Geometry of Deformation and Moduli Spaces of Complex Manifolds

复流形的变形几何和模空间

• 批准号: 1510216
• 负责人: Kefeng Liu
• 金额: $ 42.6万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510216&HistoricalAwards=false
• 关键词: Geometry; Deformation; Moduli; Spaces; Complex

AbstractAward: DMS 1510216, Principal Investigator: Kefeng LiuDeformation theory, moduli spaces and modular forms are fundamental to many subjects of mathematics and physics from geometry, topology, algebraic geometry, number theory to theoretical physics like string theory and cosmology. Many deep results in mathematics and string theory crucially rely on moduli spaces, which describe large families of geometric structures within a related geometric space, which is sometimes larger and less directly defined than the original geometry. A simple example is the collection of isometry classes of metric structures on a circle - these are determined by the circumference of the circle and so correspond to the open line of all positive real numbers; in this example a deformation would be an expansion or contraction of one circle to another of larger or smaller circumference. Understanding of deformations and moduli spaces of projective manifolds from global geometric point of view will reveal deep connections among geometry, algebra and physics, will have fundamental impacts in many research fields in mathematics and physics.The principal investigator will study the geometric and topological structures of the deformation theory, Teichmuller and moduli spaces of projective manifolds, and the modularity of certain generating series of the dimension of the tautological rings on the moduli spaces of Riemann surfaces. More precisely, the PI will study the following three important problems: (1) using the new formulas and iteration method discovered by the PI and collaborators to systemically study global deformation theory and prove deformation invariance of the dimension of pluricanonical sections of Kahler manifolds as conjectured by Siu by explicit geometric constructions; (2) proving a conjecture of Griffiths, which asserts the existence of simultaneous uniformization of all the periods for a family of projective manifolds; (3) exploring a striking relation discovered by the PI and Hao Xu between the Ramanujan mock theta-function and the dimensions of the tautological ring of moduli spaces of Riemann surfaces. In carrying out the projects the PI will train several young students and postdoctors to conduct research in these projects through collaboration, seminars, and lectures.

摘要奖：DMS 1510216，首席研究员：刘克峰变形理论、模空间和模形式是数学和物理许多学科的基础，从几何、拓扑、代数几何、数论到弦论和宇宙学等理论物理。数学和弦理论中的许多深层次结果关键依赖于模空间，模空间描述了相关几何空间内的大族几何结构，有时比原始几何更大且不太直接定义。一个简单的例子是圆上度量结构的等距类的集合 - 这些是由圆的周长决定的，因此对应于所有正实数的开线；在此示例中，变形是一个圆向更大或更小周长的另一个圆的膨胀或收缩。 从全局几何角度理解射影流形的变形和模空间将揭示几何、代数和物理学之间的深刻联系，将对数学和物理的许多研究领域产生根本性影响。首席研究员将研究射影流形的几何和拓扑结构变形理论、Teichmuller 和射影流形模空间，以及黎曼曲面模空间上同义反复环维数的某些生成级数的模性。更准确地说，课题组将研究以下三个重要问题：（1）利用课题组和合作者发现的新公式和迭代方法，系统研究全局变形理论，并证明猜想的卡勒流形多规范截面维数的变形不变性由 Siu 通过明确的几何构造； (2) 证明格里菲斯猜想，该猜想断言射影流形族的所有周期同时一致化的存在； (3) 探索 PI 和徐浩发现的拉马努金模拟 theta 函数与黎曼曲面模空间同义反复环维数之间的惊人关系。在实施这些项目的过程中，PI 将培训几名年轻学生和博士后，通过合作、研讨会和讲座的方式开展这些项目的研究。