}\omega_{\varepsilon}^{i}\gamma_{\varepsilon}^{d-i}.

Only products in the fiber direction contribute to the fiber integration, hence

f_{*}(\sum_{i=0}^{d}\omega_{\varepsilon}^{i}\gamma_{\varepsilon}^{d-i})=\int_{X_{y}}(\omega_{\varepsilon,y})^{d}=(d+1)V_{F}\varepsilon^{d}.

As a result, $T_{\varepsilon}=(1-\delta_{\varepsilon})\omega_{\rm can}+R_{\varepsilon}-\frac{\varepsilon f_{*}(\omega_{\rm SRF}^{d+1})}{(d+1)V_{F}}$ . On the other hand,

	$\displaystyle f_{*}\omega_{\varepsilon}^{n}\geq$	$\displaystyle f_{}\left(\binom{n}{d+1}\gamma_{\varepsilon}^{d+1}(1-\delta_{\varepsilon})^{m-1}f^{}\omega_{\rm can}^{m-1}+\binom{n}{m}\gamma_{\varepsilon}^{d}(1-\delta_{\varepsilon})^{m}f^{*}\omega_{\rm can}^{m}\right)$
	$\displaystyle=$	$\displaystyle\binom{n}{d}\varepsilon^{d}V_{F}[(1-\delta_{\varepsilon})^{m-1}mR_{\varepsilon}\wedge\omega_{\rm can}^{m-1}+(1-\delta_{\varepsilon})^{m}\omega_{\rm can}^{m}]$

Hence

\frac{f_{*}\omega_{\varepsilon}^{n}}{\binom{n}{m}\varepsilon^{d}V_{F}\omega_{\rm can}^{m}}\geq(1-\delta_{\varepsilon})^{m-1}\tr_{\omega_{\rm can}}R_{\varepsilon}+(1-\delta_{\varepsilon})^{m}.

Since $f_{*}\omega_{\varepsilon}^{n}\to V_{\varepsilon}f^{*}\omega_{\rm can}^{m}$ by Lemma 4.11, we get

0\leq|R_{\varepsilon}|_{\omega_{\rm can}}\leq\tr_{\omega_{\rm can}}R_{\varepsilon}\to 0.

This proves $T_{\varepsilon}\to\omega_{\rm can}$ uniformly away from singular fibers. ∎

Lemma 6.4.

There are smooth function $r_{\varepsilon}$ on $X^{\circ}$ such that for any compact $K\Subset Y^{\circ}$

\omega_{\varepsilon}=f^{*}T_{\varepsilon}+\varepsilon\omega_{\rm SRF}+\varepsilon dd^{c}r_{\varepsilon}\text{ on }X^{\circ},\qquad\|r_{\varepsilon}\|_{L^{\infty}(f^{-1}(K))}\to 0.

Proof.

For a potential $\varphi$ , recall that the energy is

E_{y}(\varphi):=\frac{1}{(d+1)V_{F}}\sum_{j=0}^{d}\int_{X_{y}}\varphi({\omega_{y}}+dd^{c}\varphi)^{j}\wedge{\omega_{y}}^{d-j}.

Here and in the following $dd^{c}$ is taken with respect to the full directions. It follows that

f_{*}(({\omega}+dd^{c}\varphi)^{d+1}-{\omega}^{d+1})=(d+1)vV_{F}dd^{c}E_{y}(\varphi)

We can write $\omega_{\varepsilon}=f^{*}(\chi_{Y}+dd^{c}\bar{\varphi}_{\varepsilon,y})+\varepsilon({\omega}+dd^{c}v_{\varepsilon,y})$ , where $\bar{\varphi}_{\varepsilon,y}=V_{F}^{-1}\int_{X_{y}}\varphi_{\varepsilon}{\omega_{y}}^{d}$ . Hence

\frac{f_{*}(\omega_{\varepsilon}^{d+1})}{(d+1)\varepsilon^{d}V_{F}}=\chi_{Y}+dd^{c}\bar{\varphi}_{\varepsilon,y}+\frac{\varepsilon f_{*}(({\omega}+dd^{c}v_{\varepsilon,y})^{d+1})}{(d+1)V_{F}}.

By the definition of $T_{\varepsilon}$ , we get

T_{\varepsilon}=\chi_{Y}+dd^{c}\bar{\varphi}_{\varepsilon,y}+\varepsilon dd^{c}(E_{y}(v_{\varepsilon,y})-E_{y}(\rho)).

Note that

|E_{y}(v_{\varepsilon,y})-E_{y}(\rho)|\leq\|v_{\varepsilon,y}-\rho\|_{L^{\infty}}\to 0.

Hence let $r_{\varepsilon}=v_{\varepsilon,y}-\rho-f^{*}(E_{y}(v_{\varepsilon,y})-E_{y}(\rho))$ , we get the desired representation and the estimate. ∎

The following $dd^{c}$ -lemma with estimate is standard.

Lemma 6.5.

Let $\mathcal{P}\subset\mathbb{C}^{N}$ be a fixed polydisc. A smooth closed real $(1,1)$ -form $\alpha$ on a neighborhood of $\overline{\mathcal{P}}$ , has a real potential $b\in C^{\infty}(\mathcal{P})$ such that

dd^{c}b=\alpha,\qquad\|b\|_{L^{\infty}(\mathcal{P})}\leq C_{\mathcal{P}}\|\alpha\|_{L^{\infty}(\mathcal{P})}.

(6.1)

Here the norms of forms are Euclidean tensor norms.

Now we can prove the metric convegence by a blow-up argument. We will use the following theorem of Cheng-Xu on the interior $W^{2,p}$ -estimate for complex Monge-Ampère equations, regardless of the boundary behavior. The addition assumption is the closeness to a smooth stricly psh function.

Theorem 6.6 (Cheng-Xu, Theorem 1.4[3]).

Suppose on $B_{1}\subset\mathbb{C}^{n}$ that

C_{0}^{-1}\omega_{\rm E}\leq dd^{c}w_{0}\leq C_{0}\omega_{\rm E},\qquad\|D^{3}w_{0}\|_{\infty}\leq C_{0}.

For every $1<P<\infty$ , there are tolerances $\delta_{0}(C_{0},n)>0$ and $\delta_{\rm MA}(n,P)>0$ such that

\begin{gathered}\phi\in C^{0}\cap\PSH(B_{1}),\qquad\|\phi-w_{0}\|_{\infty}\leq\delta_{0},\\ (1-\delta_{\rm MA})\omega_{\rm E}^{n}\leq(dd^{c}\phi)^{n}\leq(1+\delta_{\rm MA})\omega_{\rm E}^{n}\end{gathered}\quad\Longrightarrow\quad\|\phi\|_{W^{2,P}(B_{1/2})}\leq C.

Lemma 6.7.

For every compact $K\Subset Y^{\circ}$ ,

(1)

$\|\omega_{\varepsilon}-(f^{*}\omega_{\rm can}+\varepsilon\omega_{\rm SRF})\|_{f^{*}\omega_{\rm can}+\varepsilon{\omega}}\to 0$ uniformly on $f^{-1}(K)$ .
(2)

$\sup_{y\in K}\diam(X_{y},\omega_{\varepsilon}|_{X_{y}})\leq C_{K}\sqrt{\varepsilon}.$

Proof.

If $d=0$ , $f:X^{\circ}\to Y^{\circ}$ is biholomorphic. The sharp Schwarz estimate gives $\omega_{\varepsilon}\geq(1-o(1))f^{*}\omega_{\rm can}$ , while Lemma 4.11 gives $\omega_{\varepsilon}^{n}/(f^{*}\omega_{\rm can})^{n}\to 1$ . Every eigenvalue is bounded below by $1-o(1)$ and their product tends to $1$ ; hence every eigenvalue tends to $1$ . This proves (1), with the convention $\omega_{\rm SRF}={\omega}$ . The fibers are points, so (2) is immediate. Assume henceforth $d>0$ .

