Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles

TL;DR

This paper investigates the asymptotic behavior of spectral functions and Bergman kernels associated with semi-positive and big line bundles, providing detailed expansions and applications to complex geometry.

Contribution

It offers a comprehensive asymptotic expansion of spectral functions for the Kodaira Laplacian on high tensor powers of line bundles, including cases with singular metrics.

Findings

Asymptotic expansion of spectral functions on non-degenerate curvature regions

Bergman kernel asymptotics for semi-positive line bundles on complete Kähler manifolds

Asymptotics for big line bundles with singular Hermitian metrics

Abstract

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete Kaehler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.