Morse inequalities for Fourier components of Kohn–Rossi cohomology of CR covering manifolds with $S^1$-action

Rung-Tzung Huang and Guokuan Shao

Vol. 304 (2020), No. 2, 439–462

DOI: 10.2140/pjm.2020.304.439

Abstract

Let $X$ be a compact connected CR manifold of dimension $2n+1$, $n\ge 1$. Let $\tilde{X}$ be a paracompact CR manifold with a transversal CR $S^1$-action, such that there is a discrete group $\Gamma$ acting freely on $\tilde{X}$ having $X=\tilde{X}∕\Gamma$. Based on an asymptotic formula for the Fourier components of the heat kernel with respect to the $S^1$-action, we establish the Morse inequalities for Fourier components of reduced $L^2$-Kohn–Rossi cohomology with values in a rigid CR vector bundle over $\tilde{X}$. As a corollary, we obtain the Morse inequalities for Fourier components of Kohn–Rossi cohomology on $X$ which were obtained by Hsiao and Li (2016) by using Szegő kernel method.

Keywords

Kohn–Rossi cohomology, heat kernel, CR manifold

Mathematical Subject Classification 2010

Primary: 32V20

Secondary: 58J35

Milestones

Received: 13 September 2018

Revised: 2 September 2019

Accepted: 3 September 2019

Published: 12 February 2020