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{\it Ann. Scient \'{E}c. Norm. Sup}.
4${}^{\text{e}}$ s\'{e}rie, t. 34, 2001, p. 525 \`{a} 556.
SEMI-CONTINUITY OF COMPLEX SINGULARITY
EXPONENTS AND $\mathrm{K}\ddot{\mathrm{A}}$ HLER-EINSTEIN METRICS
ON FANO ORBIFOLDS
BY JEAN-PIERRE DEMAILLY AND JA'NOS KOLL\'{A}R
ABSTRACT. --We introduce complex singularity exponents of plurisubharmonic functions and prove a
general semi-continuity result for them. This concept contains as a special case several similar concepts
which have been considered e.g. by Amold and Varchenko, mostly for the study of hypersurface
singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but
it also takes into account more quantitative informations of great interest for complex analysis and complex
differential geometry. We give as an application a new derivation of criteria for the existence of K\"{a}hler-
Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication
in the technique, once the semi-continuity result is taken for granted). In this way, three new examples of
rigid K\"{a}hler-Einstein Del Pezzo surfaces with quotient singularities are obtained.
\copyright 2001 \'{E}ditions scientifques et m\'{e}dicales Elsevier SAS
$\mathrm{RE}^{r}$ SUM\'{E}. --Nous introduisons les exposants de singularit\'{e}s complexes des fonctions plurisousharmo-
niques et de'montrons un th\'{e}or\`{e}me de semi-continuite' ge'ne'ral pour ceux-ci. Le concept e'tudie' contient
comme cas particulier des concepts voisins qui ont ete conside're's par exemple par Amold et Varchenko,
principalement pour 1'e'tude des singularit\'{e}s d'hypersurfaces. La version plurisousharmonique repose en
de'finitive sur une re'duction au cas alg\'{e}brique, mais elle prend aussi en compte des informations quantita-
tives d'un grand interest pour 1 analyse complexe et la ge'ome'trie diffe'rentie $11\mathrm{e}$ complexe. Nous de'crivons
en application une nouvelle approche des crit\`{e}res existence de me'triques de K\"{a}hler-Einstein pour les va-
rie'te's de Fano, en nous inspirant des ide'es originales de Nadel-mais avec des simplifications importantes
de la technique, une fois que le r\'{e}sultat de semi-continuite' est utilis\'{e} comme outil de base. Grace \`{a} ces
crit\`{e}res, nous obtenons trois nouveaux exemples de surfaces de Del Pezzo \`{a} singularit\'{e}s quotients, rigides,
posse'dant une me'trique de K\"{a}hler-Einstein.
\copyright 2001 \'{E}ditions scientifiques et m\'{e}dicales Elsevier SAS
0. Introduction
The purpose of this work is to show how complex analytic methods (and more specifically
$L^{2}$ estimates for $\overline{\partial}$) can provide effective forms of results related to the study of complex
singularities. We prove in particular a strong form of the semi-continuity theorem for ``complex
singularity exponents'' of plurisubharmonic (psh) functions. An application to the existence of
K\"{a}hler-Einstein metrics on certain Fano orbifolds will finally be given as an illustration of this
result.
We introduce the following definition as a quantitative way of measuring singularities of a
psh function $\varphi$ (the basic definition even makes sense for an arbitrary measurable function $\varphi$,
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526
J.-P DEMAILLY AND J. KOLL\'{A}R
though it is unlikely to have any good properties in that case). Our approach is to look at the $L^{1}$
integrability of $\exp(-2c\varphi)$ in terms ofthe Lebesgue measure in some local coordinates. Several
other types of analytic or algebraic objects (holomorphic functions, coherent ideal sheaves,
divisors, currents, etc) can be treated as special cases ofthis formalism.
DEFINITION 0.1. -{\it Let} $X$ be a complex manifold and $\varphi$ be a plurisubharmonic (psh) function
on $X$. For any compact set $K\subset X$, we introduce the ``complex singularity exponent'' of $\varphi$ on $K$
to be the nonnegative number
$ c_{K}(\displaystyle \varphi)=\sup$\{ $c\geq 0:\exp(-2c\varphi)$ is $L^{1}$ on a neighborhood of $K$\},
and we define the ``Amold multiplicity'' to be $\lambda_{K}(\varphi)=c_{K}(\varphi)^{-1}$:
$\displaystyle \lambda_{K}(\varphi)=\inf$ \{A $>0:\exp(-2\lambda^{-1}\varphi)$ is $L^{1}$ on a neighborhood of $K$\}.
If $\varphi\equiv-\infty$ near some connected component of $K$, we put of course $c_{K}(\varphi)=0,\ \lambda_{K}(\varphi)=+\infty$.
The singularity exponent $c_{K}(\varphi)$ only depends on the singularities of $\varphi$, namely on the behavior
of $\varphi$ near $\mathrm{its}-\infty$ poles. Let $T$ be a closed positive current ofbidegree (1, 1) on $X$. Since $c_{K}(\varphi)$
remains unchanged ifwe replace $\varphi$ with $\psi$ such that $\psi-\varphi$ is bounded, we see that it is legitimate
to define
\begin{center}
(0.1.1)   $c_{K}(T)=c_{K}(\varphi),\ \lambda_{K}(T)=\lambda_{K}(\varphi)$
\end{center}
whenever $\varphi$ is a (local) potential of $T$, i.e. a psh function $\varphi$ such that $\mathrm{dd}^{\mathrm{c}}\varphi=T$, where
$d^{c}=(2\pi \mathrm{i})^{-1}(\partial-\overline{\partial})$. In particular, if $D$ is an effective integral divisor, we have $c_{K}([D])=$
$c_{K}(\log|g|)$ where $[D]$ is the current of integration over $D$ and $g$ is a (local) generator of $\mathcal{O}(-D)$.
When $f$ is a holomorphic function, we write simply $c_{K}(f),\ \lambda_{K}(f)$ instead of $c_{K}(\log|f|)$,
$\lambda_{K}(\log|f|)$. For a coherent ideal sheaf $\mathcal{I}=(g_{1}, \ldots, g_{N})$ we define in a similar way $c_{K}=$
$c_{K}(\log(|g_{1}|+\cdots+|g_{N}|))$. It is well known that $c_{K}(f)$ is a rational number, equal to the largest
root of the Bemstein-Sato polynomial of $|f|^{2s}$ on a neighborhood of $K$ ([30], see also [25]);
similarly $c_{K}(\mathcal{I})\in \mathbb{Q}+$ for any coherent ideal sheaf. Our main result consists in the following
semi-continuity theorem.
main THEOREM 0.2.-{\it Let} $X$ {\it be a complex manifold Let} $\mathcal{Z}_{+}^{1,1}(X)$ {\it denote the space of}
{\it closed positive currents of type} (1, 1) {\it on} $X$, {\it equipped with the weak topology, and let} $\mathcal{P}(X)$
{\it be the set of locally} $L^{1}psh$ {\it functions on} $X$, {\it equipped with the topology of} $L^{1}$ {\it convergence on}
{\it compact subsets} ( $=topology$ {\it induced by the weak topology}). {\it Then}
(1) {\it The map} $\varphi\mapsto c_{K}(\varphi)$ {\it is lower semi-continuous on} $\mathcal{P}(X)$, {\it and the map} $T\mapsto c_{K}(T)$ {\it is}
{\it lower semi-continuous on} $\mathcal{Z}_{+}^{1,1}(X)$.
(2) (``Effective version''). {\it Let} $\varphi\in \mathcal{P}(X)$ {\it be given. If} $c<c_{K}(\varphi)$ {\it and} $\psi$ {\it converges to} $\varphi$ {\it in}
$\mathcal{P}(X)$, {\it then} $\mathrm{e}^{-2c\psi}$ {\it converges to} $\mathrm{e}^{-2c\varphi}$ {\it in} $L^{1}$ {\it norm over some neighborhood} $U$ {\it of} $K$.
{\it As a special case, one gets}:
(3) {\it The map} $\mathcal{O}(X)\ni f\mapsto c_{K}(f)$ {\it is lower semi-continuous with respect to the topology of}
{\it uniform convergence on compact sets} ({\it uniform convergence on afixedneighborhood} $ofK$
{\it is of course enough}). {\it Moreover, if} $c<c_{K}(f)$ {\it and} $g$ {\it converges to} $f$ {\it in} $\mathcal{O}(X)$, {\it then} $|g|^{-2c}$
{\it converges to} $|f|^{-2c}$ {\it in} $L^{1}$ {\it on some neighborhood} $U$ {\it of} $K$.
In spite their apparent simplicity, the above statements reflect rather strong semi-continuity
properties of complex singularities under ``variation of parameters''. Such properties have been
used e.g. by Angehm-Siu [1] in their approach of the Fujita conjecture, and our arguments will
borrow some their techniques in Section 3.
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Theorem 0.2 is by nature a purely local result, which is easily seen to be equivalent to the
special case when $K=\{x\}$ is a single point and $X$ is a small ball centered at $x$. The proof
is made in several steps. The ``analytic part'' consists in a reduction of (1) and (2) to (3), and
in the proof of the effective estimates leading to the convergence statements in (2) and (3) [by
contrast, the qualitative part of (3) can be obtained in a purely algebraic way]. The reduction
to the holomorphic case (3) is based on the fact that plurisubharmonic functions can be very
accurately approximated (both from the point of view of singularities and of $L_{1\mathrm{o}\mathrm{c}}^{1}$ topology) by
special functions of the form
\begin{center}
(0.2.4)   $\alpha\log(|g_{1}|+\cdots+|g_{N}|),\ \alpha\geq 0$,
\end{center}
where the $g_{j}$ are holomorphic functions. The existence of approximations as in (0.2.4) depends
in an essential way on the Ohsawa-Takegoshi $L^{2}$ extension theorem ([35], [36]), see [11-13]
and Sections 2, 4. One is then reduced to the proof for a single holomorphic function (that is, to
a psh function ofthe form $\log|f|)$, by taking a suitable generic linear combination $f=\displaystyle \sum\alpha_{j}g_{j}$.
Another essential idea is to truncate the Taylor expansion of $f$ at $x$ at some order $k$. It can then
be shown that this affects $c_{x}(f)$ only by a perturbation that is under uniform control. In fact, the
singularity exponent $c_{x}(f)$ is subadditive on holomorphic functions:
\begin{center}
(0.2.5)   $c_{x}(f+g) \leq c_{x}(f)+c_{x}(g),\ \forall f,\ g \in \mathcal{O}_{X,x}$.
\end{center}
{\it If} $p_{k}$ is the truncation at order $k$ ofthe Taylor series, one deduces immediately from (0.2.5) that
\begin{center}
(0.2.4)   $|c_{x}(f)-c_{x}(p_{k})|\displaystyle \leq\frac{n}{k+1}$.
\end{center}
In this way, the proof is reduced to the case of polynomials of given degree. Such polynomials
only depend on finitely many coefficients, thus the remaining lower semi-continuity property to
be proved is that of the function $t\mapsto c_{x}(P_{t})$ when $P_{t}$ is a family of polynomials depending
holomorphically on some parameters $t= (t_{1}, \ldots, t_{N})$. This is indeed true, as was already
observed by Varchenko [47,48]. An algebraic proof can be given by using a $\log$ resolution of
singularities with parameters. Here, however, a special attention to effective estimates must be
paid to prove the convergence statements in (2) and (3). For instance, it is necessary to get as well
an effective version of (0.2.6); the Ohsawa-Takegoshi $L^{2}$ extension theorem is again cmcial in
that respect.
As a consequence of our main theorem, we give a more natural proof ofthe results of Siu [41,
42], Tian [46] and Nadel [33,34] on the existence of K\"{a}hler-Einstein metric on Fano manifolds
admitting a sufficiently big group of symmetries. The main point is to have sufficient control on
the ``multiplier ideal sheaves'' which do appear in case the K\"{a}hler-Einstein metric fails to exist.
This can be dealt with much more easily through our semi-continuity theorem, along the lines
suggested in Nadel's note [33] (possibly because ofthe lack of such semi-continuity results, the
detailed version [34] relies instead on a rather complicated process based on a use of ``uniform''
$L^{2}$ estimates for sequences of Koszul complexes; all this disappears here, thus providing a
substantially shorter proof). We take the opportunity to adapt Nadel's result to Fano orbifolds.
This is mostly a straightforward extension, except that we apply intersection inequalities for
currents rather than the existence of a big finite group of automorphisms to derive sufficient
criteria for the existence of K\"{a}hler-Einstein metrics. In this way, we produce 3 new ``exotic
examples'' rigid Del Pezzo surfaces with quotient singularities which admit a K\"{a}hler-Einstein
orbifold metric.
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1. Complex singularity exponent and Arnold multiplicity
Let $X$ be a complex manifold and $\varphi$ a psh function of $x$. The concepts of ``complex singularity
exponent $c_{K}(\varphi)$ and ``Amold multiplicity $\lambda_{K}(\varphi)$ {\it of} $\varphi$ along a compact set $K\subset X$ have been
defined in 0.1. An equivalent definition can be given in terms of asymptotic estimates for the
volume of sublevel sets $\{\varphi<\log r\}$.
VARIANT OF THE definition 1.1. -Let $K\subset X$ be a compact set, $U\Subset X$ a relatively
compact neighborhood of $K$, and let $\mu_{U}$ be the Riemannian measure on $U$ associated with some
choice of hermitian metric $\omega$ on {\it X. Then}
$c_{K}(\displaystyle \varphi)=\sup\{c\geq 0;r^{-2c}\mu_{U}(\{\varphi<\log r\})$ is bounded as $r\rightarrow 0$, for some $U\supset K\}$.
The equivalence with the earlier Definition 0.1 follows immediately from the elementary
inequalitie $\mathrm{s}$
$r^{-2c}\displaystyle \mu_{U}(\{\varphi<\log r\})\leq\int_{U}\mathrm{e}^{-2c\varphi}\mathrm{d}V_{\omega}\leq\mu_{U}(U)+\int_{0}^{1}2cr^{-2c}\mu_{U}(\{\varphi<\log r\})\frac{\mathrm{d}r}{r}$.
A first important observation is that $c_{K}(\varphi)$ and $\lambda_{K}(\varphi)$ depend only on the local behavior of $\varphi$:
PROPOSITION 1.2. --{\it Given a point} $x\in X$, {\it we write} $c_{x}(\varphi)$ {\it instead of} $c_{\{x\}}(\varphi)$. {\it Then}
$$
c_{K}(\varphi)=\inf_{x\in K}c_{x}(\varphi),\ \lambda_{K}(\varphi)=\sup_{x\in K}\lambda_{x}(\varphi).
$$
The statement is clear from the Borel-Lebesgue Lemma. When $x$ is a pole, that is, when
$\varphi(x)=-\infty$, the Amold multiplicity $\lambda_{x}(\varphi)$ actually measures the ``strength'' of the singularity
of $\varphi$ in a neighborhood of $x$. (It actually increases'' with the singularity, and if $x$ is not a pole, we
have $ c_{x}(\varphi)=+\infty,\ \lambda_{x}(\varphi)=0$; see Proposition 1.4 below.) We now deal with various interesting
special cases:
NOTATION $1.3.-$
(1) If $f$ is a holomorphic function on $X$, we set $c_{K}(f)=c_{K}(\log|f|)$.
(2) If $\mathcal{I}\subset \mathcal{O}_{X}$ is a coherent ideal sheaf, generated by functions $(g_{1}, \ldots, g_{N})$ on a
neighborhood of $K$, we put
$$
c_{K}(\mathcal{I})=c_{K}(\log(|g_{1}|+\cdots+|g_{N}|)).
$$
(3) If $T$ is a closed positive current ofbidegree (1, 1) on $X$ which can be written as $ T=\mathrm{dd}^{\mathrm{c}}\varphi$
on a neighborhood of $K$, we set $c_{K}(T)=c_{K}(\varphi)$.
(If no global generators exist in (2) or no global potential $\varphi$ exists in (3), we just split $K$ in
finitely many pieces and take the infimum, according to Proposition 1.2.)
(4) If $D$ is an effective divisor with rational or real coefficients, we set
$$
c_{K}(D)=c_{K}([D])=c_{K}(\mathcal{O}(-D))=c_{K}(g)\ =c_{K}(\log|g|),
$$
where $D$ is the current of integration over $D$ and $g$ is a local generator of the principal
ideal sheaf $\mathcal{O}(-D)$.
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No confusion should arise from the above definitions, especially since $c_{K}(\mathcal{I})$ does not depend
on the choice of generators of $\mathcal{I}$. We use similar conventions ofnotation for $\lambda_{K}(\varphi)$. The number
$c_{x}(f)=\displaystyle \sup\{c;|f|^{-2c}$ is $L^{1}$ on a neighborhood of $x\}=\lambda_{x}(f)^{-1}$
is clearly a measure ofthe singularities ofthe hypersurface $\{f=0\}$ at point $x$. This number came
up in the literature many times under different names. By [30], $c_{x}(f)$ is the largest root of the
Bemstein-Sato polynomial associated to the germ of $f$ around $p$. If $x$ is an isolated singularity of
$\{f=0\}$, then $c_{x}(f)=\displaystyle \min\{1, \beta_{\mathbb{C}}(f_{x})\}$ where $\beta_{\mathbb{C}}(f_{x})$ is the complex singular index as defined
in [3], vol. II, Sec. 13.1.5; the same thing is called ``complex singularity exponent'' in [49].
See [25] for a discussion these questions and for related results.
ELEMENTARY PROPERTIES {\it lA}.-{\it Let} $\mathcal{I},\ \mathcal{J}$ {\it be coherent ideals on} $X$ {\it and let} $\varphi,\ \psi$ {\it be} $psh$
{\it functions. Denote by} $x$ {\it point in} $X$ {\it and let} $K\subset X$ {\it be a compact subset}.
(1) {\it The function} $x\mapsto c_{x}(\varphi)$ {\it is lower semi-continuous for the holomorphic Zariski topology};
(2) {\it If} $\varphi\leq\psi$, {\it then} $c_{K}(\varphi)\leq c_{K}(\psi)$;
$If\mathcal{I}\subset \mathcal{J}$, {\it then} $c_{K}(\mathcal{I})\leq c_{K}(\mathcal{J})$.
(3) $\lambda_{K}(\varphi+\psi)\leq\lambda_{K}(\varphi)+\lambda_{K}(\psi)$;
$$
\lambda_{K}(\mathcal{IJ})\leq\lambda_{K}(\mathcal{I})+\lambda_{K}(\mathcal{J}).
$$
(4) $\lambda_{K}(\alpha\varphi)=\alpha\lambda_{K}(\varphi)$ {\it for all} $\alpha\in \mathbb{R}_{+}$;
$\lambda_{K}(\mathcal{I}^{m})=m\lambda_{K}(\mathcal{I})$ {\it for all integers} $m\in \mathbb{N}$.
(5) {\it Let} $\mathcal{I}=(g_{1}, \ldots, g_{N})$ {\it and let}
$\overline{\mathcal{I}}=\{f\in \mathcal{O}_{\Omega,x},\ x\in\Omega;\exists C\geq 0,\ |f|\displaystyle \leq C\max|g_{j}|$ {\it near} $x\}$
{\it be the integral closure of X. Then} $c_{K}(\overline{\mathcal{I}})=c_{K}(\mathcal{I})$.
(6) {\it If the zero variety germ} $V(\mathcal{I}_{x})$ {\it contains a p-codimensional irreducible component, then}
$$
c_{x}(\mathcal{I})\leq p,\ i.e.\ \lambda_{x}(\mathcal{I})\geq 1/p.
$$
(7) $If\mathcal{I}_{Y}$ {\it is the ideal sheaf of a p-codimensional subvariety} $ Y\subset\Omega$, {\it then} $c_{x}(\mathcal{I}_{Y})=p$ {\it at every}
{\it nonsingular point of} $Y$.
(8) {\it Define the vanishing order} $\mathrm{ord}_{x}(\mathcal{I})$ {\it of} $\mathcal{I}$ {\it at} $x$ {\it to be the supremum of all integers} $k$ {\it such}
{\it that} $\mathcal{I}_{x}\subset \mathfrak{m}_{x}^{k}$, {\it where} $\mathfrak{m}_{x}\subset \mathcal{O}_{x}$ {\it is the maximal ideal. Then}
$$
\frac{1}{n}\mathrm{ord}_{x}(\mathcal{I})\leq\lambda_{x}(\mathcal{I})\leq \mathrm{ord}_{x}(\mathcal{I}).
$$
{\it More generally} $\iota f\nu_{x}(\varphi)$ {\it is the Lelong number of} $\varphi$ {\it at} $x$, {\it then}
$$
\frac{1}{n}\nu_{x}(\varphi)\leq\lambda_{x}(\varphi)\leq\nu_{x}(\varphi).
$$
{\it Proof}. $-(1)$ Fix a point $x_{0}$ and a relatively compact coordinate ball $B :=B(x_{0}, r)\subset X$. For
every $c\geq 0$, let $\mathcal{H}_{c\varphi}(B)$ be the Hilbert space ofholomorphic functions on $B$ with finite weighted
$L^{2}$ norm
$$
\Vert f\Vert_{c}^{2}=\int_{B}|f|^{2}\mathrm{e}^{-2c\varphi}\mathrm{d}V,
$$
where $\mathrm{d}V$ is the Lebesgue volume element in $\mathrm{C}^{\mathrm{n}},\ n=\dim_{\mathrm{C}}X$. A fundamental consequence
of HOrmander's $L^{2}$ estimates (H\"{o}rmander-Bombieri-Skoda theorem [20], [6], [45]) states that
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there is an element $f\in \mathcal{H}_{c\varphi}(B)$ with $f(x)=1$ whenever $\mathrm{e}^{-2c\varphi}$ is $L^{1}$ on a neighborhood of $x$.
Hence
$$
\{x\in B;c_{x}(\varphi)\leq c_{0}\}\cap B=\bigcap_{f\in\bigcup_{c>c}\mathcal{H}_{c\varphi}(B)}f^{-1}(0)
$$
is an analytic set. This proves the holomorphic Zariski lower semi-continuity.
All other properties are direct consequences of the definitions and do not require ``hard''
analysis: (2), (4), (5) are immediate; (3) is a consequence of the HOlder inequality; (6,7) follow
from the fact that the function $(\displaystyle \sum_{j\leq p}|z_{j}|^{2})^{-c}$ {\it is} locally integrable along $z_{1}=\cdots=z_{p}=0$ if and
only if $c<p$. Finally, (8) is a well-known result of Skoda [44], depending on the basic properties
Lelong numbers and a use of standard kemel techniques. $\square $
In the case of an ideal sheaf, the following lemma reduces the computation of $c_{x}(\mathcal{I})$ to the
case of a principal ideal (possibly after raising $\mathcal{I}$ to some power $\mathcal{I}^{m}$).
PROPOSITION 1.5.-{\it Let} $(\mathrm{g}_{1}, \ldots, \mathrm{g}_{p})$ {\it be holomorphic functions defined on an open set}
$\Omega\subset \mathbb{C}^{n}$ {\it and let} $x\in V(g_{1}, \ldots, g_{p})$. {\it Then}
$$
c_{x}(\alpha_{1}g_{1}+\cdots+\alpha_{p}g_{p})\leq\min\{c_{x}(g_{1}, \ldots,g_{p}), 1\}
$$
{\it for all coefficients} $(\alpha_{1}, \ldots, \alpha_{p})\in \mathbb{C}^{p}$. {\it Moreover, the equality occurs for all} $(\alpha_{1}, \ldots, \alpha_{p})$ {\it in the}
{\it complement} $ofa$ {\it set measure zero in} $\mathbb{C}^{p}$. {\it In particular}, $\iota f\mathcal{I}$ {\it is an arbitrary ideal and} $c_{x}(\mathcal{I})\leq 1$,
{\it there is principal ideal} $(f)\subset \mathcal{I}$ {\it such that} $c_{x}(f)=c_{x}(\mathcal{I})$.
{\it Proof}-The inequality is obvious, since $c_{x}(\alpha_{1}g_{1}+\cdots+\alpha_{p}g_{p})\leq 1$ by 1.4(6) on the one hand,
and
$$
|\alpha_{1}g_{1}+\cdots+\alpha_{p}g_{p}|^{-2c}\geq(\sum|\alpha_{j}|^{2})^{-c}(\sum|g_{j}|^{2})^{-c}
$$
on the other hand. Now, fix $c<\displaystyle \min\{c_{x} (g_{1}, \ldots, g_{p}), 1\}$. There is a neighborhood $U_{c}$ of $x$ on
which
$$
\int_{|\alpha|=1}\mathrm{d}\sigma(\alpha)\int_{U_{c}}|\alpha_{1}g_{1}(z)+\cdots+\alpha_{p}g_{p}(z)|^{-2c}\mathrm{d}V(z)
$$
\begin{center}
(1.5.1)   $=A_{c}\displaystyle \int_{U_{c}}(\sum|g_{j}(z)|^{2})^{-c}\mathrm{d}V(z)<+\infty$,
\end{center}
where $\mathrm{d}\sigma$ is the euclidean area measure on the unit sphere $S^{2n-1}\subset \mathbb{C}^{n}$ and $A_{c}>0$ is a constant.
The above identity follows from the formula
$$
\int_{|\alpha|=1}|\alpha\cdot w|^{-2c}\mathrm{d}\sigma(\alpha)=A_{c}|w|^{-2c},
$$
which is obvious by homogeneity, and we have $ A_{c}<+\infty$ for $c<1$. The finiteness ofthe right
hand side of (1.5.1) implies that the left hand side is finite for all values $\alpha$ in the complement
$\mathbb{C}^{p}\backslash N_{c}$ of a negligible set. Therefore $c_{x}(\alpha_{1}g_{1}+\cdots+\alpha_{p}g_{p})\geq c$, and by taking the supremum
over an increasing sequence ofvalues $c_{l/}$ converging to $\displaystyle \min\{c_{x} (g_{1}, \ldots, g_{p}), 1\}$, we conclude that
the equality holds in Proposition 1.5 for all $\alpha\in \mathbb{C}^{p}\backslash \cup N_{c_{\nu}}$.
$$
\square 
$$
{\it Remark} 1.6. -It follows from Theorem 3.1 below that the exceptional set of values
$(\alpha_{1}, \ldots, \alpha_{p})$ occurring in Proposition 1.5 is in fact a closed algebraic cone in $\mathbb{C}^{p}$.
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The singularity exponent $c_{K}(\mathcal{I})$ of a coherent ideal sheaf $\mathcal{I}\subset \mathcal{O}_{X}$ can be computed by means
of a $\log$ resolution of $\mathcal{I}$, that is, a composition $\mu:\overline{X}\rightarrow X$ of blow-ups with smooth centers
such that $\mu^{\star}\mathcal{I}=\mathcal{O}_{\overline{X}}(-D)$ is an {\it invertible} sheaf associated with a normal crossing divisor $D$ in
$\overline{X}$ (such a $\log$ resolution always exists by Hironaka [19]). The following proposition is essentially
well known (see e.g. [24] 10.7).
PROPOSITION 1.7.-{\it Let} $X$ {\it be a complex manifold}, $\mathcal{I}\subset \mathcal{O}_{X}$ {\it a coherent ideal sheaf, and let}
$\mu:\overline{X}\rightarrow X$ {\it be a modification} ( $=proper$ {\it bimeromorphic morphism}) {\it such that} $\mu^{\star}\mathcal{I}=\mathcal{O}_{\overline{X}}(-D)$
{\it is an invertible sheaf Assume that} $\overline{X}$ {\it is normal and let} $E_{i}\subset\overline{X}$ {\it denote either an exceptional}
{\it divisor of} $\mu$ {\it or an irreducible component of D. Write}
$K_{\overline{X}}=\displaystyle \mu^{\star}K_{X}+\sum a_{i}E_{i}$ {\it and} $D=\displaystyle \sum b_{i}E_{i}$,
{\it where} $a_{i}=0\iota fE_{i}$ {\it is not a component ofthe exceptional divisor of} $\mu$ ({\it resp}. $b_{i}=0\iota fE_{i}$ {\it is not a}
{\it component of} $D$). {\it Then}:
(1) $c_{K}(\displaystyle \mathcal{I})\leq\min_{i:\mu(\underline{E}_{i})\cap K\neq\emptyset}\{(a_{i}+1)/b_{i}\}$.
(2) {\it Equality holds} $\iota fX$ {\it is smooth and} $\displaystyle \sum E_{i}$ {\it is a divisor with normal crossings}.
(3) {\it If} $g =(g_{1}, \ldots, g_{N})$ {\it are generators} $of\mathcal{I}$ {\it in a neighborhood of} $K$, {\it then for any sufficiently}
{\it small neighborhood} $U$ {\it of} $K$ {\it there is a volume estimate}
$$
C_{1}r^{2c}\leq\mu_{U}(\{|g|<r\})\leq C_{2}r^{2c}|\log r|^{n-1},\ \forall r<r_{0}
$$
{\it with} $n =\dim_{\mathbb{C}}X,\ c=c_{K}(\mathcal{I})$ {\it and} $C_{1},\ C_{2},\ r_{0}>0$.
$Proof-$ Since the question is local, we may assume that $\mathcal{I}$ is generated by holomorphic
functions $g_{1},\ \ldots,\ g_{N}\in \mathcal{O}(X)$. Then (1) and (2) are straightforward consequences ofthe Jacobian
formula for a change variable: if $U$ is an open set in $X$, the change $z=\mu(\zeta)$ yields
$$
\int_{z\in U}|g(z)|^{-2c}\mathrm{d}V(z)=\int_{\zeta\in\mu^{-1}(U)}|g\ \circ\mu(\zeta)|^{-2c}|J_{\mu}(\zeta)|^{2}\mathrm{d}\overline{V}(\zeta),
$$
where $J_{\mu}$ is the Jacobian of $\mu$, and $\mathrm{d}V,\ \mathrm{d}\overline{V}$ are volume elements of $X,\overline{X}$ respectively (embed
$\overline{X}$ in some smooth ambient space ifnecessary). Now, if $h_{i}$ is a generator of $\mathcal{O}(-E_{i})$ at a smooth
point $\overline{x}\in\overline{X}$, the divisor of $J_{\mu}$ is by definition $\displaystyle \sum a_{i}E_{i}$ and $\displaystyle \mu^{\star}\mathcal{I}=\mathcal{O}(-\sum b_{i}E_{i})$. Hence, up to
multiplicative bounded factors,
$|J_{\mu}|^{2}\displaystyle \sim\prod|h_{i}|^{2a_{i}},\ |g \displaystyle \circ\mu|^{2}\sim\prod|h_{i}|^{2b_{i}}$ near $\overline{x}$,
and $|g \circ\mu|^{-2c}|J_{\mu}|^{2}$ is $L^{1}$ near $\overline{x}$ if and only if $\displaystyle \prod|h_{i}|^{-2(cb_{i}-a_{i})}$ is $L^{1}$. A necessary
condition is that $cb_{i}-a_{i}<1$ whenever $E_{i}\ni\overline{x}$. We therefore get the necessary condition
$c<\displaystyle \min_{i:\mu(E_{i})\cap K\neq\emptyset}\{(a_{i}+1)/b_{i}\}$, and this condition is necessary and sufficient if $\displaystyle \sum E_{i}$ is a
normal crossing divisor.
For (3), we choose $(\overline{X}, \mathcal{O}(-D))$ to be a (nonsingular) $\log$ resolution of $\mathcal{I}$. The volume
$\mu_{U}(\{|g|<r\})$ is then given by integrals ofthe form
\begin{center}
(1.7.4)   $\displaystyle \int_{\mu^{-1}(U)\cap\{\zeta\in\overline{U}_{\alpha},\prod|h_{i}|^{b_{i}}<r\}}\prod|h_{i}(\zeta)|^{2a_{i}}\mathrm{d}V(\zeta)$
\end{center}
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over suitable coordinate charts $\overline{U}_{\alpha}\subset\overline{X}$. An appropriate change of variable $\zeta\mapsto w,\ w_{i}=h_{i}^{b_{i}}(\zeta)$,
$w_{j}=\zeta_{k_{j}}$ (where $i$ runs over the set indices such that $b_{i}>0$ and $j$ over a disjoint set ofindices)
and a use of a partition of unity leads to estimate (1.7.4) by a linear combination of integrals of
the form
$\displaystyle \int_{P(r)}\prod|w_{i}|^{2(a_{i}+1)/b_{i}-2}\mathrm{d}V(w)$ {\it where} $P(r)=\displaystyle \{\max|w_{i}|<1,\ \displaystyle \prod|w_{i}|<r\}$
(we assume here that a partial integration {\it with} respect to the $w_{j}$'s has already been performed).
The lower bound $C_{1}r^{2c}$ is obtained by restricting the domain of integration to a neighborhood
of a point in the unit polydisk such that only one coordinate $w_{i}$ vanishes, precisely for $i$
equal to the index achieving the minimum of $(a_{i}+1)/b_{i}$. The upper bound $C_{2}r^{2c}|\log r|^{n-1}$,
$c=\displaystyle \min(a_{i}+1)/b_{i}$, is obtained by using the inequalities
$$
\prod|w_{i}|^{2(a_{i}+1)/b_{i}-2}\leq(\prod|w_{i}|)^{2c-2}\leq r^{2c-2},\ \forall w\in P(r),
$$
$$
\mu(P(r))=\int_{\{\max(|w_{1}|,\ldots,|w_{n-1}|)<1\}}\pi\min(\frac{r^{2}}{|w_{1}|^{2}\cdots|w_{n-1}|^{2}},1)\prod_{i=1}^{n-1}\mathrm{d}V(w_{i})
$$
$$
\leq\pi\int_{\{\exists i;|w_{i}|<r\}}\prod_{i=1}^{n-1}\mathrm{d}V(w_{i})+\pi r^{2}\int_{\{\forall i;r\leq|w_{i}|<1\}}\prod_{i=1}^{n-1}\frac{\mathrm{d}V(w_{i})}{|w_{i}|^{2}}
$$
$$
\leq C_{2}r^{2}|\log r|^{n-1}.
$$
It should be observed that much finer estimates are known to exist; in fact, one can derive
rather explicit asymptotic expansions of integrals obtained by integration along the fibers of a
holomorphic function (see [5]). $\square $
2. $L^{2}$ extension theorem and inversion of adjunction
Our starting point is the following special case ofthe fundamental $L^{2}$ extension theorem due
to Ohsawa-Takegoshi ([35], [36], see also [31]).
