\magnification=1200
\vsize=20cm\hsize=13.5cm
\parindent=5mm\parskip=6pt plus 2pt minus 1pt
\def\\{\hfil\break}

\font\tenBbb=msbm10
\font\sevenBbb=msbm7
\font\fiveBbb=msbm5
\newfam\Bbbfam
  \textfont\Bbbfam=\tenBbb
  \scriptfont\Bbbfam=\sevenBbb
  \scriptscriptfont\Bbbfam=\fiveBbb
\def\Bbb{\fam\Bbbfam\tenBbb}

% Usual sets of numbers  
\def\bC{{\Bbb C}}
\def\bN{{\Bbb N}}
\def\bP{{\Bbb P}}
\def\bQ{{\Bbb Q}}
\def\bR{{\Bbb R}}
\def\bZ{{\Bbb Z}}

\font\huge=cmr10 at 17.28pt

\noindent
{\huge Report on
``The pseudo-effective cone of a compact K\"ahler manifold....'',
by S.~Boucksom, JP.~Demailly, M.~Paun, T.~Peternell}
\vskip1cm

\centerline{February 7, 2005}
\vskip7mm

There are two major parts. The first one (\S$\,$1-5) contains and proves
the main result (theorem 0.2 and its corollary 0.3). The second part
(\S$\,$8-9, very different, by the scope and techniques) shows theorems 0.4
and 9.8 (The \S$\,$6-7 deal with more special topics, of less interest, and
will not be considered).

The theorem 0.2 is of purely algebro-geometric nature, both in its
statement and proof. This fact is unfortunately hidden by the analytic
exposition given. A very clear distinction must be made between the
results obtained (purely algebro-geometric, as said, for which the
tools, definitions, and techniques developped or recalled in \S$\,$1-3
are irrelevant) and the extensions wished (for which they are, without
any doubt, absolutely pertinent).

So, I repeat my former proposal, slightly modified.

First give (in 6 pages or so) a purely algebro-geometric proof of 0.2
and its main corollary 0.3. Please, try to list precisely the
properties of volume needed, and include a proof of 4.2 in the
projective case, following [De 01, 8.5 (and not 8.4)]. This is the
main contribution, and should clearly appear as such. I personally
consider as a good new and quite remarkable that this important result
can be obtained so simply (and also that it was not obtained before).

Next, in a separate section, comment {\it as briefly as possible} on the
extension to transcendental classes and K\"ahler manifolds. Please,
give a list of properties com\-pletely parallel to the one before, and
stress that the extension would follow from (weak versions or
corollaries of) the transcendental Morse inequalities 10.1.(Here I
must agree that my former proposal to reduce this to one page was not
entirely real\-istic. But nevertheless, do not give full details, since
both definitions and properties (as for example 3.5) are already
(essentially) published elsewhere. Give references, rather than
repeating an already existing exposition. In any case, this part
should not exceed in length something like one half of the
algebro-geometric proof and exposition of theorem 0.2).

The point is , by no means, the ``exotic'' character of K\"ahler
manifolds and transcendental classes, but simply the need for a clear
distinction, not only between proved and wished results, but also
between a natural and an unnatural frame of proof and statement. (The
analytic frame originally choosen is natural only in the context of
K\"ahler manifolds, or of transcendental classes).

Next, the second part (\S$\,$8-9). What is maybe ``new'' is the
splitting in point 2 (existence of a covering by numerically
$K$-trivial curves if $K$ lies on the boundary of the pseudo-effective
cone), and point 3 (that the Kodaira dimension is then
non-negative). Point 2 seems particularly hard, since apparently by
itself a weak version of the Abundance Conjecture, as supported by
theorem 9.8. (This is indeed what is missing to establish the
uniruledness conjecture in dimension 4, as a consequence of \S$\,$8-9
of this paper).

Because these two sections are so poorly written, I would propose the
authors to publish it latter, somewhere else, separately (despite
their interest, especially theorem 9.8 and its corollary 9.9 -- where the
condition $q(X) = 0$ might be dropped).

I'm nevertheless not entirely opposed to the inclusion of this part,
provided it is rewritten in an understandable and organised way.

I shall list only some of the numerous points of \S$\,$8-9 which
obscure the exposition. First 8.3: it would be desirable to have
existence and uniqueness for a partial nef reduction with $\dim(Y) =
p(L)$, as well as a universal property. This point should be
cleared. Lemmas 8.6-7 seem out of order, here.

Next \S$\,$9. Please don't call it ``Towards abundance'' (this does
not, by far, describe its content). The very poor organisation reduces
to a collection of seemingly unrelated case-by-case studies, with no
clearly stated overall outline of proof . Give instead statements
applying to all cases, and conditions under which the general fibre of
\hbox{$f : X \to Y$}, a partial nef reduction relative to $K_X$, has
$\kappa = 0$, and under which an appropriate formula for $K_X$ exists,
ensuring that $\kappa(X)>0$. Avoid the systematic use of sending back
the reader to (presumably) simpler or similar cases, supposed to have
been already treated. Also theorem 9.8 is of general interest, as well
as its corollary 9.9 and should be proved and stated separately (maybe
before).

\end

% Local Variables:
% TeX-command-default: "TeX"
% End: