Proof. In order to lighten the notation, we will assume that $m=1$ . The proof of the general case is essentially identical. One direction of the lemma is obvious, so we need to show that boundedness of the density implies smoothness of the germ.

We can assume that we have an embedding $X\hookrightarrow \mathbb C^N$ such that for all n-tuple $I\subset \{1, \ldots , N\}$ , the linear projections $p_I:X\to \mathbb C^n$ are finite maps. It is not difficult to check that there exists a smooth function G on X such that $\omega ^n=e^G\omega _{\mathbb C^N}^n|_X$ .

Given any I, there exists a holomorphic function $\zeta _I$ on $X_{\mathrm {reg}}$ such that $dz_I|_{X_{\mathrm {reg}}} = \zeta _I \sigma $ . By normality of X, $\zeta _I$ extends to a holomorphic function on X. This shows that $f=(e^G\sum _I |\zeta _I|^2)^{-1}$ up to some positive constant. In particular, $-\log f$ is quasi-psh.

Next, if f is bounded on X, then there exists I such that $\zeta _I(x)\neq 0$ . In particular, $\zeta _I$ is nonvanishing on the germ $(X,x)$ , and $dz_I|_{X_{\mathrm {reg}}}$ is a trivialization of $K_{X_{\mathrm {reg}}}$ . In other words, the map $p_I: X\to \mathbb C^n$ is étale on $X_{\mathrm {reg}}$ , hence everywhere by purity of the branch locus. This shows that $(X,x)$ is smooth.

As for the last claim, assume that $(X,x)$ is singular. Then $\psi :=\log (\sum _I |\zeta _I|^2)$ is psh with analytic singularities and satisfies $\psi (x)=-\infty $ and by definition, we have $\mathrm {Ric} \, \omega = -dd^c(\psi +G)$ . If the Ricci curvature of $\omega $ were bounded from below, there would exist $A>0$ such that $dd^c(A\|z\|^2-\psi )\geqslant 0$ . In particular, $A\|z\|^2-\psi $ would be psh on $X_{\mathrm {reg}}$ , hence bounded above near x, which is absurd.

Proof. Pick $x\in X$ and choose a neighborhood U of x bearing a trivialization $\sigma $ of $mK_{U_{\mathrm {reg}}}$ and write $\mu _U:=i^{n^2} (\sigma \wedge \bar \sigma )^{\frac 1m}$ . Next, one can ensure that there exists $\varphi \in \mathrm {PSH}(U)\cap L^{\infty }(U)$ such that $\omega _{\mathrm {KE}}|_{U}=dd^c \varphi $ . There exist $\lambda \in {\mathbb {R}}$ and a pluriharmonic (hence smooth) function h on U such that

$$\begin{align*}\omega_{\mathrm{KE}}^n=e^{\lambda \varphi+h} \mu_U\quad \mbox{ on } U.\end{align*}$$

In particular, the domination $\omega _{\mathrm {KE}} \leqslant C\omega $ implies that the density of $\mu _U$ with respect to $\omega ^n$ is bounded. The conclusion now follows from Lemma 1.6.

Proof. The claim is an immediate application of Chern–Levine–Nirenberg inequality that ensures that T puts no mass on pluripolar sets. Alternatively, fix an open neighborhood U of a point $x\in Z$ such that $T|_U=dd^c\varphi $ . One can choose $\varphi $ such that $\varphi |_{U\setminus Z}$ is psh and locally bounded near Z, hence it extends to a psh function $\psi $ on U. Since $\varphi =\psi $ almost everywhere on U, we have $T|_U=dd^c\psi \geqslant 0$ .

Proof. We consider the smoothing $\pi :(\mathcal X,\mathcal D)\to \mathbb D$ . Up to shrinking $\mathbb D$ slightly and adding $\pi ^*dd^c |t|^2$ to $\omega _{\mathcal X}$ , one can assume that the later form is strictly positive. Since $\omega _{\mathcal X}$ is the restriction of a smooth positive $(1,1)$ -form under local embedding $\mathcal X \underset {\mathrm {loc}}{\hookrightarrow } \mathbb C^N$ , one can assume that the bisectional curvature of $\omega _{\mathcal X}|_{\mathcal X^{\mathrm {reg}}}$ is bounded above by a given constant $C_1$ , maybe up to shrinking $\mathbb D$ just a little more. This is because the bisectional curvature decreases when passing to holomorphic submanifolds. By the same argument, we see the bisectional curvature of $\omega _t=\omega _{\mathcal X}|_{X_t}$ is bounded above by $C_1$ , for any $t\in \mathbb D$ .

