#### Vol. 304, No. 2, 2020

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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
##### Abstract

Let $X$ be a compact connected CR manifold of dimension $2n+1$, $n\ge 1$. Let $\stackrel{˜}{X}$ be a paracompact CR manifold with a transversal CR ${S}^{1}$-action, such that there is a discrete group $\Gamma$ acting freely on $\stackrel{˜}{X}$ having $X=\stackrel{˜}{X}∕\Gamma$. 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 $\stackrel{˜}{X}$. 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
Primary: 32V20
Secondary: 58J35