Complex Manifold Theory and Kaehler Geometry

本课题主要研究复流形理论中的下列问题：(1)非一般类型流形的多元亏格的不变性；(2)Fujita猜想类型问题及其已知界的锐化；(3)一般类型流形的有限生成正则环；(4)具有小奇点的Levi-平坦集的复Neumann问题和不存在问题；(5)复射影空间中一般高次超曲面及其补集的双曲性。本研究将利用并进一步发展已产生很好结果的乘子理想层方法。许多几何及相关领域的问题，如数学物理、数学物理、数学物理等归结为偏微分方程解的存在性和正则性等问题。在很长一段时间里，先验估计一直是全局求解偏微分方程的主要工具。这样的先验估计通常是从偏微分方程的逐点性质得到的。在许多重要的全球几何问题中，其中一些如上所述，逐点论证是不够的。为了解决这些问题，本项目采用并进一步发展了“多倍理想捆”的方法。要估计的是在估计之前乘以一个“乘数”，这样先验估计就成立了。所有这些乘数的集合构成了“乘数理想束”。乘子理想层的整体性质，例如在某些类型的微分下的封闭性，对于某些问题，迫使函数成为乘子，该函数恒为1，从而给出偏微分方程解的期望整体解。这种方法已经解决了代数几何中一些长期悬而未决的问题。随着进一步的发展，这种利用乘子理想束来求解整体偏微分方程组的新方法应该是一个非常强大的工具，具有广泛的应用和深远的影响。

ABSTRACTThe project is to investigate the following problems in complex manifold theory.(1) Invariance of plurigenera for manifolds not of general type.(2) Fujita conjecture type problems and the sharpening of known bounds.(3) Finite generation of canonical rings for manifolds of general type.(4) Global regularity of the complex Neumann problem and nonexistence problem for Levi-flat sets with small singularities.(5) Hyperbolicity of generic high-degree hypersurfaces in the complex projective space and their complements.The investigation will use and further develop the method of multiplier ideal sheaf which has already produced very good results.Many problems in geometry and related fields, such as mathematical physics,are reduced to questions about the existence and the properties, such as regularity, of global solutions of partial differential equations. A priori estimates have for a long time been the dominant tool for globally solving partial differential equations. Such a priori estimates are usually derived from pointwise properties of the partial differential equations. In many important global geometric problems, some of which are listed above, pointwise arguments are insufficient. To solve such problems, this project uses and further develops the method of "mulitplier ideal sheaf". What is to be estimated is multiplied by a "multiplier" before estimation so that the a priori estimates hold. The set of all such multipliers forms the "multiplier ideal sheaf". Global properties of the "multiplier ideal sheaf", such as closedness under certain kinds of differentiation, for certain problems force to be a multiplier the function which is identically 1, thus giving the desired global solutions of the partial differential equations. Some long outstanding problems in algebraic geometry have already been solved by this method. With further development this new method of usingthe multiplier ideal sheaf to solve global partial differential equationsshould be a very powerful tool with broad applications and deep impact.