Higher dimensional tautological inequalities and applications

Let (X,F) be a smooth complex projective variety of dimension n endowed with a codimension 1 (possibly) singular foliation. Inspired by the celebrated Green-Griffiths conjecture and the results of McQuillan in the case of surfaces, the authors consider the problem of finding a suitable positivity condition on the foliated canonical bundle KF ensuring algebraic degeneracy of holomorphic maps f:Cn−1→X that are tangent to F. They obtain several results in that direction under various assumptions on the singularities of the foliation and on the positivity of KF and KX. For instance, when X has dimension 3, they are able to show that if KF is big and the singularities of F are canonical then there exists a proper algebraic subvariety Y⊂X containing all images of holomorphic maps f:C2→X tangent to the foliation. Their strategy is based on McQuillan's approach. They associate to any transcendental map f as above a positive closed (1,1)-current Tf. The core of their analysis is then to prove adequate tautological inequalities that give information on the positivity of the numerical class of the current Tf and of its lift to a suitable Grassmann bundle.