2.1.1 Plurisubharmonic functions

Using the local embedding j, it is possible to define the spaces of smooth forms on X as restriction of smooth forms of $\mathbb C^N$ . The notion of currents on X is defined by duality; the operators $\partial $ and $\bar{\partial}$ , d, $d^c$ and $dd^c$ are also well defined by duality (see [Reference DemaillyDem85] for more details).

Here $d=\partial+\overline{\partial}$ and $d^c=({1}/{4i\pi})(\partial-\overline{\partial})$ are real operators and $dd^c =({i}/{2\pi})\partial\overline{\partial}$ . With this normalization the function $z \in \mathbb C^n \mapsto \rho_{FS}(z)=\log[1+|z|^2] \in \mathbb R$ is smooth and plurisubharmonic (psh for short) in $\mathbb C^n$ , with

\[\int_{\mathbb C^n} (dd^c \rho_{FS})^n=1.\]

We let PSH(X) denote the set of all psh functions on X that are not identically $-\infty$ .

Recall that u is called weakly psh on X if it is locally bounded from above on X and its restriction to $X_{{reg}}$ is psh. One can extend it to X by $u^* (p) :=\limsup_{X_{{reg}} \ni y \to p}u (y)$ . Since X is locally irreducible, it follows from the work of Fornæss and Narasimhan [Reference Fornæss and NarasimhanFN80] that u is weakly psh if and only if $u^*$ is psh (see [Reference DemaillyDem85, Corollary 1.11]).

If $u \in PSH(X)$ , then u is upper semi-continuous on X and locally integrable with respect to the volume form

\[dV_X:=\omega_{\rm eucl}^n \wedge [X].\]

Here [X] denotes the current of integration along X and $\omega_{\rm eucl}:=\sum_{j=1}^N i \, dz_j\wedge d\overline{z_j}$ is the euclidean Kähler form. In particular, $dd^c u$ is a well-defined current of bidegree (1,1) which is positive.

2.1.2 Pseudoconvex domains and boundary data

Following [Reference Fornæss and NarasimhanFN80] we say that X is Stein if it admits a ${\mathcal C}^2$ -smooth strongly psh exhaustion.

Definition 2.2. A domain $\Omega \Subset X$ is strongly pseudoconvex if it admits a negative ${\mathcal{C}}^2$ -smooth strongly psh exhaustion, i.e. a function $\rho$ strongly psh in a neighborhood $\Omega'$ of $\overline{\Omega}$ such that $\Omega := \{ x \in \Omega'; \rho (x)< 0\}$ , $d\rho \neq 0$ on $\partial \Omega$ , and for any $c < 0$ ,

\[\Omega_c := \{x \in \Omega'; \rho (x) < c\} \Subset \Omega\]

is relatively compact.

We are interested in solving a Dirichlet problem for some complex Monge–Ampère equations in a bounded strongly pseudoconvex domain $\Omega=\{\rho<0\}$ , with given boundary data $\phi \in{\mathcal C}^{\infty}(\partial \Omega)$ .

Such an extension can be obtained as follows: we pick $\tilde{\phi}$ an arbitrary ${\mathcal{C}}^2$ -smooth extension to $\overline{\Omega}$ , and then consider $\phi_0:=\tilde{\phi}+A \rho$ , for A so large that $\phi_0$ is ${\mathcal C}^2$ -smooth and psh in $\overline{\Omega}$ . All quantities introduced in the remainder of the paper are essentially independent of the particular choice of the extension.

2.1.3 Monge–Ampère operators

The complex Monge–Ampère operator $(dd^c \cdot)^n$ acts on a smooth psh functions $\varphi$ . When $X=\mathbb C^n$ , it boils down to

\[(dd^c \varphi)^n=c_n \det \bigg(\frac{\partial^2 \varphi}{\partial z_j \overline{\partial} z_k}\bigg) \omega_{\rm eucl}^n,\]

where $c_n>0$ is a normalizing constant.

2.1.4 Bounded functions

Following [Reference Bedford and TaylorBT82] this operator can be extended to the class $PSH(X) \cap L^{\infty}_{\rm loc}$ by using approximation by smooth psh functions: given $\varphi \in PSH(X) \cap L^{\infty}_{\rm loc}$ , there exists a unique positive Radon measure $\mu_{\varphi}$ on X such that for any sequence $(\varphi_j)$ of smooth psh functions decreasing to $\varphi$ , one has

\[\mu_{\varphi}=\lim (dd^c \varphi_j)^n,\]

where the limit holds in the weak sense. One then sets $(dd^c \varphi)^n:=\mu_{\varphi}$ .

Definition 2.4. We set

\[\mathcal{T}^{\infty}_{\phi}(\Omega):=\{\varphi\in SPSH(\Omega)\cap {\mathcal C}^{\infty}(\overline{\Omega}): \varphi_{|\partial \Omega}=\phi\},\]

where $SPSH(\Omega)$ is the set of strictly psh functions, and

\[\mathcal{T}_{\phi}(\Omega):=\bigg\{\varphi\in PSH(\Omega)\cap {\mathcal C}^0(\overline{\Omega}):\varphi_{|\partial \Omega}=\phi,\ \int_{\Omega}(dd^{c}\varphi)^{n}<+\infty\bigg\},\]

This latter class has been introduced by Cegrell in [Reference CegrellCeg98]; it can be used as a psh version of test functions (in the sense of distributions), as well as a building block for finite-energy classes of mildly unbounded functions.

2.1.5 Mildly unbounded functions

The complex Monge–Ampère operator can be defined for mildly unbounded psh functions. We refer the reader to [Reference CegrellCeg04, Reference BłockiBlo06] for the case of smooth domains in $\mathbb C^n$ ; their analysis easily extends to our context.

Definition 2.6. We let $\mathcal{F}(\Omega)$ denote the set of all functions $\varphi\in PSH(\Omega)$ which are decreasing limit of a sequence of functions $\varphi_{j}\in \mathcal{T}_{\phi}(\Omega)$ such that

\[\sup_{j}\int_{\Omega}(dd^{c}\varphi_{j})^{n}<+\infty.\]

The operator $(dd^c \cdot )^n$ is well defined on $ \mathcal{F}(\Omega)$ , continuous along monotonic sequences, and yields Radon measures $(dd^c \varphi)^n$ which have finite mass in $\Omega$ . We endow $ \mathcal{F}(\Omega)$ with the $L^1$ -topology. Let us stress that the operator $\varphi \mapsto(dd^c \varphi)^n$ is not continuous for the $L^1$ -topology, but the class $ \mathcal{F}(\Omega)$ enjoys the following useful compactness property.

This is shown in [Reference ZeriahiZer09, Observation A.3] for smooth domains, and the same proof applies in our mildly singular context. Let us stress that the Monge–Ampère operator cannot be defined for all psh functions: there is, for example, no reasonable way to make sense of $(dd^c \log |z_1|)^n$ . A consequence of Proposition 2.7 is that one cannot approximate such a function by a decreasing sequence of psh functions with prescribed boundary values and uniformly bounded Monge–Ampère masses.

2.2.1 Log terminal singularities

Let Y be a connected normal complex variety such that $K_Y$ is $\mathbb Q$ -Cartier near $p \in Y$ . One can consider the $dd^c$ -cohomology class of $-K_Y$ , denoted by $c_1(Y)$ .

Given a log-resolution $\pi: \tilde{Y} \to Y$ of (Y,p), there exists a unique $\mathbb Q$ -divisor $\sum_i a_i E_i$ whose push-forward to Y is 0 and with

\[K_{\tilde{Y}}=\pi^*(K_Y)+\sum_i a_i E_i.\]

It is classical that this condition is independent of the choice of resolution. In the remainder of this article we assume that:

• – the singularity (X,0) is log terminal;

• – $Y=\Omega$ is a strongly pseudoconvex neighborhood of $0=p \in X$ ;

• – the canonical bundle $K_{\Omega}$ is $\mathbb Q$ -Cartier and $rK_{\Omega}=0$ for some $r\in\mathbb N$ .

Definition 2.9 [Reference Eyssidieux, Guedj and ZeriahiEGZ09, Definition 6.5]. Fix $\sigma$ a nowhere-vanishing holomorphic section of $rK_{\Omega}$ , and h a smooth hermitian metric of $ K_{\Omega}$ , then

\[\mu_{p}=\lambda \frac{(c_{n}\sigma\wedge \bar{\sigma})^{1/r}}{|\sigma|^{2/r}_{h^r}}\]

is an adapted measure, where $\lambda >0$ is a positive normalizing constant.

Observe that $\mu_p$ is independent of the choice of $\sigma$ , and

\[dd^c \log \mu_{p}=-\Theta_h(K_{\Omega})\]

is the curvature of h, as follows from the Poincaré–Lelong formula.

The measure $\mu_p$ has finite mass by [Reference Eyssidieux, Guedj and ZeriahiEGZ09, Lemma 6.4]: let $\pi: \tilde{\Omega}\to \Omega$ be a resolution of $(\Omega,0)$ , then

\[\pi^{*}\mu_{p}=\lambda \prod_{j=1}^{M}\lvert s_{E_{j}} \rvert^{2a_{j}} \, dV_{\tilde{\Omega}},\]

where $dV_{\tilde{\Omega}}$ is a smooth volume form on $\tilde{\Omega}$ , $E_{1},\ldots,E_{M}$ are exceptional divisors, $s_{E_j}$ are holomorphic sections such that $E_j=(s_{E_j}=0)$ , and

\[rK_{\tilde{\Omega}}=\pi^{*}(rK_{\Omega})+r\sum_{j=1}^{M}a_{j}E_{j}=r\sum_{j=1}^{M}a_{j}E_{j}.\]

Thus $\tilde{f}=\prod_{j=1}^{M}\lvert s_{E_{j}} \rvert^{2a_{j}}$ belongs to $L^s(dV_{\tilde{\Omega}})$ for some $s>1$ , as p is log terminal.

The results to follow are independent of this (convenient) normalization.