THEOREM 2.1 [35,36,31].-{\it Let} $\Omega\subset \mathbb{C}^{n}$ {\it be a bounded pseudoconvex domain, and let} $L$ {\it be}
{\it an affine linear subspace} $of\mathbb{C}^{n}$ {\it codimension} $p\geq 1$ {\it given by an orthonormal system} $s$ {\it of affine}
{\it linear equations} $s_{1}=\cdots=s_{p}=0$. {\it For every} $\beta<p$, {\it there exists a constant} $C_{\beta,n,\Omega}$ {\it depending}
{\it only on} $\beta,\ n$ {\it and the diameter of} $\Omega$, {\it satisf} $ing$ {\it the following property. For every} $\varphi\in \mathcal{P}(\Omega)$ {\it and}
$f\in \mathcal{O}(\Omega\cap L)$ {\it with} $\displaystyle \int_{\Omega\cap L}|f|^{2}e^{-\varphi}dV_{L}<+\infty$, {\it there exists an extension} $F\in \mathcal{O}(\Omega)$ {\it of} $f$ {\it such that}
$$
\int_{\Omega}|F|^{2}|s|^{-2\beta}\mathrm{e}^{-\varphi}\mathrm{d}V_{\mathbb{C}^{n}}\leq C_{\beta,n,\Omega\int_{\Omega\cap L}}|f|^{2}\mathrm{e}^{-\varphi}\mathrm{d}V_{L},
$$
{\it where} $\mathrm{d}V_{\mathbb{C}^{n}}$ {\it and} $\mathrm{d}V_{L}$ {\it are the Lebesgue volume elements in} $\mathbb{C}^{n}$ {\it and} $L$ {\it respectively}.
In the sequel, we use in an essential way the fact that $\beta$ can be taken arbitrarily close to $p$. It
should be observed, however, that the case $\beta=0$ is sufficient to imply the general case. In fact,
supposing $L=\{z_{1}=\cdots=z_{p}=0\}$, a substitution $(\varphi, \Omega)\mapsto(\varphi_{k}, \Omega_{k})$ with
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$$
\varphi_{k}(z_{1}, \ldots, z_{n})=\varphi(z_{1}^{k}, \ldots, z_{p}^{k}, z_{p+1}, \ldots, z_{n}),
$$
$$
\Omega_{k}=\{z\in \mathbb{C}^{n};(z_{1}^{k}, \ldots, z_{p}^{k}, z_{p+1}, \ldots, z_{n})\in\Omega\}
$$
shows that the estimate with $\beta=0$ implies the estimate with $\beta=p(1-1/k)$ (use the change of
variable $\zeta_{1}=z_{1}^{k},\ \ldots,\ \zeta_{p}=z_{p}^{k},\ \zeta_{j}=z_{j}$ for $j>p$, together with the Jacobian formula
$$
\mathrm{d}V(z)=\frac{\mathrm{C}\mathrm{o}\mathrm{n}.\mathrm{s}\mathrm{t}}{|\zeta_{1}|^{2(1-1/k)}\cdot\cdot|\zeta_{p}|^{2(1-1/k)}}\mathrm{d}V(\zeta),
$$
and take the ``trace'' of the solution $F_{k}$ on $\Omega_{k}$ to get the solution $F$ on $\Omega$). The $L^{2}$ extension
theorem readily implies the following important monotonicity result.
PROPOSITION 2.2.-{\it Let} $\varphi\in \mathcal{P}(X)$ {\it be a} $psh$ {\it function on a complex manifold} $X$, {\it and let}
$Y\subset X$ {\it be a complex submanifold such that} $\varphi_{|Y}\not\equiv-\infty$ {\it on every connected component of} $Y$.
{\it Then}, $\iota fK$ {\it is a compact subset of} $Y$, {\it we have}
$$
c_{K}(\varphi_{|Y})\leq c_{K}(\varphi).
$$
({\it Here, ofcourse}, $c_{K}(\varphi)$ {\it is computed on} $X$, {\it i. e}., {\it by means of neighborhoods of} $K$ {\it in} $X.$)
{\it Proof}.-By Proposition 1.2, we may assume that $K=\{y\}$ is a single point in $Y$. Hence,
after a change of coordinates, we can suppose that $X$ is an open set in $\mathbb{C}^{n}$ and that $Y$ is an
affine linear subspace. Let $c<c_{y}(\varphi_{|Y})$ be given. There is a small ball $B=B(y, r)$ such that
$\displaystyle \int_{B\cap Y}\mathrm{e}^{-2c\varphi}\mathrm{d}V_{Y}<+\infty$. By the $L^{2}$ extension theorem applied with $\beta=0,\ \Omega=B,\ L=Y$
and $f(z)=1$, we can find a holomorphic function $F$ on $B$ such that $F(z)=1$ {\it on} $B\cap Y$
and $\displaystyle \int_{B}|F|^{2}\mathrm{e}^{-2c\varphi}\mathrm{d}V_{B}<+\infty$. As $F(y)=1$, we infer $c_{y}(\varphi)\geq c$ and the conclusion follows.
It should be observed that an algebraic proof exists when $\varphi$ is of the form $\log|g|,\ g \in \mathcal{O}(X)$;
however that proof is rather involved. This is already a good indication of the considerable
strength ofthe $L^{2}$ extension theorem (which will be crucial in several respects in the sequel). $\square $
We now show that the inequality given by Proposition 2.2 can somehow be reversed
(Theorem 2.5 below). For this, we need to restrict ourselves to a class of psh functions which
admit a ``sufficiently good local behavior'' (such restrictions were already made in [8], [12] to
accommodate similar difficulties).
DEFINITION 2.3. -Let $X$ be a complex manifold. We denote by $\mathcal{P}_{h}(X)$ the class of all
plurisubharmonic functions $\varphi$ {\it on} $X$ such that $\mathrm{e}^{\varphi}$ is locally HOlder continuous on $X$, namely
such that for every compact set $K\subset X$ there are constants $C=C_{K}\geq 0,\ \alpha=\alpha_{K}>0$ with
$$
|\mathrm{e}^{\varphi(x)}-\mathrm{e}^{\varphi(y)}|\leq Cd(x, y)^{\alpha},\ \forall x,\ y\ \in K,
$$
where $d$ is some Riemannian metric on $X$. We say for simplicity that such a function is a HOlder
psh function.
{\it Example} 2.4.-We are mostly interested in the case of functions ofthe form
$$
\varphi=\max\log j(\sum_{k}\prod_{l}|f_{j,k,l}|^{\alpha_{jkl}})
$$
with $f_{j,k,l}\in \mathcal{O}(X)$ and $\alpha_{j,k,l}>0$. Such functions are easily seen to be HOlder psh. Especially,
if $D=\displaystyle \sum\alpha_{j}D_{j}$ is an effective real divisor, the potential $\displaystyle \varphi=\sum\alpha_{j}\log|g_{j}|$ associated with $[D]$
is a HOlder psh function.
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Theorem 2.5.-{\it Let} $H$ {\it be a smooth hypersurface} $ofX$ {\it and let} $T$ {\it be a closedpositive current}
{\it of type} (1, 1) {\it on} $X$ {\it such that its local potential functions} $\varphi$ {\it are H\"{o}lder} $psh$ {\it functions with}
$\varphi_{|H}\not\equiv-\infty$. {\it We set in this case} ({\it somewhat abusively}) $T_{|H}=\mathrm{dd}^{\mathrm{c}}\varphi_{|H}$. {\it Then for any compact}
{\it set} $K\subset H$, {\it we have}
$$
c_{K}([H]+T)\geq 1\Leftrightarrow c_{K}(T_{|H})\geq 1.
$$
In the algebraic setting (that is, when $T=[D]$ is defined by an effective divisor $D=\displaystyle \sum\alpha_{j}D_{j}$),
the above result is known as ``inversion of adjunction'', see Kolla'r et al. [23, 17.7]. One says that
the pair $(X, D)$ is lc ( $=\log$ canonical) if $c_{K}(D)\geq 1$ for every compact set $K\subset X$, i.e., if the
product $\displaystyle \prod|g_{j}|^{-2c\alpha_{j}}$ associated with the generators $g_{j}$ of $\mathcal{O}(-D_{j})$ is locally $L^{1}$ for every $c<1$.
The result can then be rephrased as
\begin{center}
(2.5.1)   $(X, H+D)$ is lc $\Leftrightarrow(H, D_{|H})$ is lc.
\end{center}
{\it Proof of Theorem 2.5}. --Since the result is purely local, we may assume that $X=D(0, r)^{n}$
is a polydisk in $\mathrm{C}^{\mathrm{n}}$, that $H$ is the hyperplane $z_{n}=0$ and $K=\{0\}$. We must then prove the
equivalence
$$
\forall c<1,\ \exists U\ni 0,\ \exp(-2c(\log|z_{n}|+\varphi(z)))\in L^{1}(U)
$$
$$
\Leftrightarrow\forall c'<1,\ \exists U'\ni 0,\ \exp(-2c'\varphi(z', 0))\in L^{1}(U'),
$$
where $z=(z', z_{n})\in \mathbb{C}^{n}$ and $U,\ U'$ are neighborhoods of $0$ in $\mathbb{C}^{n},\ \mathbb{C}^{n-1}$ respectively.
First assume that $(|z_{n}|\mathrm{e}^{\varphi(z)})^{-2c}\in L^{1}(U)$. As $\mathrm{e}^{\varphi}$ is HOlder continuous, we get
$$
\mathrm{e}^{2c\varphi(z)}\leq(\mathrm{e}^{\varphi(z',0)}+C_{1}|z_{n}|^{\alpha})^{2c}\leq C_{2}(\mathrm{e}^{2c\varphi(z',0)}+|z_{n}|^{2c\alpha})
$$
on a neighborhood of $0$, for some constants $C_{1},\ C_{2},\ \alpha>0$. Therefore the function
$$
\frac{1}{|z_{n}|^{2c}(|z_{n}|^{2c\alpha}+\mathrm{e}^{2c\varphi(z',0)})}\leq C_{2}^{-1}(|z_{n}|\mathrm{e}^{\varphi(z)})^{-2c}
$$
is in $L^{1}(U)$. Suppose that $U=U'\times D(0, r_{n})$ is a small polydisk. A partial integration with
respect to $z_{n}$ on a family of disks $|z_{n}|<\rho(z')$ with $\rho(z')=\xi;\exp(\alpha^{-1}\varphi(z', 0))(\mathrm{and}\in >0$ so
small that $\rho(z')\leq r_{n}$ for all $z'\in U'$) shows that
$$
\int_{U}\frac{\mathrm{d}V(z)}{|z_{n}|^{2c}(|z_{n}|^{2c\alpha}+\mathrm{e}^{2c\varphi(z',0)})}\geq \mathrm{Const}\int_{U}\frac{\mathrm{d}V(z')}{\mathrm{e}^{(2c-2(1-c)\alpha^{-1})\varphi(z',0)}}.
$$
Hence $\exp(-2c'\varphi(z', 0))\in L^{1}(U')$ with $c'=c-(1-c)\alpha^{-1}$ arbitrarily close to 1. Conversely,
if the latter condition holds, we apply the Ohsawa-Takegoshi extension theorem to the function
$f(z')=1$ on $L=H=\{z_{n}=0\}$, with the weight $\psi=2c'\varphi$ and $\beta=c'<1$. Since $F(z', 0)=1$,
the $L^{2}$ condition implies the desired conclusion. $\square $
{\it Remark} 2.6. -As the final part of the proof shows, the implication
$$
c_{K}([H]+T)\geq 1\Leftarrow c_{K}(T_{|H})\geq 1
$$
is still true for an arbitrary (not necessarily HOlder) psh function $\varphi$. The implication $\Rightarrow$, however,
is no longer true. A simple counterexample is provided in dimension 2 by $H=\{z_{2}=0\}$ and
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$ T=\mathrm{dd}^{\mathrm{c}}\varphi$ with
$$
\varphi(z_{1}, z_{2})=\max(\lambda\log|z_{1}|, -\sqrt{-\log|z_{2}|}),\ \lambda\ >1
$$
on the unit bidisk $D(0,1)^{2}\subset \mathbb{C}^{2}$. Then $c_{0}([H]+T)=c_{0}([H])=1$ but $c_{0}(T_{|H})=c_{0}(\lambda\log|z_{1}|)$
$=1/$ A.
PROPOSITION 2.7.-{\it Let} $X,\ Y$ {\it be complex manifolds of respective dimensions} $n,\ m$, {\it let}
$\mathcal{I}\subset \mathcal{O}_{X},\ \mathcal{J}\subset \mathcal{O}_{Y}$ {\it be coherent ideals, and let} $K\subset X,\ L\subset Y$ {\it be compact sets. Put} $\mathcal{I}\oplus \mathcal{J} :=$
$\mathrm{pr}_{1}^{\star}\mathcal{I}+\mathrm{pr}_{2}^{\star}\mathcal{J}\subset \mathcal{O}_{X\times Y}$. {\it Then}
$$
c_{K\times L}(\mathcal{I}\oplus \mathcal{J})=c_{K}(\mathcal{I})+c_{L}(\mathcal{J}).
$$
{\it Proof}-By Proposition 1.2, it is enough to show that $c_{(x,y)}(\mathcal{I}\oplus \mathcal{J})=c_{x}(\mathcal{I})+c_{y}(\mathcal{J})$ at every
point $(x, y) \in X\times Y$. Without loss generality, we may assume that $X\subset \mathbb{C}^{n},\ Y\subset \mathbb{C}^{m}$ are open
sets and $(x, y)=(0,0)$. Let $g =(g_{1}, \ldots, g_{p})$, resp. $h=(h_{1}, \ldots, h_{q})$, be systems ofgenerators of
$\mathcal{I}$ (resp. $\mathcal{J}$) on a neighborhood of $0$. Set
$$
\varphi=\log\sum|g_{j}|,\ \psi=\log\sum|h_{k}|.
$$
Then $\mathcal{I}\oplus \mathcal{J}$ is generated by the $p+q$-tuple of functions
$$
g\ \oplus h=(g_{1}(x), \ldots, g_{p}(x), h_{1}(y), \ldots, h_{q}(y))
$$
and the corresponding psh function $\Phi(x, y) =\displaystyle \log(\sum|g_{j}(x)|+\sum|h_{k}(y)|)$ has the same behavior
along the poles as $\Phi'(x, y) =\displaystyle \max(\varphi(x), \psi(y))$ (up to a term $\mathrm{O}(1)\leq\log 2$). Now, for sufficiently
small neighborhoods $U,\ V$ of $0$, we have
$$
\mu_{U\times V}\ (\{\max\ (\ \varphi(x),\ \psi(y))<\log r\})=\mu_{U}(\{\varphi<\log r\}\times\mu_{V}(\{\psi<\log r\})),
$$
hence Proposition 1.7(3) implies
(2.7.1) $ C_{1}r^{2(c+c')}\displaystyle \leq\mu_{U\times V}(\{\max$ ( $\varphi(x),\ \psi(y))<\log r\})\leq C_{2}r^{2(c+c')}|\log r|^{n-1+m-1}$
with $c=c_{0}(\varphi)=c_{0}(\mathcal{I})$ and $c'=c_{0}(\psi)=c_{0}(\mathcal{J})$. From this, we infer
$$
c_{(0,0)}(\mathcal{I}\oplus \mathcal{J})=c+c'=c_{0}(\mathcal{I})+c_{0}(\mathcal{J}).
$$
$$
\square 
$$
{\it Example} 2.8.-As $c_{0}(z_{1}^{m})=1/m$, an application of Proposition 2.7 to a quasi-homogeneous
ideal $\mathcal{I}= (z_{1}^{m_{1}}, \ldots, z_{p}^{m_{p}})\subset \mathcal{O}_{\mathbb{C}^{n},0}$ yields the value
$$
c_{0}(\mathcal{I})=\frac{1}{m_{1}}+\cdots+\frac{1}{m_{p}}.
$$
Using Proposition 2.7 and the monotonicity property, we can now prove the fundamental
subadditivity property ofthe singularity exponent.
THEOREM 2.9.-{\it Let} $f,\ g$ {\it be holomorphic on a complex manifold X. Then, for every} $x\in X$,
$$
c_{x}(f+g)\ \leq c_{x}(f)+c_{x}(g).
$$
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{\it More generally}, $\iota f\mathcal{I}$ {\it and} $\mathcal{J}$ {\it are coherent ideals, then}
$$
c_{x}(\mathcal{I}+\mathcal{J})\leq c_{x}(\mathcal{I})+c_{x}(\mathcal{J}).
$$
{\it Proof}.-Let $\triangle$ be the diagonal in $X\times X$. Then $\mathcal{I}+\mathcal{J}$ can be seen as the restriction of $\mathcal{I}\oplus \mathcal{J}$
to $\triangle$. Hence Proposition 2.2 combined with 2.7 implies
$$
c_{x}(\mathcal{I}+\mathcal{J})=c_{(x,x)}((\mathcal{I}\oplus \mathcal{J})_{1\triangle})\leq c_{(x,x)}(\mathcal{I}\oplus \mathcal{J})=c_{x}(\mathcal{I})+c_{x}(\mathcal{J}).
$$
Since $(f+g) \subset(f)+(g)$, inequality 1.4 (2) also shows that
$$
c_{x}(f+g)\ \leq c_{x}((f)+(g))\leq c_{x}(f)+c_{x}(g).
$$
$$
\square 
$$
{\it Remark} 2.10.-If $f(x_{1}, \ldots, x_{n})$, resp. $g(y_{1}, \ldots, y_{n})$, are holomorphic near $0\in \mathbb{C}^{n}$, resp.
$0\in \mathbb{C}^{m}$, and such that $f(0)=g$ (0) $=0$, we have the equality
$$
c_{0}\ (f(x_{1}, \ldots, x_{n})+g(y_{1}, \ldots, y_{m}))=\min\{1, c_{0}(f)+c_{0}(g)\}.
$$
This result is proved in [3], vol. II, sec. 13.3.5 in the case of isolated singularities. Another proof,
using the computation of $c_{0}$ via a resolution as in Proposition 1.7, is given in [25]. It can also be
reduced to Proposition 2.7 through a $\log$ resolution of either $f$ or $g$.
3. Semi-continuity of holomorphic singularity exponents
We first give a new proof (in the spirit of this work) of the semi-continuity theorem
of Varchenko [47] conceming leading zeroes of Bemstein-Sato polynomials attached to
singularities holomorphic functions (see also Lichtin [29]).
THEOREM 3.1 [47].-{\it Let} $X$ {\it be a complex manifold and} $S$ {\it a reduced complex space}.
{\it Let} $f(x, s)$ {\it be a holomorphic function on} $X\times S$. {\it Then for any} $x_{0}\in X$, {\it the function}
$s \mapsto c_{x\mathrm{o}}(f_{|X\times\{s\}})$ {\it is lower semi-continuous for the holomorphic Zariski topology on S. It even}
{\it satisfies the following much stronger property}: {\it for any} $s_{0}\in S$, {\it one has}
\begin{center}
(3.1.1)   $c_{x_{0}}(f_{|X\times\{s\}})\geq c_{x_{0}}(f_{|X\times\{s_{0}\}})$
\end{center}
{\it on a holomorphic Zariski neighborhood of} $s_{0}(i.e$. {\it the complement in} $S$ {\it of an analytic subset of}
$S$ {\it disjointfrom} $s_{0}$).
{\it Proof}.-Observe that if $f_{|X\times\{s_{0}\}}$ is identically zero, then $c_{x_{0}}(f_{|X\times\{s_{0}\}})=0$ and there is
nothing to prove; thus we only need to consider those $s$ such that $f_{|X\times\{s\}}\not\equiv 0$. We may of
course assume that $X=B$ is a ball in $\mathbb{C}^{n}$ and $x_{0}=0$. Let $Y=B\times S,\ D=\mathrm{div}f$ and $\mu:\overline{Y}\rightarrow Y$
a $\log$ resolution of $(Y, D)$. After possibly shrinking $B$ a little bit, there is a Zariski dense open
set $S_{1}\subset S$ such that if $s \in S_{1}$, the corresponding fiber
$$
\mu_{s}:\overline{Y}_{s}\rightarrow B\times\{s\}
$$
is a $\log$ resolution of $(B, \mathrm{div}f_{|B\times\{s\}})$. Moreover, we may assume that the numerical invariants
$a_{i},\ b_{i}$ attached to $\mu_{s}:\overline{Y}_{s}\rightarrow B$ as in Proposition 1.7 also do not depend on $s$. In particular,
by (1.7.2), $c_{0}(f_{|B\times\{s\}})$ is independent of $s \in S_{1}$.
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By induction on the dimension of $S$, we obtain a stratifcation $S =\cup S_{i}$ (where each $S_{i}$ is
a Zariski dense open subset of a closed complex subspace of $S$) such that $c_{0}(f_{|B\times\{s\}})$ only
depends on the stratum containing $s$. Thus (3.1.1) reduces to semi-continuity with respect to the
classical topology (considering a 1-dimensional base is enough, so we may assume the base to be
nonsingular as well). Ifwe put $\varphi=\log|f|$, this is a special case ofthe following lemma, which
is essentially equivalent to the Main Theorem of [37]. Here, we would like to point out that this
result (which we knew as early as end of 1995) can be obtained as a direct consequence of the
Ohsawa-Takegoshi theorem [35].
LEMMA 3.2.-{\it Let} $\Omega\subset \mathbb{C}^{n}$ {\it and} $S \subset \mathbb{C}^{p}$ {\it be boundedpseudoconwex open sets. Let} $\varphi(x, s)$ {\it be}
{\it a Holder} $psh$ {\it function on} $\Omega\times S$ {\it and let} $ K\subset\Omega$ {\it be a compact set. Then}
(1) $s \mapsto c_{K}(\varphi(\cdot, s))$ {\it is lower semi-continuous for the classical topology on} $S$.
(2) {\it If} $s_{0}\in S$ {\it and} $c<c_{K}(\varphi(\cdot, s_{0}))$, {\it there exists a neighborhood} $U$ {\it of} $K$ {\it and a uniform bound}
$$
\int_{U}\mathrm{e}^{-2c\varphi(x,s)}\mathrm{d}V(x)\leq M(c)
$$
{\it for} $s$ {\it in a neighborhood of} $s_{0}$.
{\it Proof}.-We use the $L^{2}$ extension theorem of [35], following an idea of Angehm and Siu [1].
However, the ``effective'' part (2) requires additional considerations. Notice that it is enough to
prove (2), since (1) is a trivial consequence. By shrinking $\Omega$ and $S$, we may suppose that $\mathrm{e}^{\varphi}$ is
HOlder continuous of exponent $\alpha$ on the whole of $\Omega\times S$ and that
$$
\int_{\Omega}\mathrm{e}^{-2c\varphi(x,s_{0})}\mathrm{d}V(x)<+\infty.
$$
Let $k$ be a positive integer. We set
$$
\psi_{k,s}(x, t)=2c\varphi(x, s +(kt)^{k}(s_{0}-s))\ on\ \Omega\times D,
$$
where $D\subset \mathbb{C}$ is the unit disk. Then $\psi$ is well defned on $\Omega\times D$ if $s$ is close enough to $s_{0}$.
Since $\psi(x, 1/k)=\varphi(x, s_{0})$, we obtain by Theorem 2.1 the existence of a holomorphic function
$F_{k,s}(x, t)$ on $\Omega\times D$ such that $F_{k,s}(x, 1/k)=1$ and
\begin{center}
(3.2.3)   $\displaystyle \int_{\Omega\times D}|F_{k,s}(x, t)|^{2}\mathrm{e}^{-\psi_{k\mathrm{s}}(x,t)}\mathrm{d}V(x)\mathrm{d}V(t)\leq C_{1}$
\end{center}
with $C_{1}$ independent of $k,\ s$ for $|s -s_{0}|<\delta k^{-k}$. As $\psi_{k,s}$ admits a global upper bound
independent of $k,\ s$, the family $(F_{k,s})$ is a normal family. It follows from the equality
$F_{k,s}(x, 1/k)=1$ that there is a neighborhood $U$ of $K$ and a neighborhood $D(\mathrm{O}, e)$ of $0$ in $\mathbb{C}$
such that $|F_{k,s}|\geq 1/2$ {\it on} $U\times D(0, \in)$ if $k$ is large enough. A change of variable $t=k^{-1}\tau^{1/k}$
in (3.2.3) then yields
$$
\int_{U\times D(0,(k\in)^{k})}\frac{\mathrm{e}^{-2c\varphi(x,s+\tau(s_{0}-s))}}{|\tau|^{2(1-1/k)}}\mathrm{d}V(x)\mathrm{d}V(\tau)\leq 4k^{4}C_{1}.
$$
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As in the {\it proof} of Theorem 2.5, we get by the HOlder continuity of $\mathrm{e}^{\varphi}$ an upper bound
$$
\mathrm{e}^{2c\varphi(x,s+\tau(s_{0}-s))}\leq C_{2}(\mathrm{e}^{2c\varphi(x,s)}+|\tau|^{2c\alpha})
$$
with a constant $C_{2}$ independent of $s$. Hence, for $ k\geq 1/\in$, we find
$$
\int_{U\times D}\frac{1}{(\mathrm{e}^{2c\varphi(x,s)}+|\tau|^{2c\alpha})|\tau|^{2(1-1/k)}}\mathrm{d}V(x)\mathrm{d}V(\tau)\leq C_{3}(k).
$$
By restricting the integration to a family of disks $|\tau|<C_{4}\mathrm{e}^{\alpha^{-1}\varphi(x,s)}$ (with $C_{4}$ so small that the
radius is $\leq 1$), we infer
$$
\int_{U}\mathrm{e}^{-2(c-1/k\alpha)\varphi(x,s)}\mathrm{d}V(x)\leq C_{5}(k).
$$
Since $ c-1/k\alpha$ can be taken arbitrarily close to $c_{K}(\varphi)$, this concludes the proof. $\square $
We can now prove the qualitative part ofthe semi-continuity theorem, in the holomorphic case.
Theorem 3.3.-{\it Let} $X$ {\it be a complex manifold and} $K\subset X$ {\it a compact subset. Then}
$f\mapsto c_{K}(f)$ {\it is lower semi-continuous on} $\mathcal{O}(X)$ {\it with respect to the topology of uniform}
{\it convergence on compact subsets. More explicitly, for every nonzero holomorphic function} $f$,
{\it for every compact set} $L$ {\it containing} $K$ {\it in its interior and every} $\xi j >0$, {\it there is a number}
$\delta=\delta(f, \xi:, K, L)>0$ {\it such that}
\begin{center}
(3.3.1)   $\displaystyle \sup_{L}|g -f|<\delta\Rightarrow c_{K}(g) \geq c_{K}(f)-\xi j$.
\end{center}
$Proof-$ As a first step we reduce (3.3.1) to the special case when $K$ is a single point. Assume
that (3.3.1) fails. Then there is a sequence of holomorphic functions $f_{i}\in \mathcal{O}(X)$ converging
uniformly to $f$ on $L$, such that
$$
c_{K}(f_{i})<c_{K}(f)-\xi j.
$$
By Proposition 1.2 we can choose for each $i$ a point $a_{i}\in K$ such that $c_{a_{i}}(f_{i})<c_{K}(f)-\xi j$. By
passing to a subsequence we may assume that the points $a_{i}$ converge to a point $a \in K$. Take a
local coordinate system on $X$ in a neighborhood of $a$. Consider the functions $F_{i}$ defined by
$$
F_{i}(x)=f_{i}(x+a_{i}-a)
$$
on a small coordinate ball $\overline{B}(a, r)\subset L^{\mathrm{o}}$. These functions are actually well defined for $i$ large
enough (choose $\xi j$ so that $\overline{B}$ ( $a,\ r+\xi:)\subset L$ and $i$ so large that $|a_{i}-a| <\xi:$). Then $F_{i}$ converges to
$f$ on $\overline{B}(a, r)$, but
$$
c_{a}(F_{i})=c_{a_{i}}(f_{i})<c_{K}(f)-\xi:\leq c_{a}(f)-\xi j.
$$
Therefore, to get a contradiction, we only need proving Theorem 3.3 in case $K=\{a\}$ is a single
point. Again we can change notation and assume that $X$ is the unit ball and that our point is the
origin $0$.
In the second step we reduce the lower semi-continuity of $c_{0}(f)$ to polynomials of bounded
degree. For a given holomorphic function $f$ let $P_{k}$ denote the degree $\leq k$ part of its Taylor
series. The subbaditivity property of Theorem 2.9 implies $|c_{0}(f)-c_{0}(p_{k})|\leq c_{0}(f-p_{k})$. As
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$|f(z)-p_{k}(z)|=\mathrm{O}(|z|^{k+1})$, the function $|f-p_{k}|^{-2c}$ is not integrable for $c\geq n/(k+1)$. From
this, it follows that $c_{0}(f-p_{k})\leq n/(k+1)$, hence
\begin{center}
(3.3.2)   $|c_{0}(f)-c_{0}(p_{k})|\displaystyle \leq\frac{n}{k+1}$.
\end{center}
Now, if $(f_{i})$ converges uniformly to $f$ on a given neighborhood $U\subset \mathrm{C}^{\mathrm{n}}$ of $0$, the degree $\leq k$
part $p_{i,k}$ converges to $p_{k}$ in the finite dimension space $\mathbb{C}[z_{1}, \ldots, z_{n}]_{k}$ of polynomials of total
degree $\leq k$. Let us view polynomials
$$
P(z, s)\ =\sum_{|\alpha|\leq k}s_{\alpha}z^{\alpha}\in \mathbb{C}[z_{1}, \ldots, z_{n}]_{k}
$$
as functions of their coefficients $s =(s_{\alpha})$. By Theorem 3.1, we know that the function $s \mapsto$
$c_{0}(P(\cdot, s))$ is lower semi-continuous. Hence we get
$c_{0}(p_{i,k})>c_{0}(p_{k})-\in\overline{2}$ for $i>i(k, \xi;)$ large enough,
and thanks to (3.3.2) this implies
$$
c_{0}(f_{i})>c_{0}(f)--\frac{2n}{k+1}>c_{0}(f)-\xi j\in\overline{2}
$$
by choosing $ k\geq 4n/\in.\ \square $
In fact, we would like to propose the following much stronger lower semi-continuity
conjecture:
CONJECTURE 3.4. {\it Notation as in Theorem} 3.3. {\it For every nonzero holomorphic function} $f$,
{\it there is a number} $\delta=\delta(f, K, L)>0$ {\it such that}
$$
\sup_{L}|g\ -f|<\delta\Rightarrow c_{K}(g)\ \geq c_{K}(f).
$$
{\it Remark} 3.5. -There is an even more striking conjecture about the numbers $c_{K}(f)$, namely,
that the set
$$
C=\{c_{0}(f)|f\in \mathcal{O}_{\mathbb{C}^{n},0}\}\subset \mathbb{R}
$$
satisfies the ascending chain condition (cf. [39]; [23], 18.16): any convergent increasing sequence
in $C$ should be stationary. This conjecture and Theorem 3.3 together would imply the stronger
form 3.4. Notice on the other hand that there do exist non stationary decreasing sequences in $C$
by (1.4.8) 1
4. Multiplier ideal sheaves and holomorphic approximations of psh singularities
The most important concept relating psh functions to holomorphic objects is the concept of
{\it multiplier ideal sheaf}, which was already considered implicitly in the work of Bombieri [6],
Skoda [44] and Siu [40]. The precise final formalization has been fixed by Nadel [33].
1 It has been recently observed by Phong and Sturm [38], $\mathrm{m}$ their study of mtegrals of the form $\displaystyle \int|f|^{-s}$, that the
ascendmg cham condition holds $\mathrm{m}$ complex dimension 2. Algebraic geometers seem to have been aware for some time
ofthe correspondmg algebraic geometric statement.
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THEOREM AND definition 4.1 ([33,34], see also [9,12]).-{\it If} $\varphi\in \mathcal{P}(X)$ {\it is a} $psh$ {\it function}
{\it on a complex manifold} $X$, {\it the multiplier ideal} $sheaf\mathcal{I}(\varphi)\subset \mathcal{O}_{X}$ {\it is defined by}
$$
\Gamma(U,\mathcal{I}(\varphi))=\{f\in \mathcal{O}_{X}(U);|f|^{2}\mathrm{e}^{-2\varphi}\in L_{1\mathrm{o}\mathrm{c}}^{1}(U)\}
$$
{\it for every open set} $U\subset X$. {\it Then} $\mathcal{I}(\varphi)$ {\it is a coherent ideal sheaf} $\mathcal{O}_{X}$.
