Chow's Theorem Revisited

TL;DR

This paper offers a new, accessible proof of Chow's theorem using Bishop's results on volumes and limits of analytic varieties, linking it to broader mathematical areas and simplifying understanding for students.

Contribution

It introduces a novel proof of Chow's theorem based on Bishop's theorems, connecting classical results in one and several complex variables.

Findings

Bishop's results imply Chow's and Remmert-Stein's theorems directly

The new proof is more economical and easier to understand

The approach links classical complex analysis results across dimensions

Abstract

We present a proof of Chow's theorem using two results of Errett Bishop retated to volumes and limits of analytic varieties. We think this approach suggested a long time ago in the beautiful book by Gabriel Stolzenberg, is very attractive and easier for students and newcomers to understand, also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's theorem. Furthermore, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this approach is more economic and just as profound as Remmert-Stein's proof. At the end of the paper there is a comparison table that explains how Bishop's theorems generalize to several complex variables classical results of one complex variable.