Limit currents and value distribution of holomorphic maps (Q442124)

Let \(\phi: X \rightarrow Y\) be a non-degenerate holomorphic map between a complex manifold \(X\) of dimension \(k\) and a compact Kähler manifold \(Y\) of dimension \(m\geq k\). The authors construct \(d\)-closed and \(dd^c\)-closed positive currents on \(Y\) associated to \(\phi\) via cluster points of normalized weighted truncated image currents. They are constructed using analogues of the Ahlfors length-area inequality in higher dimensions and are generally referred to as Ahlfors currents.NEWLINENEWLINEUsing their results, the authors give some applications to equidistribution problems in value distribution theory. That comprises a corollary about the behavior of leaves of singular holomorphic foliations of \(\mathbb{P}^m\).NEWLINENEWLINEThey also examine the size of the set of limit currents constructed in the paper using results in complex dynamics.NEWLINENEWLINEFinally, the authors relate the mass ratio conditions from the present paper to a couple of examples of classical order of growth conditions.