CR特征值估计与Kohn-Rossi上同调
CR eigenvalue estimate and Kohn-Rossi cohomology
AI总结：
本文针对弱伪凸CR流形，获得$\\(\Box_b\\)$在Fourier分量上的尖锐特征值估计，并建立Kohn-Rossi上同调的Serre对偶及渐近增长，应用于Morse不等式等。
摘要：
设$X$是一个紧致连通的CR流形，具有横截CR $S^1$-作用，实维数为$2n-1$，仅假设其为弱伪凸。令$\Box_b$为关于$T$-刚性Hermitian度量（见$T$-刚性Hermitian度量的定义3.2）的$\overline{\partial}_b$-Laplacian。$\Box_b$的特征值估计是CR几何与分析中的基本问题。在本文中，我们能够获得$\Box_b$作用于$X$上光滑$(n-1,q)$-形式的第$m$个Fourier分量时，小于或等于$\lambda$的特征值个数的尖锐估计，其中$m\in \mathbb{Z}_+$，$q=0,1,\cdots, n-1$。这里“尖锐”指的是关于$m$的增长阶是尖锐的。特别地，当$\lambda=0$时，我们得到当$m \rightarrow +\infty$时，$H^{n-1,q}_b(X)$的第$m$个Fourier分量$H^{n-1,q}_{b,m}(X)$的增长渐近估计。此外，我们建立了Kohn-Rossi上同调的Fourier分量的Serre型对偶定理，该定理具有独立的意义。作为副产品，我们建立了$m\in \mathbb{Z}_+$时Fourier分量$H^{0,q}_{b,-m}(X)$维数的渐近增长。我们还给出了主要结果的应用，包括Morse型不等式、渐近Riemann-Roch型定理、Grauert-Riemenscheider型准则，以及我们主要结果的orbifold版本，该版本回答了Hsiao告知我们的一个民间开放问题。
英文摘要：
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 $\Box_b$ 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 $\Box_b$ 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 $λ$ of $\Box_b$ acting on the $m$-th Fourier components of smooth $(n-1,q)$-forms on $X$, where $m\in \mathbb{Z}_+$ and $q=0,1,\cdots, n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $λ=0$, we obtain the asymptotic estimate of the growth for $m$-th Fourier components $H^{n-1,q}_{b,m}(X)$ of $H^{n-1,q}_b(X)$ as $m \rightarrow +\infty$. 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 appilcations of our main results, including Morse type inequalities, asymptotic Riemann-Roch type theorem, Grauert-Riemenscheider type criterion, and an orbifold version of our main results which provides an answer towards a folklore open problem informed to us by Hsiao.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
