多复变与复几何中的一些上同调群的维数的渐近估计

本项目主要研究多复变与复几何中的一些上同调群的维数的渐近估计问题. 这类问题在多复变与复几何中具有重要的意义。 我们将综合多复变与复几何中的正性的理论和技巧，结合奇点消解，Leray 谱序列，偏微分方程方法，群作用，局部化，以及多复分析的方法和理论，来研究这些问题. 我们将重点研究紧 Hermitian (Kähler) 流形上 nef (pseudo-effective) 丛的 k 次张量幂 ( k 次张量幂张量相应的乘子理想层) 的 Dobeault (Cech) 上同调群的维数的渐近估计和具有特殊群作用的弱拟凸紧 CR 流行的 Kohn-Rossi 上同调群的 Fourier 系数的维数的渐近最优估计问题。本项目拟解决上述三个问题，探究正性对于上同调群的重要影响, 进一步推动多复变与复几何的相关问题的研究。

This project is to study the asymptotic estimate of the dimensions of some cohomology groups in several complex variables and complex geometry. Problems of this kind are of great significance in several complex variables and complex geometry. We will combine the theories and techniques of positivities in several complex variables and complex geometry, with resolution of singularities, Leray spectral sequence, estimates in partial differential equation, group actions, localizations and theories and techniques in complex analysis to study these issues. We will focus on the asymptotic estimate of the dimensions of the Dobeault cohomology groups of k-th tensor power of nef line bundles on compact Hermitian or Kähler manifolds (Demailly Problem), the asymptotic estimate of the dimensions of the Cech cohomology groups of k-th tensor power of pseudo-effective line bundles tensored with the corresponding multiplier ideal sheaves on compact Hermitian or Kähler manifolds (Matsumura Problem), the sharp estimate of dimensions of the Fourier components of Kohn-Rossi cohomology groups of CR manifold with S^1-action. The project intends to solve the above three problems, to explore the great influence of positivities on the cohomology groups, further promote the study of corresponding problems in several complex variables and complex geometry.