  We study holomorphic foliations of codimension $k\geq 1$ on a complex
manifold $X$ of dimension $n+k$ from the point of view of the exceptional
minimal set conjecture. For $n\geq 2$ we show in particular that if the
holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the
foliation does not admit a compact invariant set that is a complete
intersection of $k$ smooth real hypersurfaces in $X$.
