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arXiv · 1502.02365

Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds with $S^1$ action

Chin-Yu Hsiao·Xiaoshan Li
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Abstract

Let $X$ be a compact connected CR manifold of dimension $2n-1, n\geq 2$. 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 Szeg\H{o} 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.

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Chin-Yu Hsiao, Xiaoshan Li. 2015-02-09. Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds with $S^1$ action. https://arxiv.org/abs/1502.02365

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