2.2.2 Ricci curvature

Let $\omega$ be a positive closed current of bidegree (1,1) in $\Omega$ with bounded local potentials. Its top power $\omega^n$ is well defined as explained in § 2.1.3. If $\omega^n$ is absolutely continuous with respect to $dV_X$ , then we set

\[{\rm Ric}(\omega):=-dd^c \log \omega^n.\]

Definition 2.11. We say that $\omega$ is a Kähler–Einstein metric if it satisfies

\[{\rm Ric}(\omega)=\gamma \omega\]

for some $\gamma \in \mathbb R$ .

In this article, we are mainly interested in the case when $\gamma >0$ . We choose the hermitian metric $h \equiv 1$ for $K_{\Omega}$ , so that $\Theta_h=0$ . Since

\[{\rm Ric}(\omega)={\rm Ric}(\mu_p)-dd^c \log (\omega^n/\mu_p),\]

the above Kähler–Einstein equation is equivalent, writing $\omega=dd^c \varphi$ , to

\[(dd^c \varphi)^n=e^{-\gamma \varphi} e^{w} \mu_p,\]

where w is a pluriharmonic function in $\Omega$ . Changing $\varphi$ in $\varphi-w/\gamma$ and then $\varphi$ in $t \varphi$ (observe that ${\rm Ric}(t \omega)={\rm Ric}(\omega)$ for any $t>0$ ), we can normalize $\omega$ by $\int_{\Omega} \omega^n=1$ and reduce to

\[(dd^c \varphi)^n= \frac{e^{-\gamma \varphi} \mu_p}{\int_{\Omega} e^{-\gamma \varphi} \mu_p}.\]

Seeking for a Kähler–Einstein metric thus leads one to solve $(MA)_{\gamma,\phi,\Omega}$ .

Conversely solving $(MA)_{\gamma,\phi,\Omega}$ will produce a Kähler–Einstein metric $\omega=dd^c\varphi$ , if we can establish enough regularity of the solution $\varphi$ .

2.2.3 Log canonical threshold

We consider the density $f=\mu_p/dV_X$ . It is related to the density $\tilde{f}$ in a resolution by

\[\pi^* \mu_p=f \circ \pi \cdot \pi^* \, dV_X=\tilde{f} \, dV_{\tilde{\Omega}}.\]

An analytic expression for f is obtained as follows. Recall that $dV_X=\omega_{\rm eucl}^n \wedge[X]$ , where $\omega_{\rm eucl}$ denotes the euclidean Kähler form on $\mathbb C^{N}$ . Set $dz_{I}=dz_{i_{1}}\wedge \cdots \wedge dz_{i_{n}}, $ where $1 \leq i_1< \cdots <i_n \leq N$ . There exists germs of holomorphic functions $f_{I}\in \mathcal{O}_{\Omega,0}$ such that $(dz_{I})^{r}=f_{I}\sigma $ since $\sigma$ is a local generator of $rK_{X}$ . In particular, the volume form $dV_{X}:=\omega_{\rm eucl}^{n}\wedge [\Omega]$ is comparable to $(\sum_{I}\lvert f_{I}\rvert^{{2}/{r}})\mu_{p},$ i.e.

\[\mu_{p}=f \, dV_{X},\quad \text{with}\ f\sim\bigg(\sum_{I}\lvert f_{I}\rvert^{{2}/{r}}\bigg)^{-1}.\]

The germs of holomorphic functions $f_{I}$ generate an ideal $\mathcal{I}_{p}^{r}$ , where $\mathcal{I}^{r}$ is an ideal sheaf associated to the singularities of (X,p). In particular,

\[\pi^{-1}\mathcal{I}^{r}\cdot \mathcal{O}_{\tilde{\Omega}}=\mathcal{O}_{\tilde{\Omega}}\bigg(-r\sum_{j=1}^{M}b_{j}E_{j}\bigg)\]

for coefficients $b_{j}\in\mathbb N$ such that $ f\circ \pi\sim \prod_{j=1}^{M}\lvert s_{E_{j}}\rvert^{-2b_{j}}$ .

Definition 2.12. The log canonical threshold of (X,p) is given by

\[\mathrm{lct}(X,p):=\inf_{j\in{1,\ldots,M}} \frac{a_{j}+1}{b_{j}}.\]

We let the reader check that the definition is independent of the choice of resolution, and that $\mathrm{lct}(X,\mathcal{I})\in (0,n]$ . One can equivalently use the following point of view: if $\mathcal{I}$ is a general ideal sheaf,

(2.1) \begin{equation}\mathrm{lct}(X,\mathcal{I}):=\inf_{E/X}\frac{A_{X}(E)}{\mathrm{ord}_{E}(\mathcal{I})}\end{equation}

where $A_{X}(E):=1+\mathrm{ord}_{E}(K_{Y/X}) $ is the log-discrepancy of E, and the infimum is over all prime divisors E on resolutions Y of X. When $\mathcal{I}$ is supported at p we can restrict in (2.1) to consider prime divisors centered at p.

In this case $\mathcal{I}^2=(z_1^2,\ldots,z_n^2)$ . Indeed the n-forms

\[\sigma_j:=\frac{(dz_0\wedge \cdots\wedge \widehat{dz_j}\wedge \cdots \wedge dz_n)^2}{z_j^2}=-\frac{(dz_0\wedge \cdots \wedge \widehat{dz_j}\wedge \cdots \wedge dz_n)^2}{\sum_{k\neq j} z_k^2},\]

defined on $U_j:=\{z_j\neq 0\}$ , glue together to give a local generator $\sigma$ of $2 K_X$ (note that $\sum_{j=0}^n z_j \, dz_j=0$ ). In particular, $|f_I|^{2/r}=|z_j|^2$ where $j=[0,n] \setminus I$ , $r=2$ and

\[\mu_{p} \sim \frac{1}{\sum_{j=0}^n \lvert z_j \rvert^2} \, dV_X.\]

If $\pi: \rm Bl_0 \mathbb C^{n+1}\to \mathbb C^{n+1}$ denotes the blow-up at 0, E the exceptional divisor, and F the restriction of E to Y, the strict transform of X, we obtain

\[\pi^{-1}\mathcal{I}^2\cdot \mathcal{O}_Y=\mathcal{O}_Y(-2F)\quad \text{and}\quad\pi^* \mu_p= \lvert s_F \rvert^{2(n-2)} \, dV_Y\]

for a smooth volume form $dV_Y$ . Thus, ${\rm lct}(X,p)={\rm lct}(X,\mathcal{I})=n-1$ .

We will need the following result which connects ${\rm lct}(X,p)$ and the integrability properties of the density $f=\mu_p/dV_X$ .

Proof. Let $\pi:\tilde{\Omega} \rightarrow \Omega$ be a resolution of the singularity. Recall that

\[f \circ \pi \sim \prod_{j=1}^{M}\lvert s_{E_{j}} \rvert^{-2b_{j}}\quad \text{and}\quad\tilde{f}=\prod_{j=1}^{M}\lvert s_{E_{j}} \rvert^{2a_{j}},\quad \text{hence}\quad\pi^* dV_X \sim \prod_{j=1}^{M}\lvert s_{E_{j}} \rvert^{2(a_{j}+b_j)} \, dV_{\tilde{\Omega}}.\]

It follows that $\int_{{\Omega}} f^r \,dV_X \sim \int_{\tilde{\Omega}} \prod_{j=1}^{M}\lvert s_{E_{j}} \rvert^{2(a_{j}+b_j)-2rb_j} \,dV_{\tilde{\Omega}}<+\infty $ if and only if $r<({1+a_j+b_j})/{b_j}$ for all j, which yields the statement since ${\rm lct}(X,p)=\inf_j(({1+a_j})/{b_j})$ .

3.1.1 Smooth tests

Fix $\Omega=\{\rho<0\}$ and $\phi$ as described previously, and

\[\mathcal{T}_{\phi}^{\infty}(\Omega)=\{\varphi\in SPSH(\Omega)\cap\mathcal{C}^{\infty}(\overline{\Omega}): \varphi_{\vert \partial\Omega}\equiv \phi\}.\]

Recall that $\phi_{0}\in \mathcal{C}^{\infty}(\bar{\Omega})\cap PSH(\Omega)$ denotes a smooth psh extension of $\phi$ to $\bar{\Omega}$ . We set $\omega:=dd^{c}\phi_{0}$ . This is a semi-positive form, which can be assumed to be Kähler. However, if $\phi \equiv 0$ , we can equally well take $\phi_0 \equiv 0$ and get $\omega \equiv 0$ .

While the formula depends on the choice of $\phi_0$ , it follows from Lemma 3.2 that the difference of two such relative energies is constant:

\[E_{\phi_1}(\varphi)-E_{\phi_0}(\varphi)=E_{{\phi_1}}(\phi_0).\]

For $\phi_{0}=0$ , the formula reduces to $E(\varphi):=E_{0}(\varphi)=({1}/({n+1}))\int_{\Omega}\varphi(dd^{c}\varphi)^{n}$ .

This definition is motivated by the fact the $E_{\phi}$ is a primitive of the Monge–Ampère operator for smooth psh functions with $\phi$ -boundary values.

Lemma 3.2. Fix $\varphi \in \mathcal{T}_{\phi}^{\infty}(\Omega),\ v \in {\mathcal D}({\Omega})$ . Then $\varphi+tv \in\mathcal{T}_{\phi}^{\infty}(\Omega)$ for t small, and

\[{\frac{d}{dt} }_{|t=0}E_{\phi}(\varphi+tv)=\int_{\Omega} v (dd^c \varphi)^n.\]

In particular, $\varphi \mapsto E_{\phi}(\varphi)$ is increasing.

Here ${\mathcal D}({\Omega})$ denotes the space of smooth functions with compact support in $\Omega$ .

