Geometry of Deformation and Moduli Spaces of Complex Manifolds

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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.