具有群作用强拟凹复流形与高余维数CR流形的嵌入问题

In this project, we will investigate the embedding problems of strongly pseudoconcave manifolds and high codimensional CR manifolds with group actions..First, we will make use of the Morse inequalities we have established on non-compact complex manifolds with smooth boundaries and compact CR manifolds, both of which admit with S^1 actions, to study the Henkin-Marinescu conjecture which tells us that the positive line bundle on a strongly pesudoconcave complex manifold is big. Next, we will take advantage of the methods which were used to establish weighted Szego kernel expansion and Kodaira embedding theorems on CR manifolds with S^1 actions to get the Bergman kernel expansion and as a consequence to get the Kodaira embedding theorem for strongly pseudoconcave complex manifolds..The subject of embedding problems of CR manifolds is a prominent area in complex geometry, differential geometry and partial differential equation. Most of the results are related to the embedding problems of CR manifolds of one codimension. Baouendi-Rothschild-Treves’ results tell us any arbitrary codimensional CR manifold with transvesal CR group actions can be locally embedded into complex Euclidean space. Recently, we have established the Boutet de Monvel embedding theorems and Kodaira embedding theorems for CR manifolds with circle action, real line action and torus action. .We will develop the Kohn’s L^2 estimate for Cauchy-Riemann operators on high codimensional strongly pseudoconvex CR manifolds with transversal CR group actions and take advantage of the methods which we use to get the embedding theorems for CR manifolds with S^1 action and torus action to investigate embedding problems on high codimensional CR manifolds with group actions. On the other hand, we will exploit Ohsawa-Sibony’s work on how to obtain a global solution by patching the local solutions together with the aid of successive approximations to study the embedding problems on high codimensional CR manifold foliated by strongly pseudoconvex CR manifolds of hypersurface type.

本项目计划研究具有群作用强拟凹复流形与紧致高余维数CR流形的嵌入问题。.首先，我们将利用我们在具有S^1群作用的非紧复流形与紧致CR流形上所建立Morse不等式来研究具有全纯S^1群作用的强拟凹复流形上的Henkin-Marinescu猜想：维数大于或等于3的强拟凹复流形上的正定线丛一定是big的。接下来，我们将在具有全纯S^1群作用的强拟凹复流形上研究Bergman核的渐进展开以及Kodaira嵌入定理。.我们将在具有群作用的高余维数强拟凸CR流形上发展柯西黎曼算子的平方估计，利用我们在具有S^1群作用，环面群作用超曲面型CR流形上所建立的嵌入定理的方法来研究具有李群作用高余维数强拟凸CR流形上的嵌入问题。接下来，我们还将充分发展Ohsawa-Sibony在Levi平坦CR流上所建立的连续逼近的方法来研究能被超曲面型强拟凸CR流形叶化的高余维数CR流形的嵌入问题。

非紧复流形的全纯函数空间，Bergman核以及Bergman度量的微分几何性质是多复变函数论中经典研究课题，关于此类课题的研究结果已经非常丰富。在本项目中，我们假设了紧复流形有一个全纯群作用。在该假设下，我们系统研究了非紧复流形的全纯函数空间，Bergman核以及Bergman度量，以及具有群作用紧致CR流形上的分析与几何。.. 具体来讲，在本项目中，我们研究了四个课题。它们分别是具有群作用非紧复流形上的Morse不等式，具有群作用CR流形的Kodaira嵌入定理，强拟凸Stein空间上的Bergman-Einstein度量和广义郑邵远猜想以及Stein空间上多值CR函数的全纯延拓。.. 当非紧复流形有一个全纯的S^1-群作用时，我们得到了不依赖于流形凸性的Morse不等式，这是非紧复流形上Morse不等式理论的一个重要补充。当紧致CR流形有一个横截CR的R-群作用时，我们证明了该CR流形能被正定线丛的无穷光滑的CR截面嵌入到复射影空间，该结果改进了Ohsawa和Sibony教授有关Levi平坦CR流形Kodaira嵌入定理的结果。假设强拟凸Stein空间的Bergman度量是Kahler-Einstein时，我们证明了该强拟凸Stein空间的边界局部CR同构于球面。若进一步假设边界是代数流形，我们证明了该Stein空间一定全纯同构于球商。我们证明了维数为2的强拟凸Stein空间边界上的多值CR函数能全纯延拓为Stein空间正则部分的多值全纯函数。