Asian Journal of Mathematics

Volume 22 (2018)

Number 3

Special issue in honor of Ngaiming Mok (2 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

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

Pages: 413 – 450



Chin-Yu Hsiao (Institute of Mathematics, Academia Sinica and National Center for Theoretical Sciences, Taipei, Taiwan)

Xiaoshan Li (School of Mathematics and Statistics, Wuhan University, Hubei, China; and Institute of Mathematics, Academia Sinica, Taipei, Taiwan)


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 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.


CR manifolds, $S^1$-action, Szegő kernel asymptotics, Morse inequalities

2010 Mathematics Subject Classification

32A25, 32J25, 32V20

Chin-Yu Hsiao was partially supported by Taiwan Ministry of Science of Technology project 104-2628-M-001-003-MY2, by the Golden-Jade fellowship of Kenda Foundation, and by the Academia Sinica Career Development Award.

Xiaoshan Li was supported by the National Natural Science Foundation of China (Grant No. 11501422).

Received 30 June 2016

Accepted 1 June 2017

Published 8 August 2018