Set $\omega=dd^c \phi_0$ . The function $\psi_t=\varphi-\phi_0+tv$ has zero boundary values, and

\[E_{\phi}(\varphi+tv)=\frac{1}{n+1}\sum_{j=0}^{n}\int_{\Omega}\psi_t (\omega+dd^{c}\psi_t)^{j}\wedge \omega^{n-j}.\]

It follows from Stokes theorem, as all functions involved in the integration by parts are identically zero on $\partial \Omega$ , that

\begin{align*}& (n+1) {\frac{d}{dt} }E_{\phi}(\varphi+tv) \\&\quad =\sum_{j=0}^{n}\int_{\Omega} \dot{\psi_t} (\omega+dd^{c}\psi_t)^{j}\wedge \omega^{n-j}+\sum_{j=1}^{n}\int_{\Omega} j {\psi_t} \,dd^c \dot{\psi_t} \wedge (\omega+dd^{c}\psi_t)^{j-1}\wedge \omega^{n-j} \\&\quad =\sum_{j=0}^{n}\int_{\Omega} \dot{\psi_t} (\omega+dd^{c}\psi_t)^{j}\wedge \omega^{n-j}+\sum_{j=1}^{n}\int_{\Omega} j \dot{\psi_t} \,dd^c {\psi_t} \wedge (\omega+dd^{c}\psi_t)^{j-1}\wedge \omega^{n-j} \\&\quad =\sum_{j=0}^{n}\int_{\Omega} (j+1) \dot{\psi_t} (\omega+dd^{c}\psi_t)^{j}\wedge \omega^{n-j}-\sum_{j=1}^{n}\int_{\Omega} j \dot{\psi_t} (\omega+dd^{c}\psi_t)^{j-1}\wedge \omega^{n-j+1} \\&\quad = (n+1) \int_{\Omega} \dot{\psi_t} (\omega+dd^{c}\psi_t)^n,\end{align*}

writing $dd^c \psi_t=(\omega+dd^c \psi_t)-\omega$ in the third line, and then distributing and relabelling so as to obtain a telescopic series. The formula follows for $t=0$ .

In short, the derivative of $E_{\phi}$ is the complex Monge–Ampère operator $(dd^c \varphi)^n$ which is a positive measure. It follows that $\varphi \mapsto E_{\phi}(\varphi)$ is increasing.

3.1.2 Continuous setting

The previous result extends to the case of continuous psh functions that are not necessarily strictly psh. Recall that

\[\mathcal{T}_{\phi}(\Omega):=\bigg\{\varphi\in PSH(\Omega)\cap C^{0}(\bar{\Omega}),\ \varphi_{|\partial \Omega}=\phi\text{ and } \int_{\Omega} (dd^c \varphi)^n <+\infty\bigg\}.\]

We would like to extend Lemma 3.2 to this less-regular setting. As $\varphi+tv$ is not necessarily psh, we need to project it onto the cone of all psh functions. The following result will thus be useful.

Lemma 3.3. Fix $\varphi\in \mathcal{T}_{\phi}(\Omega)$ and $f \in {\mathcal D}({\Omega})$ . Then $P(\varphi+f)\in\mathcal{T}_{\phi}(\Omega)$ where

\[P(\varphi+f):=\sup\{\psi \in PSH(\Omega),\ \psi \leq \varphi+f\}.\]

Moreover, $(dd^{c}P(\varphi+f))^{n}$ is supported on the contact set $\{P(\varphi+f)=\varphi+f\}$ .

Solving Dirichlet problems in small ‘balls’ not containing the singular point, it follows from a balayage argument that the Monge–Ampère measure of the envelope $(dd^{c}P(\varphi+f))^{n}$ is supported on the contact set $\{P(\varphi+f)=\varphi+f\}$ .

We extend $E_{\phi}(\cdot)$ to $\mathcal{T}_{\phi}(\Omega)$ by monotonicity, setting

\[E_{\phi}(\varphi):=\inf\{E_{\phi}(\psi),\ \psi \in\mathcal{T}_{\phi}^{\infty}(\Omega)\text{ and } \varphi \leq \psi\}.\]

It has been observed by Berman and Boucksom (in the setting of compact Kähler manifolds [Reference Berman and BoucksomBB10]) that $E_{\phi} \circ P$ is still differentiable, with $(E_{\phi} \circ P)'=E_{\phi}'\circ P$ . This result extends to our local singular setting.

Proposition 3.4. Fix $\varphi\in \mathcal{T}_{\phi}(\Omega)$ and $f\in\mathcal{D}(\Omega)$ . Then $t\to E_{\phi}(P(\varphi+tf))$ is differentiable and

\[{\frac{d}{dt}}_{|t=0}E_{\phi}(P(\varphi+tf)) =\int_{\Omega}f(dd^{c}\varphi)^{n}.\]

Proof. The proof is very similar to that in the compact case, we provide it as a courtesy to the reader. Set $\varphi_{t}:=P(\varphi+tf)$ . By Lemma 3.5 we have

(3.2) \begin{equation}\int_{\Omega}(\varphi_{t}-\varphi)(dd^{c}\varphi_{t})^{n}\leq E_{\phi}(\varphi_{t})-E_{\phi}(\varphi)\leq \int_{\Omega}(\varphi_{t}-\varphi)(dd^{c}\varphi)^{n}.\end{equation}

Since $\varphi_t-\varphi \leq tf$ , the second inequality yields

\[\limsup_{t\to 0^{+}}\frac{E_{\phi}(\varphi_{t})-E_{\phi}(\varphi)}{t}\leq \int_{X}f(dd^{c}\varphi)^{n},\]

and $\liminf_{t\to 0^{-}}(({E_{\phi}(\varphi_{t})-E_{\phi}(\varphi)})/{t})\geq\int_{X}f(dd^{c}\varphi)^{n}$ .

It follows from Lemma 3.3 that $(dd^{c}\varphi_{t})^{n}$ is supported on $\{\varphi_{t}=\varphi+tf\}$ , hence the first inequality in (3.2) yields

\[\int_{\Omega}\frac{\varphi_{t}-\varphi}{t}(dd^{c}\varphi_{t})^{n}=\int_{\Omega}f(dd^{c}\varphi_{t})^{n}.\]

Now $(dd^{c}\varphi_{t})^{n}\to (dd^{c}\varphi)^{n}$ weakly since $\varphi_{t}\to \varphi$ uniformly, therefore

\[\liminf_{t\to 0^{+}}\frac{E_{\phi}(\varphi_{t})-E_{\phi}(\varphi)}{t}\geq\liminf_{t\to 0^{+}}\int_{\Omega}f(dd^{c}\varphi_{t})^{n}=\int_{\Omega}f(dd^{c}\varphi)^{n},\]

and

\[\limsup_{t\to0^{-}}\frac{E_{\phi}(\varphi_{t})-E_{\phi}(\varphi)}{t}\leq\limsup_{t\to 0^{-}}\int_{\Omega}f(dd^{c}\varphi_{t})^{n}=\int_{\Omega}f(dd^{c}\varphi)^{n}.\]

Lemma 3.5. For any $\varphi_{1},\varphi_{2}\in \mathcal{T}_{\phi}(\Omega)$ ,

(3.3) \begin{equation}\int_{\Omega}(\varphi_{1}-\varphi_{2})(dd^{c}\varphi_{1})^{n}\leq E_{\phi}(\varphi_{1})-E_{\phi}(\varphi_{2})\leq\int_{\Omega}(\varphi_{1}-\varphi_{2})(dd^{c}\varphi_{2})^{n},\end{equation}

while if $\varphi_{1}\leq \varphi_{2}$ , then

(3.4) \begin{equation}E_{\phi}(\varphi_{1})-E_{\phi}(\varphi_{2})\leq \frac{1}{n+1}\int_{X}(\varphi_{1}-\varphi_{2})(dd^{c}\varphi_{1})^{n}.\end{equation}

The energy is continuous along decreasing sequence in $\mathcal{T}_{\phi}(\Omega)$ .

Proof. It follows from Stokes theorem that

\[E_{\phi}(\varphi_{1})-E_{\phi}(\varphi_{2})=\frac{1}{n+1}\sum_{j=0}^{n}\int_{\Omega}(\varphi_{1}-\varphi_{2})(dd^{c}\varphi_{1})^{j}\wedge (dd^{c}\varphi_{2})^{n-j}\]

and

\[\int_{\Omega}(\varphi_{1}-\varphi_{2})(dd^{c}\varphi_{1})^{j+1}\wedge(dd^{c}\varphi_{2})^{n-j-1}\leq \int_{\Omega}(\varphi_{1}-\varphi_{2})(dd^{c}\varphi_{1})^{j}\wedge (dd^{c}\varphi_{2})^{n-j},\]

for any $j=0,\ldots,n-1$ . The desired inequalities follow.

Let $\varphi_{j}\in \mathcal{T}_{\phi}(\Omega)$ be a decreasing sequence converging to $\varphi \in\mathcal{T}_{\phi}(\Omega)$ . We obtain

\[0\leq E_{\phi}(\varphi_{j})-E_{\phi}(\varphi)\leq\int_{\Omega}(\varphi_{j}-\varphi)(dd^{c}\varphi)^{n}\to 0\]

as $j\to +\infty$ by the monotone convergence theorem.

3.1.3 Finite energy class

Let $PSH_{\phi}(\Omega)$ denote the set of decreasing limits of functions in $\mathcal{T}_{\phi}(\Omega)$ . We extend $E_{\phi}$ to $PSH_{\phi}(\Omega)$ by monotonicity, setting

\[E_{\phi}(\varphi):=\inf\{E_{\phi}(\psi),\ \psi \in \mathcal{T}_{\phi}(\Omega)\text{ and } \varphi \leq \psi\}.\]

This ‘finite energy class’ has been introduced and studied intensively by Cegrell for smooth domains of $\mathbb C^n$ . His analysis extends to our mildly singular context. We summarize here the key facts that we shall need.

• – functions in $\mathcal{E}^1(\Omega)$ have zero Lelong numbers;

• – the sets $\mathcal{G}_b(\Omega)=\{ \varphi \in \mathcal{E}^1(\Omega),\ -b \leq E_{\phi}(\varphi) \}$ are compact for all $b \in \mathbb R$ ;

• – Lemma 3.5 holds if $\varphi_1,\varphi_2 \in \mathcal{E}^1(\Omega)$ ;

• – if $\mu$ is a non-pluripolar probability measure such that $\mathcal{E}^1(\Omega)\subset L^1(\mu)$ , then there exists a unique function $v \in \mathcal{E}^1(\Omega) \cap\mathcal{F}_1(\Omega)$ such that $\mu=(dd^c v)^n$ .

We refer the reader to [Reference CegrellCeg98, Theorems 3.8, 7.2 and 8.2] for the proof of these results when $\Omega$ is smooth.

3.2.1 Euler–Lagrange equation

The Ding functional is

\[F_{\gamma}(\varphi):=E_{\phi}(\varphi)+\frac{1}{\gamma}\log \int_{\Omega}e^{-\gamma \varphi}\,d\mu_{p}.\]

Proof. Assume that $\varphi$ maximizes $F_{\gamma}$ over $\mathcal{T}_{\phi}(\Omega)$ , fix $f \in\mathcal{D}(\Omega)$ , and set $\varphi_{t}:=P(\varphi+tf)$ . Then

\[E_{\phi}(\varphi_{t})+\frac{1}{\gamma}\log \int_{\Omega}e^{-\gamma(\varphi+tf)}\,d\mu_{p}\leq F_{\gamma}(\varphi_{t})\leq F_{\gamma}(\varphi),\]

i.e. the function $t\to E_{\phi}(\varphi_{t})+({1}/{\gamma})\log\int_{\Omega}e^{-\gamma(\varphi+tf)}\,d\mu_{p}$ reaches its maximum at $t=0$ . Combining Proposition 3.4 and Lemma 3.9, we obtain

\[0=\frac{d}{dt}\bigg(E_{\phi}(\varphi_{t})+\frac{1}{\gamma}\log\int_{\Omega}e^{-\gamma(\varphi+tf)}\,d\mu_{p}\bigg)=\int_{\Omega}f\bigg((dd^{c}\varphi)^{n}-\frac{e^{-\gamma\varphi}\,d\mu_{p}}{\int_{\Omega}e^{-\gamma\varphi}\,d\mu_{p}}\bigg),\]

i.e. $\varphi$ solves (3.1).

Lemma 3.9. Fix $\varphi \in \mathcal{T}_{\phi}(\Omega)$ , $f \in {\mathcal D}(\Omega)$ , and set $\psi_t:=\varphi+tf$ . Then

\[\frac{d}{dt}\bigg(\log \int_{\Omega}e^{-\gamma\psi_t}\,d\mu_{p}\bigg)_{\!\!|t=0}=-\gamma\frac{\int_{\Omega}fe^{-\gamma\varphi}\,d\mu_{p}}{\int_{\Omega}e^{-\gamma\varphi}\,d\mu_{p}}.\]

Proof. By the chain rule, it is enough to observe that

\[\frac{\int_{\Omega}e^{-\gamma \psi_t}\,d\mu_{p}-\int_{\Omega}e^{-\gamma \varphi}\,d\mu_{p}}{t}=-\int_{\Omega}e^{-\gamma\varphi}\bigg(\frac{1-e^{-t\gamma f}}{t} \bigg)\,d\mu_{p}\]

and to apply the Lebesgue dominated convergence theorem to conclude.

3.2.2 Coercivity

In order to solve (3.1), one is lead to try and maximize $F_{\gamma}$ . We show in § 6 that when $F_{\gamma}$ is coercive, the complex Monge–Ampère equation (3.1) admits a solution $\varphi\in \mathcal{T}_{\phi}(\Omega)$ which is smooth away from p.

Definition 3.10. The functional $F_{\gamma}$ is coercive if there exists $A,B>0$ such that

\[F_{\gamma}(\varphi)\leq A E_{\phi}(\varphi)+B\]

for all $\varphi\in \mathcal{T}_{\phi}(\Omega)$ .

We observe in Lemma 3.13 that $E_{\phi}(\varphi) \leq C(\phi_0)$ is bounded from above, uniformly in $\varphi \in \mathcal{T}_{\phi}(\Omega)$ . In particular, if $F_{\gamma}$ is coercive with slope $A>0$ , then it is coercive for any $A'\in (0,A]$ . We can thus assume, without loss of generality, that $A\in(0,1)$ . The coercivity property is then equivalent to

\[\frac{1}{\gamma}\log \int_{\Omega}e^{-\gamma\varphi}\,d\mu_{p}\leq (1-A)(-E_{\phi}(\varphi))+B,\]

or, equivalently, to the following Moser–Trudinger inequality

\[\bigg(\int_{X}e^{-\gamma \varphi}\,d\mu_{p}\bigg)^{{1}/{\gamma}} \leq C e^{(1-A)(-E_{\phi}(\varphi))}.\]

We summarize these observations in the following.

1. (i) $F_{\gamma}$ is coercive;

2. (ii) there exists $C_{\gamma}>0$ and $a\in (0,1)$ such that for all $\varphi \in\mathcal{T}_{\phi}(\Omega)$ ,

   \[\bigg(\int_{\Omega}e^{-\gamma\varphi}\,d\mu_{p}\bigg)^{\!\!1/\gamma}\leq C_{\gamma} e^{-a E_{\phi}(\varphi)}.\]

It follows from Hölder inequality (and the normalization $\mu_p(\Omega)=1$ ) that if

(3.5) \begin{equation}\bigg(\int_{\Omega}e^{-\gamma\varphi}\,d\mu_{p}\bigg)^{\!\!1/\gamma}\leq Ce^{-E_{\phi}(\varphi)}\end{equation}

holds for $\gamma>0$ , then it also holds for $\gamma'<\gamma$ . We thus introduce the following critical exponent.

Definition 3.12. We set

• – if $F_{\gamma}$ is coercive, then $\gamma \leq \gamma_{\rm crit}(X,p)$ ;

• – conversely, if $\gamma < \gamma_{\rm crit}(X,p)$ , then $F_{\gamma}$ is coercive.

Proof. Consider $\tilde{\phi_0}=P(\phi):=\sup \{ \psi, \psi \in \mathcal{T}_{\phi}(\Omega)\}$ . This is the largest psh function in $\Omega$ such that $\tilde{\phi_0} =\phi$ on $\partial \Omega$ . The reader can check that it is continuous on ${\overline \Omega}$ and satisfies $(dd^c \tilde{\phi_0})^n=0$ in $\Omega$ . If $\varphi \in \mathcal{T}_{\phi}(\Omega)$ , then $\varphi \leq \tilde{\phi_0}$ , hence $E_{\tilde{\phi_0}}(\varphi) \leq 0$ . Thus,

\[E_{{\phi_0}}(\varphi) = E_{\tilde{\phi_0}}(\varphi)+E_{\phi_0}(\tilde{\phi_0})\leq E_{\phi_0}(\tilde{\phi_0}),\]

hence $E_{{\phi_0}}(\varphi)$ is uniformly bounded from above independently of the choice of $\phi_0$ .

Similarly, the coercivity of $F_{\gamma}$ or the inequality (3.5) do not depend on the choice of $\phi_0$ . In the remainder of this proof we thus assume that $\phi_0=P(\phi)$ . Since $E_{\phi}(\varphi) \leq 0$ in this case, it follows from Proposition 3.11 that if $F_{\gamma}$ is coercive, then (3.5) holds, hence $\gamma \leq \gamma_{\rm crit}(X,p)$ .

Conversely, assume $\gamma < \gamma_{\rm crit}(X,p)$ . Fix $\gamma<\gamma'<\gamma_{\rm crit}(X,p)$ and $\lambda=\gamma/\gamma'<1$ . We can assume that $\lambda$ is close to 1. We assume first that $\phi \equiv 0$ . For $\varphi \in \mathcal{T}_{0}(\Omega)$ we observe that $\lambda \varphi \in\mathcal{T}_{0}(\Omega)$ , with $E_0(\lambda \varphi)=\lambda^{n+1} E_0(\varphi)$ . The Moser–Trudinger (3.5) applied to $(\gamma',\lambda \varphi)$ thus yields

\[\bigg(\int_{\Omega}e^{-\gamma\varphi}\,d\mu_{p}\bigg)^{\!\!1/\gamma}\leq C_{\gamma'} e^{-\lambda^n E_0(\varphi)},\]

so that $F_{\gamma}$ is coercive.

We now treat the general case, replacing the condition $\phi \equiv 0$ by $(dd^c \phi_0)^n \equiv0$ . For $\varphi \in \mathcal{T}_{\phi}(\Omega)$ we observe that $\varphi_{\lambda}=\lambda \varphi+(1-\lambda)\phi_0 \in \mathcal{T}_{\phi}(\Omega)$ , with $\varphi_{\lambda}-\phi_0=\lambda(\varphi-\phi_0) \leq 0$ and

\begin{align*}(n+1) E_{\phi_0}(\varphi_{\lambda}) &=\lambda \sum_{j=0}^{n-1} \int_{\Omega} (\varphi-\phi_0) (dd^c \varphi_{\lambda})^j \wedge (dd^c \phi_0)^{n-j} \\&= \lambda \sum_{k=1}^n \sum_{j=k}^{n}\bigg(\begin{array}{c}j \\k \end{array}\bigg)\lambda^k (1-\lambda)^{j-k}\int_{\Omega} (\varphi-\phi_0) (dd^c \varphi)^k \wedge (dd^c \phi_0)^{n-k}.\end{align*}

Now $\sum_{j=k}^{n} \big(\begin{smallmatrix} j \\ k \end{smallmatrix}\big) \lambda^k(1-\lambda)^{j-k} \leq a <1$ for all $1 \leq k \leq n$ , since $\lambda<1$ can be chosen arbitrarily close to 1. Thus, $E_{\phi_0}(\varphi_{\lambda}) \geq a \lambda E_{\phi_0}(\varphi)$ and the result follows as previously by applying the Moser–Trudinger inequality (3.5) to $\varphi_{\lambda}$ .

3.3.1 Enlarging the domain

We first reduce to the case of zero boundary values.

Proof. Consider indeed $\varphi \in {\mathcal T}_{\phi}(\Omega)$ and set

\[\varphi_2:=\sup \{u \in {\mathcal T}_0(\Omega_2),\ \text{such that}\ u \leq \varphi \text{ in } \Omega\}.\]

The family ${\mathcal F}$ of such functions is non-empty, as it contains $A \rho_2$ for some large $A>1$ , where $\rho_2$ is a psh defining function for $\Omega_2$ . Moreover, ${\mathcal F}$ is uniformly bounded from above by 0, so the upper envelope $\varphi_2$ is well defined and psh, as ${\mathcal F}$ is compact. Finally, $\varphi_2 \geq A \rho_2$ , hence $\varphi_2$ has zero boundary values, and $\varphi_2$ is lower semi-continuous, as an envelope of continuous functions, thus $\varphi_2 \in {\mathcal T}_0(\Omega_2)$ .

