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

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 $H_N^{\bullet }(M)$, can be used to distinguish non-isomorphic non-integrable almost complex structures on $M$. Another one, the $J$-cohomology $H_J^{\bullet }(M)$, 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 $\mathrm{\partial }\bar{\mathrm{\partial }}$-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 $H_J^k$ is finite-dimensional for compact integrable $(M,J)$, and use spectral sequences to establish partial results on the finite-dimensionality of $H_J^k$ in the compact non-integrable case.