Geometry of Deformation and Moduli Spaces of Complex Manifolds

摘要：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、数论到弦理论和宇宙学等理论物理。数学和弦理论中的许多深刻结果都依赖于模空间，模空间描述了相关几何空间中的大族几何结构，这些空间有时比原始几何更大，更不直接定义。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放线；在这个例子中，变形是一个圆对另一个周长较大或较小的圆的膨胀或收缩。从整体几何的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的影响。主要研究变形理论的几何结构和拓扑结构，投影流形的Teichmuller空间和模空间，以及在黎曼曲面的模空间上重言环的维数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)使用PI和合作者发现的新公式和迭代方法系统地研究全局变形理论，并通过显式几何结构证明Siu猜想的Kahler流形的多形截面的尺寸变形不变性；(2)证明了Griffiths的一个猜想，该猜想证明了射影流形族的所有周期同时均匀化的存在性；(3)探索PI和徐浩发现的拉马努金模拟函数与黎曼曲面模空间重言环的维数之间的惊人关系。在实施这些项目的过程中，PI将通过合作、研讨会和讲座等方式培训几名年轻学生和博士后进行研究。

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.