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

TL;DR

This paper constructs a new differential cohomology theory for complex manifolds, connecting Aeppli-Bott-Chern and Deligne cohomologies via Harvey-Lawson's spark complex, and computes it for Iwasawa manifolds.

Contribution

It introduces a novel cohomology framework unifying Aeppli-Bott-Chern and Deligne cohomologies through spark complexes, with applications to complex manifold classification.

Findings

Constructed a differential cohomology $\widehat{H}^*$ linking existing theories.

Established ring structures and refined Chern classes within the new framework.

Computed cohomology for Iwasawa manifolds, refining Nakamura's classification.

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 $H^*_{ABC}(X; \Z(*, *))$ that plays the role of Deligne cohomology $H^*_{\mathcal{D}}(X; \Z(*))$ for every complex manifold $X$. They fit in the short exact sequence $$ 0\rightarrow H^{k+1}_{ABC}(X; \Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{\delta_1}{\rightarrow} Z^{k+1}_I(X, p, q) \rightarrow 0$$ 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 $H^{\bullet}_{ABC}(X; \Z(\bullet, \bullet))$ inherited from $\widehat{H}^{\bullet}(X, \bullet, \bullet)$ is compatible with the one of the analytic Deligne cohomology $H^{\bullet}(X; \Z(\bullet))$. We compute $\widehat{H}^*(X, *, *)$ for $X$ the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.