Szegö kernel asymptotics and Morse inequalities on CR manifolds

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds.

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