The proofthat $\mathcal{I}(\varphi)$ is coherent is a rather simple consequence ofHOrmander's $L^{2}$ estimates,
together with the strong Noetherian property of coherent sheaves and the Krull lemma. When the
psh function $\varphi$ is defined from holomorphic functions as in 2.4, it is easy to see that $\mathcal{I}(\varphi)$ can be
computed in a purely algebraic way by means oflog resolutions. The concept ofmultiplier ideal
sheafplays a very important role in algebraic geometry, e.g. in Nadel's version ofthe Kawamata-
Viehweg vanishing theorem or in Siu's proof [43] ofthe big Matsusaka theorem.
We now recall the technique employed in [11] and [13] to produce effective bounds for the
approximation of psh functions by logarithms of holomorphic functions. The same technique
produces useful comparison inequalities for the singularity exponents of a psh function and its
associated multiplier ideal sheaves.
THEOREM 4.2.-{\it Let} $\varphi$ {\it be a plurisubharmonic function on a bounded open set} $\Omega\subset \mathbb{C}^{n}$. {\it For}
{\it every real number} $m>0$, {\it let} $\mathcal{H}_{m\varphi}(\Omega)$ {\it be the Hilbert space ofholomorphicfunctions} $f$ {\it on} $\Omega$ {\it such}
{\it that} $\displaystyle \int_{\Omega}|f|^{2}\mathrm{e}^{-2m\varphi}\mathrm{d}V<+\infty$ {\it and let} $\displaystyle \psi_{m}=\frac{1}{2m}\log\sum|g_{m,k}|^{2}$ {\it where} $(g_{m,k})$ {\it is an orthonormal}
{\it basis} $of\mathcal{H}_{m\varphi}(\Omega)$. {\it Then}:
(1) {\it There are constants} $C_{1},\ C_{2}>0$ {\it independent of} $m$ {\it and} $\varphi$ {\it such that}
$$
\varphi(z)-\frac{C_{1}}{m}\leq\psi_{m}(z)\leq\ \sup\ \varphi(\zeta)+\frac{1}{m}\log\frac{C_{2}}{r^{n}}
$$
$$
|\zeta-z|<r
$$
{\it for every} $ z\in\Omega$ {\it and} $r<d(z, \partial\Omega)$. {\it In particular}, $\psi_{m}$ {\it converges to} $\varphi$ {\it pointwise and in} $L_{1\mathrm{o}\mathrm{c}}^{1}$
{\it topology on} $\Omega$ {\it when} $ m\rightarrow+\infty$ {\it and}
(2) {\it the Lelong numbers of} $\varphi$ {\it and} $\psi_{m}$ {\it are related by}
$\displaystyle \nu(\varphi, z)-\frac{n}{m}\leq\nu(\psi_{m}, z)\leq\nu(\varphi, z)$ {\it for every} $ z\in\Omega$.
(3) {\it For every compact set} $ K\subset\Omega$, {\it the Arnold multiplicity of} $\varphi,\ \psi_{m}$ {\it and ofthe multiplier ideal}
{\it sheaves} $\mathcal{I}(m\varphi)$ {\it are related by}
$$
\lambda_{K}(\varphi)-\frac{1}{m}\leq\lambda_{K}(\psi_{m})=\frac{1}{m}\lambda_{K}(\mathcal{I}(m\varphi))\leq\lambda_{K}(\varphi).
$$
{\it Proof}. $-(1)$ Note that $\displaystyle \sum|g_{m,k}(z)|^{2}$ is the square of the norm of the evaluation linear form
$f\mapsto f(z)$ on $\mathcal{H}_{m\varphi}(\Omega)$. As $\varphi$ is locally bounded above, the $L^{2}$ topology is actually stronger
than the topology of uniform convergence on compact subsets of $\Omega$. It follows that the series
$\displaystyle \sum|g_{m,k}|^{2}$ converges uniformly on $\Omega$ and that its sum is real analytic. Moreover we have
$$
\psi_{m}(z)=\sup\underline{1}\log|f(z)|
$$
$$
f\in B(1)m
$$
where $B(1)$ is the unit ball of $\mathcal{H}_{m\varphi}(\Omega)$. For $r<d(z, \partial\Omega)$, the mean value inequality applied to
the psh function $|f|^{2}$ implies
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$$
|f(z)|^{2}\leq\frac{1}{\pi^{n}r^{2n}/n!}\int_{|\zeta-z|<r}|f(\zeta)|^{2}\mathrm{d}\lambda(\zeta)
$$
$$
\leq\frac{1}{\pi^{n}r^{2n}/n!}\exp(2m\sup_{|\zeta-z|<r}\varphi(\zeta))\int_{\Omega}|f|^{2}\mathrm{e}^{-2m\varphi}\mathrm{d}\lambda.
$$
If we take the supremum over all $f\in B(1)$ we get
$$
\psi_{m}(z)\leq_{1}\sup_{\zeta-z|<r}\varphi(\zeta)+\frac{1}{2m}\log\frac{1}{\pi^{n}r^{2n}/n!}
$$
and the right hand inequality in (1) is proved. Conversely, the Ohsawa-Takegoshi extension
theorem applied to the 0-dimensional subvariety $\{z\}\subset\Omega$ shows that for any $a \in \mathbb{C}$ there is a
holomorphic function $f$ on $\Omega$ such that $f(z)=a$ and
$$
\int_{\Omega}|f|^{2}\mathrm{e}^{-2m\varphi}\mathrm{d}\lambda\leq C_{3}|a|^{2}\mathrm{e}^{-2m\varphi(z)},
$$
where $C_{3}$ only depends on $n$ and diam $\Omega$. We fx $a$ such that the right hand side is 1. This gives
the left hand inequality
\begin{center}
(4.2.4)   $\displaystyle \psi_{m}(z)\geq\frac{1}{m}\log|a| =\displaystyle \varphi(z)-\frac{\log C_{3}}{2m}$.
\end{center}
(2) The above inequality (4.2.4) implies $\nu(\psi_{m}, z)\leq\nu(\varphi, z)$. In the opposite direction, we fnd
$$
\sup_{|x-z|<r}\psi_{m}(x)\leq\sup_{|\zeta-z|<2r}\varphi(\zeta)+\frac{1}{m}\log\frac{C_{2}}{r^{n}}.
$$
Divide by $\log r$ and take the limit as $r$ tends to $0$. The quotient by $\log r$ ofthe supremum ofa psh
function over $B(x, r)$ tends to the Lelong number at $x$. Thus we obtain
$$
\nu(\psi_{m}, x)\geq\nu(\varphi, x)-\frac{n}{m}.
$$
(3) Inequality (4.2.4) already yields $\lambda_{K}(\psi_{m})\leq\lambda_{K}(\varphi)$. Moreover, the multiplier ideal sheaf
$\mathcal{I}(m\varphi)$ is generated by the sections in $\mathcal{H}_{m\varphi}(\Omega)$ (as follows from the proof that $\mathcal{I}(m\varphi)$ is
coherent), and by the strong Noetherian property, it is generated by fnitely many functions
$(g_{m,k})_{0\leq k\leq k_{0}(m)}$ on every relatively compact open set $\Omega'\subset\Omega$. It follows that we have a lower
bound ofthe form
\begin{center}
(4.2.4)   $\displaystyle \psi_{m}(z)-C_{4}\leq\frac{1}{2m}\log\sum_{0\leq k\leq k_{0}(m)}|g_{m,k}|^{2}\leq\psi_{m}(z)$ on $\Omega'$.
\end{center}
By choosing $\Omega'\supset K$, we infer $\displaystyle \lambda_{K}(\psi_{m})=\frac{1}{m}\lambda_{K}(\mathcal{I}(m\varphi)$. If $\lambda >\lambda_{K}(\psi_{m})$, i.e., $1/m\lambda<$
$c_{K}(\mathcal{I}(m\varphi))$, and if $U\subset\Omega'$ is a sufficiently small open neighborhood of $K$, the HOlder inequality
for the conjugate exponents $ p=1+m\lambda$ and $q=1+(m\lambda)^{-1}$ yields
$\displaystyle \int_{U}\mathrm{e}^{-2mp^{-1}\varphi}\mathrm{d}V=\int_{U}(\sum_{0\leq k\leq k_{0}(m)}|g_{m,k}|^{2}\mathrm{e}^{-2m\varphi})^{1/p}(\sum_{0\leq k\leq k_{0}(m)}|g_{m,k}|^{2})^{-1/qm\lambda}\mathrm{d}V$
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\begin{center}
(4.2.6)   $\displaystyle \leq(k_{0}(m)+1)^{1/p}(\int_{U}(\sum_{0\leq k\leq k_{0}(m)}|g_{m,k}|^{2})^{-1/m\lambda}\mathrm{d}V)^{1/q}<+\infty$.
\end{center}
(The estimate in the last line uses the fact that
$$
\int_{U}|g_{m,k}|^{2}\mathrm{e}^{-2m\varphi}\mathrm{d}V\leq\int_{\Omega}|g_{m,k}|^{2}\mathrm{e}^{-2m\varphi}\mathrm{d}V=1.)
$$
This implies $c_{K}(\varphi)\geq mp^{-1}$, i.e., $\lambda_{K}(\varphi)\leq p/m=\lambda +1/m$. As $\lambda >\lambda_{K}(\psi_{m})$ was arbitrary, we
get $\lambda_{K}(\varphi)\leq\lambda_{K}(\psi_{m})+1/m$ and (3) follows. $\square $
The ``approximation theorem'' 4.2 allows to extend some results proved for holomorphic
functions to the case ofpsh functions. For instance, we have:
PROPOSITION 4.3.-{\it Let} $\varphi\in \mathcal{P}(X),\ \psi\in \mathcal{P}(Y)$ {\it bepsh functions on complex manifolds} $X,\ Y$,
{\it and let} $K\subset X,\ L\subset Y$ {\it be compact subsets. Then}:
(1) {\it For all positive real numbers} $c',\ c''$ {\it with} $c'>c_{K}(\varphi)>c''$ ({\it if any}) {\it and every sufficiently}
{\it small neighborhood} $U$ {\it of} $K$, {\it there is an estimate}
$$
C_{1}r^{2c'}\leq\mu_{U}(\{\varphi<\log r\})\leq C_{2}r^{2c''},\ \forall r<r_{0}
$$
{\it for some} $r_{0}>0$ {\it and} $C_{1}=C_{1}(c'),\ C_{2}=C_{2}(c'')$.
(2) $c_{K\times L}(\displaystyle \max(\varphi(x), \psi(y)))=c_{K}(\varphi)+c_{L}(\psi)$.
(3) {\it If} $X=Y$, {\it then} $c_{x}(\displaystyle \max(\varphi, \psi))\leq c_{x}(\varphi)+c_{x}(\psi)$ {\it for all} $x\in X$.
{\it Proof}. $-(1)$ The upper estimate is clear, since
$$
 r^{-2c''}\mu_{U}(\{\varphi<\log r\})\leq\int_{U}\mathrm{e}^{-2c''\varphi}\mathrm{d}V<+\infty
$$
{\it for} $U\subset K$ sufficiently small. In the other direction, we have an estimate
$$
\mu_{U}(\{\psi_{m}<\log r\})\geq C_{1,m}r^{2c_{K}(\psi_{m})}
$$
by Proposition 1.7 (3) and (4.2.5). As $\varphi\leq\psi_{m}+C_{2,m}$ for some constant $C_{2,m}>0$, we get
$$
\{\varphi<\log r\}\supset\{\psi_{m}<\log r-C_{2,m}\},
$$
and as $c_{K}(\psi_{m})$ converges to $c_{K}(\varphi)$ by 4.2 (3), the lower estimate of $\mu_{U}(\{\varphi<\log r\})$ follows.
(2), (3) can be derived from (1) exactly as for the holomorphic case in Proposition 2.7
and Theorem 2.9. It should be observed that 4.3(1) expresses a highly nontrivial ``regularity
property'' of the growth of volumes $\mu_{U}(\{\varphi<\log r\})$ when $\varphi$ is a psh function (when $\varphi$ is an
arbitrary measurable function, $v(r)=\mu_{U}(\{\varphi<\log r\})$ is just an arbitrary increasing function
with $\displaystyle \lim_{r\rightarrow 0}v(r)=0).\ \square $
{\it Remark} 4.4. -In contrast with the holomorphic case 1.7 (3), the upper estimate $\mu_{U}(\{\varphi<$
$\log r\})\leq C_{2}r^{2c''}$ does not hold with $c''=c_{K}(\varphi)$, when $\varphi$ is an arbitrary psh function. A simple
example is given by $\varphi(z)=\chi$ olog $|z|$ where $\chi:\mathbb{R}\rightarrow \mathbb{R}$ is a convex increasing function such that
$\chi(t)\sim t$ as $ t\rightarrow-\infty$, but $\mathrm{e}^{\chi(r)}\eta^{6}r$ as $r\rightarrow 0$, e.g. such that $\chi(t)=t-\log|t|)$ when $t<0$. On the
other hand, the lower estimate $\mu_{U}(\{\varphi<\log r\})\geq C_{1}r^{2c'}$ seems to be still true with $c'=c_{K}(\varphi)$,
although we cannot prove it.
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5. Semi-continuity of psh singularity exponents
We are now in a position to prove our main semi-continuity theorem.
5.1. {\it Proof of Theorem 0.2}. -Let $\Omega\subset \mathbb{C}^{n}$ be a bounded pseudoconvex open set and let
$\varphi_{j}\in \mathcal{P}(\Omega)$ be a sequence ofpsh functions converging to a limit $\varphi\in \mathcal{P}(\Omega)$ in the weak topology
distributions. In fact, this already implies that $\varphi_{j}\rightarrow\varphi$ almost everywhere and in $L_{1\mathrm{o}\mathrm{c}}^{1}$ topology;
to see this, we observe that the coefficients of $T_{j}:=\mathrm{dd}^{\mathrm{c}}\varphi_{j}$ are measures converging to those of
$ T=\mathrm{dd}^{\mathrm{c}}\varphi$ in the weak topology measures; moreover $\varphi_{j}$ and $\varphi$ can be recovered from $T_{j}$ and
$T$ by an integral formula involving the Green kemel; we then use the well known fact that integral
operators involving a $L^{1}$ kemel define continuous (and even compact) operators from the space
of positive measures equipped with the weak topology, towards the space of $L^{1}$ functions with
the strong $L^{1}$ topology.
Fix a compact set $ K\subset\Omega$. By the process described in Theorem 4.2, we get for each $m\in \mathbb{N}^{\star}$
an orthonormal basis $(g_{j,m,k})_{k\in \mathbb{N}}$ of $\mathcal{H}_{m\varphi_{j}}(\Omega)$, such that
\begin{center}
(5.1.1)   $\displaystyle \varphi_{j}(z)-\frac{C_{1}}{m}\leq\frac{1}{2m}\log\sum_{k\in \mathbb{N}}|g_{j,m,k}|^{2}\leq 1\zeta-z|<r\sup\varphi_{j}(\zeta)+\frac{1}{m}\log\frac{C_{2}}{r^{n}}$
\end{center}
for every $ z\in\Omega$ and $r<d(z, \partial\Omega)$. In particular, all sequences $(g_{j,m,k})_{j\in \mathbb{N}}$ are uniformly bounded
from above on every compact subset of $\Omega$. After possibly extracting a subsequence, we may
assume that all $g_{j,m,k}$ converge to a limit $g_{m,k}\in \mathcal{O}(\Omega)$ when $ j\rightarrow+\infty$. Thanks to (5.1.1) we
Iind in the limit
$$
\varphi(z)-\frac{C_{1}}{m}\leq\frac{1}{2m}\log\sum_{k\in \mathbb{N}}|g_{m,k}|^{2}\leq_{1}\sup_{\zeta-z|<r}\varphi(\zeta)+\frac{1}{m}\log\frac{C_{2}}{r^{n}}.
$$
Fix a relatively compact open subset $\Omega'\subset\Omega$ containing $K$. By the strong Noetherian property
already used for (4.2.5), there exist an integer $k_{0}(m)$ and a constant $C_{4}(m)>0$ such that
\begin{center}
$\displaystyle \varphi(z)-C_{4}(m)\leq\frac{1}{2m}\log\sum_{0\leq k\leq k_{0}(m)}|g_{m,k}|^{2}$ on $\Omega'$.
