Spectral Kernels and Holomorphic Morse Inequalities for Sequence of Line Bundles

TL;DR

This paper extends spectral kernel analysis and Morse inequalities to sequences of line bundles over possibly non-compact, non-Kähler manifolds, including cases with degenerate curvatures, revealing asymptotic behaviors of Bergman kernels.

Contribution

It generalizes the scaling method to study asymptotics of Bergman and spectral kernels for line bundles with possibly degenerate curvatures on non-compact, non-Kähler manifolds.

Findings

Derived the leading term of Bergman and spectral kernels under local convergence assumptions.

Established an asymptotic version of Demailly's holomorphic Morse inequalities.

Extended analysis to cases with negative or degenerate Chern curvatures.

Abstract

Given a sequence of Hermitian holomorphic line bundles $(L_k,h_k)$ 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 $L_k$ 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 $c_1(L_k,h_k)$, inspired by arXiv:2012.12019. The manifold $M$ may be non-K\"ahler and $c_1(L_k,h_k)$ may be negative or degenerate. Moreover, we establish the $L_k$-asymptotic version of Demailly's holomorphic Morse inequalities as an application to compact complex manifolds.