Holomorphic mappings of maximal rank into projective spaces

TL;DR

This paper establishes a refined Second Main Theorem for holomorphic mappings of maximal rank into projective spaces, providing sharper bounds on intersection counts with hyperplanes, extending classical value distribution theory.

Contribution

It introduces a new Cartan's type Second Main Theorem with truncated counting functions for holomorphic maps of maximal rank, advancing the understanding of value distribution in several complex variables.

Findings

Derived a Second Main Theorem with level $n+1-p$ truncation

Strengthened classical results of Stoll and Vitter

Interpolated key works of Cartan and Carlson-Griffiths

Abstract

Let $1\leq p\leq n$ be two positive integers. For a linearly nondegenerate holomorphic mapping $f\colon\mathbb{C}^p\rightarrow\mathbb{P}^n(\mathbb{C})$ of maximal rank intersecting a family of hyperplanes in general position, we obtain a Cartan's type Second Main Theorem in which the counting functions are truncated to level $n+1-p$. Our result strengthens the classical results of Stoll and Vitter, and interpolates the important works of Cartan and Carlson-Griffiths.