Suppose $\varepsilon_{j}\to 0$ , and choose arbitrary $x_{j}\in f^{-1}(K)$ . After passing to subsequence, $x_{j}\to x_{0}\in f^{-1}(K)$ . We fix a submersion coordinates $(y,z)\in\mathbb{C}^{m}\times\mathbb{C}^{d}$ around $x_{0}$ , such that $f(y,z)=y$ . Suppose $x_{j}=(y_{j},z_{j})$ . Define on a fixed polydisc centered at zero

D_{j}(w,\zeta)=(y_{j}+\sqrt{\varepsilon_{j}}w,z_{j}+\zeta),\quad\widetilde{\omega}_{j}=\varepsilon_{j}^{-1}D_{j}^{*}\omega_{\varepsilon_{j}}

By Lemma 6.4,

\widetilde{\omega}_{j}=\varepsilon_{j}^{-1}D_{j}^{*}(f^{*}T_{\varepsilon_{j}}+\varepsilon_{j}\omega_{\rm SRF})+dd^{c}(r_{\varepsilon_{j}}\circ D_{j}).

The reference forms converges

\varepsilon_{j}^{-1}D_{j}^{*}(f^{*}T_{\varepsilon_{j}}+\varepsilon_{j}\omega_{\rm SRF})\to\omega_{\infty}:=\operatorname{pr}_{1}^{*}\omega_{\rm can}(y_{0})+\operatorname{pr}_{2}^{*}\omega_{\rm SRF}|_{X_{y_{0}}}\quad\text{ in }L^{\infty},

where $\operatorname{pr}_{1}$ is the projection to $y$ and $\operatorname{pr}_{2}$ is the projection to $z$ .

The previous lemma provides $b_{j}$ such that

dd^{c}b_{j}=\varepsilon_{j}^{-1}D_{j}^{*}(f^{*}T_{\varepsilon_{j}}+\varepsilon_{j}\omega_{\rm SRF})-\omega_{\infty},\quad\|b_{j}\|_{L^{\infty}}\to 0.

Fix $w_{0}$ with $dd^{c}w_{0}=\omega_{\infty}$ . Then

\phi_{j}:=w_{0}+b_{j}+r_{\varepsilon_{j}}\circ D_{j},\qquad dd^{c}\phi_{j}=\widetilde{\omega}_{j},\qquad\|\phi_{j}-w_{0}\|_{L^{\infty}}\to 0.

Let $\omega_{E}=dd^{c}(|w|^{2}+|\zeta|^{2})$ , then we have the volume density convergence

\frac{(dd^{c}\phi_{j})^{n}}{\omega_{E}^{n}}=\frac{\varepsilon_{j}^{-n}D_{j}^{*}(\omega_{\varepsilon_{j}}^{n})}{\omega_{E}^{n}}\to\frac{\omega_{\infty}^{n}}{\omega_{E}^{n}},\quad\text{ in }L^{\infty}.

Choose $p>3n(n-1)$ and $P>\max\{2n,p,(n-1)p\}$ . After a linear change, we can assume $dd^{c}w_{0}=\omega_{E}(0)$ . Choose a fixed $r_{*}>0$ such taht

\left|\frac{(dd^{c}w_{0})^{n}}{\omega_{E}^{n}}-1\right|<\frac{1}{2}\delta_{\MA}(n,P)\text{ on }B_{r_{*}}.

Pull back by dilation $z\mapsto r_{*}z$ and divide the potentials by $r_{*}^{2}$ , retaining their names. The rescaled reference $w_{0}$ is fixed, so its ellipticity and third derivatives are bounded by some $C_{0}$ . Since $r_{*}$ is fixed, for all sufficiently large $j$ ,

\begin{gathered}\|\phi_{j}-w_{0}\|_{\infty}\leq\delta_{0}(C_{0},n),\\ (1-\delta_{\rm MA}(n,P))\omega_{\rm E}^{n}\leq(dd^{c}\phi_{j})^{n}\leq(1+\delta_{\rm MA}(n,P))\omega_{\rm E}^{n}.\end{gathered}

By Theorem 6.6, we get $\|\phi_{j}\|_{W^{2,P}(B_{\frac{1}{2}})}\leq C.$ Note that for any positive $(1,1)$ -form $\alpha$ ,

\tr_{\alpha}\omega_{E}\leq(\tr_{\omega_{E}}\alpha)^{n-1}\frac{\omega_{E}^{n}}{\alpha^{n}}.

Let $\alpha=dd^{c}\phi_{j}$ , then we get

\|\tr_{\omega_{E}}dd^{c}\phi_{j}\|_{L^{P}(B_{\frac{1}{2}})}+\|\tr_{dd^{c}\phi_{j}}\omega_{E}\|_{L^{P/(n-1)}(B_{\frac{1}{2}})}\leq C.

In particular both traces have bounded $L^{p}$ norm. We also have the cscK equation $R(dd^{c}\phi_{j})=r_{*}^{2}\varepsilon_{j}\underline{R}_{\varepsilon_{j}}$ . So $|R(dd^{c}\phi_{j})|+|\phi_{j}|\leq C$ . Then we can apply Chen-Cheng’s local estimate [2, Section 6], on $B_{\frac{1}{4}}$

C^{-1}\omega_{E}\leq dd^{c}\phi_{j}\leq C\omega_{E},\qquad\|\phi_{j}\|_{C^{k}}\leq C_{k},

for any $k>0$ . Hence we get the smooth convergence $\widetilde{\omega}_{j}\to\omega_{\infty}$ .

The model $\varepsilon_{j}^{-1}D_{j}^{*}(f^{*}\omega_{\rm can}+\varepsilon_{j}\omega_{\rm SRF})$ also converges to $\omega_{\infty}$ . Moreover, $\varepsilon_{j}^{-1}D_{j}^{*}(f^{*}\omega_{\rm can}+\varepsilon_{j}{\omega})$ has a positive smooth limit. Invariance of relative form norms under pullback and common scaling gives

\begin{split}&\left|\omega_{\varepsilon_{j}}-(f^{*}\omega_{\rm can}+\varepsilon_{j}\omega_{\rm SRF})\right|_{f^{*}\omega_{\rm can}+\varepsilon_{j}{\omega}}(x_{j})\\ &\quad=\left|\widetilde{\omega}_{j}-\varepsilon_{j}^{-1}D_{j}^{*}(f^{*}\omega_{\rm can}+\varepsilon_{j}\omega_{\rm SRF})\right|_{\varepsilon_{j}^{-1}D_{j}^{*}(f^{*}\omega_{\rm can}+\varepsilon_{j}{\omega})}(0)\longrightarrow 0.\end{split}

(6.2)

The centers $x_{j}$ were arbitrary, so (i) is proved.

Restriction $\omega_{\varepsilon}$ to a fiber, we have $\varepsilon^{-1}\omega_{\varepsilon}|_{X_{y}}\leq C_{K}\omega_{{\rm SRF},y}$ . The fixed compact smooth family of Ricci-flat fibers has uniformly bounded diameters. This proves (ii).

∎

7. Gromov-Hausdorff convergence from Lee–Tosatti–Zhang

The main content of this section is a summary of the work of Székelyhidi on the RCD structure on the canonical space and the argument from Lee–Tosatti–Zhang on the Gromov-Hausdorff convergence of the Kähler–Ricci flow. Those enable us to reduce the convergence to a key one-side estimate of the distance functions, Proposition 7.2.

One side of the comparision of the distance function is easier. It follows from the argument of Lee–Tosatti–Zhang [14, Section 4.1], we summarize it here:

Lemma 7.1.

For $0<m\leq n$ :

(1)

([14, Lemma 4.1]) For every $\delta>0$ , there is a compact $K\Subset Y^{\circ}$ , independent of $\varepsilon$ , such that, for all sufficiently small $\varepsilon>0$ ,

$\sup_{y\in Y}d_{\rm can}(y,K)<\delta,\qquad\sup_{x\in X}d_{\omega_{\varepsilon}}(x,f^{-1}(K))<\delta.$

(2)

([14, equation (4.14), using Lemma 4.2]) For every compact $K\Subset Y^{\circ}$ ,

\sup_{x,x^{\prime}\in f^{-1}(K)}\bigl(d_{\omega_{\varepsilon}}(x,x^{\prime})-d_{\rm can}(f(x),f(x^{\prime}))\bigr)_{+}\longrightarrow 0.

(7.1)

Proof.

The cited arguments are metric arguments and apply to the present family with the following inputs. For each $K\Subset Y^{\circ}$ ,

\begin{gathered}\|\omega_{\varepsilon}-f^{*}\omega_{\rm can}\|_{C^{0}(f^{-1}(K),{\omega})}\longrightarrow 0,\\ \sup_{y\in K}\diam(X_{y},\omega_{\varepsilon}|_{X_{y}})\longrightarrow 0.\end{gathered}

(7.2)

Theorem 2.2 gives, with constants independent of $\varepsilon$ ,

\mu_{\varepsilon}\leq C\Omega,\qquad\mu_{\varepsilon}(B_{\omega_{\varepsilon}}(x,r))\geq C^{-1}r^{a_{\rm vol}}\quad(x\in X,\ 0<r<1).