\end{center}
Now, for $c<c_{K}(\varphi)$, there is a neighborhood $U$ of $K$ on which
$$
\int_{U}(\sum_{0\leq k\leq k_{0}(m)}|g_{m,k}|^{2})^{-c/m}\mathrm{d}V\leq \mathrm{e}^{2cC_{4}(m)}\int_{U}\mathrm{e}^{-2c\varphi}\mathrm{d}V<+\infty.
$$
Take (without loss of generality) $m\geq 2c_{K}(\varphi)$. Then $c/m<1/2$ and formula 1.5.1 shows that
there is a linear combination $\displaystyle \sum_{0\leq k\leq k_{0}(m)}\alpha_{m,k}g_{m,k}$ with $\alpha=(\alpha_{m,k})$ in the unit sphere of
$\mathbb{C}^{k_{0}(m)+1}$, such that
$$
\int_{U}|\sum_{0\leq k\leq k_{0}(m)}\alpha_{m,k}g_{m,k}|^{-2c/m}\mathrm{d}V\leq C_{5}(m)\int_{U}\mathrm{e}^{-2c\varphi}\mathrm{d}V<+\infty,
$$
where $C_{5}(m)$ is a constant depending possibly on $m$. By construction,
$$
f_{j,m}=\sum_{0\leq k\leq k_{0}(m)}\alpha_{k,m}g_{j,m,k}
$$
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is an element of the unit sphere in $\mathcal{H}_{m\varphi_{j}}(\Omega)$, and $f_{j,m}$ converges uniformly on $\Omega$ {\it to} $f_{m}=$
$\displaystyle \sum\alpha_{m,k}g_{m,k}$ such that $\displaystyle \int_{U}|f_{m}|^{-2c/m}\mathrm{d}V<+\infty$. By Lemma 5.2 below, for any $c'<c$ and
$K\subset U'\subset U$, we have a uniform bound $\displaystyle \int_{U},\ |f_{j,m}|^{-2c'/m}\mathrm{d}V\leq C_{6}(m)$ for $j\geq j_{0}$ large enough.
Since $\displaystyle \int_{\Omega}|f_{j,m}|^{2}\mathrm{e}^{-2m\varphi_{j}}\mathrm{d}V=1$, the HOlder inequality for conjugate exponents $p=1+m/c'$,
$q=1+d/m$ yields
$$
\int_{U}\mathrm{e}^{-2mc'/(m+c')\varphi_{j}}\mathrm{d}V=\int_{U}(|f_{j,m}|^{2}\mathrm{e}^{-2m\varphi_{j}})^{c'/(m+c')}|f_{j,m}|^{-2_{C'}/(m+c')}\mathrm{d}V
$$
$$
\leq(\int_{U}|f_{j,m}|^{-2c'/m}\mathrm{d}V)^{m/(m+c')}\leq C_{7}(m)
$$
for $j\geq j_{0}$. Since $c,\ c'$ are arbitrary with $c'<c<c_{K}(\varphi)$, the exponent $mc'/(m+c')$ can be
taken to approach $c$ as closely as we want as $m$ gets large. Hence $c_{K}(\varphi_{j})>c_{K}(\varphi)-\xi j$ for
$j\geq j_{0}(\in)$ large enough. Moreover, by what we have seen above, if $c<c_{K}(\varphi)$ is fixed and
$0<\delta<c_{K}(\varphi)/c-1$, there exists $j_{1}(\delta)$ such that the sequence $(\mathrm{e}^{-2c\varphi_{j}})_{j\geq j_{1}(\delta)}$ is contained
in a bounded set of $L^{1+\delta}(U)$, where $U$ is a small neighborhood of $K$. Therefore
$$
U\cap\{>M\}\int_{\ominus^{-2c\varphi_{j}}}\mathrm{e}^{-2c\varphi_{j}}\mathrm{d}V\leq C_{8}M^{-\delta}
$$
for $j\geq j_{1}(\delta)$, with a constant $C_{8}$ independent of $j$. Since $\mathrm{e}^{-2c\varphi_{j}}$ converges pointwise to $\mathrm{e}^{-2c\varphi}$
on $\Omega$, an elementary argument based on Lebesgue's bounded convergence theorem shows that
$\mathrm{e}^{-2c\varphi_{j}}$ converges to $\mathrm{e}^{-2c\varphi}$ in $L^{1}(U).\ \square $
To complete the proof, we need only proving the following effective estimate for holomorphic
functions, which is a special case ofpart (3) in the Main Theorem.
LEMMA 5.2.-{\it Let} $\Omega\subset \mathbb{C}^{n}$ {\it be a bounded pseudoconvex open set, and let} $f_{i}\in \mathcal{O}(\Omega)$ {\it be a}
{\it sequence holomorphic functions converging uniformly to} $f\in \mathcal{O}(\Omega)$ {\it on every compact subset}.
{\it Fix a compact set} $ K\subset\Omega$ {\it and} $c<c_{K}(f)$. {\it Then there is a neighborhood} $U$ {\it of} $K$ {\it and a uniform}
{\it bound} $C>0$ {\it such that}
$$
\int_{U}|f_{i}|^{-2c}\mathrm{d}V\leq C
$$
{\it for} $i\geq i_{0}$ {\it sufficiently large}.
{\it Proof}.-We already know by Theorem 3.3 that $\displaystyle \int_{U}|f_{i}|^{-2c}\mathrm{d}V<+\infty$ for $U$ small enough
and $i$ large. Unfortunately, the proof given in Theorem 3.3 is not effective because it depends
(through the use of Hironaka's theorem in the proof of estimates 1.7(3) and (2.7.1)) on the use of
a sequence of $\log$ resolutions on which we have absolutely no control. We must in fact produce
an effective version of inequality (3.3.2).
The result of Lemma 5.2 is clearly local. Fix a point $x_{0}\in K$ (which we assume to be $0$ for
simplicity), real numbers $c',\ c''$ with $c<c''<c'<c_{K}(f)\leq c_{0}(f)$ and an integer $k$ so large that
$$
c<c''-\frac{n}{k+1}<c''<c'<c_{0}(f)-\frac{n}{k+1}.
$$
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Let $p_{k}$ be the truncation at order $k$ of the Taylor series of $f$ at the origin. As $ c_{0}(p_{k})\geq$
$c_{0}(f)-\displaystyle \frac{n}{k+1}>c'$ by (3.3.2), there is a small ball $B'=B$( $0$, {\it r}\'{o}) such that
$$
\int_{B}|p_{k}|^{-2c'}\mathrm{d}V<+\infty.
$$
Since the truncations $p_{i,k}$ of $f_{i,k}$ converge uniformly to $p_{k}$ on $\mathbb{C}^{n}$ as $ i\rightarrow+\infty$, Lemma 3.2
applied to the universal family of polynomials $P(z, s) =\displaystyle \sum_{|\alpha|\leq k}s_{\alpha}z^{\alpha}$ shows that for any ball
$B''\subset B'$, there is a constant $M\geq 0$ and an integer $i_{0}$ such that
$\displaystyle \int_{B'}|p_{i,k}|^{-2c''}\mathrm{d}V\leq M$ {\it for} $i\geq i_{0}$.
Let us write $p_{i,k}=f_{i}-g_{i,k}$ where $g_{i,k}$ consists of the sum of terms of degree $>k$ in the
Taylor expansion of $f_{i}$ at the origin. By the Ohsawa-Takegoshi theorem applied with the weight
function $\psi(x, y) =2c\log|f_{i}(x)-g_{i,k}(y) |$ on $B''\times B''$ and $L=$ diagonal of $\mathbb{C}^{n}\times \mathbb{C}^{n}$, there is
a holomorphic function $F_{i}$ on $B''\times B''$ such that $F_{i}(x, x)=1$ and
$$
\int_{B''\times B''}|F_{i}(x, y)\ |^{2}|f_{i}(x)-g_{i,k}(y)\ |^{-2c''}\mathrm{d}V(x)\mathrm{d}V(y)\ \leq C_{1}
$$
with a constant $C_{1}$ independent of $i$. The above $L^{2}$ estimate shows that $(F_{i})$ is bounded in $L^{2}$
norm on $B''\times B''$. Hence, there is a small ball $B=B(0, r_{0})\subset B''$ such that $|F_{i}(x, y) |\geq 1/2$
{\it on} $B\times B$ for all $i\geq i_{0}$, and
\begin{center}
(5.2.1)   $\displaystyle \int_{B\times B}|f_{i}(x)-g_{i,k}(y) |^{-2c''}\mathrm{d}V(x)\mathrm{d}V(y) \leq 4C_{1}$.
\end{center}
Moreover, we have a uniform estimate $|g_{i,k}(y)|\leq C_{2}|y|^{k+1}$ on $B$ with a constant $C_{2}$ independent
of $i$. By integrating (5.2.1) with respect to $y$ on the family of balls $|y| <(|f_{i}(x)|/2C_{2})^{1/(k+1)}$,
we find an estimate
\begin{center}
(5.2.2)   $\displaystyle \int_{B}|f_{i}(x)|^{2n/(k+1)-2c''}\mathrm{d}V(x)\leq C_{3}$.
\end{center}
As $c''-n/(k+1)>c$, this is the desired estimate. It is interesting to observe that the proofofthe
Main Theorem can now be made entirely independent of Hironaka's desingularization theorem.
In fact, the only point where we used it is in the inequality $c_{0}(p_{k})\displaystyle \geq c_{0}(f)-\frac{n}{k+1}$, which we
derived from Proposition 2.7. The latter inequality can however be derived directly from the
Ohsawa-Takegoshi theorem through estimates for $\displaystyle \int_{B\times B}|p_{k}(x)+g_{k}(y) |^{-2c}\mathrm{d}V(x)\mathrm{d}V(y).\ \square $
{\it Remark} 5.3. -It follows from the proof of Proposition 1.7 that the set of positive exponents
$c$ such that $|f|^{-2c}$ is summable on a neighborhood of a compact set $K$ is always an {\it open}
{\it interval}, namely ] $0,\ c_{K}(f)[$. We conjecture that the same property holds true more generally
for an arbitrary psh function $\varphi$ openness conjecture''); the openness conjecture is indeed true
in dimension 1, since we have the well known necessary and sufficient criterion
$$
\mathrm{e}^{-2\varphi}\in L_{1\mathrm{o}\mathrm{c}}^{1}(V(x_{0}))\Leftrightarrow\nu(\varphi, x_{0})<1
$$
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(as follows, e.g., from [44]). By using the Main Theorem, the openness conjecture would imply
the following stronger statement:
STRONG OPENNESS CONJECTURE 5.4. -{\it Let} $U'\Subset U\Subset X$ {\it be relatively compact open sets in}
{\it a complex manifold X. Let} $\varphi$ {\it be a} $psh$ {\it function on} $X$ {\it such that} $\displaystyle \int_{U}\mathrm{e}^{-\varphi}\mathrm{d}V<+\infty$. {\it Then there}
{\it exists} $\xi j =\xi j(\varphi, U, U')$ {\it such that for every} $\psi psh$ {\it on} $X$
$$
\Vert\psi-\varphi\Vert_{L^{1}(U)}<\xi j\ \Rightarrow\int_{U}\mathrm{e}^{-\psi}\mathrm{d}V<+\infty.
$$
{\it In other words, the integrability of} $\mathrm{e}^{-\varphi}$ {\it near a given compact set} $K$ {\it should be an open property}
{\it for the} $L_{1\mathrm{o}\mathrm{c}}^{1}(=weak)$ {\it topology on} $\mathcal{P}(X)$.
The main theorem only yields the weaker conclusion
$\displaystyle \int_{U}\mathrm{e}^{-(1-\delta)\psi}\mathrm{d}V<+\infty$ {\it for} $\Vert\psi-\varphi\Vert_{L^{1}(U)}<\xi j=\xi j(\varphi, U, U', \delta)$.
6. Existence of Kahler-Einstein metrics on Fano orbifolds
An {\it orbifold} is a complex variety $X$ possessing only quotient singularities, namely, every point
$x_{0}\in X$ has a neighborhood $U$ isomorphic to a quotient $\Omega/\Phi$ where $\Phi=\Phi_{x_{0}}$ is a finite group
acting holomorphically on a smooth open set $\Omega\subset \mathbb{C}^{n}$. Such an action can always be linearized,
so we may assume that $\Phi$ is a finite subgroup of GL $n(\mathbb{C})$ and $\Omega$ a $\Phi$-invariant neighborhood of $0$
(with $x_{0}$ being the image of $0$). We may also assume that the elements of $G$ distinct from identity
have a set of fixed points of codimension $\geq 2$ (otherwise, the subgroup generated by these is
a normal subgroup $N$ of $\Phi,\ \Omega/N$ is again smooth, and $\Omega/\Phi=(\Omega/N)/(\Phi/N))$. The structure
sheaf $\mathcal{O}_{X}$ (resp. the {\it m}-fold canonical sheaf $K_{X}^{\otimes m}$) is then defined locally as the direct image by
$\pi:\Omega\rightarrow U\simeq\Omega/\Phi$ ofthe subsheaf of $\Phi$-invariant sections ofthe corresponding sheaf on $\Omega$:
$$
\Gamma(V, \mathcal{O}_{X})=\Gamma(\pi^{-1}(V), \mathcal{O}_{\Omega})^{\Phi},\ \Gamma(V, K_{X}^{\otimes m})=\Gamma(\pi^{-1}(V), K_{\Omega}^{\otimes m})^{\Phi},
$$
for all open subsets $V\subset U$. There is always an integer $m_{0} (\mathrm{e}.\mathrm{g}. m_{0}=\# \Phi)$ such that $K_{\Omega}^{\otimes m_{0}}$
has $\Phi$-invariant local generating sections, and then clearly $K_{X}^{\otimes m}$ is an invertible $\mathcal{O}_{X}$-module
whenever $m$ is divisible by the lowest common multiple $\mu$ of the integers $m_{0}$ occurring in the
various quotients. Similarly, one can define on $U$ (and thus on $X$) the concepts ofK\"{a}hler metrics,
Ricci curvature form, etc, by looking at corresponding $\Phi$-invariant objects on $\Omega$. We say that a
compact orbifold $X$ is a {\it Fano orbifold} if $K_{X}^{-\mu}$ is ample, which is the same as requiring that
$K_{X}^{-\mu}$ admits a smooth hermitian metric with positive definite curvature. In that case, we define
the curvature of $K_{X}^{-1}$ to be $ 1/\mu$ times the curvature of $K_{X}^{-\mu}$. The integral of a differential form
on $X$ (say defined at least on $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$) is always computed upstairs, i.e. $\displaystyle \int_{\Omega/\Phi}\alpha=\frac{1}{\neq\Phi}\int_{\Omega}\pi^{\star}\alpha$.
DEFINITION 6.1. -A compact orbifold $X$ is said to be K\"{a}hler-Einstein if it possesses a
K\"{a}hler form $\omega =\displaystyle \frac{\mathrm{i}}{2\pi}\sum\omega_{jk}\mathrm{d}z_{j}\wedge \mathrm{d}\overline{z}_{k}$ satisfying the Einstein condition
Ricci $(\omega)=\lambda\omega$
for some real constant $\lambda$, where Ricci $(\omega)$ is the closed (1, 1)-form defined in every coordinate
patch by Ricci $(\displaystyle \omega)=-\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\log \det(\omega_{jk})$.
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Since Ricci $(\omega)$ is the curvature form of $K_{X}^{-1}=\det T_{X}$ equipped with the metric $\det\omega$, a
necessary condition for the existence of a K\"{a}hler-Einstein metric with constant $\lambda >0$ is that
$K_{X}^{-1}$ is ample, i.e., that $X$ is Fano. On the other hand, it is well known that not all Fano orbifolds
are K\"{a}hler-Einstein, even when they are smooth; further necessary conditions are required, e.g.
that the group ofautomorphisms Aut $(X)^{\mathrm{o}}$ is reductive ([32], [28]), and that the {\it Futaki invariants}
vanish [18]; for instance $\mathbb{P}^{2}$ blown up in 2 points has a non reductive group of automorphisms
and therefore is not K\"{a}hler-Einstein.
It is usually much harder to prove that a concretely given Fano orbifold is K\"{a}hler-Einstein.
Siu [41,42], and slightly later Tian [46] and Nadel [33,34], gave nice sufficient conditions
ensuring the existence ofa K\"{a}hler-Einstein metric; these conditions always involve the existence
of a sufficiently big group of automorphisms. Our goal here is to reprove Nadel's main result in
a more direct and conceptual way.
TECHNICAL SETTING 6.2.-We first briefy recall the main technical tools and notation in-
volved (see, e.g., [41] for more details). The anticanonical line bundle $K_{X}^{-1}$ is assumed to be am-
ple. Therefore it admits a smooth hermitian metric $h_{0}$ whose (1, 1)-curvature form $\displaystyle \theta_{0}=\frac{\mathrm{i}}{2\pi}D_{h_{0}}^{2}$
is positive defnite. Since $\theta_{0}\in c_{1}(X)$, the $\mathrm{Aubin}-\mathrm{Calabi}-\mathrm{Yau}$ theorem shows that there exists
a K\"{a}hler metric $\omega_{0}\in c_{1}(X)$ such that Ricci $(\omega_{0})=\theta_{0}$. [The $\mathrm{Aubin}-\mathrm{Calabi}-\mathrm{Yau}$ is still valid in
the orbifold case, because the proof depends only on local regularity arguments which can be
recovered by passing to a fnite cover, and global integral estimates which still make sense by the
remark preceding Defnition 6.1.] Since both $\theta_{0}$ and $\omega_{0}$ are in $c_{1}(X)$, we have
\begin{center}
(6.2.2)   $\displaystyle \omega_{0}=\theta_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}f$ for some $f\in C^{\infty}(X)$.
\end{center}
We look for a new K\"{a}hler form $\omega =\displaystyle \omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi$ in the same K\"{a}hler class as $\omega_{0}$, such that
Ricci $\omega =\omega$. Since Ricci $(\omega_{0})=\theta_{0}$, this is equivalent to
$-\displaystyle \frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\log(\det\omega)=\omega =\displaystyle \theta_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}(\varphi+f)=-\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\log(\det\omega_{0})+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}(\varphi+f)$,
that is,
$$
\partial\overline{\partial}(\log\frac{\det\omega}{\det\omega_{0}}+\varphi+f)=0,
$$
which in its tum is equivalent to the $\mathrm{Monge}-\mathrm{Amp}\grave{\mathrm{e}}\mathrm{re}$ equation
\begin{center}
(6.2.2)   $\displaystyle \log\frac{(\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi)^{n}}{\omega_{0}^{n}}+\varphi+f+C=0$,
\end{center}
where $C$ is a constant. Here, one can normalize $\varphi$ so that $\varphi$ is orthogonal to the 1-dimensional
space of constant functions in $L^{2}(X, \omega_{0})$, i.e., $\displaystyle \int_{X}\varphi\omega_{0}^{n}=0$. The usual technique employed to
solve (6.2.2) is the so-called ``continuity method''. The continuity method amounts to introducing
an extra parameter $t\in[0,1]$ and looking for a solution $(\varphi_{t}, C_{t})$ ofthe equation
\begin{center}
(6.2.3)   $\displaystyle \log\frac{(\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi_{t})^{n}}{\omega_{0}^{n}}+t(\varphi_{t}+f)+C_{t}=0,\ \displaystyle \int_{X}\varphi_{t}\omega_{0}^{n}=0$
\end{center}
as $t$ varies from $0$ to 1. Clearly $\varphi_{0}=0,\ C_{0}=0$ is a solution for $t=0$ and $(\varphi, C)=(\varphi_{1}, C_{1})$
provides a solution of our initial equation (6.2.2). Moreover, the linearization of the (nonlinear)
elliptic differential operator occuring in (6.2.3) is the operator
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\begin{center}
(6.2.4)   $(\displaystyle \psi, C)\frac{1}{2\pi}\triangle_{\omega_{t}}\psi+t\psi+C$,
\end{center}
where $\omega_{t}$ is the K\"{a}hler metric $\displaystyle \omega_{t}=\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi_{t}$ and $\triangle_{\omega_{t}}$ is the associated Laplace operator
(with negative eigenvalues). Eq. (6.2.3) is easily seen to be equivalent to
Ricci $(\omega_{t})=t\omega_{t}+(1-t)\theta_{0}$.
From this we infer Ricci $(\omega_{t})>t\omega_{t}$ for all $t<1$, and it then follows from the Bochner-Kodaira-
Nakano identity that all nonzero eigenvalues $\displaystyle \mathrm{of}-\frac{1}{2\pi}\triangle_{\omega_{t}}$ are $>t$ (this is clear directly for
$-\displaystyle \frac{1}{2\pi}\triangle_{\omega_{t}}$ acting on $(0,1)$-forms, and one uses the fact that $\overline{\partial}$ maps the A-eigenspace $E^{p,q}(\lambda)$
$\displaystyle \mathrm{of}-\frac{1}{2\pi}\triangle_{\omega_{t}}$ in bidegree $(p, q)$ into the corresponding eigenspace $E^{p,q+1}(\lambda))$. Then, thanks to
Schauder's estimates, (6.2.4) induces an isomorphism $C_{\perp}^{s+2}(X)\oplus \mathbb{R}\rightarrow C^{s}(X)$ where $s \in \mathbb{R}+\backslash \mathbb{N}$
and $C^{S}(X)$ (resp. $C_{\perp}^{s}(X)$) is the space real functions (resp. real functions orthogonal to con-
stants) of class $C^{s}$ on $X$. Let $\mathcal{T}\subset[0,1]$ be the set ofparameters $t$ for which (6.2.3) has a smooth
solution. By elliptic regularity for (nonlinear) PDE equations, the existence of a smooth solution
is equivalent to the existence of a solution in $C^{S}(X)$ for some $s >2$. It then follows by a standard
application ofthe implicit function theorem that $\mathcal{T}\cap[0,1$ [is an open subset ofthe interval $[0,1$ [.
Sufficient condition for closedness 6.3.-In order to obtain a solution for all times
$t\in[0,1]$, one still has to prove that $\mathcal{T}$ is {\it closed}. By the well-known theory of complex Monge-
Amp\`{e}re equations ([4], [50]), a sufficient condition for closedness is the existence of a uniform
a priori $C^{0}$-estimate $\Vert\overline{\varphi}_{t}\Vert_{C^{0}}\leq$ Const for the family of functions $\overline{\varphi}_{t}=t\varphi_{t}+C_{t},\ t\in \mathcal{T}$, occuring
in the right hand side of (6.2.3). A first observation is that
(6.3. 2) $\displaystyle \sup_{X}\varphi_{t}\leq$ Const, hence $\displaystyle \sup_{X}\overline{\varphi}_{t}\leq C_{t}+$ Const,
as follows from the conditions $\displaystyle \int_{X}\varphi_{t}\omega_{0}^{n}=0$ and $\displaystyle \frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi_{t}\geq-\omega_{0}$, by simple considerations of
potential theory. On the other hand, by [42, Prop. 2.1] or [46, Prop. 2.3], we have the Hamack-
type inequality
\begin{center}
(6.3.2)   $\displaystyle \sup_{X}(-\overline{\varphi}_{t})\leq(n +\in)\sup_{X}\overline{\varphi}_{t}+A_{\in}$,
\end{center}
where $\xi j >0$ and $A_{\in}$ is a constant depending only on $\xi j$. Hence $\displaystyle \sup_{X}(-\overline{\varphi}_{t})\leq(n +\in)C_{t}+A_{\in}'$
and we thus only need controlling the constants $C_{t}$ from above. Now, Eq. (6.2.3) implies
$$
\int_{X}\omega_{0}^{n}=\int_{X}(\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi_{t})^{n}=\int_{X}\mathrm{e}^{-\overline{\varphi}_{t}-tf}\omega_{0}^{n}.
$$
For $\gamma\in]0,1$ [, we easily infer from this and (6.3.2) that
$\displaystyle \int_{X}\omega_{0}^{n}\leq$ Const $\displaystyle \exp((1-\gamma)\sup_{X}(-\overline{\varphi}_{t}))\int_{X}\mathrm{e}^{-\gamma\overline{\varphi}_{t}}\omega_{0}^{n}$
$\leq$ Const $\displaystyle \mathrm{e}^{(1-\gamma)(n+\in)C_{t}}\int_{X}\mathrm{e}^{-\gamma\overline{\varphi}_{t}}\omega_{0}^{n}$
$\leq$ Const $\displaystyle \mathrm{e}^{-(\gamma-(1-\gamma)(n+\in))C_{t}}\int_{X}\mathrm{e}^{-\gamma t\varphi_{t}}\omega_{0}^{n}$.
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If $\displaystyle \gamma\in]\frac{n}{n+1},1$ [and $\xi j$ is small enough, we conclude that $C_{t}$ admits an upper bound ofthe form
$$
C_{t}\leq B_{\gamma}'\log\int_{X}\mathrm{e}^{-\gamma t\varphi_{t}}\omega_{0}^{n}+B_{\gamma}''
$$
where $B_{\gamma}'$ and $B_{\gamma}''$ depend only on $\gamma$. Hence closedness of $\mathcal{T}$ is equivalent to the uniform
boundedness ofthe integrals
\begin{center}
(6.3.3)   $\displaystyle \int_{X}\mathrm{e}^{-\gamma t\varphi_{t}}\omega_{0}^{n},\ t\in \mathcal{T}$,
\end{center}
for any choice of $\gamma\in$] $\displaystyle \frac{n}{n+1},1$ [.
This yields the following basic existence criterion due to Nadel [33,34].
$\mathrm{ExISTENCE}$ criterion FOR $\mathrm{KAHLER}-\mathrm{EINSTEIN}$ metrics $6.4.-LetX$ {\it be a Fano orbifold}
{\it dimension} $n$. {\it Let} $G$ {\it be a compact subgroup ofthe group complex automorphisms} $ofX$. {\it Then}
$X$ {\it admits a G-invariant Kdhler-Einstein metric, unless} $K_{X}^{-1}$ {\it possesses a G-invariant singular}
{\it hermitian metric} $h=h_{0}\mathrm{e}^{-\varphi}(h_{0}$ {\it being a smooth G-invariant metric and} $\varphi$ {\it a G-invariant}
{\it function in} $L_{1\mathrm{o}\mathrm{c}}^{1}(X))$, {\it such that thefollowingproperties occur}
(1) $h$ {\it has a semipositive curvature current}
$$
\Theta_{h}=-\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\log h=\Theta_{h_{0}}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi\geq 0.
$$
(2) $Forevery\displaystyle \gamma \mathcal{O}_{X}).\in]\frac{n}{n+1},1$ [, {\it the multiplier ideal} $sheaf\mathcal{I}(\gamma\varphi)$ {\it is nontrivial}, $(i.e.  0\neq \mathcal{I}(\gamma\varphi)\neq$
According to the general philosophy of orbifolds, the orbifold concept of a multiplier ideal
sheaf $\mathcal{I}(\gamma\varphi)$ is that the ideal sheaf is to be computed upstairs on a smooth local cover and take
the direct image ofthe subsheaf of invariant functions by the local isotropy subgroup; this ideal
coincides with the multiplier ideal sheaf computed downstairs only if we take downstairs the
volume form which is the push forward of an invariant volume form upstairs (which is in general
definitely larger than the volume form induced by a local smooth embedding ofthe orbifold).
{\it Proof}.-Let us start with a {\it G}-invariant K\"{a}hler metric $\displaystyle \omega_{0}=\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\log h_{0}^{-1}$, where $h_{0}$ and $\omega_{0}$
have the same meaning as in 6.2; indeed, if $h_{0}$ is not {\it G}-invariant, we can average it by using the
{\it G}-action, that is, we define a new metric $(h_{0}^{G})^{-1}$ on $K_{X}$ by putting
$$
(h_{0}^{G})^{-1}=\int_{g\in G}g^{\star}h_{0}^{-1}\mathrm{d}\mu(g),
$$
and we again have $\omega_{0}^{G} :=\displaystyle \frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\log(h_{0}^{G})^{-1}>0$. Now, all $\varphi_{t}$ can be taken to be {\it G}-invariant. If
the continuity process ceases to produce a solution $\varphi_{t}$ at $t=t_{0}\in[0,1]$ (thus, if $t_{0}\in\overline{\mathcal{T}}\backslash \mathcal{T}$), there
exists a sequence $t_{l/}\in \mathcal{T}$ converging to $t_{0}$ and (6.3.3) implies $\displaystyle \lim_{\mathrm{I}/\rightarrow+\infty}\int_{X}\mathrm{e}^{-\gamma t_{\nu}\varphi_{t_{\nu}}}\omega_{0}^{n}=+\infty$
for every $\gamma\in$] $\displaystyle \frac{n}{n+1},1$ [. As the space of closed positive currents contained in a given cohomology
class is compact for the weak topology, one can extract a subsequence $\displaystyle \Theta_{(p)}=\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi_{t_{\nu(p)}}$
converging weakly to a limit $\displaystyle \Theta=\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi\geq 0$. The potential $\varphi$ can be recovered from
Tr $\Theta$ by means of the Green kemel, and therefore, by the well-known properties of the Green
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kemel, we infer that $\varphi_{t_{\nu(p)}}$ converges to $\varphi$ in $L^{1}(X)$. The semicontinuity theorem in its effective
version 0.2.2 shows that
$\displaystyle \int_{X}\mathrm{e}^{-\gamma t_{0}\varphi}\omega_{0}^{n}=+\infty$ for all $\displaystyle \gamma\in]\frac{n}{n+1},1[$,
and therefore $\displaystyle \int_{X}\mathrm{e}^{-\gamma\varphi}\omega_{0}^{n}=+\infty$ for all $\gamma\in$] $\displaystyle \frac{n}{n+1},1$ [. From this we conclude that $\mathcal{I}(\gamma\varphi)\neq \mathcal{O}_{X}$.
The fact that $\mathcal{I}(\gamma\varphi)\neq 0$ is clear since $\varphi\not\equiv-\infty.\ \square $
Before going further, we need Nadel's vanishing theorem (a generalized version of the well-
known Kawamata-Viehweg vanishing theorem. It is known to be a rather simple consequence of
HOrmander's $L^{2}$ estimates, see e.g. [9], [33], [13] or [14] $)$.
NADEL's vanishing theorem 6.5. -{\it Let} ({\it X}, t{\it o}) {\it be a Kahler orbifold and let} $L$ {\it be a}
{\it holomorphic orbifold line bundle over} $X$ {\it equipped with a singular hermitian metric} $h$ {\it of}
{\it weight} $\varphi$ {\it with respect to a smooth metric} $h_{0}(i.e. h=h_{0}\mathrm{e}^{-\varphi})$. {\it Assume that the curvature form}
$\displaystyle \Theta_{h}(L)=\frac{\mathrm{i}}{2\pi}D_{h}^{2}$ {\it is positive definite in the sense of currents}, $i.e.\ \Theta_{h}(L)\geq\xi:\omega$ {\it for some} $\xi j >0$. {\it If}
$K_{X}\otimes L$ {\it is an invertible sheafon} $X$, {\it we have}
$H^{q} (X,\ K_{X}\otimes L\otimes \mathcal{I}(\varphi))=0$ {\it for all} $q\geq 1$.
Recall that an ``orbifold line bundle'' $L$, is a rank 1 sheaf which is locally an invariant direct
image of an invertible sheaf on $\Omega$ by the local quotient maps $\Omega\rightarrow\Omega/\Phi;L$ itself need not be
invertible; similarly, $\otimes$ is meant to be the orbifold tensor product, i.e., we take the tensor product
upstairs on $\Omega$ and take the direct image of the subsheaf of invariants. The proof is obtained by
the standard $L^{2}$ estimates applied on $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$ with respect to an orbifold K\"{a}hler metric on $X$. It is
crucial that $K_{X}\otimes L$ be invertible on $X$, otherwise the set of holomorphic sections of $K_{X}\otimes L$
satisfying the $L^{2}$ estimate with respect to the weight $\mathrm{e}^{-\varphi}$ might differ from the orbifold tensor
product $K_{X}\otimes L\otimes \mathcal{I}(\varphi)$ [and also, that tensor product might be equal to $K_{X}\otimes L$ even though
$\mathcal{I}(\varphi)$ is non trivial].
COROLLARY 6.6.-{\it Let} $X,\ G,\ h$ {\it and} $\varphi$ {\it be as in Criterion} 6.4. {\it Then, for all} $\gamma\in$] $\displaystyle \frac{n}{n+1},1$ [,
(1) {\it the multiplier ideal} $sheaf\mathcal{I}(\gamma\varphi)$ {\it satisfies}
$H^{q}(X,\mathcal{I}(\gamma\varphi))=0$ {\it for all} $q\geq 1$;
(2) {\it the associated subscheme} $V_{\gamma}$ {\it of structure sheaf} $\mathcal{O}_{V_{\gamma}}=\mathcal{O}_{X}/\mathcal{I}(\gamma\varphi)$ {\it is nonempty, distinct}
{\it from} $X$, {\it G-invariant and satisfies}
$H^{q}(V_{\gamma}, \mathcal{O}_{V_{\gamma}})=\left\{\begin{array}{l}
\mathbb{C} for q=0,\\
0 for q\geq 1.
\end{array}\right.$
{\it Proof}.-Apply Nadel's vanishing theorem to $L=K_{X}^{-1}$ equipped with the singular hermitian
metric $h_{\gamma}=h_{0}\mathrm{e}^{-\gamma\varphi}$. Then $\Theta_{h_{\gamma}}=\gamma\Theta_{h}+(1-\gamma)\Theta_{h_{0}}\geq(1-\gamma)\omega_{0}>0$, and (1) follows. Finally,
since $X$ is Fano, we get
$H^{q}(X, \mathcal{O}_{X})=0$ for all $q\geq 1$,
by Kodaira vanishing for $L=K_{X}^{-1}$. The exact sequence
$$
0\rightarrow \mathcal{I}(\gamma\varphi)\rightarrow \mathcal{O}_{X}\rightarrow \mathcal{O}_{V_{\gamma}}\rightarrow 0
$$
immediately implies (2). $\square $
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The strategy employed by Nadel [34] to construct K\"{a}hler-Einstein metrics is to rule out the
existence ofany {\it G}-invariant subscheme with the properties described in 6.6 (2). Ofcourse, this is
easier to achieve if $G$ is large. One uses the following observations (assuming {\it that} the closedness
property fails, so that all subschemes $V_{\gamma}$ are nontrivial).
