Complex Manifold Theory and Kaehler Geometry

该项目发展了一些先验方法，如乘子理想层，应用于代数几何和复几何中的问题，如标准环的有限生成，具有正标准线丛的紧致复流形中所有全纯曲线并的Zariski非稠密性，紧致Kaehler流形的多亏域的形变不变性，以及从Artin子模式上复Frobenius可积定理的观点，弱伪凸域的有限型条件与次椭圆估计之间的关系。本项目的研究涉及几个复变量、复代数几何、复几何和偏微分方程组之间的界面。除了应用于代数几何和复杂几何中的问题外，对乘子理想轨道，特别是由微分定义的乘子理想轨道的深入了解，将为偏微分方程组带来新的强有力的工具。乘子理想集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。它们能够提供偏微分方程解的全局条件，而不是局部条件，对于由任何科学领域提出的全局问题产生的偏微分方程将特别有用。

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.