Degenerate complex Monge-Amp\`ere equations over compact K\"ahler manifolds

^†^† Key words : Monge-Ampère equations, Kähler-Einstein metrics, Closed positive currents, Plurisubharmonic functions, Capacities, Orlicz spaces.
AMS Classification : 53C25, 53C55, 32J15.

Degenerate complex Monge-Ampère equations over compact Kähler manifolds.

Jean-Pierre Demailly and Nefton Pali

Abstract

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Ampère equations. This type of equations is precisely what is needed in order to construct Kähler-Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf.

[028R]

1 Introduction

In a celebrated paper [Yau] Yau solved the Calabi conjecture. As is well known, this problem can be formulated in terms of non degenerate complex Monge-Ampère equations as follows.

[028S]

Theorem 1

(Yau). Let $X$ be a compact Kähler manifold of complex dimension $n$ and let $\chi$ be a Kähler class. Then for any smooth density $v>0$ on $X$ such that $\int_{X}v=\int_{X}\chi^{n}$ there exists a unique $($ smooth $)$ Kähler metric $\omega\in\chi$ such that $\omega^{n}=v$ .

Another breakthrough concerning the study of complex Monge-Ampère equations has been achieved by Bedford-Taylor [Be-Te]. Their work opened the doors to the study of very degenerate complex Monge-Ampère equations. In fact Kołodziej [Kol] proved the existence of solutions of the equations of type $(\omega+i\partial\bar{\partial}\varphi)^{n}=e^{\lambda\varphi}v$ , $\lambda\geq 0$ with $\omega$ a Kähler metric and $v\geq 0$ a density in $L^{p}$ or in some complicated Orlicz spaces. However in various geometric applications it is necessary to consider $\omega$ which is merely semipositive. This difficulty has been examinated first by Tsuji [Ts]. Tsuji’s technique has been reconsidered in the recent works [Ti-Zha] and [E-G-Z]. In this paper we push further the techniques so far developed and we obtain some very general and sharp results on the existence and uniqueness of degenerate complex Monge-Ampère equations. In order to define the relevant concept of uniqueness of the solutions we need first to introduce the domain of definition of the complex Monge-Ampère operator $MA_{\chi}$ of a pseudoeffective $(1,1)$ -cohomology class $\chi$ and to prove a monotone convergence result. As a consequence of our results (see theorem 6) we derive the following generalization of Yau’s theorem.

[028T]

Theorem 2

. Let $X$ be a compact Kähler manifold of complex dimension $n$ and let $\chi$ be a big $(1,1)$ -cohomology class admitting a closed positive current with continuous local potentials. Then for any $L\log^{n+\varepsilon}L$ -density $v\geq 0$ , $\varepsilon>0$ on $X$ such that $\int_{X}v=\int_{X}\chi^{n}$ there exists a unique closed positive current $T\in MA_{\chi}$ such that $T^{n}=v$ . Moreover this current possesses continuous local potentials.

We wish to point out that the main examples of Orlicz spaces considered by Kołodziej are contained in some space $L\log^{n+\varepsilon}L$ . In the last section we prove fine regularity properties of the solution of complex Monge-Ampère equations with respect to a given degenerate metric $\omega\geq 0$ and whith right hand side possessing a density carrying complex analytic singularities (see theorem 7). This last type of equation is precisely what is needed in order to construct Kähler-Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf. This allows us also to solve generalised equations of the form $\operatorname{Ric}(\omega)=-\lambda\omega+\rho$ , $\lambda\geq 0$ . Quite recently Tian and Kołodziej [Ti-Ko] proved a very particular case of our $L^{\infty}$ -estimate. Their method, which is completely different, is based on an idea developed in [De-Pa]. Our $L^{\infty}$ -estimate allous us to completely solve a Tian’s conjecture stated in [Ti-Ko] (see the remark in the Apendix).

[028U]

2 General $L^{\infty}$ estimates for the solutions.

Let $X$ be a compact complex manifold of complex dimension $n$ and let $\gamma$ be a closed real $(1,1)$ -current with continuous local potentials or a closed positive $(1,1)$ -current with bounded local potentials. Then to any distribution $\Psi$ on $X$ such that $\gamma+i\partial\bar{\partial}\Psi\geq 0$ we can associate a unique locally integrable and bounded from above function $\psi:X\rightarrow[-\infty,+\infty)$ such that the corresponding distribution coincides with $\Psi$ and such that for any continuous or plurisubharmonic local potential $h$ of $\gamma$ the function $h+\psi$ is plurisubharmonic. In fact let $\Omega_{\alpha}$ , $\Omega_{\beta}\subset X$ two open sets such that $\gamma=i\partial\bar{\partial}h_{\alpha}$ over $\Omega_{\alpha}$ , $\gamma=i\partial\bar{\partial}h_{\beta}$ over $\Omega_{\beta}$ , with $h_{\alpha}\in C^{0}(\Omega_{\alpha})$ , $h_{\beta}\in C^{0}(\Omega_{\beta})$ in the first case or $h_{\alpha}\in\operatorname{Psh}(\Omega_{\alpha})$ , $h_{\beta}\in\operatorname{Psh}(\Omega_{\beta})$ in the second case. Consider now the distributions $U_{\alpha}:=h_{\alpha}+\Psi$ and $U_{\beta}:=h_{\beta}+\Psi$ . Then the hypothesis $\gamma+i\partial\bar{\partial}\Psi\geq 0$ implies the existence of uniques $u_{\alpha}\in\operatorname{Psh}(\Omega_{\alpha})$ , $u_{\beta}\in\operatorname{Psh}(\Omega_{\beta})$ such that the corresponding distributions are respectively $U_{\alpha}$ and $U_{\beta}$ . We set $\psi_{\alpha}:=u_{\alpha}-h_{\alpha}$ and $\psi_{\beta}:=u_{\beta}-h_{\beta}$ . The function $h_{\beta}-h_{\alpha}$ is harmonic in both cases. Then the identity $U_{\beta}=U_{\alpha}+h_{\beta}-h_{\alpha}$ over $\Omega_{\alpha}\cap\Omega_{\beta}$ implies that the plurisubharmonic functions $u_{\beta}$ and $u_{\alpha}+h_{\beta}-h_{\alpha}$ represent the same distribution so they coincide over $\Omega_{\alpha}\cap\Omega_{\beta}$ . We deduce $\psi_{\alpha}=\psi_{\beta}$ over $\Omega_{\alpha}\cap\Omega_{\beta}$ . In this way we obtain a global function $\psi:X\rightarrow[-\infty,+\infty)$ which satisfies the required properties since the function $h-h_{\alpha}$ is harmonic in both cases. Then the uniqueness of $\psi$ is obvious. The set of functions $\psi$ obtained in this way will be denoted by ${\cal P}_{\gamma}$ . We set ${\cal P}^{0}_{\gamma}:=\{\psi\in{\cal P}_{\gamma}\,\mid\,\sup_{X}\psi=0\}$ . A closed positive $(1,1)$ -current with bounded local potentials such that $\{\gamma\}^{n}:=\int_{X}\gamma^{n}>0$ , will be called big.

[028V]

Theorem 3

. Let $X$ be a compact Kähler manifold of complex dimension $n$ , let $\Omega>0$ be a smooth volume form, let $\gamma$ be a big closed positive $(1,1)$ -current with continuous local potentials. Let also $\psi\in{\cal P}_{\gamma}\cap L^{\infty}(X)$ be a solution of the degenerate complex Monge-Ampère equation $(\gamma+i\partial\bar{\partial}\psi)^{n}=f\,\Omega\,,$ with $f\in L\log^{n+\varepsilon_{0}}L(X)$ for some $\varepsilon_{0}>0$ . Then the following conclusions hold.
(A) There exist a uniform constant $C_{1}=C_{1}(\varepsilon_{0},\gamma,\Omega)>0$ such that for all $\varepsilon\in(0,\varepsilon_{0}]$ holds an estimate

\operatorname{Osc}(\psi)\leq C_{1}^{\,2n\left(1+\frac{n}{\varepsilon}\right)}\,\left[e\left(3+\frac{2n}{\varepsilon}\right)\right]^{\,\frac{\;n^{2}}{\varepsilon}}I_{\varepsilon}(f)^{\frac{n}{\varepsilon}}+\;1\,,

where

I_{\varepsilon}(f):=\{\gamma\}^{-n}\int\limits_{X}f\log^{n+\varepsilon}\left(e+\{\gamma\}^{-n}f\right)\Omega\,.

(B) Assume that the solution $\psi$ is continuous, normalized by the condition $\sup_{X}\psi=0$ and consider also a continuous solution $\varphi\in{\cal P}_{\gamma}$ , $\sup_{X}\varphi=0$ of the degenerate complex Monge-Ampère equation $(\gamma+i\partial\bar{\partial}\varphi)^{n}=g\,\Omega\,,$ with $g\in L\log^{n+\varepsilon_{0}}L(X)$ . Let $K>0$ be a constant such that $I_{\varepsilon_{0}}(f),I_{\varepsilon_{0}}(g)\leq K$ . Then there exists a constant $C_{2}=C_{2}(\varepsilon_{0},\gamma,\Omega,K)>0$ such that

	$\displaystyle\\|\varphi-\psi\\|_{{}_{C^{0}(X)}}$	$\displaystyle\leq$	$\displaystyle 2C_{2}^{{}^{\alpha_{0}}}\,\left(\log\\|\varphi-\psi\\|^{-1}_{{}_{L^{1}(X,\,\Omega)}}\right)^{{}^{-\alpha_{0}}}\,,$
	$\displaystyle\alpha_{0}$	$\displaystyle:=$	$\displaystyle\frac{1}{(n+1+n^{2}/\varepsilon_{0})}\,,$

provided that the inequality $\|\varphi-\psi\|_{{}_{L^{1}(X,\,\Omega)}}\leq\min\{1/2,e^{-C_{2}}\}$ holds.
(C) Let $(\gamma_{t})_{t>0}$ be a family of currents satisfying the same properties as $\gamma$ , fix a finite covering $(U_{\alpha})_{\alpha}$ of coordinate starshaped open sets, and let us write $\gamma_{t}=i\partial\bar{\partial}h_{t,\alpha}$ with $\sup_{U_{\alpha}}h_{t,\alpha}=0$ over $U_{\alpha}$ and $C_{1,t}:=C_{1}(\varepsilon_{0},\gamma_{t},\Omega)$ , $C_{2,t}=C_{2}(\varepsilon_{0},\gamma_{t},\Omega,K)$ . Assume
(C1) $\sup_{t>0}\max_{\alpha}\|h_{t,\alpha}\|_{L^{\infty}(U_{\alpha})}<+\infty$ and
(C2a) there exist a decomposition of the type $\gamma_{t}=\theta_{t}+i\partial\bar{\partial}u_{t}$ , whith $\theta_{t}$ smooth, $\min_{X}u_{t}=0$ , $\sup_{t>0}\max_{X}u_{t}<+\infty$ and $\theta_{t}\leq(\{\gamma_{t}\}^{n})^{1/n}\omega$ for some Kähler metric $\omega>0$ on $X$ ,
or
(C2b) the distributions $\gamma^{n}_{t}/\Omega$ are represented by functions and

\sup_{t>0}\;\;\{\gamma_{t}\}^{-n}\int\limits_{X}\log\left(e+\{\gamma_{t}\}^{-n}\gamma^{n}_{t}/\Omega\right)\gamma^{n}_{t}<+\infty\,.

Then $\sup_{t>0}C_{j,t}<+\infty$ for $j=1,2$ .

Statement (C) will follow from the arguments of the proof of theorem 3.

Remark 1. As application of his estimates, Kołodziej considers in Example 1, page 91 of [Kol] Monge-Ampère equations with non degenerate left hand side and with right hand side taking values in the Orlicz space $L^{\Psi}(X)$ , with $\Psi(t):=t\log^{n}(e+t)\log^{n+\delta}(e+\log(1+t))$ , $\delta>0$ . If we take $\varepsilon_{0}=1/k$ with an integer $k>1$ we obtain

\lim_{t\rightarrow+\infty}\frac{\log^{n+\delta}(e+\log(1+t))}{\log^{\varepsilon_{0}}(e+t)}=+\infty\,.

This implies $L^{\Psi}(X)\subset L\log^{n+\varepsilon_{0}}L(X)$ .

Let $X$ be a compact complex manifold of complex dimension $n$ , let $\gamma$ be a big closed positive $(1,1)$ -current with bounded local potentials. Set ${\cal P}_{\gamma}[0,1]:=\{\varphi\in{\cal P}_{\gamma}\,\mid\,0\leq\varphi\leq 1\}$ and

\operatorname{Cap}_{\gamma}(E):=\sup_{\varphi\in{\cal P}_{\gamma}[0,1]}\,\{\gamma\}^{-n}\int\limits_{E}\gamma^{n}_{\varphi}\,,

for all Borel sets $E\subset X$ . We remark that if $(E_{j})_{j}$ , $E_{j}\subset E_{j+1}\subset X$ is a family of Borel sets and $E=\bigcup_{j}E_{j}$ then

\displaystyle\operatorname{Cap}_{\gamma}(E)=\lim_{j\rightarrow+\infty}\operatorname{Cap}_{\gamma}(E_{j})\,.

(2.1)

In fact for all $\varepsilon>0$ let $\varphi\in{\cal P}_{\gamma}[0,1]$ such that $\{\gamma\}^{-n}\int_{E}\gamma^{n}_{\varphi}>\operatorname{Cap}_{\gamma}(E)-\varepsilon$ . Then

	$\displaystyle\operatorname{Cap}_{\gamma}(E)$	$\displaystyle\geq$	$\displaystyle\lim_{j\rightarrow+\infty}\operatorname{Cap}_{\gamma}(E_{j})$
		$\displaystyle\geq$	$\displaystyle\lim_{j\rightarrow+\infty}\,\{\gamma\}^{-n}\int\limits_{E_{j}}\gamma^{n}_{\varphi}=\,\{\gamma\}^{-n}\int\limits_{E}\gamma^{n}_{\varphi}>\operatorname{Cap}_{\gamma}(E)-\varepsilon\,.$

[028W]

Lemma 1

. Let $X$ be a connected compact complex manifold of complex dimension $n$ , let $\gamma$ be a closed real $(1,1)$ -current with continuous local potentials or a closed positive $(1,1)$ -current with bounded local potentials and let $\Omega>0$ be a smooth volume form. Then there exist constants $\alpha=\alpha(\gamma,\Omega)>0$ , $C=C(\gamma,\Omega)>0$ such that $\int_{X}-\psi\,\Omega\leq C$ and $\int_{X}e^{-\alpha\psi}\,\Omega\leq C$ for all $\psi\in{\cal P}^{0}_{\gamma}$ .

The first two integral estimates of lemma (1) are quite standard in the elementary theory of plurisubharmonic functions and the dependence of the constants $\alpha$ and $C$ on $\gamma$ is only on the $L^{\infty}$ bound of its local potentials. To be more precise concerning the uniform estimate $\int_{X}e^{-\alpha\psi}\,\Omega\leq C$ one can make the constant $\alpha$ depending only on the cohomology class of $\gamma$ as in [Ti1], but in this case the constant $C$ will depend on the $L^{\infty}$ bound of the local potentials of $\gamma$ and on the volume form $\Omega$ . One can also make $C$ depending only on the volume form $\Omega$ , but in this case $\alpha$ will depend on the $L^{\infty}$ bound of the local potentials of $\gamma$ and on the volume form $\Omega$ .

[028X]

Lemma 2

. Let $X$ be a connected compact complex manifold of complex dimension $n$ , let $\gamma$ be a big closed positive $(1,1)$ -current with bounded local potentials.
(A). If $X$ is Kähler and $\gamma$ possesses continuous local potentials then there exist a constant $C=C(\gamma)>0$ such that $\operatorname{Cap}_{\gamma}(\{\psi<-t\})\leq C/t$ for all $\psi\in{\cal P}^{0}_{\gamma}$ and $t>0$ . Moreover the constant $C$ stay bounded for pertutbations of $\gamma$ satisfying the hypothesis $(C1)$ and $(C2a)$ of the statement $(C)$ in theorem 3.
(B). If $\gamma^{n}/\Omega\in L\log L(X)$ , for a smooth volume form $\Omega>0$ then the conclusion of statement $(A)$ hold whith a constant $C=C(\gamma,\Omega)>0$ which stay bounded for pertutbations of $\gamma$ satisfying the hypothesis $(C1)$ and $(C2b)$ of the statement $(C)$ in theorem 3.

Proof. We remark first the inequality

\int\limits_{\psi<-t}\gamma^{n}_{\varphi}\;\leq\frac{1}{t}\int\limits_{X}-\psi\,\gamma^{n}_{\varphi}\,,

and we prove the following elementary claim.

[028Y]

Claim 1

. Let $\gamma$ be a closed positive $(1,1)$ -current with bounded local potentials over a compact complex manifold $X$ of complex dimension $n$ and let $\varphi,\,\psi\in{\cal P}_{\gamma}$ such that $0\leq\varphi\leq 1$ and $\psi\leq 0$ . Then

\displaystyle\int\limits_{X}-\psi\,\gamma^{n}_{\varphi}\;\leq\int\limits_{X}-\psi\,\gamma^{n}+n\int\limits_{X}\,\gamma^{n}\,.

(2.2)

Proof. The fact that the current $\gamma$ is positive implies $\psi_{c}:=\max\{\psi,c\}\in{\cal P}_{\gamma}$ , $c\in\mathbb{R}_{<0}$ , so by the monotone convergence theorem it is sufficent to prove the inequality (2.2) for $\psi\in{\cal P}_{\gamma}\cap L^{\infty}(X)$ . So assume this and let $\omega>0$ be a hermitian metric over $X$ . By a result of Greene-Wu [Gr-Wu] there exist a family of functions $(\psi_{\varepsilon})_{\varepsilon>0}$ , $\psi_{\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}\cap C^{\infty}(X)$ such that $\psi_{\varepsilon}\downarrow\psi$ as $\varepsilon\rightarrow 0^{+}$ . Consider now the integrals $I_{j}:=\int_{X}-\psi\,\gamma^{j}\wedge\gamma_{\varphi}^{n-j}\,$ for all $j=0,...,n$ . Then $I_{j}\leq I_{j+1}+\int_{X}\gamma^{n}$ . In fact by Stokes formula

$\displaystyle I_{j}$	$\displaystyle=$	$\displaystyle I_{j+1}-\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}\psi_{\varepsilon}\,\gamma^{j}\wedge i\partial\bar{\partial}\varphi\wedge\gamma_{\varphi}^{n-j-1}$
	$\displaystyle=$	$\displaystyle I_{j+1}-\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial\bar{\partial}\psi_{\varepsilon}\wedge\varphi\,\gamma^{j}\wedge\gamma_{\varphi}^{n-j-1}$
	$\displaystyle\leq$	$\displaystyle I_{j+1}+\int\limits_{X}\varphi\,\gamma^{j+1}\wedge\gamma_{\varphi}^{n-j-1}\leq I_{j+1}+\int\limits_{X}\gamma^{n}\,.$

In this way we deduce the required inequality $I_{0}\leq I_{n}+n\int_{X}\gamma^{n}$ . $\Box$

The following claim will be very useful for the rest of the paper.

[028Z]

Claim 2

. Let $(X,\omega)$ be a polarized connected compact Kähler manifold of complex dimension $n$ and let $\gamma$ , $T$ be closed positive $(1,1)$ -currents with continuous local potentials. Then for all $l=0,...,n$

C_{l}:=\sup_{\psi\in{\cal P}^{0}_{\gamma}}\;\int\limits_{X}-\psi\,T^{l}\wedge\omega^{n-l}<+\infty

and $\gamma_{\psi}\wedge T^{l}=T^{l}\wedge\gamma_{\psi}$ for all $\psi\in{\cal P}_{\gamma}$ .

