On holomorphic foliations admitting invariant CR manifolds

Abstract

We study holomorphic foliations of codimension $k\ge 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\ge 2$ 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 $X$.

to be published in Ann. Sc. Norm. Sup. Pisa. arXiv admin note: text overlap with arXiv:1804.06884