The $\mathcal L_B$-cohomology on compact torsion-free $\mathrm{G}_2$ manifolds and an application to 'almost' formality
| Autor: | Chan, Ki Fung, Karigiannis, Spiro, Tsang, Chi Cheuk |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Annals of Global Analysis and Geometry 55 (2019), 325-369 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10455-018-9629-x |
| Popis: | We study a cohomology theory $H^{\bullet}_{\varphi}$, called the $\mathcal L_B$-cohomology, on compact torsion-free $\mathrm{G}_2$-manifolds. We show that $H^k_{\varphi} \cong H^k_{\mathrm{dR}}$ for $k \neq 3, 4$, but that $H^k_{\varphi}$ is infinite-dimensional for $k = 3,4$. Nevertheless there is a canonical injection $H^k_{\mathrm{dR}} \to H^k_{\varphi}$. The $\mathcal L_B$-cohomology also satisfies a Poincar\'e duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $\mathrm{d}$ and the derivation $\mathcal L_B$, and uses both Hodge theory and the special properties of $\mathrm{G}_2$-structures in an essential way. As an application of our results, we prove that compact torsion-free $\mathrm{G}_2$-manifolds are 'almost formal' in the sense that most of the Massey triple products necessarily must vanish. Comment: 43 pages, 8 figures in colour. Version 3: Minor revisions following referee's report. Main result strengthened in the case of full G2 holonomy. Final version, to appear in Annals of Global Analysis and Geometry |
| Databáze: | arXiv |
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