Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bergman kernels and eigenvalue estimate of $\bar{\partial}$-laplacian   Let $(X,\omega)$ be a compact K\"{a}hler manifold. Let $(L,h)$ be a hermitian
holomorphic line bundle over $X$, such that $\Theta_{L,h}\geq
-\varepsilon\omega$ for a small $\varepsilon>0$, $E$ be a holomorphic line
bundle over $X$. For $k\in \mathbb{N}_+$, denote by $X_k:=(X,\omega^k)$ the
K\"{a}hler manifold $X$ with new scaled metric $\omega^k=k\omega$. Estimates of
the number of eigenvalues smaller than $\lambda$ of the $\debar$-Laplacian on
forms on $X_k$ with values in $L^k\otimes E$ are presented for $0\leq
\lambda<k$. In particular, when $\lambda=0$, we get a numeric bound for the
cohomology groups.

23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