Lemma on logarithmic derivative over directed manifolds
Peiqiang Lin
TL;DR
The paper advances value distribution theory by extending Ahlfors' Lemma on logarithmic derivatives to holomorphic tangent curves on directed projective manifolds, yielding ALLDD and GAALD. It builds a comprehensive jet-theoretic framework—encompassing Demailly-Semple jet towers, Green-Griffiths and invariant jet differentials, and Wronskian constructions—adapting it to logarithmic settings along divisors. Through Nevanlinna theory, Crofton-type averaging, and calculus lemmas, it derives quantitative LLD-type inequalities and, via linear-system transforms, a robust algebro-geometric version (AALD) with wide SMT-type applications. The resulting GAALD framework provides powerful, scalable estimates for proximity and counting functions of holomorphic curves relative to closed subschemes, with implications for hyperbolicity and Diophantine-approximation-type questions. Overall, the work unifies differential-analytic and algebro-geometric techniques to obtain second-main-theorem-type results in the directed-manifold context.
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.