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

Research paper by Jyh-;Haur Teh

Indexed on: 05 Sep '16Published on: 01 Sep '16Published in: Annals of Global Analysis and Geometry

#### 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. 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. $$\widehat{H}^*(X, *, *)$$ $$\widehat{H}^*(X, *, *)$$ $$\widehat{H}^*(X, *)$$ $$\widehat{H}^*(X, *)$$ $$H^*_{\mathrm{ABC}}(X; \mathbb Z(*, *))$$ $$H^*_{\mathrm{ABC}}(X; \mathbb Z(*, *))$$ $$H^*_{\mathcal {D}}(X; \mathbb Z(*))$$ $$H^*_{\mathcal {D}}(X; \mathbb Z(*))$$X \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} \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} $$\widehat{H}^{\bullet }(X, \bullet , \bullet )$$ $$\widehat{H}^{\bullet }(X, \bullet , \bullet )$$X $$H^{\bullet }_{\mathrm{ABC}}(X; \mathbb Z(\bullet , \bullet ))$$ $$H^{\bullet }_{\mathrm{ABC}}(X; \mathbb Z(\bullet , \bullet ))$$ $$\widehat{H}^{\bullet }(X, \bullet , \bullet )$$ $$\widehat{H}^{\bullet }(X, \bullet , \bullet )$$ $$H^{\bullet }(X; \mathbb Z(\bullet ))$$ $$H^{\bullet }(X; \mathbb Z(\bullet ))$$ $$\widehat{H}^*(X, *, *)$$ $$\widehat{H}^*(X, *, *)$$X