Since $\varphi_2 \leq \varphi$ in $\Omega$ , we observe that

\[\int_{\Omega} e^{-\gamma \varphi} \,d\mu_p \leq \int_{\Omega_2} e^{-\gamma \varphi_2} \,d\mu_p.\]

Our claim will follow if we can show that, on the other hand, $E_0(\varphi_2) \geq E_{\phi}(\varphi)$ .

If $\varphi$ is smooth one can show, by adapting standard techniques, that:

• – $\varphi_2$ is ${\mathcal C}^{1,\overline{1}}$ -smooth in $\Omega_2 \setminus \{p\}$ ;

• – $(dd^c \varphi_2)^n=0$ in $\Omega_2 \setminus \Omega$ and $(dd^c \varphi_2)^n={{\bf 1}}_{\{ \varphi_2=\varphi\}} (dd^c \varphi)^n$ in $\overline{\Omega}$ .

Assuming $\phi \geq 0$ and $\phi_0=\sup \{\psi,\ \psi \in {\mathcal T}_{\phi}(\Omega) \}$ , we infer

\begin{align*}E_0(\varphi_2) &= \frac{1}{n+1}\int_{\Omega} \varphi_2 (dd^c \varphi_2)^n= \frac{1}{n+1}\int_{\Omega} {{\bf 1}}_{\{ \varphi_2 =\varphi\}} \varphi (dd^c \varphi)^n \\&\geq \frac{1}{n+1}\int_{\Omega} \varphi (dd^c \varphi)^n\geq \frac{1}{n+1}\int_{\Omega} (\varphi-\phi) (dd^c \varphi)^n \geq E_{\phi}(\varphi).\end{align*}

To get rid of the assumption $\phi \geq 0$ , we observe that the Moser–Trudinger inequality holds for given boundary data $\phi$ if and only if does so for $\phi+c$ , for any $c \in \mathbb R$ (by changing $\varphi$ in $\varphi+c$ ).

Using Lemma 2.5, one can uniformly approximate $\varphi$ by a sequence of smooth $\varphi_{j}\in {\mathcal T}_{\phi}(\Omega)$ . The corresponding sequence $\varphi_{2,j}$ uniformly converges to $\varphi_2$ , and we obtain the desired inequality by passing to the limit in $E_0(\varphi_{2,j}) \geq E_{\phi}(\varphi_{j})$ .

3.3.2 Rescaling

We now assume that $\phi=0$ and reformulate the coercivity property after an appropriate rescaling. Observe that for any $\lambda>0$ , the map

\[\varphi\in\mathcal{T}_{0}(\Omega) \mapsto \lambda \varphi \in \mathcal{T}_{0}(\Omega)\]

is a homeomorphism. This allows us to reformulate the Moser–Trudinger inequality.

1. (a) $F_{\gamma}$ is coercive;

2. (b) there exists $C>0,\ B\in(0,1)$ such that for all $\varphi\in\mathcal{T}_{0}(\Omega)$ , $\int_{\Omega}e^{-\varphi}\,d\mu_{p}\leq Ce^{-({B}/{\gamma^{n}})E_{0}(\varphi)}$ .

In particular, we can define the following critical exponent.

Definition 3.16. We set

\[\beta_{\rm crit}:=\inf \bigg\{\beta>0;\ \sup_{\varphi\in\mathcal{T}_{0}(\Omega)}\bigg(\int_{\Omega}e^{-\varphi}\,d\mu_{p}/e^{-\beta E_{0}(\varphi)}\bigg)<+\infty\bigg\}.\]

Note that $\gamma_{\rm crit}^{n}=1/\beta_{\rm crit}$ , hence it follows from the previous analysis that $F_{\gamma}$ is coercive if and only if $\gamma< \beta_{\rm crit}^{-1/n}$ . When $p\in X$ is smooth, it has been shown in [Reference Guedj, Kolev and YeganefarGKY13, Theorem 9] and independently [Reference Berman and BerndtssonBB22, Theorem 1.5] that

\[\beta_{\rm crit}(\Omega)=\frac{1}{(n+1)^{n}},\]

or, equivalently, that $\gamma_{\rm crit}(\Omega)=n+1$ . In particular, it does not depend on $\Omega$ .

We extend this independence to the case when p is the vertex of a cone over a Fano manifold.

Recall that $K_Y=f^*K_X+aE$ , where a is the discrepancy of Y along E. The adjunction formula yields $(K_Y+E)_{|E}=K_E$ , hence $K_Z^*=(a+1)L$ . In particular, $a=-1+1/r$ and (X,p) is log terminal. The fibration $\pi:Y=L^* \rightarrow Z$ yields $K_Y=\pi^*(K_Z+L)$ , hence $f^*K_X=\pi^*(K_Z+L)-aE$ .

Since $\pi^*(K_Z+L)$ is $\mathbb C^*$ -invariant, we can cook up an adapted volume form $\mu_p=\mu_1 \cdot\mu_E$ with $D_{\lambda}^* \mu_1=\mu_1$ while $D_{\lambda}^* \mu_E=|\lambda|^{2a} \mu_E$ . For $\varphi\in {\mathcal T}_0(\Omega_{\lambda})$ we set $\varphi_{\lambda}=\varphi \circ D_{\lambda} \in {\mathcal T}_0(\Omega)$ and observe that

\[|\lambda |^{2a}\int_{\Omega} e^{-\gamma \varphi_{\lambda}} \,d\mu_p=\int_{\Omega_{\lambda}} e^{-\gamma \varphi} \,d\mu_p,\]

while $E_{\Omega,0}(\varphi_{\lambda})=E_{\Omega_{\lambda},0}(\varphi)$ . The conclusion follows.

We conjecture in § 5 that $ \gamma_{\rm crit}(X,p) \stackrel{?}{=}(({n+1})/{n}) \widehat{\mathrm{vol}}(X,p)^{1/n} $ and give partial results towards this equality, which again suggest that $\gamma_{\rm crit}(X,p) $ should be independent of $(\Omega,\phi)$ . In the whole article, we therefore use the notation $\gamma_{\rm crit}(X,p) $ instead of the more precise, and heavy, $\gamma_{\rm crit}(X,p,\Omega,\phi)$ .

5.1.1 Entropy

We let $\mathcal{P}(\Omega)$ denote the set of probability measures on $\Omega$ . Given two measures $\mu,\nu\in\mathcal{P}(\Omega)$ , the relative entropy of $\nu$ with respect to $\mu$ is

\[H_{\mu}(\nu):=\int_{\Omega}\frac{d\nu}{d\mu}\log \frac{d\nu}{d\mu} \,d\mu=\int_{\Omega}\log\frac{d\nu}{d\mu} \,d\nu\]

if $\nu$ is absolutely continuous with respect to $\mu$ , and as $H_{\mu}(\nu):=+\infty$ otherwise.

Given $\mu\in\mathcal{P}(X)$ , the relative entropy $H_{\mu}(\cdot)$ is the Legendre transform of the convex functional $g \in \mathcal{C}^{0}(\Omega)\cap L^{\infty}(\Omega) \mapsto \log\int_{\Omega}e^{g}\,d\mu \in \mathbb R$ , i.e.

\[H_{\mu}(\nu)=\sup_{g\in\mathcal{C}^{0}(\Omega)\cap L^{\infty}(\Omega)}\bigg(\int_{\Omega}g\,d\nu-\log\int_{\Omega}e^{g}\,d\mu\bigg).\]

We shall need the following duality result.

Lemma 5.2 [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ19, Lemma 2.11]. Fix $\mu\in\mathcal{P}(\Omega)$ . Then

\[\log\int_{\Omega}e^{g}\,d\mu=\sup_{\nu\in\mathcal{P}(\Omega)}\bigg(\int_{\Omega}g \,d\nu-H_{\mu}(\nu)\bigg)\]

for each lower semi-continuous function $g:\Omega\to \mathbb R\cup \{+\infty\}$ .

Recall that we have normalized the adapted volume form so that $\mu_{p}\in\mathcal{P}(\Omega)$ .

Corollary 5.3. Fix $0<\alpha<\alpha(X,\mu_p)$ . Then there exists $C_{\alpha}>0$ such that

\[H_{\mu_{p}}(\nu)\geq -\alpha\int_{\Omega}\varphi \,d\nu -C_{\alpha}\]

for all $\varphi\in\mathcal{F}_1(\Omega)$ and for all $\nu\in\mathcal{P}(\Omega)$ such that $H_{\mu_{p}}(\nu)<+\infty$ .

This corollary shows, in particular, that $\mathcal{F}_{1}(\Omega)\subset L^{1}(\nu)$ for any probability measure $\nu\in\mathcal{P}(X)$ with finite $\mu_p$ -entropy. Since the measure $\nu$ is moreover non-pluripolar, the following result is a consequence of Theorem 3.7.

