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 ${\widehat{H}}^{\ast }(X,\ast ,\ast )$ that plays the role of Harvey-Lawson spark group ${\widehat{H}}^{\ast }(X,\ast )$, and a cohomology $H^_{ABC}(X; \Z(, ))$thatplaystheroleofDelignecohomology$H^{$\mathcal{D}$}(X; \Z(*))$foreverycomplexmanifold$X$\mathrm{.}Theyfitintheshortexactsequence$$ 0\rightarrow H^{k+1}{ABC}(X; \Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{\delta_1}{\rightarrow} Z^{k+1}_I(X, p, q) \rightarrow 0$$and$\widehat{H}^{\bullet}(X, \bullet, \bullet)$ 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 $H_{ABC}^{\bullet }(X;\mathbb{Z}(\bullet ,\bullet ))$ inherited from ${\widehat{H}}^{\bullet }(X,\bullet ,\bullet )$ is compatible with the one of the analytic Deligne cohomology $H^{\bullet }(X;\mathbb{Z}(\bullet ))$. We compute ${\widehat{H}}^{\ast }(X,\ast ,\ast )$ for $X$ the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.