%% Date: jan 22, 2005
%% To: kng@math.umd.edu
%% Subject: Submission Boucksom/Demailly/Paun/peternell to Acta Mathematica

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Dear Professor Karsten Grove,

We would like to thank you for your letter dated December 12, 2004 and for the 
referee reports. Unfortunately surface mail turned out to be very slow, and
as a result your letter arrived to us mid-january 2005 and we were 
unable to react before.

While we agree with some of the remarks and suggestions given in the
two expert opinions and in the more technical report, we disagree on
some fundamental points which may have resulted in an incomplete
appraisal of our work -- and therefore in a rather drastic underevaluation
of some of our results (especially what the experts call the "second
part" of our work). We try to explain our arguments below and ask the 
Editorial Board and the experts to take them into consideration before any
attempt on our side to produce a revised version.

{\bf A}. The expert opinion 1 states that Theorem 0.2 was known in
dimension 3 as a consequence of Mori's program and its solution in
dimension 3. This is not correct. Even in dimension 3, only a very small
part of theorem 0.2 was known to be true, namely the special case
$L=K_X$.

{\bf B}. All three experts seem to consider that the last sections starting
with section 7 or 8 are not so much tightly related to the first sections 
and are of less value. It is certainly true that they are less general since
we are able to treat only the case of varieties of small dimensions (up to 4),
and that the techniques involve different ideas. Nevertheless, in our
point of view, a complete proof of the very important conjecture that
$\kappa(X)=-\infty$ implies $X$ uniruled (which is one of our main
targets) can be split in three steps, according to the ``new
philosophy'' that we tried to introduce in this paper~:

1. $K_X$ not pseudo-effective implies $X$ uniruled [this is OK by our paper].

2. If $K_X$ is on the boundary of the pseudo-effective cone, then there should
exist a covering family $(C_t)$ of curves with $K_X\cdot C_t=0$. [This point is
still to be answered].

3. If there is a covering family as in 2, then $\kappa(X)\geq 0$. [we are 
able to address partly this question]

Namely , an important point is that we are able to answer Part 3 when $\dim
X\le 4$ (section 9).  Hence, it is probably not just ``what the method gave'' 
as expert opinion~1 states, and what the other two experts seem to think
implicitly. Of course, it is possible that this misunderstanding (or
underevaluation) is due to the fact that our introduction was
insufficiently detailed, and we would therefore suggest to clarify
these aspects by giving more explanations in a revised version.

Splitting the paper in two parts would result in the fact that the second 
part would become harder to read and to understand, and it would require
anyway a long introduction where the results of part 1 would have to be 
re-explained. Worse, the perspective of what we think is a promising 
approach for solving the important ``$\kappa(X)=-\infty$'' conjecture 
would be lost (since that approach was proved to be successful in 
dim~4) . Our ``new philosophy'' which is stressed by expert 
opinion 1 as one of the main features of our work would become an 
incomplete one.

{\bf C}. The more detailed report argues that we could have proved 
Theorem 0.2 by using only algebraic language and algebraic proofs, and that
the analytic arguments should be dropped or condensed in at most one 
page of additional remarks. 

The expert is right when he writes that we could have written a purely
algebraic proof of Theorem 0.2, and that this could have made the
article easier to read to more algebraically minded people. On the
other hand, we disagree with his comments which tend to imply that the
``K\"ahler case'' is somehow an exotic curiosity which could be left
as a remark.

The point is not really, or not only, whether one is able to extend the
main result to the case of K\"ahler non projective varieties -- which, in fact,
we achieved only in certain situations -- but whether one would be able to 
treat arbitrary transcendental $(1,1)$-classes, already in the case of 
projective algebraic varieties. Even the latter case requires transcendental 
methods. An example which illustrates well this important issue is the case 
of a projective variety with Picard number $1$, say a generic algebraic 
hyperk\"ahler manifold (in which case we did prove the general conjecture
about transcendental classes). Then the algebraic part of our result is 
uninteresting since the duality statement is obvious for dimension
reasons (the cones involved are half-lines in a real line), but the
corresponding result for transcendental classes remains highly non
trivial and interesting, since it is a duality statement for cones in
high dimensional vector spaces.

Besides this, the transcendental cones are quite ``natural'' as they are
moving along rather regularly with respect to the Gauss-Manin connection,
while the algebraic part of them may exhibit an ``erratic'' arithmetic 
behaviour depending on the variation of Hodge structure. Just the same
as when one moves continuously a point in a euclidean space, rational 
points are not sufficient to perform the move, similarly algebraic varieties 
are very far from being ``complete'' in a picture which involves 
deformations.

We should probably add that even for purely arithmetic/algebraic
questions, e.g.\ the study of algebraic foliations on projective
algebraic varieties, the work of McQuillan and others shows that
transcendental classes occur e.g.\ as characteristic classes of dim 1
currents coming from integral curves. Recent work shows that these
classes play a role in the study of hyperbolic algebraic varieties,
related to number theoretic questions through the Lang-Vojta approach.

Here we can suggest to rewrite the first sections in a way which would
make the language and proof of theorem 0.2 purely algebraic, and then
start to extend the results to transcendental classes. We fear that this
would not make the paper substantially shorter, but it is true that pure
algebraists would probably feel more comfortable that way.

Additionally, we would like to inform the experts that our results
have been presented by C.~Voisin in her plenary talk at the last
European Congress, and that they will be the main theme of a
forthcoming Bourbaki seminar by O.~Debarre. Also, since R.~Lazarfeld
was informed privately of our results, of which a preliminary version
circulated as early as september 2002, some of the ideas presented in 
his book actually come from us (and from previous joint work of one of the
author with him).

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