Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part II

Abstract

In Part I of this paper we introduced the notion of the projective linking number Link(M,Z) of a compact oriented real submanifold M of dimension 2p-1 in complex projective n-space P^n with an algebraic subvariety Z of codimension p in P^n - M. It is shown here that a basic conjecture concerning the projective hull of real curves in P^2 implies the following result: M is the boundary of a positive holomorphic p-chain T of mass M(T) < K in P^n if and only if Link(M,Z) > -K p!deg(Z) for all algebraic subvarieties Z of codimension p in P^n - M. An analogous result is implied in any projective manifold X. The above theorem implies a duality between relative and absolute homology classes which are represented by positive holomorphic chains.