(7.3)

Since $f^{-1}(D)$ is proper analytic, $\Omega(f^{-1}(D))=0$ . Consequently, for a compact exhaustion $K_{j}\Subset Y^{\circ}$ ,

\sup_{\varepsilon}\mu_{\varepsilon}(X\setminus f^{-1}(K_{j}))\leq C\int_{X\setminus f^{-1}(K_{j})}\Omega\longrightarrow 0.

Together with the theorem of Szekelyhidi 2.1, these are the inputs to [14, Lemma 4.1]. The regular path construction in [14, Lemma 4.2] and (7.2) give [14, equation (4.14), in the proof of Lemma 4.3]. No evolution equation is used in these steps. ∎

The key to the global Gromov-Hausdorff convergence is the following reverse inequality.

Proposition 7.2.

For $0<m\leq n$ ,

\sup_{x,x^{\prime}\in X}\big(d_{\rm can}(f(x),f(x^{\prime}))-d_{\omega_{\varepsilon}}(x,x^{\prime})\big)_{+}\longrightarrow 0.

We postpone the proof of this key proposition to next section, and show how the Gromov-Hausdorff convergence can be proved from it.

Proposition 7.3.

For $0<m\leq n$ :

(1)

$\sup_{x,x^{\prime}\in X}\left|d_{\omega_{\varepsilon}}(x,x^{\prime})-d_{\rm can}(f(x),f(x^{\prime}))\right|\longrightarrow 0.$
(2)

$d_{\rm GH}\bigl((X,d_{\omega_{\varepsilon}}),(Y,d_{\rm can})\bigr)\longrightarrow 0.$

Proof.

Fix $\delta>0$ , and choose $K\Subset Y^{\circ}$ by Lemma 7.1(i). By Lemma 7.1(ii) and Proposition 7.2, for all sufficiently small $\varepsilon>0$ ,

	$\displaystyle d_{\omega_{\varepsilon}}(z,z^{\prime})$	$\displaystyle\leq d_{\rm can}(f(z),f(z^{\prime}))+\delta$		$\displaystyle(z,z^{\prime}\in f^{-1}(K)),$
	$\displaystyle d_{\rm can}(f(x),f(x^{\prime}))$	$\displaystyle\leq d_{\omega_{\varepsilon}}(x,x^{\prime})+\delta$		$\displaystyle(x,x^{\prime}\in X).$

For arbitrary $x,x^{\prime}\in X$ , choose $z,z^{\prime}\in f^{-1}(K)$ with

d_{\omega_{\varepsilon}}(x,z)<\delta,\qquad d_{\omega_{\varepsilon}}(x^{\prime},z^{\prime})<\delta.

The lower distance estimate then gives

d_{\rm can}(f(x),f(z))<2\delta,\qquad d_{\rm can}(f(x^{\prime}),f(z^{\prime}))<2\delta.

Hence

	$\displaystyle d_{\omega_{\varepsilon}}(x,x^{\prime})$	$\displaystyle\leq 2\delta+d_{\omega_{\varepsilon}}(z,z^{\prime})$
		$\displaystyle\leq 3\delta+d_{\rm can}(f(z),f(z^{\prime}))$
		$\displaystyle\leq 7\delta+d_{\rm can}(f(x),f(x^{\prime})).$

First $\varepsilon\to 0$ , then $\delta\downarrow 0$ , yields

\sup_{x,x^{\prime}\in X}\bigl(d_{\omega_{\varepsilon}}(x,x^{\prime})-d_{\rm can}(f(x),f(x^{\prime}))\bigr)_{+}\longrightarrow 0.

(7.4)

Combining this with Proposition 7.2 proves (i). This is the metric reduction of [14, Lemma 4.3], with the uniform lower estimate also controlling the endpoints outside $f^{-1}(K)$ .

For (ii), the surjectivity of $f$ makes $\mathcal{R}_{f}=\{(x,f(x)):x\in X\}$ a correspondence. Thus

\begin{aligned} d_{\rm GH}\bigl((X,d_{\omega_{\varepsilon}}),(Y,d_{\rm can})\bigr)&\leq\tfrac{1}{2}\operatorname{dis}(\mathcal{R}_{f})\\ &=\tfrac{1}{2}\sup_{x,x^{\prime}\in X}|d_{\omega_{\varepsilon}}(x,x^{\prime})-d_{\rm can}(f(x),f(x^{\prime}))|\longrightarrow 0.\end{aligned}\qed

8. Proof of Proposition 7.2

The key proposition can be reduced to the following lemma on Lipschitz approximation on the lift:

Lemma 8.1.

For any $v\in\Lip(Y,d_{\rm can})$ with $\Lip_{d_{\rm can}}(v)\leq 1$ , and any $\delta>0$ , there exits smooth functions $v_{\varepsilon}$ such that $\limsup_{\varepsilon\to 0}\|v_{\varepsilon}-f^{*}v\|_{L^{\infty}}\leq\delta$ and $\Lip_{d_{\omega_{\varepsilon}}}(v_{\varepsilon})\leq 1+\delta$ .

Lemma 8.1 implies Proposition 7.2.

Fix $\delta>0$ and a finite $\delta$ -net $\{z_{1},\ldots,z_{N}\}$ in $(Y,d_{\rm can})$ . For $v_{j}(z)=d_{\rm can}(z,z_{j})$ ,

\Lip(v_{j})\leq 1,\qquad d_{\rm can}(z,z^{\prime})\leq 2\delta+\max_{j}|v_{j}(z)-v_{j}(z^{\prime})|.

Choose $v_{\varepsilon,j}$ from the preceding lemma, and set

	$\displaystyle a_{\varepsilon}$	$\displaystyle=\max_{j}\bigl(\\|v_{\varepsilon,j}-f^{*}v_{j}\\|_{\infty}-\delta\bigr)_{+}\to 0,$
	$\displaystyle b_{\varepsilon}$	$\displaystyle=\max_{j}\bigl(\Lip_{\omega_{\varepsilon}}(v_{\varepsilon,j})-1-\delta\bigr)_{+}\to 0.$

For all $x,x^{\prime}\in X$ ,

	$\displaystyle d_{\rm can}(f(x),f(x^{\prime}))$	$\displaystyle\leq 2\delta+\max_{j}\|v_{j}(f(x))-v_{j}(f(x^{\prime}))\|$
		$\displaystyle\leq 4\delta+2a_{\varepsilon}+\max_{j}\|v_{\varepsilon,j}(x)-v_{\varepsilon,j}(x^{\prime})\|$
		$\displaystyle\leq 4\delta+2a_{\varepsilon}+(1+\delta+b_{\varepsilon})d_{\omega_{\varepsilon}}(x,x^{\prime}).$

Using $\diam(X,\omega_{\varepsilon})\leq C$ from Theorem 2.2,

\limsup_{\varepsilon\to 0}\sup_{x,x^{\prime}\in X}\bigl(d_{\rm can}(f(x),f(x^{\prime}))-d_{\omega_{\varepsilon}}(x,x^{\prime})\bigr)_{+}\leq(4+C)\delta.

Let $\delta\downarrow 0$ . ∎

We aim to prove Lemma 8.1. Before moving to the proof, we introduce a zero-capacity lemma due to Sturm, which turned out to be important in the study of singular Kähler space, see [20, Lemma 3.7]

Lemma 8.2.

There are $\xi_{j}\in C_{c}^{\infty}(Y^{\circ})$ with $0\leq\xi_{j}\leq 1$ , $\xi_{j}\to 1$ locally on $Y^{\circ}$ , and

\int_{Y}\bigl((1-\xi_{j})^{2}+|\nabla\xi_{j}|_{\omega_{\rm can}}^{2}\bigr)d\nu_{\rm can}\to 0.

The cited construction applies to the analytic set $D=Y\setminus Y^{\circ}$ : the current $\omega_{\rm can}$ has bounded local potentials and is smooth on $Y^{\circ}$ . The comparison-flow trace bound and its local limit give $\omega_{\rm can}\geq C^{-1}\chi_{Y}$ on $Y^{\circ}$ ; bounded local potentials extend this inequality across $D$ . Thus the Kähler-current hypothesis of the cited lemma holds.