For $t\in \mathbb D$ , we consider $\widetilde {\omega }_{t}=\omega _t+dd^c \varphi _t$ the unique Kähler–Einstein metric of the pair $(X_t, D_t)$ . For $t\neq 0$ , existence and uniqueness of $\widetilde {\omega }_{t}$ is due to [Reference KołodziejKoł98] when $\lambda \geqslant 0$ and to [Reference Spotti, Sun and YaoSSY16, Reference Li, Wang and XuLWX19], cf also [Reference Pan and TrusianiPT23] when $\lambda <0$ . In the latter case, we need t to be small enough. The Kähler–Einstein metric solves

(2.4) $$ \begin{align} (\omega_t+dd^c\varphi_t)^n=e^{\lambda \varphi_t} f_t \omega_t^n \end{align} $$

for some nonnegative function $f_t$ on $X_t$ satisfying $\int _{X_t} f_t^p \omega _t^n\leqslant C(p,\omega )$ for some $p>1$ and some constant $C(p,\omega )$ independent of $t\in \mathbb D$ . This is proved in [Reference Di Nezza, Guedj and GuenanciaDNGG23, Lemma 4.4] assuming additionally $X_0$ has canonical singularities, but the klt case can be proved with minimal changes, cf also Equation (3.6) in the next section.

If $\lambda =0$ , we normalize $\varphi _t$ by $\sup _{X_t} \varphi _t =0$ . We claim that up to shrinking $\mathbb D$ , there exists a constant $C_2>0$ such that

(2.5) $$ \begin{align} \|\varphi_t\|_{L^{\infty}(X_t)} \leqslant C_2 \end{align} $$

for any $t\in \mathbb D$ .

If $\lambda>0$ , this is a consequence of [Reference Di Nezza, Guedj and GuenanciaDNGG23, Theorem E]. If $\lambda =0$ , this is a consequence of the proof of [Reference Di Nezza, Guedj and GuenanciaDNGG23, Theorem 6.1]. Indeed, in loc. cit. the fibers are assumed to be canonical but this is a mostly cosmetic assumption since going from canonical to klt singularities leaves the proof resting on [Reference Di Nezza, Guedj and GuenanciaDNGG23, Lemma 4.4] unchanged, as explained above. Finally, if $\lambda <0$ , the estimate (2.5) is proved in [Reference Spotti, Sun and YaoSSY16, Proposition 2.23], [Reference Li, Wang and XuLWX19, Theorem 1.2 (iii)] when $\mathcal D$ is plurianticanonical and in [Reference Pan and TrusianiPT23, Theorem 5.10] in general.

Set $E:=\mathrm {Supp}(\mathcal D)$ . We choose a smooth Hermitian metric h on $\mathcal O_{\mathcal X}(E)$ , a section s of that line bundle cutting out E, and we introduce the function $\psi :=\log |s|^2_{h}$ . It satisfies $dd^c \psi =-i\Theta _h( E)$ on $\mathcal X \setminus E$ , hence there exists a constant $C_3>0$ such that

(2.6) $$ \begin{align} \psi \leqslant C_3 \quad \mbox{and} \quad dd^c \psi \geqslant -C_3 \omega \quad \mbox{on } \mathcal X \setminus E. \end{align} $$

We introduce a number $\varepsilon>0$ (meant to go to zero), and we assume wlog that $\varepsilon C_3 \leqslant 1$ . We also define $v_t:=\log \mathrm {tr}_{\widetilde {\omega }_{t}} \omega _t$ , which is smooth on $X_t \setminus E$ and globally bounded (for $t\neq 0$ ). We want to quantify that bound.

On $X_t\setminus E$ , we have $\mathrm {Ric} \, \widetilde {\omega }_{t} \geqslant -\widetilde {\omega }_{t}$ and $\mathrm {Bisec}(X_t,\omega _t) \leqslant C_1$ . By Chern–Lu inequality, that is, Proposition 2.6, we get

$$\begin{align*}\Delta_{\widetilde{\omega}_{t}} v_t \geqslant -1 -2C_1 e^{v_t}\end{align*}$$

Using the identity $\widetilde {\omega }_{t}=\omega _t+dd^c \varphi _t$ and the inequality (2.6), we get

$$\begin{align*}\Delta_{\widetilde{\omega}_{t}} (v_t-A \varphi_t+\varepsilon \psi) \geqslant e^{v_t}-C_4,\end{align*}$$

where $A=2(C_1+1)$ and $C_4=An+1$ . The maximum of the term inside the Laplacian is attained on $X_t\setminus E$ , and an easy application of the maximum principle shows

$$\begin{align*}v_t(x) \leqslant C_5-\varepsilon \psi(x)\end{align*}$$

for any $x\in X_t \setminus E$ and $\varepsilon>0$ , and where $C_5=\log C_4+2AC_2+1$ . Passing to the limit when $\varepsilon \to 0$ , we find

(2.7) $$ \begin{align} \widetilde{\omega}_{t} \geqslant C_{5}^{-1} \omega_t \quad \mbox{on } X_t\setminus E, \,\, \mbox{for any} \, t\in \mathbb D^*. \end{align} $$

Next, we choose a continuous family of smooth maps $F_t:X_{\mathrm {reg}} \to X_t$ inducing a diffeomorphism onto their image and such that $F_0=\mathrm {Id}_{X_{\mathrm {reg}}}$ . We claim that $F_t^*\widetilde {\omega }_{t}$ converges locally smoothly to $\omega _{\mathrm {KE}}$ on $X_{\mathrm {reg}}\setminus E$ when $t\to 0$ . Thanks to Equation (2.7), this would imply that

$$\begin{align*}\omega_{\mathrm{KE}} \geqslant C_5^{-1} \omega \quad \mbox{on }X_{\mathrm{reg}}\setminus E,\end{align*}$$

hence everywhere on X by Lemma 2.5.

