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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