Proposition 5.4. Fix $\nu\in\mathcal{P}(\Omega)$ such that $H_{\mu_{p}}(\nu)<+\infty$ . Then there exists a unique $v\in \mathcal{F}_{1}(\Omega) \cap \mathcal{E}^1(\Omega)$ such that

\[\nu=(dd^{c}v)^{n}.\]

5.1.2 Proof of Theorem 5.1

The proof is similar to the derivation of the Moser–Trudinger inequality from Brezis–Merle inequality by Berman and Berndtsson, see [Reference Berman and BerndtssonBB22, § 4.2]. Fix $\varphi\in\mathcal{T}_{\phi}(\Omega)$ and $0<\alpha<\alpha(X,\mu_p)$ . By Lemma 5.2 for any $\epsilon>0$ there exists $\nu_{\epsilon}\in\mathcal{P}(\Omega)$ such that $H_{\mu_{p}}(\nu_{\epsilon})<+\infty$ and

(5.1) \begin{equation}\log\int_{\Omega}e^{-(({n+1})/{n})\alpha \varphi}\,d\mu_{p}\leq \epsilon-\frac{n+1}{n}\alpha\int_{\Omega}\varphi \,d\nu_{\epsilon}-H_{\mu_{p}}(\nu_{\epsilon}).\end{equation}

Proposition 5.4 ensures the existence of $v_{\epsilon}\in\mathcal{F}_{1}(\Omega)\cap \mathcal{E}^1(\Omega)$ such that $\nu_{\epsilon}=(dd^{c}v_{\epsilon})^{n}$ . It follows, moreover, from Corollary 5.3 thatd

(5.2) \begin{equation}H_{\mu_{p}}(\nu_{\epsilon})\geq -\alpha\int_{\Omega}v_{\epsilon} \,d\nu_{\epsilon}-C_{\alpha}.\end{equation}

Combining (5.1) and (5.2) we obtain

(5.3) \begin{equation}\log\int_{\Omega}e^{-(({n+1})/{n})\alpha \varphi}\,d\mu_{p}\leq \epsilon+C_{\alpha}-\frac{n+1}{n}\alpha\int_{\Omega}\varphi \,d\nu_{\epsilon}+\alpha\int_{\Omega}v_{\epsilon} \,d\nu_{\epsilon}.\end{equation}

We observe that

\begin{align*}& -\frac{n+1}{n}\alpha\int_{\Omega}\varphi \,d\nu_{\epsilon}+ \alpha\int_{\Omega}v_{\epsilon}\,d\nu_{\epsilon}= \frac{n+1}{n}\alpha \int_{\Omega}(v_{\epsilon}-\varphi) (dd^c v_{\epsilon})^n-\frac{\alpha}{n} \int_{\Omega} v_{\epsilon} (dd^c v_{\epsilon})^n \\&\quad \leq -\frac{n+1}{n}\alpha E_{\phi}(\varphi)+\frac{\alpha}{n} \bigg\{(n+1) E_{\phi}(v_{\varepsilon})- \int_{\Omega} v_{\epsilon} (dd^c v_{\epsilon})^n\bigg\}\end{align*}

by using Lemma 3.5 (the latter has been stated for functions in ${\mathcal T}_{\phi}(\Omega)$ , it easily extends to the class ${\mathcal F}_{1}(\Omega)\cap\mathcal{E}^1(\Omega)$ by approximation). Since $v_{\varepsilon} \leq \phi_0$ and $E_{\phi}(\phi_0)=0$ , the same lemma ensures

\[(n+1) E_{\phi}(v_{\varepsilon})- \int_{\Omega} v_{\epsilon} (dd^c v_{\epsilon})^n\leq -\int_{\Omega} \phi_0 (dd^c v_{\varepsilon})^n \leq -\inf_{\Omega} \phi_0,\]

using that $\nu_{\epsilon}=(dd^c v_{\epsilon})^n$ is a probability measure. Altogether this yields

\[\log \int_{\Omega}e^{-(({n+1})/{n})\alpha\varphi}\,d\mu_{p} \leq \epsilon+C_{\alpha} -\frac{\alpha}{n} \inf_{\Omega} \phi_0-\frac{n+1}{n}\alpha E_{\phi}(\varphi).\]

Letting $\epsilon\searrow 0$ we conclude that

\[\bigg(\int_{\Omega}e^{-(({n+1})/{n})\alpha\varphi}\,d\mu_{p}\bigg)^{\!\!{n}/({(n+1)\alpha}) }\leq C_{\alpha}' e^{-E_{\phi}(\varphi)}\]

for any function $\varphi\in\mathcal{T}_{\phi}(\Omega)$ . Thus, $\gamma_{\rm crit}(X,p) \geq(({n+1})/{n})\alpha(X,\mu_p)$ .

5.2.1 Bounding the $\alpha$ -invariant by the normalized volume

For any $\epsilon>0$ and $\mathcal{I}$ coherent ideal sheaf supported at 0, the function

\[\psi_{\mathcal{I},\epsilon}:=\psi_{\mathcal{I},\lambda,\epsilon},\quad \text{with}\ \lambda=\bigg(\frac{1-\epsilon}{e(X,\mathcal{I})}\bigg)^{\!\!1/n}\]

given by Lemma 5.7, belongs to $\mathcal{F}_{1}(\Omega)$ and yields

\[\tilde{\alpha}(X,\mu_p)\leq c(\psi_{\mathcal{I},\epsilon})=\frac{1}{(1-\epsilon)^{1/n}}\mathrm{lct}(X,\mathcal{I})e(X,\mathcal{I})^{1/n}.\]

The latter equality is a consequence of Proposition 4.1. We conclude the proof by taking the infimum over all ${\mathcal I}$ and letting $\epsilon\searrow 0$ .

1. (i) $\psi_{\mathcal{I},\lambda,\epsilon}=\lambda\log(\sum_{j=1}^{m}\lvert f_{j}\rvert^{2})$ near 0 for local generators $f_{1},\ldots,f_{m}$ of $\mathcal{I}$ ;

2. (ii) $\lambda^{n}e(X,\mathcal{I}) \leq\int_{\Omega}(dd^{c}\psi_{\mathcal{I},\lambda,\epsilon})^{n} \leq\lambda^{n}e(X,\mathcal{I})+\epsilon$ .

Fix $f_{1},\ldots,f_{m}$ local generators of the ideal $\mathcal{I}$ and set $\psi_{\lambda}:=\lambda\log (\sum_{j=1}^{m}\lvert f_{j} \rvert^{2})$ . We can assume without loss of generality that the $f_j$ are well defined in $\Omega$ and normalized so that $\psi_{\lambda}\leq \phi_0-1$ in $\Omega$ . For $r>0$ , we consider

\[\varphi_{r}:=\sup \{u \in PSH(\Omega),\, u \leq \psi_{\lambda} \text{ in } B(r)\ \text{and}\, u \leq \phi_0 \text{ in } \Omega\}.\]

The corresponding family of psh functions is non-empty as it contains $\psi_{\lambda}$ . For $A>1$ large enough, the function

\[w_r=\left\{\begin{array}{ll}\psi_{\lambda} & \text{ in } B(r) ,\\\max(\psi_{\lambda}, A \rho +\phi_0) & \text{ in } \Omega \setminus B(r),\end{array}\right.\]

is psh and coincides with $A \rho+\phi_0$ near $\partial \Omega$ . It follows that:

• – $\varphi_r \in PSH(\Omega)$ with $\varphi_r =\phi$ on $\partial \Omega$ ;

• – $\varphi_r \equiv \psi_{\lambda}$ in B(r), hence $\lambda^{n}e(X,\mathcal{I}) \leq\int_{\Omega} (dd^{c} \varphi_r)^n$ ;

• – $(dd^{c} \varphi_r)^n=0$ in $\Omega \setminus \overline{B}(r)$ (balayage argument).

The family $r \mapsto \varphi_r$ increases, as $r>0$ decreases to 0, to some psh limit $\varphi$ whose Monge–Ampère measure $(dd^c \varphi)^n$ is concentrated at the origin. It follows from Bedford–Taylor continuity theorem that $(dd^c \varphi)^n $ is the weak limit of $(dd^c \varphi_r)^n \geq\lambda^{n}e(X,\mathcal{I}) \delta_0$ , hence $(dd^c \varphi)^n \geq \lambda^{n}e(X,\mathcal{I})\delta_0$ . Conversely, $\psi_{\lambda} \leq \varphi$ near 0, hence Demailly’s comparison theorem ensures that

\[(dd^c \varphi)^n(0) \leq (dd^c \psi_{\lambda})^n(0) \leq \lambda^{n}e(X,\mathcal{I}),\]

whence equality. Thus, $\phi_{\mathcal{I},\lambda,\epsilon}:=\varphi_{r_{\varepsilon}}$ satisfies the required properties.

5.2.2 Normalized volume versus uniform integrability

We refer the reader to Appendix A for a more algebraic approach based on [Reference Boucksom, de Fernex and FavreBdFF12], which moreover provides a slightly stronger result.

When (X,p) is smooth, it follows from [Reference Demailly and KollàrDK01] that $\tilde{\alpha}(X,\mu_p)={\alpha}(X,\mu_p)$ . The situation is, however, more subtle in the singular context (see § 5.3.2).

In a log resolution $\pi:\tilde{\Omega}\to \Omega$ , this boils down to $\int_{\tilde{\Omega}}e^{-\alpha \varphi\circ \pi}\prod_{i=1}^M \lvert s_i\rvert_{h_i}^{2a_i}\,dV<+\infty$ , where $s_i$ are holomorphic sections defining simple normal crossing exceptional divisors $E_1,\ldots,E_M$ , $K_{\tilde{\Omega}/\Omega}=\sum_{j=1}^M a_i E_i$ and where dV is a smooth volume form. The log terminal condition ensures that $a_i>-1$ for all $i=1,\ldots,M$ .

