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 ${\hat{H}}^{\ast }(X,\ast ,\ast )$ that plays the role of Harvey-Lawson spark group ${\hat{H}}^{\ast }(X,\ast )$, and a cohomology $H_{ABC}^{\ast }(X;\mathbb{Z}(\ast ,\ast ))$ that plays the role of Deligne cohomology $H_{\mathcal{D}}^{\ast }(X;\mathbb{Z}(\ast ))$ for every complex manifold $X$. They fit in the short exact sequence $$0\to H_{ABC}^{k+1}(X;\mathbb{Z}(p,q))\to {\hat{H}}^k(X,p,q)\stackrel{{\delta }_1}{\to }{Z}_I^{k+1}(X,p,q)\to 0$$ and ${\hat{H}}^{\bullet }(X,\bullet ,\bullet )$ 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 ${\hat{H}}^{\bullet }(X,\bullet ,\bullet )$ is compatible with the one of the analytic Deligne cohomology $H^{\bullet }(X;\mathbb{Z}(\bullet ))$. We compute ${\hat{H}}^{\ast }(X,\ast ,\ast )$ for $X$ the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.

arXiv comment20 pages