Proof. The proof of the convergence of the constants $C_{l}$ goes by induction on $l=0,...,n$ . The statement is true for $l=0$ by the first integral estimate of lemma 1. So we assume it is true for $l$ and prove it for $l+1$ . Let $\psi_{c}:=\max\{\psi,c\}\in{\cal P}_{\gamma}$ , $c\in\mathbb{R}_{<0}$ . The inductive hypothesis allows us to apply the decreasing monotone convergence theorem in order to deduce

\lim_{c\rightarrow-\infty}\,\int\limits_{X}(\psi_{c}-\psi)T^{l}\wedge\omega^{n-l}=0\,,

which means the converegence in mass $\|(\psi_{c}-\psi)T^{l}\|_{\omega}(X)\rightarrow 0$ as $c\rightarrow-\infty$ , in particular $\psi_{c}T^{l}\rightarrow\psi T$ weakly as $c\rightarrow-\infty$ . So by the weak continuity of the $i\partial\bar{\partial}$ operator we deduce $i\partial\bar{\partial}\psi_{c}\wedge T^{l}\rightarrow i\partial\bar{\partial}\psi\wedge T$ and

\displaystyle\gamma_{\psi_{c}}\wedge T^{l}\longrightarrow\gamma_{\psi}\wedge T^{l}\,,

(2.3)

weakly as $c\rightarrow-\infty$ . By the result of Greene-Wu [Gr-Wu] let $(\psi_{c,\varepsilon})_{\varepsilon>0}$ , $\psi_{c,\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}\cap C^{\infty}(X)$ such that $\psi_{c,\varepsilon}\downarrow\psi_{c}$ as $\varepsilon\rightarrow 0^{+}$ and write $T=\theta+i\partial\bar{\partial}u$ , whith $\theta$ smooth, $\theta\leq K\omega$ and $u$ continuous whith $\min_{X}u=0$ . By using the monotone convergence theorem, Stokes formula and (2.3), we expand the integral

			$\displaystyle\int\limits_{X}-\psi\,T^{l+1}\wedge\omega^{n-l-1}=\lim_{c\rightarrow-\infty}\;\lim_{\varepsilon\rightarrow 0^{+}}\;\int\limits_{X}-\psi_{c,\varepsilon}\,T^{l+1}\wedge\omega^{n-l-1}$
		$\displaystyle=$	$\displaystyle\lim_{c\rightarrow-\infty}\;\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}-\psi_{c,\varepsilon}\,\theta\wedge T^{l}\wedge\omega^{n-l-1}-\int\limits_{X}\psi_{c,\varepsilon}\,i\partial\bar{\partial}u\wedge T^{l}\wedge\omega^{n-l-1}\right]$
		$\displaystyle\leq$	$\displaystyle\lim_{c\rightarrow-\infty}\;\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}-\psi_{c,\varepsilon}\,T^{l}\wedge K\omega^{n-l}-\int\limits_{X}u\,i\partial\bar{\partial}\psi_{c,\varepsilon}\wedge T^{l}\wedge\omega^{n-l-1}\right]$
		$\displaystyle=$	$\displaystyle\int\limits_{X}-\psi\,T^{l}\wedge K\omega^{n-l}-\int\limits_{X}u\,i\partial\bar{\partial}\psi\wedge T^{l}\wedge\omega^{n-l-1}$
		$\displaystyle\leq$	$\displaystyle KC_{l}-\int\limits_{X}u\,\gamma_{\psi}\wedge T^{l}\wedge\omega^{n-l-1}+\int\limits_{X}u\,\gamma\wedge T^{l}\wedge\omega^{n-l-1}$
		$\displaystyle\leq$	$\displaystyle KC_{l}+\max_{X}u\,\int\limits_{X}\gamma\wedge T^{l}\wedge\omega^{n-l-1}<+\infty\,,$

by the inductive hypothesis. Concerning the symmetry of the exterior product we remark that the weak continuity of the $i\partial\bar{\partial}$ operator implies by induction on $l$

T^{l}\wedge\gamma_{\psi_{c}}\longrightarrow T^{l}\wedge\gamma_{\psi}\,,

weakly as $c\rightarrow-\infty$ . This combined with (2.3) implies $\gamma_{\psi}\wedge T^{l}=T^{l}\wedge\gamma_{\psi}$ . $\Box$

In the particular case $T=\gamma$ the constant

0<C(\gamma):=\sup_{\psi\in{\cal P}^{0}_{\gamma}}\;\;\{\gamma\}^{-n}\int\limits_{X}-\psi\,\gamma^{n}<+\infty

satisfies the capacity estimate of the statement (A) of the lemma 2, by the claim 1. Moreover the previous induction give us

C_{1}\leq K\int\limits_{X}-\psi\,\omega^{n}+\max_{X}u\,\int\limits_{X}\gamma\wedge\omega^{n-1}\leq K\int\limits_{X}-\psi\,\omega^{n}+RK\,\int\limits_{X}\omega^{n}\,,

where $R\geq\max_{X}u$ and in general

C_{l+1}\leq KC_{l}+R\int\limits_{X}\gamma^{l+1}\wedge\omega^{n-l-1}\leq KC_{l}+RK^{l+1}\,\int\limits_{X}\omega^{n}\,.

We deduce $C_{n}\leq K^{n}\int_{X}-\psi\,\omega^{n}+nRK^{n}\int_{X}\omega^{n}$ . Thus if $(\gamma_{t})_{t>0}$ is a family satisfying the hypothesis $(C1)$ and $(C2a)$ of the statement $(C)$ in theorem 3 and $K_{t}=(\{\gamma_{t}\}^{n})^{1/n}$ then the constant $C(\gamma)$ satisfies the stability properties of the statement (A) of the lemma 2. We prove now the statement (B) of the lemma 2. In fact let $f:=\{\gamma\}^{-n}\gamma^{n}/\Omega\geq 0$ . Then the uniform estimate for the integral

\{\gamma\}^{-n}\int\limits_{X}-\psi\,\gamma^{n}=\frac{1}{\alpha}\,\int\limits_{X}-\alpha\psi f\,\Omega

follows from the elementary inequality $-\alpha\psi f\leq e^{-\alpha\psi}-1+f\log(1+f)$ combined with the uniform estimate $\int_{X}e^{-\alpha\psi}\Omega\leq C$ of lemma 1. In this case the required stability properties of the constant $C(\gamma,\Omega)>0$ in the capacity estimate are obvious. $\Box$

[0290]

Lemma 3

(Degenerate Comparison Principle). Let $X$ be a compact Kähler manifold of complex dimension $n$ , let $\gamma$ be a closed real $(1,1)$ -current with continuous local potentials or a closed positive $(1,1)$ -current with bounded local potentials, and consider $\varphi,\,\psi\in{\cal P}_{\gamma}\cap L^{\infty}(X)$ . Then

\int\limits_{\varphi<\psi}\gamma^{n}_{\psi}\;\leq\int\limits_{\varphi<\psi}\gamma^{n}_{\varphi}\;.

Proof.
Step I. We assume first $\varphi,\,\psi\in{\cal P}_{\gamma}\cap C^{0}(X)$ . We will denote by $\partial S$ the boundary in $X$ of a set $S\subset X$ . By the continuity of the functions $\varphi,\,\psi$ we deduce:
1) the set $\{\varphi<\psi\}$ is open and $\partial\{\varphi<\psi\}\subset\{\varphi=\psi\}$ ,
2) for all $\varepsilon>0$ there exists an open neighborhood ${\cal V}\subset X$ of the set $\{\varphi\geq\psi\}$ such that $\max\{\varphi+\varepsilon,\psi\}=\varphi+\varepsilon$ over ${\cal V}$ .
So $\partial\{\varphi<\psi\}\subset{\cal V}$ and the Stokes formula implies the equality

\displaystyle\int\limits_{\varphi<\psi}\gamma^{n}_{\varphi}\;=\int\limits_{\varphi<\psi}(\gamma+i\partial\bar{\partial}\max\{\varphi+\varepsilon,\psi\})^{n}\,,

for all $\varepsilon>0$ . Moreover by the monotone convergence theorem in pluripotential theory we deuce that the current $(\gamma+i\partial\bar{\partial}\max\{\varphi+\varepsilon,\psi\})^{n}$ converges weakly to the current $\gamma_{\psi}^{n}$ over the open set $\{\varphi<\psi\}$ as $\varepsilon\rightarrow 0^{+}$ . Thus

\displaystyle\int\limits_{\varphi<\psi}\gamma_{\varphi}^{n}\;=\,\liminf\limits_{\varepsilon\rightarrow 0^{+}}\int\limits_{\varphi<\psi}(\gamma+i\partial\bar{\partial}\max\{\varphi+\varepsilon,\psi\})^{n}\geq\int\limits_{\varphi<\psi}\gamma_{\psi}^{n}\,.

Step II. For the general case we use the following well known fact.

[0291]

Claim 3

Let $f:X\rightarrow\mathbb{R}$ be a function over a topological space $X$ and let $S,\,S_{k}\subset X$ , $k\in I$ , be a locally finite family of closed subsets such that $S\subset\bigcup_{k}S_{k}$ and $f\in C^{0}(S_{k})$ for all $k$ . Then $f\in C^{0}(S)$ . $\Box$

Let now $(U_{\alpha})_{\alpha=1}^{N}$ be a finite open covering of $X$ such that $\gamma=i\partial\bar{\partial}h_{\alpha}$ over $U_{\alpha}$ with $\inf_{U_{\alpha}}h_{\alpha}=0$ . By the quasicontinuity of plurisubharmonic functions, for every $\delta>0$ there exists an open set $G_{\delta,\alpha}\subset U_{\alpha}$ such that $u_{\alpha}:=h_{\alpha}+\varphi,\,h_{\alpha}\in C^{0}(U_{\alpha}\smallsetminus G_{\delta,\alpha})$ and $\operatorname{Cap}(G_{\delta,\alpha},U_{\alpha})<\delta$ . In particular $\varphi\in C^{0}(U_{\alpha}\smallsetminus G_{\delta,\alpha})$ . Consider the open set $G_{\delta}:=\bigcup_{\alpha}G_{\delta,\alpha}\subset X$ . Then the inclusion $X\smallsetminus G_{\delta}\subset\bigcup_{\alpha}(U_{\alpha}\smallsetminus G_{\delta,\alpha})$ implies $\varphi\in C^{0}(X\smallsetminus G_{\delta})$ by claim 3. We can also assume $\psi\in C^{0}(X\smallsetminus G_{\delta})$ . Set also $v_{\alpha}:=h_{\alpha}+\psi$ . Let $\omega$ be a Kähler metric over $X$ and $g_{\alpha}$ smooth functions over $U_{\alpha}$ such that $\omega=i\partial\bar{\partial}g_{\alpha}$ , $\inf_{U_{\alpha}}g_{\alpha}=0$ . By the result of Greene-Wu [Gr-Wu] there exists a sequence $(\varepsilon_{j})_{j}\subset(0,\varepsilon)$ , $\varepsilon_{j}\downarrow 0$ and $\psi_{j},\,\varphi_{j}\in{\cal P}_{\gamma+\varepsilon_{j}\omega}\cap C^{\infty}(X)$ , with $\psi_{j}\downarrow\psi$ and $\varphi_{j}\downarrow\varphi$ . We can assume $0\leq\inf_{X}\varphi$ , $0\leq\inf_{X}\psi$ and we set $u_{j,\alpha}:=h_{\alpha}+g_{\alpha}+\varphi_{j}$ , $v_{j,\alpha}:=h_{\alpha}+g_{\alpha}+\psi_{j}$ , $M:=\max_{\alpha}\{\|u_{1,\alpha}\|_{L^{\infty}(U_{\alpha})},\|v_{1,\alpha}\|_{L^{\infty}(U_{\alpha})}\}$ . Then by step I

\displaystyle\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi_{j})^{n}\leq\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}\;.

(2.4)

Let $f\in C^{0}(X)$ such that $f=\psi$ over $X\smallsetminus G_{\delta}$ . Then the set $\{\varphi_{k}<f\}$ is open and $\{\varphi_{k}<f\}\cup G_{\delta}=\{\varphi_{k}<\psi\}\cup G_{\delta}$ . Thus

			$\displaystyle\int\limits_{\varphi_{k}<\psi}\gamma_{\psi}^{n}\;\leq\int\limits_{\varphi_{k}<f}\gamma_{\psi}^{n}+\int\limits_{G_{\delta}}\gamma_{\psi}^{n}\,\leq\int\limits_{\varphi_{k}<f}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi)^{n}+\sum_{\alpha}\;\int\limits_{G_{\delta,\alpha}}(i\partial\bar{\partial}v_{\alpha})^{n}$
		$\displaystyle\leq$	$\displaystyle\liminf_{j\rightarrow+\infty}\int\limits_{\varphi_{k}<f}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi_{j})^{n}+M^{n}\sum_{\alpha}\;\int\limits_{G_{\delta,\alpha}}(i\partial\bar{\partial}M^{-1}\,v_{\alpha})^{n}$
		$\displaystyle\leq$	$\displaystyle\liminf_{j\rightarrow+\infty}\left(\;\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi_{j})^{n}+\int\limits_{G_{\delta}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi_{j})^{n}\right)+M^{n}N\delta$
		$\displaystyle\leq$	$\displaystyle\liminf_{j\rightarrow+\infty}\left(\;\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}+M^{n}\sum_{\alpha}\;\int\limits_{G_{\delta,\alpha}}(i\partial\bar{\partial}M^{-1}\,v_{j,\alpha})^{n}\right)$
		$\displaystyle+$	$\displaystyle M^{n}N\delta\qquad\qquad\mbox{(by \eqref{CompPrincC0}) }$
		$\displaystyle\leq$	$\displaystyle\lim_{j\rightarrow+\infty}\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}+2M^{n}N\delta$
		$\displaystyle=$	$\displaystyle\int\limits_{\varphi_{k}\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}+2M^{n}N\delta\,.$

Then by letting $k\rightarrow+\infty$ we get

\displaystyle\int\limits_{\varphi<\psi}\gamma_{\psi}^{n}\;\leq\,\limsup_{k\rightarrow+\infty}\int\limits_{\varphi_{k}\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}+2M^{n}N\delta\,.

(2.5)

Now the set $\{\varphi\leq\psi\}\smallsetminus G_{\delta}$ is closed by the continuity of $\varphi$ and $\psi$ over $X\smallsetminus G_{\delta}$ . Thus

			$\displaystyle\int\limits_{\varphi\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi)^{n}\geq\int\limits_{\{\varphi\leq\psi\}\smallsetminus G_{\delta}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi)^{n}$
		$\displaystyle\geq$	$\displaystyle\limsup_{k\rightarrow+\infty}\int\limits_{\{\varphi\leq\psi\}\smallsetminus G_{\delta}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}$
		$\displaystyle\geq$	$\displaystyle\limsup_{k\rightarrow+\infty}\left(\;\,\int\limits_{\varphi\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}-\int\limits_{G_{\delta}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}\right)$
		$\displaystyle\geq$	$\displaystyle\limsup_{k\rightarrow+\infty}\left(\;\,\int\limits_{\varphi\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}-M^{n}\sum_{\alpha}\;\int\limits_{G_{\delta,\alpha}}(i\partial\bar{\partial}M^{-1}\,u_{k,\alpha})^{n}\right)$
		$\displaystyle\geq$	$\displaystyle\limsup_{k\rightarrow+\infty}\int\limits_{\varphi_{k}\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}-M^{n}N\delta\,.$

So by (2.5) we derive

	$\displaystyle\int\limits_{\varphi<\psi}\gamma_{\psi}^{n}\;$	$\displaystyle\leq$	$\displaystyle\int\limits_{\varphi\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi)^{n}+3M^{n}N\delta$
		$\displaystyle\leq$	$\displaystyle\int\limits_{\varphi\leq\psi}\gamma_{\varphi}^{n}\;+\;\sum_{l=1}^{n}{n\choose l}\int\limits_{X}\varepsilon^{l}\omega^{l}\wedge\gamma^{n-l}+3M^{n}N\delta\,.$

Then letting $\varepsilon\rightarrow 0$ and $\delta\rightarrow 0$ we get

\int\limits_{\varphi<\psi}\gamma_{\psi}^{n}\;\leq\int\limits_{\varphi\leq\psi}\gamma_{\varphi}^{n}\;.

Now the conclusion follows by replacing $\varphi$ by $\varphi+t$ , $t>0$ in the previous formula and letting $t\rightarrow 0$ . $\Box$

We recall now the following lemma due to Kołodziej [Kol], (see also [Ti-Zhu1], [Ti-Zhu2]).

[0292]

Lemma 4

. Let $a:(-\infty,0]\rightarrow[0,1]$ , be a monotone non decreasing function such that for some $B>0$ , $\delta>0$ the inequality

t\,a(s)\leq B\,a(s+t)^{1+\delta}

holds for all $s\leq 0,\,t\in[0,1],\,s+t\leq 0$ . Then for all $S<0$ such that $a(S)>0$ and all $D\in[0,1],\,S+D\leq 0$ we have the estimate

D\leq e(3+2/\delta)B\,a(S+D)^{\delta}\,.

The following lemma is a simple application of the main result in Bedford-Taylor [Be-Te].

[0293]

Lemma 5

. Let $X$ be a connected compact complex manifold of complex dimension $n$ , let $\gamma$ be a big closed positive $(1,1)$ -current with bounded local potentials and let $\Omega>0$ be a smooth volume form. Then there exist constants $\alpha=\alpha(\gamma,\Omega)>0$ , $C=C(\gamma,\Omega)>0$ such that

\displaystyle\int\limits_{V}\Omega\leq e^{\alpha}Ce^{-\alpha/\operatorname{Cap}_{\gamma}(V)^{1/n}}\,,

for all open sets $V\subset X$ .

Proof. In order to prove this estimate, it is sufficient by (2.1) to show the inequality

\displaystyle\int\limits_{U}\Omega\leq e^{\alpha}Ce^{-\alpha/\operatorname{Cap}_{\gamma}(\,\overline{U}\,)^{1/n}}

for an arbitrary relatively compact open set $U\subset\subset V$ . For this purpose, consider the function

\Psi_{U}(x):=\sup\{\varphi(x)\,\mid\,\varphi\in{\cal P}_{\gamma}\,,\,\varphi_{\mid_{U}}\leq 0\}\geq 0\,.

Remark that $(\Psi_{U})_{\mid_{U}}\equiv 0$ since $0\in{\cal P}_{\gamma}$ . If $U\not=\emptyset$ there exists a constant $C_{U}>0$ such that $\sup_{X}\,\varphi\leq C_{U}$ for all $\varphi\in{\cal P}_{\gamma}\,,\,\varphi_{\mid_{U}}\leq 0$ . In fact let $S_{U}:=\{\varphi\in{\cal P}_{\gamma}\,\mid\,\varphi_{\mid_{U}}\leq 0\}$ and set $\tilde{\varphi}:=\varphi-\sup_{X}\varphi$ . By contradiction we would get $\varphi_{j}\in S_{U}$ such that $\sup_{X}\varphi_{j}\rightarrow+\infty$ . This implies $\sup_{U}\tilde{\varphi}_{j}\rightarrow-\infty$ and so $\int_{U}-\tilde{\varphi}_{j}\,\Omega\geq-C\sup_{U}\tilde{\varphi}_{j}\rightarrow+\infty$ , which contradicts the first integral estimate of lemma 1.
Then it follows from quite standard local arguments that the upper regularization $\Psi^{*}_{U}\in{\cal P}_{\gamma}$ . Moreover $\Psi^{*}_{U}\in L^{\infty}(X)$ , $\Psi^{*}_{U}\geq 0$ and $\Psi^{*}_{U}\equiv 0$ over $U$ . We recall now the following well known consequence of a result of Bedford and Taylor [Be-Te].

[0294]

Theorem 4

. Let $\varphi\in{\cal P}_{\gamma}\cap L^{\infty}(X)$ and let $B$ be an open coordinate ball. Then there exists $\hat{\varphi}\in{\cal P}_{\gamma}\cap L^{\infty}(X)$ , $\hat{\varphi}\geq\varphi$ such that $\gamma^{n}_{\hat{\varphi}}=0$ on $B$ and $\hat{\varphi}=\varphi$ on $X\smallsetminus B$ . Moreover if $\varphi_{1}\leq\varphi_{2}$ , then $\hat{\varphi}_{1}\leq\hat{\varphi}_{2}$ .

This implies the following corollary.

[0295]

Corollary 1

. The extremal function $\Psi^{*}_{U}\in{\cal P}_{\gamma}\cap L^{\infty}(X)$ satisfies $\Psi^{*}_{U}\geq 0$ over $X$ , $\Psi^{*}_{U}\equiv 0$ over $U$ and $\gamma^{n}_{\Psi^{*}_{U}}=0$ over $X\smallsetminus\overline{U}$ .

Proof. By the classical Choquet lemma there exists a sequence $(\varphi_{j})_{j}\subset S_{U}$ , $\varphi_{j}\geq 0$ such that $\Psi^{*}_{U}=(\sup_{j}\varphi_{j})^{*}$ . We can assume that this sequence is increasing. Otherwise, set $\tilde{\varphi}_{1}:=\varphi_{1}$ and $\tilde{\varphi}_{j}:=\max\{\varphi_{j},\tilde{\varphi}_{j-1}\}\in S_{U}$ . Let $B$ be an open coordinate ball in $X\smallsetminus\overline{U}$ and let $\hat{\varphi}_{j}\in S_{U}$ be a solution of the Dirichlet problem $\gamma^{n}_{\hat{\varphi}_{j}}=0$ over $B$ as in theorem 4. Thus the sequence $(\hat{\varphi}_{j})_{j}\subset S_{U}$ is still increasing and $\Psi^{*}_{U}=(\sup_{j}\hat{\varphi}_{j})^{*}$ . Remember also that the plurisubharmonicity implies that $\Psi^{*}_{U}=\lim_{j}\hat{\varphi}_{j}$ almost everywhere. By the monotone increasing theorem from classical pluripotential theory, we derive $\gamma^{n}_{\Psi^{*}_{U}}=0$ on $B$ , and the conclusion follows from the fact that $B$ is arbitrary. $\Box$

By using the second integral estimate of lemma 1 we get

\int\limits_{U}\Omega=\int\limits_{U}e^{-\alpha\,\Psi^{*}_{U}}\,\Omega\,\leq\,\int\limits_{X}e^{-\alpha\,\Psi^{*}_{U}}\,\Omega\,\leq\,Ce^{-\alpha\sup_{X}\Psi^{*}_{U}}\,.