As $\alpha<\widehat{\mathrm{vol}}(X,p)^{1/n}\leq n$ , the integrability condition is equivalent to show that for any point $x\in \bigcup_{i=1}^M E_i$ there exists a small ball B(x,r) such that

(5.4) \begin{equation}\int_{B(x,r)}e^{-\alpha \varphi\circ \pi}\prod_{i=1}^M \lvert s_i\rvert_{h_i}^{2a_i} \,dV<+\infty.\end{equation}

Set $U:=\sum_{i: a_i\geq 0} a_i\log \lvert s_i \rvert_{h_i}^2$ , $V:=\alpha \varphi\circ \pi$ and $W:=-\sum_{i: a_i<0}a_i\log \lvert s_i\rvert_{h_i}^2$ . By [Reference Berman, Boucksom and JonssonBBJ21, Theorem B.5] the condition (5.4) holds if and only if there exists $\epsilon>0$ such that

(5.5) \begin{equation}\nu(U\circ g, F)+A_{\tilde{\Omega}}(F)\geq (1+\epsilon) \nu(V\circ g, F)+(1+\epsilon)\nu(W\circ g, F)\end{equation}

for any F prime divisor over $\tilde{\Omega}$ with center in a small ball $\overline{B(x,r')}\subset B(x,r)$ , i.e. $F\subset \Omega'$ for $g:\Omega'\to\tilde{\Omega}$ modification. Observe that

\begin{align*}\nu(U\circ g,F)+A_{\tilde{\Omega}}(F)-\nu(W\circ g, F)&= \mathrm{ord}_F( g^*K_{\tilde{\Omega}/\Omega})+1+\mathrm{ord}_F( K_{\Omega'/\tilde{\Omega}}) \\&= 1+\mathrm{ord}_F (K_{\Omega'/\Omega})=A_{\Omega}(F).\end{align*}

Thus, (5.5) becomes

(5.6) \begin{equation}\alpha(1+\epsilon)\leq \frac{A_{\Omega}(F)-\epsilon\nu(W\circ g,F)}{\nu(\varphi\circ \pi\circ g,F)}.\end{equation}

As $a_i>-1$ for all i, [Reference Berman, Boucksom and JonssonBBJ21, Theorem B.5] ensures the existence of $a>0$ such that $A_{\tilde{\Omega}}(F)\geq (1+a)\nu(W\circ g,F) $ for any prime divisor F over $\tilde{\Omega}$ as above. Thus,

\[A_{\tilde{\Omega}}(F)\leq A_\Omega(F)+\nu(W\circ g, F)\leq \frac{1}{1+a}A_{\tilde{\Omega}}(F)+A_\Omega(F),\]

and $\nu(W\circ g,F)\leq ({1}/({1+a}))A_{\tilde{\Omega}}(F)\leq ({1}/{a})A_\Omega(F)$ . Therefore, (5.6) holds if

(5.7) \begin{equation}\alpha(1+\epsilon)\leq \frac{a-\epsilon}{a}\frac{A_\Omega(F)}{\nu(\varphi\circ \pi\circ g,F)}.\end{equation}

Since $\varphi \in {\mathcal F}_1(\Omega)$ , it follows from the comparison theorem of Demailly [Reference DemaillyDem85, Theorem 4.2] that for a coherent ideal sheaf $\mathcal{I}$ supported at $p\in \Omega$ ,

(5.8) \begin{equation}1\geq \int_\Omega (dd^c \varphi)^n\geq \nu_\mathcal{I}(\varphi,p)^n\int_{\mathbb C^N}(dd^cf_\mathcal{I})^n\wedge [X]=\nu_\mathcal{I}(\varphi,p)^n e_p(\mathcal{I}),\end{equation}

where $f_\mathcal{I}= \log (\sum_i\lvert f_i\rvert^2)$ for generators $\{f_i\}_i$ of $\mathcal{I}$ and

\[\nu_\mathcal{I}(\varphi,p):=\sup\{s>0: \varphi\leq s f_\mathcal{I}+ O(1)\}=\min_{G}\frac{\nu(\varphi\circ\pi\circ g,G)}{\mathrm{ord}_G \mathcal{I}},\]

where $\pi\circ g: \Omega'\to \Omega$ is a log resolution for $\mathcal{I}$ and the minimum is over all exceptional divisors of $\Omega'\to \Omega$ . Lemma 5.9 ensures that for any prime divisor F and $\delta>0$ there exists an ideal $\mathcal{I}$ such that

\begin{align*}\frac{A_\Omega(F)}{\nu(\varphi\circ \pi\circ p,F)}&\geq (1-\delta)\frac{A_\Omega(F)}{\mathrm{ord}_F\mathcal{I}}(\nu_\mathcal{I}(\varphi,p))^{-1}\geq(1-\delta) \frac{A_\Omega(F)}{\mathrm{ord}_F\mathcal{I}} e_p(\mathcal{I})^{1/n} \\&\geq (1-\delta){\rm lct}(X,\mathcal{I})e_p(\mathcal{I})^{1/n}\geq (1-\delta)\widehat{\mathrm{vol}}(X,p)^{1/n}.\end{align*}

Thus, (5.7) holds if $\alpha(1+\epsilon)\leq(({(a-\epsilon)(1-\delta)})/{a})\widehat{\mathrm{vol}}(X,p)^{1/n}$ , concluding the proof.

Lemma 5.9. Fix $\varphi\in \mathcal{F}_1(\Omega)$ and $F\subset \Omega'$ prime divisor such that $\pi\circ g(F)=p$ . For any $\epsilon>0$ , there exists a coherent ideal sheaf $\mathcal{I}$ supported at p such that

\[\nu_\mathcal{I}(\varphi,p)\geq(1-\epsilon)\frac{\nu(\varphi\circ \pi\circ g,F)}{\mathrm{ord}_F (\mathcal{I})}.\]

Proof. Let $c:=\nu(\varphi\circ \pi\circ g,F)$ and for $c^{\prime}\in \mathbb Q, c^{\prime}\leq c$ , set

\[\mathcal{A}_{mc^{\prime}}(F):=\{f\in \mathcal{O}_{X,p}: \mathrm{ord}_F (f\circ \pi\circ g)\geq mc^{\prime}\}\]

for $m\in \mathbb N$ divisible enough. Then $\mathcal{A}_{mc^{\prime}}(F)$ is an ideal sheaf and

(5.9) \begin{equation}\limsup_{m\to +\infty} \frac{\mathrm{ord}_F (\mathcal{A}_{mc^{\prime}}(F))}{m}=c^{\prime}.\end{equation}

In particular, if $\varphi_{mc^{\prime}}\in \mathrm{PSH}(B(p,r))$ has algebraic singularities along $\mathcal{A}_{mc^{\prime}}(F)$ , then for any $\epsilon>0$ , $\varphi_{mc^{\prime}}$ is less singular than $({mc^{\prime}}/({c-\epsilon}))\varphi$ around p if $m\geq m_1(\epsilon)\gg 1$ . For any G exceptional divisor on $\Omega'$ and $m\geq m_1(\epsilon)$ we infer

(5.10) \begin{equation}\frac{\nu(\varphi\circ \pi\circ g,G)}{\mathrm{ord}_{G}(\mathcal{A}_{mc^{\prime}}(F))/m}=\frac{\nu(\varphi\circ \pi\circ g,G)}{\nu(\varphi_{mc^{\prime}}\circ \pi\circ g,G)/m}\geq \frac{c-\epsilon}{c^{\prime}}.\end{equation}

On the other hand, (5.9) implies that there exists $m_0(\epsilon)\geq m_1(\epsilon)\gg 1$ with

(5.11) \begin{equation}\frac{\nu(\varphi\circ \pi\circ g,F)}{\mathrm{ord}_F(\mathcal{A}_{m_0c^{\prime}}(F))/m_0}\leq \frac{c}{c^{\prime}-\epsilon}.\end{equation}

Combining (5.10) and (5.11) we obtain

\begin{align*}\min_{G}\frac{\nu(\varphi\circ \pi\circ g, G)}{\mathrm{ord}_G(\mathcal{A}_{m_0c^{\prime}}(F))/m_0}\geq \frac{c-\epsilon}{c^{\prime}} &\geq \bigg(1-\epsilon\frac{c+c^{\prime}}{cc^{\prime}}\bigg)\frac{c}{c^{\prime}-\epsilon} \\&\geq \bigg(1-\epsilon\frac{c+c^{\prime}}{cc^{\prime}}\bigg)\frac{\nu(\varphi\circ \pi\circ g,F)}{\mathrm{ord}_F(\mathcal{A}_{m_0c^{\prime}}(F))/m_0}.\end{align*}

Since $c^{\prime}$ and $\epsilon$ are arbitrary, and $x \mapsto f(x)=({c+x})/{cx}$ is decreasing, we deduce that for any $\epsilon>0$ there exists $c^{\prime}\in\mathbb Q$ and $m_0=m_0(c,c^{\prime},\epsilon)$ such that

\[\nu_{\mathcal{A}_{m_0c^{\prime}}(F)}(\varphi,p)\geq (1-\epsilon)\frac{\nu(\varphi\circ \pi\circ g,F)}{\mathrm{ord}_F(\mathcal{A}_{m_0c^{\prime}}(F))}.\]

Setting $\mathcal{A}:=\mathcal{A}_{m_0c^{\prime}}(F)$ concludes the proof.

5.3.1 Using projections on n-planes

A result of Skoda ensures that $e^{- \varphi}$ is integrable if the Lelong numbers of $ \varphi$ are small enough (see [Reference Guedj and ZeriahiGZ17, Theorem 2.50]). This has been largely extended by Demailly and Zeriahi who provided uniform integrability results for functions $\varphi \in {\mathcal F}_1(\Omega)$ (see [Reference DemaillyDem09, Reference Åhag, Cegrell, Kołodziej, Pham and ZeriahiACKPZ09]). In this section, we extend these results to our singular setting.

