解析Hilbert模的本质正规性与渐进表示

本项目主要研究单位球和多圆盘上解析函数Hilbert模的本质正规性. 这是Hilbert模几何分析的中心问题之一，与代数几何、C*-扩张理论、BDF理论、指标理论等数学分支有深刻的联系. 单位球上Bergman子模的本质正规性与渐近表示有着本质的联系. 本项目拟首先对此展开考察以进一步深化对Hilbert模的本质正规性的研究，从而找到更广泛类型的本质正规子模. 其次，多圆盘上Hilbert模的本质正规性与对应零簇的边界性质有内在的联系，本项目拟通过维数理论与Berezin变换等理论工具对此作进一步的考察，并尽可能地刻画乘法算子换位子的紧性以及相应的K-同调元的非平凡性.

The major object of this project is to study the essential normality of analytic function Hilbert modules over the unit balls and the polydiscs. Essential normality is one of the central problems of geometric analysis on Hilbert modules, and is in deep link to areas of mathematics such as algebraic geometry, C*-extension theory, BDF-theory, index theory, etc. Essential normality of Bergman submodules over the unit ball is intrinsically related to their approximate representations. This project will first make investigations on approximate representation, to deepen the study on essential normality of Hilbert modules, and to find more general classes of essentially normal submodules. Secondly, essential normality of Hilbert modules over the polydiscs is intrinsically connected to boundary properties of their algebraic variety, and we will make deep investigations on it by tools of dimension theory and Berezin transform etc., and make best efforts to characterize the compactness of commutators of multiplication operators and non-triviality of the K-homology elements.