On entire holomorphic maps tangent to foliations on threefolds

We consider complex projective threefolds endowed with a codimension one holomorphic foliation. Let us assume that there exists a holomorphic map from $\mathbb{C}^2$ to our threefold such that its image is Zariski dense and tangent to the foliation. Under these assumptions we want to study the implications for the birational geometry of the threefold. The main conjecture is that the threefold cannot be of general type. This statement can be seen as a particular instance of the Green–Griffiths–Lang conjecture as well as a generalization of a celebrated result of M. McQuillan from 1998. In my talk I will describe a new strategy towards the above conjecture, based on the study of positivity of the conormal bundle to the foliation