Boundaries of Positive Holomorphic Chains and the Relative Hodge Question

We characterize the boundaries of positive holomorphic chains in an arbitrary complex manifold. We then consider a compact oriented real submanifold of dimension 2p − 1 in a compact Kähler manifold X and address the question of which relative homology classes inH2p(X,M ; Z) are represented by positive holomorphic chains. Specifically, we define what it means for a class τ ∈ H2p(X,M ; Z) to be of type (p,p) and positive. It is then shown that τ has these properties if and only if τ = [T+S] where T is a positive holomorphic chain with dT = ∂τ and S is a positive (p,p)-current with dS = 0. §