Real rectifiable currents, holomorphic chains and algebraic cycles

We study some fundamental properties of real rectifiable currents and give a generalization of King’s theorem to characterize currents defined by positive real holomorphic chains. Our main tool is Siu’s semi-continuity theorem and our proof largely simplifies King’s proof. A consequence of this result is a sufficient condition for the Hodge conjecture.

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 ℋ2k-locally finite support. This result is applied to study homology classes represented by algebraic cycles.