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

TL;DR

This paper explores cohomologies on almost complex manifolds, introducing new tools to distinguish non-integrable structures and extend classical lemmas like the $ar{ ext{d}}$-lemma to non-integrable cases.

Contribution

It defines and studies the $N$-cohomology and $J$-cohomology, extending their applicability to non-integrable almost complex structures and relating them to the $ ext{d} ext{L}_J$-lemma.

Findings

$H^{ullet}_N (M)$ distinguishes non-isomorphic non-integrable structures.

$H^{ullet}_J (M)$ encodes the $ar{ ext{d}}$-lemma and its non-integrable analogue.

Finite-dimensionality of $H^k_J$ shown for compact integrable cases, with partial results for non-integrable cases.

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 $H^{\bullet}_N (M)$, 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.