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