Asian Journal of Mathematics
Volume 22 (2018)
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
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