CR eigenvalue estimate and Kohn–Rossi cohomology

Abstract

$\def\mth{ m\textrm{-th}}$ Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of real dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $\square_b$ be the $\overline{\partial}_b$-Laplacian, with respect to a $T$-rigid Hermitian metric (see Definition 3.2 of $T$-rigid Hermitian metric). Eigenvalue estimate of $\square_b$ is a fundamental issue both in CR geometry and analysis. In this paper, we are able to obtain a sharp estimate of the number of eigenvalues smaller than or equal to $\lambda$ of $\square_b$ acting on the $\mth$ Fourier components of smooth $(n-1,q)$-forms on $X$, where $m \in \mathbb{Z}_{+}$ and $q = 0, 1, \dotsc , n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $\lambda = 0$, we obtain the asymptotic estimate of the growth for $\mth$ Fourier components $H^{n-1,q}_{b,m} (X)$ of $H^{n-1,q}_b (X)$ as $m \to + \infty$. Furthermore, we establish a Serre type duality theorem for Fourier components of Kohn–Rossi cohomology which is of independent interest. As a byproduct, the asymptotic growth of the dimensions of the Fourier components $H^{0,q}_{b,-m} (X)$ for $m \in \mathbb{Z}_{+}$ is established. We also give applications of our main results, including Morse type inequalities, asymptotic Riemann–Roch type theorem, Grauert–Riemenschneider type criterion, and an orbifold version of our main results which provides an answer towards a folklore open problem informed to us by Hsiao.