Analytic Disks and the Projective Hull

Let X be a complex manifold and γ a simple closed curve inX . We address the question: What conditions on γ insure the existence of a 1-dimension complex variety Σ with boundary γ in X . When X = C, an answer to this question involves the polynomial hull of gamma. When X = P, complex projective space, the projective hull γ̂ of γ, a generalization of polynomial hull, comes into play. One always has Σ ⊂ γ̂, and for analytic γ they conjecturally coincide. In this paper we establish an approximate analogue of this idea which holds without analyticity. We characterize points in γ̂ as those which lie on a sequence of analytic disks whose boundaries converge down to γ. This is in the spirit of work of Poletsky and of Larusson-Sigurdsson, whose results are essential here. The results are applied to construct a remarkable example of a closed curve γ ⊂ P, which is real analytic at all but one point, and for which the closure of γ̂ is W ∪ L where L is a projective line and W is an analytic (non-algebraic) subvariety of P − L. Furthermore, γ̂ itself is the union of W with two points on L. Introduction. Let γ be a simple closed real curve in a complex manifold X . Consider the problem of finding conditions which guarantee that γ forms the boundary of a complex analytic subvariety in X . When X is C (or, more generally, Stein), there is a solution [W] which involves the polynomial hull of γ. When X is P (or, more generally, projective), there is a notion of the projective hull of γ, denoted γ̂, which is related to the polynomial hull and has the following property. If f : Σ → X is a map of a compact Riemann surface with boundary, which is holomorphic on IntΣ and continuous up to the boundary with f(∂Σ) = γ, then f(Σ) ⊆ γ̂. Partially supported by the N.S.F. Partially supported by the Institute Mittag-Leffler