A non-integrated defect relation for holomorphic maps into algebraic varieties

TL;DR

The paper generalizes Fujimoto's non-integrated defect to holomorphic maps into smooth projective varieties by defining $\delta_{\mu_0}^{f,A}(D)$ for effective divisors $D$ and using jet differential techniques to establish SMT-type defect bounds. It develops the Green–Griffiths jet framework, including logarithmic jets and the Demailly–Semple tower, and proves a Main Lemma giving $L^t$ estimates for jet differential evaluations $\mathcal{P}(j_k(f))$, which underpin the defect inequalities. By combining these SMT results with jet differential constructions tied to divisors $D_1,\dots,D_q$, the authors derive explicit defect bounds $\sum_j\delta_{k}^{f,A}(D_j)$ and their starred variants, extending Fujimoto and Yamanoi–Cartan-type theorems to projective varieties. In the setting of high-degree generic hypersurfaces in $\mathbb{P}^n$, they obtain sharp defect bounds $\delta_1^{f,A,*}(D)\le d-c$ (and $\delta_1^{f,A}(D)\le d-(c-2\rho)$ under growth hypotheses), yielding strong rigidity statements for maps omitting such hypersurfaces.

Abstract

In 1983, relating to the study of value distribution of the Guass maps of complete minimal surfaces in ${\Bbb R}^m$, H. Fujimoto introduced the notion of the non-integrated defect for holomorphic maps of an open Riemann surface into $\mathbb{P}^n(\mathbb{C})$ and obtained some results analogous to the Nevanlinna-Cartan defect relation. This paper establishes the non-integrated defect relation for holomorphic maps into projective varieties.

Theorems & Definitions (22)

• Theorem 1.1: Fujimoto Fu1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1: book:minru2, Proposition 4.2.1
• Lemma 2.2: Main Lemma for jet differentials
• proof
• Proposition 2.3