Theorem 5.10. One has

\[\alpha(X,\mu_p) \geq \frac{n}{{\rm mult}(X,p)^{1-1/n}} \frac{{\rm lct}(X,p)}{1+{\rm lct}(X,p)}.\]

Proof. Recall that $\mu_p=fdV_X$ with $f \in L^r(dV_X)$ . The exponent $r>1$ has been estimated in Lemma 2.14. Using Hölder inequality, we thus obtain

\[\alpha(X,\mu_p) \geq \frac{{\rm lct}(X,p)}{1+{\rm lct}(X,p)} \alpha(\Omega,dV_X).\]

The remainder of the proof consists of establishing the lower bound

\[\alpha(\Omega,dV_X) \geq\frac{n}{{\rm mult}(X,p)^{1-1/n}}.\]

Recall that $dV_X=\omega_{\rm eucl}^n \wedge [X]$ , where $\omega_{\rm eucl}$ denotes the euclidean Kähler form. Thus, $dV_X=\sum_I (\pi_I)^*(dV_I)$ , where $I=(i_1,\ldots,i_n)$ is a n-tuple, $\pi_I:\mathbb C^N\rightarrow \mathbb C_I^n$ denotes the linear projection on $\mathbb C_I^n$ , and $dV_I$ is the euclidean volume form on $\mathbb C_I^n$ . We choose coordinates in $\mathbb C^N$ so that each projection map $\pi_I: \Omega\rightarrow \Omega_I \subset \mathbb C^n$ is proper. For $\varphi \in {\mathcal F}_1(\Omega)$ , we obtain

\[\int_{\Omega} e^{-\alpha \varphi} \,dV_X = \sum_I \int_{\Omega_I} (\pi_I)_* (e^{-\alpha \varphi}) \,dV_I\leq {\rm mult}(X,p) \sum_I \int_{\Omega_I} e^{-\alpha (\pi_I)_* \varphi} \,dV_I.\]

We assume here, without loss of generality, that $\varphi \leq 0$ , and use the (sub-optimal) inequality $(\pi_I)_* (e^{-\alpha \varphi}) \leq {\rm mult}(X,p) e^{-\alpha (\pi_I)_* \varphi}$ . The function $\varphi_I:=(\pi_I)_* \varphi$ is psh in $\Omega_I=\pi_I(\Omega)$ , with boundary values $(\pi_I)_*(\phi)$ . We claim that

(5.12) \begin{equation}\int_{\Omega_I} (dd^c \varphi_I)^n \leq {\rm mult}(X,p)^{n-1}.\end{equation}

Once this is established, it follows from the main result of [Reference Åhag, Cegrell, Kołodziej, Pham and ZeriahiACKPZ09] that for all $0<\varepsilon$ small enough, there exists $C_{\varepsilon}>0$ independent of $\varphi$ such that

\[\int_{\Omega_I} e^{- (({n-\varepsilon})/{{\rm mult}(X,p)^{1-1/n}}) \varphi_I} \,dV_I \leq C_{\varepsilon},\]

which yields the desired lower bound $\alpha(\Omega,dV_X) \geq ({n}/({{\rm mult}(X,p)^{1-1/n}}))$ .

It remains to check (5.12). We decompose $\varphi_I(z)=\sum_{i=1}^m \varphi(x_i)$ , where $m={\rm mult}(X,p)$ and $x_1,\ldots,x_m$ denote the preimages of z counted with multiplicities. The assumption on the Monge–Ampère mass of $\varphi$ reads

\[\sum_{i=1}^m \int (dd^c \varphi)^n(x_i) \leq 1.\]

We set $a_i^n:=\int (dd^c \varphi)^n(x_i)$ and use [Reference CegrellCeg04, Corollary 5.6] to estimate

\[\int (dd^c \varphi_I)^n = \sum_{i_1,\ldots,i_n=1}^m \int dd^c \varphi(x_{i_1}) \wedge\cdots \wedge dd^c \varphi (x_{i_n}) \leq \sum_{i_1,\ldots,i_n=1}^m a_{i_1}\cdots a_{i_n}=\bigg(\sum_{i}^m a_i\bigg)^{\!\!n}.\]

The latter sum is maximized when $a_1=\cdots=a_m=m^{-1/n}$ , yielding (5.12).

Example 5.11. Let $X=\{z \in \mathbb C^{n+1},\ F(z)=0\}$ be the $A_k$ -singularity, where $F(z)=z_0^{k+1}+z_1^2+\cdots z_n^2$ . Arguing as we have done for the ODP $(k=1)$ , one can check that $\mu_p \sim{dV_X}/{\|F'\|^2}$ so that ${\rm mult}(X,p)=2$ and ${\rm lct}(X,p)=n-2+{2}/({k+1})$ . Now

\[\widehat{\mathrm{vol}}(A_k,p)^{1/n}=\begin{cases}2^{1/n}\bigg(\dfrac{n-2}{n-1}\bigg)^{\!\!1-1/n}n & \mathrm{if}\ \dfrac{k+1}{2}\geq \dfrac{n-1}{n-2},\\[16pt](k+1)^{1/n}\bigg(\dfrac{(n-2)(k+1)+2}{k+1}\bigg) & \mathrm{if}\ \dfrac{k+1}{2}<\dfrac{n-1}{n-2},\end{cases}\]

as computed by Li in [Reference LiLi18, Example 5.3]. For $n \gg 1$ , the lower bound provided by Theorem 5.10 is thus short of a factor $2={\rm mult}(X,p)$ by comparison with the expected lower bound $\widehat{\mathrm{vol}}(A_k,p)^{1/n}$ .

5.3.2 Using resolutions

Proposition 5.12. Let $\pi:\tilde{\Omega}\to \Omega$ be a resolution of singularities with simple normal crossing, and let $\{a_i\}_{i=1,\ldots,M}$ be the discrepancies. Then

\[\alpha(X,\mu_p)\geq \frac{\widehat{\mathrm{vol}}(X,p)^{1/n}}{1+(\max_i a_i)_+}.\]

In particular, if the singularity is ‘admissible’, then $\alpha(X,\mu_p)=\widehat{\mathrm{vol}}(X,p)^{1/n}$ .

Following [Reference Li, Tian and WangLTW21, Definition 1.1] we say here that (X,p) is an admissible singularity if there exists a resolution $\pi:\tilde{X} \rightarrow X$ (with snc exceptional divisor $E=\sum_jE_j$ and $\pi$ -ample divisor $-\sum b_j E_j$ , $b_j \in \mathbb Q^+$ ) such that the discrepancies $a_i \in(-1,0]$ are all non-positive. Recall that:

• – any two-dimensional log terminal singularity is admissible;

• – the vertex of the affine cone over a Fano manifold embedded in a projective space by the linear system associated to a multiple of the anticanonical bundle is admissible (cf. the proof of Proposition 3.17);

• – (X,p) is admissible if it is $\mathbb Q$ -factorial and admits a crepant resolution.

Theorem B from the introduction follows from the combination of Proposition 5.6, Theorem 5.10 and Proposition 5.12.

Proof. We seek $\alpha>0$ such that

(5.13) \begin{equation}\sup_{\varphi\in \mathcal{F}_1(\Omega)}\int_{\tilde{\Omega}} e^{-\alpha \varphi\circ \pi}\prod_{i=1}^M \lvert s_i\rvert_{h_i}^{2a_i} \,dV<+\infty.\end{equation}

If all the $a_i$ are non-positive we can use [Reference Demailly and KollàrDK01, Main Theorem] to show that $\alpha(X,\mu_p)=\tilde{\alpha}(X,\mu_p)$ , hence $\alpha(X,\mu_p)=\widehat{\mathrm{vol}}(X,p)^{1/n}$ by Proposition 5.8. Indeed, assume that there exists $\gamma>0$ such that $\alpha(X,\mu_p)<\gamma<\tilde{\alpha}(X,\mu_p)$ . By definition, we can find $\psi_j \in \mathcal{F}_1(\Omega)$ such that $\int_{\Omega} e^{-\gamma \psi_j}\,d\mu_p \rightarrow +\infty$ . Extracting and relabelling, we can assume that $\psi_j \rightarrow \psi$ in $L^1$ with $c(\psi)>\gamma$ . The psh functions $\varphi_j=\psi_j+\gamma^{-1}\sum_{i=1}^M(-a_i) \log\lvert s_i\rvert_{h_i}^2$ converge to $\varphi=\psi+\gamma^{-1}\sum_{i=1}^M (-a_i) \log \lvert s_i\rvert_{h_i}^2$ in $L^1(\tilde{\Omega})$ and $c(\varphi)>\gamma$ . It follows therefore from [Reference Demailly and KollàrDK01, Theorem 0.2.2] that

\[\int_{\Omega} e^{-\gamma \psi_j} \,d\mu_p=\int_{\tilde{\Omega}} e^{-\gamma \varphi_j} \,dV \longrightarrow\int_{\tilde{\Omega}} e^{-\gamma \varphi} \,dV <+\infty,\]

contradicting the assumption $\int_{\Omega} e^{-\gamma \psi_j} \,d\mu_p \rightarrow +\infty$ .

In general, we set $U:=\sum_{i:a_i>0}a_i\log \lvert s_i\rvert^2_{h_i}$ and $W:=-\sum_{i:a_i\leq 0}a_i \log \lvert s_i \rvert^2_{h_i}$ . Using [Reference Demailly and KollàrDK01, Main Theorem], we obtain

(5.14) \begin{equation}\alpha(X,\mu_p)\geq \inf_{\varphi\in \mathcal{F}_1(\Omega)}c_W(\varphi\circ \pi),\end{equation}

where

\[c_W(\varphi\circ\pi):=\sup\bigg\{\alpha>0: \int_{\tilde{\Omega}}e^{-\alpha\varphi\circ \pi- W}\,dV<+\infty\bigg\}\]

is the twisted complex singularity exponent. It then remains to estimate $c_W(\varphi\circ\pi)$ for a fixed $\varphi\in \mathcal{F}_1(\Omega)$ . As $\pi^*d\mu_p=e^{U-W}dV$ , Hölder inequality yields

(5.15) \begin{equation}\int_{\tilde{\Omega}}e^{-\alpha \varphi\circ \pi-W}\,dV\leq \bigg(\int_{\tilde{\Omega}}e^{-p'\alpha\varphi\circ \pi}\pi^{*}\,d\mu_p\bigg)^{\!\!1/p'}\bigg(\int_{\tilde{\Omega}}e^{(1-q')U-W}\,dV\bigg)^{\!\!1/q'}.\end{equation}

Set $A:=(\max_i a_i)_+>0$ . The second factor on the right-hand side of (5.15) is finite for any $q'<(({A+1})/{A})$ , while the first factor on the right-hand side gives the condition $p'\alpha< \tilde{\alpha}=\widehat{\mathrm{vol}}(X,p)^{1/n}$ . We infer $ c_W(\varphi\circ \pi)\geq({\widehat{\mathrm{vol}}(X,p)^{1/n}})/({1+A})$ , which concludes the proof.

As the proof shows, the main obstruction to proving the equality $\alpha(X,\mu_p)=\tilde{\alpha}(X,\mu_p)=\widehat{\mathrm{vol}}(X,p)^{1/n}$ is the lack of a Demailly–Kollàr result on complex spaces. Resolving the singularities, one ends up with a twisted version of Demailly and Kollàr’s problem on a smooth manifold. It is known that the general form of such a problem has a negative answer [Reference PhamPha14, Remark 1.3].