Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A logarithmic $\bar{\partial}$-equation on a compact K\"ahler manifold
  associated to a smooth divisor   In this paper, we solve a logarithmic $\bar{\partial}$-equation on a compact
K\"ahler manifold associated to a smooth divisor by using the cyclic covering
trick. As applications, we discuss the closedness of logarithmic forms,
injectivity theorems and obtain a kind of degeneration of spectral sequence at
$E_1$, and we also prove that the pair $(X,D)$ has unobstructed deformations
for any smooth divisor $D\in|-2K_X|$. Lastly, we prove some logarithmic
vanishing theorems for $q$-ample divisors on compact K\"ahler manifolds.

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The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
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The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th