Spectral kernels and holomorphic Morse inequalities for sequence of line bundles

Given a sequence of Hermitian holomorphic line bundles (Lₖ,hₖ) over a complex manifold M which may not be compact, we generalize the scaling method in [5] to study the asymptotic behavior of the Bergman kernels and spectral kernels with respect to the Kodaira Laplacian ₖ on the space of sections of Lₖ with $(0,q)$-forms. We derive the leading term of the Bergman and spectral kernels under the local convergence assumption in the sequence of Chern curvatures c₁(Lₖ,hₖ), inspired by [6]. The manifold M may be non-Kähler and c₁(Lₖ,hₖ) may be negative or degenerate. Moreover, we establish the Lₖ-asymptotic version of Demailly's holomorphic Morse inequalities as an application to compact complex manifolds.