Cohomologies on almost complex manifolds and the $\partial \overline{\partial}$-lemma
| Autor: | Spiro Karigiannis, Ki Fung Chan, Chi Cheuk Tsang |
|---|---|
| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Asian Journal of Mathematics. 23:561-584 |
| ISSN: | 1945-0036 1093-6106 |
| DOI: | 10.4310/ajm.2019.v23.n4.a2 |
| Popis: | We study cohomologies on an almost complex manifold $(M, J)$, defined using the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as vector-valued forms on $M$. We show how one of these, the $N$-cohomology $H^{\bullet}_N (M)$, can be used to distinguish non-isomorphic non-integrable almost complex structures on $M$. Another one, the $J$-cohomology $H^{\bullet}_J (M)$, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The $J$-cohomology encodes whether a complex manifold satisfies the $\partial \bar{\partial}$-lemma, and more generally in the non-integrable case the $J$-cohomology encodes whether $(M, J)$ satisfies the $\mathrm{d} \mathcal{L}_J$-lemma, which we introduce and motivate in this paper. We discuss several explicit examples in detail, including a non-integrable example. We also show that $H^k_J$ is finite-dimensional for compact integrable $(M, J)$, and use spectral sequences to establish partial results on the finite-dimensionality of $H^k_J$ in the compact non-integrable case. Comment: 23 pages. Version 4: Changes made after publication: We corrected a misstatement in Theorem 3.23 and improved the original notation of Theorem 3.32 which caused unnecessary confusion. The authors thank Scott Wilson for alerting us to these issues |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|