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 arXiv:2310.08048 to study the asymptotic behavior of the Bergman kernels and spectral kernels with respect to the space of global holomorphic 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 arXiv:2012.12019. The manifold M may be non-K\"ahler 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.