Geometry of Deformation and Moduli Spaces of Complex Manifolds

摘要奖：DMS 1510216，首席研究员：刘克峰变形理论、模空间和模形式是许多数学和物理学科的基础，从几何、拓扑、代数几何、数论到理论物理，如弦论和宇宙学。数学和弦论中的许多深层结果关键依赖于模空间，模空间描述了相关几何空间中的大族几何结构，有时比原始几何空间更大，更不直接定义。一个简单的例子是圆上的度量结构的等距类的集合-这些由圆的周长确定，因此对应于所有正实数的开线；在这个例子中，变形是一个圆到另一个圆周更大或更小的扩张或收缩。从整体几何的角度理解射影流形的形变和模空间，将揭示几何、代数和物理之间的深刻联系，将在数学和物理的许多研究领域产生根本性的影响。主要研究人员将研究射影流形的形变理论、TeichMuller空间和模空间的几何和拓扑结构，以及黎曼曲面的模空间上重言环的某些生成级数的模性。更确切地说，PI将研究以下三个重要问题：(1)利用PI和合作者发现的新的公式和迭代方法，系统地研究整体形变理论，并通过显式几何构造证明Siu猜想的Kahler流形的多正则截面的维度的形变不变性；(2)证明Griffiths的猜想，该猜想断言对于一族射影流形，所有周期的同时一致化的存在；(3)探索Pi和Hao Xu发现的Ramanujan模拟theta函数与Riemann曲面的模空间的重言环的维度之间的显著关系。在实施这些项目时，PI将培训几名年轻的学生和博士后，通过合作、研讨会和讲座进行这些项目的研究。

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.