Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cohomologies on almost complex manifolds and the $\partial
  \bar{\partial}$-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
$H^{\bullet}_N (M)$, 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.

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The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
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The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
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The categories of this paper are: eess
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs