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

Abstract

By comparing Deligne complex and Aeppli–Bott-Chern complex, we construct a differential cohomology $\widehat{H}^*(X, *, *)$ that plays the role of Harvey–Lawson spark group $\widehat{H}^*(X, *)$, and a cohomology $H^*_{\mathrm{ABC}}(X; \mathbb Z(*, *))$ that plays the role of Deligne cohomology $H^*_{\mathcal {D}}(X; \mathbb Z(*))$ for every complex manifold X. They fit in the short exact sequence

$$\begin{aligned} 0\rightarrow H^{k+1}_{\mathrm{ABC}}(X; \mathbb Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{\delta _1}{\rightarrow } Z^{k+1}_I(X, p, q) \rightarrow 0 \end{aligned}$$

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 $H^{\bullet }_{\mathrm{ABC}}(X; \mathbb Z(\bullet , \bullet ))$ 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 $\widehat{H}^*(X, *, *)$ for X the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.