Connections and the Second Main Theorem for Holomorphic Curves

TL;DR

This paper introduces a geometric approach using $C^ abla$-connections to prove a general second main theorem for holomorphic curves, providing new proofs and results in complex geometry.

Contribution

It presents a novel geometric proof of Cartan's second main theorem and extends second main theorems to product spaces of the Riemann sphere.

Findings

Geometric proof of Cartan's second main theorem

Second main theorems for $( ext{P}^1)^2$

Method applicable to various complex spaces

Abstract

By means of $C^\infty$-connections we will prove a general second main theorem and some special ones for holomorphic curves. The method gives a geometric proof of H. Cartan's second main theorem in 1933. By applying the same method, we will prove some second main theorems in the case of the product space $(\pone)^2$ of the Riemann sphere.