In this subsection, gradients and integrals are taken on the smooth manifold $(Y^{\circ},\omega_{\rm can})$ , with measure $\nu_{\rm can}$ . The complement $D$ has measure and capacity zero. Thus we are in the almost smooth setting of [22, Definition 5], up to the constant normalization of volume; the smooth open set is $Y^{\circ}$ . In particular,

W^{1,2}(Y,\omega_{\rm can},\nu_{\rm can})=\overline{C^{\infty}_{c}(Y^{\circ})}^{\|\cdot\|_{W^{1,2}}},\qquad\|u\|_{W^{1,2}}^{2}=\int_{Y^{\circ}}(u^{2}+|\nabla u|_{\omega_{\rm can}}^{2})d\nu_{\rm can}.

Indeed, for $u\in\Lip(Y)$ ,

	$\displaystyle\\|(1-\xi_{j})u\\|_{L^{2}}$	$\displaystyle\leq\\|u\\|_{\infty}\\|1-\xi_{j}\\|_{L^{2}}\longrightarrow 0,$
	$\displaystyle\\|d((1-\xi_{j})u)\\|_{L^{2}}$	$\displaystyle\leq\\|(1-\xi_{j})du\\|_{L^{2}}+\\|u\\|_{L^{\infty}}\\|d\xi_{j}\\|_{L^{2}}\longrightarrow 0.$

Smooth approximation of $\xi_{j}u$ in coordinate balls, followed by density of Lipschitz functions in the metric Sobolev space, proves the equality. The same cutoff estimate applies to any bounded $u\in W^{1,2}_{\rm loc}(Y^{\circ})$ with $du\in L^{2}$ . in the following, without ambiguity, we write $W^{1,2}$ for $W^{1,2}(Y,\omega_{\rm can},\nu_{\rm can}),$ write $\omega,\nu$ for $\omega_{\rm can},\nu_{\rm can}$ and write $\Delta$ for $\Delta_{\omega_{\rm can}}$ .

For $u\in W^{1,2}$ , and $g\in L^{\infty}$ , $u$ is a said to be a weak solution of $\Delta u=g$ if $\int_{Y^{\circ}}\langle\nabla u,\nabla v\rangle_{\omega_{\rm can}}d\nu_{\rm can}+\int_{Y^{\circ}}gvd\nu_{\rm can}=0$ for all $v\in W^{1,2}$ . For $\lambda>0$ , $g\in L^{\infty}$ , by the standard Lax-Milgram theorem, there is a unique $h\in W^{1,2}$ such that $\Delta h=\lambda h+g$ . Moreover, $\|h\|_{L^{\infty}}\leq\lambda^{-1}\|g\|_{L^{\infty}}$ , and if $g\in C^{\infty}(Y^{\circ})$ , then $h\in C^{\infty}(Y^{\circ}).$ Indeed, to see the $L^{\infty}$ -bound, just take the test function to be $v=(h-\lambda^{-1}\|g\|_{L^{\infty}})_{+}$ .

Lemma 8.3.

Let $h\in W^{1,2}$ be the unique solution to $\Delta h=\lambda h+g$ for $\lambda>0$ , and $g\in L^{\infty}$ . Then $\Lip(h)\leq C_{\lambda}\|g\|_{L^{\infty}}.$

Proof.

Szekelyhidi’s theorem shows that the $(Y,d_{\rm can},\nu_{\rm can})$ is a $RCD(-1,2m)$ space. [11, Theorem 2.33](see also [13]) shows the quantitative Lipschitz regularity of solution of Poisson equations over balls. Cover $Y$ by finite balls gives

|\nabla h|(x)\leq C(\|h\|_{L^{\infty}}+\|\Delta h\|_{L^{\infty}})\leq C_{\lambda}\|g\|_{L^{\infty}},

for $\nu_{\rm can}$ -almost every $x$ . Such $h$ has a unique $\Lip$ -representative; see the proof of [22, Proposition 9]. ∎

Lemma 8.4.

For $\lambda>1$ and $g\in\Lip(Y)$ . The solution to $\Delta h=\lambda h+g$ satisfies the sharper $\Lip$ -bound:

\Lip(h)\leq\frac{1}{\lambda-1/2}\Lip(g).

Moreover, $\|\lambda h+g\|_{L^{\infty}}\to 0$ as $\lambda\to\infty.$

Proof.

From local regularity, $h\in W^{3,p}_{loc}(Y^{\circ})$ for any $p>1$ , and hence locally $C^{2}$ . The previous lemma gives $|\nabla h|^{2}\in L^{\infty}$ . Since $\Ric(\omega_{\rm can})\geq-\omega_{\rm can}$ , we have, on $Y^{\circ}$ ,

\Delta|\nabla h|_{\omega}^{2}\geq\sum_{i,j}(|h_{ij}|^{2}+|h_{i\bar{j}}|^{2})+(2\lambda-1)|\nabla h|_{\omega}^{2}+2\operatorname{Re}h_{i}g_{\bar{i}}\geq(2\lambda-1)|\nabla h|_{\omega}(|\nabla h|_{\omega}-L),

where we take $L=(2\lambda-1)^{-1}\sqrt{2}\Lip(g)$ . Let $w=|\nabla u|_{\omega}^{2}$ and $z=(w-L^{2})_{+}$ . Test the Bochner formula with $\xi_{j}^{2}z$ , where $\xi_{j}$ is the cut-off with capacity tends to $0$ . Integration by parts and Young’s inequality give

	$\displaystyle\int\xi_{j}^{2}\|\nabla z\|_{\omega}^{2}+(2\lambda-1)\int\xi_{j}^{2}z\sqrt{w}(\sqrt{w}-L)$	$\displaystyle\leq-\int\xi_{j}z\langle\nabla\xi_{j},\nabla w\rangle$
		$\displaystyle\leq\frac{1}{4}\int\xi_{j}^{2}\|\nabla z\|^{2}+\\|z\\|_{\infty}^{2}\int\|\nabla\xi_{j}\|^{2}.$

Absorb the first term on the right. By Fatou’s lemma,

(\lambda-1)\int z\sqrt{w}(\sqrt{w}-L)=0\quad\Longrightarrow\quad z=0\quad\Longrightarrow\quad\Lip(h)\leq\frac{\Lip(g)}{\lambda-1/2}.

For the uniform convergence, test the equation with $\lambda h+g$ , we get

\|\lambda h+g\|_{L^{2}}\leq\frac{1}{2\sqrt{\lambda}}\|\nabla g\|_{L^{2}}\to 0.

After scaling, assume $\Lip(g)\leq 1$ . For $\lambda>2$ , $\Lip(\lambda h+g)\leq 3$ . Hence if $\|\lambda h+g\|_{L^{\infty}}\geq a>0$ ,

\|\lambda h+g\|_{L^{2}}^{2}\geq\frac{a^{2}}{4}\inf_{y\in Y}\nu_{\rm can}(B_{d_{\rm can}}(y,a/6))>0.

A contradiction. ∎

Lemma 8.5.

For $\lambda>0$ , $g\in L^{\infty}(Y)$ , and $h\in C^{\infty}(Y^{\circ})\cap L^{\infty}(Y^{\circ})$ , if $h$ satisfies $\Delta h=\lambda h+g$ pointwise on $Y^{\circ}$ , then $h\in W^{1,2}$ and is the weak solution to $\Delta h=\lambda h+g$ on $Y$ .

Proof.

Let $\xi_{j}$ be the sequence of cut-offs with capacity tends to $0$ . Choose test function $\xi_{j}^{2}h$ , then

	$\displaystyle\int\xi_{j}^{2}\|\nabla h\|^{2}+\lambda\int\xi_{j}^{2}h^{2}$	$\displaystyle=-\int g\xi_{j}^{2}h-2\int\xi_{j}h\langle\nabla h,\nabla\xi_{j}\rangle$
		$\displaystyle\leq\\|g\\|_{L^{\infty}}\\|h\\|_{L^{\infty}}+\frac{1}{2}\int\xi_{j}^{2}\|\nabla h\|^{2}+2\\|h\\|_{L^{\infty}}^{2}\int\|\nabla\xi_{j}\|^{2}.$

Here all integrals use $d\nu_{\rm can}$ , and gradients are taken with respect to the metric $\omega_{\rm can}$ . By Fatou’s lemma $\nabla h\in L^{2}$ . It follows that $h\in W^{1,2}$ . The weak equation holds globally by taking limit of test functions in $W^{1,2}$ . ∎

Lemma 8.6.

For any $v\in\Lip(Y)$ with $\Lip(v)\leq 1$ , and every $\delta>0$ , there exists $\lambda>1$ and a smooth function $g$ on $\mathbb{C}P^{N}$ , such that the solution to the equation $\Delta h=\lambda h+g|_{Y}$ satifies

h\in C^{\infty}(Y^{\circ})\cap\Lip(Y),\quad\|h-v\|_{L^{\infty}}<\delta,\quad\Lip(h)<1+\delta.

Proof.

By Lemma 8.4, we can choose $\lambda>1$ such that the solution $\Delta h_{v}=\lambda h_{v}+v$ , satisfies

\Lip(\lambda h_{v})\leq\frac{\lambda}{\lambda-1/2}\Lip(v)\leq 1+\frac{\delta}{3},\qquad\|\lambda h_{v}+v\|_{L^{\infty}}\leq\frac{\delta}{3}.

By Stone-Weierstrass theorem, the smooth function on $\mathbb{C}P^{N}$ are dense in $C(Y)$ . So we can choose such $g_{0}\in C^{\infty}(\mathbb{C}P^{N})$ such that $\|g_{0}-v\|_{L^{\infty}}<\min\{\delta/3,\delta/(3\lambda C_{\lambda})\},$ and set $g=-\lambda g_{0}.$ Then we have

	$\displaystyle\\|h-v\\|_{L^{\infty}}$	$\displaystyle\leq\\|h+\lambda h_{v}\\|_{L^{\infty}}+\\|\lambda h_{v}+v\\|_{L^{\infty}}\leq\lambda C_{\lambda}\\|g_{0}-v\\|_{L^{\infty}}+\frac{\delta}{3}\leq\frac{3\delta}{3},$
	$\displaystyle\Lip(h)$	$\displaystyle\leq\Lip(\lambda h_{v})+\lambda C_{\lambda}\\|g_{0}-v\\|_{\infty}<1+2\delta/3.\qed$

Next, we show that the same method can be carried out also on $(X,\omega_{\varepsilon})$ , thanks to the previous uniform estimates. The next two lemmas use only a Ricci-potential identity, a Sobolev inequality, and first-derivative bounds. We state these assumptions explicitly so that the lemmas also apply to the Kähler–Ricci flow. And in the next two lemmas, we use $\omega$ to denote any Kähler form on $X$ with the assumptions stated in its statement.

Lemma 8.7.

Let $\omega$ vary among Kähler metrics on $X$ and put $\mu=\omega^{n}/\int_{X}\omega^{n}$ . Suppose, with common constants,

\begin{gathered}\Ric(\omega)=-\chi-dd^{c}F,\qquad\chi\geq 0,\qquad\tr_{\omega}\chi+|\nabla F|_{\omega}^{2}\leq C,\\ \|a\|_{L^{2q}(\mu)}^{2}\leq C_{S}\int_{X}(|\nabla a|_{\omega}^{2}+a^{2})\,d\mu,\qquad q>1.\end{gathered}

Fix $\lambda>0$ . If $\Delta_{\omega}H=\lambda H+G$ and $\|G\|_{L^{\infty}}+\|\nabla G\|_{L^{\infty}}\leq C$ , then

\|H\|_{L^{\infty}}+\|\nabla H\|_{L^{\infty}}\leq C^{\prime}.

The constant $C^{\prime}$ depends only on $X,C$ and $\lambda$ .

Proof.

The maximum principle and the test function $H$ imply

\|H\|_{\infty}\leq\lambda^{-1}\|G\|_{\infty},\qquad\int_{X}(|\nabla H|^{2}+\lambda H^{2})d\mu=-\int_{X}GHd\mu\leq\lambda^{-1}\|G\|_{\infty}^{2}.

(8.1)

Write $u=|\nabla H|_{\omega}^{2}$ . At an $\omega$ -unitary frame, with repeated indices summed, the Bochner identity is

\Delta_{\omega}u=|H_{ij}|^{2}+|H_{i\bar{j}}|^{2}+2\lambda u+2\operatorname{Re}(G_{i}H_{\bar{i}})-\chi_{i\bar{j}}H_{j}H_{\bar{i}}-F_{i\bar{j}}H_{j}H_{\bar{i}}.

(8.2)

For a nonnegative real test function $\psi$ , integration in the $\bar{j}$ direction gives the invariant identity

\begin{split}I_{\psi}&:=\int_{X}\psi F_{i\bar{j}}H_{j}H_{\bar{i}}\,d\mu\\ &=-\operatorname{Re}\int_{X}F_{i}\bigl(\psi_{\bar{j}}H_{j}H_{\bar{i}}+\psi H_{j\bar{j}}H_{\bar{i}}+\psi H_{j}H_{\bar{i}\bar{j}}\bigr)\,d\mu.\end{split}

Since $|\nabla F|+|\Delta H|\leq C$ ,

|I_{\psi}|\leq C\int_{X}\bigl(u|\nabla\psi|+\psi\sqrt{u}+\psi\sqrt{u}\,|H_{ij}|\bigr)\,d\mu.

(8.3)

Put $v=1+u$ and take $\psi=v^{p-1}$ , $p\geq 2$ . The two terms requiring absorption satisfy

\begin{split}C(p-1)v^{p-1}|\nabla v|&\leq\tfrac{p-1}{4}v^{p-2}|\nabla v|^{2}+Cp\,v^{p},\\ Cv^{p-1/2}|H_{ij}|&\leq\tfrac{1}{2}v^{p-1}|H_{ij}|^{2}+Cv^{p}.\end{split}

Also $\chi_{i\bar{j}}H_{j}H_{\bar{i}}\leq Cu$ and $|G_{i}H_{\bar{i}}|\leq C\sqrt{u}$ . Testing (8.2), using (8.3), and retaining the positive Hessian term therefore gives

(p-1)\int v^{p-2}|\nabla v|^{2}\leq Cp\int v^{p},\qquad\int|\nabla v^{p/2}|^{2}\leq Cp^{2}\int v^{p}.

The common Sobolev inequality yields $\|v\|_{L^{pq}}\leq(Cp^{2})^{1/p}\|v\|_{L^{p}}$ . Iteration with $p=2q^{j}$ gives $\|v\|_{L^{\infty}}\leq C\|v\|_{L^{2}}$ . Finally

\|v\|_{L^{2}}^{2}\leq\|v\|_{L^{\infty}}\|v\|_{L^{1}},\qquad\|v\|_{L^{1}}=1+\int ud\mu\leq C

by (8.1). This bounds $\|v\|_{L^{\infty}}$ . ∎

We want to show the sharp estimates for the gradient. The idea is similar to the those in section 3, that using Moser iteration to improve weak convergence to strong convergence. The difference is that we need to handle the term involving $dd^{c}F$ in the differential inequality of the quantity.

Lemma 8.8.

Under the hypotheses in the preceding lemma, let $c\geq 0$ be fixed and set $z=(|\nabla H|_{\omega}^{2}-c)_{+}$ . Then

\|z\|_{L^{\infty}}\leq C_{c}\left(\int_{X}z\,d\mu\right)^{\alpha},\qquad\alpha=\frac{q-1}{3q-1}>0.

In particular, $\int_{X}z\,d\mu\to 0$ implies $\|z\|_{L^{\infty}}\to 0$ .

Proof.

The preceding lemma gives $0\leq u\leq C$ and $0\leq z\leq C$ . Test (8.2) with $\psi=z^{p-1}$ , $p\geq 3$ . Use regularized positive parts and pass to the limit if necessary. On $\{z>0\}$ , $\nabla z=\nabla u$ . The first term in (8.3) is bounded by

C(p-1)z^{p-2}|\nabla z|\leq\tfrac{p-1}{4}z^{p-2}|\nabla z|^{2}+Cp\,z^{p-2}.

The last term is absorbed by

Cz^{p-1}|H_{ij}|\leq\tfrac{1}{2}z^{p-1}|H_{ij}|^{2}+Cz^{p-1}.

All other terms are bounded by $Cz^{p-1}$ . Since $z^{p-1}+z^{p}\leq Cz^{p-2}$ with a constant independent of $p$ , we obtain

\int_{X}\bigl(|\nabla z^{p/2}|^{2}+z^{p}\bigr)\,d\mu\leq Cp^{2}\int_{X}z^{p-2}\,d\mu.

(8.4)

Thus

\|z\|_{L^{pq}}\leq(Cp^{2})^{1/p}\|z\|_{L^{p-2}}^{(p-2)/p}.

Let $p_{0}=1$ and $p_{j+1}=q(p_{j}+2)$ . Applying the last inequality with $p=p_{j}+2$ gives

\|z\|_{L^{p_{j+1}}}\leq[C(p_{j}+2)^{2}]^{1/(p_{j}+2)}\|z\|_{L^{p_{j}}}^{p_{j}/(p_{j}+2)}.

Here

p_{j}=q^{j}\left(1+\frac{2q}{q-1}\right)-\frac{2q}{q-1},\qquad\prod_{j=0}^{k-1}\frac{p_{j}}{p_{j}+2}=\frac{q^{k}}{p_{k}}\longrightarrow\alpha.

The product of the constants is bounded because $\sum_{j}\log(C(p_{j}+2)^{2})/(p_{j}+2)<\infty$ . Letting $k\to\infty$ proves the estimate. ∎

Next, we fix $\lambda>0$ , and a smooth function $g\in C^{\infty}(\mathbb{C}P^{N})$ . Let $h_{\varepsilon}$ be the solution to $\Delta_{\omega_{\varepsilon}}h_{\varepsilon}=\lambda h_{\varepsilon}+f^{*}g$ . The trace bound gives $\|f^{*}g\|+|\nabla f^{*}g|_{\omega_{\varepsilon}}\leq C$ . All hypotheses of Lemma 8.7 hold by Sections 2–3; hence $|h_{\varepsilon}|+|\nabla h_{\varepsilon}|_{\omega_{\varepsilon}}\leq C$ . Let $h\in W^{1,2}$ be the unique solution to $\Delta_{\omega_{\rm can}}h=\lambda h+g$ .

Lemma 8.9.

For any $p\in[1,\infty)$ , $\|h_{\varepsilon}-f^{*}h\|_{L^{p}(X,\Omega)}\to 0$

Proof.

Maximum principle and testing with $h_{\varepsilon}$ give $\|h_{\varepsilon}\|_{L^{\infty}}\leq\lambda^{-1}\|g\|_{L^{\infty}}$ and $\int_{X}(|\nabla h_{\varepsilon}|_{\omega_{\varepsilon}}^{2}+\lambda h_{\varepsilon}^{2})d\mu_{\varepsilon}\leq\lambda^{-1}\|g\|_{L^{\infty}}.$ So we can apply Lemma 4.2 to conclude that passing to a subsequence, $h_{\varepsilon}\to f^{*}v$ in every $L^{p}(X,\Omega)$ , and $\|v\|_{L^{\infty}}<\infty.$ For $\varphi\in C^{\infty}_{c}(Y^{\circ})$ , then we have

\int_{X}h_{\varepsilon}(\Delta_{\omega_{\varepsilon}}f^{*}\varphi-\lambda f^{*}\varphi)d\mu_{\varepsilon}=\int_{X}f^{*}(g\varphi)d\mu_{\varepsilon}.

Since $\varphi$ is supported in a compact subset of $Y^{\circ}$ , the metric $\omega_{\varepsilon}\to f^{*}\omega_{\rm can}+\varepsilon\omega_{\rm SRF}$ uniformly there. Therefore,

\int_{X}f^{*}v(f^{*}(\tr_{\omega_{\rm can}}dd^{c}\varphi)-\lambda f^{*}\varphi)e^{f^{*}\varphi_{\rm can}}\Omega=\int_{X}f^{*}(g\varphi)e^{f^{*}\varphi_{\rm can}}\Omega,

which implies

\int_{Y}v(\Delta_{\omega_{\rm can}}\varphi-\lambda\varphi)\omega_{\rm can}^{m}=\int_{Y}g\varphi\omega_{\rm can}^{m}.

Interior elliptic regularity implies $v\in C^{\infty}_{loc}(Y^{\circ})$ and satisfies $\Delta_{\omega_{\rm can}}v=\lambda v+g$ . Since $v\in L^{\infty}$ , by Lemma 8.5, $v$ can be extended to $Y$ and $v\in W^{1,2}$ and satisfies the global weak equation. From the uniqueness of solution, every subsequence has the same limit, and the lemma follows. ∎

Lemma 8.10.

(1)

$\|h_{\varepsilon}-f^{*}h\|_{L^{\infty}}\to 0$ .
(2)

For any $K\Subset Y^{\circ}$ ,

$\int_{f^{-1}(K)}|\nabla(h_{\varepsilon}-f^{*}h)|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}\to 0.$

Proof.

For simplicity, put $e_{\varepsilon}:=h_{\varepsilon}-f^{*}h$ .

(1) Let $d_{FS}$ be the ambient Fubini-Study metric restricted to $Y$ . By lemma 3.3 and the normalization $f^{*}\omega_{FS}=\ell\chi,$

0\leq f^{*}\omega_{FS}\leq\ell(\tr_{\omega_{\varepsilon}}\chi)\omega_{\varepsilon}\leq\ell C\omega_{\varepsilon}.

Therefore, for any cuver $\gamma$ in $X$ , aftering taking infumum,

d_{FS}(f(x),f(x^{\prime}))\leq\inf_{\gamma}\operatorname{Length}_{FS}(f\circ\gamma)\leq\inf_{\gamma}\sqrt{\ell C}\operatorname{Length}_{\omega_{\varepsilon}}(f\circ\gamma)=\sqrt{\ell C}d_{\omega_{\varepsilon}}(x,x^{\prime}).

Apply Lemma 8.7, we get $|\nabla h_{\varepsilon}|_{\omega_{\varepsilon}}\leq C$ for a uniform constant $C$ .

Let $r=d_{\omega_{\varepsilon}}(x,x^{\prime})$ , then

|e_{\varepsilon}(x)-e_{\varepsilon}(x^{\prime})|\leq Cr+\sup_{\begin{subarray}{c}y,y^{\prime}\in Y\\ d_{FS}(y,y^{\prime})\leq\sqrt{\ell C}\,r\end{subarray}}|h(y)-h(y^{\prime})|

Since $(\overline{Y^{\circ}},d_{\rm can})$ is homeomorphic to $(Y,d_{FS})$ , $h$ is continuous on $Y$ . Hence, by compactness of $Y$ , for each $\delta>0$ , there exits $r_{\delta}\in(0,1)$ , which is independent of $\varepsilon$ , such that

|e_{\varepsilon}(x)|\geq\delta\quad\Longrightarrow\quad|e_{\varepsilon}|\geq\delta/2\text{ on }B_{\omega_{\varepsilon}}(x,r_{\delta}).

By the volume noncollapsing estimate in Theorem 2.2,

\|e_{\varepsilon}\|_{L^{2}(\mu_{\varepsilon})}^{2}\geq\frac{\delta^{2}}{4}\mu_{\varepsilon}(B_{\omega_{\varepsilon}}(x,r_{\delta}))\geq\frac{\delta^{2}r_{\delta}^{a_{\rm vol}}}{4C}>0.

However, Lemma 8.10 gives $e_{\varepsilon}\to 0$ in $L^{2}(\Omega)$ , and also in $L^{2}(\mu_{\varepsilon})$ since $|F_{\varepsilon}|\leq C$ .

Hence $\|e_{\varepsilon}\|_{L^{\infty}}\to 0.$

(2) Choose $\xi\in C^{\infty}_{c}(Y^{\circ})$ with $0\leq\xi\leq 1$ and $\xi=1$ near $K$ ; write $\xi$ also for $f^{*}\xi$ . On $\operatorname{supp}\xi$ , one has

\Delta_{\omega_{\varepsilon}}e_{\varepsilon}=\lambda e_{\varepsilon}+f^{*}\Delta_{\omega_{\rm can}}h-\Delta_{\omega_{\varepsilon}}h.

$\tr_{\omega_{\varepsilon}}\chi\leq C$ implies $|\nabla\xi|_{\omega_{\varepsilon}}\leq C,|\Delta_{\omega_{\varepsilon}}f^{*}h|\leq C$ , $|\Delta_{\omega_{\varepsilon}}e_{\varepsilon}|\leq C$ . Therefore

	$\displaystyle\int_{X}\xi^{2}\|\nabla e_{\varepsilon}\|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}=$	$\displaystyle-\int_{X}\xi^{2}e_{\varepsilon}\Delta_{\omega_{\varepsilon}}e_{\varepsilon}d\mu_{\varepsilon}-2\int_{X}\xi e_{\varepsilon}\langle\nabla\xi,\nabla e_{\varepsilon}\rangle_{\omega_{\varepsilon}}d\mu_{\varepsilon}$
	$\displaystyle\leq$	$\displaystyle C\\|e_{\varepsilon}\\|_{L^{\infty}}+\frac{1}{2}\int_{X}\xi^{2}\|\nabla e_{\varepsilon}\|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}+2\int_{X}e_{\varepsilon}\|\nabla\xi\|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}.$

As a reusult,

\int_{f^{-1}(K)}|\nabla e_{\varepsilon}|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}\leq C(\|e_{\varepsilon}\|_{L^{\infty}}+\|e_{\varepsilon}\|_{L^{\infty}}^{2})\to 0.

∎

Now we can show the sharp gradient estimate.

Lemma 8.11.

$\limsup_{\varepsilon\to 0}\Lip_{d_{\omega_{\varepsilon}}}h_{\varepsilon}\leq\Lip_{d_{\rm can}}(h)$ .

Proof.

Let $L_{h}=\Lip_{d_{\rm can}}(h)$ and fix $K\Subset Y^{\circ}$ . Note that our notation is $|\nabla h|_{\omega}^{2}=\frac{1}{2}|\nabla h|_{g_{\omega}}^{2}$ where $g_{\omega}$ is the Riemannian metric of $\omega$ . From the sharp Schwarz type estimate,

|\nabla f^{*}h|_{\omega_{\varepsilon}}^{2}\leq(1+o_{K}(1))f^{*}|\nabla h|_{\omega_{\rm can}}^{2}\leq L_{h}^{2}/2+o_{K}(1)\quad\text{ on }f^{-1}(K).

Note that

(|\nabla h_{\varepsilon}|^{2}-L_{h}^{2}/2)_{+}\leq|\nabla e_{\varepsilon}|_{\omega_{\varepsilon}}^{2}+2|\nabla e_{\varepsilon}|_{\omega_{\varepsilon}}|\nabla f^{*}h|_{\omega_{\varepsilon}}+(|\nabla f^{*}h|_{\omega_{\varepsilon}}^{2}-L_{h}^{2}/2)_{+}.

By Cauchy-Schwarz inequality and previous lemma,

\int_{f^{-1}(K)}(|\nabla h_{\varepsilon}|^{2}-L_{h}^{2}/2)_{+}d\mu_{\varepsilon}\leq\int_{f^{-1}(K)}|\nabla e_{\varepsilon}|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}+C_{K}\left(\int_{f^{-1}(K)}|\nabla e_{\varepsilon}|^{2}d\mu_{\varepsilon}\right)^{1/2}+o_{K}(1)\longrightarrow 0

Since $\|e_{\varepsilon}\|_{L^{\infty}}\to 0$ , and $|\nabla h_{\varepsilon}|_{\omega_{\varepsilon}}$ is uniformly bounded,

\limsup_{\varepsilon\to 0}\int_{X}(|\nabla h_{\varepsilon}|^{2}-L_{h}^{2}/2)_{+}d\mu_{\varepsilon}\leq C\Omega\bigl(X\setminus f^{-1}(K)\bigr).

Let $K\uparrow Y^{\circ}$ . Since $\Omega(f^{-1}(D))=0$ ,

\int_{X}(|\nabla h_{\varepsilon}|^{2}-L_{h}^{2}/2)_{+}d\mu_{\varepsilon}\longrightarrow 0.

Thus with Lemma 8.8, we get the desired estimate. ∎

Proof of Lemma 8.1.

Lemma 8.6 provides $g\in C^{\infty}(\mathbb{C}P^{N})$ , such that the solution $\Delta_{\omega_{\rm can}}h=\lambda h+g$ satisfies $\|h-v\|_{L^{\infty}}<\delta$ , and $\Lip(h)<1+\delta$ . Set $v_{\varepsilon}=h_{\varepsilon}$ , where $h_{\varepsilon}$ is the solution to $\Delta_{\omega_{\varepsilon}}h_{\varepsilon}=\lambda h_{\varepsilon}+f^{*}g$ . The preceding lemma shows the desired approximation property. ∎

Completion of the proof of Theorem 1.1.

The case $m=0$ was proved in Section 1. For $m>0$ , Lemma 6.7 proves the local metric assertion. Lemma 8.1 gives Proposition 7.2; Proposition 7.3 then proves Gromov–Hausdorff convergence. ∎

9. A new proof of Gromov-Hausdorff convergence for Kähler–Ricci flow

Based on our previous discussion on the Lipschitz approximation, we can give an independent new proof of the theorem of Lee-Tosatti-Zhang [14] on the Gromov-Haudorff convergence of the long-time Kähler–Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle.

Let ${\tilde{\omega}}(t)$ solve the normalized Kähler–Ricci flow

\partial_{t}{\tilde{\omega}}(t)=-\Ric({\tilde{\omega}}(t))-{\tilde{\omega}}(t),\qquad{\tilde{\omega}}(0)={\omega}.

(9.1)

We retain $f:X\to Y$ , $m=\dim_{\mathbb{C}}Y$ , $d=n-m$ , $\chi$ , and $\Omega$ from Section 2. In particular, $\Ric(\Omega)=-\chi$ and $\int_{X}\Omega=1$ . The cohomology class is

[{\tilde{\omega}}(t)]=(1-e^{-t})2\pi c_{1}(K_{X})+e^{-t}[{\omega}].

Write

\begin{gathered}\chi(t)=(1-e^{-t})\chi+e^{-t}{\omega},\qquad{\tilde{\omega}}(t)=\chi(t)+dd^{c}{\tilde{\varphi}}(t),\\ {\dot{\tilde{\varphi}}}=\log\frac{e^{dt}{\tilde{\omega}}(t)^{n}}{\binom{n}{m}V_{F}V_{Y}\Omega}-{\tilde{\varphi}},\qquad{\tilde{\varphi}}(0)=0,\\ V(t)=\int_{X}{\tilde{\omega}}(t)^{n}=\int_{X}\chi(t)^{n},\qquad d\mu_{t}=\frac{{\tilde{\omega}}(t)^{n}}{V(t)}=e^{F(t)}\Omega.\end{gathered}

(9.2)

Thus

F(t)={\dot{\tilde{\varphi}}}(t)+{\tilde{\varphi}}(t)-\log(e^{dt}V(t)),\qquad\Ric({\tilde{\omega}}(t))=-\chi-dd^{c}F(t).

(9.3)

We summarize some facts along the NKRF in the next lemma.

Lemma 9.1.

There exist $q>1$ and constants independent of $t\geq 0$ such that:

(1)

$C^{-1}\leq e^{F(t)}\leq C,\qquad\tr_{{\tilde{\omega}}(t)}\chi+|\nabla F(t)|_{{\tilde{\omega}}(t)}^{2}\leq C.$

(2)

For every $v\in W^{1,2}(X,{\tilde{\omega}}(t))$ ,

\left(\int_{X}|v|^{2q}\,d\mu_{t}\right)^{1/q}\leq C\int_{X}\bigl(|\nabla v|_{{\tilde{\omega}}(t)}^{2}+v^{2}\bigr)\,d\mu_{t}.

(9.4)

(3)

For $x\in X$ and $0<r\leq 1$ ,

$\mu_{t}(B_{{\tilde{\omega}}(t)}(x,r))\geq C^{-1}r^{2q/(q-1)},\qquad\diam(X,{\tilde{\omega}}(t))\leq C.$ (9.5)

Proof.

(1) The estimates of Song–Tian [19, Lemma 2.1 and Propositions 2.1, 2.2, 3.1] give

|{\tilde{\varphi}}|+|{\dot{\tilde{\varphi}}}|\leq C,\qquad\tr_{{\tilde{\omega}}(t)}\chi\leq C,\qquad|\nabla({\dot{\tilde{\varphi}}}+{\tilde{\varphi}})|_{{\tilde{\omega}}(t)}^{2}\leq C.

The last inequality is [19, equation (3.19)], obtained by the maximum principle. The cohomological volume satisfies

V(t)=\sum_{k=0}^{m}\binom{n}{k}(1-e^{-t})^{k}e^{-(n-k)t}\int_{X}\chi^{k}\wedge{\omega}^{n-k},\qquad C^{-1}\leq e^{dt}V(t)\leq C.

