On foliations with nef anti-canonical bundle

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