Boundaries of Analytic Varieties

Abstract

Let γ be a simple closed oriented curve in ℂ². Under what conditions does γ bound an analytic variety of complex dimension one? More precisely, when does there exist an analytic variety Σ in some open set in ℂ² such that the closure of Σ is compact and γ is the boundary of Σ? Here we take “boundary” in the sense of Stokes’ Theorem; i.e.,

$$ \smallint _\gamma \omega = \smallint _\sum d\omega , $$

, for every smooth 2-form ω on ℂ². This is stronger than being a boundary in a point-set theoretical sense and in particular takes orientation into account.