Aeppli-Bott-Chern cohomology and Deligne cohomology from a viewpoint of Harvey-Lawson's spark complex

By comparing Deligne complex and Aeppli-Bott-Chern complex, we construct a differential cohomology (H) over cap*(X, *, *) that plays the role of Harvey-Lawson spark group (H) over cap*(X, *), and a cohomology H-ABC*(X; Z(*, *)) that plays the role of Deligne cohomology H-D* (X; Z(*)) for every complex manifold X. They fit in the short exact sequence 0 -> H-ABC(k+1)(X; Z(p, q)) -> (H) over cap (k)(X, p, q) -> Z(1)(k+1)(X, p, q) -> 0 and (H) over cap*(X, center dot, center dot) possess ring structure and refined Chern classes, acted by the complex conjugation, and if some primitive cohomology groups of X vanish, there is a Lefschetz isomorphism. Furthermore, the ring structure of H-ABC center dot(X; Z(center dot, center dot)) inherited from H-center dot(X, center dot, center dot) is compatible with the one of the analytic Deligne cohomology H-center dot(X; Z(center dot)). We compute (H) over cap*(X, *, *) for X the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.