在這項計畫中主持人計劃研究在高餘維緊緻CR流形上具有局部自由CR緊緻李群作用的解析扭率的漸近展開。最近Finski將Bismut-Vasserot漸近公式推廣到緊緻複orbifold上。這計畫應與S. Finski關於Bismut-Vasserot的緊緻複orbifold的漸近公式的結果密切相關。其他可能應用為緊緻複流形上正向量叢對稱冪的Bismut-Vasserot漸近公式的推廣。另一個項目是與中研院蕭欽玉和邵國寬（中國中山大學珠海分校）合作，我們在一個緊緻的CR流形上利用李群作用G來研究G等變Szego核。我們將建立G等變的Boutet de Monvel-Sjostrand型定理。當CR流形是具有S^1作用的強擬凸CR流形時，我們計劃計算G等變Szego核函數的漸近展開的前幾項的係數。我們的主要工具是Hormander的固定相公式。我們期望函數的G等變Szego核函數展開的前幾個較低階項的係數將包含一些幾何量。 ;In this project the PI plan to study the asymptotic expansion of the analytic torsion on a compact CR manifold of high codimension with a compact Lie group locally free CR action. Recently S. Finski generalized the Bismut-Vasserot asymptotic formula to setting of compact complex orbifolds. This project should be closely related to the results of Finski on Bismut-Vasserot's asymptotic formula on compact complex orbifolds. One other possible application will be a generalization of the Bismut-Vasserot asymptotic formula for symmetric powers of a positive vector bundle over a compact complex manifold.Another project is jointly with Chin-Yu Hsiao (Academia Sinica) and Guokuan Shao (Sun Yat-Sen University (Zhuhai), China), we study G-equivariant Szego kernels on a compact CR manifold with compact Lie group action G associated with all equivalent classes of irreducible unitary representations of G. We shall establish G-equivariant Boutet de Monvel-Sjostrand type theorems. When the CR manifold is strongly pseudoconvex with S^1 action, we shall compute coefficients of the first few terms of the asymptotic expansions of G-equivariant Szego kernel functions. The main tool of our appraoch is Hormander's stationary phase formula. We expect that the coefficients of the first few lower order terms of the G-equivalent Szego kernel function's expansion for functions will contain some geometric quantities.
