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The Second Main Theorem with moving hypersurfaces in subgeneral position

Qili Cai, Chin-Jui Yang

TL;DR

This work extends Nevanlinna theory to moving targets by proving a Second Main Theorem for holomorphic curves into $\mathbb{P}^N(\mathbb{C})$ with slowly moving hypersurfaces in $m$-subgeneral position. It develops a universal-field framework and employs the distributive constant $\Delta$ and a Bézout-type intersection property on the associated variety $V_f$ to quantify interactions of the moving divisors. A two-case proof strategy yields the main inequality with the sharp factor $\frac{3}{2}$ in terms of $T_f(r)$, generalizing prior results of Heier–Levin and Ru–Stoll to moving targets in subgeneral position. The results provide refined defect relations for moving targets in higher-dimensional projective spaces and broaden the applicability of the Second Main Theorem in value distribution theory.

Abstract

In this paper, we prove a second main theorem for a holomorphic curve $f$ into $\mathbb P^N (\mathbb C)$ with a family of slowly moving hypersurfaces $D_1,...,D_q$ with respect to $f$ in $m$-subgeneral position, proving an inequality with factor $3 \over 2$. The motivation comes from the recent result of Heier and Levin.

The Second Main Theorem with moving hypersurfaces in subgeneral position

TL;DR

This work extends Nevanlinna theory to moving targets by proving a Second Main Theorem for holomorphic curves into with slowly moving hypersurfaces in -subgeneral position. It develops a universal-field framework and employs the distributive constant and a Bézout-type intersection property on the associated variety to quantify interactions of the moving divisors. A two-case proof strategy yields the main inequality with the sharp factor in terms of , generalizing prior results of Heier–Levin and Ru–Stoll to moving targets in subgeneral position. The results provide refined defect relations for moving targets in higher-dimensional projective spaces and broaden the applicability of the Second Main Theorem in value distribution theory.

Abstract

In this paper, we prove a second main theorem for a holomorphic curve into with a family of slowly moving hypersurfaces with respect to in -subgeneral position, proving an inequality with factor . The motivation comes from the recent result of Heier and Levin.
Paper Structure (2 sections, 5 theorems, 30 equations)

This paper contains 2 sections, 5 theorems, 30 equations.

Table of Contents

  1. introduction
  2. Proof of Theorem \ref{['T13']}

Key Result

Theorem 1.1

Let $f: \Bbb C\rightarrow \Bbb P^N (\Bbb C)$ be a holomorphic map, and let ${\mathcal{H}}:=\{H_1, ... ,H_q\}$ be a family of slowly moving hyperplanes with respect to $f$ located in $m$-subgeneral position. Assume that $f$ is linearly nondegenerate over ${\mathcal{K}}_{{\mathcal{H}}}$. Then, for any where “$\leq_{exc}$” means that the above inequality holds for all $r$ outside a set with finite Le

Theorems & Definitions (13)

  • Theorem 1.1: Ru and Stoll RuS2
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: Main Theorem
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: S.D. Quang, Lei Shi, Qiming Yan and Guangsheng Yu SYY
  • ...and 3 more