Set $X^\circ :=X_{\mathrm {reg}} \setminus E$ . In order to show the convergence, we claim successively:

• ○ The family of functions $F_t^*\varphi _t$ is precompact in the $\mathcal C^2_{\mathrm {loc}}(X^\circ )$ -topology.

• ○ Each cluster value $\omega _{\infty }=\omega +dd^c \varphi _{\infty }$ of $F_t^*\widetilde {\omega }_{t}$ solves the Monge–Ampère equation

  $$\begin{align*}(\omega+dd^c \varphi_{\infty})^n=e^{\lambda \varphi_{\infty} }f_0\omega^n \quad \mbox{on } X^\circ.\end{align*}$$

• ○ Each cluster value $\varphi _{\infty }$ is globally bounded on $X^\circ $ , hence its unique $\omega $ -psh extension to X solves the equation above on X.

By uniqueness of the Kähler–Einstein metric, it would then follow that $\omega _{\infty }=\omega _{\mathrm {KE}}$ , concluding the proof of the convergence. Let us briefly justify each of the items above.

The first point follows from the local Laplacian estimate on $X^\circ $ obtained from Equation (2.7). Indeed, once the Laplacian estimates are obtained, one can invoke Evans–Krylov and Schauder estimates since $F_t^*\omega _t$ (resp. $F_t^*J_t, F_t^*f_t$ ) converges locally smoothly on $X^\circ $ to $\omega $ (resp $J, f_0$ ) when $t\to 0$ .

The second item is an immediate consequence of the first one. As for the last one, it is a consequence of Equation (2.5).

Proof. Let H be an arbitrary smooth extension of h to ${\mathcal X}$ . For A large enough, the function $v:=A\rho -H$ is a smooth psh function near $\overline {{\mathcal X}}$ such that $v_t=-h_t$ on $\partial X_t$ . Thus, $u_t+v_t$ is psh in $X_t$ with zero boundary values. It follow from the maximum principle that $u_t+v_t \leqslant 0$ in $X_t$ . Using the subsolution constructed in the previous subsection, we obtain a two-sided bound

$$ \begin{align*}\underline {u}_t \leqslant u_t \leqslant -v_t, \end{align*} $$

with $\underline {u}_t=u_t=-v_t=h_t$ on $\partial X_t$ . The desired uniform ${\mathcal C}^1$ -bound on $\partial X_t$ follows.

Proof. It follows from Proposition 3.3 and Lemma 3.7 that the assumptions of Theorem 6.3 in the preprint version are met and that we have uniform upper bounds on $\|u_t\|_{{\mathcal C}^1(\overline {V_t})}$ , $\|v_t\|_{{\mathcal C}^2(\overline {V_t})}$ , $\|f_t\|_{{\mathcal C}^1(\overline {V_t})}$ , $\|h\|_{{\mathcal C}^4(\partial X_t)}$ , $\varepsilon ^{-1},\delta ^{-1}$ . The proposition follows.

End of the proof of Theorem 3.4

Pick a Kähler metric $\widehat \omega $ on $\mathcal X$ , and set $\widehat \omega _t:=\widehat \omega |_{X_t}$ . Since $(dd^c u_t)^n$ is uniformly comparable to $\widehat \omega _t^n$ in a small neighborhood of $\partial X_t$ , the uniform bound $\|\Delta u_t\|_{L^{\infty }(\partial X_t)} \leqslant C_2$ actually yields a uniform constant $c_2>0$ such that

(3.14) $$ \begin{align} c_2^{-1} \widehat \omega_t \leqslant dd^c u_t \leqslant c_2\widehat \omega_t \; \; \text{ on } \partial X_t. \end{align} $$

Indeed, let $\sigma :=\inf _{t} \inf _{p\in \partial X_t} \liminf _{z\to p, z\in \partial X_t} \frac {\theta _t^n(z)}{\widehat \omega _t^n(z)}$ ; we have $\sigma>0$ since $\pi $ is smooth along $\partial {\mathcal {X}}$ and $\|u_t\|_{L^{\infty }(X_t)}$ is uniformly bounded below by Theorem 3.6. Given $p\in \partial X_t$ , we have $\limsup _{z\to p, z\in X_t} \mathrm {tr}_{\widehat \omega _t}\theta _t(z) \leqslant C$ by our boundary Laplacian estimate, hence

$$\begin{align*}\limsup_{z\to p, z\in X_t} \mathrm{tr}_{\theta_t}\widehat \omega_t(z)\leqslant \sigma^{-1} C^{n-1},\end{align*}$$

and Equation (3.14) follows.

Arguing as in the proof of Theorem 2.8, we consider $v_t=\log \mathrm {tr}_{{\theta _t}}(\widehat \omega _t)$ , where $\theta _t=dd^c u_t$ and deduce from the Chern–Lu formula that

(3.15) $$ \begin{align} \Delta_{\theta_t} (v_t-A u_t) \geqslant e^{v_t}-C_3, \end{align} $$

for uniform constants $A,C_3>0$ . We infer that $v_t \leqslant C_4$ is uniformly bounded from above. Indeed, either the maximum of the function $v_t-A u_t$ is reached in $X_t$ , and the bound follows from Equation (3.15) and the uniform bound on $u_t$ or the maximum of $v_t-A u_t$ is reached on $\partial X_t$ and we conclude from Equation (3.14).

