Complex Manifold Theory and Kaehler Geometry

摘要本课题发展了乘子理想束等超越方法，将其应用于代数几何和复几何中的问题，如正则环的有限生成、紧实复流形中具有正正则线束的所有全纯曲线的并的Zariski非密度、紧实Kaehler流形的多胚变形不变性、从Artinian子方案上的复Frobenius可积性定理出发，讨论弱伪凸域的有限型条件与亚椭圆估计的关系。本课题的研究主要集中在几个复杂变量、复杂代数几何、复杂微分几何和偏微分方程之间的界面。除了在代数几何和复杂几何问题上的应用外，对乘法器理想束的理解，特别是那些由微分定义的乘法器理想束，将为偏微分方程引入新的有力工具。乘数理想束识别估计失败的高阶微分方向，然后将它们粘合在具有丰富属性的全局几何实体中。它们能够为偏微分方程的可解性提供全局条件，而不是局部条件，对于任何科学领域提出的全局问题的偏微分方程特别有用。

ABSTRACTThe project develops transcendental methods, such as multiplier ideal sheaves, to apply to problems in algebraic geometry and complex geometry, such as the finite generation of canonical rings, the Zariski nondensity of the union of all entire holomorphic curves in a compact complex manifold with positive canonical line bundle, the deformational invariance of plurigenera for compact Kaehler manifolds, and the relation between finite type condition of weakly pseudoconvex domains and subelliptic estimates from the viewpoint of complex Frobenius integrability theorem over Artinian subschemes.The investigations in this project are in the interface between severalcomplex variables, complex algebraic geometry, complex differential geometry, and partial differential equations. Besides applications to problems in algebraic geometry and complex geometry, a good understanding of multiplier ideal sheaves, especially those defined by differentiation, will introduce to partial differential equations new powerful tools. Multiplier ideal sheaves identify the higher-order directions of differentiation where estimates fail and then glue them together in global geometric entities with rich properties. They are able to provide global conditions for the solvability of partial differential equations instead of local conditions and will be especially useful for partial differential equations from global problems posed by any scientific field.