Morse不等式和CR流形上的热核

The Atiyah-Singer index theorem is one of the most important results in mathematics of the 20th century. The index theorem has found broad applications in the fields of differential geometry, topology, algebraic geometry, representation theory and mathematical physics. While carrying on the research of the immersion formula of the Quillen metric, Bismut and Lebeau developed analytic localization techniques in the local index theory, and these techniques have found many applications and has gradually become an important tool in the study of the global geometric properties of manifolds.. This program intends to study the following applications of the Bismut-Lebeau analytic localization techniques: (1). whether there exists Morse-Bott inequalities on covering manifolds; (2). to calculate the coefficients of the asymptotic expansion of the heat kernel of the pseudoconformal Laplacian on CR manifolds. Solutions to the these problems will help us to better interpret the global geometric properties of the corresponding underlying manifolds, especially the solution to the problem (2), will also assist us to better understand some critical problems in conformal geometry.

Atiyah-Singer 指标定理是二十世纪纯粹数学领域中的辉煌成果之一。至今，指标定理在微分几何、拓扑学、代数几何、表示论以及数学物理中有着广泛的应用。在研究Quillen度量的浸入公式时，Bismut和Lebeau建立了一套局部指标理论中的解析局部化技巧，现在这套技巧被发现有越来越广泛的应用，逐渐成为研究流形的整体几何性质的重要工具。. 本项目拟研究Bismut-Lebeau解析局部化技巧在如下两个方面的应用：（1）、研究在覆叠流形上Morse-Bott不等式的存在性及其证明；（2）、研究在CR流形上伪共形Laplace算子的热核渐近展开式的系数计算问题。这些问题的研究，将有助于我们更深入理解对应底流形的整体几何性质，特别是第(2)个问题的研究，还将有助于我们理解共形几何中的若干重要问题。

Atiyah-Singer指标理论是二十世纪纯粹数学领域重要的数学成果之一。Bismut和Lebeau从中发展了一套极具理论价值的解析局部化技巧。本项目主要研究Bismut-Lebeau解析局部化技巧的几何应用：主要包括Morse不等式及Bergman核的渐近展开等两个方面。我们证明了带边覆叠流形上的Morse不等式；证明了紧致带边流形上的Thom-Smale-Witten定理；在辛流形上，利用广义Bergman核的渐近展开式，我们得到了Donaldson Q-算子的一个估计，这个估计在Hermite数量曲率的L2范数的下界估计方面起到重要作用；另外，我们研究了对应于正则化Bochner算子的小特征值空间的Berezin-Toeplitz量子化。