Workshop on birational geometry October 30, 2018 18:00–19:00, Moscow, Laboratory of Algebraic Geometry and its applications, Higher School of Economics

On entire holomorphic maps tangent to foliations on threefolds

Abstract:
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