Set $K_{U}:=\sup_{X}\Psi^{*}_{U}$ . If $K_{U}>1$ set $\varphi:=K_{U}^{-1}\Psi^{*}_{U}$ . Then $0\leq\gamma_{\Psi^{*}_{U}}\leq K_{U}\gamma_{\varphi}$ and so $\varphi\in{\cal P}_{\gamma}[0,1]$ . By corollary 1 we deduce

\{\gamma\}^{n}K_{U}^{-n}=K_{U}^{-n}\int\limits_{\overline{U}}\gamma^{n}_{\Psi^{*}_{U}}\,\leq\,\int\limits_{\overline{U}}\gamma^{n}_{\varphi}\,\leq\,\{\gamma\}^{n}\operatorname{Cap}_{\gamma}(\,\overline{U}\,)\,,

thus $-\alpha K_{U}\leq-\alpha/\operatorname{Cap}_{\gamma}(\,\overline{U}\,)^{1/n}$ . If $K_{U}\leq 1$ then $\Psi^{*}_{U}\in{\cal P}_{\gamma}[0,1]$ and so

1=\{\gamma\}^{-n}\int\limits_{\overline{U}}\gamma^{n}_{\Psi^{*}_{U}}\,\leq\,\operatorname{Cap}_{\gamma}(\,\overline{U}\,)\leq\operatorname{Cap}_{\gamma}(X)=1\,.

In both cases we reach the required conclusion. $\Box$

Proof of theorem 3, part A.
We can assume $\sup_{X}\psi=0$ . The fact that the current $\gamma$ has continuous local potentials implies that the function $\psi$ is upper semicontinuous, so the set $U_{s}:=\{\psi<s\}$ , $s\leq 0$ is open. Let $t\in[0,1]$ , $s+t\leq 0$ , $\varphi\in{\cal P}_{\gamma}[-1,0]$ and set $V:=\{\psi-s-t<t\varphi\}$ . Then the inclusions $U_{s}\subset V\subset U_{s+t}$ hold. Using the Degenerate Comparison Principle we get

t^{n}\int\limits_{U_{s}}\gamma_{\varphi}^{n}\;\leq\int\limits_{U_{s}}\gamma_{t\varphi}^{n}\;\leq\int\limits_{V}\gamma_{t\varphi}^{n}\;\leq\int\limits_{V}\gamma_{\psi}^{n}\;\leq\int\limits_{U_{s+t}}\gamma_{\psi}^{n}\,,

thus combining this with Hölder’s inequality in Orlicz spaces [Iw-Ma] and lemma 5 we obtain

$\displaystyle t^{n}\operatorname{Cap}_{\gamma}(U_{s})$	$\displaystyle\leq$	$\displaystyle\{\gamma\}^{-n}\int\limits_{U_{s+t}}\gamma_{\psi}^{n}\;=\,\{\gamma\}^{-n}\int\limits_{U_{s+t}}f\,\Omega$
	$\displaystyle\leq$	$\displaystyle\{\gamma\}^{-n}C_{\varepsilon_{0}}\\|f\\|_{L\log^{n+\varepsilon}L(X)}\cdot\\|1\\|_{\operatorname{Exp}^{\frac{1}{n+\varepsilon}}L(U_{s+t})}$
	$\displaystyle=$	$\displaystyle\frac{\{\gamma\}^{-n}C_{\varepsilon_{0}}\\|f\\|_{L\log^{n+\varepsilon}L(X)}}{\log^{n+\varepsilon}\left(1+1/\operatorname{Vol}_{\Omega}(U_{s+t})\right)}$
	$\displaystyle\leq$	$\displaystyle\frac{\{\gamma\}^{-n}C_{\varepsilon_{0}}\\|f\\|_{L\log^{n+\varepsilon}L(X)}}{\log^{n+\varepsilon}\left(1+e^{-\alpha}C^{-1}e^{\alpha/\operatorname{Cap}_{\gamma}(U_{s+t})^{1/n}}\right)}$
	$\displaystyle\leq$	$\displaystyle C_{\varepsilon_{0}}(k/\alpha)^{n+\varepsilon}\{\gamma\}^{-n}\\|f\\|_{L\log^{n+\varepsilon}L(X)}\operatorname{Cap}_{\gamma}(U_{s+t})^{(n+\varepsilon)/n}\,.$

(Here $k>0$ is a constant such that $k^{-1}\alpha/x\leq\log(1+e^{-\alpha}C^{-1}e^{\alpha/x})$ for all $x\in(0,1]$ ). So if we set $\delta:=\varepsilon/n$ and

B:=C_{\varepsilon_{0}}^{1/n}(k/\alpha)^{1+\varepsilon/n}I_{\varepsilon}(f)^{1/n}\,,

we deduce that the function $a(s):=\operatorname{Cap}_{\gamma}(U_{s})^{1/n}$ , $s\leq 0$ , satisfies the hypothesis of lemma (4). Consider the function $\kappa(t):=K_{\delta}B\,t^{\delta}$ , with constant $K_{\delta}:=e(3+2/\delta)$ . Remember also the uniform capacity estimate $a(s)\leq C\,(-s)^{-1/n}$ of lemma (2). Let now $\eta>1$ be arbitrary. We claim that $a(S_{\eta})=0$ for

-S_{\eta}=C^{n}(K_{\delta}B\,\eta)^{n/\delta}+1\,.

The fact that the function $a$ is left continuous (by formula (2.1)) will imply that $a(S_{1})=0$ also. Remark that $S_{\eta}$ is a solution of the equation

C(-S_{\eta}-1)^{-1/n}=\kappa^{-1}(\eta^{-1})\,,

where $\kappa^{-1}$ is the inverse of the function $\kappa$ . So if by absurd $a(S_{\eta})>0$ we deduce by lemmas (4) and (1)

1\leq\kappa(a(S_{\eta}+1))\leq\kappa(C(-S_{\eta}-1)^{-1/n})=\eta^{-1}<1\,,

which is a contradiction. Thus if we set $-I:=\max\{s\leq 0\,\mid\,a(s)=0\}$ we obtain $I\leq-S_{1}\leq C^{n}(K_{\delta}B)^{n/\delta}+1$ , which by arranging the coefficients yelds to the right hand side of the estimate in the statement of the theorem 3. Moreover by definition $\operatorname{Cap}_{\gamma}(U_{-I})=0$ . In order to prove that $U_{-I}=\emptyset$ we will show that every relatively compact open set $V\subset\subset U_{-I}$ is empty. We know that $\operatorname{Cap}_{\gamma}(\,\overline{V}\,)=0$ . We recall that $V\not=\emptyset$ implies $\sup_{X}\Psi_{V}<+\infty$ , by the remark in the begining of lemma 5. So $\sup_{X}\Psi_{V}=+\infty$ implies $V=\emptyset$ . Moreover $K_{V}:=\sup_{X}\Psi^{*}_{V}=+\infty$ if and only if $\sup_{X}\Psi_{V}=+\infty$ . We recall also that $K_{V}\leq 1$ implies $\operatorname{Cap}_{\gamma}(\,\overline{V}\,)=1$ , by the proof of lemma 5. This is equivalent to say that $\operatorname{Cap}_{\gamma}(\,\overline{V}\,)<1$ implies $K_{V}>1$ . In this case $K_{V}\geq\operatorname{Cap}_{\gamma}(\,\overline{V}\,)^{-1/n}$ . We deduce $K_{V}=+\infty$ , thus $V=\emptyset$ . $\Box$

Proof of part B.
Set $a:=\max\{\|\varphi\|_{L^{\infty}}(X),\,\|\psi\|_{L^{\infty}}(X)\}$ , consider $\theta\in{\cal P}_{\gamma}[0,1]$ , $s\geq 0$ , $t\in[0,1]$ and set

V:=\left\{\varphi<\frac{t}{1+a}\,\theta+\left(1-\frac{t}{1+a}\right)\psi-s-t\right\}\,.

Then the elementary inequality $0\leq-\frac{t}{1+a}\psi\leq\frac{at}{1+a}$ implies the inclusions $\{\varphi-\psi<-s-t\}\subset V\subset\{\varphi-\psi<-s\}$ . Thus by applying the Degenerate Comparison Principle we obtain

	$\displaystyle\frac{t^{n}}{(1+a)^{n}}\int\limits_{\varphi-\psi<-s-t}\gamma_{\theta}^{n}$	$\displaystyle\leq$	$\displaystyle\int\limits_{V}\left[\frac{t}{1+a}\,\gamma_{\theta}+\left(1-\frac{t}{1+a}\right)\gamma_{\psi}\right]^{n}$
		$\displaystyle\leq$	$\displaystyle\int\limits_{V}\gamma_{\varphi}^{n}\;\;\leq\;\;\int\limits_{\varphi-\psi<-s}\gamma_{\varphi}^{n}\,.$

By inverting the roles of $\varphi$ and $\psi$ in the previous inequality and by summing up we get

\frac{t^{n}}{(1+a)^{n}}\,\int\limits_{|\varphi-\psi|>s+t}\gamma_{\theta}^{n}\;\;\leq\;\;\int\limits_{|\varphi-\psi|>s}(f+g)\,\Omega\,.

By taking the sup over $\theta$ we obtain the capacity estimate

\displaystyle t^{n}\operatorname{Cap}_{\gamma}(|\varphi-\psi|>s+t)\leq(1+a)^{n}\{\gamma\}^{-n}\int\limits_{|\varphi-\psi|>s}(f+g)\,\Omega\,,

(2.6)

for all $s\geq 0$ , $t\in[0,1]$ . The fact that the solutions $\varphi$ and $\psi$ are continuous implies that the sets $U_{s}:=\{|\varphi-\psi|>s\}\subset X$ are open. Thus by combining lemma 5 with a computation similar to that in the proof of part A we obtain

	$\displaystyle t^{n}\operatorname{Cap}_{\gamma}(U_{s+t})$	$\displaystyle\leq$	$\displaystyle(1+a)^{n}\{\gamma\}^{-n}C_{\varepsilon_{0}}\\|f+g\\|_{L\log^{n+\varepsilon_{0}}L(X)}\operatorname{Cap}_{\gamma}(U_{s})^{(n+\varepsilon)/n}$
		$\displaystyle\leq$	$\displaystyle B^{n}\operatorname{Cap}_{\gamma}(U_{s})^{(n+\varepsilon)/n}\,,$

where the constant $B>0$ depends on the same quantities as the constant $C_{2}$ in the statement B of theorem 3. We deduce that the function $a(s):=\operatorname{Cap}_{\gamma}(U_{-s})^{1/n}$ , $s\leq 0$ , satisfies the hypothesis of lemma (4) with $\delta=\varepsilon_{0}/n$ . On the other hand, the capacity estimate (2.6) combined with Hölder’s inequality in Orlicz spaces implies for all $t\in[0,1]$ the inequalities

$\displaystyle t^{n}\operatorname{Cap}_{\gamma}(\|\varphi-\psi\|>2t)$	$\displaystyle\leq$	$\displaystyle(1+a)^{n}\{\gamma\}^{-n}\int\limits_{\|\varphi-\psi\|>t}(f+g)\,\Omega$	(2.7)
	$\displaystyle\leq$	$\displaystyle\frac{(1+a)^{n}\{\gamma\}^{-n}}{t}\int\limits_{X}\|\varphi-\psi\|(f+g)\,\Omega$
	$\displaystyle\leq$	$\displaystyle\frac{2(1+a)^{n}\{\gamma\}^{-n}}{t}\\|\varphi-\psi\\|_{\operatorname{Exp}L(X)}\\|f+g\\|_{L\log L(X)}$
	$\displaystyle\leq$	$\displaystyle\frac{4K(1+a)^{n}}{t}\\|\varphi-\psi\\|_{\operatorname{Exp}L(X)}\,.$

[0296]

Claim 4

. If $\|\varphi-\psi\|_{L^{1}(X)}\leq 1/2$ , then there exists a constant $C_{a}>0$ such that

\|\varphi-\psi\|_{\operatorname{Exp}L(X)}\leq C_{a}/\log\|\varphi-\psi\|^{-1}_{L^{1}(X)}\,.

Proof. We assume $\|\varphi-\psi\|_{L^{1}(X)}>0$ , otherwise there is nothing to prove. Set $C_{k,a}:=k(e^{2a/k}-1)/(2a)$ , $k>0$ . Then for all $k>0$ and all $x\in[0,2a/k]$ the inequality $e^{x}-1\leq C_{k,a}\,x$ holds. Then the inequality $|\varphi-\psi|/k\leq 2a/k$ implies

\int\limits_{X}\left(e^{|\varphi-\psi|/k}-1\right)\Omega\leq C_{k,a}\int\limits_{X}\frac{|\varphi-\psi|}{k}\,\Omega\,.

We get from there the implication

\displaystyle\|\varphi-\psi\|_{L^{1}(X)}=k/C_{k,a}\quad\Longrightarrow\quad\|\varphi-\psi\|_{\operatorname{Exp}L(X)}\leq k\,,

(2.8)

since by definition

\|\varphi-\psi\|_{\operatorname{Exp}L(X)}:=\inf\left\{k>0\,\mid\;\int\limits_{X}\left(e^{|\varphi-\psi|/k}-1\right)\Omega\leq 1\,\right\}\,.

So if we set $\mu(k):=k/C_{k,a}>0$ we deduce by the implication (2.8)

\displaystyle\|\varphi-\psi\|_{\operatorname{Exp}L(X)}\leq\mu^{-1}\left(\|\varphi-\psi\|_{L^{1}(X)}\right)\,,

(2.9)

where $\mu^{-1}:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ is the inverse function of $\mu$ . Explicitly $\mu^{-1}(y)=2a/\log(1+2a/y)$ , for all $y>0$ . Now there exists a constant $C_{a}>0$ such that $\mu^{-1}(y)\leq C_{a}/\log(1/y)$ for all $y\in(0,1/2]$ . This combined with (2.9) implies the conclusion. $\Box$

Combining claim 4 with the estimate (2.7) we infer the capacity estimate

\displaystyle a(-t)\leq\frac{C}{t^{1+1/n}}\left(\log\|\varphi-\psi\|^{-1}_{L^{1}(X)}\right)^{-1/n}\,,

(2.10)

where the constant $C>0$ depends on the same quantities as the constant $C_{2}$ in statement B. Set now $C_{2}:=C^{n}(2K_{\delta}B)^{n/\delta}>0$ (with $K_{\delta}>0$ as in the proof of part A) and define

t:=C_{2}^{\alpha_{0}}\left(\log\|\varphi-\psi\|^{-1}_{L^{1}(X)}\right)^{-\alpha_{0}}\,.

The hypothesis $t\in(0,1]$ combined with the hypothesis of claim 4 forces the condition $\|\varphi-\psi\|_{L^{1}(X)}\leq\min\{1/2,e^{-C_{2}}\}$ . Moreover $t$ is solution of the equation

\frac{C}{t^{1+1/n}}\left(\log\|\varphi-\psi\|^{-1}_{L^{1}(X)}\right)^{-1/n}=\kappa^{-1}\left(\frac{t}{2}\right)\,,

where $\kappa^{-1}$ is the inverse of the function $\kappa$ introduced in the proof of part A. We claim that $a(-2t)=0$ . Otherwise, by lemma (4) and inequality (2.10), we infer

0<t\leq\kappa(a(-t))\leq\kappa(\kappa^{-1}(t/2))=t/2\,,

which is absurd. By using the argument already explained at the end of the proof of part A, we deduce that the set $\{|\varphi-\psi|>2t\}\subset X$ is empty, which implies the desired conclusion. $\Box$

[0297]

3 The domain of definition of the complex Monge-Ampère operator.

We start with a few definitions.

[0298]

Definition 1

Let $X$ be a compact complex manifold of complex dimension $n$ , let $\chi\in H^{1,1}(X,\mathbb{R})$ be a pseudoeffective class, let $\omega>0$ be a hermitian form and let $(U_{\alpha})_{\alpha}$ be a finite covering of coordinate starshaped open sets. We denote by $MA_{\chi}$ the set of closed positive $(1,1)$ -currents $\gamma\in\chi$ such that

-\int\limits_{U_{\alpha}}h_{\alpha}\gamma^{k}\wedge\omega^{n-k}<+\infty\,,

with $\gamma=i\partial\bar{\partial}h_{\alpha}$ , $\sup_{U_{\alpha}}h_{\alpha}=0$ and $\gamma^{k}:=i\partial\bar{\partial}(h_{\alpha}\gamma^{k-1})$ over $U_{\alpha}$ , for all $k=1,...,n-1$ .

It is clear by the definition that the closed positive currents $\gamma^{k}$ , for $k=1,...,n$ are globally well defined. Consider now $\gamma\geq 0$ be a closed positive $(1,1)$ -current with continuous local potentials. We define

\hat{\cal P}_{\gamma}:=\left\{\varphi\in{\cal P}_{\gamma}\,|\,\gamma+i\partial\bar{\partial}\varphi\in MA_{\{\gamma\}}\right\}\,.

Let $\varphi\in\hat{\cal P}_{\gamma}$ with zero Lelong numbers. It is well known from the first author work (which becomes drastically simple in this particular case) the existence of a family $(\varphi_{\varepsilon})_{\varepsilon>0}$ , $\varphi_{\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}\cap C^{\infty}(X)$ , such that $\varphi_{\varepsilon}\downarrow\varphi$ as $\varepsilon\downarrow 0^{+}$ . In the case $\varphi\in{\cal P}_{\gamma}\cap C^{0}(X)$ the convergence of $\varphi_{\varepsilon}$ is also uniform. We have the following crucial result.

[0299]

Theorem 5

(Degenerate monotone convergence result).
Let $(X,\omega)$ be a polarized compact Kähler manifold of complex dimension $n$ and let $\gamma$ , $T$ be closed positive $(1,1)$ -currents with continuous local potentials. Then the following statements hold true.
A) For all $\varphi\in\hat{\cal P}_{\gamma}$ , $\varphi\leq 0$ and $k,l\geq 0$ , $k+l\leq n$ , $k\leq n-1$

\int\limits_{X}-\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}<+\infty\,,

B) Let $\varphi\in\hat{\cal P}_{\gamma}$ , $\varphi\leq 0$ with zero Lelong numbers and $\varphi_{\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}\cap C^{\infty}(X)$ , such that $\varphi_{\varepsilon}\downarrow\varphi$ as $\varepsilon\rightarrow 0^{+}$ . Then for all $k,l\geq 0$ , $k+l\leq n$ , $k\leq n-1$

	$\displaystyle\varphi_{\varepsilon}\,(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k}\wedge T^{l}\longrightarrow\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\,,$		(3.1)

	$\displaystyle(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k+1}\wedge T^{l}\longrightarrow\gamma_{\varphi}^{k+1}\wedge T^{l}\,,$		(3.2)

weakly as $\varepsilon\rightarrow 0^{+}$ . Moreover $\gamma_{\varphi}^{k}\wedge T^{l}=T^{l}\wedge\gamma_{\varphi}^{k}$ for all $\varphi\in\hat{\cal P}_{\gamma}$ and $k,l\geq 0$ , $k+l\leq n$ .

As follows immediately from the proof, the statement of this theorem still holds if we replace $T^{l}$ with a product $T_{1}\wedge....\wedge T_{l}$ , where the currents $T_{j}$ have the same properties as $T$ . As a matter of fact, we wrote the statement in the previous special case only for the sake of notation simplicity. However during the proof it is useful to consider that statements concerning terms involving $T^{l}$ are still valid if we replace $T^{l}$ with $\gamma^{r}\wedge T^{l-r}$ .

Proof. Statement (3.2) follows from (3.1) by using the weak continuity of the $i\partial\bar{\partial}$ operator and an induction on (3.2). We remark that claim 2 asserts statement A) in full generality for $k=0$ . We denote by $A_{k,l}$ the assertion A) in the statement of the theorem for the relative indices $(k,l)$ . For all $k=0,...,n-1$ and $l=0,...,n-k$ we define the following statement $B_{k,l}$ : for all $p=0,...,k$

	$\displaystyle\varphi_{\varepsilon}\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}\longrightarrow\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\,,$		(3.3)

	$\displaystyle i\partial\bar{\partial}\varphi_{\varepsilon}\wedge\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}\longrightarrow i\partial\bar{\partial}\varphi\wedge\gamma_{\varphi}^{k}\wedge T^{l}\,,$		(3.4)

	$\displaystyle\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p+1}\wedge T^{l}\longrightarrow\gamma_{\varphi}^{k+1}\wedge T^{l}\,,$		(3.5)

	$\displaystyle\varphi\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}\longrightarrow\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\,,\,,$		(3.6)

weakly as $\varepsilon\rightarrow 0^{+}$ and

\displaystyle\gamma_{\varphi}^{k+1}\wedge T^{l}=T^{l}\wedge\gamma_{\varphi}^{k+1}\,.

(3.7)

We remark that (3.4) follows from (3.3) by the weak continuity of the $i\partial\bar{\partial}$ operator. By combining (3.4) with the weak continuity of the $i\partial\bar{\partial}$ operator we obtain

(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)\wedge\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}\longrightarrow\gamma_{\varphi}^{k+1}\wedge T^{l}\,,

weakly as $\varepsilon\rightarrow 0^{+}$ . On the other hand $\eqref{SymWeg}_{p-1,\bullet}$ implies

	$\displaystyle(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)\wedge\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}$	$\displaystyle=$	$\displaystyle(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p+1}\wedge T^{l}\wedge\gamma_{\varphi}^{p}$
		$\displaystyle=$	$\displaystyle\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p+1}\wedge T^{l}\,.$

In this way we deduce (3.5). The symmetry identity (3.7) follows from (3.5) for $p=0$ and from the fact that

T^{l}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k+1}\longrightarrow T^{l}\wedge\gamma_{\varphi}^{k+1}\,,

weakly as $\varepsilon\rightarrow 0^{+}$ . This last convergence statement follows by combining (3.5) for $p=l=0$ with an induction on $l$ by means of the weak continuity of the $i\partial\bar{\partial}$ operator.
We now prove simultaneously the statements $A_{k,l}$ and $B_{k,l}$ , $l=0,...,n-k$ by using an induction on $k=0,...,n-1$ . For the moment we assume that the potential $\varphi$ in statement A) also has zero Lelong numbers, but we will get rid of this hypothesis at the end. Statements $A_{0,\bullet}$ and $B_{0,\bullet}$ are true by claim 2 and its proof. So we assume that these statements hold for $j\leq k-1$ and we prove them for $k$ . The induction process is divided in two main steps.

Step I. This step consists in proving the

[029A]

Claim 5

. If $A_{j,\bullet}$ and $B_{j,\bullet}$ hold true for all $j=0,...,k-1$ , then $A_{k,l}$ implies $B_{k,l}$ , with $l=0,...,n-k$ .

As pointed out before in order to prove $B_{k,l}$ is sufficient to show (3.3) and (3.6). The proof of (3.6) is quite similar to the proof of (3.3) that we now explain. We first prove by induction on $s=0,...,k-p$ the inequality

	$\displaystyle\int\limits_{X}-\varphi_{\varepsilon}\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}\wedge\omega^{n-k-l}$	(3.8)
$\displaystyle\leq$	$\displaystyle\int\limits_{X}-\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s}\wedge T^{l}\wedge\omega^{n-k-l}$
$\displaystyle+$	$\displaystyle\sum_{r=0}^{s-1}\;\int\limits_{X}(\varphi_{\varepsilon}-\varphi)\,\gamma_{\varphi}^{p+r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-r-1}\wedge\gamma\wedge T^{l}\wedge\omega^{n-k-l}$
$\displaystyle-$	$\displaystyle\sum_{r=0}^{s-1}\;\int\limits_{X}\varepsilon\varphi\,\gamma_{\varphi}^{p+r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-r-1}\wedge T^{l}\wedge\omega^{n-k-l+1}.$

Inequality (3.8) is obviously true for $s=0$ . (Here we adopt the usual convention of neglecting a sum when it runs over an empty set of indices.) Before procedding to the proof of inequality (3.8), we need to point out two useful remarks.

1) Let $\alpha$ be a smooth closed real $(q,q)$ -form, $R$ e a closed positive $(r,r)$ -current, $v\geq 0$ be a measurable function such that $\int_{X}vR\wedge\omega^{n-r}<+\infty$ . This implies that the currents $i\partial\bar{\partial}v\wedge R:=i\partial\bar{\partial}(v\,R)$ and $i\partial\bar{\partial}v\wedge\alpha\wedge R:=i\partial\bar{\partial}(v\alpha\wedge R)$ are well defined. Then the Leibnitz formula implies

\displaystyle\alpha\wedge i\partial\bar{\partial}v\wedge R=i\partial\bar{\partial}v\wedge\alpha\wedge R\,.

(3.9)

2) Thanks to the inductive hypothesis $A_{j,\bullet}$ , $j\leq k-1$ we have

\int\limits_{X}-\varphi\,\gamma_{\varphi}^{p+r}\wedge\gamma^{h}\wedge T^{l}\wedge\omega^{n-p-r-h-l}<+\infty

for all $h=0,...,k-p-r-1$ . By (3.9) this implies

\int\limits_{X}-\varphi\,\gamma_{\varphi}^{p+r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-r-1}\wedge T^{l}\wedge\omega^{n-k-l+1}<+\infty\,,

so the current

S:=\varphi\,\gamma_{\varphi}^{p+r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-r-1}\wedge T^{l}

is well defined and we can define the current

i\partial\bar{\partial}\varphi\wedge\gamma_{\varphi}^{p+r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-r-1}\wedge T^{l}:=i\partial\bar{\partial}S\,.

Then the integration by parts formula

\int\limits_{X}i\partial\bar{\partial}\varphi_{\varepsilon}\wedge S\wedge\omega^{n-k-l}\;=\;\int\limits_{X}\varphi_{\varepsilon}\,i\partial\bar{\partial}S\wedge\omega^{n-k-l}

writes explicitly as

			$\displaystyle\int\limits_{X}i\partial\bar{\partial}\varphi_{\varepsilon}\wedge\varphi\,\gamma_{\varphi}^{p+r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-r-1}\wedge T^{l}\wedge\omega^{n-k-l}$		(3.10)
		$\displaystyle=$	$\displaystyle\;\int\limits_{X}\varphi_{\varepsilon}\,i\partial\bar{\partial}\varphi\wedge\gamma_{\varphi}^{p+r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-r-1}\wedge T^{l}\wedge\omega^{n-k-l}\,.$		(3.10)

We suppose now inequality (3.8) true for $s$ and we prove it for $s+1$ . We start by expanding, thanks to formula (3.9), the integral

$\displaystyle I$	$\displaystyle:=$	$\displaystyle\int\limits_{X}-\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle=$	$\displaystyle\int\limits_{X}-\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma+\varepsilon\omega)\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle+$	$\displaystyle\int\limits_{X}-\varphi\,\gamma_{\varphi}^{p+s}\wedge i\partial\bar{\partial}\varphi_{\varepsilon}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle=$	$\displaystyle\int\limits_{X}-\varepsilon\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l+1}$
	$\displaystyle-$	$\displaystyle\int\limits_{X}\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge\gamma\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle-$	$\displaystyle\int\limits_{X}i\partial\bar{\partial}\varphi_{\varepsilon}\wedge\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l}\,.$

By applying the integration by parts formula (3.10) to the last integral we deduce

$\displaystyle I$	$\displaystyle=$	$\displaystyle\int\limits_{X}-\varphi_{\varepsilon}\,\gamma_{\varphi}^{p+s+1}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle+$	$\displaystyle\int\limits_{X}\varphi_{\varepsilon}\,\gamma\wedge\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle-$	$\displaystyle\int\limits_{X}\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge\gamma\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle-$	$\displaystyle\int\limits_{X}\varepsilon\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l+1}\,.$

By combining the main $(k-1)$ -inductive hypothesis $\eqref{SymWeg}_{j,\,\bullet}$ , in $B_{j,\,\bullet}$ , for $j\leq k-1$ with formula (3.9) we get

$\displaystyle\gamma\wedge\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}$	$\displaystyle=$	$\displaystyle\gamma\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\gamma_{\varphi}^{p+s}$
	$\displaystyle=$	$\displaystyle(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge\gamma\wedge T^{l}\wedge\gamma_{\varphi}^{p+s}$
	$\displaystyle=$	$\displaystyle\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge\gamma\wedge T^{l}\,.$

By plugging this into the previous expression of $I$ we obtain

$\displaystyle I$	$\displaystyle=$	$\displaystyle\int\limits_{X}-\varphi_{\varepsilon}\,\gamma_{\varphi}^{p+s+1}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle+$	$\displaystyle\int\limits_{X}(\varphi_{\varepsilon}-\varphi)\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge\gamma\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle-$	$\displaystyle\int\limits_{X}\varepsilon\varphi\,\gamma_{\varphi}^{p+s}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p-s-1}\wedge T^{l}\wedge\omega^{n-k-l+1}$

which implies inequality (3.8) for $s+1$ . The inequality (3.8) for $s=k-p$ rewrites as

			$\displaystyle\int\limits_{X}-\varphi_{\varepsilon}\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}\wedge\omega^{n-k-l}\;\leq\;\int\limits_{X}-\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}$
		$\displaystyle+$	$\displaystyle\sum_{r=p}^{k-1}\;\int\limits_{X}(\varphi_{\varepsilon}-\varphi)\,\gamma_{\varphi}^{r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-r-1}\wedge\gamma\wedge T^{l}\wedge\omega^{n-k-l}$
		$\displaystyle-$	$\displaystyle\sum_{r=p}^{k-1}\;\int\limits_{X}\varepsilon\varphi\,\gamma_{\varphi}^{r}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-r-1}\wedge T^{l}\wedge\omega^{n-k-l+1}\,.$

By using the convergence inductive hypothesis $\eqref{Mcv3}_{j,\bullet}$ , $\eqref{Mcv4}_{j,\bullet}$ in $B_{j,\bullet}$ for $j\leq k-1$ we deduce

	$\displaystyle\limsup_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}-\varphi_{\varepsilon}\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}\wedge\omega^{n-k-l}$

	$\displaystyle\leq\int\limits_{X}-\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}<+\infty\,,$		(3.11)

since we suppose $A_{k,l}$ true. (We can always arrange $\varphi_{\varepsilon}\leq 0$ for all $\varepsilon\in(0,1)$ by changing $\varphi$ into $\varphi-C$ .) Thus by weak compactness of the mass there exists a sequence $(\varepsilon_{j})_{j}$ , $\varepsilon_{j}\downarrow 0^{+}$ and a current of order zero $\Theta\in{\cal D}^{\prime}_{n-k-l,n-k-l}(X)$ such that

\varphi_{\varepsilon_{j}}\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon_{j}}}+\varepsilon_{j}\,\omega)^{k-p}\wedge T^{l}\longrightarrow\Theta\,,

weakly as $j\rightarrow+\infty$ . So for any strongly positive $(n-k,n-k)$ -form $\alpha$ , we have

\varphi_{\varepsilon_{j}}\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon_{j}}}+\varepsilon_{j}\,\omega)^{k-p}\wedge T^{l}\wedge\alpha\longrightarrow\Theta\wedge\alpha\,,

weakly as $j\rightarrow+\infty$ . The fact that $\varphi_{\varepsilon_{j}}\downarrow\varphi$ and

\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon_{j}}}+\varepsilon_{j}\,\omega)^{k-p}\wedge T^{l}\wedge\alpha\longrightarrow\gamma_{\varphi}^{k}\wedge T^{l}\wedge\alpha\,,

weakly as $j\rightarrow+\infty$ , by the convergence inductive hypothesis $\eqref{Mcv32}_{k-1,l}$ , implies

\Theta\wedge\alpha\leq\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\alpha\,,

thanks to lemma (3.9), page 189 in [Dem1]. Thus $\Theta\leq\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}$ . Combining this with the inequality (3.11) we obtain

$\displaystyle\int\limits_{X}\Theta\wedge\omega^{n-k-l}$	$\displaystyle\leq$	$\displaystyle\int\limits_{X}\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle\leq$	$\displaystyle\liminf_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}\varphi_{\varepsilon}\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k-p}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle\leq$	$\displaystyle\lim_{j\rightarrow+\infty}\int\limits_{X}\varphi_{\varepsilon_{j}}\,\gamma_{\varphi}^{p}\wedge(\gamma_{\varphi_{\varepsilon_{j}}}+\varepsilon_{j}\,\omega)^{k-p}\wedge T^{l}\wedge\omega^{n-k-l}$
	$\displaystyle=$	$\displaystyle\int\limits_{X}\Theta\wedge\omega^{n-k-l}\,.$

We deduce $\operatorname{Tr}_{\omega}(\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}-\Theta)=0$ , which implies $\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}=\Theta$ since $0\leq\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}-\Theta$ . This proves statement $B_{k,l}$ .

Step II. This step consists in proving

[029B]

Claim 6

. If $A_{j,\bullet}$ and $B_{j,\bullet}$ hold true for all $j=0,...,k-1$ , then $A_{k,l}$ hold also true for all $l=0,...,n-k$ .

We prove this claim by induction on $l=0,...,n-k$ . For $l=0$ the conclusion follows from the hypothesis $\varphi\in\hat{\cal P}_{\gamma}$ . So we assume $A_{k,l-1}$ and we prove $A_{k,l}$ . For this purpose set $T=\theta+i\partial\bar{\partial}u$ , with $\theta$ smooth and $u$ continuous and expand the integral

			$\displaystyle\int\limits_{X}-\varphi_{\varepsilon}(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k}\wedge T^{l}\wedge\omega^{n-k-l}$
		$\displaystyle=$	$\displaystyle\int\limits_{X}-\varphi_{\varepsilon}(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k}\wedge T^{l-1}\wedge\theta\wedge\omega^{n-k-l}$
		$\displaystyle-$	$\displaystyle\int\limits_{X}u\,i\partial\bar{\partial}\varphi_{\varepsilon}\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k}\wedge T^{l-1}\wedge\omega^{n-k-l}$
		$\displaystyle=$	$\displaystyle\int\limits_{X}-\varphi_{\varepsilon}(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k}\wedge T^{l-1}\wedge\theta\wedge\omega^{n-k-l}$
		$\displaystyle-$	$\displaystyle\int\limits_{X}u\,(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k+1}\wedge T^{l-1}\wedge\omega^{n-k-l}$
		$\displaystyle+$	$\displaystyle\int\limits_{X}u\,(\gamma+\varepsilon\omega)\wedge(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k}\wedge T^{l-1}\wedge\omega^{n-k-l}\,.$

Hypothesis $A_{k,l-1}$ implies by step I

(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k+1}\wedge T^{l-1}\longrightarrow\gamma_{\varphi}^{k+1}\wedge T^{l-1}\,,

weakly as $\varepsilon\rightarrow 0^{+}$ . Thus by taking the limit as $\varepsilon\rightarrow 0^{+}$ in the previous identity and by combining the inductive hypothesis $\eqref{Mcv32}_{k-1,l-1}$ for $p=0$ with the weak continuity of the $i\partial\bar{\partial}$ operator, we deduce

			$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}-\varphi_{\varepsilon}(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k}\wedge T^{l}\wedge\omega^{n-k-l}$
		$\displaystyle=$	$\displaystyle\int\limits_{X}-\varphi\,\gamma_{\varphi}^{k}\wedge T^{l-1}\wedge\theta\wedge\omega^{n-k-l}$
		$\displaystyle-$	$\displaystyle\int\limits_{X}u\,\gamma_{\varphi}^{k+1}\wedge T^{l-1}\wedge\omega^{n-k-l}$
		$\displaystyle+$	$\displaystyle\int\limits_{X}u\,\gamma\wedge\gamma_{\varphi}^{k}\wedge T^{l-1}\wedge\omega^{n-k-l}<+\infty\,.$

By weak compactness of the mass we infer the existence of a sequence $(\varepsilon_{j})_{j}$ , $\varepsilon_{j}\downarrow 0^{+}$ and a current of order zero $\Xi\in{\cal D}^{\prime}_{n-k-l,n-k-l}(X)$ such that

\varphi_{\varepsilon_{j}}(\gamma_{\varphi_{\varepsilon_{j}}}+\varepsilon_{j}\,\omega)^{k}\wedge T^{l}\longrightarrow\Xi\,,

weakly as $j\rightarrow+\infty$ . In particular

\varphi_{\varepsilon_{j}}(\gamma_{\varphi_{\varepsilon_{j}}}+\varepsilon_{j}\,\omega)^{k}\wedge T^{l}\wedge\omega^{n-k-l}\longrightarrow\Xi\wedge\omega^{n-k-l}\,,

weakly as $j\rightarrow+\infty$ . The fact that $\varphi_{\varepsilon_{j}}\downarrow\varphi$ and

(\gamma_{\varphi_{\varepsilon_{j}}}+\varepsilon_{j}\,\omega)^{k}\wedge T^{l}\wedge\omega^{n-k-l}\longrightarrow\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}\,,

weakly as $j\rightarrow+\infty$ , by the convergence $(k-1)$ -inductive hypothesis $\eqref{Mcv32}_{k-1,l}$ , $p=0$ in the statement $B_{k-1,l}$ , implies

\Xi\wedge\omega^{n-k-l}\leq\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}\,,

thanks to lemma (3.9), page 189 in [Dem1]. We conclude

\displaystyle\int\limits_{X}-\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}\leq-\int\limits_{X}\Xi\wedge\omega^{n-k-l}<+\infty\,.

End of the proof. In the case the Lelong numbers of $\varphi$ are not zero we replace in the previous computations $\gamma$ with $\gamma+R\omega$ and $\varepsilon\omega$ with $0$ . Here $R>0$ is chosen sufficiently big such that $0\leq\gamma+R\omega+i\partial\bar{\partial}\varphi_{\varepsilon}$ for all $\varepsilon\in(0,1)$ and $\varphi_{\varepsilon}\downarrow\varphi$ as $\varepsilon\rightarrow 0^{+}$ . Then the previous arguments still work and statement A) rewrites as

+\infty>\int\limits_{X}-\varphi\,(\gamma_{\varphi}+R\omega)^{k}\wedge T^{l}\wedge\omega^{n-k-l}\geq\int\limits_{X}-\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}\geq 0\,.

Statement B) of the theorem rewrites as

	$\displaystyle\varphi_{\varepsilon}\,(\gamma_{\varphi_{\varepsilon}}+R\omega)^{k}\wedge T^{l}\longrightarrow\varphi\,(\gamma_{\varphi}+R\omega)^{k}\wedge T^{l}\,,$		(3.12)

	$\displaystyle(\gamma_{\varphi_{\varepsilon}}+R\omega)^{k+1}\wedge T^{l}\longrightarrow(\gamma_{\varphi}+R\omega)^{k+1}\wedge T^{l}\,,$		(3.13)

weakly as $\varepsilon\rightarrow 0^{+}$ and $(\gamma_{\varphi}+R\omega)^{k}\wedge T^{l}=T^{l}\wedge(\gamma_{\varphi}+R\omega)^{k}$ for the relative indices $(k,l)$ . The last inequality implies $\gamma_{\varphi}^{k}\wedge T^{l}=T^{l}\wedge\gamma_{\varphi}^{k}$ . In fact this follows by expanding by linearity the term $(\gamma_{\varphi}+R\omega)^{k}$ and using an induction by means of formula (3.9). The base of the induction follows from claim 2. $\Box$

We consider also the subset $\check{\cal P}_{\gamma}:=\{\varphi\in\hat{\cal P}^{0}_{\gamma}\,\mid\,\int_{X}-\varphi\,\gamma_{\varphi}^{n}<+\infty\}+\mathbb{R}\subset\hat{\cal P}_{\gamma}$ . Without changes in the proof of theorem 5 we get the following corollary.

[029C]

Corollary 2

For all $\varphi\in\check{\cal P}_{\gamma}$ the assertions A $)$ , B $)$ and (3.12), (3.13) of theorem 5 hold for all $k=0,...,n$ .

Let now $\Theta$ be a closed positive $(n-1,n-1)$ -current and consider the $L^{2}$ -space

\displaystyle L^{2}(X,\Theta):=\left\{\alpha\in\Gamma(X,\Lambda^{1,0}T_{X}^{*})\;\mid\;\int\limits_{X}i\alpha\wedge\bar{\alpha}\wedge\Theta<+\infty\right\}_{\Big/\Theta-a.e}\,,

equipped with the hermitian product $\left<\alpha,\beta\right>_{\Theta}:=\int_{X}i\alpha\wedge\bar{\beta}\wedge\Theta$ , which is well defined by the polarization identity. The $\Theta$ -almost everywhere equality relation is defined by : $\alpha\sim\beta$ iff

\int\limits_{X}i(\alpha-\beta)\wedge\overline{(\alpha-\beta)}\wedge\Theta=0\,.

Let $\alpha_{k},\,\alpha\in L^{2}(X,\Theta)$ . We say that the sequence $\alpha_{k}$ converges $L^{2}(X,\Theta)$ -weakly to $\alpha$ if

\int_{X}i\alpha\wedge\bar{\beta}\wedge\Theta=\lim_{k\rightarrow+\infty}\int_{X}i\alpha_{k}\wedge\bar{\beta}\wedge\Theta\,,

for all $\beta\in L^{2}(X,\Theta)$ . Let $\varphi\in{\cal P}^{0}_{\gamma}$ such that $\int_{X}-\varphi\,\Theta\wedge\,\omega<+\infty$ . Then one can define $\partial\varphi\wedge\Theta:=\partial(\varphi\Theta)$ . We write $\partial\varphi\in L^{2}(X,\Theta)$ if there exists $\alpha\in L^{2}(X,\Theta)$ such that $\partial(\varphi\Theta)=\alpha\wedge\Theta$ in the sense of currents. In this case we write

\int\limits_{X}i\partial\varphi\wedge\bar{\partial}\varphi\wedge\Theta:=\int\limits_{X}i\alpha\wedge\bar{\alpha}\wedge\Theta\,.

