Lemma on logarithmic derivative over directed manifolds

TL;DR

This paper extends Ahlfors' lemma on logarithmic derivatives to directed projective manifolds, deriving new algebro-geometric versions and applications to second main theorem type results for holomorphic curves.

Contribution

It generalizes Ahlfors' lemma to directed manifolds and introduces algebro-geometric versions with transformations and applications to value distribution theory.

Findings

Derived Algebro-Geometric Ahlfors' Lemma on Logarithmic Derivative (AALD)

Established a general form of GAALD for directed projective manifolds

Obtained second main theorem type results for holomorphic curves

Abstract

In this paper, we generalize Ahlfors' lemma on logarithmic derivative to holomorphic tangent curves of directed projective manifolds intersecting closed subschemes. As a consequence, we obtain Algebro-Geometric Ahlfors' Lemma on Logarithmic Derivative (AALD for short) and General form of Algebro-Geometric Version of Ahlfors' Lemma on Logarithmic Derivative (GAALD for short) for holomorphic tangent curves of directed projective manifolds. We also get a transform of AALD and GAALD with respect to a linear system. Finally, we get the Second Main Theorem type results for holomorphic curves as the applications of GAALD and its transform.