Complex Manifold Theory and Kaehler Geometry

本文主要研究复流形理论中的以下问题。(1)非一般型流形的多属不变性。(2)藤田猜想型问题与已知界的锐化。(3)一般型流形的标准环的有限生成。(4)复Neumann问题的全局正则性与不存在性 问题的列维平坦集与小奇点。(5)复形中一般高次超曲面的双曲性 射影空间及其补空间的解的存在性问题，研究将利用并进一步发展已经产生很好结果的乘子理想层方法，几何及数学物理等相关领域中的许多问题都归结为偏微分方程整体解的存在性及其正则性等性质的问题。 长期以来，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来自于偏微分方程的逐点性质。 在许多重要的整体几何问题中，其中一些是上面列出的，逐点论证是不够的。 为了解决这类问题，本项目使用并进一步发展了“乘理想层”方法。 在估计之前，要估计的内容乘以“乘数”，以便先验估计保持不变。 所有这些乘子的集合形成“乘子理想层”。 “乘子理想层”的全局性质，例如在某些类型的微分下的封闭性，对于某些问题，强制使同为1的函数成为乘子，从而给出偏微分方程的期望全局解。 代数几何中一些长期悬而未决的问题已经用这种方法解决了。 随着乘子理想层方法的进一步发展，它将成为一个具有广泛应用和深远影响的强有力的工具.

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.