  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.