Therefore, $dd^c u_t \geqslant C_4^{-1}\widehat \omega _t$ in $X_t$ for all $t \neq 0$ . Similarly to what we have done at the end of the proof of Theorem 2.8, we conclude by letting $t \rightarrow 0$ that $dd^c u_0 \geqslant C_4^{-1} \widehat \omega _0$ , hence $\omega _{\mathrm {KE}} =dd^c u_0$ is a Kähler current. The proof of Theorem 3.4 is complete.

Proof. We work near an isolated singular point a. We let B denote a small strictly pseudoconvex neighborhood of a in X, isomorphic to the trace of a ball in some local embedding in ${\mathbb {C}}^N$ , and let $\rho $ denote a local smooth potential for $\omega =dd^c \rho $ in B.

Recall from [Reference Eyssidieux, Guedj and ZeriahiEGZ09, Reference PăunPău08] that the Kähler–Einstein potential $\varphi _{\mathrm {KE}}$ is smooth in $B \setminus \{a\}$ . Define $\lambda =1$ if $K_X$ is ample, and $\lambda =0$ if $K_X \equiv 0$ , and set $F=\lambda \rho $ . The local theory recalled in Section 1.2 shows that $\psi =\rho +\varphi _{\mathrm {KE}}$ is the unique solution of the Dirichlet problem

$$ \begin{align*}(dd^c w)^n=e^{\lambda w+F} \mu_{(X,a),h} \,\, \text{ in } B, \; \; \text{ with } \; \; w_{|\partial B}=\psi_{|\partial B}. \end{align*} $$

We assume that $(X,a)$ is a smoothable singularity in the sense of Definition 3.2, and we let $\pi : {\mathcal X} \rightarrow \mathbb D$ denote a smoothing so that $B=\pi ^{-1}(0)$ and $X_t=\pi ^{-1}(t)$ is smooth for all $t \neq 0$ . We let h denote a smooth extension of $(\rho +\varphi _{\mathrm {KE}})_{| \partial B}$ to $\partial {\mathcal X}$ and still denote by F a smooth extension of F to ${\mathcal X}$ . It follows from Proposition 3.3 that there exists a unique function $u_t \in \mathrm {PSH}(X_t) \cap {\mathcal C}^0(\overline {X_t})$ such that

(4.1) $$ \begin{align} \begin{cases} (dd^c u_t)^n = e^{\lambda u_t+F_t} \mu_t & \mbox{on } X_t\\ u_t|_{\partial X_t}=h_t \end{cases}. \end{align} $$

together with a uniform bound $\|u_t\|_{L^{\infty }(X_t)} \leqslant C_0$ . We can thus apply Theorem 3.4 and conclude that $dd^c u_0=dd^c \psi =\omega _{KE}$ is a Kähler current near a.

Proof of Theorem 4.5

By assumption, one can find an integer $m>0$ and sections $s_i\in H^0(X,\mathcal O_X(m\Delta _i))$ such that $\mathrm {div}(s_i)=m\Delta _i$ . We pick Hermitian metrics $h_i$ on the ${\mathbb {Q}}$ -line bundle $O_X(\Delta _i)$ and define $|s_i|^2:=|s_i|^2_{h_i^{\otimes m}}$ . Since $K_X$ is ${\mathbb {Q}}$ -Cartier (as a difference $K_X=(K_X+\Delta )-\Delta )$ of ${\mathbb {Q}}$ -Cartier divisors), one can find a metric $h_X$ on $K_X$ such that $h=h_X\otimes \bigotimes _i h_i^{\otimes a_i}$ . Setting $b_i:=\frac {a_i} m$ , one can rewrite the Monge-Ampère equation solved by $\varphi $ as

$$\begin{align*}(\omega_X+dd^c \varphi)^n=e^{\lambda\varphi+F} \frac{d\mu_{X,h_X}}{\prod_i |s_i|^{2b_i}}.\end{align*}$$

We consider the unique (normalized) solution $\varphi _\varepsilon $ of the regularized equation

$$\begin{align*}(\omega_X+dd^c \varphi_\varepsilon)^n=e^{\lambda\varphi_\varepsilon+F} \frac{d\mu_{X,h_X}}{\prod_i (|s_i|^2+\varepsilon^2)^{b_i}}.\end{align*}$$

and we set $\omega _\varepsilon :=\omega _X+dd^c \varphi _\varepsilon $ . It follows from [Reference Eyssidieux, Guedj and ZeriahiEGZ09, Reference PăunPău08] that $\omega _\varepsilon $ is a Kähler form on $X_{\mathrm {reg}}$ for any $\varepsilon>0$ , and moreover there is a uniform constant $C_0>0$ such that

(4.2) $$ \begin{align} \|\varphi_\varepsilon\|_{L^{\infty}(X)}\leqslant C_0 \quad \mbox{and} \quad \varphi_\varepsilon \underset{\varepsilon \to 0}{\longrightarrow} \varphi \quad \mbox{in } L^1(X). \end{align} $$

thanks to [Reference Guedj and ZeriahiGZ12, Theorem C] applied to a desingularization of X. Therefore, we can reduce the proof of the theorem to showing that there exists a constant $C>0$ independent of $\varepsilon $ such that

(4.3) $$ \begin{align} \omega_\varepsilon \geqslant C^{-1}\omega_X \quad \mbox{on } X. \end{align} $$

Claim 4.8. The uniform inequality (4.3) holds if for any $\varepsilon>0$ , one has the qualitative inequality

(4.4) $$ \begin{align} \omega_\varepsilon \geqslant C_\varepsilon^{-1}\omega_X \quad \mbox{on } X, \end{align} $$

where $C_\varepsilon>0$ is a positive constant that may depend on $\varepsilon $ .

Proof of Claim 4.8

Here again, the key tool is Chern–Lu inequality, Proposition 2.6. The bisectional curvature of $\omega _X$ is bounded above since $\omega _X$ can be extended to a Kähler metric in local embeddings in $\mathbb C^N$ , and we need to bound $\mathrm {Ric} \, \omega _\varepsilon $ from below.

A classical computation shows that if s is a holomorphic section cutting out a divisor D and if $h_D$ is a smooth Hermitian metric on $L:=\mathcal O_X(D)$ , then

$$\begin{align*}\Theta_{h_D}(L)+dd^c \log(|s|^2_{h_D}+\varepsilon^2) = \frac{\varepsilon^2}{(|s|_{h_D}^2+\varepsilon^2)^2} \cdot |D's\wedge \overline{D's}|^2_{h_D}-\frac{\varepsilon^2}{|s|_{h_D}^2+\varepsilon^2} \cdot \Theta_{h_D}(L).\end{align*}$$

In particular, if one choses a constant $C_D>0$ such that $\Theta _{h_D}(L)\leqslant C_D\omega _X$ , then

(4.5) $$ \begin{align} \Theta_{h_D}(L)+dd^c \log(|s|^2_{h_D}+\varepsilon^2)\geqslant -C_D\omega_X. \end{align} $$

Since

$$\begin{align*}\mathrm{Ric} \, \omega_\varepsilon = -\lambda \omega_\varepsilon+\lambda \omega_X-dd^c F+\sum_i b_i \left( \Theta_{h_i^{\otimes m}}(\mathcal O_X(m\Delta_i))+dd^c \log(|s_i|^2+\varepsilon^2)\right)\end{align*}$$

and F is smooth (so that its Hessian is bounded with respect to $\omega _X$ ), inequality (4.5) ensures that one can find $C_1>0$ such that

$$\begin{align*}\mathrm{Ric} \, \omega_\varepsilon = -\omega_\varepsilon-C_1\omega_X.\end{align*}$$

Finally, let $\psi \in \mathrm {PSH}(X,\omega _X)$ be such that $(\psi =-\infty )=X_{\mathrm {sing}}$ and $\psi |_{X_{\mathrm {reg}}}\in \mathcal C^{\infty }(X_{\mathrm {reg}})$ . For any $\delta>0$ , the smooth quantity

$$\begin{align*}\log \mathrm{tr}_{\omega_\varepsilon}\omega_X+\delta \psi\end{align*}$$

on $X_{\mathrm {reg}}$ tends to $-\infty $ near $X_{\mathrm {sing}}$ thanks to our assumption (4.4). In particular, the quantity above achieves its maximum on $X_{\mathrm {reg}}$ . Moreover, our curvature estimates coupled with Proposition 2.6 yield a constant $C_2>0$ such that

$$\begin{align*}\Delta_{\omega_\varepsilon} \left[\log \mathrm{tr}_{\omega_\varepsilon}\omega_X+\delta \psi\right] \geqslant -1 -C_2\mathrm{tr}_{\omega_\varepsilon}\omega_X \end{align*}$$

on $X_{\mathrm {reg}}$ . Since $-\Delta _{\omega _\varepsilon } \varphi _\varepsilon \geqslant -n+\mathrm {tr}_{\omega _\varepsilon }\omega _X$ , we infer that

$$\begin{align*}\Delta_{\omega_\varepsilon} \left[\log \mathrm{tr}_{\omega_\varepsilon}\omega_X+\delta \psi-(C_2+1)\varphi_\varepsilon \right] \geqslant \mathrm{tr}_{\omega_\varepsilon}\omega_X-C_3 \end{align*}$$

where $C_3=-1+n(C_2+1)$ . A classical application of the maximum principle shows that

$$\begin{align*}\mathrm{tr}_{\omega_\varepsilon}\omega_X \leqslant C_4 e^{\delta(\sup_X \psi- \psi)}\end{align*}$$

with $C_4=C_3e^{2C_0(C_2+1)}$ . Passing to the limit when $\delta \to 0$ , we get

$$\begin{align*}\omega_\varepsilon \geqslant C_4^{-1} \omega_X\quad \mbox{on } X_{\mathrm{reg}},\end{align*}$$

hence everywhere on X by Lemma 2.5. The claim follows.

In order to prove the theorem, we are left to establishing the qualitative estimate (4.4). In order to achieve that, we use a local deformation argument. From now on, $\varepsilon>0$ is fixed and all subsequent constants are allowed to depend on $\varepsilon $ . Since $\omega _\varepsilon $ is smooth on $X_{\mathrm {reg}}$ , it is enough to work in small neighborhoods of the finitely many singular points in X. From now on, we pick a small Stein neighborhood $U'$ of x admitting a quasi-étale cover $p:V'\to U'$ such that $p^{-1}(x)=\{y\}$ is a singleton and $(V',y)$ admits a smoothing $\mathcal V\to \mathbb D$ whose fibers $V_t$ satisfy $V_0\simeq V'$ .

One can assume that $U'$ is small enough so that $\omega _X|_{U'}=dd^c \rho $ for some smooth strictly psh function $\rho $ on $U'$ . Next, since p is quasi-étale, we have $K_{V'}=p^*K_{U'}$ and the smooth Hermitian metric $h_{U'}:=h_X|_{U'}$ on $K_{U'}$ pulls back to a smooth Hermitian metric $h_{V'}$ on $K_{V'}$ satisfying $p^*\left (d\mu _{U',h_X}\right )=d\mu _{V',h_{V'}}$ .

Next, we fix $U\Subset U'$ strongly pseudoconvex and set $V:=p^{-1}(U)$ . The function $v_\varepsilon :=p^*(\rho +\varphi _\varepsilon )$ satisfies

$$\begin{align*}\begin{cases} (dd^c v_\varepsilon)^n = e^{\lambda v_\varepsilon+p^*(F-\lambda \rho)} \frac{d\mu_{V,h_V}}{\prod_i (|p^*s_i|^2+\varepsilon^2)^{b_i}} & \mbox{on } V\\ v_\varepsilon|_{\partial V}=p^*((\rho+\varphi_\varepsilon)|_{\partial U}), \end{cases} \end{align*}$$

hence by Theorem 3.4 the current $dd^cv_\varepsilon $ is a Kähler current. In particular, it dominates a multiple of $p^*(\omega _X)|_U$ . Pushing forward, we get that in restriction to U, we have $\omega _\varepsilon \geqslant C_{\varepsilon }^{-1} \omega _X$ and the theorem is proved.

Proof. Let $p:Y\to X$ be the quasi-étale index one cover of X. Clearly, $p^*\omega _{\mathrm {KE}}$ is the Kähler–Einstein metric in the Kähler class $p^*[\omega _X]$ so that if we prove the statement for Y, it will follow for X by push-forward (since any Kähler form on Y dominates a multiple of $p^*\omega _X$ ).

From now on, one can assume that $K_X\sim _{{\mathbb {Z}}} \mathcal O_X$ and, in particular, X has canonical singularities. Let us consider a terminalization $\pi : \widehat X \to X$ of X, cf [Reference Kollár and MoriKM98, Theorem 6.23]. The complex space $\widehat X$ is Kähler, and it has terminal singularities, hence isolated singularities. Moreover, one has $K_{\widehat X}=\pi ^*K_X$ , that is, $\pi $ is crepant (but in general, the exceptional locus of $\pi $ might have codimension one components). This implies that $K_{\widehat X}\sim _{{\mathbb {Z}}} \mathcal O_{\widehat X}$ so that $\widehat X$ has smoothable singularities by Theorem 5.1. Moreover, since $\pi $ is crepant, it is automatically isomorphic over $X_{\mathrm {reg}}$ .

We choose $\widehat \omega $ a Kähler metric on $\widehat X$ , and one considers the singular Ricci flat metric $\omega _\varepsilon \in [\pi ^*\omega _X+\varepsilon \widehat \omega ]$ . One can write $\omega _\varepsilon = \pi ^*\omega _X+\varepsilon \widehat \omega +dd^c \varphi _\varepsilon $ with $\sup _{\widehat X}\varphi _\varepsilon =0$ , where $\varphi _{\varepsilon }$ is a solution of $(\pi ^*\omega _X+\varepsilon \widehat \omega +dd^c \varphi _{\varepsilon })^n=c_{\varepsilon } \mu _{\hat {X}}=c_{\varepsilon } \hat {f} \widehat \omega ^n$ , where $c_\varepsilon =\frac {[\pi ^*\omega _X+\varepsilon \widehat \omega ]^n}{ \mu _{\widehat X}(\widehat X)}$ and $\hat {f} \in L^{1+\delta }(\widehat X,\widehat \omega ^n)$ with $\delta>0$ since $\hat {X}$ has terminal singularities. It follows from the techniques in [Reference Eyssidieux, Guedj and ZeriahiEGZ08, Reference Demailly and PaliDP10] that