With this notations we have the following corolary of theorem 5.

[029D]

Corollary 3

. Let $(X,\omega)$ be a polarized compact Kähler manifold of complex dimension $n$ and let $\gamma$ , $T$ be closed positive $(1,1)$ -currents with continuous local potentials, let $\Theta$ be a closed positive $(n-1,n-1)$ -current and consider $\varphi\in\check{\cal P}_{\gamma}$ , $\varphi\leq 0$ , $\psi\in{\cal P}_{\gamma}\cap C^{0}(X)$ , $\psi\leq 0$ . Then for all $k,l\geq 0$ , $k+l\leq n-1$ ,

	$\displaystyle\int\limits_{X}i\partial\varphi\wedge\bar{\partial}\varphi\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}<+\infty\,,$		(3.14)

	$\displaystyle\int\limits_{X}i\partial\psi\wedge\bar{\partial}\psi\wedge\Theta<+\infty\,.$		(3.15)

Moreover let $(\varphi_{\varepsilon})_{\varepsilon>0}$ , $(\psi_{\varepsilon})_{\varepsilon>0}\subset C^{\infty}(X)$ , $\varphi_{\varepsilon}\in{\cal P}_{\gamma+R\omega}$ , $\psi_{\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}$ such that $\varphi_{\varepsilon}\downarrow\varphi$ , $\psi_{\varepsilon}\downarrow\psi$ as $\varepsilon\rightarrow 0^{+}$ . Then

	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial(\varphi_{\varepsilon}-\varphi)\wedge\bar{\partial}(\varphi_{\varepsilon}-\varphi)\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}=0\,,$		(3.16)

	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial(\psi_{\varepsilon}-\psi)\wedge\bar{\partial}(\psi_{\varepsilon}-\psi)\wedge\Theta=0\,.$		(3.17)

Proof. By integrating by parts we obtain

			$\displaystyle\int\limits_{X}i\partial\varphi_{\varepsilon}\wedge\bar{\partial}\varphi_{\varepsilon}\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$
		$\displaystyle=$	$\displaystyle-\int\limits_{X}\varphi_{\varepsilon}\,i\partial\bar{\partial}\varphi_{\varepsilon}\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$
		$\displaystyle=$	$\displaystyle\int\limits_{X}\varphi_{\varepsilon}\,(\gamma+R\omega)\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$
		$\displaystyle-$	$\displaystyle\int\limits_{X}\varphi_{\varepsilon}\,(\gamma_{\varphi_{\varepsilon}}+R\omega)\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}\,.$

By the proof of theorem 5 we can take the limit, so

	$\displaystyle 0$	$\displaystyle\leq$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial\varphi_{\varepsilon}\wedge\bar{\partial}\varphi_{\varepsilon}\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$		(3.18)
		$\displaystyle=$	$\displaystyle\int\limits_{X}\varphi\,(\gamma-\gamma_{\varphi})\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}<+\infty\,.$		(3.18)

On the other hand the weak convergence of the sequence

\varphi_{\varepsilon}\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}\longrightarrow\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}\,,

combined with the weak continuity of the $\partial$ operator implies

\partial\varphi_{\varepsilon}\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}\longrightarrow\partial\varphi\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}\,,

weakly as $\varepsilon\rightarrow 0^{+}$ . Then the $L^{2}(X,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1})$ -weak compactness implies (3.14) and the $L^{2}(X,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1})$ -weak convergence $\partial\varphi_{\varepsilon}\rightarrow\partial\varphi$ as $\varepsilon\rightarrow 0^{+}$ , which implies

			$\displaystyle\int\limits_{X}i\partial\varphi\wedge\bar{\partial}\varphi\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial\varphi_{\varepsilon}\wedge\bar{\partial}\varphi\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}-\varphi_{\varepsilon}\,i\partial\bar{\partial}\varphi\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}-\varphi_{\varepsilon}\,(\gamma-\gamma_{\varphi})\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$

		$\displaystyle=$	$\displaystyle\int\limits_{X}-\varphi\,(\gamma-\gamma_{\varphi})\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial\varphi_{\varepsilon}\wedge\bar{\partial}\varphi_{\varepsilon}\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}\,,$

by identity (3.18). This implies (3.16) by elementary facts about Hilbert spaces. The proof of (3.15) and (3.17) is quite similar. $\Box$

The conclusion of the corollary 3 still holds true if we replace the current $\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}$ with a sum of currents

\Xi:=\sum_{k+l\leq n-1}C_{k,l}\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}\,,

where $C_{k,l}\in\mathbb{R}$ such that $\Xi\geq 0$ . We infer the linearity formula

\int\limits_{X}i\partial\varphi\wedge\bar{\partial}\varphi\wedge\Xi\;=\;\sum_{k+l\leq n-1}C_{k,l}\,\int\limits_{X}i\partial\varphi\wedge\bar{\partial}\varphi\wedge\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l-1}\,.

[029E]

4 Existence and uniqueness of the solutions.

[029F]

Theorem 6

. Let $X$ be a compact Kähler manifold of complex dimension $n$ , let $\Omega>0$ be a smooth volume form, let $\theta$ be a continuous closed positive $(1,1)$ -form such that $\{\theta^{n}=0\}$ is a set of measure $0$ , let $\gamma$ be a closed positive $(1,1)$ -current with continuous local potentials such that $\gamma\geq\eta\,\theta$ , $\eta>0$ . Let also $f\in L\log^{n+\delta}L(X)$ , $\delta>0$ such that $\int_{X}\gamma^{n}=\int_{X}f\,\Omega$ and $\lambda\geq 0$ be a real number. Then there exists a unique solution $\psi\in\hat{\cal P}_{\gamma}$ of the degenerate complex Monge-Ampère equation

(\gamma+i\partial\bar{\partial}\psi)^{n}=f\,e^{\lambda\,\psi}\Omega\,,

which in the case $\lambda=0$ is normalized by $0=\max_{X}\psi$ . The solution is continuous and satisfies the $C^{0}$ -estimate $\|\psi\|_{C^{0}(X)}\leq C(\delta,\gamma,\Omega)\,I_{\delta}(f)^{\frac{n}{\delta}}+1$ , with

I_{\delta}(f):=\{\gamma\}^{-n}\int\limits_{X}f\log^{n+\delta}\left(e+\{\gamma\}^{-n}f\right)\Omega\,.

Moreover the constant $C(\delta,\gamma,\Omega)>0$ stays bounded for perturbations of $\gamma\geq 0$ as in the statement (C) of theorem 3.

Proof. Let $\omega>0$ be a Kähler metric and write $\gamma=\theta+i\partial\bar{\partial}u$ whith $\theta$ smoth and $u$ continuous. There exist a sequence $\gamma_{j}=\theta+i\partial\bar{\partial}u_{j}$ , whith $u_{j}\in C^{\infty}(X)$ , such that $u_{j}\downarrow u$ uniformly and $0<\gamma_{j}+\varepsilon_{j}\omega$ whith $0<\varepsilon_{j}\downarrow 0$ . We consider also a regularizing family $(f_{j})_{j}\subset C^{\infty}(X)$ , $f_{j}>0$ of $f$ in $L\log^{n+\delta}L(X)$ . We can assume

\int\limits_{X}(\gamma_{j}+\varepsilon_{j}\omega)^{n}=\int\limits_{X}f_{j}\,\Omega\,,

otherwise we multiply $f_{j}$ by a constant $c_{j}>0$ which converges to $1$ by the normalising condition $\int_{X}\gamma^{n}=\int_{X}f\,\Omega$ . We distinguish two cases.

Case $\lambda=0$ . By Yau’s solution of the Calabi conjecture there exists a unique family $(\psi_{j})_{j}$ , $\psi_{j}\in{\cal P}^{0}_{\gamma_{j}+\varepsilon_{j}\omega}$ of smooth solutions of the complex Monge-Ampère equations

(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\psi_{j})^{n}=f_{j}\,\Omega\,.

The hypothesis (C1) and (C2a) of statement (C) of theorem 3 are obviously satisfied for the family $(\gamma_{j}+\varepsilon_{j})_{j}$ . We deduce that the constant $C_{1}=C_{1}(\delta\,,\,\gamma_{j}+\varepsilon_{j}\omega\,,\,\Omega)>0$ in the statement of theorem 3,A does not blow up as $j\rightarrow+\infty$ . Moreover the uniform estimate

\displaystyle\|f_{j}\|_{L\log^{n+\delta}L(X)}\leq C^{\prime}\|f\|_{L\log^{n+\delta}L(X)}=:K\,,

(4.1)

holds for all $j$ . (See [Ra-Re] page 364 or [Iw-Ma], theorem 4.12.2, page 79.) Thus by theorem 3, A we obtain the uniform estimate $\|\psi_{j}\|_{C^{0}(X)}\leq C$ . On the other hand we have a priori a uniform estimate $\int_{X}-\psi_{j}\,\Omega\leq C^{\prime}$ since the local potentials of the family $(\gamma_{j}+\varepsilon_{j})_{j}$ stay uniformly bounded. Thus, by elementary properties of plurisubharmonic functions, (see [Dem1], chapter 1) there exists a $L^{1}$ -convergent subsequence $(\psi_{j})_{j}$ (which by abuse of notations we denote in the same way). We can apply theorem 3, B to the complex Monge-Ampère equation in consideration since we dispose of the estimate (4.1) and the constant $C_{2}=C_{2}(\delta\,,\,\gamma_{j}+\varepsilon_{j}\omega\,,\,\Omega,K)>0$ in the relative statement is uniformly bounded in $j$ thanks to the same considerations concerning the constant $C_{1}$ . We infer that the sequence $(\psi_{j})_{j}$ is a Cauchy sequence in the $C^{0}$ -topology, thus convergent to some $\psi\in{\cal P}_{\gamma}\cap C^{0}(X)$ . This implies the convergence of the weak limits

(\gamma+i\partial\bar{\partial}\psi)^{n}=\lim_{j\rightarrow+\infty}(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\psi_{j})^{n}=\lim_{j\rightarrow+\infty}f_{j}\,\Omega=f\,\Omega\,,

thus $\psi$ is the required solution of our degenerate complex Monge-Ampère equation. We normalise the solution $\psi$ whith the condition $\max_{X}\psi=0$ . We prove now the uniqueness of the solutions in the case $\lambda=0$ . Let $\varphi\in\hat{\cal P}^{0}_{\gamma}$ be another solution. Then the identity $\gamma_{\varphi}^{n}=\gamma_{\psi}^{n}$ implies $\varphi\in\check{\cal P}^{0}_{\gamma}$ by claim 2 in the proof of theorem 5. Let $\varphi_{\varepsilon}$ , $\psi_{\varepsilon}$ be as in the statement of corollary 3 and set $u:=\psi-\varphi$ , $u_{\varepsilon}:=\psi_{\varepsilon}-\varphi_{\varepsilon}$ . Let us also recall the formula

\alpha^{k}-\beta^{k}=(\alpha-\beta)\wedge\sum_{l=0}^{k-1}\alpha^{l}\wedge\beta^{k-l-1}\,.

From there we deduce

$\displaystyle 0$	$\displaystyle=$	$\displaystyle\int\limits_{X}-u(\gamma^{n}_{\psi}-\gamma^{n}_{\varphi})=\lim_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}-u_{\varepsilon}(\gamma^{n}_{\psi}-\gamma^{n}_{\varphi})$	(4.2)
	$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\sum_{l=0}^{n-1}\,\int\limits_{X}-u_{\varepsilon}\,i\partial\bar{\partial}u\wedge\gamma^{l}_{\psi}\wedge\gamma^{n-l-1}_{\varphi}$
	$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\sum_{l=0}^{n-1}\,\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}u\wedge\gamma^{l}_{\psi}\wedge\gamma^{n-l-1}_{\varphi}$
	$\displaystyle=$	$\displaystyle\sum_{l=0}^{n-1}\,\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{l}_{\psi}\wedge\gamma^{n-l-1}_{\varphi}=:I\,,$

since $\partial u_{\varepsilon}\rightarrow\partial u$ in $L^{2}(X,\gamma^{l}_{\psi}\wedge\gamma^{n-l-1}_{\varphi})$ by corollary 3. Inspired by an idea of S. Blocki [Blo1], we will prove by induction on $k=0,...,n-1$ that

\displaystyle\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{r}_{\psi}\wedge\gamma^{s}_{\varphi}\wedge\gamma^{k}=0

(4.3)

for all $r,s\geq 0$ , $r+s=n-k-1$ . For $k=0$ this follows from (4.2). So we assume (4.3) for $k-1$ and we prove it for $k$ . In fact consider the identity

\gamma^{k}=\gamma^{k}_{\psi}-i\partial\bar{\partial}\psi\wedge\sum_{l=0}^{k-1}\gamma^{l}_{\psi}\wedge\gamma^{k-l-1}\quad\mbox{and set }\quad\Xi:=\gamma^{r}_{\psi}\wedge\gamma^{s}_{\varphi}\wedge\sum_{l=0}^{k-1}\gamma^{l}_{\psi}\wedge\gamma^{k-l-1}\,.

By applying several times corollary 3 and by integrating by parts we derive

			$\displaystyle\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{r}_{\psi}\wedge\gamma^{s}_{\varphi}\wedge\gamma^{k}\;=\;\lim_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}u\wedge\gamma^{r}_{\psi}\wedge\gamma^{s}_{\varphi}\wedge\gamma^{k}$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}(u\gamma^{r+k}_{\psi}\wedge\gamma^{s}_{\varphi})\;-\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}(u\,i\partial\bar{\partial}\psi\wedge\Xi)\;\right]$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}u\wedge\gamma^{r+k}_{\psi}\wedge\gamma^{s}_{\varphi}\;+\;\int\limits_{X}u_{\varepsilon}\,i\partial\bar{\partial}u\wedge i\partial\bar{\partial}\psi\wedge\Xi\;\right]$
		$\displaystyle=$	$\displaystyle\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{r+k}_{\psi}\wedge\gamma^{s}_{\varphi}\;-\;\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}u_{\varepsilon}\,i\partial\bar{\partial}\psi\wedge(\gamma_{\varphi}-\gamma_{\psi})\wedge\Xi$
		$\displaystyle\leq$	$\displaystyle I+\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}\left[\,\psi\,(\gamma_{\varphi}-\gamma_{\psi})\wedge\Xi\,\right]$

		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}\psi\wedge\gamma_{\varphi}\wedge\Xi\;-\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}\psi\wedge\gamma_{\psi}\wedge\Xi\;\right]$		(4.4)
		$\displaystyle=$	$\displaystyle\int\limits_{X}i\partial u\wedge\bar{\partial}\psi\wedge\gamma_{\varphi}\wedge\Xi\;-\;\int\limits_{X}i\partial u\wedge\bar{\partial}\psi\wedge\gamma_{\psi}\wedge\Xi\,.$		(4.4)

Set $\chi=\varphi$ or $\chi=\psi$ . Then the Cauchy-Schwarz inequality implies

			$\displaystyle\left\|\;\int\limits_{X}i\partial u\wedge\bar{\partial}\psi\wedge\gamma_{\chi}\wedge\Xi\;\right\|$
		$\displaystyle\leq$	$\displaystyle\left(\;\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma_{\chi}\wedge\Xi\;\right)^{1/2}\left(\;\int\limits_{X}i\partial\psi\wedge\bar{\partial}\psi\wedge\gamma_{\chi}\wedge\Xi\;\right)^{1/2}=0\,,$

by the inductive hypothesis. This combined with (4.4) implies (4.3) for $k$ . So at the end of the induction we get

0=\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{n-1}\geq\eta^{n-1}\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\theta^{n-1}\geq 0\,,

which implies $\varphi=\psi$ by elementary properties of plurisubharmonic functions.

Case $\lambda>0$ . We start by proving the following lemma, which is a particular case of a more general result due to Yau. (See [Yau], sect. 6, page 376).

[029G]

Lemma 6

. Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$ , let $h$ be a smooth function such that $\int_{X}\omega^{n}=\int_{X}e^{h}\omega^{n}$ and $\varphi\in{\cal P}_{\omega}$ a solution of the complex Monge-Ampère equation

\displaystyle(\omega+i\partial\bar{\partial}\varphi)^{n}=e^{h+\lambda\varphi}\omega^{n}\,,

(4.5)

$\lambda>0$ . Consider also two solutions $\varphi^{\prime}\,\varphi^{\prime\prime}\in{\cal P}_{\omega}$ of the complex Monge-Ampère equation $(\omega+i\partial\bar{\partial}\varphi)^{n}=e^{h}\omega^{n}$ such that $\min_{X}\varphi^{\prime}=0=\max_{X}\varphi^{\prime\prime}$ . Then $\varphi^{\prime\prime}\leq\varphi\leq\varphi^{\prime}$ .

Proof. The argument is a simplification, in our particular case, of Yau’s original argument for the proof of thm. 4, sect. 6 in [Yau]. Set $\varphi^{\prime}_{0}:=\varphi^{\prime},\,\varphi^{\prime\prime}_{0}:=\varphi^{\prime\prime}$ and consider the solutions $\varphi^{\prime}_{j},\,\varphi^{\prime\prime}_{j}$ of the complex Monge-Ampère equations given by the iteration

	$\displaystyle(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}$	$\displaystyle=$	$\displaystyle e^{h+(\lambda+1)\varphi^{\prime}_{j}-\varphi^{\prime}_{j-1}}\,\omega^{n}\,,$		(4.6)
	$\displaystyle(\omega+i\partial\bar{\partial}\varphi^{\prime\prime}_{j})^{n}$	$\displaystyle=$	$\displaystyle e^{h+(\lambda+1)\varphi^{\prime\prime}_{j}-\varphi^{\prime\prime}_{j-1}}\,\omega^{n}\,.$		(4.7)

Notice that we can solve this equations even if the terms $e^{h-\varphi^{\prime}_{j-1}}$ , $e^{h-\varphi^{\prime\prime}_{j-1}}$ are not normalized, see lem. 2 page 378 in [Yau]. Set $k=\lambda+1$ and consider

(\omega+i\partial\bar{\partial}\varphi^{\prime}_{1})^{n}=e^{h+k(\varphi^{\prime}_{1}-\varphi^{\prime}_{0})+\lambda\varphi^{\prime}_{0}}\omega^{n}\geq e^{k(\varphi^{\prime}_{1}-\varphi^{\prime}_{0})}e^{h}\omega^{n}=e^{k(\varphi^{\prime}_{1}-\varphi^{\prime}_{0})}(\omega+i\partial\bar{\partial}\varphi^{\prime}_{0})^{n}\,.

At a maximum point of $\varphi^{\prime}_{1}-\varphi^{\prime}_{0}$ we have the inequality

(\omega+i\partial\bar{\partial}\varphi^{\prime}_{0})^{n}\geq(\omega+i\partial\bar{\partial}\varphi^{\prime}_{1})^{n}\,.

By plugging this into the previous one, we deduce $\varphi^{\prime}_{1}\leq\varphi^{\prime}_{0}$ . We now prove by induction the inequality $\varphi^{\prime}_{j}\leq\varphi^{\prime}_{j-1}$ . In fact by dividing $(\ref{MAith1})_{j}$ with $(\ref{MAith1})_{j-1}$ we get

\frac{(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}}{(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j-1})^{n}}=e^{k(\varphi^{\prime}_{j}-\varphi^{\prime}_{j-1})-(\varphi^{\prime}_{j-1}-\varphi^{\prime}_{j-2})}\geq e^{k(\varphi^{\prime}_{j}-\varphi^{\prime}_{j-1})}\,.

At a maximum point of $\varphi^{\prime}_{j}-\varphi^{\prime}_{j-1}$ we again find the inequality

(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}\leq(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j-1})^{n}\,.

Combining this with the previous one we deduce $\varphi^{\prime}_{j}\leq\varphi^{\prime}_{j-1}$ . We also prove by induction the inequality $\varphi^{\prime\prime}_{j}\leq\varphi^{\prime}_{j}$ , which is true by definition in the case $j=0$ . By dividing $(\ref{MAith1})_{j}$ with $(\ref{MAith2})_{j}$ we get

\frac{(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}}{(\omega+i\partial\bar{\partial}\varphi^{\prime\prime}_{j})^{n}}=e^{k(\varphi^{\prime}_{j}-\varphi^{\prime\prime}_{j})-(\varphi^{\prime}_{j-1}-\varphi^{\prime\prime}_{j-1})}\leq e^{k(\varphi^{\prime}_{j}-\varphi^{\prime\prime}_{j})}\,,

by the induction hypothesis $\varphi^{\prime\prime}_{j-1}\leq\varphi^{\prime}_{j-1}$ . At a minimum point of $\varphi^{\prime}_{j}-\varphi^{\prime\prime}_{j}$ we get

(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}\geq(\omega+i\partial\bar{\partial}\varphi^{\prime\prime}_{j})^{n}\,,

hence $\varphi^{\prime\prime}_{j}\leq\varphi^{\prime}_{j}$ . As a conclusion, we have proved the sequence of inequalities

\displaystyle\varphi^{\prime\prime}_{0}\leq\varphi^{\prime\prime}_{j-1}\leq\varphi^{\prime\prime}_{j}\leq\varphi^{\prime}_{j}\leq\varphi^{\prime}_{j-1}\leq\varphi^{\prime}_{0}\,.

