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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 Kähler 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.

ABSTRACT. Let be an abstract compact orientable CR manifold of dimension ¾Ò ½, Ò ¾, and let Ä be the -th tensor power of a CR complex line bundle Ä over . We assume that condition ´Õµ holds at each point of . In this paper we obtain a scaling upper-bound for the Szegö kernel on´¼ Õµ-forms with values in Ä , for large . After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convexconcave manifolds.

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