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We propose a version of the Hodge conjecture in Bott-Chern cohomology andusing results from characterizing real holomorphic chains by real rectifiablecurrents to provide a proof for this question. We define a Bott-Cherndifferential cohomology and use atomic section theory of Harvey and Lawson toconstruct refined Bott-Chern classes for holomorphic vector bundles in thisdifferential cohomology. These refined Bott-Chern classes transform naturallyto standard Chern classes, Bott-Chern classes and Cheeger-Simons' refined Chernclasses.

Bott-Chern homology, Bott-Chern differential cohomology and the Hodge conjecture