(2) Fix a smooth reference Kähler form with volume one. Part (1) implies that the density of $\mu_{t}$ relative to its volume form is bounded above and below by fixed positive constants. Consequently all its Nash–Yau entropies are bounded, and the lower density function in [10, Theorem 2.1] can be chosen to be a positive constant. Moreover,

[{\tilde{\omega}}(t)]\cdot[{\omega}]^{n-1}=(1-e^{-t})A\cdot B^{n-1}+e^{-t}B^{n}\leq C.

The cited theorem gives a uniform mean-zero Sobolev inequality for some $q>1$ . Since $\mu_{t}(X)=1$ and $|\int_{X}v\,d\mu_{t}|^{2}\leq\int_{X}v^{2}\,d\mu_{t}$ , it implies (9.4).

(3) It is a standard consequence of the uniform Sobolev inequality from part (2). ∎

Fix $\lambda>0$ and $g\in C^{\infty}(\mathbb{C}P^{N},\mathbb{R})$ . Define $h_{t},h$ to be the unique solution to

\Delta_{{\tilde{\omega}}(t)}h_{t}=\lambda h_{t}+f^{*}g,\qquad\Delta_{\omega_{\rm can}}h=\lambda h+g.

(9.6)

The trace bound $\tr_{{\tilde{\omega}}(t)}\chi\leq C$ gives

\|f^{*}g\|_{L^{\infty}}+\|\nabla^{{\tilde{\omega}}(t)}(f^{*}g)\|_{L^{\infty}}\leq C.

By Lemmas 9.1 and 8.7(note that all properties we used for $\omega_{\varepsilon}$ in Lemma 8.7 are $\tr_{\omega_{\varepsilon}}\chi\leq C$ and $|\nabla F_{\varepsilon}|_{\omega_{\varepsilon}}\leq C$ ), we have

\|h_{t}\|_{L^{\infty}}+\|\nabla^{{\tilde{\omega}}(t)}h_{t}\|_{L^{\infty}}\leq C.

(9.7)

The constants may depend on the fixed $\lambda$ and $g$ .

Lemma 9.2.

As $t\to\infty$ ,

h_{t}\longrightarrow f^{*}h\quad\text{locally uniformly on }X^{\circ}\quad\text{and in }L^{p}(X,\Omega),\qquad 1\leq p<\infty.

Proof.

Take any sequence $t_{j}\to\infty$ . On a compact subset of $X^{\circ}$ , the bound ${\tilde{\omega}}(t_{j})\leq C_{K}{\omega}$ and (9.7) give equicontinuity in the fixed smooth metric. Arzelà–Ascoli and a compact exhaustion give a subsequence converging locally uniformly to a bounded continuous function. On each fiber over $K\Subset Y^{\circ}$ ,

\operatorname{osc}_{X_{y}}h_{t_{j}}\leq C\diam(X_{y},{\tilde{\omega}}(t_{j})|_{X_{y}})\to 0.

Thus the limit is $f^{*}h_{\infty}$ , for a bounded continuous function $h_{\infty}$ on $Y^{\circ}$ .

For $\psi\in C^{\infty}_{c}(Y^{\circ})$ , integration by parts gives

\int_{X}h_{t_{j}}\bigl(\lambda f^{*}\psi-\Delta_{{\tilde{\omega}}(t_{j})}f^{*}\psi\bigr)d\mu_{t_{j}}=\int_{X}f^{*}(g\psi)d\mu_{t_{j}}.

All integrands are supported in a fixed compact subset of $X^{\circ}$ . The local uniform convergence implyies

\int_{Y}h_{\infty}(\lambda\psi-\Delta^{\mathbb{R}}_{\omega_{\rm can}}\psi)d\nu_{\rm can}=\int_{Y}g\psi\,d\nu_{\rm can}.

Interior elliptic regularity gives $h_{\infty}\in C^{\infty}(Y^{\circ})$ . By Lemma 8.5, $h_{\infty}=h$ . Every subsequential limit is therefore $f^{*}h$ , which proves local uniform convergence for the whole family. Since $h_{t},h$ are bounded and $\Omega(X\setminus X^{\circ})=0$ , dominated convergence gives the finite- $L^{p}$ limits. ∎

Lemma 9.3.

Put $e_{t}=h_{t}-f^{*}h$ . Then:

(1)

$\|e_{t}\|_{\infty}\to 0$ .
(2)

For every $K\Subset Y^{\circ}$ ,

$\int_{f^{-1}(K)}|\nabla e_{t}|_{{\tilde{\omega}}(t)}^{2}\,d\mu_{t}\longrightarrow 0.$

Proof.

The proof is identical to the proof of 8.10. ∎

Lemma 9.4.

\limsup_{t\to\infty}\Lip_{{\tilde{\omega}}(t)}(h_{t})\leq\Lip_{d_{\rm can}}(h).

Proof.

Just identical to the proof of Lemma 8.6. ∎

The lower distance estimate replaces the reduced-length argument of [14, Sections 4.2–5]. The inputs used here are the Song–Tian flow estimates, the static Sobolev inequality, regular-locus convergence, and the analysis on the canonical model; neither reduced length nor reduced-volume monotonicity enters this proof.

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	$\displaystyle\|\nabla_{V}u_{\varepsilon}\|_{{\omega_{y}}}^{2}$	$\displaystyle\leq\|\nabla h_{\varepsilon}\|_{\omega_{\varepsilon}}^{2}\tr_{{\omega_{y}}}(\omega_{\varepsilon}\|_{X_{y}}),$
	$\displaystyle\int_{f^{-1}(K)}\|\nabla_{V}u_{\varepsilon}\|_{{\omega_{y}}}\Omega$	$\displaystyle\leq C_{K}\left(\int_{X}\|\nabla u_{\varepsilon}\|_{\omega_{\varepsilon}}^{2}d\mu_{\varepsilon}\right)^{1/2}\varepsilon^{1/2}\leq C_{K}\sqrt{\varepsilon}.$

	$\displaystyle\left\|\int_{X}(\varphi_{\varepsilon}-f^{*}u)\,d\mu_{\varepsilon}\right\|$	$\displaystyle\leq C\\|\varphi_{\varepsilon}-f^{*}u\\|_{L^{1}(\Omega)}\longrightarrow 0,$
	$\displaystyle\left\|\int_{X}f^{}(u-\varphi)\bigl(e^{F_{\varepsilon}}-e^{f^{}F}\bigr)\Omega\right\|$	$\displaystyle\leq\\|u-\varphi\\|_{\infty}\\|e^{F_{\varepsilon}}-e^{f^{*}F}\\|_{L^{1}(\Omega)}\longrightarrow 0,$

	$\displaystyle 2\xi v^{p-1}\|\nabla\xi\|\,\|\nabla v\|$	$\displaystyle\leq\frac{p-1}{4}\xi^{2}v^{p-2}\|\nabla v\|^{2}+\frac{C}{p-1}v^{p}\|\nabla\xi\|^{2},$
	$\displaystyle C\xi v^{p-1}\|\nabla\xi\|$	$\displaystyle\leq Cv^{p}\|\nabla\xi\|^{2}+C\xi^{2}v^{p-2},$
	$\displaystyle C(p-1)\xi^{2}v^{p-2}\|\nabla v\|$	$\displaystyle\leq\frac{p-1}{4}\xi^{2}v^{p-2}\|\nabla v\|^{2}+C(p-1)\xi^{2}v^{p-2}.$

	$\displaystyle\\|(1-\xi_{j})u\\|_{L^{2}}$	$\displaystyle\leq\\|u\\|_{\infty}\\|1-\xi_{j}\\|_{L^{2}}\longrightarrow 0,$
	$\displaystyle\\|d((1-\xi_{j})u)\\|_{L^{2}}$	$\displaystyle\leq\\|(1-\xi_{j})du\\|_{L^{2}}+\\|u\\|_{L^{\infty}}\\|d\xi_{j}\\|_{L^{2}}\longrightarrow 0.$

	$\displaystyle\\|h-v\\|_{L^{\infty}}$	$\displaystyle\leq\\|h+\lambda h_{v}\\|_{L^{\infty}}+\\|\lambda h_{v}+v\\|_{L^{\infty}}\leq\lambda C_{\lambda}\\|g_{0}-v\\|_{L^{\infty}}+\frac{\delta}{3}\leq\frac{3\delta}{3},$
	$\displaystyle\Lip(h)$	$\displaystyle\leq\Lip(\lambda h_{v})+\lambda C_{\lambda}\\|g_{0}-v\\|_{\infty}<1+2\delta/3.\qed$