On a Conjecture of Harvey and Lawson

TL;DR

This paper investigates the properties of the projective hull of a smooth closed curve in complex projective space, exploring its relation to complex analytic subvarieties and Stein subdomains.

Contribution

It establishes the connection between the projective hull of a curve and the existence of Stein subdomains containing it, advancing understanding of complex hulls in projective spaces.

Findings

The projective hull minus the curve forms a complex analytic subvariety.

Existence of a Stein subdomain containing the hull is characterized.

Conditions under which the hull relates to complex analytic structures are identified.

Abstract

We consider complex projective space P^{n} and a smooth closed curve gamma in P^{n}. Harvey and Lawson have defined the notion of the projective hull \hat{K} of a compact subset K in P^n. This concept is an analogue of the polynomial hull of compact subsets of C^{n}. In the present note we study the relation between the following two properties of the curve gamma: (1) \hat{gamma} - gamma is a one-dimensional complex analytic subvariety of P^{n} - gamma, and (2) There exists a Stein subdomain of P^{n} which contains the projective hull \hat{gamma} of gamma.