Errett Bishop theorems on complex analytic sets: Chow’s theorem revisited and foliations of compact leaves on Kähler 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, extensions, and limits of analytic varieties. We start with a brief introduction to the tools developed by Bishop and show their usefulness by proving Chow’s theorem via a technique suggested a long time ago in a beautiful book by Gabriel Stolzenberg, then we show some of the relationships between the theory of analytic subsets and classical results of complex-analytic functions. We finish with the original contributions of the paper which consist of applications of these tools to the theory of holomorphic foliations with alternative and, we believe, simpler proofs to Edwards, Millet, and Sullivan’s impactful result for foliations with compact leaves in the case of complex foliations in Kähler manifolds and J. V. Pereira’s global stability result for holomorphic foliations on compact Kähler manifolds.