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{\bf Expert opinion 1:}

{\leftskip=10mm
 \parindent=0mm
{Dear Professor Grove

The paper has 2 main parts. I very much like Thm 0.2.
Earlier this was known assuming Mori's program (proved only in
dim 3 so far) and was assumed to need the full force of Mori's program.
A direct proof was completely unexpected and it is a real
change of point of view. This is a first rate result.

I am less impressed by Thm 0.4. This is clearly what the method gave,
and not what one would like to get.

I would unhesitatingly recommend a paper half as long containing Thm
0.2 only.  I am less certain with the current form. The two parts may
be more connected than I realize since I have not read it in
detail. So it may not make sense to split it up.}
\vskip 0pt}
\vskip 1cm

{\bf Expert opinion 2:}

{\leftskip=10mm
{\parindent=5mm
The first part of the paper - essentailly through Section 5 or 6 -- is
really wonderful. In this part of the paper the authors analyze what
is the dual of the cone of effective divisors on a projective
variety. This part of the paper obtains optimal results, and makes a
fundamental and lasting contribution. If the paper ended after at that
point, it would be a good catch for any journal.

The later sections prove special cases of some further conjectures of
the authors. I find this material very much less natural, and I feel
that it greatly detracts from the overall appeal of the paper.

If you could convince the authors to remove the later sections, then
it would be great for Acta, or anywhere else. If they insist on
keeping the whole paper intact, then the case becomes less clearcut.
\vskip 0pt
}
\vfill\eject

\centerline{November 15, 2004}
\bigskip

{\bf Report on the article:}\\
The pseudo-effective cone of a compact K\"ahler manifold~...\\
submitted to Acta Mathematica

by S.~Boucksom, JP.~Demailly, M.~Paun, and T.~Peternell

The main results of this paper are (2.2) and its corollaries (2.4) and
(2.6); (2.2) states that, on a projective manifold X, an $\bR$-line
bundle is pseudo effective iff its intersection number with every
movable curve is nonnegative. Its corollary (2.4) saj's that the
closed cones generated by movable and strongly movable curves
coincide. The second corollary (2.6) says that $K_X$, the canonical
bundle of~$X$, is pseudo-effective if $X$ is not uniruled.

These results, as well as their proofs are very interesting. They will
certainly be of current use in the future. In my opinion they deserve
publication in the Acta Mathematica. (The other results of the paper
are not very tightly connected with this theme, and should rather
appear separately. The interesting and not trivial result of sections
7, 8 and 9 on the four dimensional case need maybe to be completed).

Unfortunately, the exposition prevents the publication in the present form.

Indeed, although the results can be stated and proved in an elementary
way and a few pages along the very same lines, using the standard
tools and concepts of algebraic geometry, the language and techniques
systematically chosen are those of transcendental methods of algebraic
geometry (currents), in the context of compact K\"ahler manifolds. These
here unnatural tools considerably lengthen and obscure the exposition.

For example, the first section gives in the K\"ahler context the
definition of the cones involved, and states their equivalence with
the standard corresponding notions of algebraic geometry. This is
unnecessary, since one can directly use the algebro-geometric notions
in the proofs of sections 4 and 5.

Next, the proof of (2.2) is in fact given in three pages
(pp.~13-16). It does not use the preceding-sections (neither
(Demailly-Paun (2.1)), nor the results of Boucksom (3.5) and Demailly
(3.1). For lemma 4.2, the authors refer to the holomorphic Morse
inequalities, although a very simple proof in the algebro-geometric
context exists ( [De01,~8.5], but the given reference is [De01,~8.4]!)

My suggestion is thus that the authors resubmit a short, clear and
concise text (something like 5 or 6 pages), written in the language of
algebraic geometry, containing the proofs of 4.1, 2.2 and its
corollaries 2.4, 2.6 and 2.7 (explaining why normal is sufficient in
this last result). There should be an introductory section stating and
showing (in the algebro-geometric context) the needed facts on volume
of line bundles and approximate Zariski decompositions. (The book
{\it Positivity in Algebraic Geometry} by R.~Lazarsfeld, sections 2.2 and
10.3, for example might be a convenient reference for almost all of
the facts needed). If they wish, they could add in a {\it separate short}
paragraph (one page or so) comments on the K\"ahler case.

Otherwise, other expositions (concisely written in the language of
algebraic geometry) of these very same results will certainly appear
elsewhere, and become the standard references on this topic.


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