PROPOSITION 6.7. -{\it All subschemes} $V_{\gamma}$ {\it are connected Therefore}, $\iota fG$ {\it has no fixed points},
$V_{\gamma}$ {\it cannot be} 0-{\it dimensional}.
{\it Proof}.-The connectedness of $V_{\gamma}$ is a straightforward consequence of the equality
$H^{0}(V_{\gamma}, \mathcal{O}_{V_{\gamma}})=\mathbb{C}.\ \square $
PROPOSITION 6.8.-{\it If} $V_{\gamma}$ {\it contains irreducible components} $Z_{j}$ {\it of codimension 1, then the}
{\it corresponding divisor} $Z=\displaystyle \sum m_{j}Z_{j}$ {\it satisfies the numerical inequality} $[Z]\leq\gamma[K_{X}^{-1}]$ {\it in the sense}
{\it that} $\gamma[K_{X}^{-1}]-[Z]$ {\it can be represented by a closedpositive current. In particular, one always has}
{\it the inequality}
$$
(-K_{X})^{n-1}\cdot Z\leq\gamma(-K_{X})^{n}.
$$
{\it If} $K_{X}^{-1}$ {\it generates the group} $W(X)$ {\it of Weil divisors of} $X$ {\it modulo numerical equivalence, then}
$V_{\gamma}$ {\it must have codimension} $\geq 2$.
In the smooth case we have of course $W(X)=$ Pic({\it X}), but in general Pic({\it X}) is a subgroup
of finite index in $W(X)$.
{\it Proof}.-Consider the closed positive (1, 1) current $\displaystyle \Theta_{h}=\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi$ (which belongs to the
first Chem class $c_{1}(K_{X}^{-1}))$, and let
$$
\Theta_{h}=\sum\lambda_{j}[Z_{j}]+R,\ \lambda_{j}>0,\ R\ \geq 0,
$$
be the Siu decomposition of $\Theta_{h}$ (namely, the $[Z_{j}]$ 's are currents of integration over irreducible
divisors and $R$ is a closed (1, 1)-current which has nonzero Lelong numbers only in codi-
mension 2). It is then easy to see that the subscheme $V_{\gamma}$ defined by $\mathcal{I}(\gamma\varphi)$ precisely has
$[Z]=\displaystyle \sum\lfloor\gamma\lambda_{j}\rfloor[Z_{j}]$ as its 1-codimensional part (here, $\lfloor\rfloor$ denotes the integral part). Hence
$\gamma\Theta_{h}-[Z]\geq 0$ as asserted. If $K_{X}^{-1}$ generates Pic({\it X}), this implies $Z=0$, since there cannot
exist any nonzero effective integral divisor numerically smaller than $[K_{X}^{-1}].\ \square $
When $\dim X=3,\ G$ has no fixed points and $K_{X}^{-1}$ generates $W(X)$, we are only left with the
case $V_{\gamma}$ is pure dimension 1. This case can sometimes be ruled out by observing that certain
groups cannot act effectively on the curve $V_{\gamma}$ (As $H^{1}(V_{\gamma}, \mathcal{O}_{V_{\gamma}})=0,\ V_{\gamma}$ is a tree of rational
curves; see Nadel [34, Th. 4.1, 4.2 and Cor. 4.1] $)$.
Further a priori inequalities can be derived for certain components of the multiplier ideal
subschemes $V_{\gamma}$. Especially, for components of codimension 2, we have the following simple
bound, based on a use of a self-intersection inequality for the current $\displaystyle \Theta=\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi$.
PROPOSITION 6.9. {\it Assume that} $W(X)$ {\it is generated by} $K_{X}^{-1}$ {\it and that} $a$ {\it is a nonnegative}
{\it number such that the orbifold vector bundle} $T_{X}\otimes \mathcal{O}(-aK_{X})$ {\it is numerically effective. Then the}
{\it codimension} 2 {\it components} $Z_{j}$ {\it of} $V_{\gamma}$ {\it satisfy the inequality}
$$
\sum\frac{1}{\delta_{j}}\nu_{j}(\nu_{j}-1)(-K_{X})^{n-2}\cdot Z_{j}\leq(1+a)(-K_{X})^{n},
$$
{\it where} $\nu_{j}\geq 1/\gamma$ {\it is the generic Lelong number of} $\displaystyle \Theta=\omega_{0}+\frac{\mathrm{i}}{2\pi}\partial\overline{\partial}\varphi$ {\it along} $Z_{j}$, {\it and} $\delta_{j}$ {\it is the order}
{\it ofthe local isotropy group ofthe orbifold at a genericpoint in} $Z_{j}$. {\it Especially}, $\iota f\gamma$ {\it is taken to be}
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{\it sufficiently close to} $\displaystyle \frac{n}{n+1}$, {\it we have}
$$
\sum(-K_{X})^{n-2}\cdot Z_{j}\leq\frac{n^{2}}{n+1}\delta(1+a)(-K_{X})^{n},
$$
{\it where} $\delta$ {\it is the maximum ofthe orders ofthe isotropy groups}.
$Proof-$ Since $V_{\gamma}$ is of codimension 2 for $\gamma$ arbitrarily close to 1, the generic Lelong
number of $\varphi$ must be $\leq 1$ along all components of codimension 1 in the Lelong sublevel
sets $E_{c}(\varphi)=E_{c}(\Theta)$ [again, Lelong numbers and Lelong sublevel sets are to be interpreted
upstairs, on a smooth finite cover]. If a codimension 2 component $Z_{j}$ occurs in $\mathcal{I}(\gamma\varphi)$, the
generic Lelong number $\gamma\nu_{j}$ of $\gamma\varphi$ along that component must be $\geq 1$, hence $\nu_{j}\geq 1/\gamma$. We
now apply the regularization theorem for closed (1, 1)-current ([11], Main Theorem). For every
$c>1$ we obtain a current $\Theta_{h,c}$ cohomologous to $\Theta$ (hence in the class $c_{1}(K_{X}^{-1})$), which
is smooth on $X\backslash E_{c}(\Theta)$, thus smooth except on an analytic set of codimension $\geq 2$, such
that $\Theta_{h,c}\geq-(ca+\in)\omega_{0}$ and such that the Lelong numbers of $\Theta_{h,c}$ are shifted by $c$, i.e.
$\nu_{x}(\Theta_{h,c})=(\nu_{x}(\Theta)-c)_{+}$. The intersection product $\Theta\wedge(\Theta_{c}+(ca +\in)\omega_{0})$ is well defined,
belongs to the cohomology class $(1+ca +\xi:)(-K_{X})^{2}$ and is larger than $\displaystyle \sum\frac{1}{\delta_{j}}\nu_{j}(\nu_{j}-c)[Z_{j}]$ as
a current. Hence, by taking the intersection with the class $(-K_{X})^{n-2}$ we get
$$
\sum\frac{1}{\delta_{j}}\nu_{j}(\nu_{j}-c)(-K_{X})^{n-2}\cdot Z_{j}\leq(1+ca +\xi:)(-K_{X})^{n}.
$$
[The extra factor $1/\delta_{j}$ occurs because we have to divide by $\delta_{j}$ to convert an integral on a finite
cover $\Omega$ to an integral on the quotient $\Omega/\Phi.$] As $c$ tends to 1$+$0 and $\xi j$ tends to 0$+$, we get the
desired inequality. The last observation comes from the fact that $\mathcal{I}(V_{\gamma})$ must be constant on some
interval] $\displaystyle \frac{n}{n+1},\ \displaystyle \frac{n}{n+1}+\delta[$, by the Noetherian property of coherent sheaves. $\square $
{\it Example} 6.10.-Let $\mathbb{P}_{a}=\mathbb{P}^{3}(a_{0}, a_{1}, a_{2}, a_{3})$ be the weighted projective 3-space with weights
$a_{0}\leq a_{1}\leq a_{2}\leq a_{3}$ such that the components $a_{i}$ are relatively prime 3 by 3. It is equipped with
an orbifold line bundle $\mathcal{O}_{X}(1)$ which, in general, is not locally free. Let $t=a_{0}+a_{1}+a_{2}+a_{3}$
and
$$
X=\{P(x_{0}, x_{1}, x_{2}, x_{3})=0\}
$$
be a generic surface of weighted degree $d$ in $\mathbb{P}_{a}$. It is known (see Fletcher [16]) that $X$ has an
orbifold structure (i.e., is quasi-smooth in the terminology of Dolgachev [15]), if and only if the
following conditions are satisfied:
(i) For all $j$, there exists a monomial $x_{j}^{m}x_{k(j)}$ of degree $d$;
(ii) For all distinct $j,\ k$, either there exists a monomial $x_{j}^{m}x_{k}^{p}$ of degree $d$, or there exist
monomials $x_{j}^{m_{1}}x_{k^{1}}^{p}x\ell_{1},\ x_{j}^{m_{2}}x_{k^{2}}^{p}x\ell_{2}$ of degree $d$ with $\ell_{1}\neq\ell_{2}$;
(iii) For all $j$, there exists a monomial of degree $d$ which does not involve $x_{j}$.
Moreover, $-K_{X}=\mathcal{O}_{X}(t-d) ($and hence $(-K_{X})^{2}=d(t-d)^{2}/(a_{0}a_{1}a_{2}a_{3}))$ if and only if the
following condition also holds:
(iv) For every $j,\ k$ such that $a_{j}$ and $a_{k}$ are not relatively prime, there exists a monomial $x_{j}^{m}x_{k}^{p}$
of degree $d$.
We would like to use the conditions ofPropositions 6.8 and 6.9 to show that $X$ carries a K\"{a}hler-
Einstein metric.
Proposition 6.8 clearly applies ifwe can prove that $(-K_{X})\displaystyle \cdot Z>\frac{2}{3}(-K_{X})^{2}$ for every effective
curve on $X$. This is not a priori trivial in the examples below since the Picard numbers will
always be bigger than 1. Using the torus action, every curve on a weighted projective space can
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be degenerated to a sum of lines of the form $(x_{i}=x_{j}=0)$. Thus $(-K_{X})\cdot Z$ is bounded from
below by $(t-d)/(a_{2}a_{3})$. Thus $(-K_{X})\displaystyle \cdot Z>\frac{2}{3}(-K_{X})^{2}$ holds if
\begin{center}
$\displaystyle \frac{t-d}{a_{2}a_{3}}>\frac{2}{3}\frac{d(t-d)^{2}}{a_{0}a_{1}a_{2}a_{3}}$, i.e. $a_{0}a_{1}>\displaystyle \frac{2}{3}d(t-d)$.
\end{center}
In the examples we give at the end, which all concem the case $d=t-1$, this is always satisfied.
In order to apply Proposition 6.9, we need to determine $T_{X}$. We have exact sequences
$$
0\rightarrow \mathcal{O}_{\mathbb{P}_{a}}\rightarrow\oplus \mathcal{O}_{\mathbb{P}_{a}}(a_{i})\rightarrow T_{\mathbb{P}_{a}}\rightarrow 0,
$$
$$
0\rightarrow T_{X}\rightarrow T_{\mathbb{P}_{a}|X}\rightarrow \mathcal{O}_{X}(d)\rightarrow 0,
$$
and we get from there a surjective arrow
$$
\oplus \mathcal{O}_{X}(a_{i})\rightarrow \mathcal{O}_{X}(d)
$$
given explicitly by the matrix $(\partial P/\partial x_{i})$. From the above exact sequences, we find a sequence of
surjective arrows
$$
\bigoplus_{i<j}\mathcal{O}_{X}(a_{i}+a_{j})\rightarrow \mathcal{O}_{X}(\Lambda^{2}T_{\mathbb{P}_{a}|X})\rightarrow T_{X}\otimes \mathcal{O}_{X}(d).
$$
(Of course, formally speaking, we are dealing with orbifold vector bundles, which can be
considered as locally free sheaves only when we pass to a finite Galois cover.) Moreover,
$$
\bigoplus_{i\neq k\neq j}\mathcal{O}_{X}(a_{i}+a_{j})\rightarrow T_{X}\otimes \mathcal{O}_{X}(d)
$$
is surjective over the open set where $x_{k}\neq 0$. This proves that, as an orbifold vector bundle,
$T_{X}\otimes \mathcal{O}_{X}(d-a_{0}-a_{2})$ is nef ifthe line $(x_{0}=x_{1}=0)$ is not contained in $X$.
The maximal order $\delta$ of the isotropy groups is less than $a_{3}$ -which is indeed the maximum
for $\mathbb{P}_{a}$ itself-resp. $a_{2}$ if $a_{3}$ divides $d$, since in that case a generic surface of degree $d$ does not
pass through the point $[0 : 0 : 0 : 1]$. This shows that we can take $a =(d-a_{0}-a_{2})/(t-d)$ in
Proposition 6.9, and as the $Z_{j}$ are points and $n =2$, we find the condition
$$
1\leq\frac{4}{3}a_{3}(1+\frac{d-a_{0}-a_{2}}{t-d})\frac{d(t-d)^{2}}{a_{0}a_{1}a_{2}a_{3}},
$$
with the initial $a_{3}$ being replaced by $a_{2}$ if $a_{3}$ divides $d$. We thus compute the ratio
\begin{center}
$\displaystyle \rho_{a}=\frac{4}{3}\frac{d(t-d)(t-a_{0}-a_{2})}{a_{0}a_{1}a_{2}}$ if $a_{3}\sqrt{}^{/}d$,
$\displaystyle \rho_{a}=\frac{4}{3}\frac{d(t-d)(t-a_{0}-a_{2})}{a_{0}a_{1}a_{3}}$ if $a_{3}|d$,
\end{center}
and when $\rho_{a}<1$ we can conclude that the Del Pezzo surface is K\"{a}hler-Einstein. Clearly, this
is easier to reach when $t-d$ is small, and we concentrated ourselves on the case $d=t-1$.
It is then easy to check that $\rho_{a}$ is never less than 1 when $a_{0}=a_{1}=1$. On the other hand, a
computer check seems to indicate that there is only a finite list weights with $a_{0}>2$ satisfying
the Fletcher conditions, which all satisfy $a_{0}\leq 142$ Among these, 2 cases lead to $\rho_{a}<1$, namely
2 {\it Added after proof}: this has actually been shown to be true $\mathrm{m}[21]$.
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$a =(11,49,69,128),\ d=256,\ \rho_{a}\simeq 0.875696,\ x_{0}^{17}x_{2}+x_{0}x_{1}^{5}+x_{1}x_{2}^{3}+x_{3}^{2}=0$,
$a =(13,35,81,128),\ d=256,\ \rho_{a}\simeq 0.955311,\ x_{0}^{17}x_{1}+x_{0}x_{2}^{3}+x_{1}^{5}x_{2}+x_{3}^{2}=0$.
It tums out that there are no other monomials of degree $d$ than those occurring in the above
equations. As a result, the above K\"{a}hler-Einstein Del Pezzo surfaces are {\it rigid} as weighted
hypersurfaces.
There are several ways to improve the estimates. For instance, $T_{X}\otimes \mathcal{O}_{X}(d-a_{1}-a_{2})$ is nef
except possibly along the irreducible components of the curve $(x_{0}=0)\subset X$. The restriction of
the tangent bundle to these curves can be computed by hand. This improvement is sufficient to
conclude that Propositions 6.8 and 6.9 also apply in one further case:
$$
a\ =(9,15,17,20),\ d=60,\ x_{0}^{5}x_{1}+x_{0}x_{2}^{3}+x_{1}^{4}+x_{3}^{3}=0.
$$
This is again a rigid weighted hypersurface. We would like to thank P. Boyer and K. Galicki for
pointing out a numerical error which had been committed in an earlier version ofthis work, where
a further (incorrect) example $a =(11,29,39,49),\ d=127$ was claimed. In [7], it is shown that
the three above examples lead to the construction ofnon regular Sasakian-Einstein 5-manifolds.
Acknowledgements
We would like to thank R.R. Simha for useful discussions which got us started with the idea of
simplifying Nadel's approach. We also thank Mongi Blel for sharing several viewpoints on the
semicontinuity properties of psh functions, and Jeff McNeal for pointing out a slight inaccuracy
in our original calculation of volumes of analytic tubes.
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(Manuscript received November 17, 1999.)
Jean-Pierre Demailly
Laboratoire de math\'{e}matiques,
Institut Fourier,
Universit\'{e} de Grenoble-1,
B $\mathrm{P}74$,
38402 Samt-Martm-d'H \`{e}res $\mathrm{cedex}_{9}$ France
E-mail: demailly@fourier ujf-grenoble. fr
J\'{a}nos KOLLAR
Department of Mathematics,
Prmceton University,
Prmceton, NJ 08544-1000, USA
E-mail: kollar\copyright math.prmceton.edu
4${}^{\text{e}}$ SERIE--TOME $34-2001-\mathrm{N}^{\mathrm{o}}4$

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