Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dolbeault Harmonic $(1,1)$-forms on $4$-dimensional compact quotients of
  Lie Groups with a left invariant almost Hermitian structure   We study Dolbeault harmonic $(1,1)$-forms on compact quotients
$M=\Gamma\backslash G$ of $4$-dimensional Lie groups $G$ admitting a left
invariant almost Hermitian structure $(J,\omega)$. In this case, we prove that
the space of Dolbeault harmonic $(1,1)$-forms on $(M,J,\omega)$ has dimension
$b^-+1$ if and only if there exists a left invariant anti self dual
$(1,1)$-form $\gamma$ on $(G,J)$ satisfying $id^c\gamma=d\omega$. Otherwise,
its dimension is $b^-$. In this way, we answer to a question by Zhang.

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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph