One of the most mysterious and intriguing phenomenon that appears in the classification of foliations on surfaces is the existence of non-abundant foliations, i.e. foliations for which the numerical Kodaira dimension and the Kodaira dimension differ. Non-abundant foliations on surfaces are understood and have remarkable properties. They are foliations obtained as quotients by
irreducible lattices of the natural fibrations on polydisks which have an unexpected behavior in at least one other aspect: their reduction to positive characteristic is algebraically integrable for a Zariski dense set of primes, and non-algebraically integrable also for a Zariski dense set of primes.
Besides these two-dimensional examples and their natural generalizations to higher dimensions/codimensions, there are not many other examples thoroughly studied in the literature. Nevertheless, work by Mok on bounded symmetric domains gives a wealth of examples of non-abundant foliations. This research in
pairs aims to study this family of foliations discovered by Mok from several points of view. For instance, it would be interesting to investigate the following topics:
• Hyperbolicity: behaviour of entire curves tangent to them
• Foliation Theory: existence of transverse structures
• Arithmetic: behaviour of the reduction modulo primes of these