Szegö kernel asymptotics and Morse inequalities on CR manifolds with $S^1$ action

Abstract

Let $X$ be a compact connected CR manifold of dimension $2n-1,n\ge 2$. We assume that there is a transversal CR locally free $S^1$ action on $X$. Let $L^k$ be the $k$-th power of a rigid CR line bundle $L$ over $X$. Without any assumption on the Levi-form of $X$, we obtain a scaling upper-bound for the partial Szegő kernel on $(0,q)$-forms with values in $L^k$. After integration, this gives the weak Morse inequalities. By a refined spectral analysis, we also obtain the strong Morse inequalities in CR setting. We apply the strong Morse inequalities to show that the Grauert-Riemenschneider criterion is also true in the CR setting.

34 pages, final version, to appear in the Asian Journal of Mathematics. arXiv admin note: text overlap with arXiv:1204.4810, arXiv:1401.6647

Cited by in corpus (6)

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• The asymptotics of the analytic torsion on CR manifolds with $S^1$ action
• Local Demailly-Bouche's holomorphic Morse inequalities
• CR eigenvalue estimate and Kohn-Rossi cohomology
• The anomaly formula of the analytic torsion on CR manifolds with $S^1$ action