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\centerline{\bf Open Problem suggested by J.P. Demailly}
\centerline{\bf in the US-USSR Algebraic Geometry Symposium}
\centerline{\bf (Chicago, July 1989)}
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The well-known Hodge conjecture asserts that every rational
cohomology class of bidegree $(p,p)$ on a projective $n$-fold
$X$ is the cohomology class of an algebraic cycle of codimension
$p$ with rational coefficients. One of the main difficulties  for
a proof of the conjecture is the arithmetic nature of the rationality 
assumption. Even for abelian varieties, the conjecture remains unsettled.
Questions (1), (2) below are an attempt to split the difficulties in 
several steps by taking apart the arithmetic considerations.

Let $\fr$ be a foliation on $X$ with complex analytic leaves of
codimension $p$.  If $\fr$ has a transversal smooth volume form, then
this form can be seen as a closed positive $(p,p)$-current $\alpha$ with
$T\fr$ in its kernel.  Conversely, every smooth closed positive
$(p,p)$-form $\alpha$ which is decomposable at each point is associated 
to a foliation $\fr$ as above and will be called a {\sl foliated
$(p,p)$-current} (according to D.~Sullivan).  When $X$ is a complex
torus ${\bf C}^n/\Gamma$, it is clear that $H^{p,p}(X)$ is generated by
the cohomology classes $\{\alpha\}$ of constant decomposable
$(p,p)$-forms, which are associated to linear foliations of the
torus.  Moreover, it is easy to check in this case that $\{\alpha\}\in
H^{2p}(X,{\bf R})$ is rational if and only if the foliation has compact
leaves. When $X$ is an arbitrary projective $n$-fold, we are thus led 
to the following questions. A positive answer to these would imply
the Hodge conjecture for $X$.
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\noindent(1) {\sl Is $H^{p,p}(X)$ generated by cohomology
classes of currents $\pi_\star\alpha$ which are direct images of a
foliated $(q,q)$-current $\alpha$ on $Y$ by some morphism
$\pi:Y\to X$, where $Y$ is projective and $q-p=\dim Y-\dim X$~?~
Is $H^{p,p}(X)\cap H^{2p}(X,{\bf Q})$ generated by those classes
$\{\pi_\star\alpha\}$ for which $\{\alpha\}$ is rational~?}
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\noindent(2) {\sl If $\alpha$ is a foliated $(p,p)$-current with
cohomology class in $H^{2p}(X,{\bf Q})$, can one say that
the leaves of the foliation $\fr$ associated to $\alpha$ are compact~?}
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Note that taking direct images in (1) cannot be avoided in general
(except perhaps if highly singular foliations were admitted).  For
instance ${\bf CP}^n$ has no foliated $(1,1)$-current: if $\alpha$ is
decomposable, we have $\alpha^n=0$, hence $\{\alpha\}=0$.  In that case,
however, we can take $Y$ to be the flag manifold of pairs $(x,S)$,
$x\in S$, where $S$ is a linear subspace of codimension $p$ in 
${\bf CP}^n$ and take $\pi:Y\longrightarrow{\bf CP}^n$ to be the obvious
projection.  Then $H^{p,p}({\bf CP}^n,{\bf Q})$ is generated by the
foliated current given by the fibers of the projection of $Y$ onto the
Grassmannian $G$ of codimension $p$ subspaces, with the unitary
invariant volume form of $G$ as transversal volume form. 

The answer to (2) is negative in general, e.g. for some Hilbert modular
surfaces, but statement (2) can be weakened by requiring only that 
the foliation $\fr$ associated to $\alpha$ can be deformed (within
the same cohomology class $\{\alpha\}$) into a foliation with compact 
leaves.

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