Higher dimensional tautological inequalities and applications

Abstract

Abstract. We study the degeneracy of holomorphic mappings tangent to holomorphic foliations on projective manifolds. Using Ahlfors currents in higher dimension, we obtain several strong degeneracy statements.

Toward classification of codimension 1 foliations on threefolds of general type

“…Entire curves tangent to foliations on surfaces have been studied in connection with the Green-Griffiths conjecture (see [10,23,36]). In higher dimensions, we have, for example, the following theorem [25,Theorem F].…”

“…Entire curves tangent to foliations on surfaces have been studied in connection with the Green–Griffiths conjecture (see [10, 23, 36]). In higher dimensions, we have, for example, the following theorem [25, Theorem F]. Theorem Let $$\operatorname{\mathcal {F}}$$ be a codimension 1 foliation with canonical singularities on a smooth projective threefold X .…”

Brunella-Khanedani-Suwa variational residues for invariant currents

“…McQuillan, in his proof of the Green-Griffiths conjecture (for a projective surface X with c 2 1 (X) > c 2 (X) ), [22], showed that if X is a complex surface of general type and F is a holomorphic foliation on X, then F has no entire leaf which is Zariski dense. See [12,23,16,13] for more details about the Green-Griffiths conjecture and generalizations. M. Brunella in [1] provided an alternative proof of McQuillan's result by showing that if [T f ] is the Ahlfors current associated to a Zariski dense entire curve f : C → X which is tangent to F , then…”