CR eigenvalue estimate and Kohn-Rossi cohomology (Q6561319)

The study of Kohn-Rossi cohomology and CR eigenvalue estimate is one of the central topics in CR geometry, which has many remarkable applications to the theory of singularities, complex Plateau problem and embedding problem etc. Most works focus on strongly pseudoconvex CR manifolds.\N\NIn this paper, the authors study the difficult case on weakly pseudoconvex CR manifolds. Assume \(X\) is a compact connected weakly pseudoconvex CR manifold admitting a transversal CR \(S^1\)-action. A sharp growth order estimate of the number of eigenvalues smaller than or equal \(\lambda\) of the Kohn Laplacian is established, which is a significant progress in the eigenvalue estimate problem.\N\NThe authors also obtain a Serre-type duality theorem for Fourier components of Kohn-Rossi cohomology with respect to the \(S^1\)-action. Then they deduce an estimate of the dimensions of the Fourier components of the Kohn-Rossi cohomology as the component degree tends to \(-\infty\), which greatly improves the result by \textit{C.-Y. Hsiao} and \textit{X. Li} [Math. Z. 284, No. 1--2, 441--468 (2016; Zbl 1352.32013)]. Moreover, they generalize the eigenvalue estimate result to CR orbifold case and establish isomorphism of Grauert-type theorem in the orbifold level. As a consequence, Berndtsson's estimate is extended to the orbifold case, which answers a folklore open question. Other important applications of the main results include Morse-type inequalities, asymptotic Riemann-Roch type theorem and Grauert-Riemenschneider type criterion.\N\NBy using BRT coordinate, the authors reduce the CR estimate to the complex geometry setting with deep insight. The scaling technique is also developed in this paper.