[Paper Review] Morse inequalities for Fourier components of Kohn-Rossi cohomology of CR manifolds with $S^1$-action

Chin-Yu Hsiao, Xiaoshan Li|arXiv (Cornell University)|Jun 22, 2015

Holomorphic and Operator Theory15 references6 citations

TL;DR

This paper establishes Morse inequalities for the Fourier components of Kohn-Rossi cohomology on compact CR manifolds of dimension $2n-1$ ($n \geq 2$) equipped with a transversal CR $S^1$-action. By analyzing the Szegő kernel of these components, the authors derive asymptotic estimates for the dimensions of positive Fourier modes, proving that weakly pseudoconvex CR manifolds with strong pseudoconvexity at a point possess abundant CR functions, resolving a key question on the growth of $H^0_{b,m}(X)$.

ABSTRACT

Let $X$ be a compact connected CR manifold of dimension $2n-1, n\geq 2$ with a transversal CR $S^1$-action on $X$. We study the Fourier components of the Kohn-Rossi cohomology with respect to the $S^1$-action. By studying the Szegö kernel of the Fourier components we establish the Morse inequalities on $X$. Using the Morse inequalities we have established on $X$ we prove that there are abundant CR functions on $X$ when $X$ is weakly pseudoconvex and strongly pseudoconvex at a point.

Motivation & Objective

• To understand the asymptotic growth of the dimensions of positive Fourier components of Kohn-Rossi cohomology on CR manifolds with $S^1$-action.
• To address the question of when $\dim H^0_{b,m}(X) \approx m^{n-1}$ for large $m$, which is central to CR embedding problems.
• To prove the existence of abundant CR functions on weakly pseudoconvex CR manifolds that are strongly pseudoconvex at a point, via Morse-type estimates.

Proposed method

• The authors analyze the Szegő kernel of the Fourier components of the $\overline{\partial}_b$-complex to derive spectral estimates.
• They apply the scaling technique in local coordinates to study the behavior of the Szegő kernel near the base point of the $S^1$-action.
• The proof relies on canonical local coordinates adapted to the $S^1$-action and the use of the scaling limit to extract leading-order asymptotics.
• The authors establish weak and strong Morse inequalities for the $m$-th Fourier components of Kohn-Rossi cohomology as $m \to \infty$, using spectral theory and microlocal analysis.
• They analyze the structure of the fixed point set of the $S^1$-action and prove that the set of points with nontrivial isotropy has measure zero and is nowhere dense.
• The key technical step involves showing that $\lim_{m \to \infty} m^{-(n-1)} |\alpha_{m,1}(p)|^2 = \frac{|\lambda_1(p) \cdots \lambda_{n-1}(p)|}{2\pi^n}$, which yields the precise asymptotic growth of the Szegő kernel.

Experimental results

Research questions

• RQ1 Under what conditions does $\dim H^0_{b,m}(X) \approx m^{n-1}$ hold for large $m$?
• RQ2 Can Morse inequalities be established for the Fourier components of Kohn-Rossi cohomology on CR manifolds with $S^1$-action?
• RQ3 Does the existence of a strongly pseudoconvex point in a weakly pseudoconvex CR manifold with $S^1$-action imply the abundance of CR functions in positive Fourier modes?
• RQ4 How does the $S^1$-action influence the spectral properties of the $\overline{\partial}_b$-complex on CR manifolds?
• RQ5 What is the precise asymptotic behavior of the Szegő kernel for the $m$-th Fourier component as $m \to \infty$?

Key findings

• The authors establish weak and strong Morse inequalities for the $m$-th Fourier components of Kohn-Rossi cohomology on compact CR manifolds with transversal $S^1$-action.
• For large $m$, the dimension of the $m$-th positive Fourier component of CR functions satisfies $\dim H^0_{b,m}(X) \sim c \cdot m^{n-1}$, where $c$ is a constant depending on the Levi-form at strongly pseudoconvex points.
• The limit $\lim_{m \to \infty} m^{-(n-1)} |\alpha_{m,1}(p)|^2 = \frac{|\lambda_1(p) \cdots \lambda_{n-1}(p)|}{2\pi^n}$ is proven, giving the exact leading-order asymptotic of the Szegő kernel at a strongly pseudoconvex point $p$.
• The set of points with nontrivial $S^1$-isotropy has measure zero and is nowhere dense, ensuring the genericity of regular points in the asymptotic analysis.
• The paper proves that a compact weakly pseudoconvex CR manifold with a transversal CR $S^1$-action and a point of strong pseudoconvexity admits an abundance of CR functions in positive Fourier modes.
• The asymptotic growth of the Szegő kernel is shown to be controlled by the eigenvalues of the Levi form, confirming the geometric significance of the curvature at strongly pseudoconvex points.