%latex2e
\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb,color,srctex}

\title{Referee's report on  manuscript
" Semicontinuity of complex singularity exponents... "
by Demailly and Koll\'ar, submitted to
Annales de l'Ecole Normale Sup\'erieure}
\date{\today}





\begin{document}
\maketitle






\section{Comments for the authors}

I have compiled a short list of misprints or remarks:

\begin{enumerate}
\item I believe the Anghern-Siu 1995 article should be cited in the
introduction
 since
this article is arguably one of the origins
 of semicontinuity techniques in effective
complex algebraic geometry .
\item p.4, in the 1.1 Variant of Definition , $U$
should be taken relatively compact.
\item p.5, l. 20, in 4.3 (3) \lq .., acoording to Prop (1.2)...\rq.
\item p.7, l. 5,  $\{ x\in B; c_x(\phi)\le c_0\} $ and $ f\in \cup_{c>c_0}
H_{c\phi} (B) $.
\item p.7, l. 22, proof of prop 1.5. $ c_x(\alpha_1 g_1 +\ldots)\le 1$ by
1.4.6.
\item p.6 l.11 l.18. The symbol $\mu^*$ is used for $\mu^* I = \mu^{-1} I .
O_{\tilde X}$,
this is not the convention of Hartshorne's algebraic geometry who uses $\mu^*$
for pull back of sheaves (in $\mu^* K_X$ for instance). This does not cause
any
\cyan{confusion however.}
\item p.15, l 24,  \lq ... independant of $k,s$ .. \rq. Perhaps one should
make explicit the implicit condition $|s-s_0| < \delta k^{-k}$.
\item p. 18,  l. 35, in 4.2 (3), $\lambda _K (I(m))$.
\item p.18, in 4.2 (3), it's worth emphasizing that $C_1, C_2$ are
independant of $\phi$.
\item p. 21 l.18. The actual counterexample for
$\mu_U(\{ \phi< \log r\} ) \le C_2 r^{2c} \log ^{n-1} r$ is of course
$\log |z|-\log \log |z|^{-1}$. The way the remark is written
seems to imply that it is $\log \log |z|-\log \log \log |z|^{-1}$.
\item p.32, l. 24. There is a conflit of notation. $k$
is sometimes used as a dummy index and sometimes as $k=a_0+a_1+a_2+a_3$.
Furthermore we have $-K_X.Z \ge (k-d)/ a_2 a_3$.
I understand that the examples satisfy  $k-d=1$.
\item p.33 l. 10. typographical error: in  $\oplus_{i<j} O_X( \Lambda^2 T)$,
there is no need for the $\oplus$.
\item  Having few working expertise on
weighted hypersurfaces, I find it difficult to read example 6.10.
 I have not been able to find the
Fletcher unpublished manuscript in my department's
library on weighted complete intersections.
 It is easier to find
the well known  paper of Dolgachev in Springer Lecture Notes {\bf 956}
and perhaps it could be used as an alternate reference.
For instance,  the first example
is quasismooth in the terminology of the
aforementioned Dolgachev article hence an orbifold.
 Probably the other ones
are also quasismooth. Also the fact that $O_X(1)$
is an orbifold line bundle is worth emphasizing.



\end{enumerate}


\end{document}
