Contents Online

# Asian Journal of Mathematics

## Volume 23 (2019)

### Number 4

### Cohomologies on almost complex manifolds and the $\partial \overline{\partial}$-lemma

Pages: 561 – 584

DOI: https://dx.doi.org/10.4310/AJM.2019.v23.n4.a2

#### Authors

#### Abstract

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 $H^{●}_N (M)$, can be used to distinguish non-isomorphic non-integrable almost complex structures on $M$. Another one, the $J$-cohomology $H^{●}_J (M)$, 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 $\partial \overline{\partial}$-lemma, and more generally in the non-integrable case the $J$-cohomology encodes whether (M, J) satisfies the $d\mathcal{L}_J$-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.

#### Keywords

derivations, almost complex manifolds, cohomology, $\partial \overline{\partial}$-lemma, non-integrable

#### 2010 Mathematics Subject Classification

32Q60, 53C55, 53C56, 55T25

Received 16 October 2017

Accepted 22 March 2018

Published 7 January 2020