Value Distribution Theory for Parabolic Riemann Surfaces

TLDR: In this paper, Nevanlinna's theory for holomorphic maps f from Riemann surfaces to the projective line has been studied and a generalization of Bloch Theorem concerning the Zariski closure of maps f with values in a complex torus is presented.

Abstract: We survey several results in value distribution theory for parabolic Riemann surfaces. Let Y be a parabolic Riemann surface, i.e. subharmonic functions defined on Y are constant. We discuss Nevanlinna's theory for holomorphic maps f from Y to the projective line. The results we obtain parallel the classical case Y is the complex line, as we describe now. Let X be a manifold of general type, and let A be an ample line bundle on X. It is known that there exists a holomorphic jet differential P (of order k) with values in the dual of A. If the map f has infinite area and if Y has finite Euler characteristic, then we show that f satisfies the differential relation induced by P. As a consequence, we obtain a generalization of Bloch Theorem concerning the Zariski closure of maps f with values in a complex torus. An interesting corollary of these techniques is a refined Ax-Lindemann theorem, for which we give a quick proof. We then study the degree of Nevanlinna's current T[f] associated to a parabolic leaf of a foliation F by Riemann surfaces on a compact complex manifold. We show that the degree of T[f] on the tangent bundle of the foliation is bounded from below in terms of the counting function of f with respect to the singularities of F, and the Euler characteristic of Y. In the case of complex surfaces of general type, we obtain a complete analogue of McQuillan's result: a parabolic curve of infinite area and finite Euler characteristic tangent to F is not Zariski dense.