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

## TL;DR

This paper characterizes real holomorphic chains on complex manifolds through rectifiability, closedness, and support conditions, and applies this to understanding homology classes represented by algebraic cycles.

## Contribution

It provides a new characterization of real holomorphic chains and applies this to the study of algebraic cycle representations of homology classes.

## Key findings

- Characterization of real holomorphic chains via rectifiability and closedness.
- Application to homology classes represented by algebraic cycles.
- Establishment of conditions for a current to be a real holomorphic chain.

## Abstract

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.