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Defect relation for holomorphic maps from complex discs into projective varieties and hypersurfaces

Si Duc Quang

TL;DR

The paper addresses defect relations for holomorphic maps from complex discs into smooth projective varieties intersecting (moving) hypersurfaces. It develops a refined second main theorem by combining Nevanlinna theory with algebraic-geometric weight bounds (Chow and Hilbert weights) to achieve explicit truncation levels and coefficients that do not depend on the number of hypersurfaces $q$, for both moving and fixed hypersurfaces. The main contribution is Theorem 1.1, which provides SMT inequalities with truncated counting terms $N^{[L-1]}_{Q_j(f)}(r)$ and an explicit error term tied to the growth index $c_f$ and the distributive constant $\Delta_V$, improving prior results (Theorem A and Theorem B) and yielding a practical truncated defect relation. The results extend to holomorphic maps into a variety $V$ with finite growth index and advance the defect relation literature for hypersurface targets in projective varieties, including moving hypersurfaces in weakly $\ell$-subgeneral position. These findings enhance the applicability of SMTs in higher-dimensional Nevanlinna theory and complex algebraic geometry.

Abstract

In this paper, we establish a second main theorem for holomorphic maps with finite growth index on complex discs intersecting arbitrary families of hypersurfaces (fixed and moving) in projective varieties, which gives an above bound of the sum of truncated defects. Our result also is generalizes and improves many previous second main theorems for holomorphic maps from $\mathbb C$ intersecting hypersurfaces (moving and fixed) in projective varieties.

Defect relation for holomorphic maps from complex discs into projective varieties and hypersurfaces

TL;DR

The paper addresses defect relations for holomorphic maps from complex discs into smooth projective varieties intersecting (moving) hypersurfaces. It develops a refined second main theorem by combining Nevanlinna theory with algebraic-geometric weight bounds (Chow and Hilbert weights) to achieve explicit truncation levels and coefficients that do not depend on the number of hypersurfaces , for both moving and fixed hypersurfaces. The main contribution is Theorem 1.1, which provides SMT inequalities with truncated counting terms and an explicit error term tied to the growth index and the distributive constant , improving prior results (Theorem A and Theorem B) and yielding a practical truncated defect relation. The results extend to holomorphic maps into a variety with finite growth index and advance the defect relation literature for hypersurface targets in projective varieties, including moving hypersurfaces in weakly -subgeneral position. These findings enhance the applicability of SMTs in higher-dimensional Nevanlinna theory and complex algebraic geometry.

Abstract

In this paper, we establish a second main theorem for holomorphic maps with finite growth index on complex discs intersecting arbitrary families of hypersurfaces (fixed and moving) in projective varieties, which gives an above bound of the sum of truncated defects. Our result also is generalizes and improves many previous second main theorems for holomorphic maps from intersecting hypersurfaces (moving and fixed) in projective varieties.
Paper Structure (3 sections, 7 theorems, 73 equations)

This paper contains 3 sections, 7 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Some auxiliary results
  3. Proof of Theorem \ref{['1.1']}

Key Result

Theorem 1.1

Let $f$ be a nonconstant holomorphic map of $\Delta (R)$ into an $n$-dimension smooth projective subvariety $V\subset{\mathbb{P}}^N(\mathbf{C})$ with finite growth index $c_f$. Let $\{Q_i\}_{i=1}^q$ be a family slow (with respect to $f$) moving hypersurfaces with the distributive constant $\Delta_{V where with $u=\lceil 2\Delta_{V}(2n+1)(n+1)d^n\deg V(\Delta_{V}(n+1)+\epsilon)\epsilon^{-1}\rceil$

Theorems & Definitions (8)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1: see EF01
  • Lemma 2.2: see Q22b
  • Theorem 2.3: reformulation of RS
  • Lemma 2.4: see Q22c
  • Claim 3.1
  • Theorem 3.11