Complex Manifold Theory and Kaehler Geometry

复流形理论和凯勒几何

• 批准号: 0500964
• 负责人: Yum-Tong Siu
• 金额: --
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500964&HistoricalAwards=false
• 关键词: Complex; Manifold; Theory; Kaehler; Geometry

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.

该项目发展了超越方法，如乘子理想层，以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Zebrski非稠密性，紧致Kaehler流形的多属变形不变性，从复Frobenius可积性定理的观点出发，研究了弱伪凸域的有限型条件与次椭圆估计之间的关系，本项目的研究内容是多复变函数、复代数几何、复微分几何和偏微分方程之间的接口。 除了在代数几何和复几何中的应用外，对乘子理想层，特别是由微分定义的乘子理想层的良好理解，将为偏微分方程引入新的强大工具。乘子理想层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。他们能够提供全球条件的可解性偏微分方程，而不是当地的条件，将是特别有用的偏微分方程从全球问题所提出的任何科学领域。