(5.1) $$ \begin{align} \|\varphi_\varepsilon\|_{L^{\infty}(\widehat X)} \leqslant C_0, \quad \mbox{and} \quad \omega_\varepsilon \underset{\varepsilon \to 0}{\longrightarrow} \pi^*\omega_{\mathrm{KE}} \quad \mbox{weakly,} \end{align} $$

where $C_0>0$ is independent of $\varepsilon $ . The discerning reader will have noticed that the situation is slightly different from the one in loc. cit. since X is singular. To patch this little gap, one could, for example, appeal to [Reference Di Nezza, Guedj and GuenanciaDNGG23, Theorem A] applied to a desingularization of $\widetilde X\to \widehat X$ with reference form the pull-back of $\pi ^*\omega _X+\varepsilon \widehat \omega $ to $\widetilde X$ , and this would yield the uniform estimate. As for the stability statement, it is a classical consequence of the uniform estimate: One first gets higher-order estimates locally on $X_{\mathrm {reg}}$ using Tsuji’s trick and then one uses uniqueness of the Kähler–Einstein metric to conclude that the relatively compact family $(\varphi _\varepsilon )_{\varepsilon>0}$ has a single cluster value in $L^1(\widehat X)$ which is nothing but $\varphi _0$ .

The main point to establish is that there is a constant $C>0$ independent of $\varepsilon $ such that

(5.2) $$ \begin{align} \omega_\varepsilon \geqslant C^{-1} \pi^*\omega_X. \end{align} $$

Indeed, passing to the limit when $\varepsilon \to 0$ would then imply that $\pi ^*\omega _{\mathrm {KE}}\geqslant C^{-1} \pi ^*\omega _X$ and the theorem would follow by pushing forward by $\pi $ .

Now, we know that inequality (5.2) holds for any $\varepsilon>0$ with a constant $C=C_\varepsilon $ depending a priori from $\varepsilon $ , as this is the content of Theorem 4.1. As before, we will use Chern–Lu inequality to make this qualitative control quantitative.

Choose a function $\psi \in \mathrm {PSH}(X,\omega _X)$ be such that $(\psi =-\infty )=X_{\mathrm {sing}}$ and $\psi |_{X_{\mathrm {reg}}}\in \mathcal C^{\infty }(X_{\mathrm {reg}})$ . For any $\delta>0$ , the smooth quantity

$$\begin{align*}\log \mathrm{tr}_{\omega_\varepsilon}\pi^*\omega_X+\delta \psi\end{align*}$$

on $\pi ^{-1}(X_{\mathrm {reg}}) \subset \widehat X \setminus \mathrm {Exc}(\pi )$ tends to $-\infty $ near $\pi ^{-1}(X_{\mathrm {sing}})$ since $\omega _\varepsilon $ is a Kähler form. In particular, the quantity above achieves its maximum on $X_{\mathrm {reg}}$ .

Next, the bisectional curvature of $\pi ^*\omega _X$ is well defined and bounded above uniformly on $\pi ^{-1}(X_{\mathrm {reg}})$ , while $\mathrm {Ric} \,\omega _\varepsilon = 0$ . By Chern–Lu inequality (Proposition 2.6), we find a constant $C_2>0$ such that

$$\begin{align*}\Delta_{\omega_{\varepsilon}} \left[\log {\mathrm{tr}}_{\omega_{\varepsilon}}\pi^{*}\omega_{X}+\delta \psi\right] \geqslant -C_2{\mathrm{tr}}_{\omega_\varepsilon}\pi^{*}\omega_{X} \end{align*}$$

on $\pi ^{-1}(X_{\mathrm {reg}})$ . Since $-dd^c \varphi _\varepsilon =-\omega _\varepsilon +\pi ^{*}\omega _X+\varepsilon \widehat \omega $ , we have

$$ \begin{align*} -\Delta_{\omega_{\varepsilon}} \varphi_{\varepsilon} \geqslant -n+{\mathrm{tr}}_{\omega_{\varepsilon}}\pi^*\omega_{X}, \end{align*} $$

hence

$$\begin{align*}\Delta_{\omega_\varepsilon} \left[\log {\mathrm{tr}}_{\omega_\varepsilon}\pi^{*}\omega_X+\delta \psi-(C_2+1)\varphi_\varepsilon \right] \geqslant {\mathrm{tr}}_{\omega_\varepsilon} \pi^{*}\omega_X-C_3, \end{align*}$$

where $C_3=-1+n(C_2+1)$ . Using Equation (5.1), we obtain in the end

$$\begin{align*}\mathrm{tr}_{\omega_\varepsilon}\pi^{*}\omega_X \leqslant C_4 e^{\delta(\sup_X \psi- \psi)}\end{align*}$$

with $C_4=C_3e^{C_0(C_2+1)}$ , holding on $\pi ^{-1}(X_{\mathrm {reg}})$ for any $\delta>0$ . Equivalently, we have $\omega _\varepsilon \geqslant C_4 e^{-\delta (\sup _X \psi - \psi )}\, \pi ^*\omega _X$ . After passing to the limit when $\delta \to 0$ , one can appeal to Lemma 2.5 to obtain Equation (5.2), hence the theorem.

