Limit currents and value distribution of holomorphic maps

Higher dimensional tautological inequalities and applications

“…One remarks that the above definition can easily be generalized to associate currents of any bidimension (k, k), 1 ≤ k ≤ m to f : C m → X. But doing this, one loses for k = 1 the closedness property as discussed in [7] and recently in [3]. Moreover, for our problem, Green-Griffiths' philosophy suggests that the geometry of these holomorphic maps should be determined by a line bundle, K X .…”

“…In the second section, we generalize the construction of such currents for arbitrary nondegenerate holomorphic mappings f : C p → X in compact Kähler manifolds and show how classical Nevanlinna theory (see [10] or [25]) translate into intersection theory for such currents. The problem of associating to several variables holomorphic maps currents with suitable properties has been recently considered by de Thélin and Burns-Sibony in [7,3]. We refer the reader to these interesting papers for more details on their motivations and for applications in other directions.…”

“…We denote by [Φ] ∈ H n−1,n−1 (X, R) the cohomology class of Φ. A consequence of the First Main Theorem of Nevanlinna theory (see [25] p.63) is (see also [3,Theorem 3.6] for the same observation).…”

Some Open Problems on Holomorphic Foliation Theory

“…A Riemann surface Y is parabolic if all bounded subharmonic functions on it are constant. If a leaf of a foliation is parabolic, then the foliation admits a directed positive closed current of bi-dimension (1, 1), see [7]. It follows that when there is no such current as in the case described by Brunella, the leaves are not parabolic and hence admit Green functions.…”

Generalized Hénon Mappings and Foliation by Injective Brody Curves

“…See [3], [17] for producing a Brody curve from an entire curve. The reader is referred to the references for the space of Brody curves mentioned in the introduction.…”