A characterization of real holomorphic chains and applications in representing homology classes by algebraic cycles

We show that a $2k$-current $T$ on a complex manifold is a real holomorphic $k$-chain if and only if $T$ is locally real rectifiable, $d$-closed and has $\mathcal{H}^{2k}$-locally finite support. This result is applied to study homology classes represented by algebraic cycles.