Errett Bishop theorems on Complex Analytic Sets: Chow's Theorem Revisited and Foliations with all leaves Compact on K\"ahler Manifolds

TL;DR

This paper revisits classical theorems in complex analysis using Bishop's approach, providing simpler proofs and extending results to complex foliations on K"ahler manifolds, connecting various fundamental theorems.

Contribution

It offers a unified, accessible approach to proving Chow's theorem, extends Bishop's results to several complex variables, and provides an alternative proof for foliations with compact leaves on K"ahler manifolds.

Findings

Chow's theorem proved via Bishop's volume and limit techniques

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

Alternative proof of foliations with all leaves compact on K"ahler manifolds

Abstract

In this paper we present a series of seemingly unrelated results of Complex Analysis which are in fact connected via a different approach to their proofs using the results of Errett Bishop of volumes and limits of analytic varieties. We start by proving Chow's theorem by a technique suggested long time ago in the beautiful book by Gabriel Stolzenberg. We think this approach 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 result. In addition, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this view is simpler and just as profound as Remmert-Stein's proof. After that, we give a comparison table that explains how Bishop's theorems generalize to several complex variables classical results of one complex variable and prove Montel's compactness theorem using the techniques presented here. Finally we give an alternative proof of a theorem of Edwards, Millet and Sullivan of foliations with compact leaves for the case of complex foliations in K\"ahler manifolds.