Proof. The reduction to the terminal case is a bit more involved here. More precisely, we cannot easily assume that X has canonical singularities without resorting to a local argument which would collapse since the singularities of X are not isolated. Instead, we use a ${\mathbb {Q}}$ -factorial terminalization $\pi :\widehat X\to X$ whose existence (in any dimension) is guaranteed by [Reference Birkar, Cascini, Hacon and McKernanBCHM10, Corollary 1.4.3]. The map $\pi $ is such that there exists an effective divisor $\widehat \Delta $ on $\widehat X$ such that $K_{\widehat X}+\widehat \Delta =\pi ^*K_X$ and $(\widehat X, \widehat \Delta )$ is a terminal pair, hence $\widehat X$ has isolated ${\mathbb {Q}}$ -smoothable singularities by Theorem 5.1. Moreover, $\widehat X$ is ${\mathbb {Q}}$ -factorial.

Next, let $\widehat \omega $ be a Kähler metric on $\widehat X$ and let us consider the twisted Kähler–Einstein metric $\omega _\varepsilon \in c_1(\pi ^*K_X)+\varepsilon [\widehat \omega ]$ , that is, the solution of

$$\begin{align*}\mathrm{Ric} \, \omega_\varepsilon = -\omega_\varepsilon+\varepsilon \widehat \omega+[\widehat \Delta].\end{align*}$$

If $\omega _X\in c_1(K_X)$ is a Kähler metric, say the curvature of an Hermitian metric h on $K_X$ , one can write $\omega _\varepsilon = \pi ^*\omega _X+\varepsilon \widehat \omega +dd^c \varphi _\varepsilon $ , where $\varphi _{\varepsilon }$ is a solution of

$$\begin{align*}(\pi^*\omega_X+\varepsilon \widehat \omega+dd^c \varphi_{\varepsilon})^n=e^{\varphi_{\varepsilon}} \mu_{(\hat{X},\widehat \Delta), \pi^*h}=e^{\varphi_{\varepsilon}} \hat{f} \widehat \omega^n,\end{align*}$$

with $\hat {f} \in L^{1+\delta }(\widehat X,\widehat \omega ^n)$ with $\delta>0$ since $\hat {X}$ has terminal singularities. As in the Calabi–Yau case, one has

(5.3) $$ \begin{align} \|\varphi_\varepsilon\|_{L^{\infty}(\widehat X)} \leqslant C, \quad \mbox{and} \quad \omega_\varepsilon \underset{\varepsilon \to 0}{\longrightarrow} \pi^*\omega_{\mathrm{KE}} \quad \mbox{weakly,} \end{align} $$

where $C>0$ is independent of $\varepsilon $ . The only difference with the Calabi–Yau case is that we first need to show that there exists $C>0 $ independent of $\varepsilon $ such that

(5.4) $$ \begin{align} \sup_{\widehat X}\varphi_\varepsilon \leqslant C. \end{align} $$

This is certainly classical, but we recall the argument for the reader’s convenience. First, one can assume without loss of generality that $\mu :=\mu _{(\hat {X},\widehat \Delta ), \pi ^*h}$ is a probability measure. By Jensen’s inequality, we infer that

$$\begin{align*}\int_{\widehat X} \varphi_\varepsilon d\mu \leqslant \log \int_{\widehat X}(\pi^*\omega_X+\varepsilon\widehat \omega)^n \leqslant C_{1}\end{align*}$$

for some $C_{1}>0$ independent of $\varepsilon $ . Let $p:\widetilde X\to \widehat X$ be a resolution of singularities, and let $\omega _{\widetilde X}$ be a Kähler metric on $\widetilde X$ . Up to scaling the latter metric, one can assume that $p^*(\pi ^*\omega _X+\varepsilon \widehat \omega )\leqslant \omega _{\widetilde X}$ and $p^*\mu \leqslant \omega _{\widetilde X}^n$ , where the latter follows since $\widehat X$ has terminal (hence canonical) singularities. By the usual compactness properties of $\omega _{\widetilde X}$ -psh functions, we find a constant $C_{2}$ such that

$$\begin{align*}\int_{\widetilde X}(\sup_{\widetilde X}\psi-\psi)\omega_{\widetilde X}^n \leqslant C_{2}\end{align*}$$

for any $\psi \in \mathrm {PSH}(\widetilde X, \omega _{\widetilde X})$ . Now,

$$ \begin{align*} \sup_{\widehat X} \varphi_\varepsilon &= \int_{\widehat X}(\sup_{\widehat X} \varphi_\varepsilon-\varphi_\varepsilon)d\mu+\int_{\widehat X} \varphi_\varepsilon d\mu\\ & \leqslant \int_{\widehat X}(\sup_{\widetilde X} p^*\varphi_\varepsilon-p^*\varphi_\varepsilon)\omega_{\widetilde X}^n+C_1\\ & \leqslant C_1+C_2 \end{align*} $$

since $p^*\varphi _\varepsilon \in \mathrm {PSH}(\widetilde X, \omega _{\widetilde X})$ . Hence, Equation (5.4) follows.

Coming back to proof of the main result, the remaining point to establish is that there is a constant $C>0$ independent of $\varepsilon $ such that

(5.5) $$ \begin{align} \omega_\varepsilon \geqslant C^{-1} \pi^*\omega_X. \end{align} $$

Thanks to Theorem 4.5, inequality (5.5) holds for any $\varepsilon>0$ with a constant $C=C_\varepsilon $ depending a priori from $\varepsilon $ . In order to make this bound quantitative, the exact same strategy as in the Calabi–Yau case applies.