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

Ki Fung Chan (Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong)

Spiro Karigiannis (Department of Pure Mathematics, University of Waterloo, Ontario, Canada)

Chi Cheuk Tsang (Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong)

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

Full Text (PDF format)

Received 16 October 2017

Accepted 22 March 2018

Published 7 January 2020