Bergman kernels and eigenvalue estimate of $\bar{\partial}$-laplacian

Let $(X,ω)$ be a compact Kähler manifold. Let $(L,h)$ be a hermitian holomorphic line bundle over $X$, such that $Θ_{L,h}\geq -\varepsilonω$ for a small $\varepsilon>0$, $E$ be a holomorphic line bundle over $X$. For $k\in \mathbb{N}_+$, denote by $X_k:=(X,ω^k)$ the Kähler manifold $X$ with new scaled metric $ω^k=kω$. Estimates of the number of eigenvalues smaller than $λ$ of the $\debar$-Laplacian on forms on $X_k$ with values in $L^k\otimes E$ are presented for $0\leq λ<k$. In particular, when $λ=0$, we get a numeric bound for the cohomology groups.

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