(4.8)

These inequalities imply $0<2n+\Delta_{\omega}\varphi^{\prime}_{j}\leq C\,B_{j}$ , where $B_{j}>0$ satisfies the uniform estimate

\displaystyle 0\geq C_{1}\,B_{j}^{\frac{1}{n-1}}-\left(2n+\max_{X}\Delta_{\omega}\varphi^{\prime}_{j-1}\right)B_{j}^{-1}-C_{0}\,,

(4.9)

and $C_{0},\,C_{1}>0$ , which are obtained by applying the maximum principle in a way similar to Yau’s proof of the second order estimate for the solution of the Calabi conjecture [Yau]. (In the case $n=1$ the uniform estimate $0<2n+\Delta_{\omega}\varphi^{\prime}_{j}\leq C^{\prime}$ follows immediately from the inequalities (4.8).) Fix now a constant $C_{3}>0$ such that the inequality

C_{1}\,x^{1+\frac{1}{n-1}}\geq(C_{0}+2C_{2})x-C_{3}\,,

hold for all $x\geq 0$ . This implies by (4.9) the estimate

2(2n+\Delta_{\omega}\varphi^{\prime}_{j})\leq 2C_{2}\,B_{j}\leq\left(2n+\max_{X}\Delta_{\omega}\varphi^{\prime}_{j-1}\right)+C_{3}\,,

thus

2n+\max_{X}\Delta_{\omega}\varphi^{\prime}_{j}\leq 2^{-j}\left(2n+\max_{X}\Delta_{\omega}\varphi^{\prime}_{j-1}\right)+C_{3}\,,

by iteration. By taking the derivate in the Green Formula (see [Aub], Th. 4.13 page 108) we get the identity

d_{x}\varphi^{\prime}_{j}=-\int\limits_{X}d_{x}G_{\omega}(x,\cdot)\,\Delta_{\omega}\varphi^{\prime}_{j}\,\omega^{n}\,,

which implies the estimate $|\nabla_{\omega}\varphi^{\prime}_{j}|_{\omega}\leq C_{\omega}\max_{X}\Delta_{\omega}\varphi^{\prime}_{j}\leq C$ . By applying the complex version of the Evans-Krylov theory [Ti2] we deduce the uniform estimate $\|\varphi^{\prime}_{j}\|_{C^{2,\alpha}(X)}\leq C^{\prime}$ . This implies that the sequence $(\varphi^{\prime}_{j})_{j}$ converges in the $C^{2,\alpha}$ -topology to the unique solution $\varphi$ of the complex Monge-Ampère equation (4.5). Then the conclusion follows from inequalities (4.8). $\Box$

We consider now the unique family $(\psi_{j})_{j}$ , $\psi_{j}\in{\cal P}_{\gamma_{j}+\varepsilon_{j}\omega}$ of smooth solutions of the complex Monge-Ampère equations

(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\psi_{j})^{n}=f_{j}\,e^{\lambda\,\psi_{j}}\Omega\,,

given by the Aubin-Yau’s solution of the Calabi conjecture. Consider also the solutions $\psi^{\prime}_{j}$ , $\psi^{\prime\prime}_{j}$ , $\min_{X}\psi^{\prime}_{j}=0=\max_{X}\psi^{\prime\prime}_{j}$ of the complex Monge-Ampère equation

(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\varphi)^{n}=f_{j}\,\Omega\,.

By applying lemma 6 we deduce $\psi^{\prime\prime}_{j}\leq\psi_{j}\leq\psi^{\prime}_{j}$ for all $j$ . By the same argument in the case $\lambda=0$ , we deduce $\|\psi^{\prime}_{j}\|_{C^{0}(X)},\,\|\psi^{\prime\prime}_{j}\|_{C^{0}(X)}\leq C$ , thus $\|\psi_{j}\|_{C^{0}(X)}\leq C$ and so

\|f_{j}\,e^{\lambda\psi_{j}}\|_{L\log^{n+\delta}L(X)}\leq C^{\prime}e^{\lambda C}\|f\|_{L\log^{n+\delta}L(X)}\,.

This fact allows us to apply theorem 3, B as in the case $\lambda=0$ in order to get a sequence of solutions $(\psi_{j})_{j}$ convergent in the uniform topology to some $\psi\in{\cal P}_{\gamma}\cap C^{0}(X)$ . This implies the convergence of the weak limits

(\gamma+i\partial\bar{\partial}\psi)^{n}=\lim_{j\rightarrow+\infty}(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\psi_{j})^{n}=\lim_{j\rightarrow+\infty}f_{j}e^{\lambda\psi_{j}}\,\Omega=fe^{\lambda\psi}\,\Omega\,.

The integral estimate in the statement of theorem 6 follows immediately from theorem 3, A and from the inequalities $\psi^{\prime\prime}_{j}\leq\psi_{j}\leq\psi^{\prime}_{j}$ . We prove now the uniqueness of the solutions. Let $\varphi\in\hat{\cal P}_{\gamma}$ be another solution. The fact that $\varphi\leq C$ implies

\|f\,e^{\lambda\varphi}\|_{L\log^{n+\delta}L(X)}\leq e^{\lambda C}\|f\|_{L\log^{n+\delta}L(X)}\,,

which allows us to solve the degenerate complex Monge-Ampère equation

(\gamma+i\partial\bar{\partial}u)^{n}=f\,e^{\lambda\varphi}\Omega\,,

with $u\in{\cal P}^{0}_{\gamma}\cap C^{0}(X)$ . By the uniqueness result in the case $\lambda=0$ we deduce $u=\varphi-\sup_{X}\varphi$ , thus $\varphi\in{\cal P}_{\gamma}\cap C^{0}(X)$ . By applying the comparison principle we get

\int\limits_{\varphi<\psi}\gamma^{n}_{\psi}\,\leq\,\int\limits_{\varphi<\psi}\gamma^{n}_{\varphi}\,=\,\int\limits_{\varphi<\psi}e^{\lambda(\varphi-\psi)}\gamma^{n}_{\psi}\,,

which implies $\int_{\varphi<\psi}\gamma^{n}_{\psi}=0$ since $e^{\lambda(\varphi-\psi)}<1$ . This implies $\varphi\geq\psi$ $\gamma^{n}_{\psi}$ -almost everywhere, thus $\gamma^{n}_{\varphi}=e^{\lambda(\varphi-\psi)}\gamma^{n}_{\psi}\leq\gamma^{n}_{\psi}$ $\gamma^{n}_{\psi}$ -almost everywhere. By symmetry we also deduce $\gamma^{n}_{\psi}\leq\gamma^{n}_{\varphi}$ $\gamma^{n}_{\varphi}$ -almost everywhere. The fact that $\varphi,\,\psi$ are solutions of our complex Monge-Ampère equation implies that a property holds $\gamma^{n}_{\psi}$ -almost everywhere if and only if it holds $\Omega$ -almost everywhere and the same for $\gamma^{n}_{\varphi}$ . We thus infer $\gamma^{n}_{\psi}=\gamma^{n}_{\varphi}$ , which implies $\psi=\varphi$ by the expression of the Monge-Ampère equation. $\Box$

The following lemma gives us an important class of functions for the right hand side of the degenerate complex Monge-Ampère equation.

[029H]

Lemma 7

. Let $X$ be a compact complex manifold, let $\Omega>0$ be a smooth volume form and let $\sigma_{j}\in H^{0}(X,E_{j})$ , $\tau_{r}\in H^{0}(X,F_{r})$ , $j=1,...,N$ , $r=1,...,M$ be, non identically zero, holomorphic sections of some holomorphic vector bundles over $X$ such that the integral condition

\displaystyle\int\limits_{X}\,\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}}\,\Omega<+\infty

holds for some real numbers $l_{j}\geq 0,\,h_{r}\geq 0$ . Then the integrand function belongs to some $L^{p}$ space, $p>1$ and the family of functions

\displaystyle G_{\varepsilon}:=\prod\limits_{j=1}^{N}(|\sigma_{j}|^{2}+\varepsilon^{A})^{l_{j}}\cdot\prod\limits_{r=1}^{M}(|\tau_{r}|^{2}+\varepsilon)^{-h_{r}}\,,

$\varepsilon\in[0,1)$ , $A:=(\sum_{r}h_{r}+1)/(\min_{j}l_{j})$ , converges in $L^{p}$ -norm to function $G_{0}$ when $\varepsilon\rightarrow 0$ .

Proof. We define the coherent complex analytic sheaves ${\cal J}_{\sigma_{j}}:=\sum_{p}{\cal O}_{X}\sigma_{j,p}$ and ${\cal J}_{\tau_{r}}:=\sum_{q}{\cal O}_{X}\tau_{r,q}\subset{\cal O}_{X}$ , with $\sigma_{j}=(\sigma_{j,p})_{p}$ , $\tau_{r}=(\tau_{r,q})_{q}$ for some local holomorphic trivializations of the vector bundles $E_{j}$ , $F_{r}$ . Clearly the definition is independent of the local trivialization, thus this sheaves are globally well defined. By the Hironaka desingularization theorem [Hir] we can find a proper bimeromorphic morphism $\mu:\tilde{X}\rightarrow X$ of compact complex manifolds such that there exists a family $(H_{s})_{s}$ , $H_{s}\subset\tilde{X}$ of smooth hypersurfaces with normal crossing in $\tilde{X}$ such that $\mu\left(\bigcup_{s}H_{s}\right)=\bigcup_{j}V(\sigma_{j})\cup\bigcup_{r}V(\tau_{r})$ ,

K_{\tilde{X}}=\mu^{*}K_{X}+\sum_{s}\,a_{s}\,H_{s}\,,\quad a_{s}\in\mathbb{N}\,,

and $\mu^{*}{\cal J}_{\sigma_{j}}={\cal O}_{\tilde{X}}\left(-\sum_{s}\,b_{j,s}\,H_{s}\right)$ , $\mu^{*}{\cal J}_{\tau_{r}}={\cal O}_{\tilde{X}}\left(-\sum_{s}\,c_{r,s}\,H_{s}\right)$ , $b_{j,s}$ , $c_{r,s}\in\mathbb{N}$ for all $j,r$ . The fact that $\mu$ is a holomorphic map implies $d\mu\otimes_{{}_{\mathbb{R}}}\mathbb{I}_{{}_{\mathbb{C}}}=\partial\mu\oplus\overline{\partial\mu}$ , with $\partial\mu\in H^{0}(\tilde{X},\Lambda^{1,0}T^{*}_{\tilde{X}}\otimes\mu^{*}T^{1,0}_{X})$ . Thus the divisor of the Jacobian $\Lambda^{n}\partial\mu\in H^{0}(\tilde{X},K_{\tilde{X}}-\mu^{*}K_{X})$ of $\mu$ is by definition $\sum_{s}\,a_{s}\,H_{s}$ . On the other hand the invariance of the integral by orientation preserving diffeomorphisms implies

	$\displaystyle I_{p}$	$\displaystyle:=$	$\displaystyle\int\limits_{X}\,\prod\limits_{j=1}^{N}\|\sigma_{j}\|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}\|\tau_{r}\|^{-2h_{r}p}\,\Omega$
		$\displaystyle=$	$\displaystyle\int\limits_{\tilde{X}}\,\prod\limits_{j=1}^{N}\|\sigma_{j}\circ\mu\|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}\|\tau_{r}\circ\mu\|^{-2h_{r}p}\,(\Omega\circ\mu)\cdot(\Lambda^{n}\partial\mu)\wedge\overline{(\Lambda^{n}\partial\mu)}\,.$

For any open set $U\subset\tilde{X}$ we denote by $\{s_{1},...,s_{m}\}:=\{s\,|\,H_{s}\cap U\not=\emptyset\}$ . For any point in $\tilde{X}$ one can find a coordinate neighborhood $U:=\{|z_{k}|<1\}$ such that $H_{s_{t}}=\{z_{t}=0\}$ , $t=1,...,m\leq n$ . With respect to this coordinates, we have

(\Omega\circ\mu)\cdot(\Lambda^{n}\partial\mu)\wedge\overline{(\Lambda^{n}\partial\mu)}=\rho\prod\limits_{t=1}^{m}|z_{t}|^{2a_{s_{t}}}\,i^{n^{2}}dz_{1}\wedge...\wedge dz_{n}\wedge d\bar{z}_{1}\wedge...\wedge d\bar{z}_{n}\,,

$\rho>0$ , $\rho\in C^{\infty}(U)$ and $\sigma_{j,p}\circ\mu=f_{j,p}\prod_{t=1}^{m}z_{t}^{b_{j,s_{t}}}$ , $\tau_{r,q}\circ\mu=g_{r,q}\prod_{t=1}^{m}z_{t}^{c_{r,s_{t}}}$ , $f_{j,p},\,g_{r,q}\in{\cal O}(U)$ . Then modulo factors that are bounded away from $0$ and $\infty$ , we find

	$\displaystyle\|\sigma_{j}\circ\mu\|^{2}$	$\displaystyle\thicksim$	$\displaystyle\left(\sum_{p}\|f_{j,p}\|^{2}\right)\prod_{t=1}^{m}\|z_{t}\|^{2b_{j,s_{t}}}\,,$
	$\displaystyle\|\tau_{r}\circ\mu\|^{2}$	$\displaystyle\thicksim$	$\displaystyle\left(\sum_{q}\|g_{r,q}\|^{2}\right)\prod_{t=1}^{m}\|z_{t}\|^{2c_{r,s_{t}}}\,,$

with $\sum_{p}|f_{j,p}|^{2}>0$ , $\sum_{q}|g_{r,q}|^{2}>0$ . The latter nonvanishing property follows from the fact that the terms $|\sum_{s}\,b_{j,s}\,H_{s}|=\mu^{-1}V(\sigma_{j})$ , $|\sum_{s}\,c_{r,s}\,H_{s}|=\mu^{-1}V(\tau_{r})$ , $(\sigma_{j,p}\circ\mu)_{p}$ correspond to local generators of the sheaf ${\cal O}_{\tilde{X}}\left(\sum_{s}\,b_{j,s}\,H_{s}\right)$ and $(\tau_{r,q}\circ\mu)_{q}$ are local generators of the sheaf ${\cal O}_{\tilde{X}}\left(\sum_{s}\,c_{r,s}\,H_{s}\right)$ . We infer

\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}}\thicksim\prod_{t=1}^{m}|z_{t}|^{-2d_{s_{t}}}\,,

with $d_{s}:=C_{s}H-B_{s}L$ , $L:=\sum_{j}l_{j}$ , $H:=\sum_{r}h_{r}$ , $B_{s}:=\sum_{j}b_{j,s}$ and $C_{s}:=\sum_{r}c_{r,s}$ . Set also $T:=\{t\in\{1,...,m\}\,|\,0<d_{s_{t}}-a_{s_{t}}\}$ . The hypothesis $I_{0}<+\infty$ implies $d_{s_{t}}<1+a_{s_{t}}$ for all $t\in T$ . Thus there exists $p_{U}>1$ such that $p_{U}d_{s_{t}}<1+a_{s_{t}}$ for all $t\in T$ . If $(U_{\alpha})_{\alpha}$ is a finite covering of $\tilde{X}$ , with $U_{\alpha}$ as $U$ and $p:=\min_{\alpha}p_{U_{\alpha}}>1$ then $I_{p}<+\infty$ . This proves the first claim in the statement of lemma 7. In order to prove the convergence in the $L^{p}$ norm of the functions $G_{\varepsilon}$ we distinguish two cases. In the case where $l_{j}=0$ for all $j$ , the claim follows imediately from the monotone convergence theorem. The other possible case is $l_{j}>0$ for all $j$ . In this case we set $l:=\min_{j}l_{j}>0$ . Then our setting implies $A=(H+1)/l$ . For all $\varepsilon,\rho\in(0,1)$ consider the sequence of inequalities

			$\displaystyle\int\limits_{X}\,\prod\limits_{j=1}^{N}\|\sigma_{j}\|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}(\|\tau_{r}\|^{2}+\varepsilon)^{-h_{r}p}\,\Omega\;\leq\;\int\limits_{X}G_{\varepsilon}^{p}\,\Omega$
	$\displaystyle=$		$\displaystyle\int\limits_{\bigcup_{t=1}^{N}\{\rho\|\sigma_{t}\|^{2}\,<\,\varepsilon^{A}\}}G_{\varepsilon}^{p}\,\Omega\;+\;\int\limits_{\bigcap_{t=1}^{N}\{\rho\|\sigma_{t}\|^{2}\,\geq\,\varepsilon^{A}\}}G_{\varepsilon}^{p}\,\Omega$
	$\displaystyle\leq$		$\displaystyle\sum_{t=1}^{N}\;\int\limits_{\{\rho\|\sigma_{t}\|^{2}\,<\,\varepsilon^{A}\}}G_{\varepsilon}^{p}\,\Omega\;+\;\int\limits_{\bigcap_{t=1}^{N}\{\rho\|\sigma_{t}\|^{2}\,\geq\,\varepsilon^{A}\}}G_{\varepsilon}^{p}\,\Omega$
	$\displaystyle\leq$		$\displaystyle\sum_{t=1}^{N}\;\int\limits_{\{\rho\|\sigma_{t}\|^{2}\,<\,\varepsilon^{A}\}}\varepsilon^{-Hp}\,\prod\limits_{j=1}^{N}(\|\sigma_{j}\|^{2}+\varepsilon^{A})^{l_{j}p}\,\Omega$
	$\displaystyle+$		$\displaystyle\int\limits_{\bigcap_{t=1}^{N}\{\rho\|\sigma_{t}\|^{2}\,\geq\,\varepsilon^{A}\}}\,\prod\limits_{j=1}^{N}(1+\rho)^{l_{j}p}\|\sigma_{j}\|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}(\|\tau_{r}\|^{2}+\varepsilon)^{-h_{r}p}\,\Omega$
	$\displaystyle\leq$		$\displaystyle\sum_{t=1}^{N}(1+\rho^{-1})^{Lp}\,\varepsilon^{(Al-H)p}\int\limits_{\{\rho\|\sigma_{t}\|^{2}\,<\,\varepsilon^{A}\}}\,\prod\limits_{j\not=t}(\|\sigma_{j}\|^{2}+\varepsilon^{A})^{l_{j}p}\,\Omega$
	$\displaystyle+$		$\displaystyle(1+\rho)^{Lp}\int\limits_{X}\,\prod\limits_{j=1}^{N}\|\sigma_{j}\|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}\|\tau_{r}\|^{-2h_{r}p}\,\Omega$
	$\displaystyle\leq$		$\displaystyle(1+\rho^{-1})^{Lp}\,\varepsilon^{p}C+(1+\rho)^{Lp}\int\limits_{X}G^{p}_{0}\,\Omega\,,$

where $C>0$ is a constant uniform in $\varepsilon$ . Thus by letting $\varepsilon\rightarrow 0^{+}$ and by applying the increasing monotone convergence theorem to the first integral in the previous inequalities, we obtain

\int\limits_{X}G_{0}^{p}\,\Omega\leq\liminf_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}G_{\varepsilon}^{p}\,\Omega\leq\limsup_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}G_{\varepsilon}^{p}\,\Omega\leq(1+\rho)^{Lp}\int\limits_{X}G_{0}^{p}\,\Omega\,.

Then the conclusion follows by letting $\rho\rightarrow 0^{+}$ and by the fact that $G_{\varepsilon}$ converges pointwise almost everywhere to $G_{0}$ as $\varepsilon\rightarrow 0^{+}$ . $\Box$

[029I]

5 Higher order regularity of the solutions.

We now prove the following theorem.

[029J]

Theorem 7

. Let $X$ be a compact Kähler manifold of complex dimension $n$ , let $\omega\geq 0$ be a big closed smooth $(1,1)$ -form such that $\{\omega^{n}=0\}$ is a set of zero measure and let $\Omega>0$ be a smooth volume form. Consider also $\sigma_{j}\in H^{0}(X,E_{j})$ , $\tau_{r}\in H^{0}(X,F_{r})$ , $j=1,...,N$ , $r=1,...,M$ be non identically zero holomorphic sections of some holomorphic vector bundles over $X$ , such that the intregral condition

\displaystyle\int\limits_{X}\,\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}}\,\Omega=\int\limits_{X}\omega^{n}

(5.1)

holds for some real numbers $l_{j}\geq 0,\,h_{r}\geq 0$ . Then there exists a unique solution $\varphi\in\hat{\cal P}_{\omega}$ of the degenerate complex Monge-Ampère equation

\displaystyle(\omega+i\partial\bar{\partial}\varphi)^{n}=\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}}\,e^{\lambda\varphi}\,\Omega\,,\quad\lambda\geq 0\,.

(5.2)

Moreover there exists a complex analytic set $\Sigma_{\omega}\subset X$ depending only on the $(1,1)$ -cohomology class of $\omega$ possessing the following properties.

(A). The set $\Sigma_{\omega}$ is empty if and only if the class of $\omega$ is Kähler.

(B). If we define the complex analytic sets

\displaystyle S^{\prime}_{0}:=\bigcup_{r}\{\tau_{r}=0\}\,,\quad S_{0}:=S^{\prime}_{0}\cup(\bigcup_{j}\{\sigma_{j}=0\})\,,

\displaystyle S^{\prime}:=\Sigma_{\omega}\cup S^{\prime}_{0}\,,\quad S:=S^{\prime}\cup(\bigcup_{j}\{\sigma_{j}=0\})\,,

then in the case $n\geq 2$

\varphi\in{\cal P}_{\omega}\cap C^{0}(X)\cap C^{\alpha}(X\smallsetminus S^{\prime})\cap C^{\infty}(X\smallsetminus S)\,,

for all $\alpha\in(0,1)$ . If $n=1$ then $\varphi\in{\cal P}_{\omega}\cap C^{0}(X)\cap C^{\alpha}(X\smallsetminus S^{\prime}_{0})\cap C^{\infty}(X\smallsetminus S_{0})$ and the class of of $\omega$ is Kähler.

Proof.
Step I. We first assume the existence of an effective divisor $D$ and $\delta>0$ such that $\{\omega\}-\delta\{D\}$ is a Kähler class. This is certainly the case if $X$ is projective and $\{\omega\}\in H^{1,1}(X,\mathbb{Q})$ . So by using the Lelong-Poincaré formula we deduce $0<\omega_{\delta}=\omega-2\pi\delta[D]+\delta\,i\partial\bar{\partial}\log|s|^{2}$ with $\operatorname{div}(s)=D$ . By convention we will put $\delta=0$ if $\omega>0$ and by abusing notations we will denote by $D$ the support $|D|$ of the divisor $D$ .

IA) Setup of Step I.
We first consider the case $n\geq 2$ . We can assume without any lost of generality $N=M=1$ . Let $\alpha>0$ be a Kähler metric, let $\varepsilon\in(0,1)$ and let $c_{\varepsilon}$ be a normalizing constant for the integral condition

\displaystyle e^{c_{\varepsilon}}\,\int\limits_{X}\frac{(|\sigma|^{2}+\varepsilon^{A})^{l}}{(|\tau|^{2}+\varepsilon)^{h}}\;\Omega=\int\limits_{X}(\omega+\varepsilon\alpha)^{n}\,,

(5.3)

with $A:=(h+1)/l$ . The condition (5.1) combined with lemma 7 implies $c_{\varepsilon}\rightarrow 0$ , when $\varepsilon\rightarrow 0^{+}$ . Consider the standard solutions $\varphi_{\varepsilon}\in C^{\infty}(X)$ of the complex Monge-Ampère equations

\displaystyle(\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon})^{n}=e^{c_{\varepsilon}}\,\frac{(|\sigma|^{2}+\varepsilon^{A})^{l}}{(|\tau|^{2}+\varepsilon)^{h}}\;e^{\lambda\varphi_{\varepsilon}}\,\Omega\,,

(5.4)

given by Yau’s solution of the Calabi conjecture. Notice that the integral condition (5.3) implies that a non identicaly zero solution $\varphi_{\varepsilon}$ changes signs in the case $\lambda>0$ . By combining lemma 7 with the estimate of theorem 6 we deduce a uniform bound for the oscillations $\operatorname{Osc}(\varphi_{\varepsilon})\leq C$ . Set now $\omega_{\delta,\varepsilon}:=\omega_{\delta}+\varepsilon\alpha$ and $\psi_{\varepsilon}:=\varphi_{\varepsilon}-\delta\log|s|^{2}$ . Then

\displaystyle 0<\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon}=\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}

(5.5)

over $X\smallsetminus D$ , and the equation (5.4) rewrites as

\displaystyle(\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon})^{n}=e^{F^{\varepsilon}+\lambda\delta\log|s|^{2}+\lambda\psi_{\varepsilon}}\,\omega_{\delta,\varepsilon}^{n}

(5.6)

on $X\smallsetminus D$ , with $F^{\varepsilon}:=f^{\varepsilon}+l\cdot a^{\varepsilon}-h\cdot b^{\varepsilon}$ , and with

f^{\varepsilon}:=c_{\varepsilon}+\log(\Omega/\omega_{\delta,\varepsilon}^{n})\,,\quad a^{\varepsilon}:=\log(|\sigma|^{2}+\varepsilon^{A})\,,\quad b^{\varepsilon}:=\log(|\tau|^{2}+\varepsilon)\,.

(Here the supscripts in $\varepsilon$ are indices and not powers.) Consider now the function $\gamma_{1}^{\delta,\varepsilon}:X\rightarrow\mathbb{R}$ defined by the formula

\gamma_{1}^{\delta,\varepsilon}(x):=\min_{\xi\in T^{\otimes 2}_{X,x}\smallsetminus 0_{x}}{\cal C}^{\omega_{\delta,\varepsilon}}_{{}_{X,J}}(\xi,\xi)|\xi|_{\omega_{\delta,\varepsilon}}^{-2}.

So $\gamma_{1}^{\delta,\varepsilon}(x)$ is the smallest eigenvalue of the Chern curvature form ${\cal C}^{\omega_{\delta,\varepsilon}}_{{}_{X,J}}(x)$ of the metric $\omega_{\delta,\varepsilon}>0$ . It is well known (see [Kat], chap II, sect. 5.1, theorem 5.1, page 107) that the function $\gamma_{1}^{\delta,\varepsilon}$ is continuous. We observe that the family of metrics $(\omega_{\delta,\varepsilon})_{\varepsilon}$ has bounded geometry. In particular for all $\varepsilon\in(0,1)$

\gamma_{1}^{\delta,\varepsilon}\geq C_{\delta}\,,\quad|f^{\varepsilon}|\leq K_{0,\delta}\,,\quad\lambda(\omega_{\delta}-\omega)+i\partial\bar{\partial}f^{\varepsilon}\geq-K_{0,\delta}\,\omega_{\delta,\varepsilon}\,.

IB) The Laplacian estimate.
This estimate is obtained as a combination of ideas of Yau, Blocki and Tsuji, [Yau], [Blo2], [Ts]. Consider the continuous function $\Lambda_{\varepsilon}:X\rightarrow(0,+\infty)$ given by the maximal eigenvalues of $\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}$ with respect to the Kähler metric $\omega_{\delta,\varepsilon}$ ,

\Lambda_{\varepsilon}(x):=\max_{\xi\in T_{X,x}\smallsetminus 0_{x}}(\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon})(\xi,J\xi)|\xi|_{\omega_{\delta,\varepsilon}}^{-2}\,,

i.e. we extend $\Lambda_{\varepsilon}$ over $D$ by continuity, as is permitted by (5.5). Consider also the continuous function over $X\smallsetminus D$ ,

A_{\varepsilon}:=\log\Lambda_{\varepsilon}-k\psi_{\varepsilon}+hb^{\varepsilon}\,,

with $0<k:=2(1+h\,K_{0,\delta}/2-K_{1})$ and

K_{1}:=\min\{-[\lambda+(1+l+h)K_{0,\delta}/(2n)]\,,\,C_{\delta}\}<-\lambda\,.

The singularity of the function $\psi_{\varepsilon}$ imply the existence of a maximum of the function $A_{\varepsilon}$ at a certain point $x_{\varepsilon}\in X\smallsetminus D$ . Let $g$ be a smooth real valued function in a neighborhood of $x_{\varepsilon}$ in $X\smallsetminus D$ such that $\omega_{\delta,\varepsilon}=\frac{i}{2}\partial\bar{\partial}g$ , and let $u:=g+2\psi_{\varepsilon}$ . Then

\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}=\frac{i}{2}\partial\bar{\partial}u\,.

In the following calculations we use the notation $u_{l,\bar{r}}:=\frac{\partial^{2}u}{\partial z_{l}\partial\bar{z}_{r}}$ . Let $(z_{1},\ldots,z_{n})$ be $\omega_{\delta,\varepsilon}$ -geodesic holomorphic coordinates with center the point $x_{\varepsilon}$ such that the metric $\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}$ can be writen in diagonal form in $x_{\varepsilon}$ . Explicitly $\omega_{\delta,\varepsilon}=\frac{i}{2}\sum_{l,r}g_{l,\bar{r}}\,dz_{l}\wedge d\bar{z}_{r}$ , with

	$\displaystyle\displaystyle{g_{l,\bar{r}}=\delta_{l,r}-\sum_{j,k}C^{j,\bar{k}}_{r,l}z_{j}\bar{z}_{k}+O(\|z\|^{3})\,,\quad g_{j,\bar{k},l,\bar{r}}(x_{\varepsilon})=-C^{j,\bar{k}}_{r,l}\,,}$
	$\displaystyle\displaystyle{{\cal C}^{\omega_{\delta,\varepsilon}}_{{}_{X,J}}(x_{\varepsilon})=\sum_{j,k,l,r}C^{j,\bar{k}}_{r,l}\,dz_{j}\otimes dz_{l}\otimes d\bar{z}_{k}\otimes d\bar{z}_{r}\,.}$

and $\frac{i}{2}\partial\bar{\partial}u=\frac{i}{2}\sum_{l}u_{l,\bar{l}}\,dz_{l}\wedge d\bar{z}_{l}$ , with $0<u_{1,\bar{1}}\leq...\leq u_{n,\bar{n}}$ at the point $x_{\varepsilon}$ . For every $\zeta\in\mathbb{C}^{n}$ we set $g_{\zeta,\bar{\zeta}}:=\sum_{l,r}g_{l,\bar{r}}\,\zeta_{l}\,\bar{\zeta}_{r}$ . Then

\Lambda_{\varepsilon}(x)=\max_{\xi\in T_{X,x}\smallsetminus 0_{x}}\,\frac{\partial\bar{\partial}u(\xi^{1,0},\xi^{0,1})}{\partial\bar{\partial}g(\xi^{1,0},\xi^{0,1})}=\max_{|\zeta|=1}\,\frac{u_{\zeta,\bar{\zeta}}}{g_{\zeta,\bar{\zeta}}}\,,

and so $\Lambda_{\varepsilon}(x_{\varepsilon})=u_{n,\bar{n}}(x_{\varepsilon})$ , $\frac{u_{n,\bar{n}}}{g_{n,\bar{n}}}\leq\Lambda_{\varepsilon}$ . We also set

\tilde{A}_{\varepsilon}:=\log\frac{u_{n,\bar{n}}}{g_{n,\bar{n}}}-k\psi_{\varepsilon}+hb^{\varepsilon}\,.

Then $\tilde{A}_{\varepsilon}\leq A_{\varepsilon}$ , with $\tilde{A}_{\varepsilon}(x_{\varepsilon})=A_{\varepsilon}(x_{\varepsilon})$ . This implies that $\tilde{A}_{\varepsilon}$ also reaches a maximum at $x_{\varepsilon}$ , thus $\Delta_{\psi_{\varepsilon}}\tilde{A}_{\varepsilon}(x_{\varepsilon})\leq 0$ , where $\Delta_{\psi_{\varepsilon}}$ is the Laplacian respect to the metric $\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}$ . All the subsequent computations in this part of the proof will be made at point $x_{\varepsilon}$ . By the local expressions for the Ricci tensor we obtain

	$\displaystyle\partial^{2}_{n,\bar{n}}\log\operatorname{det}(u_{j,\bar{k}})$	$\displaystyle=$	$\displaystyle\sum_{l,p}\Big(u_{n,\bar{n},l,\bar{p}}-\sum_{s,t}u_{n,l,\bar{s}}\,u^{s,\bar{t}}\,u_{\bar{n},t,\bar{p}}\Big)u^{p,\bar{l}}$
		$\displaystyle=$	$\displaystyle\sum_{p}\frac{u_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}}-\sum_{p,q}\frac{\,\|u_{n,p,\bar{q}}\|^{2}}{u_{p,\bar{p}}\,u_{q,\bar{q}}}\,,$

and in a similar way $\partial^{2}_{n,\bar{n}}\log\operatorname{det}(g_{j,\bar{k}})=\sum_{p}\,g_{n,\bar{n},p,\bar{p}}$ . Then by differentiating with respect to $\partial^{2}_{n,\bar{n}}$ the identity (5.6), which rewrites as

\log\operatorname{det}(u_{j,\bar{k}})=F^{\varepsilon}+\lambda\delta\log|s|^{2}+\lambda(u-g)/2+\log\operatorname{det}(g_{j,\bar{k}})\,,

we obtain

	$\displaystyle\sum_{p}\frac{u_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}}-\sum_{p,q}\frac{\,\|u_{n,p,\bar{q}}\|^{2}}{u_{p,\bar{p}}\,u_{q,\bar{q}}}$	$\displaystyle=$	$\displaystyle f^{\varepsilon}_{n,\bar{n}}+\lambda[(\omega_{\delta})_{n,n}-\omega_{n,n}]/2$
		$\displaystyle+$	$\displaystyle la^{\varepsilon}_{n,\bar{n}}-hb^{\varepsilon}_{n,\bar{n}}+\lambda(u_{n,\bar{n}}-1)/2+\sum_{p}\,g_{n,\bar{n},p,\bar{p}}\,.$

Combining this with the inequality $\Delta_{\psi_{\varepsilon}}\tilde{A}_{\varepsilon}(x_{\varepsilon})\leq 0$ , we get

$\displaystyle 0$	$\displaystyle\geq$	$\displaystyle\sum_{p}\frac{\tilde{A}_{p,\bar{p}}}{u_{p,\bar{p}}}=\sum_{p}\left(\frac{u_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}\,u_{n,\bar{n}}}-\frac{\,\|u_{n,\bar{n},p}\|^{2}}{u_{p,\bar{p}}\,u^{2}_{n,\bar{n}}}+\frac{k/2+hb^{\varepsilon}_{p,\bar{p}}-g_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}}\right)-nk/2$
	$\displaystyle=$	$\displaystyle\sum_{p,q}\frac{\,\|u_{n,p,\bar{q}}\|^{2}}{u_{p,\bar{p}}\,u_{q,\bar{q}}\,u_{n,\bar{n}}}-\sum_{p}\frac{\,\|u_{n,\bar{n},p}\|^{2}}{u_{p,\bar{p}}\,u^{2}_{n,\bar{n}}}$
	$\displaystyle+$	$\displaystyle\frac{f_{n,\bar{n}}+\lambda[(\omega_{\delta})_{n,n}-\omega_{n,n}-1]/2+la^{\varepsilon}_{n,\bar{n}}-hb^{\varepsilon}_{n,\bar{n}}}{u_{n,\bar{n}}}$
	$\displaystyle+$	$\displaystyle\sum_{p}\left(\frac{g_{n,\bar{n},p,\bar{p}}}{u_{n,\bar{n}}}+\frac{k/2+hb^{\varepsilon}_{p,\bar{p}}-g_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}}\right)-(nk-\lambda)/2\,.$

We use now (see the Appendix) the existence of smooth $(1,1)$ -forms $iT^{a^{\varepsilon}}\geq 0$ , $iT^{b^{\varepsilon}}\geq 0$ , on $X$ such that

$\displaystyle-hi\partial\bar{\partial}b^{\varepsilon}$	$\displaystyle\geq$	$\displaystyle-hK_{0,\delta}\,\omega_{\delta,\varepsilon}-hiT^{b^{\varepsilon}}\,,$
$\displaystyle hi\partial\bar{\partial}b^{\varepsilon}$	$\displaystyle\geq$	$\displaystyle-hK_{0,\delta}\,\omega_{\delta,\varepsilon}+hiT^{b^{\varepsilon}}\,,$
$\displaystyle li\partial\bar{\partial}a^{\varepsilon}$	$\displaystyle\geq$	$\displaystyle-lK_{0,\delta}\,\omega_{\delta,\varepsilon}\,.$

By plugging these inequalities in the previous computations we get

	$\displaystyle 0\geq-\frac{hT_{n,\bar{n}}^{b^{\varepsilon}}}{u_{n,\bar{n}}}+\sum_{p}\left(\frac{K_{1}-C_{p,p}^{n,\bar{n}}}{u_{n,\bar{n}}}+\frac{-K_{1}+C_{p,p}^{n,\bar{n}}}{u_{p,\bar{p}}}+\frac{1}{u_{p,\bar{p}}}+\frac{hT_{p,\bar{p}}^{b^{\varepsilon}}}{u_{p,\bar{p}}}\right)-(nk-\lambda)/2$

	$\displaystyle\geq\sum_{p}\frac{(C_{p,p}^{n,\bar{n}}-K_{1})(u_{n,\bar{n}}-u_{p,\bar{p}})}{u_{p,\bar{p}}\,u_{n,\bar{n}}}+\sum_{p}\frac{1}{u_{p,\bar{p}}}-(nk-\lambda)/2\,.\qquad\qquad\qquad\qquad\;\,$

Denote by $(x_{1},...,x_{n})$ the real part of the complex coordinates $(z_{1},...,z_{n})$ . Then the inequality $C^{n,\bar{n}}_{p,p}={\cal C}^{\omega_{\delta,\varepsilon}}_{{}_{X,J}}(\frac{\partial}{\partial x_{n}}\otimes\frac{\partial}{\partial x_{p}},\frac{\partial}{\partial x_{n}}\otimes\frac{\partial}{\partial x_{p}})(x_{0})\geq\gamma_{1}^{\delta,\varepsilon}(x_{0})\geq C_{\delta}$ implies

\displaystyle 0\geq\sum_{p}\frac{1}{u_{p,\bar{p}}}-C_{0}\geq\left(\frac{u_{n,\bar{n}}}{\prod_{p}u_{p,\bar{p}}}\right)^{\frac{1}{n-1}}-C_{0}=e^{\frac{-\lambda\psi_{\varepsilon}-\lambda\delta\log|s|^{2}-F^{\varepsilon}}{n-1}}\,u_{n,\bar{n}}^{\frac{1}{n-1}}-C_{0}\,,

where $C_{0}>0$ and all the following constants are indipendents of $\varepsilon$ . Consider now the function $B_{\varepsilon}:=e^{A_{\varepsilon}}=\Lambda_{\varepsilon}\,e^{-k\psi_{\varepsilon}+hb^{\varepsilon}}$ . Then $x_{\varepsilon}$ is also a maximum point for $B_{\varepsilon}$ over $X\smallsetminus D$ and the previous inequality rewrites as

	$\displaystyle 0$	$\displaystyle\geq$	$\displaystyle e^{\frac{(k-\lambda)\psi_{\varepsilon}-\lambda\delta\log\|s\|^{2}-hb^{\varepsilon}-F^{\varepsilon}}{n-1}(x_{\varepsilon})}\,\,B_{\varepsilon}(x_{\varepsilon})^{\frac{1}{n-1}}-C_{0}$
		$\displaystyle=$	$\displaystyle e^{\frac{(k-\lambda)\varphi_{\varepsilon}-\delta k\log\|s\|^{2}-la^{\varepsilon}-f^{\varepsilon}}{n-1}(x_{\varepsilon})}\,\,B_{\varepsilon}(x_{\varepsilon})^{\frac{1}{n-1}}-C_{0}\,.$

Then by the inequalities $k-\lambda>0,\;|s|^{2}\leq C$ , $a^{\varepsilon}\leq C$ and $|f^{\varepsilon}|\leq K_{0,\delta}$ , it follows the estimate

0\geq C_{1}\,e^{\frac{(k-\lambda)}{n-1}\min_{X}\varphi_{\varepsilon}}\,\,B_{\varepsilon}(x_{\varepsilon})^{\frac{1}{n-1}}-C_{0}\,.

In conclusion we have found over $X\smallsetminus D$ the estimates

	$\displaystyle 0$	$\displaystyle<$	$\displaystyle 2n+\Delta_{\omega_{\delta,\varepsilon}}\varphi_{\varepsilon}-\delta\Delta_{\omega_{\delta,\varepsilon}}\log\|s\|^{2}=\operatorname{Tr}_{\omega_{\delta,\varepsilon}}(\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon})$
		$\displaystyle\leq$	$\displaystyle 2n\Lambda_{\varepsilon}\leq 2n\,e^{k\psi_{\varepsilon}-hb^{\varepsilon}}B_{\varepsilon}(x_{\varepsilon})\leq\frac{C_{2}\,e^{k\varphi_{\varepsilon}-(k-\lambda)\min_{X}\varphi_{\varepsilon}}}{\|s\|^{2\delta k}(\|\tau\|^{2}+\varepsilon)^{h}}\leq\frac{C_{2}\,e^{k\operatorname{Osc}(\varphi_{\varepsilon})}}{\|s\|^{2\delta k}\,\|\tau\|^{2h}}\,.$

The last inequality follows from the fact that $\lambda\min_{X}\varphi_{\varepsilon}\leq 0$ , since a non identicaly zero solution $\varphi_{\varepsilon}$ changes signs in the case $\lambda>0$ . Then using the inequality

\left|\delta\Delta_{\omega_{\delta,\varepsilon}}\log|s|^{2}\right|=|\operatorname{Tr}_{\omega_{\delta,\varepsilon}}(\omega-\omega_{\delta})|\leq C

over $X\smallsetminus D$ we deduce the singular estimate

-C<2n+\Delta_{\omega_{\delta,\varepsilon}}\varphi_{\varepsilon}\leq\frac{C_{2}\,e^{k\operatorname{Osc}(\varphi_{\varepsilon})}}{|s|^{2\delta k}\,|\tau|^{2h}}+C\,.

IC) Higher order estimates
An elementary computation yields the singular estimate

			$\displaystyle\kern-30.0ptC_{3}^{-1}\,\|s\|^{2\delta k(n-1)}\,\|\sigma\|^{2l}\,\|\tau\|^{2h(n-2)}\,e^{-kn\operatorname{Osc}(\varphi_{\varepsilon})}\,\omega_{\delta,\varepsilon}$		(5.7)
		$\displaystyle\leq$	$\displaystyle\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon}\leq C_{3}\,\|s\|^{-2\delta k}\,\|\tau\|^{-2h}\,e^{k\operatorname{Osc}(\varphi_{\varepsilon})}\,\omega_{\delta,\varepsilon}\,.$		(5.7)

Morover the fact that $\varphi_{\varepsilon}\in{\cal P}_{\omega+\varepsilon\alpha}$ implies

2|\partial\bar{\partial}\varphi_{\varepsilon}|_{\omega_{\delta,\varepsilon}}\leq\Delta_{\omega_{\delta,\varepsilon}}\varphi_{\varepsilon}+2\operatorname{Tr}_{\omega_{\delta,\varepsilon}}(\omega+\varepsilon\alpha)\,.

We set first

\displaystyle\Sigma_{\omega}:=\bigcap_{\{\omega\}-\delta\{D\}>0\atop D\geq 0,\;\delta>0}D\,.

Then by the standard Schauder estimates [Gi-Tru] we find that for any coordinate open set $K\subset X\smallsetminus S^{\prime}$ there are uniform constants $C_{K}>0$ such that

\max_{K}|\nabla\varphi_{\varepsilon}|\leq C_{K}\left(\max_{K}\Delta\varphi_{\varepsilon}+\max_{K}|\varphi_{\varepsilon}|\right)\,.

Therefore, we can apply the complex version of Evans-Krylov theory [Ti2] on every compact set $K\subset X\smallsetminus S$ to get uniform constants $C_{K,2}>0$ such that $\|\varphi_{\varepsilon}\|_{C^{2,\alpha}(K)}\leq C_{K,2}$ . Let now $U\subset X\smallsetminus S$ be an open set and $\xi\in{\cal O}(T^{1,0}_{X})(U)$ . By deriving with respect to the complex vector field $\xi$ the complex Monge-Ampère equation (5.4), which we rewrite under the form

\displaystyle(\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon})^{n}=e^{H_{\varepsilon}+\lambda\varphi_{\varepsilon}}\alpha^{n}\,,

with

H_{\varepsilon}:=c_{\varepsilon}+\log(\Omega/\alpha^{n})+la^{\varepsilon}-hb^{\varepsilon}\,,

we obtain (see the proof of formula 11 in [Pal])

\displaystyle\Delta_{\varphi_{\varepsilon}}(\xi\,.\,\varphi_{\varepsilon})-2\lambda\,\xi\,.\,\varphi_{\varepsilon}=-\operatorname{Tr}_{\varphi_{\varepsilon}}L_{\xi}(\omega+\varepsilon\alpha)+\operatorname{Tr}_{\alpha}L_{\xi}\alpha+2\xi\,.\,H_{\varepsilon}\,,

(5.8)

where $\Delta_{\varphi_{\varepsilon}}$ and $\operatorname{Tr}_{\varphi_{\varepsilon}}$ are respectively the Laplacian and the trace operators with respect to the Kähler metric $\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon}>0$ . By the uniform estimates (5.7) and $\|\varphi_{\varepsilon}\|_{C^{2,\alpha}(K)}\leq C_{K,2}$ it follows that the operator $\Delta_{\varphi_{\varepsilon}}$ is uniformly elliptic with coefficients uniformly bounded in $C^{\alpha}$ -norm at least, over any compact set $K\subset U$ . The right hand side of equation (5.8) is also uniformly bounded in $C^{\alpha}$ -norm at least. By the standard regularity theory for linear elliptic equations [Gi-Tru] we deduce $\|\xi\,.\,\varphi_{\varepsilon}\|_{{C^{2,\alpha}(K)}}\leq C^{\prime}_{K}$ for all $\varepsilon>0$ . By conjugation the same hold for $\bar{\xi}\,.\,\varphi_{\varepsilon}$ . Thus we obtain the uniform estimate $\|\varphi_{\varepsilon}\|_{C^{3,\alpha}(K)}\leq C_{K,3}$ .
In its turn this estimate implies that the coefficients of the Laplacian $\Delta_{\varphi_{\varepsilon}}$ and the right hand side of equation (5.8) are uniformly bounded in $C^{1,\alpha}$ -norm at least. By iteration we get the uniform estimates $\|\varphi_{\varepsilon}\|_{C^{r,\alpha}(K)}\leq C_{K,r}$ for all $\varepsilon>0$ and $r\in\mathbb{N}$ . We deduce that the family $(\varphi_{\varepsilon})_{\varepsilon>0}\subset C^{\infty}(X\smallsetminus S)$ is precompact in the smooth topology.
On the other hand the uniqueness result of theorem 6, combined with the arguments which showed the existence of a continuous solution $\varphi$ imply the uniform convergence of the family $\varphi_{\varepsilon}$ towards $\varphi$ , thus this convergence is achieved in the $C^{\infty}$ topology over $X\smallsetminus S$ . In this way we get smoothness of the solution $\varphi$ over $X\smallsetminus S$ . The regularity statement in the case $n=1$ follows immediately from the latter considerations.

Step II.
IIA) Setup of step II.
We start with a few definitions adapted to our situation.

[029K]

Definition 2

Let ${\cal J}\subset{\cal O}_{X}$ be a coherent ideal sheaf over a compact complex manifold.
A) A modification of $X$ is a bimeromorphic morphism $\mu:\tilde{X}\rightarrow X$ of compact complex manifolds with connected fibers.
B) A log resolution of the sheaf ${\cal J}$ is a modification $\mu:\tilde{X}\rightarrow X$ such that $\mu^{*}{\cal J}={\cal O}(-D)$ , with $D$ an effective divisor with normal crossing and such that the restriction $\tilde{X}\smallsetminus|D|\rightarrow X\smallsetminus V({\cal J})$ is a biholomorphism.

Let now $T$ be a closed positive $(1,1)$ -current. There are two ways to associate an ideal sheaf to $T$ . One can define the ideal sheaf ${\cal J}(T)\subset{\cal O}_{X}$ of germs $f\in{\cal O}_{x}$ such that $\int|f|^{2}e^{-\varphi}<+\infty$ , where $\varphi$ is a local plurisubharmonic potential of $T$ in a neighborhood of $x$ . One can also define the ideal sheaf ${\cal J}^{T}\subset{\cal O}_{X}$ of germs $f\in{\cal O}_{x}$ such that $|f|^{2}e^{-\varphi}\leq C$ . The sheaf ${\cal J}(T)$ is coherent by a result of Nadel [Nad] and $V({\cal J}^{T})=V({\cal J}^{kT})$ for all $k\in\mathbb{R}_{>0}$ .

[029L]

Definition 3

. A closed positive $(1,1)$ -current $T$ over a compact complex manifold $X$ possesses analytic singularities if there exists $k\in\mathbb{N}_{>0}$ such that;
a) the ideal sheaf ${\cal J}^{kT}$ is coherent,
b) if $\mu:\tilde{X}\rightarrow X$ a modification such that $\mu^{*}{\cal J}^{kT}={\cal O}(-D)$ , with $D$ an effective divisor with normal crossing, then $\mu^{*}T=\lambda[D]+\alpha$ , with $\lambda\in\mathbb{R}_{>0}$ and $\alpha$ smooth.

Consider now a polarized compact Kähler manifold $(X,\omega)$ and a class $\chi\in H^{1,1}(X,\mathbb{R})$ . We define the set of Kähler currents

\chi_{>0}:=\{T\in\chi\,\mid\,\exists\varepsilon>0\,:\,T\geq\varepsilon\omega\,\}\,.

If $\chi$ is nef and big then $\chi_{>0}\not=\emptyset$ by a result in [De-Pa]. By the regularization theorem in [Dem1] we deduce that for all $T\in\chi_{>0}$ and for all integers $k\gg 0$ there exists $T_{k}\in\chi_{>0}$ such that ${\cal J}^{kT}=\overline{{\cal J}(kT)}$ and $\mu^{*}T_{k}=2\pi k^{-1}[D]+\alpha$ , with $\alpha$ smooth, for any modification $\mu$ such that $\mu^{*}{\cal J}(kT)={\cal O}(-D)$ , with $D$ an effective divisor with normal crossing. Notice that $\mu^{*}\overline{{\cal J}(kT)}={\cal O}(-D)$ . We deduce that the subset $\chi^{as}_{>0}\subset\chi_{>0}$ of Kähler currents with analytic singularities is also non empty. By the proof of theorem 3.4 in [De-Pa] we deduce the following generalization of Kodaira’s lemma.

[029M]

Lemma 8

. Let $X$ be a compact Kähler manifold and $\chi\in H^{1,1}(X,\mathbb{R})$ a class which is nef and big. Then for all $T\in\chi^{as}_{>0}$ there exists a log resolution $\mu$ of the coherent ideal sheaf ${\cal J}^{kT}$ , $\mu^{*}{\cal J}^{kT}={\cal O}(-D)$ , an effective divisor $D^{\prime}$ with $|D^{\prime}|=|D|$ and $r\in\mathbb{N}_{>0}$ such that the class $\mu^{*}\chi-2\pi r^{-1}\{D^{\prime}\}$ is Kähler.

We define in the end the complex analytic set

\Sigma_{\chi}:=\bigcap_{T\in\chi^{as}_{>0}}V({\cal J}^{T})\,.

IIB) Application of step I.
Back to our situation, fix $T\in\chi^{as}_{>0}$ and consider a log resolution $\mu:\tilde{X}\rightarrow X$ of the coherent ideal sheaf ${\cal J}^{kT}$ . Let $f\geq 0$ be the integrand in the first inegral in (5.1). Then the integral condition (5.1) implies

0<\int\limits_{\tilde{X}}\mu^{*}\omega^{n}=\int\limits_{\tilde{X}}(f\Omega\circ\mu)\cdot(\Lambda^{n}\partial\mu)\wedge\overline{(\Lambda^{n}\partial\mu)}=\int\limits_{\tilde{X}}(f\circ\mu)\left|\Lambda^{n}\partial\mu\right|^{2}_{\alpha,\beta}\,\Omega^{\prime}\,

where $\alpha,\beta>0$ are hermitian forms respectively over $X$ and $\tilde{X}$ ,

\left|\Lambda^{n}\partial\mu\right|^{2}_{\alpha,\beta}:=\frac{\alpha^{n}\cdot(\Lambda^{n}\partial\mu)\wedge\overline{(\Lambda^{n}\partial\mu)}}{\beta^{n}}\,\qquad\mbox{and}\qquad\Omega^{\prime}=\left(\frac{\Omega}{\alpha^{n}}\circ\mu\right)\beta^{n}>0\,.

Therefore by the generalised Kodaira’s lemma 8 we can solve the degenerate complex Monge-Ampère equation

(\mu^{*}\omega+i\partial\bar{\partial}\Phi)^{n}=(f\circ\mu)\left|\Lambda^{n}\partial\mu\right|^{2}_{\alpha,\beta}e^{\lambda\Phi}\,\Omega^{\prime}\,,

with the method of step I, so as to obtain a solution

\Phi\in{\cal P}_{\mu^{*}\omega}\cap C^{0}(\tilde{X})\cap C^{\alpha}(\tilde{X}\smallsetminus(D\cup\mu^{-1}S^{\prime}_{0}))\cap C^{\infty}(\tilde{X}\smallsetminus(D\cup\mu^{-1}S_{0}))\,.

Remark in fact that $|\operatorname{div}(\Lambda^{n}\partial\mu)|\subset|D|$ by our definition of a log resolution. Let now $j_{q}:\mu^{-1}(q)\hookrightarrow\tilde{X}$ , $q\in V({\cal J}^{T})$ be the inclusion map. By hypothesis $\mu^{-1}(q)$ is compact and connected and obviously $j_{q}^{*}\mu^{*}\omega=0$ . Then by the hypothesis $\mu^{*}\omega+i\partial\bar{\partial}\Phi\geq 0$ we deduce $i\partial\bar{\partial}(\Phi\circ j_{q})\geq 0$ , which implies that $\Phi$ is constant along the fibers $\mu^{-1}(q)$ . Therefore we can define $\varphi:=\pi_{*}\Phi\in{\cal P}_{\omega}\cap C^{0}(X)$ . The fact that $\varphi$ is bounded implies that the current $(\omega+i\partial\bar{\partial}\varphi)^{n}$ does not carry any mass on complex analytic sets. Thus $\varphi$ is the solution of the complex Monge-Ampère equation (5.2) with the required regularity over the set $X\smallsetminus V({\cal J}^{T})$ . Then the conclusion about the regularity of the solution $\varphi$ follows by letting $T\in\chi^{as}_{>0}$ vary. We finally remark that if $\Sigma_{\omega}$ is empty then the class of $\omega$ is Kähler. In fact chose the volume form $\Omega>0$ such that $\int_{X}\omega^{n}=\int_{X}\Omega$ . By the previous arguments we can find a unique solution $\varphi$ of the equation $(\omega+i\partial\bar{\partial}\varphi)^{n}=\Omega>0$ , which is smooth, thus $\omega+i\partial\bar{\partial}\varphi>0$ is a Kähler metric. In the case $n=1$ the solution $\varphi$ of the equation $\omega+i\partial\bar{\partial}\varphi=\Omega>0$ is allways smoth. $\Box$

[029N]

6 Appendix

Let $\sigma\in H^{0}(X,E)$ be a holomorphic section of a holomorphic hermitian vector bundle $(E,h)$ and set $S_{\varepsilon}:=\log(|\sigma|^{2}+\varepsilon)$ , for some $\varepsilon>0$ . We denote by $\{\cdot,\cdot\}$ the exterior product of $E$ -valued forms respect to the hermitian metric $h$ . We have

i\partial S_{\varepsilon}=\frac{i\{\partial_{h}\sigma,\sigma\}}{|\sigma|^{2}+\varepsilon}\,,

since $\sigma$ is a holomorphic section. We compute now the complex hessian

$\displaystyle i\partial\bar{\partial}S_{\varepsilon}$	$\displaystyle=$	$\displaystyle-\bar{\partial}\,\frac{i\{\partial_{h}\sigma,\sigma\}}{\|\sigma\|^{2}+\varepsilon}$
	$\displaystyle=$	$\displaystyle\frac{-i\{\bar{\partial}\partial_{h}\sigma,\sigma\}+i\{\partial_{h}\sigma,\partial_{h}\sigma\}}{\|\sigma\|^{2}+\varepsilon}\,+\,i\{\partial_{h}\sigma,\sigma\}\wedge\bar{\partial}\left(\frac{1}{\|\sigma\|^{2}+\varepsilon}\right)$
	$\displaystyle=$	$\displaystyle\frac{i\{\partial_{h}\sigma,\partial_{h}\sigma\}\,-\,\{i{\cal C}_{h}(E)\sigma,\sigma\}}{\|\sigma\|^{2}+\varepsilon}\,-\,\frac{i\{\partial_{h}\sigma,\sigma\}\wedge\{\sigma,\partial_{h}\sigma\}}{(\|\sigma\|^{2}+\varepsilon)^{2}}$
	$\displaystyle=$	$\displaystyle\underbrace{\frac{(\|\sigma\|^{2}+\varepsilon)i\{\partial_{h}\sigma,\partial_{h}\sigma\}-i\{\partial_{h}\sigma,\sigma\}\wedge\{\sigma,\partial_{h}\sigma\}}{(\|\sigma\|^{2}+\varepsilon)^{2}}}_{iT(S_{\varepsilon})}\,-\,\frac{\{i{\cal C}_{h}(E)\sigma,\sigma\}}{\|\sigma\|^{2}+\varepsilon}\,.$

We show that the $(1,1)$ -form $iT(S_{\varepsilon})$ is nonnegative. In fact by using twice the Lagrange inequality

i\{\partial_{h}\sigma,\sigma\}\wedge\{\sigma,\partial_{h}\sigma\}\leq|\sigma|^{2}\,i\{\partial_{h}\sigma,\partial_{h}\sigma\}\,,

(which is an equality in the case of line bundles) we get

\displaystyle iT(S_{\varepsilon})\geq\frac{\varepsilon i\{\partial_{h}\sigma,\partial_{h}\sigma\}}{(|\sigma|^{2}+\varepsilon)^{2}}\geq\frac{\varepsilon i\{\partial_{h}\sigma,\sigma\}\wedge\{\sigma,\partial_{h}\sigma\}}{|\sigma|^{2}(|\sigma|^{2}+\varepsilon)^{2}}=\frac{\varepsilon}{|\sigma|^{2}}\,i\partial S_{\varepsilon}\wedge\bar{\partial}S_{\varepsilon}\geq 0\,.

Observe that the last form is smooth. Consequently, we find the inequalities

	$\displaystyle i\partial\bar{\partial}S_{\varepsilon}$	$\displaystyle\geq$	$\displaystyle\frac{\varepsilon}{\|\sigma\|^{2}}\,i\partial S_{\varepsilon}\wedge\bar{\partial}S_{\varepsilon}\,-\,\frac{\{i{\cal C}_{h}(E)\sigma,\sigma\}}{\|\sigma\|^{2}+\varepsilon}$
		$\displaystyle\geq$	$\displaystyle\frac{\varepsilon}{\|\sigma\|^{2}}\,i\partial S_{\varepsilon}\wedge\bar{\partial}S_{\varepsilon}\,-\,\\|{\cal C}_{h}(E)\\|_{h,\omega}\,\frac{\|\sigma\|^{2}}{\|\sigma\|^{2}+\varepsilon}\,\omega$

where $\omega$ is a positive $(1,1)$ -form.

Remark. Let $(X,\omega_{X})$ be a polarised compact Kähler manifold of complex dimension $n$ , let $(Y,\omega_{Y})$ be a compact irreducible Kähler space of complex dimension $m\leq n$ , let $\pi:X\rightarrow Y$ be a surjective holomorphic map and let $0\leq f\in L\log^{n+\varepsilon}L(X,\omega^{n}_{X})$ , for some $\varepsilon>0$ such that $1=\int_{X}f\omega^{n}_{X}$ . Set $K_{t}:=\{\pi^{*}\omega_{Y}+t\omega_{X}\}^{n}>0$ for $t\in(0,1)$ . Consider the complex Monge-Ampère equations $(\pi^{*}\omega_{Y}+t\omega_{X}+i\partial\bar{\partial}\psi_{t})^{n}=K_{t}\,f\,\omega^{n}_{X}$ . The hypothesis $(C1)$ of statement $(C)$ in theorem 3 is obviously satisfied. The hypothesis $(C2b)$ is also satisfied since

\lim_{t\rightarrow 0}\,\frac{(\pi^{*}\omega_{Y}+t\omega_{X})^{n}}{K_{t}\,\omega^{n}_{X}}=\left(\;\int\limits_{y\in Y}\omega^{m}_{Y}(y)\cdot\int\limits_{z\in\pi^{-1}(y)}\omega^{n-m}_{X}\right)^{-1}\frac{\pi^{*}\omega_{Y}\wedge\omega^{n-m}_{X}}{\omega^{n}_{X}}<+\infty.

We deduce $\operatorname{Osc}(\psi_{t})\leq C<+\infty$ for all $t\in(0,1)$ by the statements $(C)$ and $(A)$ of theorem 3. This solves in full generality a Tian’s conjecture [Ti-Ko].

Acknowledgments. The second named author is grateful to Professor Gang Tian for bringing this type of problems to his attention. He expresses also his gratitude to the members of Institut Fourier for providing an excellent research environment. In particular he thanks Adrien Dubouloz, Hervé Pajot, Olivier Lablée and Eric Dumas for useful conversations.

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Jean-Pierre Demailly
Université de Grenoble I, Département de Mathématiques
Institut Fourier, 38402 Saint-Martin d’Hères, France
E-mail: demailly@fourier.ujf-grenoble.fr

Nefton Pali
Université Paris Sud, Département de Mathématiques
Bâtiment 425 F91405 Orsay, France
E-mail: nefton.pali@math.u-psud.fr

	$\displaystyle\|\sigma_{j}\circ\mu\|^{2}$	$\displaystyle\thicksim$	$\displaystyle\left(\sum_{p}\|f_{j,p}\|^{2}\right)\prod_{t=1}^{m}\|z_{t}\|^{2b_{j,s_{t}}}\,,$
	$\displaystyle\|\tau_{r}\circ\mu\|^{2}$	$\displaystyle\thicksim$	$\displaystyle\left(\sum_{q}\|g_{r,q}\|^{2}\right)\prod_{t=1}^{m}\|z_{t}\|^{2c_{r,s_{t}}}\,,$