On foliations with nef anti-canonical bundle

In this paper we prove that the anti-canonical bundle of a holomorphic foliation $\mathcal{F}$ on a complex projective manifold cannot be nef and big if either $\mathcal{F}$ is regular, or $\mathcal{F}$ has a compact leaf. Then we address codimension one regular foliations whose anti-canonical bundle is nef with maximal Kodaira dimension.