Abstract
By comparing Deligne complex and Aeppli–Bott-Chern complex, we construct a differential cohomology (Formula presented.) that plays the role of Harvey–Lawson spark group (Formula presented.), and a cohomology (Formula presented.) that plays the role of Deligne cohomology (Formula presented.) for every complex manifold X. They fit in the short exact sequence (Formula presented.)and (Formula presented.) 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 (Formula presented.) inherited from (Formula presented.) is compatible with the one of the analytic Deligne cohomology (Formula presented.). We compute (Formula presented.) for X the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.