Analytic Disks and the Projective Hull

TL;DR

This paper explores conditions under which a closed curve in a complex manifold bounds a complex subvariety, introducing an approximate approach to projective hulls that extends previous analytic results.

Contribution

It establishes an approximate characterization of the projective hull using sequences of analytic disks, applicable without the analyticity assumption on the curve.

Findings

Characterization of points in the projective hull via analytic disks.

Construction of a curve in P^2 with a non-algebraic hull.

Identification of the hull as a union of an analytic variety and two points.

Abstract

Let X be a complex manifold and c a simple closed curve in X. We address the question: What conditions on c ensure the existence of a 1-dimensional complex subvariety V with boundary c in X. When X = C^n, an answer to this question involves the polynomial hull of gamma. When X = P^n, complex projective space, the projective hull hat{c} of c comes into play. One always has V contained in hat{c}, and for analytic curves they conjecturally coincide. In this paper we establish an approximate analogue of this idea which holds without the analyticity of c. We characterize points in hat{c} as those which lie on a sequence of analytic disks whose boundaries converge down to c. This is in the spirit of work of Poletsky and of Larusson-Sigurdsson, whose work is essential here. The results are applied to construct a remarkable example of a closed curve c in P^2, which is real analytic at all but one point, and for which the closure of hat{c} is W \cup L where L is a projective line and W is an analytic (non-algebraic) subvariety of P^2 - L. Furthermore, hat{c} itself is the union of W with only two points on L.