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\centerline{\bf Problems suggested by J.-P. Demailly.}

\vskip.5 cm\noindent



\noindent PROBLEM 1: {\it Kawamata-Viehweg vanishing theorem for compact
K\"ahler manifolds}.\\
Let $L$ a line bundle over a compact K\"ahler manifold $(X,\omega)$ of
complex dimension $n$.

\noindent {\bf Definition.} {\it
 $L$ is a numerically effective line bundle, nef for short, if for every
$\epsilon>0$,
there is a smooth metric $h_{\epsilon}$ on $L$ such that, its hermitian
curvature $i\Theta_{h_{\epsilon}}$ satisfies $i\Theta_{h_{\epsilon}}\geq
-\epsilon\omega$.\ \
 Let $L$ a nef line bundle on $X$. One defines the numerical dimension of
$L$ to be 
$$\displaystyle{\nu(L)=\max\{k\in N\,;\; c_1(L)^{k}\not=0\ {\rm in}\  H^{2k}
(X,\bR)\}}$$
where $K_X$ denotes the canonical bundle of $X$.}

If ${D=\sum\alpha_j D_j\geq 0}$ is an effective $\bQ$-divisor, we define 
the multiplier ideal sheaf ${\cal I}(D)$ to be equal to ${\cal I}(\varphi)$
(for a precise definition of the multiplier ideal sheaf ${\cal I}
(\varphi)$, we refer to the quoted article of J.-P. Demailly) where
$\displaystyle
{\varphi=\sum\alpha_j\log|g_j|}$ is the locally defined plurisubharmonic
function defined by generators $g_j$ of ${\cal O}(D_j)$.

Let $(X,\omega)$ be a K\"ahler manifold and $F$ a line bundle over $X$ such
that some positive multiple $mF$ can be written $mF=L\otimes D$, where $L$ is
nef and $D$ is an effective divisor.

Is it true that $H^q(X,K_X\otimes F\otimes {\cal I}(m^{-1}D))=0$ for
all $q>n-\nu(L)$~? For projective manifolds, this is a consequence of
the Kawamata-Viehweg vanishing theorem and a trivial slicing argument
by hyperplane sections (see e.g.\ Theorem 6.12 in J.-P.  Demailly,
Lecture Notes in Math.\ n${}^\circ\,$1646).

\vskip .5 cm \noindent
\noindent PROBLEM 2: {\it Towards a Mori theory for K\"ahler varieties}.\\
Assume that the K\"ahler manifold $(X,\omega)$ is such that its canonical 
bundle $K_X$ is not nef. Does there exist a rational curve $C$ in $X$ such 
that $-K_X\cdot C\leq n+1\,$?
Again, if $X$ is projective, the answer is known to be positive.

\vskip.5 cm \noindent
PROBLEM 3: {\it Deformations of varieties in the class ${\cal C}$}.\\
Recall that a compact complex space is said to be in the class ${\cal C}$ if
it is bimeromorphic to some compact complex K\"ahler manifold.
It follows from the theory of Kodaira and Spencer (see Ann.\ of Math.\ 71,
1960) that small deformations of a compact K\"ahler manifold are again 
K\"ahler. However an example of F.~Campana (see F.~Campana, Math.\ 
Ann.\ 290, (1991)) shows that there exists a Moishezon (hence in the 
class ${\cal C}$), non K\"ahler threefold, wich has arbitrarily small 
deformations wich are not in the class~${\cal C}$.

Let ${\cal X}\rightarrow \Delta$ a deformation of compact complex manifolds
over the unit disc in $\bC$. Assume that $X_s$, the fiber over $s$, is 
K\"ahler for every $s\in\Delta\setminus \{0\}$ . Does this imply that 
$X_0$ belongs to the class ${\cal C}$ ?

A positive answer is known under some restrictive assumptions on 
the family of K\"ahler metrics $\omega_s$ on the fibers $X_s$,
$s\in\Delta\setminus 0$. Namely, the metrics should remain
uniformly bounded, while their volume is bounded away from~$0$ when
approaching the central fiber. Can one remove these restrictive 
assumptions~?\vskip 1cm

\end
