  We consider a compact Kaehler manifold whose dual Kaehler cone contains a
rational interior point. The general problem we have in mind is how far the
manifold is from being projective; i.e. we ask for the algebraic dimension. We
prove e.g. that if the interior point is represented by an effective curve in
dimension 3, then the manifold has algebraic dimension at least 2 unless the
variety is a so-called simple non-Kummer 3-fold. We also investigate 3-folds
containing curves with ample normal bundle. It seems likely that examples with
algebraic dimension 2 really exist.
