解析局部化技术的若干应用研究

In the project we intend to study holomorphic Morse inequalities and Bergman kernels, which have found so far a large number of important applications in complex geometry, symplectic geometry and several complex variables theory, for example, the existence of Kähler metrics of constant scalar curvature, the solution to Grauert-Riemenschneider conjecture, Berezin-Toeplitz quantization and the analytic proof of Kodaira embedding theorem etc. Therefore, such research in the project is of great theoretic meaning. .The applicant has made intensive research on the subjects and has obtained some meaningful results. Based on the research works, in the project we will continue to study the G-invariant holomorphic Morse inequalities on compact complex manifolds with nonempty boundary and the asymptotic expansion of Bergman kernels on compact symplectic manifolds with nonempty boundary. The solutions to these problems, especially to the second problem, will assist us to better understand related problems in local index theory, for example, the problem of Berezin-Toeplitz quantization on symplectic manifolds with boundary.

本项目拟研究全纯Morse不等式和Bergman核。这两方面的内容在复几何、辛几何及多复变函数论中有重要的应用,比如常数量曲率Kähler度量的存在性、Grauert-Riemenschneider猜想的证明、Berezin-Toeplitz量子化及Kodaira嵌入定理的解析证明等。因此，对这两方面内容的深入研究具有重要的理论意义。.项目申请人已经在这一领域进行了深入的研究，取得了一些有意义的研究成果。在此基础上，本项目拟进一步研究紧致带边复流形上紧群作用下的全纯Morse不等式及紧致带边辛流形上的Bergman核的渐近展开式。这些问题的解决，特别是第二个问题的解决，将为局部指标理论中相关问题的研究提供新的观点，比如紧致带边辛流行上的Berezin-Toeplitz量子化问题。

本项目主要研究Bismut-Lebeau解析局部化技术的几何应用。在紧致辛流形上，我们证明了由Kodaira嵌入定理诱导的Bergman形式收敛到辛形式的速度可以达到1/p^2，其中参数p表示预量化线丛的高次幂；在紧致Kähler流形上任意一列正线丛情形，在一个自然的曲率假设下，我们得到了Bergman核的渐近展开式；进一步，我们还研究了这种情形下的Berezin-Toeplitz量子化问题。