We develop some foundational results in a higher-dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman–Mori cone of curves in terms of the numerical properties of $K_{{\mathcal{F}}}$ for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program (MMP) for rank 2 foliations on threefolds.

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

A foliation on a non-singular projective variety is algebraically integrable if all leaves are algebraic subvarieties. A non-singular hypersurface X in a non-singular projective variety M equipped with a symplectic form has a naturally defined foliation, called the characteristic foliation on X. We show that if X is of general type and dim M≥4, then the characteristic foliation on X cannot be algebraically integrable. This is a consequence of a more general result on Iitaka dimensions of certain invertible sheaves associated with algebraically integrable foliations by curves. The latter is proved using the positivity of direct image sheaves associated to families of curves.

We show that holomorphic singular codimension one foliations of the complex projective space with a Kupka singular set of radial type and verifying some global hypotheses have rational first integral. The generic elements of such pencils are Calabi–Yau.