Morse inequalities for fourier components of kohn-rossi cohomology of CR covering manifolds with s¹-action

Let X be a compact connected CR manifold of dimension 2_n+1, n ≥ 1. Let X be a paracompact CR manifold with a transversal CR S¹-action, such that there is a discrete group Γ acting freely on X having X = X=Γ. Based on an asymptotic formula for the Fourier components of the heat kernel with respect to the S¹-action, we establish the Morse inequalities for Fourier components of reduced L²-Kohn-Rossi cohomology with values in a rigid CR vector bundle over 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 Szegó kernel method.